text
stringlengths 0
2.11M
| id
stringlengths 33
34
| metadata
dict |
|---|---|---|
Research Article a]M.Marko Voutilainencor1[label=e1][email protected] [type=corresp,id=cor1]Corresponding author. b]L.Lauri Viitasaari[label=e2][email protected] a]P.Pauliina Ilmonen[label=e3][email protected][a]Department of Mathematics and Systems AnalysisAalto University School of ScienceP.O. Box 11100, FI-00076 Aalto, Finland [b]Department of Mathematics and StatisticsUniversity of HelsinkiP.O. Box 68, FI-00014 University of Helsinki, Finland M. Voutilainen et al.On model fitting and estimation of strictly stationary processes Stationary processes have been extensively studied in the literature. Their applications include modeling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, we challenge this conventional approach. Instead of fitting an ARMA model, we apply an AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.AR(1) representation asymptotic normality consistency estimation strictly stationary processes [2010] 60G10 62M09 62M10 60G18 13 9 201722 11 201725 11 201722 12 2017 § INTRODUCTIONStochastic processes are widely used in modeling and forecasting numerous real life phenomena such as natural disasters, activity of the sun, sales of a company and market movements, to mention a few. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, in this paper, we challenge this conventional approach. Instead of fitting an ARMA model, we apply the AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter. One of the reasons why ARMA processes have been in a central role in modeling of time-series data is that for every autocovariance function γ(·) vanishing at infinity and for every n∈ℕ there exists an ARMA process X such that γ(k) = γ_X(k) for every k = 0,1,..,n. For a general overview of the theory of stationary ARMA processes and their estimation, the reader may consult for example <cit.> or <cit.>.ARMA processes, and their extensions, have been studied extensively in the literature. A direct proof of consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible ARMA processes was given in <cit.>. The result was originally obtained, using asymptotic properties of the Whittle estimator, in <cit.>. The estimation of the parameters of strictly stationary ARMA processes with infinite variances was studied in <cit.>, again, by using Whittle estimators. Portmanteau tests for ARMA models with stable Paretian errors with infinite variance were introduced in <cit.>. An efficient method for evaluating the maximum likelihood function of stationary vector ARMA models was presented in <cit.>. Fractionally integrated ARMA models with a GARCH noise process, where the variance of the error terms is also of ARMA form, was studied in <cit.>. Consistency and asymptotic normality of the quasi-maximum likelihood estimators of ARMA models with the noise process driven by a GARCH model was shown in <cit.>. A least squares approach for ARMA parameter estimation has been studied at least in <cit.> by contrasting its efficiency with the maximum likelihood estimation. Also estimators of autocovariance and their limiting behavior have been addressed in numerous papers. See for example <cit.> and <cit.>.Modeling an observed time-series with an ARMA process starts by fixing the orders of the model. This is often done by an educated guess, but there also exists methods for estimating the orders, see e.g. <cit.>. After the orders are fixed, the related parameters can be estimated, for example, by using the maximum likelihood or least squares estimators. These estimators are expressed in terms of optimization problems and do not generally admit closed form representations. The final step is to conduct various diagnostic tests to determine whether the estimated model is sufficiently good or not. These tests are often designed to recognize whether the residuals of the model support the underlying assumptions about the error terms. Depending on whether one considers strict or weak stationarity, the error process is usually assumed to be an IID process or white noise, respectively. If the tests do not support the assumptions about the noise process, then one has to start all over again. Tests for the goodness of fit of ARMA modelshave been suggested e.g. in <cit.>.The approach taken in this paper is based on the discrete version of the main theorem of <cit.> leading to an AR(1) characterization for (any) strictly stationary processes. Note that this approach covers, but is not limited to, strictly stationary ARMA processes.It was stated in <cit.> that a process is strictly stationary if and only if for every fixed 0<H<1 it can be represented in the AR(1) form with ϕ= e^-H and a unique, possibly correlated, noise term. Although the representation is unique only after H is fixed, we show that in most of the cases, given just one value of the autocovariance function of the noise, one is able to determine the AR(1) parameter and, consequently, the entire autocovariance function of the noise process. It is worth emphasizing that since the parameter–noise pair in the AR(1) characterization is not unique, it is natural that some information about the noise has to be assumed. Note that conventionally, when applying ARMA models, we have assumptions about the noise process much stronger than being IID or white noise. That is, the autocovariance function of the noise is assumed to be identically zero except at the origin. When founding estimation on the AR(1) characterization, one does not have to select between different complicated models. In addition, there is only one parameter left to be estimated. Yet another advantage over classical ARMA estimation is that we obtain closed form expressions for the estimators. The paper is organized as follows. We begin Section <ref> by introducing some terminology and notation. After that, we give a characterization of discrete time strictly stationary processes as AR(1) processes with a possibly correlated noise term together with some illustrative examples. The AR(1) characterization leads to Yule–Walker type equations for the AR(1) parameter ϕ. In this case, due to the correlated noise process, the equations are of quadratic form in ϕ. For the rest of the section, we study the quadratic equations and determine ϕ with as little information about the noise process as possible. The approach taken in Section <ref> leads to an estimator of the AR(1) parameter. We consider estimation in detail in Section <ref>. The end of Section <ref> is dedicated to testing the assumptionswe make when constructing the estimators. A simulation study to assess finite sample properties of the estimators is presented in Section <ref>. Finally, we end the paper with three appendices containing a technical proof, detailed discussion on some special cases and tabulated simulation results.§ ON AR(1) CHARACTERIZATION IN MODELING STRICTLY STATIONARY PROCESSESThroughout the paper we consider strictly stationary processes. Assume that X = (X_t)_t∈ℤ is a stochastic process. If (X_t+n_1, X_t+n_2,…,X_t+n_k ) law= (X_n_1, X_n_2,…,X_n_k ) for all k∈ℕ and t, n_1, n_2,…,n_k ∈ℤ, then X is strictly stationary.Assume that G = (G_t)_t∈ℤ is a stochastic process and denote Δ_t G = G_t - G_t-1. If (Δ_t G )_t∈ℤ is strictly stationary, then the process G is a strictly stationary increment process. The following class of stochastic processes was originally introduced in <cit.>.Let H > 0 be fixed and let G= (G_t)_t∈ℤ be a stochastic process. If G is a strictly stationary increment process with G_0 = 0 and if the limit lim_k→-∞∑_t=k^0 e^tHΔ_t G exists in probability and defines an almost surely finite random variable, then G belongs to the class of converging strictly stationary increment processes, and we denote G∈𝒢_H. Next, we consider the AR(1) characterization of strictly stationary processes. The continuous time analogy was proved in <cit.> together with a sketch of a proof for the discrete case. For the reader's convenience, a detailed proof of the discrete case is presented in Appendix <ref>.Let H >0 be fixed and let X = (X_t)_t∈ℤ be a stochastic process. Then X is strictly stationary if and only if lim_t→-∞e^tHX_t = 0 in probability and Δ_t X = (e^-H - 1 )X_t-1 + Δ_t G for a unique G∈𝒢_H.Let H>0 be fixed. Then every discrete time strictly stationary process (X_t)_t∈ℤ can be represented as X_t - ϕ^(H) X_t-1 = Z_t^(H), where ϕ^(H) = e^-H and Z_t^(H) = Δ_t G is another strictly stationary process. It is worth to note that the noise Z in Corollary <ref> is unique only after the parameter H is fixed. The message of this result is that every strictly stationary process is an AR(1) process with a strictly stationary noise that may have a non-zero autocovariance function. The following examples show how some conventional ARMA processes can be represented as an AR(1) process.Let X be a strictly stationary AR(1) process defined by X_t - φ X_t-1 = ϵ_t, ( ϵ_t)∼ IID(0, σ^2) with φ>0. Then we may simply choose ϕ^(H) = φ and Z_t^(H) = ϵ_t. Let X be a strictly stationary ARMA(1,q) process defined by X_t - φ X_t-1 = ϵ_t + θ_1ϵ_t-1+⋯ +θ_q ϵ_t-q,(ϵ_t )∼ IID(0, σ^2) with φ>0. Then we may set ϕ^(H) = φ, and Z_t^(H) then equals to the MA(q) process.Consider a strictly stationary AR(1) process X with φ<0. Then X admits an MA(∞) representation X_t = ∑_k=0^∞φ^k ϵ_t-k. From this it follows that Z_t^(H) = ϵ_t + ∑ _k=0^∞φ^k (φ-ϕ ^(H))ϵ_t-1-k Hence Z_t^(H) = ∑ _k=0^∞φ^kϵ_t-k - ϕ^(H)∑_k=0^∞φ^k ϵ_t-1-k=ϵ_t + ∑_k=0^∞φ^k+1ϵ_t-1-k - ϕ ^(H)∑ _k=0^∞φ^k ϵ_t-1-k=ϵ_t + ∑_k=0^∞φ^k (φ-ϕ^(H)) ϵ_t-1-k. Let var(ϵ_t) = σ^2. Then, for t>0 it holds that cov(Z_t^(H), Z_0^(H))= φ^t-1 (φ- ) σ^2 + cov(∑_k=t^∞φ^k (φ- )ϵ_t-1-k, ∑ _k=0^∞φ^k (φ- ) ϵ_-1-k). The covariance on the on right-hand side can be written as cov(∑ _n=-1^-∞φ^t-n-1 (φ- ) ϵ_n, ∑_n=-1^-∞φ^-n-1 (φ- )ϵ_n ). = φ^t-2 (φ- )^2σ^2 ∑ _n=1^∞φ^2n. Since X was stationary, the sum above converges and we obtain that and cov(Z_t^(H), Z_0^(H)) = φ^t-2 (φ- )σ^2 (φ+ (φ-)∑_n=1^∞( φ^2)^n ). Hence in the case of an AR(1) process with a negative parameter, the autocovariance function of the noise Z of the representation (<ref>) is non-zero everywhere. Next we show how to determine the AR(1) parameter ϕ^(H) in (<ref>) provided that the observed process X is known. In what follows, we omit the superindices in (<ref>). We assume that the second moments of the considered processes are finite and that the processes are centered. That is, (X_t) = (Z_t) = 0 for every t∈ℤ. Throughout the rest of the paper, we use the notation cov(X_t, X_t+n) = γ(n) and cov(Z_t,Z_t+n) = r(n) for every t,n∈ℤ.Let centered (X_t)_t∈ℤ be of the form (<ref>). Then ϕ^2 γ(n) - ϕ(γ(n+1) + γ(n-1) ) + γ (n) - r(n) = 0 for every n∈ℤ. Let n∈ℤ. By multiplying both sides of X_n - ϕ X_n-1 = Z_n with Z_0 = X_0 - ϕ X_-1 and taking expectations, we obtain (X_n(X_0- ϕ X_-1) )- ϕ(X_n-1(X_0 - ϕ X_-1) ) = ϕ^2γ(n) - ϕ(γ(n+1) + γ(n-1))+ γ(n) = r(n). γ(k) - ϕγ(k-1)= (Z_tX_t-k) = (Z_t(Z_t-k+ϕ X_t-k-1)) = r(k) + ϕ(Z_t(Z_t-k-1 + ϕ X_t-k-2) ) =∑_i=0^∞ϕ^i r(i+k). By using k=n+2 in the equation above and multiplying the both sides with ϕ, we obtain ϕγ(n+2) - ϕ^2γ(n+1) = ∑ _i=1^∞ϕ^i r(i+n+1). Let us next subtract the equation above from (<ref>) with k = n+1 yielding ϕ^2 γ(n+1) -ϕ(γ(n+2) + γ(n)) + γ(n+1) = r(n+1). Let centered (X_t)_t∈ℤ be of the form (<ref>) and let N∈ℕ be fixed. (1) If γ(N) ≠0, then either ϕ = γ(N + 1)+ γ(N - 1)+ √( (γ(N + 1) + γ (N - 1) )^2-4γ(N)(γ(N)-r(N)))/2γ(N) or ϕ = γ(N + 1)+ γ(N - 1)- √( (γ(N + 1) + γ (N - 1) )^2-4γ(N)(γ(N)-r(N)))/2γ(N).(2) If γ(N) = 0 and r(N) ≠0, then ϕ= -r(N)/γ(N+1) + γ(N-1). Note that if γ(N) = r(N) = 0, then Lemma <ref> yields only γ(N+1) + γ(N-1) = 0 providing no information about the parameter ϕ. As such, in order to determine the parameter ϕ, we require that either γ(N) ≠ 0 or r(N) ≠0. If the variance r(0) of the noise is known, then (<ref>) and (<ref>) reduces to ϕ= γ(1) ±√(γ(1)^2 - γ(0) (γ(0) - r(0) ))/γ(0). At first glimpse it seems that Corollary <ref> is not directly applicable. Indeed, in principle it seems like there could be complex-valued solutions although representation (<ref>) together with (<ref>) implies that there exists a solution ϕ∈(0,1). Furthermore, it is not clear whether the true value is given by (<ref>) or (<ref>). We next address these issues. We start by proving that the solutions to (<ref>) cannot be complex. At the same time we are able to determine which one of the solutions one should choose.The discriminants of (<ref>) and (<ref>) are always non-negative.Let k∈ℤ. By multiplying both sides of (<ref>) with X_t-k, taking expectations, and applying (<ref>) repeatedly we obtain γ(k) - ϕγ(k-1)= (Z_tX_t-k) = (Z_t(Z_t-k+ϕ X_t-k-1)) = r(k) + ϕ(Z_tX_t-k-1 ) = r(k) + ϕ(Z_t(Z_t-k-1 + ϕ X_t-k-2) ) =r(k) + ϕ r(k+1) + ϕ^2 (Z_t X_t-k-2 ). Proceeding as above l times we get γ(k) - ϕγ(k-1) = ∑_i=0^l-1ϕ^i r(k+i) + ϕ ^l(Z_t(ϕ X_t-k-l-2)). Letting l approach infinity leads to γ(k) - ϕγ(k-1) = ∑_i=0^∞ϕ^i r(k+i), where the series converges as r(k+i)≤ r(0) and 0<ϕ<1. It now follows from (<ref>) that γ(N)= ϕγ(N-1) + ∑ _i=0^∞ϕ^i r(N+i) =ϕγ(N-1) +r(N)+ϕ∑_i=1^∞ϕ^i-1r(N+i) =ϕγ(N-1) +r(N)+ϕ∑_i=0^∞ϕ^ir(N+i+1) = ϕγ(N-1) + ϕ(γ(N+1) - ϕγ(N) )+r(N). Denote the discrimant of (<ref>) and (<ref>) by D. That is, D = (γ(N-1)+γ(N+1) )^2 - 4γ(N) (γ (N)-r(N) ). By using the equation above we observe that D= (γ(N) + ϕ^2γ(N)-r(N)/ϕ)^2 - 4γ(N) (γ(N)-r(N)). Denoting a_N = r(N)/γ(N), multiplying by ϕ ^2/γ(N)^2, and using the identity (a+b)^2 -4ab = (a-b)^2 yields ϕ^2/γ(N)^2D = (1+ϕ^2-a_N )^2 - 4ϕ^2(1-a_N) = (ϕ ^2-1+a_N)^2 ≥0. This concludes the proof. Note that if r(N) = 0, as ϕ<1, the discriminant is always positive. Let a_N = r(N)/γ(N). The proof above now gives us the following identity ϕ= 1/2ϕ(1+ϕ^2-a_N±|γ(N)|/γ (N)ϕ^2-1+a_N ). This enables us to consider the choice between (<ref>) and (<ref>). Assume that γ(N) > 0. If ϕ^2 - 1 + a_N > 0, then ϕ is given by (<ref>) (as ϕ∈(0,1)). Similarly, if ϕ^2 - 1 +a_N<0, then ϕ is determined by (<ref>). Finally, contrary conclusions hold in the case γ(N) < 0. In particular, we can always choose between (<ref>) and (<ref>) provided that either a_N ≤0 or a_N ≥1. Moreover, from (<ref>) it follows that r(N)/γ(N) = r(N+k)/γ(N+k) if and only if γ(N+1)+γ(N-1)/γ(N) = γ(N+1+k)+γ (N-1+k)/γ(N+k), provided that the denominators differ from zero. Since (<ref>) and (<ref>) can be written as ϕ = γ(N+1)+γ(N-1)/2γ(N)±1/2sgn(γ(N))√((γ(N+1)+γ(N-1)/γ (N))^2-4 (1-r(N)/γ(N))), we observe that one can always rule out one of the solutions (<ref>) and (<ref>) provided that a_N ≠ a_N+k. Therefore, it always suffices to know two values of the autocovariance r such that a_N ≠ a_N+k, except the worst case scenario where a_j = a ∈(0,1) for every j∈ℤ. A detailed analysis of this particular case is given in Appendix <ref>. Consider a fixed strictly stationary process X. If we fix one value of the autocovariance function of the noise such that Corollary <ref> yields an unambiguous AR(1) parameter, then the quadratic equations (<ref>) will unravel the entire autocovariance function of the noise process. In comparison, conventionally, the noise is assumed to be white — meaning that the entire autocovariance function of the noise is assumed to be known a priori. We end this section by observing that in the case of vanishing autocovariance function of the noise, we get the following simplified form for the AR(1) parameter.Let centered (X_t)_t∈ℤ be of the form (<ref>) and let N∈ℕ be fixed. Assume that r(m)=0 for every m≥ N. If γ(N-1)≠0, then for every n≥ N, we have ϕ= γ(n)/γ(n-1). In particular, γ admits an exponential decay for n≥ N.Let γ(N-1)≠0. It follows directly from (<ref>) and the assumptions that γ(n) = ϕγ(n-1)for every n≥ N. The condition γ(N-1)≠0 now implies the claim. Recall that the representation (<ref>) is unique only after H is fixed. As a simple corollary for Theorem <ref> we obtain the following result giving some new information about the uniqueness of the representation (<ref>). Let X be a strictly stationary process with a non-vanishing autocovariance. Then there exists at most one pair (H, G) satisfying (<ref>) such that the non-zero part of the autocovariance function of the increment process (Δ_t G)_t∈ℤ is finite.Assume that there exists H_1, H_2 > 0, and G_1∈𝒢_H_1 and G_2∈𝒢_H_2 such that the pairs (H_1, G_1) and (H_2, G_2) satisfy (<ref>) and the autocovariances of (Δ_t G_1)_t∈ℤ and (Δ_t G_2)_t∈ℤ have cut-off points. From Theorem <ref> it follows that H_1 = H_2 and since for a fixed H the process G in (<ref>) is unique, we get G_1 = G_2. § ESTIMATIONCorollary <ref> gives natural estimators for ϕ provided that we have been able to choose between (<ref>) and (<ref>), and that a value of r(n) is known. We emphasize that in our model it is sufficient to know only one (or in some cases two) of the values r(n), whereas in conventional ARMA modeling much stronger assumptions are required. (In fact, in conventional ARMA modeling the noise process is assumed to be white noise.) It is also worth to mention that, generally, estimators of the parameters of stationary processes are not expressible in a closed form. For example, this is the case with the maximum likelihood and least squares estimators of conventionally modeled ARMA processes, see <cit.>. Within our method, the model fitting is simpler.Finally, it is worth to note thatassumption of one known value of r(n) is a natural one and cannot be avoided. Indeed, this is a direct consequence of the fact that the pair (ϕ,Z) in representation (<ref>) is not unique. In fact, for practitioner, it is not absolutely necessary to know any values of r(n). The practitioner may make an educated guess and proceed in estimation. If the obtained estimate then turns out to be feasible, the practitioner can stop there. If the obtained estimate turns out to be unreasonable (not on the interval (0,1)), then the practitioner have to make another educated guess. The process is similar to selecting p and q in traditional ARMA(p,q) modeling.Throughout this section, we assume that (X_1, …, X_T) is an observed series from a centered strictly stationary process that is modeled using the representation (<ref>). We use (n) to denote an estimator of the corresponding autocovariance γ(n). For example, (n) can be given by (n) = 1/T∑_t=1^T-n X_tX_t+n, or more generally (n) = 1/T∑_t=1^T-n (X_t-X̅ ) (X_t+n-X̅ ), where X̅ is the sample mean of the observations. For this estimator the corresponding sample covariance (function) matrix is positive semidefinite. On the other hand, the estimator is biased while it is asymptotically unbiased. Another option is to use T-n-1 as a denominator. In this case one has an unbiased estimator, but the sample covariance (function) matrix is no longer positive definite. Obviously, both estimators have the same asymptotic properties. Furthermore, for our purposes it is irrelevant how the estimators (n) are defined, as long as they are consistent, and the asymptotic distribution is known.We next consider estimators of the parameter ϕ arising from characterization (<ref>). In this context, we pose some assumptions related to the autocovariance function of the observed process X. The justification and testing of these assumptions are discussed in Section <ref>. From a priori knowledge that ϕ∈(0,1) we enforce also the estimators to the corresponding closed interval. However, if one prefers to use unbounded versions of the estimators, one may very well do that. The asymptotic properties are the same in both cases. We begin by defining an estimator corresponding to the second part (2) of Corollary <ref>.For any h∈ℕ the asymptotic joint distribution of √(T)((0)-γ(0), (1)-γ(1), …, (h)-γ (h) ) is multivariate normal with zero expectation.The estimator related to <ref> is asymptotically normal with the rate of convergence of √(T). Let us denote X_T = √(T)(γ̂_T(n+1) - γ_T(n+1)),Y_T = √(T)( γ(n) - γ̂_T(n)). By (**) the asymptotic joint distribution of (X_T, Y_T) is normal and hence, we write (X_T, Y_T) law⟶𝒩([ 0; 0 ],[ σ_X^2σ_XY;σ_XY σ_Y^2 ]) Now let us study the asymptotic behavior of √(T)(γ̂_T(n+1)/(n)- γ (n+1)/γ(n)) = 1/(n)√(T)((n+1) - γ(n+1)) + √(T)(γ(n+1)/(n) - γ(n+1)/γ(n)) = 1/(n)( √(T)((n+1) - γ(n+1)) + γ(n+1)/γ(n)√(T)(γ(n)- (n) ) ) = 1/(n)(X_T + γ(n+1)/γ(n)Y_T ), where (X_T, γ(n+1)/γ(n)Y_T ) law⟶𝒩([ 0; 0 ],[ σ_X^2 γ(n+1)/γ(n)σ_XY; γ(n+1)/γ(n)σ_XY γ(n+1)^2/γ (n)^2σ_Y^2 ]) and hence X_T + γ(n+1)/γ(n)Y_T law⟶𝒩(0, σ_X^2 + γ (n+1)^2/γ(n)^2σ_Y^2 + 2γ(n+1)/γ(n)σ _XY). By Slutsky's theorem, we may conclude that 1/(n)(X_T + γ(n+1)/γ(n)Y_T ) law⟶𝒩(0, σ _X^2/γ(n)^2 + γ(n+1)^2/γ(n)^4σ_Y^2 + 2 γ(n+1)/γ(n)^3σ_XY).Assume that γ(N) = 0. Then we define ϕ̂_T = -r(N)/γ̂_T(N+1) + γ̂_T(N-1)1_(N+1)+(N-1) ≠0 whenever the right-hand side lies on the interval [0,1]. If the right-hand side is below zero, we set ϕ̂_T = 0 and if the right-hand side is above one, we set ϕ̂_T = 1. Assume that γ(N) =0 and r(N)≠0. If the vector-valued estimator[(N+1), (N-1) ]^⊤ is consistent, then ϕ̂_T is consistent.Since γ(N) = 0 and r(N) ≠0, Equation (<ref>) guarantees that γ(N+1) + γ(N-1) ≠0. Therefore consistency of ϕ̂_T follows directly from the continuous mapping theorem. Let ϕ̂_T be given by (<ref>), and assume that γ(N) = 0 and r(N) ≠0. Set γ =[γ (N+1), γ(N-1) ]^⊤ and γ̂_T =[ (N+1),(N-1) ]^⊤. If l(T) (γ̂_T - γ )law⟶𝒩 (0,) for some covariance matrixand some rate function l(T), then l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, ∇ f(γ)^⊤∇ f(γ) ), where ∇ f(γ) is given by ∇ f(γ) = -r(N)/ (γ(N+1) + γ (N-1) )^2·[ 1; 1 ].For the simplicity of notation, in the proof we use the unbounded version of the estimator ϕ̂_T. Since the true value of ϕ lies strictly between 0 and 1, the very same result holds also for the bounded estimator of Definition <ref>. Indeed, this is a simple consequence of the Slutsky's theorem. To begin with, let us define an auxiliary function f by f(x) = f(x_1,x_2) = r(N)/x_1 + x_21_x_1+x_2 ≠0. If x_1+x_2≠0, the function f is smooth in a neighborhood of x. Since γ(N) = 0 together with r(N) ≠0 implies that γ(N+1) + γ(N-1) ≠0, we may apply the delta method at x = γ to obtainl(T) (ϕ̂_T - ϕ ) = -l(T) (f(γ̂_T) - f(γ) )law⟶𝒩(0, ∇ f(γ)^⊤∇ f(γ) ), where ∇ f(γ) is given by (<ref>). This concludes the proof.By writing =[ σ_X^2σ_XY;σ_XY σ_Y^2 ] the variance of the limiting random variable reads r(N)^2/ (γ(N+1) + γ(N-1) )^4(σ_X^2 + 2 σ_XY + σ_Y^2).In many cases the convergency rate is the best possible, that is l(T) = √(T). However, our results are valid with any rate function. One might, for example in the case of many long memory processes, have other convergency rates for the estimators (n). We continue by defining an estimator corresponding to the first part (1) of the Corollary <ref>. For this we assume that, for reasons discussed in Section <ref>, we have chosen the solution (<ref>) (cf. Remark <ref> and Section <ref>). As above, we show that consistency and asymptotic normality follow from the same properties of the autocovariance estimators. In the sequel we use a short notation g(x) = g(x_1,x_2,x_3) = (x_1+x_3 )^2-4x_2 (x_2-r(N)). In addition, we denote γ = [γ(N+1), γ(N), γ(N-1) ]^⊤ and γ̂_T = [(N+1),(N),(N-1) ]^⊤. Assume that γ(N) ≠0. We define an estimator for ϕ associated to (<ref>) by ϕ̂_T = γ̂_T(N+1) + γ̂_T(N-1) + √(g (γ̂_T ))1_g (_T )>0/2γ̂_T(N)1_(N) ≠0 whenever the right-hand side lies on the interval [0,1]. If the right-hand side is below zero, we set ϕ̂_T = 0 and if the right-hand side is above one, we set ϕ̂_T = 1. Assume that γ(N)≠0 and g(γ)>0. Furthermore, assume that ϕ is given by (<ref>). If γ̂_T is consistent, then ϕ̂_T is consistent.As g(γ)>0, the result is again a simple consequence of the continuous mapping theorem. Before proving the asymptotic normality, we present some short notation. We set C_N = γ(N+1)+γ(N-1)+√(g(γ))/γ(N) and _ϕ = 1/4γ(N)^2( (∇√(g(γ)))^⊤∇√(g(γ))+ 2[1; -C_N;1 ]^⊤∇√(g(γ))..+[1; -C_N;1 ]^⊤[1; -C_N;1 ]), where ∇√(g(γ)) = 1/√(g(γ))[ γ(N+1) + γ(N-1); 2(r(N) - 2γ(N)); γ(N+1) + γ(N-1) ]. Let the assumptions of Theorem <ref> prevail. If l(T) (γ̂_T - γ )law⟶𝒩 (0,) for some covariance matrixand some rate function l(T), then l(T) (ϕ̂_T - ϕ ) is asymptotically normal with zero mean and variance given by (<ref>).The proof follows the same lines as the proof of Theorem <ref> but for the reader's convenience, we present the details. Furthermore, as in the proof of Theorem <ref>, since the true value of ϕ lies strictly between 0 and 1, for the notational simplicity, we may and will use the unbounded version of the estimator. Indeed, the asymptotics for the bounded version then follow directly from the Slutsky's theorem. We have (γ̂_T(N+1)1_(N) ≠0/(N)- γ(N+1)/γ(N)) =1/(N)((N+1)1_(N) ≠0 - γ (N+1) ) + (γ(N+1)/(N) - γ (N+1)/γ(N)) = 1/(N)( (N+1)1_(N) ≠0 - γ (N+1) - γ(N+1)/γ(N)((N)- γ(N) ) ). Similarly (γ̂_T(N-1)1_(N) ≠0/(N)- γ(N-1)/γ(N)) = 1/(N)( (N-1)1_(N) ≠0 - γ (N-1) - γ(N-1)/γ(N)((N)- γ(N) ) ) and (√(g(γ̂_T))1_g ( _T )>01_(N) ≠0/(N)- √(g(γ))/γ(N)) = 1/(N)( √(g(γ̂_T))1_g (_T )>01_(N) ≠0 - √(g(γ)) - √(g(γ))/γ(N)((N)- γ(N) ) ). For C_N given in (<ref>) we have l(T) (ϕ̂_T-ϕ ) = l(T)/2γ̂(N)((N+1)- γ(N+1) + (N-1)- γ(N-1) -C_N ((N)-γ(N) ) + √(g(γ̂_T))- √(g(γ))). By defining h(x) = h(x_1,x_2,x_3) = (x_1 + x_3 + √(g(x))1_g(x) >0)1_x_2≠0 - C_N x_2 we have l(T) (ϕ̂_T-ϕ ) = l(T)/2γ̂_T(N)(h (γ̂_T ) - h(γ) ). If x_2 ≠0 and g(x) > 0, the function h is smooth in a neighborhood of x. Therefore we may apply the delta method at x = γ to obtain l(T) (h (γ̂_T ) - h(γ) ) law⟶𝒩(0, ∇ h(γ)^⊤∇ h(γ) ), where ∇ h(γ)^⊤∇ h( γ)= ([1; -C_N;1 ]+ ∇√(g(γ)))^⊤([1; -C_N;1 ]+ ∇√(g(γ))) = (∇√(g(γ)))^⊤∇√(g(γ)) + 2[1; -C_N;1 ]^⊤∇√(g(γ)) +[1; -C_N;1 ]^⊤[1; -C_N;1 ]. Hence (<ref>) and Slutsky's theorem imply that l(T) (ϕ̂_T - ϕ ) is asymptotically normal with zero mean and variance given by (<ref>). One straightforwardly observes the same limiting behavior as in Theorems <ref> and <ref> for the estimator related to (<ref>). This fact also can be used to determine which one of Equations (<ref>) and (<ref>) gives the correct ϕ (cf. Section <ref>). If γ(N) ≠0 and g(γ) = 0 we may define an estimator ϕ̂_T = (N+1) + (N-1)/2(N). Assuming that l(T) (γ̂_T - γ )law⟶𝒩 (0,) it can be shown similarly as in the proofs of Theorems <ref> and <ref> that l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, 1/4γ(N)^2[ 1; -γ(N+1) + γ(N-1)/γ(N); 1 ]^⊤[ 1; -γ(N+1) + γ(N-1)/γ(N); 1 ])The estimator related to Theorem <ref> reads ϕ̂_T = γ̂_T(n+1)/γ̂_T(n)1_(n) ≠0, where we assume that γ(n)≠0. By using the same techniques as earlier, it can be shown that if l(T) (γ̂_T(n+1) - γ(n+1), γ̂_T(n)- γ (n) ) law⟶𝒩( 0,[ σ_X^2σ_XY;σ_XY σ_Y^2 ]), then l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, σ_X^2/γ (n)^2 + γ(n+1)^2/γ(n)^4σ_Y^2 - 2γ (n+1)/γ(n)^3σ_XY). Note that the asymptotics given in Remarks <ref> and <ref> hold also if one forces the corresponding estimators to the interval [0,1] as we did in Definitions <ref> and <ref>. The asymptotics of the square root term of the estimator is given by √(T)(g(γ̂_T) - g(γ) ) law⟶𝒩(0, ∇ g(γ)^⊤∑∇ g(γ) )§.§ Testing the underlying assumptions When choosing the estimator that corresponds the situation at hand, we have to make assumptions related to the values of γ(N) (for some N) and g(γ). In addition, we have to consider the question of the choice between (<ref>) and (<ref>).Let us first discuss how to test the null hypothesis that γ(N) = 0. If the null hypothesis holds, then by asymptotic normality of the autocovariances, we have that l(T)(N)law⟶𝒩(0,σ^2 ) with some σ^2. Hence we may use (N) ∼_a 𝒩(0, σ^2/l(T)^2) as a test statistics. A similar approach can be applied also when testing the null hypothesis that g(γ) = 0, where g is defined by (<ref>). The alternative hypothesis is of the form g(γ) > 0. Assuming that the null hypothesis holds, we obtain by the delta method that l(T) (g(γ̂_T) - g(γ) ) law⟶𝒩(0, σ̃^2 ) for some σ̃^2 justifying the use of g(γ̂_T) ∼_a 𝒩(0, σ̃^2/l(T)^2) as a test statistics. If the tests above suggest that γ(N) ≠ 0 and g(γ)> 0, then the choice of the sign can be based on the discussion in Section <ref>. Namely, if for the ratio a_N = r(N)/γ(N) it holds that a_N ≤0 or a_N ≥1, then the sign is unambiguous. The sign of γ(N) can be deduced from the previous testing of the null hypothesis γ(N) = 0. By (<ref>), if necessary, one can test the null hypothesis γ(N) = r(N) using the test statistics (N) ∼_a 𝒩(r(N), σ^2/l(T)^2), where the alternative hypothesis is of the form r(N)/γ(N) < 1. Finally, assume that one wants to test if the null hypothesis a_N = a_k holds. By the delta method we obtain that l(T) (â_N-â_k - a_N +a_k )law⟶𝒩(0, σ̅^2 ) for some σ̅^2 suggesting that â_N-â_k ∼_a 𝒩(0, σ̅^2/l(T)^2) could be utilized as a test statistics.§ SIMULATIONS We present a simulation study to assess the finite sample performance of the estimators. In the simulations, we apply the estimator corresponding to the first part (1) of Corollary <ref>.We simulate data from AR(1) processes and ARMA(1,2) processes with θ_1 = 0.8 and θ_2 = 0.3 as the MA parameters. (Note that these processes correspond to Examples <ref> and <ref>.) We assess the effects of the sample size T, AR(1) parameter φ, and the chosen lag N. We consider the sample sizes T=50, 500, 5000, 50000, lags N=1, 2, 3, …, 10, and the true parameter values φ= 0.1, 0.2, 0.3, …, 0.9. For each combination, we simulate 1000 draws. The sample means of the obtained estimates are tabulated in Appendix <ref>.Histograms given in Figures <ref>, <ref> and <ref> reflect the effects of the sample size T, AR(1) parameter φ, and the chosen lag N, respectively. In Figure <ref>, the parameter φ= 0.5 and the lag N = 3. In Figure <ref>, the sample size T = 5000 and the lag N = 3. In Figure <ref>, the parameter φ= 0.5 and the sample size T = 5000. The summary statistics corresponding to the data displayed in the histograms are given in Appendix <ref>. Figure <ref> exemplifies the rate of convergence of the estimator as the number of observations grows. One can see that with the smallest sample size, the lower bound is hit numerous times due to the large variance of the estimator. In the upper series of the histograms, the standard deviation reduces from 0.326 to 0.019, whereas in the lower series it reduces from 0.250 to 0.008. The faster convergence in the case of ARMA(1,2) can be explained with the larger value of γ(3) reducing the variance in comparison to the AR(1) case. The same phenomenon recurs also in the other two figures.Figure <ref> reflects the effect of the AR(1) parameter on the value of γ(3) and consequently on the variance of the estimator. The standard deviation reduces from 0.322 to 0.020 in the case of AR(1) and from 0.067 to 0.009 in the case of ARMA(1,2).In Figure <ref> one can see how an increase in the lag increases the variance of the estimator. In the topmost sequence, the standard deviation increases from 0.014 to 0.326 and in the bottom sequence from 0.015 to 0.282.We wish to emphasize that in general smaller lag does not imply smaller variance, since the autocovariance function of the observed process is not necessarily decreasing. In addition, although the autocovariance γ(N) appears to be the dominant factor when it comes to the speed of convergence, there are also other possibly significant terms involved in the limit distribution of Theorem <ref>.§ PROOF OF THEOREM <REF>We provide here a detailed proof of Theorem <ref>. The continuous time version of the theorem was recently proved in <cit.> and we loosely follow the same lines in our proof for the discrete time version. Let H>0. A discrete time stochastic process Y = (Y_e^t)_t∈ℤ with lim_t→-∞ Y_e^t = 0 is H-self-similar if (Y_e^t+s )_t∈ℤlaw=(e^sH Y_e^t)_t∈ℤ for every s∈ℤ in the sense of finite-dimensional distributions.Let H>0. In addition, let X = (X_t)_t∈ℤ and Y = (Y_e^t)_t∈ℤ be stochastic processes. We define the discrete Lamperti transform by (ℒ_H X)_e^t = e^tHX_t and its inverse by (ℒ_H^-1 Y)_t = e^-tHY_e^t. If X = (X_t)_t∈ℤ is strictly stationary, then (ℒ_H X)_e^t is H-self-similar. Conversely, if Y= (Y_e^t)_t∈ℤ is H-self-similar, then (ℒ_H^-1 Y)_t is strictly stationary. Assume that (X_e^n)_n∈ℤ is H-self-similar and let us denote Δ_n X_e^n = X_e^n - X_e^n-1. Then, for every n, l∈ℤ it holds that Δ_n-l X_e^n-llaw= e^-HlΔ_n X_e^n.Since there is one-to-one correspondence between H-self-similar and stationary processes, for some stationary U it holds that X_e^n - X_e^n-1 = e^HnU_n - e^H(n-1)U_n-1law= e^Hn U_n-1 - e^H(n-1) U_n-2= e^H ( X_e^n-1 - X_e^n-2). Repeating the step above l times concludes the proof. Let H>0 and assume that (Y_e^t)_t∈ℤ is H-self-similar. Let us denote Δ_t Y_e^t = Y_e^t - Y_e^t-1. Then the process (G_t)_t∈ℤ defined by G_t = {[∑_k=1^t e^-kHΔ_k Y_e^k,t≥1; 0,t=0; -∑_k=t+1^0 e^-kHΔ_k Y_e^k, t≤-1 ]. belongs to 𝒢_H. By studying the cases t≥2, t=1, t=0 and t≤-1 separately, it is straightforward to see that Δ_t G = e^-tHΔ_t Y_e^tfor every t∈ℤ. Now lim_k→-∞∑_t = k^0 e^tHΔ_t G = lim_k→ -∞∑ _t=k^0 Δ_t Y_e^t = Y_e^0 - lim_k→-∞ Y_e^k and since Y is self-similar, we have Y_e^k e^kH Y_e^0. Thus lim_k→-∞ Y_e^k = 0 in distribution, and hence also in probability. This implies that ∑_t = -∞^0 e^tHΔ_t G is an almost surely finite random variable. Next we show that G has strictly stationary increments. For this, assume that t,s,l ∈ℤ with t>s are arbitrary. Then G_t - G_s= ∑ _k=s+1^t Δ_k G= ∑ _k=s+1^t e^-kHΔ_k Y_e^k = ∑_j = s+l+1^t+l e^-(j-l)HΔ_j-l Y_e^j-llaw=∑_j = s+l+1^t+l e^-jHΔ_j Y_e^j = G_t+l - G_s+l, where the equality in law follows from H-self-similarity of (Y_e^t)_t∈ℤ. Treating n-dimensional vectors similarly concludes the proof.Assume first that X is strictly stationary. In this case X clearly satisfies the limit condition. In addition, there exists a H-self-similar Y such that Δ_t X= e^-tH Y_e^t - e^-(t-1)HY_e^t-1= (e^-H - 1 )e^-(t-1)H Y_e^t-1 + e^-tH(Y_e^t - Y_e^t-1) = (e^-H - 1 )X_t-1 + e^-tHΔ_t Y_e^t. Defining the process G as in Lemma <ref> completes the proof of the `if' part. For the proof of the `only if' part, assume that G∈𝒢_H. From (<ref>) it follows that X_t= e^-HX_t-1 + Δ_tG = e^-2HX_t-2 + e^-HΔ_t-1G + Δ_t G =∑_j=0^n e^-jHΔ_t-j G + e^-(n+1)HX_t-n-1= e^-tH(∑_k = t-n^t e^kHΔ_k G + e^(t-n-1)HX_t-n-1) for every n∈ℕ. Since G∈𝒢_H and lim_m→ -∞ e^mH X_m = 0 in probability, we obtain that X_t = e^-tH∑ _k= -∞^t e^kHΔ_k G for every t∈ℤ. Now, by strictly stationary increments of G, we have e^-tH∑_j=-M^t e^jHΔ_j+s G law= e^-tH∑_j=-M^t e^jHΔ_j G. for every t, M∈ℤ such that -M ≤ t. Since the sums above converge as M tends to infinity, we obtain X_t+s = e^-(t+s)H∑ _ j =-∞^t e^(j+s)HΔ_j+s G law= e^-tH∑_j=-∞^t e^jHΔ_j G = X_t. Treating multidimensional distributions similarly we thus observe that X is strictly stationary. Finally, to prove the uniqueness assume there exist G_1, G_2∈𝒢_H such that e^tH X_t = ∑ _k=-∞^t e^kHΔ_k G_1 = ∑_k=-∞^t e^kHΔ_k G_2 for every t∈ℤ. Then e^tH X_t - e^(t-1)HX_t-1 = e^tHΔ_t G_1 = e^tHΔ_t G_2. Hence Δ_t G_1 = Δ_t G_2 for every t∈ℤ implying that G_1 = G_2 + c. Since both processes are zero at t=0, it must hold that c = 0.Corollary <ref> is almost trivial. However, it is well motivated by Theorem <ref>. On the other hand, Theorem <ref> is far away from trivial as it states both sufficient and necessary conditions. We prove Theorem <ref> using discrete Lamperti transform. In principle, one could consider proving Theorem <ref> by starting from Corollary <ref>. However, at this point, we have not assumed any moment conditions, and thus it is not clear whether a process G constructed from Z^(H) of Corollary <ref> would satisfy G∈𝒢_H. Indeed, a counter example is provided in <cit.>. See also <cit.>, where moment conditions are discussed. § DISCUSSION ON SPECIAL CASES In this appendix we take a closer look at “worst case scenario” processes related to the choice between (<ref>) and (<ref>). These are such processes that, for some 0<a<1, a_j = a for every j∈ℤ. By (<ref>) this is equivalent to γ(j+1)+γ(j-1)/γ(j) = b for every j∈ℤ, where ϕ<b<ϕ+ 1/ϕ. In order to study processes of this form, we consider formal power series. Let f(x) = ∑_n=0^∞ c_nx^n be a formal power series in x. We now define the coefficient extractor operator [·]{*} by [x^m]{f(x)} = c_m Setting j=0 in (<ref>) we obtain that γ(1) = b/2γ (0). This leads to the following recursion. γ(n) = bγ(n-1) - γ(n-2)for n≥2. It follows immediately from the first step of the recursion that b>2 does not define an autocovariance function of a stationary process. Note also that for b=2 Equation (<ref>) implies that γ(n)=γ(0) for every n∈ℤ. This corresponds to the completely degenerate process X_n = X_0. We next study the case 0 <b<2. For this, we define a generating function regarded as a formal power series by f(x) = ∑_n=0^∞γ(n) x^n. Then the coefficients of f(x) satisfy [x^n]{f(x)} = b [x^n-1]{f(x)} - [x^n-2]{f(x)}=[x^n]{bxf(x)} - [x^n] {x^2f(x)}= [x^n]{bxf(x)-x^2f(x)} for n≥2. For simplicity, we assume that γ(0) = 1. By taking the constant and the first order terms into account we obtain f(x) = bxf(x)-x^2f(x)-bx+1+b/2x, which implies f(x) = 1-b/2x/x^2-bx+1. Since the function above is analytic at x=0, the corresponding power series expansion is (<ref>). Furthermore, since the recursion formula is linear, for a general γ(0) it holds that γ(n) = γ(0)[x^n] {(1- b/2x )∑_n=0^∞(bx-x^2)^n }.§ TABLESThe simulation results highlighted in Section <ref> are chosen from a more extensive set of simulations. All the simulation results are given in a tabulated form in this appendix. The two processes considered in the simulations are AR(1) and ARMA(1,2). The used MA parameters are θ_1 = 0.8 and θ_2 = 0.3. The tables represent the efficiency dependence of the estimator on the AR(1) parameter φ and the used lag N. We have varied the column variable AR(1) parameter from 0.1 to 0.9 and the row variable lag from 1 to 10. The tables display the sample means of the estimates from 1000 iterations with different sample sizes. At the end of this appendix, we provide summary statistics tables corresponding to the histograms presented in Section <ref>. 18 brockwell Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991). mr=1093459 Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991) structpybdavis1986limit Davis, R., Resnick, S.: Limit theory for the sample covariance and correlation functions of moving averages. The Annals of Statistics 14(2), 533–558 (1986). mr=0840513 Davis, R., Resnick, S.: Limit theory for the sample covariance and correlation functions of moving averages. The Annals of Statistics 14(2), 533–558 (1986) structpybfrancq2004maximum Francq, C., Zakoïan, J.-M.: Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10(4), 605–637 (2004) Francq, C., Zakoïan, J.-M.: Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10(4), 605–637 (2004) structpybfrancq2005diagnostic Francq, C., Roy, R., Zakoïan, J.-M.: Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100(470), 532–544 (2005) Francq, C., Roy, R., Zakoïan, J.-M.: Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100(470), 532–544 (2005) structpybhamilton1994time Hamilton, J.D.: Time Series Analysis, 1st edn. Princeton university press, Princeton (1994). mr=1278033 Hamilton, J.D.: Time Series Analysis, 1st edn. Princeton university press, Princeton (1994) structpybhannan1980estimation Hannan, E.J.: The estimation of the order of an ARMA process. The Annals of Statistics 8(5), 1071–1081 (1980). mr=0585705 Hannan, E.J.: The estimation of the order of an ARMA process. The Annals of Statistics 8(5), 1071–1081 (1980) structpybhannan1973asymptotic Hannan, E.J.: The asymptotic theory of linear time-series models. Journal of Applied Probability 10(1), 130–145 (1973) Hannan, E.J.: The asymptotic theory of linear time-series models. Journal of Applied Probability 10(1), 130–145 (1973) structpybhorvath2008sample Horváth, L., Kokoszka, P.: Sample autocovariances of long-memory time series. Bernoulli 14(2), 405–418 (2008). mr=2544094 Horváth, L., Kokoszka, P.: Sample autocovariances of long-memory time series. Bernoulli 14(2), 405–418 (2008) structpybkoreisha1990generalized Koreisha, S., Pukkila, T.: A generalized least-squares approach for estimation of autoregressive moving-average models. Journal of Time Series Analysis 11(2), 139–151 (1990) Koreisha, S., Pukkila, T.: A generalized least-squares approach for estimation of autoregressive moving-average models. Journal of Time Series Analysis 11(2), 139–151 (1990) structpyblamperti1962semi Lamperti, J.: Semi-stable stochastic processes. Transactions of the American mathematical Society 104(1), 62–78 (1962) Lamperti, J.: Semi-stable stochastic processes. Transactions of the American mathematical Society 104(1), 62–78 (1962) structpyblevy2011robust Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S., Reisen, V.A.: Robust estimation of the scale and of the autocovariance function of Gaussian short-and long-range dependent processes. Journal of Time Series Analysis 32(2), 135–156 (2011) Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S., Reisen, V.A.: Robust estimation of the scale and of the autocovariance function of Gaussian short-and long-range dependent processes. Journal of Time Series Analysis 32(2), 135–156 (2011) structpyblin2008portmanteau Lin, J.-W., McLeod, A.I.: Portmanteau tests for ARMA models with infinite variance. Journal of Time Series Analysis 29(3), 600–617 (2008) Lin, J.-W., McLeod, A.I.: Portmanteau tests for ARMA models with infinite variance. Journal of Time Series Analysis 29(3), 600–617 (2008) structpybling1997fractionally Ling, S., Li, W.: On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92(439), 1184–1194 (1997) Ling, S., Li, W.: On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92(439), 1184–1194 (1997) structpybmauricio1995exact Mauricio, J.A.: Exact maximum likelihood estimation of stationary vector ARMA models. Journal of the American Statistical Association 90(429), 282–291 (1995) Mauricio, J.A.: Exact maximum likelihood estimation of stationary vector ARMA models. Journal of the American Statistical Association 90(429), 282–291 (1995) structpybmcelroy2012subsampling McElroy, T., Jach, A.: Subsampling inference for the autocovariances and autocorrelations of long-memory heavy-tailed linear time series. Journal of Time Series Analysis 33(6), 935–953 (2012) McElroy, T., Jach, A.: Subsampling inference for the autocovariances and autocorrelations of long-memory heavy-tailed linear time series. Journal of Time Series Analysis 33(6), 935–953 (2012) structpybmiko Mikosch, T., Gadrich, T., Kluppelberg, C., Adler, R.J.: Parameter estimation for ARMA models with infinite variance innovations. The Annals of Statistics 23(1), 305–326 (1995) Mikosch, T., Gadrich, T., Kluppelberg, C., Adler, R.J.: Parameter estimation for ARMA models with infinite variance innovations. The Annals of Statistics 23(1), 305–326 (1995) structpybasymptotics Viitasaari, L.: Representation of stationary and stationary increment processes via Langevin equation and self-similar processes. Statistics & Probability Letters 115, 45–53 (2016) Viitasaari, L.: Representation of stationary and stationary increment processes via Langevin equation and self-similar processes. Statistics & Probability Letters 115, 45–53 (2016) structpybyao2006gaussian Yao, Q., Brockwell, P.J.: Gaussian maximum likelihood estimation for ARMA models. I. time series. Journal of Time Series Analysis 27(6), 857–875 (2006) Yao, Q., Brockwell, P.J.: Gaussian maximum likelihood estimation for ARMA models. I. time series. Journal of Time Series Analysis 27(6), 857–875 (2006) structpyb | http://arxiv.org/abs/1708.07446v2 | {
"authors": [
"Marko Voutilainen",
"Lauri Viitasaari",
"Pauliina Ilmonen"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170824150104",
"title": "On model fitting and estimation of strictly stationary processes"
} |
Broad pore lifetime distributions: A fundamental concept for cell electroporationJulie V. Stern^1, Thiruvallur R. Gowrishankar^1,Kyle C. Smith^1, and James C. Weaver^1,*, ^1Harvard-MIT Division of Health Sciences and Technology, Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA, USA;^*Corresponding authorAbstractWe describe a concept that has the potential to change how we think about the underlying mechanisms of cell membrane electroporation (EP). Prior experimental, theoretical and modeling have emphasized a single pore lifetime as adequate for particular conditions. Here we introduce a much more complex response:The rapid creation of many types of pore structures, of which some are traditional transient lipidic pores (TPs), but the great majority are complex pores (CPs) based on both lipids and other molecules or molecular segments.At the inner leaflet of the cell plasma membrane (PM) non-lipidic molecules come from the over-crowded cytoplasm. At the outer leaflet they originate from the extracellular medium and extracellular matrix. Some partially or fully insert into TPs during or shortly after TP formation, or bind to the membrane nearby. This process is complex, leading to mostly short-lived structures, with relatively few lasting for long times.We speculate that the characteristic pore lifetimes range from ∼100 ns to 1,000 s, based on implications from experiments.The frequency-of-occurrence probably falls off extremely rapidly with increasing lifetime, τ_𝐂𝐏, which implies that most are inaccessible to traditional experimental methods. It also suggests that unexpected behavior can occur early in pulsing,vanishing before post-pulse observations begin.IntroductionDistinguishing cell electroporation (EP) from EP in other membrane systems We present a hypothesis that involves greatly increased complexity of pore creation, evolution and eventual destruction. We believe this is necessary to explain experimental results for cell electroporation (EP), which we explicitly distinguish from vesicular and artificial planar bilayer membrane EP.Although investigation and application of cell EP has been pursued for decades <cit.>, it is generally accepted that the basic mechanisms are poorly understood.For perspective we briefly describe EP generally, and then focus on our hypothesis that cell EP should be distinguished from EP in simpler systems. It is generally agreed that in all membrane systems there are three stages: (1) pore creation during pulsing, with a highly non-linear dependence on transmembrane voltage, Δϕ_m, (2) pore expansion/contraction, also affected by Δϕ_m, and (3) pore persistence and eventual destruction, usually considered for Δϕ_m = 0 (full depolarization), with post-pulse membrane resealing attributed to pore lifetimes.Of these pore creation is reasonably well understood, including agreement between molecular dynamics (MD) and continuum models for transmembrane voltages and associated membrane fields where these very different methods overlap <cit.>.Behavior during pulsing is only occasionally measured <cit.> due to significant technical difficulties, and has therefore received little attention.Pore destruction and associated lifetimes We emphasize pore lifetime, a basic feature of pore destruction.Itis usually considered as the mean value of an exponentially decaying membrane conductance or permeability.A short perspective <cit.> makes the case that the traditional view of pure lipidic pores vanishing stochastically with a long lifetime is not correct.The basic argument is that molecular dynamics (MD) publications show that a basic assumption of the traditional model is invalid, and in addition shows many examples of lipidic pores vanishing with a ∼100 ns lifetime.This has a major implication. For example, if there are ∼ 10^6 pores during a pulse, within about 2μ s after pulsing the probable number of pores is ∼0.001 (essentially zero) <cit.>. Important MD examples include an illustrative progression of pore structures leading to pore destruction <cit.>, an associated pore energy landscape as a function of pore radius that shows a negligible barrier at all to pore destruction <cit.> and an exponential decay fit to a series of pore closures (destructions) that supports a ∼100 ns lifetime <cit.>.The concept of complex cell membrane pores In Figure 1 we show an initial, incomplete picture of a broad lifetime distribution that we propose for cell membrane pores created by large physical perturbations. Although we are mainly concerned with applied electric fields, mechanical perturbations that increase membrane tension may also be involved<cit.>.We envision involvement of both purely lipidic pores (TPs), the traditional view, and also complex pores (CPs) that involve other types of molecules.Presently we do not know what CP structures emerge, as there is a huge, “biologically large”, number of combinations of biological molecules in various states, configurations and binding strengths. <cit.>.For very large electric field strengths a collection of unusual membrane openings and other structures has been explicitly suggested <cit.>. Overall, the response of the plasma membrane (PM) to two complex environments, each with a large number of molecules, should be markedly different than a pure lipid membrane contacting only aqueous electrolyte-based media, even if some have full or residual content from cell growth media. The most complex, and realistic, condition is that of the PM in a living tissue. The inner leaflet is in direct contact with the overcrowded cytoplasm, and the outer leaflet is intimately exposed to the extracellular medium and also contacts the extracellular matrix.It is known that the cortex (cytoskeleton) interacts with the PM during mitosis,and that the cytoskeleton is suggested to limit expanded pore size <cit.>. These and other structures near and or contacting cell membranes provide our general motivation to consider CPs with diverse behavior even if we know little about their structures. Although we argue that traditional TPs vanish within ∼ 2μ s after a well defined pulse <cit.>, we extend our thinking to the possibility for cell membranes close to many molecules of different size, shape (including branching) and charge. This leads to the expectation than large local fields associated with pulsing are heterogeneous on a nanoscale, and these together with fluctuations can lead to a wide variety of membrane openings, complex pores (CPs).We suggest that CPs can contribute to post-pulse behavior for times well beyond the 2μ s estimate for TPs. CPs are “accidental” pores, in analogy with accidental cell death Further, CPs are “accidental”, with presently unknown structures, perhaps rapidly changing, due to random encounters due to the large physical perturbation. The CP structures are not the result of evolution, but are accidental in the same sense that accidental cell death is not regulated, but due instead to large physical interactions <cit.>.This unregulated process is likely to lead to a wide distribution in binding or association energies, with most weak, corresponding to short lifetimes.A consequence is an extremely broad range of CP lifetimes (Figure 1). Experiments that suggest CPs existImportantly, there is experimental evidence that CPs can exist for varying times. These are characteristic times with little information regarding how many pores are involved. The bold upper-case letter A - E in the caption of Figure 1 are associated with experiments, discussed below.Several experiments support the orders of magnitude in B through E. There are additional experiments which report evidence for pores that cannot be traditional TPs<cit.>, and also some conjecture for CP existence.This diverse evidence is presented chronologically.In 2007 an important experiment described the post-pulse recovery of the PM, reporting that the conductance returned to (within experimental noise) its initial value in ∼ 15 m (900 s) <cit.>. We conservatively assume that 5 exponential lifetimes corresponds to essentially full recovery, and this yields a recovery (resealing) lifetime of ∼ 180 s. This is a very long lifetime compared to MD.In 2008 rather different experiments <cit.> reported delayed increases of single cell PM permeability following a single pulse for 40μ s for various electric field strengths. The permeability increases were inferred, based on quantified propidium uptake, at post-pulse times of order ∼ 300 ±∼ 300 s, a huge range in which measurement times extended to 1,800 s. Additional results by the same group were included in a later publication <cit.>, with and without peptide-altered behavior.In 2009 experiments were reported <cit.> with the interpretation that “...nanopores can form a stable, ion channel-like conduction pathway in [a] cell membrane” by nsPEF pulsing protocols. This is an explicit claim that CPs are created, not traditional lipidic pores. In 2011 additional experiments <cit.> reported delayed uptake of propidium, but in this case the perturbation/stimulus was a pulse train with strengths of 1.8 to 12.3 kV/cm, durations of 60 ns to 9μ s, and 2 to 3,750 pulses applied sequentially. Strikingly, these experiments report delayed permeability increases similar to those found earlier for single, 40μ s pulses <cit.>.In 2015 an experimental study reported significant recovery between the individual pulses of a pulse train. This provides evidence that some long-lived pores are involved, with their longevity related to the capability to create more pores after the inter-pulse interval of ∼ 10^-2s new pores are created <cit.>. The concept of evaluating experiments in terms of more pores created after a gap between pulses may provide a powerful approach to quantitative examination of CP lifetimes, perhaps limited mainly by technical issues such as the constraints on separations of the individual members of a pulse train, and the likelihood that between the individual pulsesΔϕ_m is not zero (full depolarization is not realized). We have reported a modeling approach for assessing the degree of recovery between the individual members of a pulse train <cit.>, which should be applicable to this proposed approach for determining CP lifetime ranges at particular times after pulsing.Overall, the use of pulse trains with a wide range of intra-pulse spacings may provide a powerful approach to determining part of a broad lifetime distribution, at least approximately. Recently a rather different type of experiment reported that post-pulse contains a contribution from active transport, not only diffusion <cit.>. The conventional view is that post-pulse the PM is fully depolarized, so that post-pulse transport is thereafter purely passive. The 2017 report shows otherwise. It also provides support for one or more types of long-lived permeability states that exist long after TPs have vanished. SummarySingle lifetimes reported from experiments can be long, but reporting the single value distracts the reader from the possibility that there may be a lifetime distribution.The important distinction is that the broad lifetime distribution (BLD) hypothesis allows early behavior by numerous short lifetime CPs to contribute to behavior (e.g. solute transport that persists), but these CPs are not observed because of their fleeting existence. They can, however, possibly contribute to molecular transport, leaving evidence of their existence and their potential importance to understanding cell electroporation by determining the molecular and ionic transport these CPs allowed. AcknowledgmentsThis work was supported by AFOSR MURI grant FA9550-15-1-0517 on Nanoelectropulse-Induced Electromechanical Signaling and Control of Biological Systems, administered through Old Dominion University.We thank P. T. Vernier,E. Sözer, R. S. Son and A. E. Esser for stimulating discussions,and K. G. Weaver for computer support.References unsrt10NeumannSowersJordan_Book_ElectroporationElectrofusionInCellBiology_Pl enum1989 E. Neumann, A. E. Sowers, and C. A. Jordan, editors. Electroporation and Electrofusion in Cell Biology. Plenum Press, New York, 1989.ChangEtAlEporeBook1992 D. C. Chang, B. M. Chassy, J. A. Saunders, and A. E. Sowers, editors. Guide to Electroporation and Electrofusion. Academic Press, New York, 1992.PakhomovMiklavcicMarkov_AdvancedElectroporationTechniquesBiologyMedic ine_CRC2010 A. G. Pakhomov, D. Miklavčič, and M. S. Markov, editors. Advanced electroporation techniques in biology and medicine. CRC Press, Boca Raton, 2010.VasilkoskiEtAl_ElectroporationAbsouteRateEquationNanosecondPoreCreati on_PhysRevE2006 Z. Vasilkoski, A. T. Esser, T. R. Gowrishankar, and J. C. Weaver. Membrane electroporation: The absolute rate equation and nanosecond timescale pore creation. Phys. Rev. E, 74:021904, 2006.KinositaEtAl_ElectroporationVisualizedPulseLaserFluorescenceMicroscop e_BPJ1988 K. Kinosita, I. Ashikawa, N. Saita, H. Yoshimura, H. Itoh, K. Nagayma, and A. Ikegami. Electroporation of cell membrane visualized under a pulsed-laser fluorescence microscope. Biophys. J., 53:1015–1019, 1988.Frey_PlasmaMembraneVoltageChangesDuringNanosecondPulsedElectricFields _BPJ2006 W. Frey, J. A. White, R. O. Price, P. F. Blackmore, R. P. Joshi, R. Nuccitelli, S. J. Beebe, K. H. Schoenbach, and J. F. Kolb. Plasma membrane voltage changes during nanosecond pulsed electric field exposures. Biophys. J., 90:3608–3615, 2006.FlickingerEtAlFrey_PlantCellsProtoplastsVoltageSensitiveDyeMembraneVo ltages_Protoplasma2010 B. Flickinger, T. Berghöfer, P. Hohenberger, C. Eing, and W. Frey. Transmembrane potential measurements on plant cells using the voltage-sensitive dye ANNINE-6. Protoplasma, 247:3–12, 2010.draftPoreLifetimesShortDiscussionJUL2017_Bchem2017 J. C. Weaver and P. T. Vernier. Pore lifetimes in electroporation: Complex dark pores? (in preparation).LevineVernier_LifeCyclePoreStepsCreationAnnihilation_JMemBiol2010 Z. A. Levine and P. T. Vernier. Life cycle of an electropore: Field-dependent and field-independent steps in pore creation and annihilation. J. Memb. Biol., 236:27–36, 2010.WohlertEtAl_FreeEnergyPoreLinearTermMD_JChemPhys2006 J. Wohlert, W. K. den Otter, O. Edholmc, and W. J. Brielsd. Free energy of a trans-membrane pore calculated from atomistic molecular dynamics simulations. J. Chem. Phys., 124:154905–1–154905–9, 2006.BennettEtAlTieleman_AtomisticSimulationsPoreFormationClosure_BLM_BPJ2 014 W. F. D. Bennett, N. Sapay, and D. P. Tieleman. Atomistic simulations of pore formation and closure in lipid bilayers. Biophysical J., 106:210–219, 2014.Evans_NewMembraneConceptAppliedToFluidShear-LyticTension_BPJ1973 E. A. Evans. New membrane concept applied to the analysis of fluid shear - and micropipette-deformed red blood cells. Biophysical J., 13:941–954, 1973.EvansSmith_KineticsHoleNucleation-BiomembraneRupture_NewJPhys2011 E. Evans and B. A Smith. Kinetics of hole nucleation in biomembrane rupture. New J. Phys., 13:095010, 2011.Frauenfelder_BiologicallyLargeNumbers-GreaterThanAstronomicallyLargeN umbers_Proteins-ParadigmsOfComplexity_PNAS2002 H. Frauenfelder. Proteins: Paradigms of complexity. Proc. Nat. Acad. Sci., 99 Suppl. 1:2479–2480, 2002.PliquettEtAl_HighElectricFieldEMicellsCellMembranes_BChem2007 U. Pliquett, R. P. Joshi, V. Sridhara, and K. H. Schoenbach. High electric field effects on cell membranes. Bioelectrochemistry, 70:275–282, 2007.KrassenPliquettNeumann_SingleCHOcellNonlinearIVcurveElectroporation60 nmPore_BChem2007 H. Krassen, U. Pliquett, and E. Neumann. Nonlinear current - voltage relationship of the plasma membrane of single CHO cells. Bioelectrochemistry, 70:71–77, 2007.PakhomovEtAlPakhomova_MultipleNanosecondPulsesIncreaseNumber-NotLongL ivedPoreSize_BBA2015 A. G. Pakhomov, E. Gianulis, P. T. Vernier, I. Semenov, S. Xiao, and O. Pakhomova. Multiple nanosecond electric pulses increase the number but not the size of long-lived nanopores in the cell membrane. Biochim. Biophys. Acta, 1848:958–966, 2015.SmithKC_CellModelWithDyamicPoresAndElectrodiffusionofChargedSpecies_D octorateThesisMIT_2011 K. C. Smith. A unified model of electroporation and molecular transport. Massachusetts Institute of Technology, http://dspace.mit.edu/bitstream/handle/1721.1/63085/725958797.pdf.CanatellaEtAlPrausnitz_ElectroporationUptakeViabilityQuantitativeStud y_BPJ2001 P. J. Canatella, J. F. Karr, J. A. Petros, and M. R. Prausnitz. Quantitative study of electroporation-mediated molecular uptake and cell viability. Biophysical J., 80:755–764, 2001.PakhomovEtAl_nsPEF_LongLastingSinglePulsePermeabilizationPM_BEMS2007 A. G. Pakhomov, J. F. Kolb, J. A. White, R. P. Joshi, S. Ziao, and K. H. Schoenbach. Long-lasting membrane permeabilzation in mammalian cells by nanosecond pulsed electric field (nsPEF). Bioelectromagnetics., 28:655–663, 2007.KennedyEtAlBooske_QuantificationElectroporationKineticsPI_FinalVersio n_BPJ2008 S. M. Kennedy, Z. Ji, J. C. Hedstrom, J. H. Booske, and S. C. Hagness. Quantitation of electroporation uptake kinetics and electric field heterogeneity effects in cells. Biophys. J., 94:5018–5027, 2008.PakhomovaEtAlPakhomov_Electroporationn-InducedElectrosensitization_PL oS_ONE2011 O. N. Pakhomova, B. W. Gregory, V. A. Khorokhorina, A. M. Bowman, S. Xiao, and A. G. Pakhomov. Electroporation-induced electrosensitization. PLoS ONE, 6:e17100, 2011.GalluzziEtAlKroemer_Essential-vs-AccessoryAspectsCellDeath_Recommenda tions-NCCD2015_SHORT_CellDeathDiff2015 L. Galluzzi, J. M. Bravo-San Pedro, and G. Kroemer. (108 authors total) Essential versus accessory aspects of cell death: recommendations of the NCCD 2015. Cell Death Diff., 22:58–73, 2015.KennedyEtAlMurphy_CationicPeptideEnhancesElectroporation-IRE-Motivati on_PLoS-ONE2014 S. M. Kennedy, E. J. Aiken, K. A. Beres, A. R. Hahn, S. J. Kamin, S. C. Hagness, J. H. Booske, and W. L. Murphy. Cationic peptide exposure enhances pulsed-electric-field-mediated membrane disruption. PLoS ONE, 9:e92528, 2014.PakhomovEtAl_LongLivedLipidNanoporesIonChannel-Like_BBRC2009 A. G. Pakhomov, A. M. Bowman, B. L. Ibey, F. M. Andre, O. N. Pakhomova, and K. H. Schoenbach. Lipid nanopores can form a stable, ion channel-like conduction pathway in cell membrane. Biochem. Biophys. Res. Commun., 385:181–186, 2009.SonEtAl_PulseTrainConventionalEP-20FoldDecreaseThenSawtooth-CalciumUp ake-Electrosensitization_TBME2015 R. S. Son, T. R. Gowrishankar, K. C. Smith, and J. C. Weaver. Modeling a conventional electroporation pulse train: decreased pore number, cumulative calcium transport and an example of electrosensitization. Transactions in Biomedical Engineering, 63:571–580, 2016.SozerLevineVernier_QuantitativeLimits_SmallMoleculeTransport-Electrop ermeome_MeasuringModeling-SingleNanosecondPerturbations_SciRep2017 E. Sözer, Z. A. Levine, and P. T. Vernier. Quantitative limits on small molecule transport via the electropermeome — measuring and modeling single nanosecond perturbations. Sci. Reports, 7:(57–1)–(57–13), 2017. | http://arxiv.org/abs/1708.07613v1 | {
"authors": [
"Julie V. Stern",
"Thiruvallur R. Gowrishankar",
"Kyle C. Smith",
"James C. Weaver"
],
"categories": [
"physics.bio-ph"
],
"primary_category": "physics.bio-ph",
"published": "20170825050117",
"title": "Broad pore lifetime distributions: A fundamental concept for cell electroporation"
} |
Clique-based Method for Social Network ClusteringGuang Ouyang, Dipak K. Dey, Panpan Zhang===================================================================================================================== In this article, we develop a clique-based method for social network clustering. We introduce a new index to evaluate the quality of clustering results, and propose an efficient algorithm based on recursive bipartition to maximize an objective function of the proposed index. The optimization problem is NP-hard, so we approximate the semi-optimal solution via an implicitly restarted Lanczos method. One of the advantages of our algorithm is that the proposed index of each community in the clustering result is guaranteed to be higher than some predetermined threshold, p, which is completely controlled by users. We also account for the situation that p is unknown. A statistical procedure of controlling both under-clustering and over-clustering errors simultaneously is carried out to select localized threshold for each subnetwork, such that the community detection accuracy is optimized. Accordingly, we propose a localized clustering algorithm based on binary tree structure. Finally, we exploit the stochastic blockmodels to conduct simulation studies and demonstrate the accuracy and efficiency of our algorithms, both numerically and graphically.Keywords: Clique-score index, localized clustering algorithm, modularity, social network, spectral analysis,stochastic block model § INTRODUCTION Networks are proliferating all around us, and they appear in different forms, such as (hardwired) electrical grids or (virtual) social relationships. Networked systems spread in various scientific and applied disciplines, for example, the Internet, the World Wide Web, metabolic networks, neural networks, food webs and social networks, etc. In this paper, we place our focus on social networks. Social network analysis is a branch of the social science which is an academic discipline studying a society and the behavior of entities therein. In sociometric or other quantitative studies, social networks are usually modeled by graph structures consisting of a set of nodes or vertices connected by directed arcs or undirected edges. More specifically, the actors in a social network are represented by nodes. For each pair of nodes, if there appears some pattern of ties, they will be connected by an edge. Edges can be directed or undirected, depending on the feature and interpretation of the network. There are a variety of social networks, such as the Facebook <cit.>, marriage networks <cit.> and smart business networks <cit.>.Due to solid theoretical foundation of statistics, statistical methodologies have been a significant input to the studies of social networks. A plethora of statistical methods have been established and developed to uncover relational structure of social networks, and dyadic ties between actors therein. In this paper, we focus on a significant research topic in social networks—clustering. More precisely, our goal is to develop some statistical methods for accurately clustering actors in a social network into mutually exclusive communities. This process, in many literatures, is called “community detection.” Roughly speaking, the fundamental principle underlying social network clustering is that a group of actors who are excessively connected are more likely to form a community. Technically speaking, the formation of a community requires that the connections of actors within the community are significantly higher than the connections between actors from different communities. From the sociological point of view, the occurrence of high connection density between actors in a community is usually due to some kind of homology or homogeneity of actors. For instance, consider a friendship network on Facebook. Intrinsically, students from the same department of a college are more likely form a friendship community, as they have a very high probability to know and friend each other. The homogeneities are reflected in the location parameter (i.e., college) and the academic parameter (i.e., department). On the other hand, students with different education, social or geographic background are much less likely to be connected. Past research on social network clustering can be summarized into two categories. One approach is to propose a (parametric or nonparametric) probabilistic graphical model (PGM) which characterizes the community structures of a social network. Pioneering work of such model-based approach were the p_1 model <cit.> and the stochastic blockmodel (SBM) <cit.>. Several successful models were proposed for community detection in the last two decades, including an extension of the SBM <cit.>, a latent position model <cit.>, a latent position cluster model <cit.>, and a mixed membership SBM in <cit.>, etc. We refer the interested readers to <cit.> for a complete and comprehensive review of PGMs for social network clustering. Another approach is to consider a metric that can be used to quantitatively evaluate the quality of social network clustering. The task is to specify an objective function based on the proposed metric, and the ultimate goal is to design an efficient algorithm to optimize the objective function over all potential clustering strategies. In this article, we call this class of approaches metric-based methods.Precursory work in this direction traced back to <cit.>, in which a measure called cluster coefficient was proposed to evaluate mutual acquaintance between actors in a social network. We refer the interested readers to <cit.> and <cit.> for extensive discussions about cluster coefficient. While cluster coefficient is the metric that is often used for random graphs or dynamic network models, we place our focus on clustering problems of static networks in this manuscript. Representative and popular methods for static network clustering are the spectral clustering method <cit.>, the normalized cut approach <cit.>, and the modularity maximization method <cit.>, etc.The contribution of this paper is as follows: * We propose a novel clustering method based on graph cliques such that the clique score (a new index defined to evaluate clustering outcomes) of each identified community is higher than some threshold p. * When threshold p is unknown, we develop a systematic strategy to select localized thresholds p (for each subnetwork which requires further subdivision) by controlling over-clustering and under-clustering simultaneously. * The two algorithms proposed in this manuscript are efficient, easy to implement, and provide consistently reliable clustering results. We would like to point out that even though undirected binary social networks are considered in this manuscript for the sake of interpretation, our method can be extended to directed or weighted networks in a similar manner, done mutatis mutandis. The rest of this manuscript is organized as follows. In Section <ref>, we introduce some notations that will be used throughout the paper. In Section <ref>, we briefly review two classical clustering methods. Section <ref> is divided into two subsections. In Section <ref>, we propose a new metric to quantitatively measure the quality of clustering results, and then establish an objective function based on it, followed by an algorithm which is designed upon the idea of recursive bipartition to optimize the objective function. In Section <ref>, we consider the situation that the global parameter p is not predetermined, and develop a method to compute localized parameter p so as to maximize the detection accuracy of our algorithm. We further propose a localized algorithm (corresponding to localized parameter p), with some modifications on the algorithm developed in Section <ref>. We present several simulation examples in Section <ref> to demonstrate the efficiency of our algorithms and accuracy and consistency of clustering results. In addition, we numerically compare the performance of the proposed algorithm and the traditional modularity maximization algorithm. Lastly, we address some concluding remarks and potential future studies in Section <ref>.§ NOTATIONS In this section, we introduce the notations that will be used throughout the manuscript. Let G = G(V, ) be an undirected graph to model a social network of|V| = n actors, whereis an n × n adjacency matrix such that= (a_ij)_n × n,in which a_ij =1,, 0,,for i, j ∈ V. Letbe an n × n diagonal matrix such that = [ (1) 0 ⋯ 0; 0 (2) ⋯ 0; ⋮ ⋮ ⋱ ⋮; 0 0 ⋯ (n) ],where (i) represents the degree of node i ∈ V. Consider the matrixdefined as=- .The normalized Laplacian matrix of G <cit.> is then given by= ^-1/2^-1/2 =- ^-1/2^-1/2,wheredenotes the identity matrix of rank n.Two graph invariants on which many classical network clustering methods depend on are volume and cut. The volume of a graph G, denoted (G), is the total number of degrees of the nodes in G, i.e., (G) = ∑_i = 1^n(i). The cut, on the other hand, is defined on subgraphs of G. Let G_1 and G_2 be two disjoint subgraphs of G, the cut of G_1 and G_2, denoted (G_1, G_2), is the number of edges linking G_1 and G_2, i.e.,(G_1, G_2) = ∑_i ∈ G_1, j ∈ G_2 a_ij.§ PREREQUISITES In this section, we give brief reviews of the spectral clustering method with a concentration on the normalized cut and the modularity maximization approach as prerequisites. §.§ Spectral network clustering We first look at the spectral network clustering algorithm developed in <cit.> and <cit.>, inspired from the spectral graph theory <cit.>. Generally speaking, a measure called the normalized cut was adopted as an objective function to quantify the number of edges across different communities. Suppose that a social network G is split into h communities G_1, G_2, …, G_h. The normalized cut is given by(G_1, G_2, …, G_h) = ∑_k = 1^h(G_k, G_k)/(G_k),where G_k denotes the complement of G_k in G, for k = 1, 2, …, h. Consider an n × h community indicating matrix = (p_ik)_n × h, in which each entry is p_ik = 1/√((G_k)) for node i in community G_k. The optimization problem corresponding to Equation (<ref>) is, in fact, a discrete trace minimization problem, which can be approximated by a standard trace minimization problem with a relaxation of the discreteness condition as follows:min_G_1, G_2, …, G_h(^⊤) ≈min_∈ℝ^n × h(^⊤),subject to ^⊤ =. Let = ^1/2. The optimization problem on the right hand side of Equation (<ref>) is equivalent tomin_∈ℝ^n × h(^⊤),subject to ^⊤ =. This is a standard Rayleigh quotient problem, and the solution ofis composed of the first h eigenvectors of ; see <cit.> and <cit.>. Thus, the solution to the right hand side of Equation (<ref>) can be obtained by solving the first h eigenvectors in a generalized eigenvalue system = λ, which can be done via an adaptive algorithm based on bipartition proposed in <cit.>. §.§ Modularity maximization Another popular algorithm for social network clustering, known as the modularity maximization, was first introduced in <cit.>. In essence, the underlying principle is to partition a social network into mutually exclusive communities such that the number of edges across different communities is significantly less than the expectation, whereas the number of edges within each community is significantly greater than the expectation. In <cit.>, a bipartition situation was considered, and the clustering outcome was evaluated by a measure—modularity—defined asQ = 1/2 (G)^⊤,whereis an n × 1 column indication vector such thats_i = 1,; s_i = -1,,and = (b_ij)_n × n is an n × n matrix with entiresb_ij = a_ij - (i)(i)/(G),for i = 1, 2, …, n. Subdivisions for existing communities are available by repeatedly implementing the proposed modularity algorithm. The decision of whether or not subdividing an existing community G_k of size n^G_k depends on an associated modularity matrix ^G_k = (b^G_k_ij)_n^G_k× n^G_k with entriesb^G_k _ij = b_ij - δ(i, j) ∑_l ∈ G_k b_il,where δ(·, ·) denotes the Kronecker delta function. Subdivision for G_k is terminated if the largest eigenvalue of ^G_k is zero. § CLIQUE-BASED CLUSTERING ALGORITHM In this section, we propose a novel algorithm for social network clustering. More specifically, we define a new measure called the p-clique index to quantitatively evaluate the quality of clustering outcome. The section is divided into two subsections. We introduce a clique-based clustering algorithm in Section <ref>, and then extend it to a localized clique-based clustering algorithm in Section <ref>. The localized algorithm enables us to update the threshold p for each subnetwork so as to well control the potential over-clustering or under-clustering problems. We will discuss the details in the sequel. §.§ Clique-based clustering algorithm In this section, We give a clique-based clustering algorithm, inspired from the modularity maximization algorithm proposed in <cit.>. In graph theory, a clique is defined as a complete graph on a set of nodes, i.e., each pair of nodes is connected by an edge. A clique is called maximal if it cannot be extended to any larger-size clique by including any adjacent node. The ideal clustering outcome is that each community in a network is a maximal clique. In reality, this is hard to achieve. For many real world social networks, it is even difficult to guarantee that each community forms a clique. Therefore, an appropriate measure to assess the degree of connectivity of a community is needed. The next graph invariant measure can be used to gauge the internal link density of each community in a network. [clique score] The clique score of a community (cluster, subnetwork) is the ratio of the number of observed ties to the number of edges in the clique over the same number of nodes.For example, a community consisting of 10 nodes and 18 internal links has clique score 18/102 = 0.4. It is obvious that a clique always has clique score one. Our goal is to develop an adaptive algorithm to maximize the overall clique scores (e.g., weighted average) of all communities in a network, subject to the clique score of each community exceeding some predetermined threshold 0 ≤ p ≤ 1. The condition of “exceedance” in our algorithm is essential as it guarantees that none of the communities in our clustering outcome performs extremely bad or does not achieve the minimum standard. In addition to this, the choice of p is flexible, depending on the users' needs or the realistic features of communities in a social network. In Section <ref>, we will discuss how to choose an appropriate value of p to optimize the performance of the proposed algorithm when no prior information of p is available. In the next definition, we bridge the gap between clique graph and p. [p-clique] A p-clique is a random graph of a set of nodes, of which each pair is connected by an edge independently with probability p, for 0 ≤ p ≤ 1.The p-clique defined in this manuscript is not novel, and structurally it is equivalent to themodel proposed by <cit.>. The definition of p-clique (c.f. Definition <ref>) adopts an alternative interpretation of themodel given by <cit.>. Since the nodes are connected independently, there is no structure of communities or clusters in p-cliques theoretically. Hence, p-cliques appear to be a proper benchmark model in our study. The expected number of edges in a p-clique on n nodes is np(1 - p).Suppose that a social network G of size n is clustered into h communities, G_1, G_2, …, G_h respectively with community sizes n_1, n_2, …, n_h such that ∑_k = 1^h n_k = n. Our task is to search for a clustering rule such that the total degree of nodes in each community is significantly larger than the expected number of edges of a p-clique of the same size, whereas the total number of links across different communities is minimized. Based on this idea, we propose a measure called the p-clique index as follows, and our ultimate goal is to design an adaptive algorithm for an optimal clustering rule where the p-clique index is maximized. [p-clique index] Let G be a social network that consists of n nodes, and C = [G_1, G_2, …, G_h] be a clustering rule which divides G into h communities. The p-clique index is given by D(, p)= 1/n(n - 1)(∑_k = 1^h((G_k) - p n_k (n_k - 1)) . - . ∑_1 ≤ k ≠ l ≤ h( Cut(G_k, G_l) - p n_k n_l ) ) = 1/n(n - 1)∑_1 ≤ i ≠ j ≤ n((a_ij - p)δ(c_i, c_j) + (p - a_ij)(1 - δ(c_i, c_j))), where = (c_1, c_2, …, c_n)^⊤ is the membership indication vector for nodes.The essence of p-clique index is to reward connected nodes in the same community (with 1 - p) and disconnected nodes from different communities (with p), but penalize connected nodes from different communities (with -(1 - p)) and disconnected nodes in the same community (with -p). Different from model-based methods, this approach is not to fit data (i.e., observation) to any type of generative model or p-clique structures. Rather, our goal is to determine a clustering rule, in which each community has a higher clique score than a predetermined threshold p.The algorithm for our clique-based clustering approach is based on the hierarchical clustering algorithm developed for modularity maximization in <cit.>. To begin with, we consider bipartition, i.e., clustering a social network into two communities 1 and 2. Define an alternative membership indication vector, = (s_1, s_2, …, s_n)^⊤, as follows:s_i = 1,, s_i = -1,,for i = 1, 2, …, n. Then, the p-clique index (c.f. Equation (<ref>)) is equivalent to D(, p) = 1/n(n - 1)∑_1 ≤ i ≠ j ≤ n (a_ij - p) s_i s_j = 1/n(n - 1)^T(p) ,where the p-clique matrix (p) is given by(p) =- p(_n × n - _n × n),in which _n × n is an n × n matrix of all ones. In what follows, the network clustering is converted to an optimization problemmax_{^⊤(p) }.This is equivalent tomax_{D(, p) - D(_n × 1, p)}= 1/n(n - 1)max_{^⊤(p)- ^⊤_n × 1(p) _n × 1}= 1/n(n - 1)max_{^⊤( (p) - (1/n∑_i, j = 1^n C(p)_ij) _n × n) },where C(p)_ij denotes the (i, j)th entry of p-clique matrix (p). Since = {-1, 1}^n is dyadic, the optimization problem is NP-hard. To relax the problem, our strategy is to allowto be any normalized real-valued vector. One solution to Equation (<ref>) is the eigenvector = (v_1, v_2, …, v_n)^⊤ corresponding to the largest eigenvalue of (p) - (1/n∑_i, j = 1^n C(p)_ij) _n × n, denoted by λ_ max. We thus obtain a natural approximation of the solution; that is, we cluster the nodes with respect to the signs of the components of :s^ approx_i = 1,v_i ≥ 0, s_i^ approx = -1,v_i < 0,for i = 1, 2, …, n.We remark that we would rather work on the optimization problem in Equation (<ref>) than that in Equation (<ref>) analytically, albeit they are mathematically identical. The underlying reason is that Equation (<ref>) provides an instinctive insight of the decision rule of partitioning an existing community. Our rule is that we do not further subdivide the existing cluster if the eigenvalue λ_ max≤ 0 and D(_n × 1, p) ≥ 0. Both of the conditions are needed since λ_ max≤ 0 indicates that D(_n × 1, p) ≥max{max_{D(, p)}, 0 } and D(_n × 1, p) ≥ 0 implies that the link density of the existing community is already higher than p.We next propose an algorithm based on recursive bipartition for our p-clique approach. In practice, we first determine the p-clique matrix, ^(G)(p) for a given network G of size n. We partition G into two clusters G_1 and G_2 with respect to the signs of the entries of the eigenvector corresponding to the largest eigenvalue of ^(G)(p), and then compute the additional contribution to the p-clique index due to the division, denoted by Δ D(p) = 1/n(n - 1)∑_i, j ∈ G(s_i C^(G)(p)_ij s_j - C^(G)(p)_ij).We continue to apply the bipartition algorithm respectively to G_1 (according to the associated p-clique matrix ^(G_1)(p), where ^(G_1)(p) is a submatrix extracted directly from ^(G)(p) only for the nodes contained in G_1) and G_2 (according to the associated p-clique matrix ^(G_2)(p) obtained in a similar manner), if at least one of the two regularity criteria is met. We terminate the algorithm until no further partition is needed. The algorithmic procedure is presented in Algorithm <ref>.1em1emTo conclude this section, we address some remarks on the proposed algorithm. The essence of Algorithm <ref> is to maximize the additional contributions to the overall p-clique index, whenever a division of an existing cluster is implemented (versus its current state) . As mentioned, the optimization problem is NP-hard, and our strategy is to relax the problem by allowingto be any real vector with ℓ_2 norm equal to one. Our goal is to determine the eigenvector associated with the largest eigenvalue of the p-clique matrix, and specifically our strategy is to exploit an implicitly restarted Lanczos method developed in <cit.>. It is worthy of noting that our algorithm will continue to be executed if ∑_i,j ∈ G^' C^(G^')_ij(p) < 0 for some G^' even when Δ D(p) ≤ 0, since, under such circumstance, the clique index of G^' is less than p. We take the quality (i.e., having clique score higher than predetermined threshold p) of each resulting community as a higher priority, bearing with non-optimal grade for the overall p-clique index. We view this as a major feature, as well as an advantage, of our algorithm, since the choice of threshold p is absolutely flexible, depending on the users' needs. On the other hand, a natural question arising from our algorithm is “ what if there is no specific requirement or prior information for p?” An arbitrary choice of p may lead to over-clustering or under-clustering problems. Having this concern in mind, we propose a modified algorithm based on Algorithm <ref> in Section <ref>.In addition, we carry out a statistical procedure for “reasonable” selection of unspecified p so as to simultaneously minimize the errors of over-clustering and under-clustering. §.§ Localized clustering algorithm We have demonstrated that the algorithm proposed in Section <ref> is advantageous for social network clustering, as it allows users to control the quality of each resulting community. In many cases, the threshold parameter p is predetermined or preselected, based on users' needs or past research experience. For the cases that p is not specified, it is needed to develop a systematic method to choose an optimal assignment of p. Probably a question that needs to be answered in advance is whether or not there exists such optimum p. An analogous issue occurs for the modularity maximization algorithm, and was discussed in <cit.> and <cit.>, where the conclusion from both was negative for the existence of such overall optimal value of p. In this section, we consider a localized clustering strategy, the core idea of which is to select different values of p for different subnetworks. In essence, our strategy for the selection of p is a process of balancing over-clustering and under-clustering. Intrinsically, a large value of p usually results in small sizes of clusters (i.e., over-clustering), whereas a small value of p sometimes fails to guarantee the quality of clusters (i.e., under-clustering). The procedure that balances between over-clustering and under-clustering is analogous to dealing with type I errors and type II errors in statistical hypothesis testings. We borrow these two terminologies in our study. Let us call the error of over-clustering as type I error and the error of under-clustering as type II error. Our goal is determine a distinct value of p for each existing cluster such that both types of errors are well controlled.Recall the p-clique (i.e., themodel) introduced in Definition <ref>. The model is constructed completely at random without any cluster structure, suggesting that we can exploit it to control type I error, and accordingly determine an upper bound of p. Consider angraph, ER(n, p_0), where n and p_0 are given. Intuitively, we need to set the threshold parameter p significantly less than p_0 so that there is a small probability to divide ER(n, p_0) into two subgraphs or more by our algorithm. Assume that α is the maximal percentage of the number of nodes to be split from thegraph under tolerance. Our goal turns to find the largest value of p such that at most α n nodes are split off of the majority. Let X denote the observed inter-across link density between the split group (i.e., α n nodes) and the remainder (i.e., (1 - α)n nodes), and F_X be its distribution. An instinctive and reasonable choice of p is thus the αth percentile of F_X. Although the expected number of inter-across edges is (1 - α)np_0 in theory, the inter-across link density is anticipated to be much smaller than the internal density within the large group of (1 - α) n nodes. In this regard, we approximate F_X by a truncated normal distribution, where the top α is curtailed; that is, X is normally distributed with mean[X] = p_0 - ϕ(z_α)/1 - α√(p_0(1 - p_0)/(1 - α)n),and variance[X] = p_0 (1 - p_0)/(1 - α)n[1 - z_αϕ(z_α)/1 - α - (ϕ(z_α)/1 - α)^2],where ϕ(·) denote the density function of the standard normal distribution, and z_α is the (1 - α)th percentile of standard normal. Thus, we obtain an upper bound for p as a function of α, which is given byp_ U(α)= max{0, [X] - z_α√([X])}= max{0, p_0 - ξ(α) √(p_0 (1 - p_0)/(1 - α)n)},where ξ(α) is a constant depending on α only.The next task is to determine the lower bound for p. Suppose that we have two independentgraphs which are respectively denoted by ER(n_1, p_1) and ER(n_2, p_2), and p_12 is the link density between the two graphs. Let β be the probability that the two graphs are merged. We once again consider normal approximation, and conclude that the two graphs are both statistically significant if we havemin{p_1, p_2} > p_12 + z_β√(p_12(1 - p_12)/n_1 n_2),the right-hand side of which in fact forms the lower bound for the parameter p. However, this lower bound is intractable in general as it requires the information of p_12, which is usually unknown for most real-world networks. Therefore, we place focus on the upper bound for p, and our strategy is to set the parameter p as large as possible under tolerance so as to minimize the probability of merging significant subnetworks.Finally, we propose a new algorithm for unspecified p based on Algorithm <ref> proposed in Section <ref>. The new algorithm is designed for controlling type I and II errors simultaneously. As mentioned, however, it seems to be unreasonable to have a global threshold p for our algorithm. We elaborate the reasonings and demonstrate our solution via the following illustrative example. Suppose that we cluster a network G of size n into two subnetworks G_1 of size n_1 and G_2 of size n_2. Initially, we adopt a threshold p^(G) which depends on the observed link density of G. However, if we continue to use p^(G) as the threshold when G_1 or G_2 or both need to be further clustered, we may have to bear with over-clustering or under-clustering risks in the subdivision processes. Alternatively, we suggest to reset the threshold(s) before subdividing G_1 or G_2 or both. In other words, we choosep^(G_1) = max{0, p_1 - ξ(α)√(p_1 (1 - p_1)/(1 - α)n_1)}andp^(G_2) = max{0, p_2 - ξ(α)√(p_2 (1 - p_2)/(1 - α)n_2)}as updated thresholds for G_1 and G_2, respectively. From then on, the threshold parameters are refreshed in this manner for all subnetworks that need to be further subdivided. As for each subnetwork to which a fresh threshold parameter is assigned, we call the new algorithm localized clustering algorithm. One of the most effective ways to illustrate this heirachical clustering process is probably to exploit binary tree. We start with a root node that represents the original network which needs to be clustered. At the first level, two child nodes are attached to the root node. The child nodes represent either subnetworks or communities. If a child node is a subnetwork which requires further subdivision, it will carry over two higher-level child nodes and itself turns to an internal node. If a child node is a community which does not need further subdivision, it becomes a terminal node in the tree. We use rectangle for internal nodes and circle for terminal nodes. When the algorithm is terminated, the clustering result is reflected in all the terminal nodes in the binary tree. An example of binary tree is given in Figure <ref>. We are now in the position of defining a new clique index, the local clique index, modified from the p-clique index given in Definition <ref>. [local clique index] Consider a network G consisting of n nodes. Let T be the binary tree that describes a hierarchical clustering procedure on G, and {v} be the collection of all internal nodes of T. The local clique index is given by LD(T) = 1/n(n - 1)∑_v ∈ T∑_i ≠ j ∈ v((a_ij - p_v)δ_ij^(v) + (p_v - a_ij)(1 - δ_ij^(v))), where p_v is the threshold of parameter p for internal node (subnetwork) v. As shown in Figure <ref>, our new algorithm is again based on recursive bipartition procedures, and the goal is to maximize the score function given in Equation (<ref>). Analogous to Algorithm <ref>, we need to determine the local clique matrix for each v in T, i.e.,^(v) = ^(v) - p_v (_n_v × n_v - _n_v × n_v),where ^(v) is the updated adjacency matrix for subnetwork v and n_v is the size of v. Besides, we need to compute the additional contribution from each bipartition of v to the overall score LD(T), i.e.,Δ LD(v) = 1/n(n - 1)∑_i ≠ j ∈ v((a_ij - p_v)δ_ij^(v) + (p_v - a_ij)(1 - δ_ij^(v))).The algorithm for hierarchical clustering process is terminated as Δ LD(v) ≤1/n(n - 1)∑_i ≠ j ∈ v (a_ij - p_v) for all existing v in T, and then all remaining internal nodes v turn out to be terminal nodes, i.e., added to the community list. We summarize our strategy in Algorithm <ref>, slightly modified from Algorithm <ref>.1em1em§ SIMULATIONS In this section, we conduct some simulation studies to evaluate the performance of two clique-based clustering algorithms proposed in the previous sections. We also show that the proposed algorithm outperforms the modularity maximization approach <cit.> for network clustering. The SBM in <cit.> is adopted to generate networks, as they allow us to predetermine the structure of simulated networks, which can be used as the ground truth for comparisons.To begin with, we briefly review the SBM. Given a network of n nodes which belong to h nonempty communities, let n_1, n_2, …, n_h be respectively the size of each community. For i = 1, 2, …, n, a mapping c: {1, 2, …, n}↦{1, 2, …, h} preserves the membership information for each node labeled with i. An h × h probability matrixdescribes the link densities within every community, as well as those between different communities, i.e., = (B_c_(i), c_(j)). Notice that we have to require B_c_(i), c_(j)≤min{B_c_(i), c_(i), B_c_(j), c_(j)} for all i, j, when predefining matrix ; otherwise, the simulated network would be against the clustering structure obtained by our algorithm(s). Suppose that(the probabilistic structure of network) is specified, we are able to simulate the entries in the adjacency matrix a_ij =Bernoulli(B_c_(i), c_(j))for i < j. Suppose that the network is undirected, we have a_ji equal a_ij by symmetry. We also assume thatall the entries on the diagonal equal 0, as loops are not considered in our study.To quantitatively evaluate the performance of algorithms, we adopt two well-defined robust measures: the normalized mutual information (NMI) and the Adjusted Rand Index (ARI), respectively proposed in <cit.> and in <cit.>. Suppose that T is the ground truth of community structure, and S is the clustering result of an algorithm, the NMI of T and S is given byNMI(T, S) = -2 ∑_k = 1^C_T∑_l = 1^C_S N_kllog(N_kl N_··/N_k ·N_· l)/∑_k = 1^C_T N_k ·log(N_k ·/N_··) + ∑_l = 1^C_S N_· llog(N_· l/N_··),where C_T and C_S are the number of communities for T and S, respectively; (N_kl) is a C_T × C_L confusion matrix, in which N_kl denotes the number of nodes that should be in community k according to the truth, but are mis-clustered into community l according to algorithm S; N_k ·, N_l ·, and N_·· are standard definitions of the sum of the kth row, the sum of the lth column, and the overall sum of the confusion matrix, respectively. We borrow the same notations and give the definition of ARI as follows: ARI(T, S) = ∑_k = 1^C_T∑_l = 1^C_SN_kl2 - [∑_k = 1^C_TN_k ·2∑_l = 1^C_SN_l ·2] /N_··2/1/2[∑_k = 1^C_TN_k ·2 + ∑_l = 1^C_SN_l ·2] - [∑_k = 1^C_TN_k ·2∑_l = 1^C_SN_l ·2] /N_··2. In addition, we propose a measure analogous to NMI.Once again, assuming that T is the true structure of the simulated network based on SBMs, and S is the analogy from our clustering algorithm, we consider two n × n binary matrices, (M^(T)_ij) and (M^(S)_ij), defined respectively as follows:M_ij^(T) =1,, 0,;and M_ij^(S) =1,, 0,.Our measure is defined on a cluster level, and in the form of an h × h matrix. The fundamental principle of our measure is simple; that is, we compute the proportion of mis-clustered nodes, from cluster to cluster. More precisely, for all nodes in different communities k and l (k ≠ l) under T, the error measure is given byϵ_kl = ∑_i, j c^(T)(i) = k, c^(T)(j) = l|M_ij^(T) - M_ij^(S)|/n_k n_l,where n_k and n_l are the sizes of communities k and l, respectively. The error measure for a single community k is defined analogously, i.e.,ϵ_kk = ∑_i, j c^(T)(i) = c^(T)(j) = k|M_ij^(T) - M_ij^(S)|/n_k (n_k - 1).Ultimately, the error matrix is given by= (ϵ_kl)_h × h.We would like to point out that our error matrix measure can be more appealing in some comprehensive analysis, as it shows where (i.e., in which community or communities) exactly a large amount of mis-clustering occurs when the clustering result far deviates from ground truth. §.§ Effect of threshold p In the first experiment, we show that a correct choice of threshold p has a significant impact on the p-clique index clustering algorithm (c.f. Algorithm <ref>). We simulate an SBM consisting of 120 nodes which are clustered into three communities of sizes 100, 10 and 10, respectively. The parameters of in-group and cross-group link densities are summarized in Table <ref>. The expected clique score of the simulated SBM1 is approximately 0.1597, which can be used as an estimate of the overall link density of the network. Let the error parameter α be equal to 0.025. According to Equation (<ref>), the upper bound for the threshold p is 0.09541. On the other hand, the lower bound for p (c.f. Equation (<ref>)) is max{0.0635, 0.0927} = 0.0927. Suppose that we choose p = 0.09, simulate 100 independent SBM1s, and compute the distance-based error matrix . Albeit the threshold p is just slightly less than the lower bound, we have a large under-clustering error for clusters 2 and 3, i.e., ϵ_23 = ϵ_32 = 0.2218 with standard error SE(ϵ_23) =SE(ϵ_32) = 0.0376, indicating that the probability of misclassifying nodes from these two clusters is about 22% in average. Suppose that the value of p is increased to 0.11 (even though p = 0.11 is greater than the upper bound), the errors ϵ_23 = ϵ_32 drop dramatically to 0.0614 with standard error SE(ϵ_23) =SE(ϵ_32) = 0.0172. As ϵ_23 = ϵ_32≈ 6 % is the largest entry in the error matrixfor p = 0.11, it seems that over-clustering does not bring too much trouble in this experiment; the reason is that all predefined in-group densities are significantly larger than cross-group densities. Due to the limit of space, we refer the interested readers to <cit.> for more analogous examples.Nevertheless, as long as the parameter p is fairly close to the proposed threshold selection interval from the above, the clustering outcomes for this experiment (c.f. Table <ref>) are under satisfactory. We choose the optimal value of p = 0.9527, and depict the result in Figure <ref>, where the three communities are clearly identified and colored by blue, red and green.§.§ Global threshold v.s. localized thresholds The next experiment is designed to compare the performance of the algorithms proposed in this manuscript. We show that the localized algorithm provides more reliable outcomes when the threshold p is unknown. We consider SBM2 with 140 nodes which are clustered into three communities of sizes 100, 20 and 20, respectively. The parameters of in-group and cross-group link densities are summarized in Table <ref>. The expected clique score of the simulated SBM2 is approximately 0.1546. We again set the error parameter α at 0.025. The associated threshold parameter p according to Equation (<ref>) is 0.0959. As the parameter p is less than the cross-group link density between communities 2 and 3, it seems to be difficult to separate the two clusters with the overall threshold p via Algorithm <ref>. To verify our conjecture, we simulate 100 independent SBM2s, compute the NMI for each replication, and take the average as an estimate. We obtain NMI = 0.8488 with standard error 0.0025. In addition, we compute the proposed block-wise distance-based measure, summarized in Table <ref>. Although community 1 is successfully identified, we observe that the estimate of error rate between communities 2 and 3 is ϵ_23 = 0.9890 = 98.90 %, which suggests that the BiPartition algorithm with the global threshold p = 0.0959 fails to separate these two communities almost surely. This is also reflected in the clustering result (via Algorithm <ref>) of the simulated SBM (c.f. Table <ref>) given in Figure <ref>. The entire network is divided into two cluster (rather than three), which are colored by red and green. We remedy this problem by applying the localized clustering algorithm, i.e., Algorithm <ref>. We start with the root node (the original simulated SBM) in the binary tree structure. We use threshold p = 0.0959 to bipartition the root node, and obtain community 1, and subnetwork 2, where the latter requires for further clustering. Subnetwork 2 consists of 40 nodes and the expected clique score is approximately 0.4026. With the same α = 0.025, the updated threshold parameter for subnetwork 2 is 0.2536. We would like to mention that in practice we also compute the threshold parameter for subnetwork 1 (community 1), and obtain 0.1231 for implementing Algorithm <ref>, where we find that no further subdivision is needed. After subdividing subnetwork 2, we continue to compute new thresholds for both subsequent subnetworks, and ultimately find that they form the other two communities in the network.Another 100 independent SBMs are simulated, and the NMI for each simulated network is computed. The mean estimate is NMI_ local = 0.9677 with standard error 0.0034. We also compute the block-wise error rates, and present them in Table <ref>, where we find a significant improvement of ϵ_23 = ϵ_32 = 0.0019 versus 0.9890. The small error rate implies that the communities 2 and 3 are significantly identifiable according to the updated threshold for subnetwork 2. Analogously, we depict the clustering result of a simulated SBM via Algorithm <ref>, shown in Figure <ref>. The network is successfully divided into three communities as predefined in Table <ref>. The communities are colored by blue, red and green.§.§ The propose algorithm v.s. modularity maximization Notice that the algorithm proposed in this manuscript is inspired from the modularity maximization algorithm. In this section, we would like to compare the proposed algorithm with the modularity approach. In the literature, there is an extensive discussion about the limitations and drawbacks of the modularity-based algorithms. One of the most significant problems of the modularity method is that the maximization of modularity score is not always consistent with the optimized clustering outcome, which was addressed by <cit.>. This limitation is related to under-clustering, as it seems that the modularity maximization algorithm intends to merge small clusters, especially in large-size networks. It is believed that the modularity-based methods undergo over-clustering problems as well. Both of these issues were stressed and discussed in <cit.> and <cit.>. To the best of our knowledge, neither of them has been systematically solved up to date. We refer the readers to <cit.> for extensive discussions about inconsistency of clustering results from the modularity maximization algorithm. On the other hand, both over-clustering and under-clustering concerns are considered and well controlled in our algorithm, which empirically ensures consistency. We present the following three simulation examples to show resistance of the proposed algorithm in this manuscript.We reconsider SBM1, a simulated network contains two extremely small communities which are extremely loosely linked; SBM2, a simulated network contains two relatively small communities which are fairly loosely linked; and SBM3, a simulated network which is partitioned into communities of approximately even size. The structural parameters are summarized in Table <ref>.We respectively apply the modularity maximization algorithm and the localized clique-based algorithm to these three networks, and evaluate the clustering results by using the mean estimates of both NMI and ARI, based on 100 independent copies of each simulated network. The results are presented in Table <ref>.Based on the experimental results, we observe that the modularity maximization algorithm performs extremely poorly for SBM1. As a matter of fact, the modularity algorithm even fails to distinguish the two smaller communities (c.f. clusters 2 and 3). As the sizes of small communities and the link density there-between increase, the modularity maximization algorithm recovers (as reflected in the results for SBM2), but apparently still does not appear as well-performed as our algorithm. For networks comprising similar-size communities (e.g., SBM3), the modularity maximization algorithm seems to outperform our algorithm, but not significantly. Having the randomness of simulated networks in mind, we thus conclude that the proposed algorithm in this manuscript is preferred due to its consistency and reliability. §.§ Time complexityIn the era of big data, researchers always concern about the efficiency of algorithms, especially the newly proposed algorithm, when the network size is large. In the last part of this section, we look into this issue numerically by implementing the localized algorithm in Python for several networks of different sizes and structure. Both mean estimates of NMI and ARI are computed. Our testing results are summarized in Table <ref>. We are convinced that the proposed algorithm is able to provide accurate clustering results in a relatively short amount of time.§ CONCLUDING REMARKS In this section, we address several remarks and discuss some possible future work. To conclude, we propose a new clique-based measure in this manuscript to evaluate the quality of network clustering, and design an algorithm to maximize an objective function based on the proposed measure. The clique score of each community in our clustering result is guaranteed to be higher than some predetermined threshold p. We also consider the situation at which the threshold p is unspecified. We develop an approach which accounts for over-clustering and under-clustering problems simultaneously to select localized p for the network and its subsequent networks. An associated localized algorithm is proposed and discussed.Studies of networks or network models usually involve big data problems. When network size or parameter space or both are large, it is always challenging to use model-based clustering methods, as many of them depend on accurate but slow Bayesian MCMC algorithms, for example <cit.>. However, the methods proposed in this manuscript attempt to convert clustering analysis to optimization problems. Therefore, many sophisticated machine learning techniques and well-developed approximation methods are ready to use to deal with big data issues.It is worthy of mentioning that the methods considered in this manuscript also applicable to sparse social network, since the computation of p-clique index is primarily based on the p-clique matrix (p) =- p(_n × n - _n × n). This matrix is not sparse even if the adjacency matrixis sparse. Numerical methods, such as the Lanczos method in <cit.>, promise that the eigenvector corresponding to the largest eigenvalue of (p) always can be determined very fast.The development of the localized clustering algorithm in this paper depends on a strong assumption that the null network is anmodel. This may not be true for many real-world networks. It is suggested in <cit.> that many networks around us follow power law, which is not the case for thegraphs. We conjecture that it may be more accurate to consider a scale-free network as null for the studies of social networks possessing power-law property, and we will consider future research in this direction. Airoldi Airoldi, E.M., Blei, D.M., Fienberg, S.E. and Xing, E.P. Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 1981–2014. (2008) Barabasi Barabási, A.-L. and Albert, R. Emergence of scaling in random networks. Science, 286, 509–512. (1999) Bickle Bickel, P.J. and Chen, A. A nonparametric view of network models and Newman-Girvan and other modularities Proceedings of the National Academy of Sciences of the United States of America, 106, 21068–21073. (2009) Calvetti Calvetti, D., Reichel, L. and Sorensen, D. An implicitly restarted Lanczos method for large symmetric eigenvalue problems. Electronic Transactions on Numerical Analysis, 2, 1–21. (1994) Chung Chung, F.R.K. Spectral Graph Theory. American Mathematical Society, Providence, RI. (1997) Erdos Erdös, P. and Rényi, A. On random graphs I. Publicationes Mathematicae, 6, 290–297. (1959) Fortunato Fortunato, S. and Barthélemy, M. Resolution limit in community detection. Proceedings of the National Academy of Sciences of the United States of America, 104, 36–41. (2007) Fred Fred, A. and Jain, A. Robust data clustering, in the Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2, 128–133. (2003) Gilbert Gilbert, E. N. Random graphs. Annals of Mathematical Statistics, 30, 1141–1144. (1959) Goldenberg Goldenberg, A., Zheng, A.X., Fienberg, S.E. and Airoldi, E.M.A survey of statistical network models. Foundations and Trends in Machine Learning, 2, 129–233. (2010) Handcock Handcock, M.S., Raftery, A.E. and Tantrum, J.M. Model-based clustering for social networks. Journal of the Royal Statistical Society, Series A, 170, 301–354. (2007) Hoff Hoff, P.D., Raftery, A.E. and Handcock, M.S. Latent space approaches to social network analysis. Journal of the American Statistical Association, 97, 1090–1098. (2002) Holland1981 Holland, P. W. and Leinhardt, S. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76, 33–50. (1981) Holland1983 Holland, P.W., Laskey, K.B. and Leinhardt, S. Stochastic blockmodels: First steps. Social Networks, 5, 109–137. (1983) Horn Horn, R.A. and Johnson, C.R. Matrix Analysis. Cambridge University Press, New York. (1985) Hubert Hubert, L. and Abrabie, P. Comparing partitions. Journal of Classification, 2, 193–218. (1985 Lancichinetti Lancichinetti, A. and Fortunato, S. Limits of modularity maximization in community detection. Physical Review E, 84, 066122. (2011) Newman2001a Newman, M.E.J. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences of the United States of America, 98, 404–409. (2001) Newman2001b Newman, M.E.J., Strogatz, S.H. and Watts, D.J.. Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64, 026118. (2001) Newman2006 Newman, M.E.J. Modularity and community structure in networks. Proceedings of the National Academy of Sciences of the United States of America, 103, 8577–8582. (2006) Ng Ng, A.Y., Jordan, M.I. and Weiss, Y. On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems, 14, 849–856. (2001) Snijders Snijders, T.A.B. and Nowicki, K. Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification, 14, 75–100. (1997) Ouyang Ouyang, G. Social Network Community Detection. Ph.D. dissertation. University of Connecticut. (2015) Pao Pao, L.-F. Discovering the dynamics of smart business networks. Computational Management Science, 1, 445–458. (2014) Pei Pei, X., Zhan, X.-X. and Jin, Z. Application of pair approximation method to modeling and analysis of a marriage network. Applied Mathematics and Computation, 294, 280–293. (2017) Reichardt Reichardt, J. and Bornholdt, S. Statistical mechanics of community detection. Physical Review E, 74, 016110. (2006) Shi Shi, J. and Malik, J. Normalized cuts and image segmentation. IEEE Transaction on Pattern Analysis and Machine Intelligence, 22, 888–905. (2000) Watts Watts, D.J. and Strogatz, S.H. Collective dynamics of “small-world” networks. Nature, 440–442. (1998) Wohlgemuth Wohlgemuth, J. and Matache, M.T. Small-wold properties of Facebook group networks. Complex Systems, 23, 197–225. (2014) | http://arxiv.org/abs/1708.07609v2 | {
"authors": [
"Guang Ouyang",
"Dipak K. Dey",
"Panpan Zhang"
],
"categories": [
"cs.SI",
"physics.soc-ph",
"stat.AP"
],
"primary_category": "cs.SI",
"published": "20170825041637",
"title": "Clique-based Method for Social Network Clustering"
} |
Extended AIC-type method , Hu, Zhang and Zhu Central China Normal Universityt1, University of Miamit2and University of Michigant3 Department of StatisticsCentral China Normal UniversityWuhan, 430079Chinae1 Department of Management ScienceUniversity of MiamiCoral Gables, FL 33124USAe2 Department of statisticsUniversity of MichiganAnn Arbor, MI 48109USAe3Estimating the number of principal components is one of the fundamental problems in many scientific fields such as signal processing (or the spiked covariance model). In this paper, we first demonstrate that, for fixed p, any penalty term of the form k'(p-k'/2+1/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n→∞ and C_n/n→0. Compared with the condition in <cit.>, i.e., C_n/loglog n→∞ and C_n/n→0, this condition is significantly weakened. Then we propose to select the number of signals k by the iterated logarithm penalty (ILP). We also extend our results to the case n,p→∞, with p/n→ c>0. In this case, for increasing k, we first investigate the limiting laws for the leading eigenvalues of the sample covariance matrix _n under the condition that λ_k>1+√(c), which extend the results in <cit.> and <cit.>. This includes the case λ_k→∞. At low SNR, since the AIC tends to underestimate the number of signals k,the AIC should be re-defined in this case. As a natural extension of the AIC for fixed p, we propose the extended AIC (EAIC), i.e., the AIC-type method with tuning parameter γ=φ(c)=1/2+√(1/c)-log(1+√(c))/c, and demonstrate that the EAIC-type method, i.e., the AIC-type method with tuning parameter γ>φ(c), can select the number of signals k consistently. As a result, in the following two cases, (1) p fixed, n→∞, (2) n,p→∞ with p/n→ 0, if the AIC is defined as the degeneration of the EAIC in the case n,p→∞ with p/n→ c>0, i.e., γ=lim_c→ 0+φ(c)=1, then we have essentially demonstrated that, the AIC tends to overestimate k. To achieve the consistency of the AIC-type method in the above two cases, γ>1 is required. On the other hand, in the case n,p→∞, with p/n→ c>0, we have φ(c)<1. Then in this case, we have actually explained why the AIC tends to underestimate k. Moreover, we show that the EAIC-type method is essentially tuning-free and outperforms the well-known KN estimator proposed in <cit.> and the BCF estimator proposed in <cit.>. Numerical studies indicate that the proposed method works well. [class=MSC] [Primary ]62E20 [; secondary ]60G05Consistency minimum description length signal-to-noise ratio spiked covariance model Tracy-Widom distribution § INTRODUCTIONDetection of the number of principal components from a noisy data is one of the fundamental problems in many scientific fields. It is often the starting point for the signal parameter estimation problem such as signal processing <cit.>, wireless communications <cit.>, array processing <cit.> and finance <cit.> to list a few and has primary importance <cit.>. The most common approach to solving this problem is by using information theoretic criteria, and in particular minimum description length (MDL) <cit.>, Bayesian information criterion (BIC) and Akaike information criterion (AIC) <cit.>.Consider the model(t)=𝐬(t)+𝐧(t)=(_1^⊤(t),_2^⊤(t))^⊤where k<p, =((Ψ_1),⋯,(Ψ_k)), 𝐬(t)=(𝐬_1(t),⋯,𝐬_k(t))^⊤ and 𝐧(t)=(𝐧_1(t),⋯,𝐧_p(t))^⊤. In (<ref>), 𝐧(t) is the noise vector distributed independent of 𝐬(t) as multivariate normal with mean vector 0 and covariance matrix σ^2 _p. 𝐬(t) is distributed as multivariate normal with mean 0 and nonsingular matrix Ω and (Ψ_i): p× 1 is a vector of functions of the elements of unknown vector Ψ_i associated with i-th signal. Then, the covariance matrix Σ of (t) is given byΣ=Ω^⊤+σ^2 _p=([ _11 _12; _21 _22; ]).We assume that (t_1),⋯, (t_n) are independent observations on (t). Let λ_1≥⋯≥λ_k>λ_k+1=⋯=λ_p=λ=σ^2 denote the eigenvalues of . Note that (<ref>) is also called the spiked covariance model in <cit.>.Using the well-known spectral representation theorem from linear algebra, we can express Σ as=∑_i=1^k(λ_i-λ)_i_i^⊤+λ_p,where _i is the eigenvector ofwith eigenvalue λ_i. Denoting by θ the parameter vector of the model, it follows thatθ^⊤=(θ_1^⊤,θ_2^⊤),where θ_1^⊤=(λ_1,⋯,λ_k,λ) and θ_2^⊤=(_1^⊤,⋯,_k^⊤). With thisparameterization we now proceed to the derivation of the information theoretic criteria for the detection problem. Since the observation are regarded as statistically independent Gaussian random vectors with zero mean, their joint probability density is given byf(|θ)=∏_i=1^nf((t_i)|θ)=∏_i=1^n1/(2π)^p/2||^1/2exp-1/2^⊤(t_i)^-1(t_i),where ^⊤=(^⊤(t_1),⋯,^⊤(t_n)).Taking the logarithm, the log-likelihood function is given bylog f(|θ) = ∑_i=1^nlog f((t_i)|θ)= log∏_i=1^n1/(2π)^p/2||^1/2exp-1/2^⊤(t_i)^-1(t_i)= -1/2nlog||-1/2n(^-1_n)-1/2nplog(2π).where _n=1/n∑_i=1^n(t_i)(t_i)^⊤=1/n^⊤ is the sample covariance matrix.Let ={λ_1,⋯,λ_p} and =^⊤, whereis orthogonal and its column _i is the eigenvector of with eigenvalue λ_i. Also,denote by d_1≥⋯≥ d_p the eigenvalues of _n. Let ={d_1,⋯,d_p} and_n=^⊤≜([ _11 _12; _21 _22; ])=1/n([ _1^⊤_1 _1^⊤_2; _2^⊤_1 _2^⊤_2;]),whereis orthogonal and its column _j is the eigenvector of _nwith eigenvalues d_j. In the signal processing literature, the AIC estimates k by maximizing ℓ(k'):k̂=max_k'ℓ(k'),whereℓ(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-k'(p-k'/2+1/2).Here λ̂_k'=1/(p-k')∑_i=k'+1^pd_i. The BIC (or the MDL) estimates k by maximizing ℓ(k'):k̂=max_k'ℓ^B(k'),whereℓ^B(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-1/2k'(p-k'/2+1/2)log n. Note that the AIC and the BIC are slightly different from that of <cit.>, where the number of unknown parameters was k'(p-k'/2) and was corrected by <cit.>.The first consistency proof of the BIC was established in <cit.>. By usingWilks' theorem,<cit.> showed that any penalty term of the form k'(p-k'/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n→∞ and C_n/n→0. Later, it was demonstrated that Wilks' theorem cannot be used in this problem in <cit.>. By using Taylor's expansion, <cit.> gave an alternative proof and showed that, any penalty term of the form k'(p-k'/2+1/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n/loglog n→∞ and C_n/n→0. Although<cit.> proved that the BIC is consistent, the BIC fails to detect signals at low signal-to-noise ratio (SNR), hence underestimating the number of signals at small sample size. In contrast, while the AIC is able to detect low SNR signals, it has a non-negligible probability to overestimate the number of signals and thus is not consistent. To remedy the shortcoming of the AIC,<cit.> proposed an modified AIC. The only difference between these two methods lies in that the penalty of the modified AIC is two times that of the AIC. In the presence of pure noise with no signals (i.e., k=0),<cit.> showed that the modified AIC had a negligible overestimation probability for large n. However, for k>0,<cit.> did not give an explanation why the modified AIC had a negligible overestimation probability. On the other hand, for small and medium n, the modified AIC tends to underestimate the number of signals k in our simulations. <cit.> and<cit.> proposed a quite different method for estimating the number of signals, via a sequence of hypothesis tests, at each step testing the significance of the k-th eigenvalue as arising from a signal. The main tools used in these two papers are recent results from random matrix theory regarding both the distribution of noise eigenvalues and of signal eigenvalues in the presence of noise. In the absence of signals, the matrix n_n follows a Wishart distribution <cit.> with parameter n, p. In the joint limit n,p→∞, with p/n→ c>0, the distribution of the largest eigenvalue of _n converges to a Tracy-Widom distribution <cit.>,<cit.>,<cit.>,<cit.>,<cit.>. For fixed p, although<cit.> proved the strong consistency of their estimator, our simulations show that when the SNR is low, for fixed p and medium n, the KN tends to underestimate the number of signals k.Before further proceeding, we mention that a similar, if not identical problem, also appears in other literatures, likelihood ratio test statistic <cit.>, Kac-Rice test <cit.>. For some other work in this topic, we refer to <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>and so on.In this paper, we first demonstrate that any penalty term of the form k'(p-k'/2+1/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n→∞ and C_n/n→0. Compared with the condition in <cit.>, i.e., C_n/loglog n→∞ and C_n/n→0, this condition is significantly weakened. Then we propose to select the number of signals k by the iterated logarithm penalty (ILP). Moreover, we also extend our results to the case n,p→∞, with p/n→ c>0. In this case, for increasing k, we first investigate the limiting laws for the leading eigenvalues of the sample covariance matrix _n under the condition that λ_k>1+√(c), which extend the results in <cit.> and <cit.>. This includes the case λ_k→∞. At low SNR, since the AIC tends to underestimate the number of signals k,the AIC should be re-defined in this case. As a natural extension of the AIC for fixed p, we propose the extended AIC (EAIC), i.e., the AIC-type method with tuning parameter γ=φ(c)=1/2+√(1/c)-log(1+√(c))/c, and demonstrate that the EAIC-type method, i.e., the AIC-type method with tuning parameter γ>φ(c), can select the number of signals k consistently. Moreover, we show that the EAIC-type method is essentially tuning-free and outperforms the well-known KN estimator proposed in <cit.> and the BCF estimator proposed in <cit.>. For the remainder of the paper, we proceed as follows. In Section <ref>, we demonstrate that any penalty term of the form k'(p-k'/2+1/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n→∞ and C_n/n→0. We extend our results to the case n,p→∞, with p/n→ c>0 in Section <ref>. The numerical studies are given in Section <ref>. Some further discussions are made in Section <ref>. All proofs are given in Section <ref>.§ FIXED P In this section, we work under the classical setting where p, k and λ_i are all fixed as n →∞. We establish the consistency of (<ref>) in the sense that it chooses the correct k with probability tending to one when n goes to infinity.We consider the following general criterion:k̂=max_k'ℓ(k'),whereℓ(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-k'(p-k'/2+1/2)C_n,where C_n→∞ and C_n/n→0. Here λ̂_k'=1/(p-k')∑_i=k'+1^pd_i.Throughout this paper, we assume that λ is known. Without loss of generality, we assume that λ=1. The signal-to-noise ratio is defined asSNR=λ_k-λ/λ=λ_k/λ-1=λ_k-1. Define λ̂=1/(p-k)∑_i=k+1^pd_i. Note that for i=1,2,⋯, k, d_i and λ̂ are the maximum likelihood estimates of λ_i and λ, respectively. The following result which shows how close is the eigenvalue of Σ to its maximum likelihood estimate. For i=1,2,⋯, p,(λ_i-d_i)^2=O_p(p/n)=O_p(1/n), (p-k)(λ-λ̂)^2=O_p(p^2/n)=O_p(1/n).Noting that for fixed p, it is difficult to demonstrate the consistency of (<ref>) directly (see the proof of Theorem <ref>). Similar to the discussion in <cit.> (see also <cit.>) , we consider an alternative method.When k' is equal to k, ℓ(k') defined in (<ref>) can be re-written as:ℓ(k)=-1/2n∑_i=1^klog d_i-1/2n∑_i=k+1^p(d_i-1)-k(p-k/2+1/2)C_n.where C_n→∞ and C_n/n→0.Then we estimate k by maximizing ℓ̃(k'):k̂=max_k'ℓ̃(k'),whereℓ̃(k')=-1/2n∑_i=1^k'log d_i-1/2n∑_i=k'+1^p(d_i-1)-k'(p-k'/2+1/2)C_n.where C_n→∞ and C_n/n→0. Next, we establish the consistency of (<ref>).Let ℓ̃(k') be the penalized likelihood function defined in (<ref>).For k'<k,P(ℓ̃(k)>ℓ̃(k'))→1.For k'>k,P(ℓ̃(k)>ℓ̃(k'))→1.Especially, we propose to select the number of signals k by the iterated logarithm penalty (ILP). That is, we estimate k by maximizing ℓ(k'):k̂=max_k'ℓ(k'),whereℓ(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-γ k'(p-k'/2+1/2)loglog n,where γ>0 is a tuning parameter.As an alternative method, we may also estimate k by maximizing ℓ̃(k'):k̂=max_k'ℓ̃(k'),whereℓ̃(k')=-1/2n∑_i=1^k'log d_i-1/2n∑_i=k'+1^p(d_i-1)-γ k'(p-k'/2+1/2)loglog n,where γ>0 is a tuning parameter. Simulations show that (<ref>) and (<ref>) behave almost the same. For simplicity and comparisons with other methods, we consider (<ref>) in our simulations. For simplicity, we may use γ=1 in our simulations which gives good performance. <cit.> demonstrated that any penalty term of the form k'(p-k'/2+1/2)C_n leads to an asymptotically consistent estimator under the condition thatC_n/loglog n→∞ and C_n/n→0. Theorem <ref> shows that this condition can be significantly weakened as C_n→∞ and C_n/n→0. To show that our results are correct, we also consider different C_n's in our simulations in Section <ref>, such as ILP^1/2, which estimates k by maximizing ℓ̃(k'):k̂=max_k'ℓ̃(k'),whereℓ̃(k')=-1/2n∑_i=1^k'log d_i-1/2n∑_i=k'+1^p(d_i-1)-γ k'(p-k'/2+1/2)(loglog n)^1/2.§ INFINITE P In this section, we allow p, k and λ_k grow as n grows. §.§ Asymptotic behavior of the spiked sample eigenvalues In this subsection, in the case n,p→∞, with p/n→ c>0, for increasing k, we investigate the limiting laws for the leading eigenvalues of the sample covariance matrix _n under the condition that λ_k>1+√(c), which extend the results in <cit.> and <cit.>. This includes the case λ_k→∞.We wish to point out that the empirical spectral distribution (ESD) F(x) ofconverges weakly to the limiting spectral distribution (LSD) H(x) as p→∞, that is,sup_x | F(x)-H(x)|=k/p,where F(x)=1/p∑_i=1^pI_(-∞,x](λ_i), H(x)=I(1≤ x) and I_A(·) is the indicator function of A.Note that the ESD of _n is given byF_n(x)=1/p∑_i=1^pI_(-∞,x](d_i).With probability 1,F_n(x)d→F_c(x).Here, for 0<c≤ 1, F_c(x) is given asF_c'(x)=f_c(x)={[ 1/2π xc√((b-x)(x-a)), ifx∈ (a,b),;0,otherwise, ].where a=(1-√(c))^2 and b=(1+√(c))^2.If c>1, F_c(x) has a point mass 1-1/c at the origin, that is, F_c(x)={[ 0,if x<0,; 1-1/c, if 0≤ x<a,; 1-1/c+∫_a^xf_c(t)dt, if a≤ x<b, ].where a and b are the same as in the case 0<c≤ 1. We remark that ∫_a^bf_c(t)dt=1 or 1/c according to 0<c≤ 1 or c>1 respectively.By a result of<cit.> (see also<cit.>),sup_x | F_n(x)-F_c(x)|=O_p(k/n).This implies that the ESD of _n also converges weakly to F_c(x) as n→∞ under the condition where the number of spikes, k, is fixed or k=o(n). From the MP law, we have the easy consequence that if i/p→α∈ (0,1), then d_ia.s.→μ_1-α, where μ_α is the α-th quantile of the MP law, that is F_c(μ_α)=α. The i-th largest eigenvalue, λ_i, is said to be a distant spiked eigenvalue if ψ'(λ_i)>0 where ψ(λ_i)=λ_i+cλ_i/λ_i-1. Equivalently, λ_i is a distant spiked eigenvalue if λ_i>1+√(c).Define x(t) as(t)=^1/2(t),where (t) is distributed with mean 0 and covariance matrix _p. Thus, the covariance matrix of (t) is given by .For fixed k, under the assumption that λ_1 is bounded,according to <cit.>, the following result holds. In the case n,p→∞ with p/n→ c>0, suppose that for any 1≤ i≤ n, 1≤ j≤ p, E(((t_i))_j^4)≤ C, λ_1 is bounded and λ=1. If λ_i is a distant spiked eigenvalue, we haved_ia.s.⟶ψ(λ_i).The above definition and result are a special case of a more general definition and result in <cit.> and <cit.>.For fixed k, under the assumption that λ_k→∞ as p→∞, a new limiting result for the distant spiked eigenvalues was also established in <cit.>.In the case n,p→∞ with p/n→ c>0, suppose that for any 1≤ i≤ n, 1≤ j≤ p, E(((t_i))_j^4)≤ C and λ=1. If λ_k→∞ as p→∞, forany 1≤ i≤ k, we haved_i/ψ(λ_i)a.s.⟶1.For fixed k, from Lemmas <ref> and <ref>, under the assumption that (1) at least one λ_i (1≤ i≤ k-1) satisfying λ_i→∞ as p→∞,(2)1+√(c)<λ_k<∞,for any 1≤ i≤ k,d_i/ψ(λ_i)a.s.⟶1.is still not established. For increasing k, under the assumption that λ_k→∞ as p→∞,<cit.> established the following result. In the case n,p→∞ with p/n→ c>0, suppose that for any 1≤ i≤ n, 1≤ j≤ p, E(((t_i))_j^4)≤ C and λ_k+1≥⋯≥λ_p are bounded. If λ_k→∞ as p→∞ and k=min{o(n^1/6), o(λ_i^-1/2)}, forany 1≤ i≤ k, we haved_i/λ_i-1=max{O_p(1/λ_i), O_p(k^4/n)}.Based on Lemma <ref>,<cit.> also established the central limit theorem for increasing k, under the assumption that λ_k→∞ as p→∞.In the case n,p→∞ with p/n→ c>0, suppose that for any 1≤ i≤ n, 1≤ j≤ p, E(((t_i))_j^4)≤ C and λ_k+1≥⋯≥λ_p are bounded, there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k→∞ as p→∞ and k=min{o(n^1/6), o(λ_i^-1/2)}, forany 1≤ i≤ k, we have√(n)(d_i-ψ(λ_i))/ψ(λ_i)d⟶ N(0, σ_i^2),where σ_i^2 is defined in <cit.>. For more work on this topic, we refer to <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and <cit.>.For increasing k, under the assumption that λ_k>1+√(c), we will extend the results in <cit.> and <cit.>.Our first result gives the limits for the spiked eigenvalues of _n, d_1≥⋯≥ d_k, which will be used to establish the consistency of the following EAIC-type method.Suppose that k=o(n^1/3) and p/n-c=O(√(k/n))(=o(n^-1/3)). If λ_k>1+√(c), for any 1≤ i≤ k,d_i/ψ(λ_i)a.s.⟶1, where ψ(λ_i)=λ_i+cλ_i/λ_i-1.To demonstrate this result, we need the following result which extends Lemma <ref>. This result can be of independent interest. For any positive semi-definite matrix _n, denoting by l_1≥⋯≥ l_n the eigenvalues of _n, then there exist some universal constant C>0 and some constant ρ such that the following inequality holdsP(||1/n_11^-1/2_1^⊤_n_1_11^-1/2-1/n(_n)||>t)≤ 2exp(Ck-nt^2/ρ∑_s=1^nl_s^2/n)for all 0<t<ρ∑_s=1^nl_s^2/n/64l_1, where ||·|| is the spectral norm. For any positive semi-definite matrix _n, denoting by l_1≥⋯≥ l_n the eigenvalues of _n, then there exist some universal constant C>0 and some constant ρ such that the following inequality holdsP(||1/n_11^-1/2_1^⊤(+_n)_1Σ_11^-1/2-1/n(+_n)||>t)≤ 2exp(Ck-nt^2/ρ∑_s=1^n(1+l_s)^2/n),for all 0<t<ρ∑_s=1^n(1+l_s)^2/n/64(1+l_1). Our second result shows that the spiked eigenvalues of _n, d_1≥⋯≥ d_k are √(n/k)-consistent, which will be used to establish Theorem <ref>.Compared with Theorem <ref>, we need one more condition: there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. This condition implies that the spiked eigenvalues are well-separated and is also used in <cit.>.Suppose that k=o(n^1/4), p/n-c=O(√(k/n))(=o(n^-3/8)), and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,d_i-ψ(λ_i)/λ_i=O_p(√(k/n)).To demonstrate this result, we need the following Lemma <ref>. We give some notations.Definem_1(d)=∫x/d-xdF_c(x), m_2(d)=∫x^2/(d-x)^2dF_c(x), m_3(d)=∫x/(d-x)^2dF_c(x), _̋n(d)=1/n_2(d -_22)^-1_2^⊤, K_n(d)=1/n_1^⊤(+_̋n(d))_1, (d)=1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2-(1+cm_1(d)).Let[ _11=(λ_1,…,λ_k)^⊤, ]whereis an orthogonal matrix.Obviously,_11^-1=(1/λ_1,…,1/λ_k)^⊤. Also, let(d_i) = (a_st)_k× k≜^⊤_11^-1/2(d_i- K_n(d_i))_11^-1/2= ^⊤(_11^-1d_i-1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2)= (d_i-(1+cm_1(d_i))λ_1/λ_1,…,d_i-(1+cm_1(d_i))λ_k/λ_k)-^⊤(d_i). According to <cit.>, for λ_i>1+√(c),we have m_1(ψ(λ_i))=1/λ_i-1. Thus, for the j-th diagonal elementwith index j≠ i,e_j≜d_i-(1+cm_1(d_i))λ_j/λ_j = d_i/λ_j-(1+cm_1(d_i))→ ψ(λ_i)/λ_j-(1+cm_1(ψ(λ_i)))= (λ_i/λ_j-1)(1+c/λ_i-1)is uniformly bounded away from 0.Let((d_i))=∏_j=1^kê_j≜∏_j=1^k(e_j+ϵ_j),where ê_j's are eigenvalues of matrix (d_i).Similar to the proof of (<ref>) in Theorem <ref>, we have∑_i=1^k|ϵ_j|=O_p(√(k^3/n)).For j≠ i, under the condition that k=o(n^1/3), ê_j is uniformly bounded away from 0 since the contributions from ^⊤(d_i) tend to O_p(√(k^3/n))=o_p(1).By (<ref>), for sufficiently large n, ((d_i))=0 if and only if ê_i=0.However, ê_i=0 does not imply that a_ii=0. For any 1≤ i≤ k, the following lemma gives the order of the i-th diagonal element of matrix (d_i),which is the key to establish Theorems <ref> and <ref>. Suppose that k=o(n^1/4), p/n-c=O(√(k/n))(=o(n^-3/8)), and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,a_ii=d_i-(1+cm_1(d_i))λ_i/λ_i-[^⊤(d_i)]_ii=O_p(k^2/n).Our third result demonstrates that the spiked eigenvalues of _n, d_1≥⋯≥ d_k are asymptotical normal.Suppose that k=o(n^1/4), p/n-c=o(n^-1/2), and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,√(n)(d_i-ψ(λ_i))/λ_ia.s.⟶T(ψ(λ_i)),whereT(ψ(λ_i))≜(1+cm_3(ψ(λ_i))λ_i)^-1[^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2]_ii,andR_n(ψ(λ_i))=1/√(n)[_1^⊤(+_̋n(ψ(λ_i)))_1-(+_̋n(ψ(λ_i)))_11]. For any 1≤ i≤ k, the limiting distribution of T(ψ(λ_i)) is normal. For more details, we refer to <cit.> and <cit.>.Especially, we consider the case _11 is diagonal as discussed in <cit.>. Similar to the discussion in <cit.>, for any 1≤ i≤ k, we have√(n)(d_i-ψ(λ_i))/λ_id⟶N(0,σ_λ_i^2),where σ_λ_i^2=2-2c/(λ_i-1)^2. Note that throughout this paper we assume that (t) is normal. Under the condition that for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, we have the following results.For any positive semi-definite matrix _n, denoting by l_1≥⋯≥ l_n the eigenvalues of _n, suppose that for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C hold,we have||1/n_11^-1/2_1^⊤_n_1_11^-1/2-1/n(_n)I||=√(1/n∑_s=1^nl_s^2)O_p(k/√(n)). For any positive semi-definite matrix _n, denoting by l_1≥⋯≥ l_n the eigenvalues of _n, suppose that for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C hold,we have||1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2-1/n(+_n)I||=√(1/n∑_s=1^n(1+l_s)^2)O_p(k/√(n)).Suppose that k=o(n^1/4), p/n-c=O(k/√(n))(=o(n^-1/4)), and for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C. If λ_k>1+√(c), for any 1≤ i≤ k,d_i/ψ(λ_i)a.s.⟶1, where ψ(λ_i)=λ_i+cλ_i/λ_i-1. Suppose that k=o(n^1/5), p/n-c=O(k/√(n))(=o(n^-3/10)), for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,d_i-ψ(λ_i)/λ_i=O_p(k/√(n)).Suppose that k=o(n^1/5), p/n-c=O(k/√(n))(=o(n^-3/10)), for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,a_ii=d_i-(1+cm_1(d_i))λ_i/λ_i-[^⊤(d_i)]_ii=O_p(k^3/n).Suppose that k=o(n^1/6), p/n-c=o(n^-1/2), for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, and there exists a positive constant C not depending on n such that λ_i-1/λ_i≥ C>1, i=1,2,…,k. If λ_k>1+√(c), for any 1≤ i≤ k,√(n)(d_i-ψ(λ_i))/λ_ia.s.⟶(1+cm_3(ψ(λ_i))λ_i)^-1[^⊤_11^-1/2 R_n(ψ(λ_i))Σ_11^-1/2]_ii,whereR_n(ψ(λ_i))=1/√(n)[_1^⊤(+_̋n(ψ(λ_i))X_1-(+_̋n(ψ(λ_i))_11].§.§ Extended AICIn this subsection, we extend our results to the case n,p→∞, with p/n→ c>0. In this case, the AIC tends to underestimate the number of signals k (see Section <ref>). It is well-known that, for fixed p, the AIC tends to overestimate the number of signals k.This implies that, in the case n,p→∞, with p/n→ c>0, the AIC does not work and should be re-defined. In view of this, we propose an extended AIC (EAIC).The EAIC estimates k by maximizing ℓ(k'):k̂=max_k'ℓ(k'),whereℓ(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-γ k'(p-k'/2+1/2),where γ=φ(c)=1/2+√(1/c)-log(1+√(c))/c. Note that when γ is equal to 1, (<ref>) is the AIC. In the following, we first show that the EAIC is a natural extension of the AIC for fixed p. In the next subsection, we further explain that, for the EAIC, why γ should be defined as φ(c)=1/2+√(1/c)-log(1+√(c))/c.In the following two cases, (1)p fixed, n→∞,(2) n,p→∞ with p/n→ 0,c is defined as c=p/n. Noting that c→ 0+, by Taylor's expansion, we get[φ(c) =1/2+√(1/c)-log(1+√(c))/c; = 1/2+√(1/c)-(√(c)-c/2(1+o(1)))/c; =1+o(1); →1. ] As a result, in the following three cases, (1) p fixed, n→∞,(2) n,p→∞ with p/n→ 0,(3) n,p→∞ with p/n→ c, where 0<c<∞,approximately, the AIC may be uniformly written as (<ref>).Note that the EAIC is different from that of <cit.> which proposes the following BCF for estimating the number of principal components k.In the case 0<c<1,<cit.> estimates k by minimizing ℓ^1(k'):k̂=min_k'ℓ^1(k'),where ℓ^1(k')=(p-k')logd̅_k'-∑_i=k'+1^plog d_i-(p-k'-1)(p-k'+2)/n, d̅_k'=∑_i=k'+1^pd_i/(p-k').Note that,the BCF defined in (<ref>) is essentially equivalent to the AIC and is referred as the AIC in <cit.>. In the next subsection, we will show that,at low SNR, the BCF defined in (<ref>) tends to underestimate the number of signals k, especially in the case c→1-(see also Section <ref>).In the case c>1,<cit.> replaces the p in (<ref>) by n-1and estimates k by minimizing ℓ^2(k'):k̂=min_k'ℓ^2(k'),where ℓ^2(k')=(n-1-k')logd̅_k'-∑_i=k'+1^n-1log d_i-(n-k'-2)(n-k'+1)/p, d̅_k'=∑_i=k'+1^n-1d_i/(n-1-k').Simulations show that, the BCF defined in (<ref>) is better than the AIC and is referred as the quasi-AIC in <cit.>. However, at low SNR, the BCF defined in (<ref>) still tends to underestimate the number of signals k, especially in the case c→1+(see also Section <ref>).§.§ Consistency of the EAIC-type method Noting that the EAIC tends to overestimate the number of signals k, we propose the EAIC-type method which chooses the correct k with probability tending to one when n goes to infinity.The EAIC-type method estimates k by maximizing ℓ(k'):k̂=max_k'ℓ(k'),whereℓ(k')=-1/2n(∑_i=1^k'log d_i+(p-k')logλ̂_k')-γ k'(p-k'/2+1/2),where γ>φ(c)=1/2+√(1/c)-log(1+√(c))/c.To achieve the consistency of (<ref>), weneed two additional conditions,ψ(λ_k)-1-logψ(λ_k)>2γ c, λ_k>1+√(c),where ψ(λ_k)=λ_k+cλ_k/λ_k-1.Since there is a tuning parameter γ in (<ref>), λ_k>1+√(c) does not implies ψ(λ_k)-1-logψ(λ_k)>2γ c. However, by γ>φ(c) and ψ(λ_k)-1-logψ(λ_k)>2γ c, we have λ_k>1+√(c).Next, we establish the consistency of (<ref>). In the case n,p→∞ with p/n→ c>0, let ℓ(k') be the penalized likelihood function defined in (<ref>). Suppose that k=o(n^1/3) and p/n-c=O(√(k/n))(=o(n^-1/3)), and the number of candidate models, q, satisfies q = o(p).For k'<k, if (<ref>) and (<ref>) hold (or if γ>φ(c) and (<ref>) hold),P(ℓ(k)>ℓ(k'))→1.For k'>k, if γ>φ(c),P(ℓ(k)>ℓ(k'))→1.where φ(c)=1/2+√(1/c)-log(1+√(c))/c. Note that throughout this paper, we assume that (t) is normal. Under the more general condition that for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, we have the following result.In the case n,p→∞ with p/n→ c>0, let ℓ(k') be the penalized likelihood function defined in (<ref>). Suppose that k=o(n^1/4), p/n-c=O(k/√(n))(=o(n^-1/4)), for any 1≤ i≤ n, 1≤ j≤ k, E((_11^-1/2_1(t_i))_j^4)≤ C, and the number of candidate models, q, satisfies q = o(p).For k'<k, if (<ref>) and (<ref>) hold (or if γ>φ(c) and (<ref>) hold),P(ℓ(k)>ℓ(k'))→1.For k'>k, if γ>φ(c),P(ℓ(k)>ℓ(k'))→1.where φ(c)=1/2+√(1/c)-log(1+√(c))/c. Note that for fixed p and n→∞, γ>φ(c) becomes γ>1. It is well-known that, for fixed p, the AIC tends to overestimate the number of signals k. To achieve the consistency of the AIC-type method, the tuning parameter γ should be larger than one. That is, in the case n,p→∞ with p/n→ c>0, the EAIC can be seen as a natural extension of the AIC for fixed p. There is a similar discussion in the case n,p→∞ with p/n→ 0.In the following two cases, (1) p fixed, n→∞,(2) n,p→∞ with p/n→ 0,if the AIC is defined as the degeneration of the EAIC in the case n,p→∞ with p/n→ c>0, i.e., γ=lim_c→ 0+φ(c)=1, then we have essentially demonstrated that, the AIC tends to overestimate k. To achieve the consistency of the AIC-type method in the above two cases, γ>1 is required. By Theorem <ref>, in the case n,p→∞ with p/n→ c>0, we have actually explained why the AIC tends to underestimate the number of signals k. We note that γ>φ(c) is required in our theoretic analysis, to avoid overestimating the number of signals k. On the other hand, (<ref>) requires that γ cannot be too large, to avoid underestimating the number of signals k. Noting that, in the case n,p→∞ with p/n→ c>0, we always have φ(c)<1. Thus, the penalty of the AIC may be too large. As a result, it is not surprising that the AIC may tend to underestimate the number of signals k.Since γ>φ(c) is required in our theoretic analysis, we set γ=1.1φ(c) for simulation studies which gives good performance. In this sense, the EAIC-type method is essentially tuning-free.For fixed k, to prove the consistency of the BCF,<cit.> gives the following two conditions, (1) in the case 0<c<1,ψ(λ_k)-1-logψ(λ_k)>2cand (2) in the case c>1,ψ(λ_k)/c-1-log(ψ(λ_k)/c)>2/c. Compared with (<ref>) and (<ref>), (<ref>) contains a tuning parameter γ. By simulations, we found that when c≥0.12, we have γ=1.1φ(c)<1. That is, when 0.12≤ c≤1, (<ref>) is weaker than (<ref>). This implies that, when 0.12≤ c≤1, in the case where (<ref>) holds while (<ref>) does not hold, the EAIC-type method is better than the BCF. When c>1, although it is difficult to compare (<ref>) with (<ref>), by simulations, we found that there do exist cases where (<ref>) holds while (<ref>) does not hold (see Section <ref>). This implies that, in the casewhere (<ref>) holds while (<ref>) does not hold, the EAIC-type method is still better than the BCF. We also note that in the case c>1, if c is close to one, (<ref>) and (<ref>) are almost the same. Thus, in the case where c is close to one, the EAIC-type method outperforms the BCF. The reason lies in that, when c=p/n is close to one, (<ref>) is weaker than (<ref>) and (<ref>). As a result, the EAIC-type method outperforms the BCF at least in the following two cases, (1) in the case 0.12≤ c≤1,(2) in the case c>1, and c is close to one.Moreover, note that (<ref>) is essentially equivalent to the AIC. Since when c≥0.12, we have γ=1.1φ(c)<1. That is, when (1) 0.12≤ c≤ 1; (2) c is larger than one and is close to one, the BCF tends to underestimate the number of signals k.On the other hand, in the case where c is larger than one, simulations show that the EAIC-type method is still comparable to the BCF (see Section <ref>).For fixed k,<cit.>established the consistency of the BCF. Along the way of the proposed method in this paper, it is possible to extend their results to increasing k. Although the BCFdefined in (<ref>) and (<ref>) is tuning-free and simulation results are encouraging,<cit.> essentially defines two different criteria in the case 0<c<1 and c>1, respectively. As a result, to achieve the consistency of the BCF, two different consistency conditions are required (i.e., (<ref>) and (<ref>)). Compared with (<ref>) and (<ref>), (<ref>) with γ>φ(c) is a natural extension of the AIC for fixed p. We also note that, both the formula (<ref>) and the consistency condition (<ref>) are more simple.§ EXPERIMENTSWe compare the performance of the ILP with the BIC, the AIC and the KN in a series of simulations with λ=σ^2=1. As suggested in <cit.>, the confidence level α=10^-4 was used in the KN. Our performance measure is the probability of the successful recovery of the number of signals,Pr(k̂=k).We restrict our attention to candidate values for the true number of signals in the range k'∈{0,1,…,min{p-1,15}} in simulations. Each simulation in this section is repeated 200 times.In our experiments, we define x(t) as(t)=^1/2(t),where ={λ_1,⋯, λ_k,λ,⋯,λ} and (t) is distributed as multivariate normal with mean 0 and covariance matrix _p. Thus, the covariance matrix of (t) is given by .§.§ Fixed p Simulation 1. In this simulaiton, we investigate how the accuracy of the ILP changes as the SNR varies. We set p=12, k=3, γ=1, λ_1=⋯=λ_k-1=1+2SNR, λ_k=1+SNR and λ=1. We let SNR=δ√(4(p-k/2+1/2)loglog n/n), where δ increases from 1 to 2.We compare the ILP with the BIC, the AIC, the modified AIC and the KN.From Table <ref>, we can see that, when δ≥1.5, the ILP outperforms other methods in general.§.§ Infinite pNote that the AIC-type method with γ>φ(c) is called the EAIC-type method in this paper. In simulations, we set γ=1.1φ(c).Simulation 2. In this simulation, in the case n,p→∞ with p<n, we investigate how the accuracy of the EAIC-type method changes as the SNR varies. We set n=500, p=200, k=10, λ_1=⋯=λ_k-1=1+2SNR, λ_k=1+SNR and λ=1. Note that when c=p/n=0.4, γ=1.1φ(c)=0.94 and √(c)=0.632. We let δ increase from 0.5 to 2.5. We also note that when δ=0.5, (<ref>) does not hold, while when 1≤δ≤2.5, (<ref>) holds. It can be seen from Table <ref> that, when δ=0.5, the success rate of the EAIC-type method is low. On the other hand, when δ≥ 1, the success rate is high. This simulation verifies that Theorem <ref> is correct, i.e., if (<ref>) and (<ref>) hold, the EAIC-type method is consistent. It can also be seen from Table <ref> that, in the case n,p→∞ with p<n, the EAIC-type method isbetter than the KN and the BCF , especially for δ≤1.Note that when δ=1, (<ref>) holds while (<ref>) does not hold. Thus, it not surprising that in this case, the EAIC-type method is better than the BCF. On the other hand, although (<ref>) holds, ψ(λ_k)-1-logψ(λ_k)-2γ c=0.017 is close to zero. As a result, when δ=1, the success rate of the EAIC-type method is not high.Simulation 3. In this simulation, in the case n,p→∞ with p>n, we investigate how the accuracy of the EAIC-type method changes as the SNR varies. We set n=200, p=500, k=10, λ_1=⋯=λ_k-1=1+2SNR, λ_k=1+SNR and λ=1. Note that when c=p/n=2.5, γ=1.1φ(c)=0.83 and √(c)=1.581. We let δ increase from 1.5 to 4.5. We also note that when δ=1.5, 2.5, (<ref>) does not hold, while when 2.68≤δ≤4.5, (<ref>) holds. It can be seen from Table <ref> that, when δ=1.5, 2.5, the success rate of the EAIC-type method is low. On the other hand, when δ≥ 3.5, the success rate is high. This simulation verifies that Theorem <ref> is correct, i.e., if (<ref>) and (<ref>) hold, the EAIC-type method is consistent. It can also be seen from Table <ref> that, in the case n,p→∞ with p>n, the EAIC-type method isbetter than the KN and the BCF, especially for δ≤3.5.Note that when δ=2.68, (<ref>) holds while (<ref>) does not hold. Thus, it not surprising that in this case, the EAIC-type method is better than the BCF. On the other hand, although (<ref>) holds, ψ(λ_k)-1-logψ(λ_k)-2γ c=0.0085 is close to zero. As a result, when δ=2.68, the success rate of the EAIC-type method is not high. Simulation 4. In this simulation, in the case n,p→∞ with p=n, we investigate how the accuracy of the EAIC-type method changes as the SNR varies. We set n=200, p=200, k=10, λ_1=⋯=λ_k-1=1+2SNR, λ_k=1+SNR and λ=1. Note that when c=p/n=1, γ=1.1φ(c)=0.89 and √(c)=1. We let δ increase from 1 to 3. We also note that when δ=1,1.5, (<ref>) does not hold, while when 2≤δ≤3, (<ref>) holds. It can be seen from Table <ref> that, when δ=1,1.5, the success rate of the EAIC-type method is low. On the other hand, when δ≥ 2, the success rate is high. This simulation verifies that Theorem <ref> is correct, i.e., if (<ref>) and (<ref>) hold, the EAIC-type method is consistent. It can also be seen from Table <ref> that, in the case n,p→∞ with p=n, the EAIC-type method is better than the KN and the BCF, especially for δ≤2.5.Note that when δ=2, (<ref>) holds while (<ref>) does not hold. Thus, it not surprising that in this case, the EAIC-type method significantly outperformsthe BCF.Simulation 5. In this simulation, in the case n,p→∞, at low SNR, i.e., SNR=2√(p/n), we further compare the EAIC-type method with the BCF. We set k=10, λ_1=⋯=λ_k-1=1+2SNR, λ_k=1+SNR and λ=1. It can be seen from Tables <ref> and<ref> that, when c=p/n is less than or close to one, the EAIC-type method outperforms the BCF. On the other hand, when c=p/n is larger than one, the EAIC-type method is still comparable to the BCF. § DISCUSSION In this paper, we have demonstrated that any penalty term of the form k'(p-k'/2+1/2)C_n may lead to an asymptotically consistent estimator under the condition that C_n→∞ and C_n/n→0. Compared with the condition in <cit.>, i.e., C_n/loglog n→∞ and C_n/n→0, this condition has been significantly weakened.We have also extended our results to the case n,p→∞, with p/n→ c>0. In this case, for increasing k, we have investigated the limiting law for the leading eigenvalues of the sample covariance matrix _n under the condition that λ_k>1+√(c). This includes the case λ_k→∞. At low SNR, since the AIC tends to underestimate the number of signals k,the AIC should be re-defined in this case. As a natural extension of the AIC for fixed p, we have proposed the extended AIC (EAIC), i.e., the AIC-type method with tuning parameter γ=φ(c)=1/2+√(1/c)-log(1+√(c))/c, and demonstrated that the EAIC-type method, i.e., the AIC-type method with tuning parameter γ>φ(c), can select the number of signals k consistently. In simulations, we set γ=1.1φ(c) which gives good performance and outperforms the KN proposed in <cit.> and the BCF proposed in <cit.>. That is, the EAIC-type method is essentially tuning-free. We have noted that in the case n,p→∞ with p/n→ c>0, if (<ref>) or (<ref>) does not hold, γ=1.1φ(c) tends to underestimate the number of of principal components k. As a result, 0<γ<1.1φ(c) may be a better choice. In this case, we may use the cross-validation method to choose the tuning parameter γ, which will be explored infuture work. § APPENDIXThis section presents auxiliary lemmas used in this paper and proofs for main results stated in this paper.§.§ Auxiliary Lemmas Let ,∈_n, assume that ,+ are both normal, let λ_1,…,λ_n be the eigenvalues ofin some given order, and let λ̂_1,…,λ̂_n be the eigenvalues of + in some given order. There is a permutation σ(·) of the integers 1,…,n such that∑_l=1^k(λ̂_σ(i)-λ_i)^2≤ ||||_2^2=(^*). Suppose =(_1,⋯, _p)' is sub-Gaussian with constant ρ>0 and with mean 0 and covariance matrix . Let ^(1),⋯, ^(n) be n independent copies of Y. Then there exist some universal constant C>0 and some constant ρ_1 depending ρ, such that the sample covariance matrix of ^(1),⋯, ^(n), , satisfiesP(->t)≤ 2exp(-nt^2ρ_1+Cp),for all 0<t<ρ_1. Here · is the spectral norm.For more work on this topic, the reader is referred to <cit.>, <cit.>, <cit.> and <cit.>.Letandbe p× p Hermitian matrices with eigenvalues ordered as λ_1≥⋯≥λ_p and μ_1≥⋯≥μ_p, respectively. Thensup_1≤ j≤ p|μ_j-λ_j|≤-,where · is the spectral norm.§.§ Proofs for Main Results By Lemmas <ref> and <ref>, for i=1,2,⋯, p,|λ_i-d_i|≤max_1≤ i≤ p|ϕ_i|=||_n-||=O_p(√(p/n)).Thus, for i=1,2,⋯, p, we have(λ_i-d_i)^2=O_p(p/n)=O_p(1/n).For fixed p, by Lemmas <ref> and <ref>,(p-k)(λ-λ̂)^2 = (p-k)(λ-1/p-k∑_i=k+1^pd_i)^2= (∑_i=k+1^p(λ-d_i))^2/(p-k)≤ pmax_1≤ i≤ p(d_i-λ_i)^2≤ pmax_1≤ i≤ p|ϕ_i|^2= p||_n-||^2= O_p(p^2/n) Noting that (p-k)(λ-λ̂)^2=O_p(1/n), by Taylor's expansion, we getlog L_k = -1/2n(∑_i=1^klog d_i+(p-k)logλ̂)= -1/2n∑_i=1^klog d_i-1/2n(p-k)log((1/p-k∑_i=k+1^pd_i-λ)+λ)= -1/2n∑_i=1^klog d_i-1/2n(p-k)log((1/p-k∑_i=k+1^pd_i-1)+1)= -1/2n∑_i=1^klog d_i-1/2n(p-k)((1/p-k∑_i=k+1^pd_i-1) -1/2(1/p-k∑_i=k+1^pd_i-1)^2(1+o(1)))= -1/2n∑_i=1^klog d_i-1/2n∑_i=k+1^pd_i+1/2n(p-k)+O_p(1)= -1/2n∑_i=1^klog d_i-1/2n∑_i=k+1^p(d_i-1)+O_p(1).Noting that C_n→∞, (<ref>) holds. Let logL̃_k=-1/2n∑_i=1^klog d_i-1/2n∑_i=k+1^p(d_i-1). Suppose k'<k. We haveℓ̃(k)-ℓ̃(k')=logL̃_k-logL̃_k'-γ(k-k')(p-(k+k')/2+1/2)C_n.By Lemmas <ref> and <ref>, for i=1,2,⋯, p,|λ_i-d_i|≤max_1≤ i≤ p|ϕ_i|=||_n-||=O_p(√(1/n)). Since λ_k>1, for large n,λ_k-1-logλ_k>ϵ,where ϵ>0 is a sufficiently small constant.Since(d_i-1-log d_i)-(λ_i-1-logλ_i)P⟶0,for large n, we haveP{ℓ̃(k)>ℓ̃(k')} = P{logL̃_k-logL̃_k'>γ(k-k')(p-(k+k')/2+1/2)C_n}= P{1/2n∑_i=k'+1^k(d_i-1-log d_i)>γ(k-k')(p-(k+k')/2+1/2)C_n}≥ P{1/2n(k-k')(λ_k-1-logλ_k)+o_p(n) >γ(k-k')(p-(k+k')/2+1/2)C_n}≥ P{1/2n(λ_k-1-logλ_k)+o_p(n)>γ(p-k/2+1/2)C_n}→ 1.Suppose k'>k. Since d_i-1=O_p(√(1/n)) for i>k, by Taylor's expansion, we getlogL̃_k-logL̃_k' = 1/2n∑_i=k+1^k'(log d_i+1-d_i)= 1/2n∑_i=k+1^k'(log (1+(d_i-1))+1-d_i)= -1/4n∑_i=k+1^k'(d_i-1)^2(1+o(1))= -O_p(1).Thus,P{ℓ̃(k)>ℓ̃(k')} = P{logL̃_k-logL̃_k'>γ(k-k')(p-(k+k')/2+1/2)C_n}> P{-O_p(1)>γ(k-k')(p-(k+k')/2+1/2)C_n}→ 1.Let Ł={l_1,⋯,l_n} and_n=Ł^⊤,whereis orthogonal and its column _j is the eigenvector of Łwith eigenvalues l_j. Without loss of generality, we assume that E(_1(t))=. Then,Let=_11^-1/2_1^⊤.For 1≤ s≤ n, the s-th column of ,_s=(_11^-1/2_1^⊤)_s∼ N(, ),and are independent and identically distributed.Noting that(1/n_11^-1/2_1^⊤(+_n)X_1_11^-1/2)_ij = (1/n(_11^-1/2_1^⊤)(+Ł)(^⊤_1_11^-1/2))_ij= (1/n(+Ł)^⊤)_ij= 1/n∑_s=1^n(1+l_s)z_isz_js,we have1/n^̌⊤_11^-1/2_1^⊤(+_n)_1_11^-1/2=̌1/n∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js),where ^̌⊤=̌1. Hence,E(1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2)_ij = 1/n∑_s=1^n(1+l_s)E(z_isz_js)= 1/n∑_s=1^n(1+l_s)I_ij.That is,E(1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2) = 1/n∑_s=1^n(1+l_s)=1/n(+_n).Noting that,1/n^̌⊤(_11^-1/2_1^⊤(I+A_n)_1_11^-1/2-∑_s=1^n(1+l_s)) = 1/n∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js)-1/n∑_s=1^n(1+l_s)^̌⊤ = 1/n∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js)-1/n∑_s=1^n(1+l_s),we haveP(1/n^̌⊤(_11^-1/2_1^⊤(+_n)_1_11^-1/2-(+_n))>̌t)= P(^̌⊤(_11^-1/2_1^⊤(+_n)_1_11^-1/2-∑_s=1^n(1+l_s))>̌nt)= ∫_∏_s=1^ndF(z_1s,…,z_ks)≤ ∫_exp(a∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js)-a∑_s=1^n(1+l_s))/exp(ant)∏_s=1^ndF(z_1s,…,z_ks)≤ ∫_exp(a∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js)-a∑_s=1^n(1+l_s))/exp(ant)∏_s=1^ndF(z_1s,…,z_ks)= exp(-ant-a∑_s=1^n(1+l_s))∏_s=1^nE(exp(a(1+l_s)∑_i,j=1^kv_iv_jz_isz_js))= exp(-ant-a∑_s=1^n(1+l_s))∏_s=1^nE(exp(a(1+l_s)W_s)),where={∑_s=1^n(1+l_s)(∑_i,j=1^kv_iv_jz_isz_js)-∑_s=1^n(1+l_s)>nt}, W_s=∑_i,j=1^kv_iv_jz_isz_js=(∑_i=1^kv_iz_is)(∑_j=1^kv_jz_js)=(∑_i=1^kv_iz_is)^2∼χ^2(1). E(exp(a(1+l_s)W_s)) = ∑_i=0^∞∫(a(1+l_s)w)^i/i!dF(w)≤ 1+a(1+l_s)+∑_i=2^∞∫(a(1+l_s)w)^i/i!dF(w)≤ 1+a(1+l_s)+∑_i=2^∞(a(1+l_s))^iE(w^i)/i!≤ 1+a(1+l_s)+∑_i=2^∞(2a(1+l_s))^i/i!Γ(1/2+i)/Γ(1/2)≤ 1+a(1+l_s)+∑_i=2^∞(2a(1+l_s))^i≤ 1+a(1+l_s)+4ρ_1a^2(1+l_s)^2,where 2a(1+l_1)<1, ρ_1≥1/1-2a(1+l_1).Hence, for any 0<a<1/2(1+l_1), we have∏_s=1^nE(exp(a(1+l_s)W_s)) ≤ e^∑_s=1^nlog(1+a(1+l_s)+4ρ_1a^2(1+l_s)^2)≤ e^a∑_s=1^n(1+l_s)+4ρ_1a^2∑_s=1^n(1+l_s)^2. That is, for any 0<a<1/2(1+l_1), we haveP(1/n^̌⊤(_11^-1/2X_1^⊤(+_n)_1_11^-1/2-(+_n))>̌t)≤ exp(-ant+4ρ_1a^2∑_s=1^n(1+l_s)^2).Let f(a)=-ant+4ρ_1a^2∑_s=1^n(1+l_s)^2. Setting the derivative f'(a) to zero gives a=nt/8ρ_1∑_s=1^n(1+l_s)^2. Thus,P(1/n^̌⊤(_11^-1/2_1^⊤(+_n)_1_11^-1/2-(+_n)I)>̌t)≤ exp(-nt^2/16ρ_1∑_s=1^n(1+l_s)^2/n). Furthermore, according to the proof of Lemma 3 in <cit.>, we haveP(||1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2-1/n(+_n)||>t)≤ P(sup_1≤ j≤5^k1/n|_̌j^⊤(_11^-1/2_1^⊤(+_n)_1_11^-1/2-∑_s=1^n(1+l_s))_̌j|>t/4)≤ ∑_j=1^5^k2P(1/n_̌j^⊤(_11^-1/2_1^⊤(+_n)_1_11^-1/2-∑_s=1^n(1+l_s))_̌j>t/4)≤2×5^kexp(-nt^2/16^2ρ_1∑_s=1^n(1+l_s)^2/n)= 2exp(Ck-nt^2/ρ∑_s=1^n(1+l_s)^2/n),where _̌j^⊤_̌j=1, ρ=16^2ρ_1.Noting that 2a(1+l_1)<1, we havet<4ρ_1∑_s=1^n(1+l_s)^2/n/1+l_1=ρ∑_s=1^n(1+l_s)^2/n/64(1+l_1). By definition, each d_i solves the equation0=| d -_n|=| d -_22|| d - K_n(d)|,whereK_n(d)=1/n_1^⊤(+_̋n(d))_1, _̋n(d)=1/n_2(d -_22)^-1_2^⊤.The d_i's then solve the determinant equation| d - K_n(d)|=0.That is, d_i's are solutions of| d-1/n_1^⊤(+_̋n(d))_1|=0.Or equivalently,| _11^-1d-1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2|=0Next, we demonstrate that the above determinant (<ref>) has k solutions inthe interval ((1+√(c))^2, ∞). For d>(1+√(c))^2, by the definition of F_c(x), it is easy to see that m_1(d)=O(1) and m_2(d)=O(1). Let β_j, j=1,…, p-k be the eigenvalues of _22=1/n_2^⊤_2. Then we have1/n(_̋n(d)) =1/n(1/n_2(d -_22)^-1_2^⊤)=1/n((d -_22)^-1_22)=1/n∑_j=1^p-kβ_j/d-β_j=p-k/n1/p-k∑_j=1^p-kβ_j/d-β_j=p-k/n∫x/d-xdF_n(x)=p-k/n∫x/d-xdF_c(x)(1+O_p(k/n))=(c+(p/n-c)-k/n)m_1(d)(1+O_p(k/n))=cm_1(d)(1+O_p(√(k/n))).Moreover, we have1/n(_̋n^2(d)) =1/n((d -_22)^-1_22(d -_22)^-1_22)=1/n∑_j=1^p-kβ_j^2/(d-β_j)^2=p-k/n1/p-k∑_j=1^p-kβ_j^2/(d-β_j)^2=p-k/n∫x^2/(d-x)^2dF_n(x)=p-k/n∫x^2/(d-x)^2dF_c(x)(1+O_p(k/n))=(c+(p/n-c)-k/n)m_2(d)(1+O_p(k/n))=cm_2(d)(1+O_p(√(k/n))). Combing (<ref>) and (<ref>), we have1/n∑_s=1^n(1+l_s)^2=1+2/n(_̋n(d))+1/n(_̋n^2(d))=1+O_p(1),where l_1≥⋯≥ l_n are the eigenvalues of _̋n(d).By Corollary <ref>, we have||1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2-1/n(+_̋n(d))I||=O_p(√(k/n)). As a result, we have the following spectral decomposition of1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2-1/n(+_̋n(d))= (d)(v_1(d),…,v_k(d))^⊤(d),where (d) is an orthogonal matrix and max_1≤ j≤ k|v_j(d)|=O_p(√(k/n)).By (<ref>), we have1/n(+_̋n(d))=1+cm_1(d)(1+O_p(√(k/n))),Combing (<ref>) and (<ref>), we have1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2-(1+cm_1(d))= (d)≜(d)(ṽ_1(d),…,ṽ_k(d))^⊤(d),where max_1≤ j≤ k|ṽ_j(d)|=O_p(√(k/n)).Note that_11^-1d-1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2= _11^-1d-(1+cm_1(d))-[1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2-(1+cm_1(d))]= _11^-1d-(1+cm_1(d))-(d).and[ _11^-1=(1/λ_1,…,1/λ_k)^⊤. ] Multiplying both sides of (<ref>) by ^⊤ from the left and byfrom the right yields|^⊤(_11^-1d-1/n_11^-1/2_1^⊤(+_̋n(d))_1_11^-1/2)|= |(d-(1+cm_1(d))λ_1/λ_1,…,d-(1+cm_1(d))λ_k/λ_k)-^⊤(d)|= 0.Let ê_σ(i)'s be the eigenvalues of the following matrix,(d-(1+cm_1(d))λ_1/λ_1,…,d-(1+cm_1(d))λ_k/λ_k)-^⊤(d),where σ(·) is a permutationof the integers 1,…,n. Then, d_i's are the solutions of∏_i=1^kê_σ(i)=0 That is, forany 1≤ i≤ k, d_i is the solution ofê_σ(i)=0. Also, let e_i's be the eigenvalues of the following matrix,(d-(1+cm_1(d))λ_1/λ_1,…,d-(1+cm_1(d))λ_k/λ_k).The following result shows how close are the eigenvalues of ê_σ(i)'s to e_i's. Noting that^⊤(d)=^⊤(d)(ṽ_1(d),…,ṽ_2(d))^⊤(d), we have((^⊤(d))^2)=((ṽ_1^2(d),…,ṽ_k^2(d)))=kO_p(k/n)=O_p(k^2/n).By Lemma <ref>, we have∑_i=1^k(ê_σ(i)-e_i)^2≤((^⊤(d))^2).Hence,∑_i=1^k|ê_σ(i)-e_i|≤√(k∑_i=1^k(ê_σ(i)-e_i)^2)=O_p(√(k^3/n)). Combining (<ref>) and (<ref>), forany 1≤ i≤ k, d_i tends to the solution ofd_i-(1+cm_1(d_i))λ_i/λ_i=0. According to <cit.>, forany 1≤ i≤ k, we haved_i/ψ(λ_i)a.s.⟶1. By definition, each d_i solves the equation0=| d -_n|=| d -_22|| d - K_n(d)|,whereK_n(d)=1/n_1^⊤(+_̋n(d))_1, _̋n(d)=1/n_2(d -_22)^-1_2^⊤.The d_i's then solve the determinant equation| d - K_n(d)|=0.That is| d_i-1/n_1^⊤(+_̋n(d_i))_1|=0.Or equivalently,| _11^-1d_i-1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2|=0Note thatΣ_11^-1d_i-1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2= Σ_11^-1d_i-(1+cm_1(d_i))-[1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2-(1+cm_1(d_i))]= _11^-1d_i-(1+cm_1(d_i))-(d_i),where (d_i)=1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2-(1+cm_1(d_i)). Similar to the proof of (<ref>) in Theorem <ref>, we have(d_i)=(d_i)(ṽ_1(d_i),…,ṽ_k(d_i))^⊤(d_i),where (d_i) is an orthogonal matrix and max_1≤ j≤ k|ṽ_j(d_i)|=O_p(√(k/n)). Note that _11^-1=(1/λ_1,…,1/λ_k)^⊤. Multiplying both sides of (<ref>) by ^⊤ from the left and byfrom the right yields|^⊤(_11^-1d_i-1/n_11^-1/2_1^⊤(+_̋n(d_i))_1_11^-1/2)|= |(d_i-(1+cm_1(d_i))λ_1/λ_1,…,d_i-(1+cm_1(d_i))λ_k/λ_k)-^⊤(d_i)|= 0. Let(d_i)=(b_st)_k× k= ^⊤(d_i)= ^⊤(d_i)(ṽ_1(d_i),…,ṽ_k(d_i))^⊤(d_i) ≜ (d_i)(ṽ_1(d_i),…,ṽ_k(d_i))^⊤(d_i),where (d_i)=^⊤(d_i)=(q_st)_k× k is an orthogonal matrix. For any 1≤ s≤ k, 1≤ t≤ k, and s≠ t, by the definitions of (d_i) and (d_i), we have a_st=b_st.Similar to the proof of (<ref>) in Theorem <ref>, we have∑_s=1^k∑_t=1^kb_st^2=(^2(d_i))=O_p(k^2/n). We also have∑_t=1^kb_it^2=∑_t=1^k(∑_l=1^kṽ_l(d_i)q_ilq_tl)^2= ∑_t=1^k∑_l=1^kṽ_l^2(d_i)q_il^2q_tl^2+∑_t=1^k∑_l=1^k∑_s≠ l^kṽ_l(d_i)ṽ_s(d_i)q_ilq_tlq_isq_ts= ∑_t=1^k∑_l=1^kṽ_l^2(d_i)q_il^2q_tl^2+∑_l=1^k∑_s≠ l^kṽ_l(d_i)ṽ_s(d_i)q_ilq_is∑_t=1^kq_tlq_ts= ∑_t=1^k∑_l=1^kṽ_l^2(d_i)q_il^2q_tl^2=∑_l=1^kṽ_l^2(d_i)q_il^2∑_t=1^kq_tl^2= ∑_l=1^kṽ_l^2(d_i)q_il^2≤max_1≤ l≤ kṽ_l^2(d_i)∑_l=1^kq_il^2= O_p(k/n). Note that((d_i))=∑_t=1^k(-1)^i+ta_itH_it=0,where the (k-1)-th order determinant H_it is the cofactor of a_it, obtained by deleting the i-th row and the t-th column of the k-th order determinant ((d_i)).LetH_ii=∏_j≠ iê_j'≜∏_j≠ i(e_j+ϵ_j'),where ê_j'(j≠ i) are eigenvalues of corresponding matrix _̋ii. Noting that _̋ii is a symmetric matrix, similar to the proof of (<ref>) in Theorem <ref>, we have∑_j≠ i^k|ϵ_j'|=O_p(√(k^3/n)). Noting that|∑_t≠ i1/e_tϵ_t'| ≤ ∑_t≠ i1/|e_t||ϵ_t'|≤ 1/min_t≠ i|e_t|∑_t≠ i|ϵ_t'|≤ 1/min_t≠ i|e_t|O_p(√(k^3/n))= O_p(√(k^3/n)),and|∑_t≠ i∑_s≠ i,1≤ s<t1/e_te_sϵ_t'ϵ_s'| ≤ ∑_t≠ i∑_s≠ i,1≤ s<t1/|e_te_s||ϵ_t'ϵ_s'|≤ 1/min_t≠ i,s≠ i,1≤ s<t|e_te_s|∑_t≠ i∑_s≠ i,1≤ s<t|ϵ_t'ϵ_s'|≤ 1/min_t≠ i,s≠ i,1≤ s<t|e_te_s|(∑_t≠ i|ϵ_t'|)^2= 1/min_t≠ i,s≠ i,1≤ s<t|e_te_s|O_p(k^3/n)= O_p(k^3/n),we haveH_ii =∏_j≠ i(e_j+ϵ_j')=∏_j≠ ie_j+∏_j≠ ie_j∑_t≠ i1/e_tϵ_t'+∏_j≠ ie_j∑_t≠ i∑_s≠ i,1≤ s<t1/e_te_sϵ_t'ϵ_s'+⋯=∏_j≠ ie_j+O_p(√(k^3/n))∏_j≠ ie_j+O_p(k^3/n)∏_j≠ ie_j+⋯=(1+O_p(√(k^3/n)))∏_j≠ ie_j. For t≠ i, H_it=∏_j≠ i, t^kê_j”ê_t”,where ê_j”(j≠ i, t) and ê_t” are eigenvalues of corresponding matrix _̋it.By Gerschgorin's circle theorem, we have|H_it|=∏_j≠ i, t^k|ê_j'||ê_t”|≤∏_j≠ i, t^k(|e_j|+∑_r=1^k|b_jr|)|ê_t”|≤∏_j≠ i, t^k(|e_j|+∑_r=1^k|b_jr|)|∑_r≠ t^k|a_tr|.Noting that∑_s≠ i,t1/|e_s|∑_r=1^k|b_sr| ≤ 1/min_s≠ i,t|e_s|∑_s≠ i,t∑_r=1^k|b_sr|≤ 1/min_s≠ i,t|e_s|√(k^2∑_s≠ i,t∑_r=1^kb_sr^2)= O_p(√(k^4/n)),and∑_s≠ i,t∑_u≠ i,t,1≤ u<s1/|e_se_u|∑_r=1^k|b_sr|∑_r=1^k|b_ur|≤ 1/min_s≠ i,t,u≠ i,t,1≤ u<s|e_se_u|∑_s≠ i,t∑_r=1^k|b_sr|∑_u≠ i,t,1≤ u<s∑_r=1^k|b_ur|= 1/min_s≠ i,t,u≠ i,t,1≤ u<s|e_se_u|O_p(k^4/n)= O_p(k^4/n),we have∏_j≠ i, t^k(|e_j|+∑_r=1^k|b_jr|)= ∏_j≠ i,t|e_j|+∏_j≠ i,t|e_j|∑_s≠ i,t1/|e_s|∑_r=1^k|b_sr|+∏_j≠ i,t|e_j|∑_s≠ i,t∑_u≠ i,t,1≤ u<s1/|e_se_u|∑_r=1^k|b_sr|∑_r=1^k|b_ur|+⋯= ∏_j≠ i,t|e_j|+O_p(√(k^4/n))∏_j≠ i,t|e_j|+O_p(k^4/n)∏_j≠ i,t|e_j|+⋯= (1+O_p(√(k^4/n)))∏_j≠ i,t|e_j|≤ 1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i|e_j|Hence,|H_it|≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i|e_j||ê_t”|.Combing (<ref>), (<ref>), (<ref>), and noting that for any 1≤ s≤ k, 1≤ t≤ k, s≠ t, a_st=b_st, we have|∑_t≠ i^ka_itH_it| ≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i^k|e_j|∑_t≠ i^k|a_itê_t”|≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i^k|e_j|√(∑_t≠ i^ka_it^2∑_t≠ i^kê_t”^2)≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i^k|e_j|√(∑_t≠ i^ka_it^2∑_t≠ i^k(∑_r≠ t^k|a_tr|)^2)≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i^k|e_j|√(∑_t≠ i^kb_it^2∑_t≠ i^k(∑_r≠ t^k|b_tr|)^2)≤1+O_p(√(k^4/n))/min_j≠ i|e_j|∏_j≠ i^k|e_j|√(∑_t≠ i^kb_it^2∑_t≠ i^kk∑_r≠ t^kb_tr^2)=1+O_p(√(k^4/n))/min_j≠ i|e_j|O_p(√(k^4/n^2))∏_j≠ i^k|e_j|.Combing (<ref>),(<ref>), and (<ref>), we have|a_ii| = |∑_t≠ i^ka_itH_it/H_ii|≤ 1+O_p(√(k^4/n))/1+O_p(√(k^3/n))1/min_j≠ i|e_j|O_p(√(k^4/n^2))= 1+O_p(√(k^4/n))/min_j≠ i|e_j|O_p(√(k^4/n^2))= O_p(√(k^4/n^2)).Thus,for any 1≤ i≤ k, the i-th diagonal element ofsatisfiesd_i-(1+cm_1(d_i))λ_i/λ_i-[^⊤(d_i)]_ii=O_p(√(k^4/n^2)).According to (<ref>) in Lemma <ref>, we have[^⊤(d_i)]_ii=b_ii=∑_l=1^kṽ_l(d_i)q_il^2=O_p(√(k/n)).By Lemma <ref>, we haved_i-(1+cm_1(d_i))λ_i/λ_i=O_p(√(k/n)).Forany 1≤ i≤ k, by Theorem <ref>, we have d_i>(1+√(c))^2. Let d_i=α_i+cα_i/α_i-1≜ψ(α_i).Then, we getα_i=-c+d_i+1+√((c-d_i-1)^2-4d_i)/2, α̣_i/ḍ_i=1/2+1/2d_i-c-1/√((c-d_i-1)^2-4d_i)>0.As a result, we have α_i>1+√(c). According to <cit.>, m_1(d_i)=m_1(ψ(α_i))=1/α_i-1. Thus, (<ref>) becomes(α_i/λ_i-1)(1+c/α_i-1)=O_p(√(k/n)).That is,α_i=λ_i(1+O_p(√(k/n))).Thus, forany 1≤ i≤ k, we haved_i-ψ(λ_i)/λ_i = α_i+cα_i/α_i-1-λ_i-cλ_i/λ_i-1/λ_i= 1+O_p(√(k/n))+c(1+O_p(√(k/n)))/λ_i(1+O_p(√(k/n)))-1-1-c/λ_i-1= O_p(√(k/n))-cO_p(√(k/n))/(λ_i(1+O_p(√(k/n)))-1)(λ_i-1)= O_p(√(k/n)).By definition, each d_i solves the equation0=| d -_n|=| d -_22|| d - K_n(d)|,whereK_n(d)=1/n_1^⊤(+_̋n(d))_1, _̋n(d)=1/n_2(d -_22)^-1_2^⊤.The d_i's then solve the determinant equation| d - K_n(d)|=0.That is| d_i-1/n_1^⊤(+_̋n(d_i))_1|=0. The random form K_n(d_i) can be decomposed as followsK_n(d_i)1/n_1^⊤(+_̋n(d_i))_1= 1/n(_1^⊤(+_̋n(d_i))_1-(+_̋n(d_i))_11)+1/n(+_̋n(d_i))_11= 1/√(n)R_n(d_i)+1/n(+_̋n(d_i))_11,whereR_n(d_i)=1/√(n)(_1^⊤(+_̋n(d_i))_1-(+_̋n(d_i))_11).Letδ_i=√(n)(d_i-ψ(λ_i))/λ_i=O_p(√(k)). Then the form d_i - K_n(d_i) can be decomposed as followsd_i - K_n(d_i)=ψ(λ_i)+1/√(n)δ_iλ_i- K_n(ψ(λ_i))-( K_n(d_i)- K_n(ψ(λ_i))). Furthermore, using ^-1-^-1=^-1(-)^-1 and treating =(ψ(λ_i)+1/√(n)δ_iλ_i)-_22, =ψ(λ_i)-_22, we have_11^-1/2[ K_n(d_i)- K_n(ψ(λ_i))]_11^-1/2= _11^-1/21/n_1^⊤1/n_2{[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1-(ψ(λ_i)-_22)^-1}_2^⊤_1_11^-1/2= -1/√(n)δ_iλ_i_11^-1/21/n_1^⊤1/n_2{[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1[ψ(λ_i)-_22]^-1}_2^⊤_1_11^-1/2.Treating_n=λ_i1/n_2[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_2^⊤,we have1/n(_n)= 1/n{λ_i1/n_2[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_2^⊤}= λ_i1/n{[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_22}= λ_i1/n∑_j=1^p-kβ_j/(ψ(λ_i)+1/√(n)δ_iλ_i-β_j)(ψ(λ_i)-β_j)= λ_ip-k/n1/p-k∑_j=1^p-kβ_j/(ψ(λ_i)+1/√(n)δ_iλ_i-β_j)(ψ(λ_i)-β_j)= λ_ip-k/n1/p-k∑_j=1^p-kβ_j/(ψ(λ_i)-β_j)^2(ψ(λ_i)-β_j)/(ψ(λ_i)+1/√(n)δ_iλ_i-β_j)= λ_ip-k/n1/p-k∑_j=1^p-kβ_j/(ψ(λ_i)-β_j)^2(1+O_p(1/√(n)δ_i))= λ_ip-k/n∫x/(ψ(λ_i)-x)^2dF_n(x)(1+O_p(1/√(n)δ_i))= λ_ip-k/n∫x/(ψ(λ_i)-x)^2dF_c(x)(1+O_p(1/√(n)δ_i))(1+O_p(k/n))= λ_i(c+(p/n-c)-k/n)m_3(ψ(λ_i))(1+O_p(1/√(n)δ_i))(1+O_p(k/n))= λ_icm_3(ψ(λ_i))(1+O_p(√(k/n))),where m_3(d)=∫x/(d-x)^2dF_c(x).Noting thatλ_im_3(ψ(λ_i)) = λ_i∫x/(ψ(λ_i)-x)^2dF_c(x)= ∫λ_i/(ψ(λ_i)-x)x/(ψ(λ_i)-x)dF_c(x)≤ λ_i/ψ(λ_i)-(1+√(c))^2∫x/ψ(λ_i)-xdF_c(x)= O(1),we have1/n(_n)=O_p(1).Note that,1/n(_n^2)= 1/n{λ_i1/n_2[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_2^⊤ λ_i1/n_2[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_2^⊤}= λ_i^21/n{[(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_22 [(ψ(λ_i)+1/√(n)δ_iλ_i)-_22]^-1(ψ(λ_i)-_22)^-1_22}= λ_i^21/n∑_j=1^p-kβ_j^2/(ψ(λ_i)+1/√(n)δ_iλ_i-β_j)^2(ψ(λ_i)-β_j)^2= λ_i^2p-k/n1/p-k∑_j=1^p-kβ_j^2/(ψ(λ_i)+1/√(n)δ_iλ_i-β_j)^2(ψ(λ_i)-β_j)^2= λ_i^2p-k/n∫x^2/(ψ(λ_i)-x)^4dF_n(x)(1+O_p(1/√(n)δ_i))= λ_i^2p-k/n∫x^2/(ψ(λ_i)-x)^4dF_c(x)(1+O_p(1/√(n)δ_i))(1+O_p(k/n))= λ_i^2(c+(p/n-c)-k/n)∫x^2/(ψ(λ_i)-x)^4dF_c(x)(1+O_p(1/√(n)δ_i))(1+O_p(k/n))= λ_i^2 cm_4(ψ(λ_i))(1+O_p(√(k/n))),where m_4(d)=∫x^2/(d-x)^4dF_c(x).Noting thatλ_i^2m_4(ψ(λ_i)) = λ_i^2∫x^2/(ψ(λ_i)-x)^4dF_c(x)= ∫λ_i^2/(ψ(λ_i)-x)^2x^2/(ψ(λ_i)-x)^2dF_c(x)≤ λ_i^2/(ψ(λ_i)-(1+√(c))^2)^2∫x^2/(ψ(λ_i)-x)^2dF_c(x)= O(1),we have1/n(_n^2)=O_p(1).Hence, we have1/n∑_s=1^nl_s^2=1/n(_n^2)=O_p(1).where l_1≥⋯≥ l_n are the eigenvalues of _n. By Lemma <ref>, we have||1/n_11^-1/2_1^⊤_n_1_11^-1/2-1/n(_n)||=O_p(√(k/n)).Combining (<ref>), (<ref>), (<ref>), and (<ref>), we have the following spectral decomposition of_11^-1/2[ K_n(d_i)- K_n(ψ(λ_i))]_11^-1/2= -1/√(n)δ_iλ_icm_3(ψ(λ_i))(1+O_p(√(k/n)))-1/√(n)δ_i(d_i)(w_1(d_i),…,w_k(d_i))^⊤(d_i),where (d_i) is orthogonal and max_1≤ j≤ k|w_j(d_i)|=O_p(√(k/n)).Noting that p/n-c=o(n^-1/2), similar to the proof of (<ref>) in Theorem <ref>, we have1/n(+_̋n(d_i))=1+cm_1(ψ(λ_i))(1+o(n^-1/2)). Combining (<ref>), (<ref>), (<ref>), and (<ref>), we have _11^-1/2(d_i - K_n(d_i))_11^-1/2= _11^-1ψ(λ_i)+1/√(n)δ_iλ_i_11^-1-[1+cm_1(ψ(λ_i))(1+o(n^-1/2))]-1/√(n)_11^-1/2 R_n(ψ(λ_i))_11^-1/2+1/√(n)δ_iλ_icm_3(ψ(λ_i))(1+O_p(√(k/n)))+1/√(n)δ_i(d_i)(w_1(d_i),…,w_k(d_i))^⊤(d_i).Note thatψ(λ_i)-(1+cm_1(ψ(λ_i)))λ_j/λ_j=(λ_i/λ_j-1)(1+c/λ_i-1),and[ _11^-1=(1/λ_1,…,1/λ_k)^⊤. ] Multiplying both sides of (<ref>) by ^⊤ from the left and byfrom the right, we have|^⊤_11^-1/2(d_i - K_n(d_i))_11^-1/2|= |((λ_i/λ_1-1)(1+c/λ_i-1),…,(λ_i/λ_k-1)(1+c/λ_i-1)) +cm_1(ψ(λ_i))o(n^-1/2)+1/√(n)δ_iλ_i(1/λ_1,…,1/λ_k) -1/√(n)^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2 +1/√(n)δ_iλ_icm_3(ψ(λ_i))(1+O_p(√(k/n))) -1/√(n)δ_i^⊤(d_i)(w_1(d_i),…,w_k(d_i))^⊤(d_i)|. Note that ^⊤(d_i) is an orthogonal matrix and max_1≤ j≤ k|w_j(d_i)|=O_p(√(k/n)). The i-th diagonal element of1/√(n)δ_i^⊤(d_i)(w_1(d_i),…,w_k(d_i))^⊤(d_i)is 1/√(n)δ_iO_p(√(k/n)). Since (λ_i/λ_i-1)(1+c/λ_i-1)=0, by Lemma <ref>, the i-th diagonal element of(d_i)=^⊤_11^-1/2(d_i - K_n(d_i))_11^-1/2satisfies1/√(n)δ_i(1+cm_3(ψ(λ_i))λ_i)-[1/√(n)^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2]_ii +1/√(n)δ_iO_p(√(k/n))+o(n^-1/2)=O_p(√(k^4/n^2)). That is,1/√(n)δ_i(1+cm_3(ψ(λ_i))λ_i)-[1/√(n)^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2]_ii +1/√(n)δ_iO_p(√(k/n))+o(n^-1/2)+O_p(√(k^4/n^2))=0. Or equivalently,|δ_i(1+cm_3(ψ(λ_i))λ_i)-[^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2]_ii +δ_iO_p(√(k/n))+o(1)+O_p(√(k^4/n))|=0. Noting thatδ_i=√(n)(d_i-ψ(λ_i))/λ_i=O_p(√(k)),under the condition k=o(n^1/4), δ_i tends to a solution of|δ_i(1+cm_3(ψ(λ_i))λ_i)-[^⊤_11^-1/2 R_n(ψ(λ_i))Σ_11^-1/2]_ii|=0.That is, for any 1≤ i≤ k,δ_ia.s.⟶(1+cm_3(ψ(λ_i))λ_i)^-1[^⊤_11^-1/2 R_n(ψ(λ_i))_11^-1/2]_ii.Similar to the proof of Corollary <ref>, by Chebyshev's inequality, we haveP(||1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2-1/n(+_n)I||>t)≤ P(max_1≤ i≤ k∑_1≤ j≤ k|1/n∑_s=1^n(1+l_s)z_isz_js-1/n∑_s=1^n(1+l_s)I_ij|>t)≤ ∑_1≤ i≤ kP(∑_1≤ j≤ k∑_s=1^n(1+l_s)|z_isz_js-I_ij|>nt)= ∑_1≤ i,j≤ k∑_1≤ j≤ k∑_s=1^n(1+l_s)^2E(z_is^2z_js^2)/n^2t^2= ∑_1≤ i,j≤ k∑_s=1^n(1+l_s)^2E(z_is^2z_js^2)/n^2t^2≤ ∑_1≤ i,j≤ k∑_s=1^n(1+l_s)^2√(E(z_is^4)E(z_js^4))/n^2t^2≤ Ck^2∑_s=1^n(1+l_s)^2/n^2t^2. Thus, we have||1/n_11^-1/2_1^⊤(+_n)_1_11^-1/2-1/n(+_n)||=√(1/n∑_s=1^n(1+l_s)^2)O_p(k/√(n)).Let log L_k'=-1/2n∑_i=1^k'log d_i-1/2n(p-k')logλ̂_k'. According to <cit.>,λ̂=1/(p-k)∑_i=k+1^pd_ia.s.⟶1.Suppose k'<k. We haveℓ(k)-ℓ(k')=log L_k-log L_k'-γ(k-k')(p-(k+k')/2+1/2). By Theorem <ref>, for k'<i≤ k,d_i/ψ(λ_i)a.s.⟶ 1. Since ψ(λ_k)-1-logψ(λ_k)>2γ c, by Taylor's expansion, we getP{ℓ(k)>ℓ(k')}= P{log L_k-log L_k'>γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n∑_i=1^klog d_i-1/2n(p-k)logλ̂+1/2n∑_i=1^k'log d_i+1/2n(p-k')logλ̂_k' >γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n(p-k)logλ̂+1/2n(p-k')logλ̂_k'-1/2n∑_i=k'+1^klog d_i >γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n(p-k')logλ̂+1/2n(p-k')logλ̂_k'+1/2n(k-k')logλ̂ -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}= P{1/2n(p-k')log(λ̂_k'/λ̂)+1/2n(k-k')logλ̂ -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}= P{1/2n(p-k')log[(1-(k-k')/(p-k'))(∑_i=k'+1^pd_i/∑_i=k+1^pd_i)]+1/2n(k-k')logλ̂ -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n(k-k')+1/2n(p-k')log[(1/p-k∑_i=k'+1^kd_i+1/p-k∑_i=k+1^pd_i)/(1/p-k∑_i=k+1^pd_i)] +o(n(k-k'))-1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}.For the case 1/p-k∑_i=k'+1^kψ(λ_i)=o(1), noting that 1/p-k∑_i=k'+1^kd_i=o_p(1), we haveP{ℓ(k)>ℓ(k')}= P{-1/2n(k-k')+1/2n(p-k')log[(1/p-k∑_i=k'+1^kd_i+1/p-k∑_i=k+1^pd_i)/(1/p-k∑_i=k+1^pd_i)] +o(n(k-k'))-1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n(k-k')+1/2n(p-k')[(1/p-k∑_i=k'+1^kd_i)/(1/p-k∑_i=k+1^pd_i)]+o(n(k-k')) -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}= P{1/2n∑_i=k'+1^k(d_i-1-log d_i)+o(n(k-k')) >γ(k-k')(p-(k+k')/2+1/2)}= P{1/2n∑_i=k'+1^k(ψ(λ_i)-1-logψ(λ_i))>γ(k-k')(p-(k+k')/2+1/2)(1+o(1))}≥ P{ψ(λ_k)-1-logψ(λ_k)>2γ(p-(k+k')/2+1/2)(1+o(1))/n}≥ P{ψ(λ_k)-1-logψ(λ_k)>2γ p(1+o(1))/n}→ 1. For the case 1/p-k∑_i=k'+1^kψ(λ_i)/C → 1, where C is a constant, noting that 1/p-k∑_i=k'+1^kd_i/Ca.s.⟶ 1, we have∏_i=k'+1^kd_i≤ (1/k-k'∑_i=k'+1^kd_i)^k-k', ∑_i=k'+1^klog d_i≤(k-k')log(1/k-k'∑_i=k'+1^kd_i)=O_p(klog p),we haveP{ℓ(k)>ℓ(k')}= P{-1/2n(k-k')+1/2n(p-k')log[(1/p-k∑_i=k'+1^kd_i+1/p-k∑_i=k+1^pd_i)/(1/p-k∑_i=k+1^pd_i)] +o(n(k-k'))-1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}≥ P{-1/2n(k-k')+1/2n(p-k')log C+o(n(k-k')) -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}→ 1.For the case 1/p-k∑_i=k'+1^kd_i/C_n→1, where C_n→∞, noting that 1/p-k∑_i=k'+1^kd_i/C_na.s.⟶1, we have∑_i=k'+1^klog d_i≤(k-k')log(1/k-k'∑_i=k'+1^kd_i)=O_p(k(log C_n+log p))we haveP{ℓ(k)>ℓ(k')}= P{-1/2n(k-k')+1/2n(p-k')log[(1/p-k∑_i=k'+1^kd_i+1/p-k∑_i=k+1^pd_i)/(1/p-k∑_i=k+1^pd_i)] +o(n(k-k'))-1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}≥ P{-1/2n(k-k')+1/2n(p-k')log C_n+o(n(k-k')) -1/2n∑_i=k'+1^klog d_i>γ(k-k')(p-(k+k')/2+1/2)}→ 1. Suppose k'>k. For k<i≤ k'=o(p),d_ia.s.⟶μ_1.According to <cit.>, μ_1=(1+√(c))^2.Since γ>1/2+√(1/c)-log(1+√(c))/c, by Taylor's expansion, we getP{ℓ(k)>ℓ(k')}= P{log L_k-log L_k'>γ(k-k')(p-(k+k')/2+1/2)}= P{-1/2n∑_i=1^klog d_i-1/2n(p-k)logλ̂+1/2n∑_i=1^k'log d_i+1/2n(p-k')logλ̂_k' >-γ(k'-k)(p-(k'+k)/2+1/2)}= P{-1/2n(p-k)logλ̂+1/2n(p-k')logλ̂_k'+1/2n∑_i=k+1^k'log d_i >-γ(k'-k)(p-(k'+k)/2+1/2)}= P{-1/2n(p-k')logλ̂+1/2n(p-k')logλ̂_k'-1/2n(k'-k)logλ̂ +1/2n∑_i=k+1^k'log d_i>-γ(k'-k)(p-(k'+k)/2+1/2)}= P{1/2n(p-k')log(λ̂_k'/λ̂)-1/2n(k'-k)logλ̂ +1/2n∑_i=k+1^k'log d_i>-γ(k'-k)(p-(k'+k)/2+1/2)}= P{1/2n(p-k')log[(1+(k'-k)/(p-k'))(∑_i=k'+1^pd_i/∑_i=k+1^pd_i)]-1/2n(k'-k)logλ̂ +1/2n∑_i=k+1^k'log d_i>-γ(k'-k)(p-(k'+k)/2+1/2)}= P{1/2n(k'-k)+1/2n(p-k')log[(1/p-k∑_i=k+1^pd_i-1/p-k∑_i=k+1^k'd_i)/(1/p-k∑_i=k+1^pd_i)] -o(n(k'-k))+1/2n∑_i=k+1^k'log d_i>-γ(k'-k)(p-(k'+k)/2+1/2)}, P{ℓ(k)>ℓ(k')}= P{1/2n(k'-k)-1/2n(p-k')[(1/p-k∑_i=k+1^k'd_i)/(1/p-k∑_i=k+1^pd_i)]-o(n(k-k')) +1/2n∑_i=k+1^k'log d_i>-γ(k'-k)(p-(k'+k)/2+1/2)}= P{-1/2n∑_i=k+1^k'(d_i-1-log d_i)>-γ(k'-k)(p-(k'+k)/2+1/2)(1+o(1))}= P{-1/2n((1+√(c))^2-1-2log(1+√(c)))>-γ(p-(k'+k)/2+1/2)(1+o(1))}= P{γ>n/(p-(k'+k)/2+1/2)(c/2+√(c)-log(1+√(c))(1+o(1))}= P{γ>p/(p-(k'+k)/2+1/2)n/p(c/2+√(c)-log(1+√(c)))(1+o(1))}→ 1. ims =14.5pt | http://arxiv.org/abs/1708.07595v7 | {
"authors": [
"Jianwei Hu",
"Jingfei Zhang",
"Ji Zhu"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170825014536",
"title": "Detection of the number of principal components by extended AIC-type method"
} |
Oleksiy Golubov [email protected] 0000-0002-5720-2282]Oleksiy Golubov Department of Aerospace Engineering Sciences, University of Colorado at Boulder 429 UCB, Boulder, CO, 80309, USA Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine Institute of Astronomy of V. N. Karazin Kharkiv National University, 35 Sumska Str., Kharkiv, 61022, Ukraine The tangential YORP effect (TYORP) plays a significant role in the dynamical evolution of asteroids, and up to now has only been studied numerically. This paper describes the first analytic model of the TYORP effect. Although the model rests on numerous physical and mathematical simplifications, the final analytic expression for TYORP is found to be in agreementwith the results of rigorous numeric simulations to the accuracy of several tens per cent.The obtained analytic expression is used to estimate the TYORP produced by the non-flat surface of regolith, – a contribution to TYORP that has never been considered. It is found that the contribution to TYORP arising from regolith can be comparable to the conventional TYORP produced by boulders.Then, the analytic expression is fitted with a log-normal function and used to integrate TYORP over all boulder sizes. The general trend of TYORP for multiple boulders appears qualitatively similar to the trend of one boulder, and also demonstrates a maximal TYORP at some particular rotation rate. The obtained expression for integrated TYORP may be instrumental for simulations of evolution of asteroids subject to TYORP.To conclude, the physical origin of TYORP is discussed in light of the constructed analytic model.§ INTRODUCTIONThe tangential YORP effect, or TYORP, appears when stones on the surface of an asteroid emit different amounts of infrared light eastward and westward, thus experiencing a net recoil force tangential to the asteroid's surface. Until now, this effect has only been studied in numeric simulations <cit.>. Although it was generally understood that the effect was due to the non-linearity of the heat emission law, the detailed physics of the effect remained obscure. Moreover, the question remained whether the whole effect could be attributed to numeric artifacts.In this article I propose a minimalistic analytic model of the effect, which is based on the following simplifying assumptions:1. Instead of solving partial differential equation for the heat conduction in a boulder, the boulder is split into two parts, the eastern part and the western part, and the mean temperature of each part is introduced. Then the mathematical description of the model boils down to a system of two ordinary differential equations for the mean temperatures.2. The incoming solar energy as a function of time is approximated by the sum of its zeroth and first order Fourier terms;all higher-order terms are neglected.3. The first order Fourier term is treated perturbatively, as if it were small compared to the zeroth order term.After these simplifications the problem can be easily solved analytically, and the TYORP drag can be calculated. The result fits the numeric simulations surprisingly well.In Section <ref>, I provide a derivation of the analytic expression for TYORP. In Section <ref>, the derived analytic expression is applied to different geometries, to test it and to make some new predictions. In Section <ref>, I simplify the analytic expression and integrate it over different boulder sizes, to evaluate the total TYORP experienced by an asteroid. In Section <ref>, I discuss how the derived analytic expression helps to better understand the physics of TYORP. § GENERAL THEORY §.§ Derivation of the heat conduction equationsHeat balance within any volume part V of a boulder is governed by the following heat conduction equation in the integral form,Cρ∫_V∂ T/∂ tdV=κ∫_S_st∂ T/∂ X_idS_i+ +(1-A)∫_S_spI_idS_i-ϵσ∫_S_spT^4dS .Here T stands for the temperature. The left-hand side describes the total heat energy increase in the volume V, while the right-hand side is the sum of the heat conduction into this volume, the direct solar heat absorbed by its open surface,and the negative heat emitted by the open surface. The surface areas S_st, S_sp, and S_reg are parts of the volume's boundary bordering respectively stone, space, and regolith, so that S_st+S_sp+S_reg=∂ V is the full boundary of the volume V(see the left-hand panel of Figure <ref>). The heat conductivity of the stone is κ,its heat capacity is C, the density is ρ,the hemispherical albedo is A, and the emissivity is ϵ. The heat conductivity of the regolith is assumed to be zero. σ is Stefan–Boltzmann's constant, and 𝐈 is the vector of the incoming solar energy flux.Now I nondimensionalize the variables. Instead of time t, I use the rotation phase ϕ=ω t, with ω being the angular velocity of the asteroid. By definition, ϕ=0 at noon. The characteristic scales of length and temperature areL_cond=κ/((1-A)Φ)^3/4(ϵσ)^1/4 , T_0=√((1-A)Φ/ϵσ) ,with Φ being the solar constant. Here T_0 is the equilibrium temperature at the subsolar point, while L_cond is the distance at which the temperature difference T_0 creates heat flux equal to AΦ. I use these two scales to introduce the dimensionless variables x_i=X_i/L_cond and τ=T/T_0. The relative importance of heat conduction with respect to heat absorption and emission is characterized by the thermal parameterθ=(Cρκω)^1/2/((1-A)Φ)^3/4(ϵσ)^1/4. With these definitions, Eqn. (<ref>) transforms intoθ^2∫_v∂τ/∂ϕdv=∫_s_st∂τ/∂ x_ids_i+∫_s_spI_i/Φds_i-∫_s_spτ^4ds .Here v and s correspond to the same volumes and areas as before, but measured in the dimensionless variables x_i instead of the dimensional variables X_i.Now I separate the boulder into the western and the eastern parts and apply Eqn. (<ref>) to each part separately. I assume that the boulder is symmetric, with the western and the eastern parts being mirror reflections of each other. Let v henceforth denote the dimensionless volume of each half of the boulder (either v_w or v_e, see the right-hand panel of Figure <ref>), and s_sp denote their equal dimensionless surface areas bordering space (either s_sp e or s_sp w in the right-hand panel of Figure <ref>). I denote the mean dimensionless temperatures of the western and the eastern parts of the boulder via τ_w and τ_e correspondingly,τ_w = 1/v∫_v_wτdv,τ_e = 1/v∫_v_eτdv .The temperature gradient at the border between the two parts of the boulder can be estimated as the temperature difference divided by the distance, (τ_w-τ_e)/l_ew,with l_ew being the typical distance between the eastern and western parts of the boulder, i.e. roughly the distance between the centers of the two parts. To estimate the last term in the right-hand side of Eqn. (<ref>), τ can be substituted by its mean value, i.e. τ_w and τ_e for the western and the eastern parts of the boulder respectively.[When interpreted literally, the assumptions of a constant temperature gradient in the body of the boulder and of a constant temperature on the two parts of its surface might seem to contradict each other. Still, these assumptions should provide an acceptable estimate for the corresponding terms in Eqn. (<ref>), and thus finally lead to a reasonable estimate for TYORP.It is possible to construct a more sophisticated model for the temperature distribution inside the boulder, but it will lead to a more complicated mathematics and a more obscure physics, while its accuracy will be anyway largely negated by the assumptions I am going to make below.] Therefore, with these simplifications in place, Eqn. (<ref>) for the two parts of the boulder assumes the following form:θ^2 v τ_w' = s_st/l_ew(τ_e-τ_w)+s_spi_w-s_spτ_w^4 ,θ^2 v τ_e' = s_st/l_ew(τ_w-τ_e)+s_spi_e-s_spτ_e^4 .Here i_w and i_e denote the dimensionless solar energy fluxes, defined asi_w = 1/Φ s_sp∫_s_sp wI_ids_i ,i_e = 1/Φ s_sp∫_s_sp eI_ids_i . I decompose i_w(ϕ) and i_e(ϕ) into a Fourier series, and disregard all of the terms except for the zeroth and first order ones. This simplification can alter the final result, but still can serve as an estimate. Even with a modified illumination function, it is still a valid physical problem, whose solution must still bear the basic properties of the tangential YORP. Thus, for insolation I substitutei_w=Ccosϕ+Ssinϕ+τ_0^4,i_e=Ccosϕ-Ssinϕ+τ_0^4,where C, S, and τ_0 are constants. Here, τ_0 has a physical meaning of the dimensionless temperature of the boulder, for which the emitted power equals the time-averaged absorbed power. Usually τ_0 is close to the mean temperature of the boulder. When writing the same τ_0 in both equations, the same coefficients for cosine and opposite coefficients for sine, I took into account that in the morning the eastern part of a symmetric boulder is illuminated in exactly the same manner as the western part is in the evening, so that i_w(ϕ)=i_e(-ϕ).Next, I introduce the coefficientsa=v/s_spl ,b=s_stl/s_spl_ew ,where l is some typical boulder size. Both coefficients a and b depend solely on the boulder shape, while the dependence on size only enters through l.Finally, substituting Eqs. (<ref>) and (<ref>) into Eqn. (<ref>), I obtainθ^2 al τ_w' = b/l (τ_e-τ_w)+Ccosϕ+Ssinϕ+τ_0^4-τ_w^4 , θ^2 al τ_e' = b/l (τ_w-τ_e)+Ccosϕ-Ssinϕ+τ_0^4-τ_e^4 .Although this system is much simpler than the exact partial differential equation describing the heat conduction, it still cannot be exactly solved analytically because of the nonlinearity τ^4. Therefore, I aim to construct its approximate analytic solution, which I do in the following subsection. §.§ Approximate solution of the heat conduction equationsI am looking for the solution of Eqs. (<ref>) in the form of a series in terms of C and S,τ_w = τ_w0+τ_w1+τ_w2+... ,τ_e = τ_e0+τ_e1+τ_e2+... ,where τ_w0 and τ_e0 are independent of C and S, τ_w1 and τ_e1 are proportional to the first powers of C and S, τ_w2 and τ_e2 are proportional to their second powers, and so on. As I am looking for a relaxed periodic solution, all the terms have to be periodic in ϕ with a period of 2π. I am going to account for the contribution to TYORP of only the first three terms, and to disregard higher-order terms. This is perfectly justified if C,S≪τ_0^4, but usually C and S are only slightly less than τ_0 (compare with Table <ref>). It implies that the decomposition in the form of Eqn. (<ref>) converges slowly, if at all. Still, making C and S a factor of few smaller would make for a good convergence, and all the further analysis would be justified. Then I can assume that the extrapolation of the approximate formulas into the domain C∼ S∼τ_0^4must give a reasonable order-of-magnitude estimate for TYORP.Note, that Eqn. (<ref>) is a Taylor series in terms of C and S,in contrast to Eqn. (<ref>), which is a Fourier series in terms ϕ. I hold Fourier terms in Eqn. (<ref>) up to the first order and Taylor terms in Eqn. (<ref>) up to the second order, in order to construct the minimal model for TYORP. With merely zeroth-order terms in either Eqn. (<ref>) or Eqn. (<ref>), TYORP would vanish. Therefore, retaining first-order terms in both expansions is absolutely necessary.Retaining also the second-order Taylor term in Eqn. (<ref>) is motivated by the fact that its contribution is of the same order as the contribution of the first-order Taylor term (as we will see later from Eqs. (<ref>) and (<ref>)). Taking more terms in either the Fourier decomposition Eqn. (<ref>) or the Taylor decomposition Eqn. (<ref>) can make the solution more precise, but at the cost of increasing complexity of the problem and of the final expression. Moreover, treating even an infinite number of terms in both the decompositions will not make the solution exact, because the initial Equation (<ref>) is already an approximation. This can turn any attempt to go beyond the minimalistic model in Eqs. (<ref>) and (<ref>) into an overkill.Thus I substitute Eqn. (<ref>) into Eqn. (<ref>), and equate the terms of the same order.In the highest (zeroth) order I getθ^2 al τ_w0' = b/l (τ_e0-τ_w0)+τ_0^4-τ_w0^4 ,θ^2 al τ_e0' = b/l (τ_w0-τ_e0)+τ_0^4-τ_e0^4 .The periodic solution of this equation is τ_w0=τ_e0=τ_0.I substitute the obtained τ_w0 and τ_e0 back into Eqs. (<ref>)and write the terms of the first order, which are linear in terms of C and S:θ^2 al τ_w1' = b/l (τ_e1-τ_w1)+Ccosϕ+Ssinϕ-4τ_0^3 τ_w1 , θ^2 al τ_e1' = b/l (τ_w1-τ_e1)+Ccosϕ-Ssinϕ-4τ_0^3 τ_e1 .This is a system of linear differential equations with a sinusoidal inhomogeneity, whose periodic solution can be found in the formτ_w1 = C_w1cosϕ+S_w1sinϕ , τ_e1 = C_e1cosϕ+S_e1sinϕ .By substituting Eqn. (<ref>) into Eqn. (<ref>) and equating the coefficients in front of the sines and cosines, I getC_w1 = 4τ_0^3C/16τ_0^6+a^2l^2θ^4-al^3θ^2S/4(b+2lτ_0^3)^2+a^2l^4θ^4 , S_w1 = alθ^2C/16τ_0^6+a^2l^2θ^4+2l(b+2lτ_0^3)S/4(b+2lτ_0^3)^2+a^2l^4θ^4 , C_e1 = 4τ_0^3C/16τ_0^6+a^2l^2θ^4+al^3θ^2S/4(b+2lτ_0^3)^2+a^2l^4θ^4 , S_e1 = alθ^2C/16τ_0^6+a^2l^2θ^4-2l(b+2lτ_0^3)S/4(b+2lτ_0^3)^2+a^2l^4θ^4 . Finally, I write down the second order terms of Eqs. (<ref>), which are quadratic in terms of C and S:θ^2 al τ_w2' = b/l (τ_e2-τ_w2)-6τ_0^2 τ_w1^2-4τ_0^3 τ_w2 ,θ^2 al τ_e2' = b/l (τ_w2-τ_e2)-6τ_0^2 τ_e1^2-4τ_0^3 τ_e2 .I subtract Eqs. (<ref>) from each other, and average the result. The left-hand side averages to 0, as τ_w2 and τ_e2 are periodic, and I am left with⟨τ_w2⟩-⟨τ_e2⟩=-3lτ_0^2/b+2lτ_0^3 (⟨τ_w1^2⟩-⟨τ_e1^2⟩) .One does not need to find the exact expressions for τ_w2 and τ_e2: as will be seen in the next subsection, Eqn. (<ref>) suffices to compute TYORP in the second order in terms of C and S. §.§ Computation of TYORPA heated surface emits light and experiences the recoil pressureP=2/3cϵσ T^4=(1-A)Φ/c2/3τ^4 .Here c is the speed of light, and the coefficient 2/3 corresponds to the light emission in accordance with Lambert's law.To get the force experienced by the boulder in the eastward direction, I integrate this pressure over the boulder's surface, and take the F_1 component of the total force, assuming that the x_1 axis is directed from west to east:F_1=-∫_S_spP dS_1 . I average this force over time and nondimensionalize it by dividing it by (1-A)Φ S_proj/c, where S_proj is the horizontal projected area of the boulder. Thus I get the dimensionless TYORP pressure,p=-2/31/s_proj∫_s_sp⟨τ^4⟩ ds_1 .Now let us recall the assumption that the boulder is separated into two parts,with the dimensionless temperatures τ_w and τ_e, and the two parts are symmetric, with the same surface area of the western and eastern parts,s_sp w=s_sp e. Assuming ds_1 to always be negative at s_sp w and always positive at s_sp e, I getp = -2/31/s_proj( ⟨τ_w^4⟩∫_s_sp wds_1 +⟨τ_e^4 ⟩∫_s_sp eds_1)= 2/31/s_proj( ⟨τ_w^4⟩∫_s_sp wd|s_1| -⟨τ_e^4 ⟩∫_s_sp ed|s_1|)= 2/3⟨τ_w^4-τ_e^4⟩1/s_proj∫_s_sp wd|s_1|= 2/3s_ew/s_proj⟨τ_w^4-τ_e^4⟩ .Here s_ew=∫_s_sp wd|s_1| denotes the area of the boulder projected onto the vertical meridianal plane. It is convenient to introduce a new coefficient,n=2s_ew/s_proj .This coefficient n characterizes steepness of the surface. If s_proj is understood as the horizontal projected area of the boulder, then n=⟨tanα_ew⟩ is the mean tangent of the slope of the surface in the east-west direction. It is zero for a flat surface, unity for a surface with 45^∘ slopes, and bigger for even steeper slopes.Now, substituting Eqn. (<ref>) into Eqn. (<ref>)and using the decomposition of τ from Eqn. (<ref>), results intop = 1/3n⟨τ_w^4-τ_e^4 ⟩= 1/3n⟨τ_0^4+4τ_0^3τ_w1+4τ_0^3τ_w2+6τ_0^2τ_w1^2+...-τ_0^4-4τ_0^3τ_e1-4τ_0^3τ_e2-6τ_0^2τ_e1^2-... ⟩ .As τ_w1 and τ_e1 are sinusoidal, their means are 0. The means of τ_w2 and τ_e2 are obtained from Eqn. (<ref>).Thus Eqn. (<ref>) results inp=2nbτ_0^2/b+2lτ_0^3 (⟨τ_w1^2⟩-⟨τ_e1^2⟩) .Using Eqs. (<ref>) for τ_w1 and τ_e1 leads to⟨τ_w1^2⟩-⟨τ_e1^2⟩=1/2(C_w1^2+S_w1^2-C_e1^2-S_e1^2) .Finally, substituting this expression into Eqn. (<ref>) and using Eqn. (<ref>), I obtainp=8nab^2τ_0^2l^2θ^2CS/(b+2lτ_0^3)(16τ_0^6+a^2l^2θ^4)(4(b+2lτ_0^3)^2+a^2l^4θ^4) . This is the analytic expression of TYORP that I was looking for. It allows one to approximately estimate TYORP using the shape and thermal properties of the boulder. In the following section, I apply it to several different boulder shapes.§ APPLICATION OF ANALYTIC TYORP EXPRESSION TO BOULDERS OF DIFFERENT SHAPES §.§ One-dimensional wallLet us start testing this analytic expression with the simplest and historically first model of TYORP: one-dimensional heat conductivity in a wall by <cit.>. The high long wall stands on regolith, going from the north to the south, and is illuminated by the sun alternatively from the east or the west (panel (a) in Figure <ref>). Given that the dimensionless thickness of the wall is d and considering its patch of dimensionless surface area s, I separate the wall into the eastern and the western slabs with the thicknesses d/2. Then the volume of each slab is v=sd/2 and the typical distance between the slabs is l_ew=d/2. Thus I go on, filling in the geometric properties in the first row of Table <ref>, until I get a, b and n. (Note though, that for s_proj I take not a horizontal projected surface area, but a vertical one, which is the only meaningful definition for a very high vertical wall. It also renders n different from its conventional meaning of ⟨tanα_ew⟩.)To compute the last three columns in Table <ref>, I need the insolation function, for which I takei_w(ϕ)=2sinϕ,0<ϕ<π/2.The coefficient 2 is due to the assumed mirror reflection of light by the regolith. Decomposing i_w(ϕ) into the Fourier series, I finish filling in the first row of Table <ref>.Now I substitute the obtained coefficients a, b, n, τ_0, C and S into Eqn. <ref>, and in Figures <ref> and <ref> compare the resulting analytic expression for TYORP pressure p with the numeric simulations by <cit.> and <cit.>. The agreement is almost too good, given the approximations made while constructing the theory. For the values of l and θ that correspond to big p, the value of p is predicted with an accuracy of a few tens per cent (Figure <ref>), although far from the maximal p the accuracy is worse. The area where p is big, is very similar in the analytic theory and in the simulations (Figure <ref>). The qualitative behavior of p as a function of l and θ is also reproduced correctly. This is as good as one could expect from a simple estimate done by such an approximate theory.A few warnings are still to be made. In Figure <ref>, I disregard negative simulated values of p, observed by <cit.> for big l and θ. <cit.> suggested that these negative values could be numeric artifacts, and this was later confirmed by <cit.>, who only obtained positive p in all their simulations. Furthermore, <cit.> proved that mirror reflection of light by the regolith is a bad approximation (worse, in fact, than disregarding reflected light completely). Still, this approximation enters both the theory and the simulations,so they should be wrong in the same way, and thus agree with each other, which they do. §.§ Spherical boulderThe second model I want to study is a particular case from <cit.>: a spherical boulder of radius r lying half buried in the regolith on the equator of an asteroid (panel (b) in Figure <ref>). I separate the boulder into the eastern and the western hemispheres, and list all the necessary geometric properties in Table <ref>. For the typical distance between the hemispheres l_ew I take r, which is half the distance between their most remote points. Measuring the distance between their centers of mass instead would result into 3/4r, and an argument could be also made for using this value instead, although the difference should not matter much given the crudeness of all the previously made assumptions. For s_proj I take the horizontal projection of the boulder, π r^2. This is different from <cit.>, who considered a regular array of boulders and used for s_proj the surface area per boulder. Therefore, I re-normalize their results. Note that the choice of s_proj does not influence the observable physical values: s_proj enters the definition of p inversely, but also enters the transformation factor from p to F directly, and thus cancels out.For the insolation I try two possibilities. First, assuming mirror reflection of light by the regolith, I get (see Figure <ref> for explanation)i_w(ϕ)=1/2(1+sinϕ),-π/2<ϕ<π/2.Second, assuming full absorption of light by the regolith, and thus accounting for only the direct solar irradiation, I geti_w(ϕ)=1/2(1+sinϕ),if -π/2<ϕ<0,i_w(ϕ)=1/2(cosϕ+sinϕ),if 0<ϕ<π/2. I treat the two cases separately in the second and third rows of Table <ref>.In Figure <ref>, I compare the numeric simulations of <cit.> (solid lines) with the two analytic models. Qualitatively both of them work well, but the model including only the direct light has a better quantitative agreement with the simulations than the model with mirror reflection (similarly to findings of <cit.> for a wall). In Figure <ref>, I compare the better analytical model with the numeric simulation. One must also keep in mind that the simulations of <cit.> were conducted for an array of boulders partially shadowing each other. Subtracting the shadowing should increase TYORP by a few tens per cent, as seen in the middle right panel of Figure 4 in <cit.>. This could somewhat improve the agreement between the presented analytical model and the simulations. §.§ Wave in the regolith Now, having tested the model for already studied cases, let us move to an as-yet unknown terrain. Consider a regular array of hills and dales, with flat slopes angled at α, with a dimensionless wavelength λ, all positioned on the equator in the north-south direction (panel (c) in Figure <ref>). The model is intended as a proxy for small bumps and pits on the regolith, whose contribution to TYORP has never yet been considered. What I am going to do here is but a crude estimate, yet to be tested by thorough numeric simulations. Unlike for a boulder with a well-specified volume, now the volume in which heat conduction occurs has no well-defined borders, and estimating geometric properties for Table <ref> gets more complicated. Obviously, considering an infinitely deep volume below the surface is unsatisfactory, as an infinitely large volume will have infinite heat capacity, zero temperature oscillations, and thus zero TYORP.Differently directed slopes cause uneven heating of the surface, and thus temperature variations in the east-west direction. The principal Fourier harmonics of this horizontal temperature oscillation has the wavelength λ, and the corresponding wave vector k=2π/λ. The temperature oscillation will dampen with depth, and one can estimate the penetration depth of these temperature oscillations to be of the order of 1/k=λ/2π. Thus I consider heat conduction in a slab of depth λ/2π and surface area s. Although for λ or θ very different from unity the typical depth can be different and determined by other physical processes, and thus the model can produce a bigger error than in the previous two cases, even now it must at least give a reasonable order-of-magnitude estimate of TYORP for λ∼θ∼ 1. The slab under consideration borders two other slabs, to the east and to the west. These two slabs have the same temperature, and one can account for having two of them by just doubling their area of contact, s_st.Having said all this, I fill in the last row in Table <ref>. I assume that α is small, thus substituting cosα≈ 1, tanα≈sinα≈α. For the illumination function I takei_w(ϕ)=cos(ϕ-α),-π/2+α<ϕ<π/2-α.When Fourier-decomposing i_w(ϕ), I also keep only the principal terms in terms of α.Substituting the last row from Table <ref> into Eqn. <ref>, I get Figure <ref>. The figure was produced assuming slope angles α=0.1. For other slope angles the TYORP pressure will scale as α^2.The maximal TYORP for regolith in Figure <ref> is about 30 times smaller than the maximal TYORP for spherical boulders in Figure <ref>. Therefore, at first sight, it may seem that regolith is unimportant. Still, one must bear in mind several reservations. Firstly, non-cracked boulders have big thermal parameters, θ=10→ 100, while for regolith θ∼ 1 <cit.>. This will substantially suppress TYORP for boulders, while leaving TYORP for regolith close to its maximum. Secondly, a much bigger fraction of surface is covered by regolith than is by boulders. Lastly, the typical slopes α for asteroids are utterly unknown, and altering α can drastically decrease or increase TYORP. The length scales I am referring to are millimeters to centimeters, which is far below the resolution of even in situ observations of asteroids.Summing up, one should acknowledge that regolith may have a substantial contribution to TYORP, which deserves a more detailed analysis than this rough estimate. Still, such an analysis would greatly divert us from the main focus of this paper.§ INTEGRATION OF TYORP OVER ALL BOULDER SIZES §.§ Derivation of the integral for the overall TYORP All the previous analysis was done for boulders of only one particular size. Let us now integrate TYORP over all sizes of boulders on the asteroid, thus generalizing the numeric results obtained by <cit.> for 25143 Itokawa. For the size distribution of boulders, I take a power law: it is simple, reasonably precise, and the most widely used distribution.The interested reader is referred to <cit.> for an excellent review of the available measurements of the boulder size distribution for different asteroids.Consider a power-law distribution of boulder sizes,dN/dl=N_0 l^-γ ,where dN is the number of boulders with sizes between l and l+dl lying on the entire surface area of the asteroid s_ast, while N_0 and γ are constant. For us it is more convenient to re-write this distribution not in terms of the number of boulders, but in terms of the area covered by them:dn/dl=n_0 l^2-γ .Here dn=dS/S_ast is the fraction of the surface area covered by boulders with sizes between l and l+dl, while n_0=N_0 s_proj/s_astl^2 is a new constant.(Remember that for geometrically similar boulders of different sizes s_proj∝ l^2.)Of course, a power law in the form of Eqs. (<ref>) or (<ref>) can only be an approximation. This distribution would necessarily diverge for either small or big boulders, and would necessarily eventually start disagreeing with the data for both small and big boulders. Still, I assume that deviations from the power law happen only for such small and such big boulders that their contribution to TYORP is negligible anyway.Let us determine the total dimensionless TYORP drag p_tot experienced by the surface as the dimensional TYORP drag force divided by (1-A)Φ S/c. Then p_tot can be obtained from p(l) by integrating it over fractions of the surface area dn covered by boulders of each size:p_tot=∫ p(l) dn=∫_0^∞ n_0 p(l)l^2-γ dlGiven asymptotics of Eqn. (<ref>) p∝ l^2 for l→ 0 and p∝ l^-5 for l→∞, this integral converges for γ∈(-2;5). Typical power indices for small boulders indeed lie in this range <cit.>. For γ outside this interval, finiteness of the integral has to be provided by setting the minimal or the maximal size of boulders, starting from which the power-law size distribution breaks.From Eqn. (<ref>) one can see the problem: the rational expression for p provided by Eqn. (<ref>) does not look frightening only up until the moment one tries to integrate it over l. Having spent many pleasant hours doing contour integration in the complex plane, I must acknowledge that the final result is too complicated to be of any practical use despite it being a closed form algebraic expression. Moreover, when written in terms of real variables, it is far too lengthy to be accommodated into this subsection.To get a more practical result from Eqn. (<ref>), p should be transformed into a simpler form. Still, one can not neglect either term in Eqn. (<ref>): a, b, τ_0, l and θ are all of the order of unity, and thus all comparable with each other. Therefore, I choose not to simplify the equation, but to brute-forcedly interpolate it. This will cause some loss of accuracy, but as we will see, this loss is not much bigger than the errors already caused by the previous simplifications. §.§ Approximate expression for TYORP In this subsection I do a detour to find the best fit for p as a function of l and θ as given by Eqn. (<ref>). In Figures <ref> and <ref> one can see the three-dimensional surface p(l,θ) from above, and notice that in the l-θ plane lines of constant p are roughly elliptical and roughly similar to each other. In Figures <ref> and <ref> one can see that vertical cross-sections of this three-dimensional surface look similar to Gaussian functions of the same width. These observations allow us to guess that a decent approximation to p(l,θ) will be given by a two-dimensional log-normal function,p=p_0e^A_llnl+A_θlnθ+1/2A_llln^2l+A_lθlnllnθ+1/2A_θθln^2θ. There are several ways to choose the coefficients in this fitting function, and in Figure <ref> I explore different ways of fitting. The least squares fit to the numeric solution is presented with the orange lines, the least squares fit to the approximate solution Eqn. <ref> is plotted in blue. Both fits are constructed for the same range of r and θ as presented in Figure <ref>. The purple lines are constructed by requiring Eqs. (<ref>) and (<ref>) to have the same values in the point l=θ=1, as well as the same first and second partial derivatives. For brevity, I call this approach the Taylor fit, as I have indeed requested equality of the zeroth, first and second order terms in the Taylor series for p(l,θ) and its fit. The analytically computed coefficients of the Taylor fit are presented in Table <ref>. Although cumbersome, they can still be instrumental to estimations of TYORP for complex-shaped bodies.The exact numeric solution is overplotted with a red line, and the approximate solution given by Eqn. <ref> with a green line. One can see that the least squares fit to the numeric result agrees with them very well, as do the two other fits agree with the approximate solution Eqn. <ref>. The Taylor fit to Eqn. <ref> is by construction perfect in the vicinity of the point l=θ=1, but far away from this point it also works quite well.Coefficients of the three considered fittings are given in Table <ref>.As a minor side result, in the same table I present α_0=tan(1/2arctan2A_lθ/A_ll-A_θθ), which determines orientation of the ellipsis with constant p for the log-normal distribution. In the l-θ plane decrease of p is slowest along the line θ∝ l^α_0. We see, that α_0 obtained from different models are consistent with each other, and roughly consistent with the elongation of the red and yellow areas in Figure <ref>.Now, having obtained the coefficients of the log-normal fit, I am ready to finish computing the integral for the overall TYORP. §.§ Computing the integral for the overall TYORPReturning to the main focus of this section, I still need to integrate the simplified expression for TYORP over all boulder sizes. Substituting Eqn. (<ref>) into Eqn. (<ref>) and computing the Gaussian integral, I getp_tot=n_0μe^-(lnθ-lnθ_0)^2/ν^2,where three new constants have been introduced:μ = p_0√(2π/-A_ll)e^2A_θ A_lθ(A_l-γ+3)-A_θθ(A_l-γ+3)^2-A_θ^2A_ll/2(A_θθA_ll-A_lθ^2),ν = √(-2A_ll/A_θθA_ll-A_lθ^2),lnθ_0 = A_lθ(A_l-γ+3)-A_θ A_ll/A_θθA_ll-A_lθ^2. The mathematical expressions are again somewhat lengthy, but they are worth the agreement between the different estimates of p_tot we see in Figure <ref>. Even the substantial discrepancies between the numeric and the analytic solutions observed in Figure <ref> are mostly eliminated by integration in Figure <ref>. The resulting discrepancies between the exact results and the different approximations lie within about 20% of the maximal value on the plot, thus certifying that both of our approximations are good for estimating the TYORP effect. The constants Eqn. (<ref>) are also listed in Table <ref>, and one can see that they are close to each other.I now apply these results to asteroid 25 143 Itokawa, both to estimate the TYORP and to appease myself by certifying the uncertainties in physical properties of even such a well-studied asteroid cause a much bigger error in the TYORP acceleration than the simplifications underlying my analytic model. The assumed physical properties of boulders taken from <cit.> are listed in Table <ref>. Following <cit.>, I use two different values of the heat conductivity of rock: bigger κ_1 presumably corresponds to solid rock, while smaller κ_2 could correspond to cracked rock, which is assumed to be widely present on atmosphereless bodies <cit.>. Respectively, I get two different thermal parameters θ_1=13 and θ_2=4.1. I use the size distribution of boulders from <cit.>, dN=(14± 9)· 10^3 L^-3.0± 0.2dL, where dL is in meters. For simplicity, I choose the power index γ=3. In this case n_0 is independent of L_cond.Nondimensionalizing L, transforming from the number of boulders dN into their relative surface area dn, assuming roughly circular projections of boulders, and taking for the surface area of Itokawa 3.93· 10^5 m^2 <cit.>, I get n_0=0.028± 0.018. From Figure <ref>, I find corresponding pressures P(θ_1)≈ 0.001 and P(θ_2)≈ 0.005. Estimating the dimensionless TYORP of the whole asteroid as τ_z≈ 9sT(θ) <cit.>, I get for solid rocks τ_z1=0.00025± 0.00015 and for cracked rocks τ_z2=0.0013± 0.0007. This value should be compared to the difference Δτ_z= 0.003± 0.002 <cit.> between the observed accelerating torque acting on Itokawa and the predicted normal YORP torque, as one may expect Δτ_z to be equal to Itokawa's TYORP. One can see that the cracked rock gives a value consistent with Δτ_z, while the solid rock gives a value an order of magnitude smaller than Δτ_z. As a side remark, I note that the tens-of-per-cent uncertainty of the theoretical model is much smaller than an order of magnitude uncertainty caused by the uncertainty of the physical properties of the asteroid. The most important takeaway message from this estimate for 25 143 Itokawa is that for realistic properties of boulders and their realistic shape distribution, Eqn. (<ref>) predicts a significant effect, comparable to the normal YORP.§ DISCUSSION Reviewing the derivation of the analytic expression of TYORP (Eqn. (<ref>)), one can single out three major ingredients sufficient for TYORP to appear:1. Heating pattern with a day–night asymmetry (C≠ 0) and an east–west asymmetry (S≠ 0).2. Thermal inertia (0<θ<∞, 0<a<∞) and thermal conduction (0<b<∞).3. Non-linearity of the heat emission law.The non-linearity enters the equations for TYORP in two related ways: through the boundary condition of the heat conduction equation (Eqn. (<ref>)) and through the definition of TYORP pressure (Eqn. (<ref>)). The former's contribution is negative, the latter's contribution is positive, and the latter wins, as can be seen from the substitution of Eqn. (<ref>) into Eqn. (<ref>).Acknowledging the generality of conditions 1–3, one must conclude that TYORP is a very general feature, which is not bound to any particular boulder shape or illumination pattern. More or less any boulder shapes can be substituted into Eqn. (<ref>), while Eqn. (<ref>) presents a most generic illumination function, which is still capable of causing TYORP of a realistic magnitude, although boulders with steeper slopes (bigger n) and illuminated in a more asymmetric way (bigger S) are subject to systematically bigger TYORP, as can be seen from Eqn. (<ref>).In my model, C is always positive, as the boulder is illuminated in the day, not at night. S is usually positive, as all realistic boulders are more illuminated from the east before noon, and more illuminated from the west after noon. Thus all terms in Eqn. (<ref>) are positive, and the resulting TYORP derived from the analytic model should accelerate the rotation of the asteroid under any realistic conditions, not decelerate it. This fact has already been observed for different geometries of boulders by <cit.> and <cit.> in their numerical simulations.This holds not only for convex structures like boulders or mounds, but also for concave structures, like grooves or pits on the surface of the asteroid. In the latter case the mechanism is the same as usual: east-facing slopes are better illuminated in the morning, west-facing slopes in the afternoon, and the heat conduction occurs between them.The problem is again approximately described by Eqs. (<ref>) and (<ref>) with all constants being positive, and the solution again leads to Eqn. (<ref>). It means that the west-facing slopes again emit on average more light, push the surface more in the eastern direction, and again accelerate the rotation of the asteroid.Thus most imaginable surface structures should be to a certain extent subject to positive TYORP: crater rims, crater pits, grooves and cracks on the surface, small mounds and summits, or just wavy patterns of regolith, rough surfaces of stones, lone boulders and boulder fields... A heap of pebbles can contribute to TYORP either as a single body, or via individual pebbles, or via individual pocks and wedges of individual pebbles, and which of these descriptions is the most appropriate depends on the rotation rate of the asteroid, the heat conductivity of rock, as well as other parameters. Singling out the most important contributors to TYORP and accounting for them all is an important task for future research.§ CONCLUSIONS I have constructed an approximate fully analytic model of TYORP, in which I used an averaging procedure to transform a partial differential equation into an ordinary differential equation, then used perturbation theory to transform the latter into algebraic equations, and finally arrived at an algebraic expression for TYORP. Along the way the shape of the boulder was boiled down to just a few geometric coefficients, and the illumination pattern – to its few Fourier harmonics. Despite the simplicity, the analytic expression for TYORP was found to be in good agreement with numeric simulations for a stone wall <cit.> and a spherical boulder <cit.>.I used the analytic model to estimate the TYORP produced by the non-smoothness of regolith on an asteroid. On the one hand, regolith should be much flatter than boulders, which diminishes its TYORP. On the other hand, it has a more favorable thermal parameter θ and presumably covers a bigger fraction of the surface, which increases its TYORP contribution. Given the utter uncertainty of the unevenness of regolith on different scales, it is hard to say whether or not its contribution is significant, but neglecting it now seems unsafe.I then have integrated TYORP over boulders of all different sizes, assuming a power-law size distribution for boulders, and doing a two-dimensional log-normal fit to the analytic TYORP drag. The additional error introduced by this fit was generally not bigger than the error already caused by the assumptions underlying the analytical model. Moreover, the error substantially diminished after integration over the boulder sizes. The resulting TYORP drag of the entire assembly of boulders appeared log-normal in terms of the thermal parameter θ, and concurrently in terms of rotation rate of the asteroid. It means that even for a broad distribution of boulders over sizes, TYORP still has a relatively narrow peak, a corresponding rotation rate with the biggest TYORP, and a relatively fast decrease of TYORP for faster or slower rotation. The integrated TYORP drags for the log-normal approximation, for the full analytic solution and for the numeric simulation were found to be in good agreement with each other.The proposed analytic model demonstrates that TYORP should appear for most realistic shapes of boulders or regolith, and should be positive in most realistic cases.§ ACKNOWLEDGEMENTSI am very grateful to Daniel J. Scheeres for discussing with me the astronomical content of the problem, to Uliana Pyrohova for counseling me with mathematical issues, to Vlad Unukovych for reviewing the physics of the paper, and to Alexander Kostenko for improving the language style. Without the help of either of them the paper would have been stuck at a certain point.99 [Delbo et al.2014]delbo14 Delbo M., Libourel G., Wilkerson J., et al., 2014, Nature 508, 233[Demura et al.2006]demura06 Demura H., Kobayashi S., Nemoto E., et al., 2006, Science 312, 1347 [Golubov & Krugly2011]golubov11 Golubov O., Krugly Yu. N., 2011, Planetary Defence Conference, From Threat to Action, held 09-12 May 2011, in Bucharest, Romania[Golubov & Krugly2012]golubov12 Golubov O., Krugly Yu. N., Tangential component of the YORP effect, 2012, ApJL 752, L11 [Golubov et al.2014]golubov14 Golubov O., Scheeres D. J., Krugly Yu. N., A three-dimensional model of tangential YORP, 2014, ApJ 794, 22 [Murdoch et al.2015]asteroids4 Murdoch N., Sánchez P., Schwartz S. R., Miyamoto H., 2015, Asteroids IV, 767 [Ševeček et al.2015]sevecek15 Ševeček P., Brož M., Čapek D., & Ďurech, J., The thermal emission from boulders on (25143) Itokawa and general implications for the YORP effect, 2015, MNRAS 450, 2104 [Ševeček et al.2016]sevecek16 Ševeček, Golubov O., Scheeres D.J., Krugly Yu. N., Obliquity dependence of the tangential YORP, 2016, A&A 592, 115 | http://arxiv.org/abs/1708.07988v1 | {
"authors": [
"Oleksiy Golubov"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170826155631",
"title": "Analytic model for tangential YORP"
} |
The shape derivative of the Gauss curvature Partially supported by CONICET through grants PIP 112-2011-0100742 and 112-2015-0100661, by Universidad Nacional del Litoral through grants CAI+D 501 201101 00476 LI, by Agencia Nacional de Promoción Científica y Tecnológica, through grants PICT-2012-2590 and PICT-2014-2522 (Argentina) Aníbal Chicco-Ruiz, Pedro Morin, and M. Sebastian Pauletti UNL, Consejo Nacional de Investigaciones Científicas y Técnicas, FIQ, Santiago del Estero 2829, S3000AOM, Santa Fe, Argentina. December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================ We introduce new results about the shape derivatives ofscalar- and vector-valued functions.They extend the results from <cit.>to more general surface energies. In <cit.>Doğan and Nochetto considersurface energies defined as integrals over surfaces of functionsthat can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature(a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of anyfinite dimension. As an application of the results, with relevance for numerical methods in applied problems, we introduce a new scheme of Newton-type to approximate a minimizer of a shape functional. It is a mathematically sound generalization of the method presented in <cit.>.We finally find the particular formulas for the first and second order shape derivative of the area and the Willmore functional, which are necessary for the Newton-type method mentioned above.2010 Mathematics Subject Classification. 65K10, 49M15, 53A10, 53A55. Keywords. Shape derivative, Gauss curvature, shape optimization,differentiation formulas.§ INTRODUCTION Energies that depend on the domain appear in applications in many areas, frommaterials science, to biology, to image processing.Examples when the domain dependence of the energy occurs through surfaces include the minimal surface problem, the study of the shape of droplets (surface tension), image segmentation and shape of biomembranes, to name a few.In the language of the shape derivative theory <cit.>, these energies are called shape functionals. This theory provides a solid mathematical framework to pose and solve minimization problems for such functionals. For most of the problems of interest, the energy (shape functional) can be cast as ∫_Γ F(“geometrical quantities”), where “geometrical quantities” stands for quantities such as the normal ,the mean curvature κ, the Gauss curvature κ_g, or in general any quantity that is well definedfor a surface Γ as a geometric object, i.e.,independent of theparametrization.For example, F=1 in the case of minimal surface, F=F(x,) is used in the modeling of crystals <cit.> in materials science. The Willmore functional corresponds to F=1/2κ^2<cit.>—where κ is the mean curvature—and the related spontaneous curvature functional to F=1/2(κ-κ_0)^2; they are used in models for the bending energy of membranes, particularly in the study of biological vesicles <cit.>. The modified form of the Willmore functional, which corresponds to F=g(x)κ^2, is applied to model biomembranes when the concentration or composition of lipids changes spatially <cit.>.The minimization of these energies requires the knowledge of their (shape) derivatives with respect to the domain and has motivated researchers to seek formulas for the shape derivative of the normal and the mean curvature. The shape derivative of the normal is simple and can be found in <cit.> among other references. Particular cases of F=F(x,n) are derived in <cit.>. The shape derivative of the mean curvature or particular cases of F=F(κ) can also befound in <cit.>, where the shape derivative is computed from scratch; some using parametrizations,others in a more coordinate-free setting using the signed distance function, but in general the same computations are repeated each time a new functionaldependent on the mean curvature appears.A more systematic approach to the computations is found in <cit.>, where Doğan and Nochetto propose a formula for the shape derivative of a functional of the form F=F(x,,κ), that relies on knowing the shape derivatives ofand κ. They rightfully assert that by having this formula at hand, it wouldn't be necessary toredo all the computations every time a new functional depending on these quantities appears. The main motivation of this article is to find such a formula when F alsodepends on the Gauss curvature κ_gwhich, as far as we know, has not been provided elsewhere. In fact, we let F also depend on differential operators of basic geometric quantities such as ∇_Γκ or Δ_Γκ. These are important when second order shape derivatives are necessary in Newton-type methods for minimizing functionals.Our new results (Section <ref>) allow us to develop a more systematic approachto compute shape derivatives of integrands that are functional relations of geometric quantities. The method, starting from the shape derivative of the normal, provides a formula for the shape derivative of higher order tangential derivatives ofgeometrical quantities. In particular we give a nice formula for the shape derivative of the gaussiancurvature and extend the results of <cit.> to more generals integrands.These results are also instrumental to develop a relevant numerical method in applied problems. More precisely, we introduce a new scheme of the Newton-type to find a minimizer of a surface shape functional. It is a mathematically sound generalization of the method used in <cit.>.In Section <ref> we state some preliminary concepts and elements ofbasic tangential calculus. In Section <ref> we recall the concept of shape differentiablefunctionals through the velocity method. In Section <ref> we motivate and introduce the concept of shapederivative of functions involved in the definition of shape functionals throughintegrals over the domain. In Section <ref> we motivate and introduce the concept ofshape derivative of functions involved in the definition of shape functionalsthrough integrals over the boundaries of domains. In Section <ref> we explore the relationship between theshape derivative of domain functions and the classical derivative operators. In Section <ref> we explore the relationship between theshape derivative of boundary functions and the tangential derivative operators. These last sections set the foundations for Section <ref> where theshape derivatives of the tangential derivatives of geometric quantities areobtained. We end with Sections <ref> and <ref> where weapply the results to obtain the shape derivatives of the Gauss curvature, thegeometric invariants and introduce a quasi Newton method in the language ofshape derivatives whose formula is then computed for the Willmorefunctional. § PRELIMINARIES§.§ General concepts Our notation follows closely that of <cit.>. Adomain is an open and bounded subset Ω of [N], and aboundary is the boundary of some domain, i.e., Γ=∂Ω. An N-1 dimensional surface in [N] can be thought of as areasonable subset of a boundary in [N]. If a boundary Γ is smooth, we denote the normal vector fieldby n and assume that it points outward of Ω.The principal curvatures, denoted by κ_1, …, κ_N-1,are the eigenvalues of the second fundamental form of Γ, which are allreal. The mean curvature κ and Gaussian curvature κ_g are κ = ∑_i=1^N-1κ_iand κ_g =∏_i=1^N-1κ_i.We will obtain analogous and more useful definitions ofκ and κ_g using tangential derivatives of the normal; see(<ref>) and (<ref>) for N=3, and Section<ref> for any dimension, where we introduce the geometricinvariants of a surface, following Definition 3.46 of <cit.>.In the scope of this work a tensor S is a bounded, linear operatorfrom a normed vector space 𝕍 to itself.The set of tensors is denoted by (𝕍). If (𝕍)=N, S can be represented by an N× N matrixS_ij. We will mainly consider 𝕍=[N]. For a vector space with a scalar product, the tensor productof twovectors u and v is the tensor uv whichsatisfies(uv)w =(v·w)u. Thetrace of a tensor S is (S) = ∑_i Se_i ·e_i,with {e_i} any orthonormal basis of 𝕍. The trace of atensor uv is (uv)=u·v.The scalar product of tensors S and T is given by S:T=(S^T T),where S^T is the transpose of S, which satisfies Su·v=u· S^Tv, and the tensor norm is |S|=√(S:S).From <cit.> we have the following properties:[Tensor Properties] For vectors u, v, a,b∈𝕍, and tensorsS, T, P ∈(𝕍), we have:* S(uv)=Suv and(uv)S=u S^Tv,* I:S=tr(S),* ST:P=S:PT^T = T:S^TP,* S: uv=u· Sv,* (ab):(uv)=(a·u)(b·v),* S:T=S:T^T = 1/2 S:(T+T^T) if S is symmetric.§.§ The signed distance function For a given domain Ω⊂[N], the signed distance functionb=b(Ω):[N]→ is given by b(Ω)(x)=d_Ω(x)-d_[N]∖Ω(x), whered_Ω(x)=inf_y∈Ω|y-x|. If Ω is of class 𝒞^1,1 (see Definitions 3.1 and 3.2 in<cit.>) then for each x∈Γ=∂Ω there exists aneighborhood W(x) such thatb∈𝒞^1,1(W(x)),|∇ b |^2 = 1 in W(x) and ∇ b = n∘ p, where n isthe unit normal vector field of Γ and p=p_Γ is the projection ontoΓ which is well defined for y ∈ W(x) as p(y) =min_z∈Γ|z-y|(see Theorem 8.5 of <cit.>).Moreover, if Ω is a 𝒞^2 domain with compact boundary Γthen there exists a tubular neighborhood S_h(Γ) such that b∈𝒞^2(S_h(Γ)) (<cit.>), andΓ is a 𝒞^2-manifold of dimension N-1. Therefore, ∇ b is a 𝒞^1 extension for n(Γ) whichsatisfies |∇ b|^2 ≡ 1inS_h(Γ).This Eikonal equation readily impliesD^2b ∇ b ≡ 0.Also, if Ω is 𝒞^3, we can differentiate (<ref>)to obtain (D^2b)·∇ b =-|D^2b|^2 where we have used the product rule formula(S^Tv)=S:∇v+v· S where S and v are tensor and vectorvalued differentiable functions, respectively,with S=D^2b and v=∇b (see <cit.>, page 30). The divergence S of a tensor valuedfunction is a vector which satisfies S ·e = (S^Te)for any vector e.Applying the well known identity <cit.>(Dv^T)=∇(v),to v=∇ b we can write (<ref>) as follows: ∇Δ b ·∇ b =-|D^2b|^2.Since n(Γ)= ∇ b |_Γ, we can obtain from b more geometricinformation about Γ. Indeed, the N eigenvalues of D^2 b|_Γ are the principal curvatures κ_1, κ_2,…,κ_N-1 of Γ and zero <cit.>. The mean curvature of Γ, given by (<ref>), can also be obtained asκ = D^2b = Δ b (on Γ). Also, |D^2b|^2= (D^2b)^2= ∑κ_i^2, the sum of the square of theprincipal curvatures so that the Gaussian curvature is κ_g=1/2[(Δ b)^2 - |D^2b|^2]; notice that the right-hand side of this last identity makes sense in S_h(Γ) whereas the left-hand side is defined only on Γ, so that the equality holds on Γ. Moreover, from (<ref>)we obtain that ∂Δ b /∂ n = - ∑κ_i^2,and with a slight abuse of notation we may say that∂κ/∂ n = - ∑κ_i^2.The projection of a point x ∈ S_h(Γ) onto Γ is given by <cit.>p(x)=x-b(x)∇ b(x),and also, for any x∈ S_h(Γ),the orthogonal projection operator of a vector of [N] onto thetangent plane T_p(x)(Γ), for Γ∈𝒞^1, is given byP(x)=I-∇ b(x) ⊗∇ b(x). Note that the tensor P(x) issymmetric and P=I-nnon Γ. It will be useful to know that the Jacobian of the projection vector fieldp(x) is given, for Γ∈𝒞^2, byD p(x)=P(x) - b(x) D^2b(x)and satisfies Dp|_Γ = P because b=0 on Γ.§.§ Elements of tangential calculus Following <cit.> we will introduce some basic elements ofdifferential calculus on a 𝒞^1-submanifold of codimension 1 denotedby Γ. This approach avoids local bases and coordinates by using intrinsictangential derivatives.All proofs can be found in the cited book, except for Lemmas<ref> and <ref>, which are proved below.[Tangential Derivatives] Assume that Γ⊂∂Ωand there exists a tubular neighborhood S_h(Γ) such thatb=b(Ω) ∈𝒞^1(S_h(Γ)). For a scalar field f∈𝒞^1(Γ) and a vector field w∈𝒞^1(Γ,[N]) we define the tangential derivative operators asf(I-) ∇ F; w D W - D Wn⊗n;wW - D Wn·n,where F and W are 𝒞^1-extensions to a neighborhood ofΓ of the functions f and w, respectively. For a scalar function f∈𝒞^2(Γ),the second order tangential derivative is given by ^2 f =( f),which is not a symmetric tensor, and the Laplace-Beltrami operator (or tangential laplacian) is given byf =f. Using the orthogonal projection operator P given by(<ref>), we can write f =(P ∇ F)|_Γ,w=(DW P)|_Γ, w =(P:DW)|_Γ.As it was proved in the cited book <cit.> these definitions areintrinsic, that is, they do not depend on the chosen extensions of f andw outside Γ. Among all extensions of f, there is one, for Γ∈𝒞^2, thatsimplifies the calculation of f. That extension is f∘ p, wherep is the projection given by (<ref>), and we call it thecanonical extension.The following properties of the canonical extensions are provedin <cit.>[Canonical extension] For Γ, f and w satisfying the assumptions ofDefinition <ref>, considerF=f∘ p, andW=w∘ p, the canonical extensions of f and w,respectively, where p is the projection given by (<ref>).Then∇(f∘ p) =[I-b D^2b] f∘ p,D(w∘ p) =w∘ p [I-b D^2b], (w∘ p) =[I-b D^2b]:w∘ p = w∘ p -b D^2b: w∘ p.In particular,f =∇(f∘ p)|_Γ, w =D(w∘ p)|_Γ,w =(w∘ p)|_Γ.The tangential divergence of a tensor valued function S is defined as S ·e =(S^Te), for any vector e. The expressions (<ref>) of tangential derivatives given bycanonical extensions allow us to prove directly the following product ruleformulas, already known for classical derivatives <cit.>.[Product Rule for tangential derivatives] Let α, u, v and S be smooth fields in Γ,with α scalar valued, u andv vector valued, and S tensor valued. Then2 * (φu)= u⊗∇_Γφ +φu,* (φu)=φu + u·∇_Γφ* (u·v) =u^Tv + v^Tu* (u⊗v)=uv+D_Γu v,* (S^Tu)=S: u + u· S.* (α S)=Sα + αS.It will be very useful for us to write the geometric invariants (seeSection <ref>) of Γ in terms of tangential derivatives ofthe normal vector field n.The tensor -n(x) (see <cit.>) defined from thetangent plane T_x(Γ) to itself, is called the Weingarten map andis associated to the second fundamental form of Γ.Since n∘ p=∇ b, (<ref>) implies n =D(∘ p)|_Γ = D^2b|_Γ, so thatκ = Δ b|_Γ = (D^2b|_Γ) = () = n,and ∑κ_i^2 =|D^2b|_Γ|^2 = |n|^2 whence, for N=3, κ_g= 1/2(κ^2 -|n|^2).In, particular, as we will see in Section <ref>, any geometricinvariant can be written in terms of I_p(n^p).The Divergence Theorem for surfaces whose proof can be found in (Prop. 15 of <cit.>) is the following[Tangential Divergence Theorem] If Γ =∂Ω is 𝒞^2 andw∈𝒞^1(Γ, [N]), then∫_Γw = ∫_Γκ w·n,where κ is the mean curvature of Γ and n its normalfield. If Γ⊊∂Ω, then∫_Γw =∫_Γκ w·n+ ∫_∂Γw·n_s, where n_s is the outward normal to ∂Γ which is alsonormal to n.The following Lemma is new and extends formula (<ref>) fortangential derivatives.If Γ is 𝒞^3 and w∈𝒞^3(Γ,[N]),we havew = P w^T-nw^Tn, where P= I-nn is the orthogonal projection operatorgiven by (<ref>). We use formula (<ref>) to write tangential derivativesusing the projection function p: w = ∇ (w∘ p) |_Γ= ∇ ( (w∘ p)∘ p )|_Γ. Then we use successively the chain rule, the derivative of p given by(<ref>) and the property of classicalderivatives(<ref>): w= Dp^T |_Γ∇ (w∘ p)|_Γ = P ∇ (w∘ p) |_Γ = P(D(w∘ p)^T) |_Γ. Note that Lemma <ref> implies D(w∘ p)^T=w^T∘ p - bD^2b(w^T∘ p), and the product rule (α S)= αS + S∇αimplies (D(w∘ p)^T)= (w^T∘ p)-b (D^2b w^T∘ p) -D^2b w^T∘ p ∇ b. Then, after restricting to Γ we have(D(w∘ p)^T)|_Γ =(w^T)-nw^T n,which implies, from (<ref>), the desired result.Applying Lemma (<ref>) to w= we obtain forκ =∇_Γκ = ∇_Γ= P ∇_Γ - ^T = P because ^T= 0 and = ∇_Γ.§ SHAPE FUNCTIONALS AND DERIVATIVES A shape functional is a function J:𝒜→ defined on aset 𝒜=𝒜(𝐃) of admissiblesubsets of a hold-all domain 𝐃⊂[N].Let the elements of 𝒜 be smooth domains and for each Ω∈𝒜, let y(Ω) be afunction in W(Ω) some Sobolev space over Ω.Then the shape functional given by J(Ω)= ∫_Ω y(Ω)(x) dx = ∫_Ω y(Ω) is called a domain functional.For example the volume functional is obtained with y(Ω) ≡ 1,but the domain function y(Ω) could be something more involved such as the solution of a PDEin Ω.Our main interest in this work are the boundary functionalsgiven by J(Γ)= ∫_Γ z(Γ)(x) dΓ = ∫_Γ z(Γ),where z is a function that for each surface Γ in a family ofadmissible surfaces 𝒜 assigns a function z(Γ) ∈ W(Γ),withW(Γ) some Sobolev space on Γ. The area functional corresponds toz(Γ) ≡ 1, but more interesting functionals are obtained when the boundary function z(Γ) depends on the mean curvature κ of Γ or on thegeometric invariants I_p(Γ)=(n^p), with p a positive integer,or any real function which involves the normal field n or higher ordertangential derivatives on Γ.§.§ The velocity MethodOn a hold-all domain(not necessarily bounded), we call anautonomous velocityto a vector field v∈ V^k() 𝒞_0^k(,[N]),the set of all 𝒞^k functions f such thatD^α f has compact support contained in , for 0≤|α|≤ k; hereafter we assume that k is a fixed positive integer. A (nonautonomous) velocity fieldV∈𝒞([0,ϵ],V^k()),(Theorem 2.16 of <cit.>)induces a trajectory x=x_V∈𝒞^1([0,ϵ], V^k()), through the system of ODEẋ(t)=V(t)∘ x(t), t∈[0,ϵ], x(0)=id,where we use a point to denote derivative respect to the time variable t. [Initial velocity] We call v to the velocity field at t=0, namelyv = V(0). In the autonomous case, V(t)= v for any t,with v∈ V^k(), and the trajectory x(t) is given byẋ(t)=v∘ x(t), t∈[0,ϵ], x(0)=id.§.§ Shape Differentiation Given a velocity field V and a subset S⊂, the perturbedset at time t is given by S_t=x(t)(S),where x(t) is the trajectory given by (<ref>). For a shape functional J:→, whereis a familyof admissible sets S (domains or boundaries),and a velocity field V∈𝒞([0,T], V^k()), theEulerian semiderivative of Jat S in the direction V is given byd J(S; V) = lim_t↘ 0J(S_t)-J(S)/t,whenever the limit exist.[Shape differentiable] We say that J is shape differentiableat S when the Eulerian semiderivative (<ref>) exists for anyV in the vector space of velocities 𝒞([0,T], V^k()),and the functional V→ d J(S; V) is linear and continuous.[Hadamard differentiable] By Theorem 3.1 of <cit.>, if a functional J is shapedifferentiablethen dJ(S,V) depends only on vV(0) (that is,J is Hadamard differentiable atS in the direction v). [Shape derivative] If J is shape differentiable we call dJ its shape derivative. [Taylor formula] Given V∈𝒞([0,T], V^k()) we can define S+V to be S_t for t=1 provided it is admissible. Then if J is shape differentiable (see <cit.>) it follows that J(S + V) = J(S)+d J(S; V) + o(|V|).§.§ The Structure Theorem One of the main results about shape derivatives is the(Hadamard-Zolesio) Structure Theorem (Theorem 3.6 of <cit.>). It establishes that, if a shape functional J is shape differentiable atthe domain Ω with boundary Γ, then the only relevant part of thevelocity field V in dJ(Ω, V)is v_n V(0)·n|_Γ. In other words, ifV(0)·n=0 in Γ, thendJ(Ω, V)=0. More precisely,[Structure Theorem] Let Ω∈ be a domain with 𝒞^k+1-boundaryΓ,k≥ 0 integer, and let J:→ be a shape functional which is shape differentiable at Ω with respect to V^k(). Thenthere exists a functional g(Γ)∈(𝒞^k(Γ))'(called the shape gradient) such that dJ(Ω, V) =⟨ g(Γ),v_n⟩_𝒞^k(Γ), where v_n = V(0)·n. Moreover, if the gradient g(Γ)∈ L^1(Γ), then dJ(Ω, V)=∫_Γ g(Γ) v_n.§ SHAPE DERIVATIVES OF DOMAIN FUNCTIONS In this section and the following, we find specialized formulas for the shape derivatives of domain and boundary functionals.These, in turn, will induce definitions for the shape derivatives of domain andboundary functions.§.§ Shape differentiation of a domain functional Consider avelocity field V∈𝒞([0,ϵ], V^k()), k ≥ 1, with trajectories x∈𝒞^1([0,ϵ], V^k())satisfying (<ref>), and note that the Eulerian semiderivative(<ref>) can be written as d J(Ω;V)=d/dt^+J(Ω_t)|_t=0, where Ω_t=x(t)(Ω) andΩ_0=Ω.Then, from a well known change of variables formula<cit.>, we haveJ(Ω_t)=∫_Ω_ty(Ω_t)dΩ_t = ∫_Ω[y(Ω_t)∘ x(t)] γ(t) dΩwhere γ(t) Dx(t), with Dx(t) denoting derivativewith respect to the spatial variable X. Note that if y(Ω_t) ∈ W^r,p(Ω_t) for each t theny(Ω_t)∘ x(t)∈W^r,p(Ω) (if 0≤ r ≤ k). The following Lemma (Theorem 4.1 in Ch. 9, p. 482 of<cit.>) provides some insight on the nature of γ̇.[Time derivative of γ] If V∈𝒞([0,ϵ], V^1()) thenγ∈𝒞^1([0,ϵ], V^1()), and its (time) derivative is given by γ̇(t) =γ(t) [V(t) ∘ x(t)]. In particular, γ̇(0) = v, where v=V(0). In order to motivate the definition of material and shape derivative, consider the following situation. Let y be a domain function which assigns a function y(Ω) ∈ W(Ω) to each domain Ω in a classof admissible smooth domains. Suppose that the function f:[0,ϵ]→ L^1(Ω) given byf(t) = y(Ω_t)∘ x(t) is differentiable at t=0 in L^1(Ω), that is, there existsḟ(0)∈ L^1(Ω) such that lim_t↘ 0f(t)-f(0)/t-ḟ(0)_L^1(Ω)=0. Then we can differentiate inside the integral (<ref>) toobtaind/dtJ(Ω_t)|_t=0= ∫_Ωḟ(0) γ(0) +f(0)γ̇(0). Finally, using Lemma <ref>, and that γ(0)=1 andf(0)=y(Ω), we obtaindJ(Ω, V)= ∫_Ωḟ(0) + y(Ω) v,where v=V(0).§.§ Material and shape derivatives [Material Derivative (Def. 2.71,p. 98 of <cit.>)] Consider a velocity vector field V∈𝒞([0,ϵ],V^k()), with k≥ 1, an admissible set S⊂ (domain or boundary) of class𝒞^k, and a function y(S) ∈ W^r, p(S),with r∈(0,k]∩ℤ. Suppose there exists y(S_t) ∈ W^r,p(S_t) for all 0<t<ϵ, where S_t=x(t)(S) is the perturbation set of S by the trajectoriesx(t) given by V.The material derivative of y(S) at S in the directionV is the functionẏ(S, V)∈ W^r-1, p(S), given by ẏ(S, V) = d/dt^+ [y(S_t)∘ x(t)]_t=0=lim_t↘ 0y(S_t)∘ x(t) - y(S)/t,whenever the limit exists in the sense of W^r-1, p(S).In this case we say that the material derivative of y(S) exists at Sin W^r-1, p(S) in the direction V. We can replace the space W^r, p(S) by 𝒞^r(Ω),1≤ r ≤ k, obtainingẏ(S, V)∈𝒞^r-1(Ω).With this definition, the existence of material derivative of y(Ω)∈W^1,1(Ω) implies the differentiability of f(t) at t=0 inL^1(Ω), which was the assumption needed for equation(<ref>) to hold. Then, for J(Ω)=∫_Ωy(Ω),we havedJ(Ω, V) = ∫_Ωẏ(Ω, V) +y(Ω)v.[Autonomous dependence] If J(Ω)=∫_Ω y(Ω) is shape differentiable atΩ andẏ(Ω, V) exists for any velocity V, we obtainfrom Remark <ref> and equation (<ref>) that ẏ(Ω, V)=ẏ(Ω, v), wherev=V(0). As a particular case, suppose that y(Ω) is independent of thegeometry, namely: y(Ω)=ϕ|_Ω, withϕ∈ W^1,1().Then, by the chain rule, ẏ(Ω, V) = ∇ϕ·v, and we have dJ(Ω, V) = ∫_Ω∇ϕ·v +ϕ v =∫_Ω(ϕ v). If the boundary Γ=∂Ω∈𝒞^1 then, theDivergence Theorem yields d J(Ω; V) = ∫_Γϕv_n dΓ.This dependence of dJ(Ω,V) on V onlytrough the normal component of v in Γ was expected by theStructure Theorem.In the general case, when y(Ω)∈ W^1,1(Ω) depends on thegeometry of Ω, from (<ref>) we can writedJ(Ω, V) = ∫_Ω[ẏ(Ω, V) - ∇y(Ω) ·v] + (y(Ω)v). Which leads to the following definition of shape derivative of adomain function.[Shape derivative of a domainfunction] Given a velocity field V∈𝒞([0,ϵ], V^k()),if there exists the material derivative ofy(Ω)∈ W^r,p(Ω), with 1≤ r≤ k, then thedomain shape derivative y'(Ω, V)∈ W^r-1,p(Ω) is given byy'(Ω, V) = ẏ(Ω, V) - ∇ y(Ω)·v,where v=V(0).If y'(Ω,V) ∈ W^r-1,p(Ω) exists, then we have d J(Ω; V) = ∫_Ω y'(Ω, V) +(y(Ω)V(0))=∫_Ω y'(Ω, V) + ∫_Γ y(Ω) v_n,whenever Γ is 𝒞^1. § SHAPE DERIVATIVE OF BOUNDARY FUNCTIONS Consider now a boundary functional of the form J(Γ)=∫_Γ z(Γ) dΓ, where Γ =∂Ωis a boundary which is also a𝒞^k-manifold, and for each admissible Γ, z(Γ)∈W^r,p(Γ), with 1≤ r ≤ k. Below we derive a formula for the shape derivative dJ(Γ, V)analogous to (<ref>) which uses the concepts from tangential calculus given in Section<ref>.§.§ Shape differentiation of a boundary functional For Γ_t x(t)(Γ), we have thatJ(Γ_t)=∫_Γ_t z(Γ_t) dΓ_t= ∫_Γ [z(Γ_t)∘ x(t)] ω(t)dΓ,where ω(t)= M(Dx(t))n=γ(t)Dx(t)^-Tn, with γ(t)=Dx(t) and M(Dx(t))= γ(t)Dx(t)^-T is thecofactor matrix of the Jacobian Dx(t). This well known change of variableformula can be found in Proposition 2.47 of <cit.>. We can also find there a formula for the derivative of ω(t) att=0, which we now cite using the notation of tangential divergence.[Derivative of ω (Lemma 2.49, p. 80, of <cit.>)] The mapping t→ω(t) is differentiable from[0,ϵ] into 𝒞^k-1(Γ),and the derivative at t=0 is given byω̇(0)=v,wherev=v - Dvn·n is thetangential divergence of v=V(0).Assuming that the material derivative ż(Γ, V), given byDefinition <ref>, exists in L^1(Γ), we can differentiate inside the integral in(<ref>) and then obtaindJ(Γ, V)=∫_Γż(Γ, V) +z(Γ) v.Using the product rule formula (zv)=z v+v· z (Lemma <ref>) we can writedJ(Γ, V)=∫_Γ[ż(Γ, V)-z(Γ)·v] + (z(Γ)v).We are now in position to introduce the concept of shape derivativeof a boundary function. [Shape Derivative of a boundary function] Let z be a boundary function which satisfiesz(Γ) ∈W^r,p(Γ) for all Γ in an admissible set 𝒜 ofboundaries of class 𝒞^k. If the material derivative ż(Γ,V) exists inW^r-1,p(Γ) (Definition <ref>) for a velocity V∈𝒞([0,ϵ], V^k()), then the shape derivative z'(Γ, V)∈ W^r-1,p(Γ)is given by z'(Γ, V) = ż(Γ, V) -z(Γ)·v, where v=V(0). Note that ż(Γ, V) and z'(Γ, V) are intrinsictoΓ, because they do not depend on any extension of z(Γ) to an openset containing Γ. With this notation, forJ(Γ)=∫_Γz(Γ) dΓ we have d J(Γ; V) =∫_Γ z'(Γ, V) +(z(Γ)v).If Γ∈𝒞^2, then the tangential divergence formula(<ref>)of Lemma <ref> gives us theexpressiond J(Γ; V) =∫_Γ z'(Γ, V) + κ z(Γ) v_n ,whereκ=κ(Γ) is the mean curvature function on Γ,and v_n = v·n.For a surface Γ⊂∂Ω, we can consider thespace of velocity fieldsV_Γ^k() = V^k()⋂{v:v|_∂Γ=0},in order to obtain (<ref>) by applying formula(<ref>) ofLemma <ref>. §.§ Relation between domain and boundary function Suppose that z(Γ) is the restriction of a domain function y(Ω) toits boundary, that is: z(Γ) = y(Ω)|_Γ,where ∂Ω = Γ and suppose that y(Ω) is shapedifferentiable at Ω in the direction V.The material derivatives of both z(Γ) and y(Ω) are the same, as is established in the following Lemma.[Proposition 2.75 of<cit.>]Assume that the material derivative of a domain function y(Ω)∈W^r,p(Ω) exists at the domain Ω of class 𝒞^k in thedirection of a velocity field V∈𝒞([0,ϵ], V^k()), and thatẏ(Ω,V)∈ W^r-1,p(Ω). Then for r>1/p+1,thereexists the material derivative of z(Γ)=y(Ω)|_Γ at Γ in the direction V, and ż(Γ,V)= ẏ(Ω,V)|_Γ∈W^r-1-1/p,p(Ω) However, the shape derivatives of domain and boundary functions are notgenerally equal for the same function, as is shown in the next lemma. [Domain and boundaryfunction] Consider a domain Ω with boundary Γ, and functions z(Γ)andy(Ω) such thatz(Γ) = y(Ω)|_Γ. Then, z(Γ) isshape differentiable in Γ at direction V∈𝒞([0,ϵ], V^k()) ify(Ω) is shapedifferentiable in Ω at direction V, andz'(Γ, V)= y'(Ω, V)|_Γ + ∂y(Ω)/∂n v_n,where v_n = v·n. Since y(Ω) is an extension to Ω of z(Γ), we have z= ∇ y|_Γ-∂ y/∂ nn. Then, using Lemma <ref> and comparing Definitions<ref>and <ref>,weobtain the desired result.Going back to formula (<ref>), ifz(Γ) admits an extension y(Ω), Lemma <ref> gives usd J(Γ; V) =∫_Γ y'(Ω, V)|_Γ + (∂ y(Ω)/∂n +κ z(Γ))v_n. § PROPERTIES OF SHAPE DERIVATIVES OF DOMAIN FUNCTIONS We first extend the Definition <ref> ofdomainshape derivative to vector and tensor valued functions.[Vector and tensor valued domainfunctions]Consider a vector field w(Ω)∈ W^r,p(Ω, [N])whichexists for all admissible domains Ω∈(). For a given velocity field V, we say that w(Ω) is shape differentiable at Ω in the direction of V if thereexists the material derivative ẇ(Ω, V) = d/dt [w(Ω_t)∘x(t)]_t=0 in W^r-1,p(Ω, [N]).In that case the (domain) shape derivativebelongs toW^r-1,p(Ω, [N]) and is given by w'(Ω, V)= ẇ(Ω, V) - Dw(Ω)v,where v=V(0).If N=1, we consider Dw=(∇w)^T.For a tensor valued function A(Ω):Ω→(𝕍),wesay that A is shape differentiable at Ω in the direction of Vifso is the vector valued function Ae, for any vector e∈𝕍. The shape derivative A'(Ω, V) is the tensor valuedfunction which satisfiesA'(Ω, V)e =(Ae)'(Ω, V) for any e∈𝕍.Throughout this paper we consider 𝕍=[N].For a given admissible domain Ω⊂ with boundary𝒞^k, k≥ 2, a velocity field V∈𝒞([0,ϵ], V^k()) anda shape differentiable functiony(Ω)∈ W^r,p(Ω), 1≤ r ≤ k, 1≤ p<∞,we have the following properties: * If the mapping V→ẏ(Ω, V) iscontinuous from 𝒞([0,ϵ], V^k()) into W^r-1,p(Ω),then y'(Ω, V)=y'(Ω, v), where v=V(0) (Proposition 2.86 of <cit.>).* Suppose that V→ẏ(Ω, V) is continuousfrom 𝒞([0,ϵ], V^k()) into W^r-1,p(Ω). If thevelocity fields V_1 and V_2 are such that V_1(0)·n= V_2(0)·n on Γ=∂Ω, then y'(Ω,V_1)=y'(Ω, V_2) (Proposition 2.87 of <cit.>).* If y(Ω)=ϕ|_Ω, for ϕ∈ W^r,p(), then yisshape differentiable in Ω for any direction V, and y'(Ω,V)=0 (Proposition 2.72 of <cit.>).[Lemma 4 from <cit.>]Suppose that y(Ω) ∈ H^3/2+ϵ(Ω) satisfiesy(Ω)|_Γ = 0 for all domains Ω∈𝒜 and that the shape derivative y'(Ω;V) exists in H^1/2 +ϵ(Ω) for some ϵ > 0. Then, we havey'(Ω, V)|_Γ =-∂ y/∂nv_nwhere v_n=v·n and v=V(0). This Lemma is proved in<cit.>. However, it can be alsodemonstrated if we consider the boundary function z(Γ)y(Ω)|_Γ. In fact, by hypothesis, z(Γ_t)≡ 0 for all smallt≥0. This gives us ż(Γ, V)=0 and z(Γ)=0,sothat z'(Γ, V)=0. The claim thus follows from Lemma <ref>. The following Lemma states that shape derivatives commute with linear transformations, both for domain and boundary functions. The proof is straightforward from the definitions. Let F∈(_1,_2), with _1 and _2 two finite dimensional vector or tensor spaces, and let w(S)∈𝒞^k(S, _1) for any admissible domain or boundary S⊂ and k≥ 1. If w(S) is shape differentiable at S in the direction V, then F∘ w(S) ∈𝒞^k(S, _2) is also shape differentiable at S in the direction V, and its shape derivative is given by (F∘ w)'(S, V) =F∘ w'(S, V).The next lemma states a chain rule combining usual derivatives with shape derivatives. [Chain rule] Consider two finite dimensional vector or tensor spaces _1 and _2, a function F∈𝒞^1(_1,_2) and a domain (or boundary) function y(S) ∈𝒞^1(S, _1), where S is an admissible domain (boundary) in ⊂^N with a 𝒞^1 boundary. If y(Ω) is shape differentiable at Ω in the direction V, then the function F∘ y(Ω)∈𝒞^1(Ω, _2) is also shape differentiable at Ω in the direction V, and its shape derivative is given by (F∘ y)'(Ω, V) =DF∘ y(Ω)[ y'(S, V)]. Since F is differentiable, for every X∈_1 there exists a linear operator DF(X) ∈(_1,_2) such that lim_u__1→ 0F(X+u)-F(X) -DF(X)[u]__2/u__1=0; where DF(X)[u] denotes the application of the linear operator DF(X)∈(_1,_2) to u ∈_1. With this notation, the chain rule applied to F ∘ y reads D(F∘ y)[v]=DF∘ y[Dy[v]] (∀v∈^N), so that from Definition <ref>, (F∘ y)'(Ω, V) =(F∘ y)^(Ω, V)- D(F∘ y)[V] = (F∘ y)^(Ω, V)- DF∘ y[Dy[V]] , in Ω,with (F∘ y)^(Ω, V) denoting the material derivative of F∘ y. Then we only need to prove the chain rule for the material derivative of y(Ω)∈𝒞^1(Ω, _1) in the direction V, i.e., (F∘ y)^(Ω, V) = DF ∘ y[ẏ(Ω,V)]. If we recall that (F∘ y)^(Ω, V) = d/dt^+[ F∘ y(Ω_t)∘ x(t)]_t=0, the proof of this last equality is straightforward from usual chain rule applied to the mapping t → F∘(y(Ω_t)∘ x(t)). The details are left to the reader. [Product rule for shape derivatives] The product rules for domain shape derivatives follow directly fromDefinitions <ref> and<ref>.The following Lemma allows us to swap shape derivatives with classicalderivatives of domain functions.It is worth noting that this is not true for boundary functions and tangential derivatives.This will be discussed in Section <ref>, where the main results ofthisarticle are presented.[Mixed shape and classical derivatives] The following results about interchanging classical and shapederivatives are satisfied. * If y(Ω)∈𝒞^2(Ω)is shape differentiable atΩ in the directionV∈𝒞([0,ϵ], V^k()), k≥ 2,then ∇ y(Ω)∈𝒞^1(Ω, [N])is also shape differentiable at Ω and (∇ y)'(Ω, V) = ∇ y'(Ω, V) .* If w(Ω)∈𝒞^2(Ω, [N])is shapedifferentiable at Ω in the direction V∈𝒞([0,ϵ], V^k()),k≥ 2,then D w(Ω)∈𝒞^1(Ω,ℝ^N× N) and w(Ω)∈𝒞^1(Ω)is also shape differentiable and (D w)'(Ω, V) = D w'(Ω, V),(w)'(Ω, V) = w'(Ω, V). * If y(Ω)∈𝒞^3(Ω)is shape differentiableat Ωin the direction V∈𝒞([0,ϵ], V^k()), k≥ 3,then Δ y(Ω)∈𝒞^1(Ω)is also shape differentiable at Ω and (Δ y)'(Ω, V) = Δ y'(Ω, V) . We will prove the first assertion. The other ones are analogous.First note that∇(y(Ω_t)∘ x(t)) =Dx(t)^T ∇y(Ω_t)∘ x(t) in Ω.Differentiating with respect to t and evaluating at t=0 we have ∂/∂ t∇(y(Ω_t)∘ x(t))|_t=0= ∂/∂ tDx(t)^T|_t=0∇ y(Ω) +∇̇ y(Ω, V)where we have used that x(0)=id, Dx(0)=I and Definition<ref>, denoting with ∇̇ y(Ω, V) the material derivative of ∇ y(Ω).Sincex ∈𝒞^1([0,ϵ],V^k()), k≥ 1, and recalling that ẋ(0) =∂/∂tx(t, ·)|_t=0 =v, we have that, uniformlylim_t↘ 0 D^α( x(t)- x(0)/t)=D^αv for any0≤ |α|≤ k,so that∂/∂ tDx(t)|_t=0 = Dv pointwise in . Analogously, the existence of the material derivative ẏ(Ω, V)in 𝒞^1(Ω) implies that, uniformly,lim_t↘ 0∂/∂ X_i ( y(Ω_t)∘x(t)-y(Ω)/t) =∂/∂ X_i ẏ(Ω, V),for 1≤ i ≤ N, and then∂/∂ t∇(y(Ω_t)∘ x(t))|_t=0= ∇ẏ(Ω, V)pointwise in Ω.Replacing with (<ref>) and (<ref>) in(<ref>), we obtain∇ẏ(Ω, V)= Dv^T∇ y(Ω)+ ∇̇y(Ω, V). By Definition <ref> of shape derivative we have∇ y'(Ω, V)= ∇ẏ(Ω, V)-∇(∇y(Ω)·v)=∇ẏ(Ω, V)-Dv^T∇ y(Ω) -D^2y(Ω)^Tv= ∇̇ y(Ω, V) - D^2y(Ω)^Tv=(∇ y)'(Ω, V),where we have used Definition <ref> and the fact that D^2 y(Ω)^T is symmetric because y(Ω)∈𝒞^2(Ω). §.§ Shape derivative of the signed distance functionConcerning the shape derivative of the signed distance functionb=b(Ω), we cite <cit.> and <cit.>. Since b|_Γ=0,we can use Lemma <ref> to obtainb'(Ω, V)|_Γ = - v·n = -v_n,where we have used that ∂ b/∂n=∇ b ·∇ b |_Γ = 1. Note that this is the shape derivative ofb(Ω) only on Γ. In order to find the shape derivative of w(Ω)∇b(Ω), we derive the Eikonal Equation (<ref>) to obtain (∇ b )'(Ω, V) ·∇ b = 0. Thismeans that, when restricted to Γ, (∇ b )'(Ω, V)isorthogonal to the normal vector field n. But also from Lemma<ref> (∇ b )'(Ω, V)=∇b'(Ω,V)so that (∇ b )'(Ω, V)|_Γ= (b'(Ω, V))= b'(Ω, V) .Finally, using formula (<ref>), we obtain(∇ b )'(Ω, V)|_Γ = -v_n,where v_n=v·n.This identity will be used later to obtain the shape derivative of the boundary (vector valued) function n.§ PROPERTIES OF THE SHAPE DERIVATIVE OF BOUNDARYFUNCTIONS We first extend Definition <ref> to vector and tensor valued functions defined on a boundary Γ=∂Ω. To consider shape derivatives in a fixed surface Γ⊊∂Ω, we consider the space of velocities V_Γ() instead of V() as it was done in Remark <ref>.[Vector Boundary Functions]Consider a vector field w(Γ)∈ W^r,p(Γ, [N]) whichexists for all admissible domains Ω⊂ with boundary Γ∈𝒞^k, with 1≤r ≤ k.For a given velocity fieldV∈𝒞([0,ϵ], V^k()),we say that w(Γ) is shape differentiable in Γ in the direction of V if thereexists the material derivative ẇ(Γ, V) = d/dt [w(Γ_t)∘x(t)]_t=0 in W^r-1,p(Γ, [N]).In that case the (boundary) shape derivative belongs toW^r-1,p(Γ, [N]) and is given by w'(Γ, V)= ẇ(Ω, V) - wv,where v=V(0).Analogously to tensor valued domain functions, for a tensor valued boundary function A(Γ):Γ→ℝ^N×N the shape derivative A'(Γ, V) is the tensor valued function which satisfies (<ref>), with Ωreplaced by Γ. We now extend Lemma <ref> to vectorvalued functions. The proof is straightforward from the fact thatw(Ω,V)·e = (w·e)'(Ω,V) forany fixed vector e, because w→w·e is a lineartransformation.[Extension] Let Ω be a domain with a 𝒞^1 boundary Γ. Ifthe boundary function w(Γ) ∈𝒞^1(Γ, [N]) admitsan extension W(Ω) which is shape differentiable at Ω in thedirection V∈𝒞([0,ϵ], V^k()), k≥ 1, then theshape derivative of w(Γ) in Γ at V is given byw'(Γ, V)= W'(Ω, V)|_Γ +DWnv_n.[Comparison with Lemma <ref>] The first two assertions of Lemma <ref> are also validfor boundary functions (the proofs can be found in <cit.>). Instead ofthe third assertion, we have the following one, which is an immediate consequence of Lemma <ref>.[Shape derivative of ϕ|_Γ]If z(Γ)=ϕ|_Γ, for ϕ∈ W^r,p(),r-1/p≥ 1, then z is shape differentiable in Γ for anydirection V, and y'(Γ, V)=∂ϕ/∂ nv_n. §.§ The shape derivatives of n and κSince the normal vector field n at Γ has the gradient of thesigned distance function b(Ω) as an extension, we can use the results of Section 7.1 to obtain the shape derivative of n.[Shape derivative of ] Let Γ be the boundary of a C^2-domain Ω and letn(Γ) be the unit normal vector fieldof Γ. Then n isshape differentiable and n'(Γ, V) = - v_n, where v_n = V(0) ·n, for all V∈𝒞([0,ϵ], V^1()). Since n(Γ)= ∇ b |_Γ,where b is the signed distance function of Ω, we use Lemma<ref> to obtain n'(Γ, V)=(∇ b)'(Ω, V)|_Γ+ D^2b|_Γnv_n.Since by equation (<ref>)D^2b ∇ b= 0 everywhere, thesecondterm vanishes and (<ref>) yields the desired result. Concerningthe mean curvature κ =κ(Γ) given by(<ref>),we recall that Δ b (Ω)= (D^2 b) is an extension of κ toa tubular neighborhood of Γ (see Section <ref>). Then, on the one hand, the shape derivative of κ can be obtained using Lemma<ref>, as we will do in the proof of thenext Lemma. On the other hand, since D^2b|_Γ =n, we can also express the meancurvature using tangential derivatives, as κ = n. Then, Corollary <ref> gives us quickly the same formula for the shapederivative of κ.[Shape derivative of κ]If κ is the mean curvature ofΓ, the boundary of a𝒞^3 domain Ω, then κ is shape differentiable inΓ and κ'(Γ, V) =- v_n - |n|^2 v_n,where v_n=V(0)·n,|n|^2=n:n=tr(n^2) and f = fis the Laplace-Beltrami operator of f. Since κ =Δ b|_Γ, by Lemma<ref> κ'(Γ, V)= (Δ b)'(Ω, V)|_Γ + (∇Δ b ·∇ b )|_Γ v_n.The second term is equal to -|n|^2 v_n using equation(<ref>), and the fact that n=D^2 b |_Γ. For the first term, we use Lemma <ref> and the definition oftangential divergence (<ref>) to obtain(Δ b)'(Ω, V) = (∇ b)'(Ω, V) =(∇ b)'(Ω, V) +D^2b|_Γn·n=-( v_n) = - v_n,where we have used (<ref>) and the fact that D^2b|_Γn= D^2b|_Γ∇ b|_Γ = (D^2b ∇ b)|_Γ = 0.§ THE SHAPE DERIVATIVES OF TANGENTIAL OPERATORS We are now in position to present the main results of this paper, namely, formulas for the shape derivatives of boundary functions that are tangential derivatives of boundary functions.More precisely, we will find the shape derivatives of boundary functions ofthe formz, w, w and z, when z(Γ) andw(Γ)are shape differentiable boundary functions, scalar and vector valued, respectively. Examples of important applications will be presented in the two subsequentsections. It is worth noting the difference with Lemma <ref> where we established that standard differential operators commute with the shape derivative of domain functions.[Shape derivative of surface derivatives] For any admissible boundary Γ = ∂Ω, where Ωis a 𝒞^2 domain in ⊂[N], consider a real functionz(Γ)∈𝒞^2(Γ). If z is shape differentiable at Γ in the directionV∈𝒞([0,ϵ], V^2()), then z is shapedifferentiable at Γ in the direction V, and( z )'(Γ, V) =z'(Γ, V) +( n v_n- v_nn)z,where v_n= V(0)·n. Before proceeding to the proof it is worth noticing the differences withLemma <ref> where we have shown that the shape derivative ofdomain integrands commutes with the space derivatives.Let y=y(Ω) be an extension of z(Γ) to Ω, i.e. z(Γ)=y(Ω)|_Γ, then by definition z(Γ) = ∇ y|_Γ- ∂_n y n = ( ∇ y - (∇ y ·∇ b) ∇ b )|_Γ,because ∂_n y= ∂ y/∂ n =∇ y ·n. Then Φ (Ω) ∇ y - (∇ y ·∇ b)∇ b is an extension to Ω of z (Γ). Due to Lemma <ref>these shape derivatives satisfy( z )'(Γ, V)= Φ'(Ω,V)|_Γ +DΦ(Ω)|_Γn v_n. We now compute the domain shape derivative of Φ(Ω). Using the product rule we haveΦ'(Ω, V)= (∇ y)'(Ω, V)- (∇y)'(Ω, V)·∇ b ∇ b-∇ y · (∇ b)'(Ω, V) ∇ b - ∇ y ·∇ b(∇ b)'(Ω, V)= (I-∇ b∇ b) ∇ y'(Ω,V)-∇ y · (∇ b)'(Ω, V) ∇ b - ∇ y ·∇ b(∇ b)'(Ω, V),where we have used Lemma <ref> to commute the shapederivative and the gradient of y. Restricting to Γ, using the definition of tangential gradient andformula(<ref>) for (∇ b)'(Ω, V)|_Γ, we obtainΦ'(Ω, V)|_Γ =y'(Ω, V) + (n v_n)z +∂_n yv_n,where we have used ∇ y(Ω)|_Γ· v_n n =z · v_n n = (n v_n)z. From Lemma <ref> y'(Ω,V)|_Γ = z'(Γ, V)-∂_n yv_nand the product rule for tangential derivative yieldsΦ'(Ω, V)|_Γ= z'(Γ, V) -(∂_n yv_n) +(n v_n)z + ∂_n yv_n =z'(Γ, V) + (n v_n)z -v_n (∂_n y). Then, from (<ref>), to complete the proof of(<ref>), we need to show that DΦ(Ω)|_Γn - (∂_n y) = - n z. Applying the product rule of classical derivatives to Φ(Ω) = ∇ y - (∇ y ·∇ b) ∇ b, we obtain, using = ∇ b|_Γ,DΦ(Ω)|_Γn = D^2y|_Γn -( n∇(∇ y ·∇ b)|_Γ)n - ∂_n y D^2b ∇ b |_Γ=D^2y|_Γn - ∂_n(∇ y ·∇ b)n,because D^2b ∇ b=0. Besides, (∂_n y)= ∇(∇ y ·∇ b)|_Γ -∂_n (∇ y ·∇ b) n= D^2y|_Γn - D^2b ∇ y|_Γ - ∂_n (∇ y·∇ b) n= DΦ(Ω)|_Γn - n z,where we have used that D^2b ∇ y|_Γ = n y = n z. From this equation we obtain (<ref>) and the claim follows.[For vector fields] If the functions z(Γ)∈𝒞^2(Γ) andw(Γ)∈𝒞^2(Γ,[N]) are shape differentiable atΓ∈𝒞^2 in the direction V∈𝒞([0,ϵ],V^2()), then w and w are also shape differentiable atΓ in the direction V and (w)'(Γ, V)= w'(Γ, V) +w[ v_nn - v_nn], (w)'(Γ, V)= w'(Γ, V) + [n v_n - v_nn]:w,where S:T=(S^T T) denote the scalar product of tensors.In order to obtain (<ref>), note thatw^Te_i= w_i, where w_i=w·e_i, with{e_1, ..., e_N} being the canonical basis of [N]. Bydefinition, the shape derivative of the tensor w^T must satisfy (w^T)'(Γ, V)e_i= (w^Te_i)'(Γ,V)= ( w_i)'(Γ, V).Applying (<ref>) to z(Γ)=w_i=w·e_i, we obtain(w^T)'(Γ, V)e_i= ( w_i)'(Γ, V)=w_i'(Γ, V) + [n v_n - v_nn] w_i= (w'(Γ, V)^T + [n v_n- v_nn]w^T)e_i.The linearity of the transpose operator andLemma <ref> yield the desired result.Finally, we recall that (w)'(Γ, V)=(w)'(Γ, V) and(ab): S = a· S b. Therefore (<ref>) implies(w)'(Γ, V)= w'(Γ, V) +w v_n ·n- v_nn:w,and (<ref>) follows. We end this section establishing the shape derivative of theLaplace-Beltrami operator of a boundary function, which is more involvedbecauseit is of second order.[Shape derivative of Lapace-Beltrami] If z=z(Γ)∈𝒞^3(Γ) is shape differentiable at a𝒞^3-boundary Γ in the directionV∈𝒞([0,ϵ], V^3()),then zzis also shape differentiable at Γ in the direction V, andits shape derivative is given by( z)'(Γ, V)=z'(Γ, V) -2v_nn:^2z +( κ v_n- 2n v_n- v_nκ)· z =z'(Γ, V)-v_n(2n:^2z +κ· z) + v_n ·( κ z - 2nz) .In order to simplify the calculation, we denote M = nv_n-v_nn.Using successively the formulas for the shape derivative of a tangentialdivergence (Corollary <ref>) and for a tangential gradient (Theorem<ref>), we have( z)'(Γ, V)= ( z)'(Γ, V) =( ( z)'(Γ,V)) + M:z =[ z'(Γ, V)+ M z] + M:^2z = z'(Γ, V)+ ( M z)+ M:^2z. Using the product rule (v) of Lemma <ref> we obtain( z)'(Γ, V) =z'(Γ, V)+M^T:^2z+M^T· z+ M:^2z =z'(Γ, V)+(M+M^T):^2z+ M^T· z.Since D_Γ^T = D_Γ, the second term of the right-hand side readsM+M^T= n v_n+v_n n -2 v_nn.Using the tensor property (ab): S = a· Sb and that ^2 zn=0, we obtain(M+M^T):^2z =n·^2zv_n - 2 v_nn: ^2 z.Observe that differentiating n· z=0 leads to^2z^Tn=-n z which impliesn·^2z v_n = -n v_n· z. Then(M+M^T):^2z =-n v_n· z - 2 v_nn:^2 z. The third term of (<ref>) contains M^T which can be computed with the product rules of Lemma <ref> to obtain M^T= ( v_n n)-(v_n n)=v_n · +v_n-v_n -v_n()=κ v_n - n v_n - v_n n,where we have used that κ=n and v_n=^2v_nn=0. Since n· z = (Pn)· z, where P is the orthogonal projection to the tangent plane, equation (<ref>) yields (Pn)=κ whenceM^T· z =κ v_n· z- n v_n· z-v_n κ· z. Finally we add equations (<ref>) and (<ref>) and replace in(<ref>) to obtain ( z)'(Γ, V)=z'(Γ, V)-2n v_n· z - 2 v_nn: ^2 z+κ v_n· z-v_n κ· z,which completes the proof. [Shape derivative of the first fundamentalform]For a 𝒞^3 surface Γ and a smooth velocity field V,the shape derivatives of the mean curvature κ=n and the tensorn at V are given by κ'(Γ, V) =- v_n - |n|^2 v_n,and(n)'(Γ, V) = -^2v_n+ n v_nn-v_nn^2. Since, by Lemma <ref>, n'(Γ, V) = -v_n, we will use corollary <ref> to obtain the shape derivatives ofn and κ=n. From (<ref>) we have(n)'(Γ, V)= n'(Γ, V) +n[ v_nn - v_nn]=-^2v_n+ n v_nn -v_nn^2,whence (<ref>) holds. To obtain the shape derivative of κ=n we can use equation(<ref>) or observe that (n)'(Γ,V)=(n)'(Γ, V) and use (<ref>). In both cases, note that v_n = (^2 v_n)=v_n and n:n =(n^2)= |n|^2. Also, since nn =0, we have(D_Γ v_n ) = n v_n :n = n v_n ·n=0. We have thus obtained (n)'(Γ, V)= κ'(Γ, V)= - v_n -|n|^2 v_n. § GEOMETRIC INVARIANTS AND GAUSSIAN CURVATUREThe geometric invariants of a 𝒞^2-surface Γ allow us to define itsintrinsic properties.They are defined as the geometric invariants of the tensor , which, in turn, are the coefficients of its characteristic polynomial p(λ) (see <cit.>).The geometric invariants of Γ, i_j(Γ): Γ→, j= 1,…, N,thus satisfyp(λ)=(n(X)-λ I)=λ^N + i_1 λ^N-1+i_2 λ^N-2+ ... + i_N-1λ + i_N,and can also be expressed using the eigenvalues of thetensor n, whichare zero and the principal curvaturesκ_1,…,κ_N-1, namelyi_1(Γ) = ∑_j=1^N-1κ_j, i_2(Γ) = ∑_j_1≠ j_2κ_j_1κ_j_2,…, i_N-1(Γ)= κ_1 …κ_N-1, i_N(Γ)=0. We can observe from definitions (<ref>) that the first invarianti_1(Γ) is the mean curvature κ and the last nonzero invariant i_N-1(Γ) is the Gaussiancurvatureκ_g.The invariant i_k(Γ), for 2≤ k ≤ N-2, is the so-called k-thmean curvature <cit.>. The geometric invariants of Γ can also be defined through thefunctions I_p(Γ):Γ→, given by I_p(Γ)=(n^p), p=1, ..., N-1.In particular, the first 4 invariants arei_1 = I_1=n, i_2 = 1/2!(I_1^2-I_2)=1/2(κ^2-|n|^2), i_3 = 1/3!(I_1^3-3I_1I_2+2I_3), i_4 = 1/4!(I_1^4-6I_1^2I_2+3I_2^2+8 I_1I_3 -6I_4).We will now establish the shape derivatives of the functionsI_p(Γ)=(^p), which are also intrinsic to the surface Γand will lead to the shape derivatives of the geometric invariantsi_k(Γ). [Shape derivatives of the invariants]Let Γ be a 𝒞^2-boundary in ^N and p a positiveinteger.The shape derivative of the scalar valued boundary functionI_p(Γ)(n^p) at Γ in the directionV∈𝒞([0,ϵ], V^2()) is given by (I_p)'(Γ, V) = -p(^2v_n : n^p-1+v_n I_p+1),where v_n=V(0)·n.For the proof of this proposition we will need the following Lemma.Let A(Γ):Γ→(𝕍) be a symmetrictensor valued function and let p be a positive integer. If A(Γ) isshape differentiable at Γ in the direction V, thenthe shape derivative of A^p(Γ) satisfies (A^p)'(Γ, V):A^j =p(A'(Γ, V):A^j+p-1), for any integer j≥ 0.We will proceed by induction. It is trivial to see that equation(<ref>) holds for p=1 and any integer j≥ 0.Assuming thatequation (<ref>) holds forp≥ 1 and any j≥ 0, we want to prove (A^p+1)'(Γ, V):A^j =(p+1)(A'(Γ,V):A^j+p), for any integerj≥ 0. Applying the product rule for the shape derivative toA^p+1=A^pA, we have(A^p+1)'(Γ, V):A^j = (A^p)'(Γ, V)A:A^j+A^p A'(Γ, V):A^j. The tensor product property BC:D=B:DC^T = C:B^TD and the factthat the tensor A is symmetric, yield (A^p+1)'(Γ, V):A^j =(A^p)'(Γ,V):A^j+1+A'(Γ, V):A^j+p. The inductive assumption for p and j+1 implies(A^p)'(Γ, V):A^j+1 =p(A'(Γ,V):A^j+p). Using this in equation (<ref>), weobtain the desired result (<ref>).First note that I_p'(Γ,V)= (n^p)'(Γ, V)=(n^p)'(Γ, V) : n^0. Then Lemma <ref> with j=0 and A=n, which is a symmetric tensor, lead to I_p'(Γ, V)= p(n'(Γ,V):n^p-1).From formula (<ref>) we have that(n)'(Γ, n):n^p-1 = -^2v_n:n^p-1- v_n I_p+1(Γ), where we have used thatn v_n n:n^p-1=0 for any integer p≥ 1. This completes the proof.We now obtain the shape derivatives of the geometric invariants, which willgive us, as particular cases, the shape derivatives of the Gaussian and meancurvatures.The goal is to obtain them in terms of the geometric invariants.We start with i_1 = κ: i_1'(Γ, v) = I_1'(Γ, v) = -^2v_n :n^0-v_nI_2 = -v_n -v_n I_2,which is consistent with the previous result (<ref>).Since I_2=i_1^2-2i_2, i_1'(Γ, v)=-v_n -v_n i_1^2(Γ) +2v_ni_2(Γ). For the second invariant, note thatI_2'(Γ, v) = -2(^2v_n : n+v_n I_3).Since i_2=1/2(I_1^2-I_2), we have i_2'(Γ, V)= I_1 I_1'(Γ, V)-1/2I_2'(Γ,V) = -I_1 v_n-v_n I_1 I_2+ ^2v_n : n+v_n I_3= -I_1v_n + ^2v_n : n + v_n(I_3-I_1I_2). To obtain a formula only involving the invariants i_k, observe thati_31/3! (I_1^3- 3 I_1 I_2 + 2 I_3)=1/3(I_3-I_1I_2 +i_1 i_2), as can be checked by replacing i_1 byI_1 and i_2 by 1/2 (I_1^2-I_2) in the right-hand side, whence i_2'(Γ, V)=-i_1v_n +^2v_n : n +v_n(3i_3-i_1i_2).If N=3, the Gaussian curvature κ_g is the secondinvarianti_2(Γ). Then, on the one hand, from (<ref>), we have the following expression for theshape derivative of the mean curvature κ in terms of κ_g:κ'(Γ, v)=-v_n -v_n κ^2 +2v_n κ_g.On the other hand, since i_3=0 for N=3, we obtain from (<ref>) thefollowing formula for the shape derivative of the Gaussian curvature.[Shape derivative of the Gauss curvature] For a 𝒞^2-surface Γ in [3], the shape derivativeof the Gaussian curvature κ_g is given byκ_g'(Γ, V)=-κ v_n +^2v_n : n -v_nκκ_g,where κ is the mean or additive curvature, n the normalvectorfield and v_n=V(0)·n.§ APPLICATION: A NEWTON-TYPE METHODMost of shape optimization problems consist in finding a minimum of somefunctional restricted to a family of admissible sets (domains or surfaces), e.g.,Γ_*=_Γ∈ J(Γ) For example, given a regular curve γ in [3], a minimal surfaceΓ with boundary γ is a solution of (<ref>) with J(Γ) =∫_Γ dΓ, the area functional, and the admissiblefamily =(γ) consisting of all regular 2 dimensional surfaces in[3] with boundary γ. If J is shape differentiable in , and Γ_* is a minimizer, then dJ(Γ_*, v)=0 for all v∈, whereis a vector space of admissible autonomous velocities, forexample =𝒞^k_c(, [N]).We thus focus our attention in the following alternative problem: Find Γ_*∈ : dJ(Γ_*, v)= 0, for allv∈:= 𝒞^k_c(, [N]). An interesting scheme to approximate the solutions of (<ref>) forsurfaces of prescribed constant mean curvature was presented in <cit.>.There, results from numerical experiments document its performance and fast convergence. The scheme was a variation of the Newton algorithm, which needs the computation of second derivatives of the shape functional. The computations there were tailored to the specific problem of prescribed mean curvature, and based on variational calculus using parametrizations, rather than using shape calculus.We first observe that, due to the structure Theorem (Theorem <ref>), Problem (<ref>) is equivalent to the following: Find Γ_*∈ : dJ(Γ_*, v∇ b_Γ_*)= 0, forall v∈_* { w ∈ : .∂ w/∂_*|_Γ_* = 0 },where b_Γ_* b(Ω_*) is the signed distance function corresponding to the domain Ω_* whose boundary is Γ_*, = 𝒞^k_c() and n_* = ∇ b_Γ_* is the normal vector to Γ_*. We now present a scheme to approximate the solution of (<ref>)using a Newton-type method that generalizes the idea of <cit.> in at least two ways. First, it uses the language of shape derivatives and secondly, ithas the potential to work for a large class of shape functionals, not just the area functional.We start by defining, for each Γ∈ and v∈, the functional J_v(Γ) = dJ(Γ, v ∇ b_Γ), so that the solution Γ_* satisfies J_v(Γ_*) = 0 for all v ∈_*. Assume now that Γ_0∈ is sufficiently close to the solution Γ_* so that there exists u∈ (small, in some sense) such that Γ_* := Γ_0+u, in the sense of Remark <ref>; this Remark also implies that J_v (Γ_*) = J_v(Γ_0 + u) = J_v (Γ_0) +dJ_v(Γ_0,u) + o(|u|).The goal of finding Γ_* = Γ_0 + u such that J_v(Γ_0 + u)=0 is now switched to a simplified problem of finding u_0 such that the linear approximation of J_v around Γ_0 vanishes at Γ_1 Γ_0 + u_0, i.e., J_v (Γ_0) +dJ_v(Γ_0,u_0) = 0. Another simplification arises when asking this equality to hold for all v ∈_0 { w ∈ : ∂ w/∂|_Γ_0 = 0 } (instead of _* or _1). Since dJ_v(Γ_0,u_0) only depends on the normal component of u_0 on Γ_0, this last problem has multiple solutions, so we restrict it by considering normal velocities of the form u_0 = u_0 _0 with u_0 ∈ V(Γ_0) = C^k(Γ_0) (and _0 the normal vector to Γ_0), and arrive at the following problem:Findu_0 ∈ V(Γ_0) : J_v (Γ_0) +dJ_v(Γ_0,u_0_0)=0 ∀v∈_0.Finally, define Γ_1 = Γ_0 + u_0 _0. This sets the basis for an iterative method that will be implemented and further investigated in forthcoming articles. §.§ Area and Willmore functionals We end this paper with the precise form of the second order shape derivatives of two important functionals: the area functional and the Willmore functional.For a shape differentiable boundary functional J(Γ)=∫_Γ z(Γ), and a function v ∈ C_c^k(), the functional J_v(Γ) := dJ(Γ, v ∇ b_Γ) is given by J_v (Γ)= ∫_Γ z_v(Γ)+κ z(Γ) v, where z_v(Γ)=z'(Γ;v ∇ b_Γ). Hence (<ref>) yieldsdJ_v (Γ,u) = ∫_Γ z_v'(Γ,u)+(κ'(Γ,u) z(Γ) + κ(Γ) z'(Γ,u))v+ κ(Γ) z(Γ)v'(Γ,u)+ (z_v(Γ)+ κ(Γ) z(Γ) v) κ(Γ) u, where u=u·∇ b_Γ. Since v does not depend on Γ, Definitions <ref> and <ref> yield v'(Γ, u)=v̇(Γ,u) -v·u = ∇ v ·u -v ·u = ∂ v/∂ nu. Recall from (<ref>) that κ'(Γ, u)=- u -u |n|^2. Using the second invariant i_2(Γ)=1/2(κ^2 - |D_Γ|^2), we can write dJ_v (Γ,u)=∫_Γ 2i_2(Γ) z(Γ) uv +v (z_u(Γ)κ(Γ)- uz(Γ))+ u κ(Γ)(∂ v/∂ n z(Γ) +z_v(Γ)) + z_v'(Γ, u).We now apply formula (<ref>) to two examples of boundaryfunctionals which make use of the results of previous sections to obtainuseful formulas for J_v(Γ) and dJ_v(Γ, u).§.§.§ Area Functional For the area functional J(Γ)= ∫_Γ dΓ, we havez(Γ)≡ 1, z_v(Γ) ≡ 0 and z_v'(Γ,u) ≡ 0. ThenJ_v(Γ)=∫_Γκ(Γ) v and by (<ref>)dJ_v(Γ, u)=∫_Γ 2i_2(Γ) u v + v· u + u∂ v/∂ nκ(Γ),where we have used an integration by parts formula, to replace ∫_Γ - u v by ∫_Γ u · v.§.§.§ Willmore functionalFor the Willmore functionalJ(Γ)=∫_Γ1/2κ(Γ)^2 we havez(Γ)=1/2κ(Γ)^2 and by the product rule for shape derivatives (Remark <ref>) z_v(Γ)= z'(Γ,v∇ b_Γ)=κ(Γ) κ'(Γ,v∇ b_Γ) = -κ(Γ) ( v +v I_2(Γ)).In order to apply formula (<ref>) we need to compute z_v'(Γ,u) = -κ'(Γ,u)( v +v I_2(Γ)) - κ(Γ) (( v)'(Γ,u) +v'(Γ,u) I_2(Γ)+v I_2'(Γ,u)) .Recall that κ'(Γ,u)=- u -u |n|^2, I_2'(Γ,u)=-2(^2u : n+u I_3(Γ)) by Proposition <ref>,v'(Γ,u)=u∂ v/∂ n, and that the shape derivative of v is, by Theorem<ref>,( v)'(Γ, u)= (u∂ v/∂ n)-u(2n:^2v +κ· v) + u ·( κ v - 2nv).Putting all these ingredients together we obtaindJ_v(Γ,u) =∫_Γ i_2(Γ)κ(Γ)^2 uv- κ(Γ)^2 u v- 2κ(Γ)^2 |D_Γ|^2 u v- κ(Γ)^2/2u v+ κ(Γ)^3/2u ∂ v/∂ n - κ(Γ)^2 u v + (u + | D_Γ|^2 u) (v + | D_Γ|^2 v)-κ(Γ) (u ∂ v/∂ n) + 2κ(Γ) u D_Γ^2v:D_Γ+ κ(Γ) uv ·κ(Γ) - κ(Γ)^2u · v + 2 κ(Γ)u · D_Γ v - κ(Γ) | D_Γ|^2 u ∂ v/∂ n + 2 κ(Γ) v D_Γ^2u : D_Γ + 2 κ(Γ)I_3(Γ) u v.plain | http://arxiv.org/abs/1708.07440v1 | {
"authors": [
"Aníbal Chicco-Ruiz",
"Pedro Morin",
"M. Sebastian Pauletti"
],
"categories": [
"math.OC",
"math.NA",
"65K10, 49M15, 53A10, 53A55"
],
"primary_category": "math.OC",
"published": "20170824144112",
"title": "The shape derivative of the Gauss curvature"
} |
The Status and Initial Results of the Majorana Demonstrator Experiment [ December 30, 2023 ====================================================================== We consider compactness characterizations of large cardinals.Based on results of Benda <cit.>, we study compactness for omitting types in various logics.In _κ, κ, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic.In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle. § INTRODUCTION Large cardinals typically have many equivalent formulations: elementary embeddings, ultrafilters or systems of ultrafilters, combinatorial properties, etc.We investigate various characterizations in terms of logical compactness.These formulations have a long history.Weakly and strongly compact cardinals were first isolated with generalizations of the compactness theorem to infinitary languages. [<cit.>] * κ > ω is weakly compact iff every <κ-satisfiable theory of size κ in _κ, κ is satisfiable. * κ > ω is strongly compact iff every <κ-satisfiable theory in _κ, κ is satisfiable. Measurable cardinals also have such a characterization, this time in terms of chain compactness of _κ, κ.This result is interesting because it seems to have been well-known in the past (evidenced by the fact that it appears as an exercise in Chang and Keisler's Model Theory <cit.>), but seems to have fallen out of common knowledge even among researchers working in the intersection of set theory and model theory (at least among the younger generation)[This is based on the author's personal impressions.Although the statement seems forgotten, the proof is standard: if T = ∪_α < κ T_α and M_α T_α, then set M_κ : = ∏ M_α /U for any κ-complete, nonprincipal ultrafilter U on κ.Łoś' Theorem for _κ, κ implies M_κ T.].κ is measurable iff every theory T ⊂_κ, κ that can be written as a union of an increasing κ-sequence of satisfiable theories is itself satisfiable. Magidor <cit.> showed that extendible cardinals are the compactness cardinals of second-order logic, and Makowsky <cit.> givesan over-arching result that Vopěnka's Principle is equivalent to the existence of a compactness cardinal for every logic (see Fact <ref> below).This seems to situate Vopěnka's Principle as an upper bound to the strength of cardinals that can be reached by compactness characterizations.However, this is not the case.Instead, a new style of compactness is needed, which we call compactness for omitting types (Definition <ref>).Recall that a type p(x) in a logicis a collection of -formulas in free variable x.A model M realizes p if there is a ∈ M realizing every formula in p, and omits p if it does not realize it.Explicitly, this means that for every a ∈ M, there is ϕ(x) ∈ p such that M ϕ(a).Although realizing a type incan be coded in the same logic by adding a new constant, omitting a type is much more difficult.Thus, while in first-order logic types can be realized using a simple compactness argument, finding models omitting a type is much more difficult.This is the inspiration for Gerald Sacks' remark[Several colleagues have suggested that Sacks is quoting himself here, as he considers himself a well-known recursion theorist and not a well-known model theorist.] <cit.> “A not well-known model theorist once remarked: `Any fool can realize a type, but it takes a model theorist to omit one.' ”Compactness for omitting types was first (and seemingly uniquely) used by Benda to characterize supercompact cardinals in _κ, κ. Tragically, Benda's fantastic result is even less well-known than the characterization of measurable cardinals.In the almost 40 years since its publication, the publisher reports zero citations of Benda's paper.Let κ≤λ.κ is λ-supercompact iff for every _κ, κ-theory T and type p(x, y) = {ϕ_i(x, y) | i < λ}, if there are club-many s ∈_κλ such that there is a model of T ∪{∃ x ( ⋀_i ∈ s∃ y ϕ_i(x,y) ∧∃ y ⋀_i ∈ sϕ_i(x, y)) }then there is a model of T ∪{∃ x ( ⋀_i < λ∃ y ϕ_i(x,y) ∧∃ y ⋀_i < λϕ_i(x, y)) }The final model omits the type q(y) = {ϕ_i(a, y) | i < λ}, where a is the witness to the existential.Phrased in these terms, this property says that if every small part of a type can be omitted, then the whole type can be omitted.One complicating factor is that monotonicity for type omission works in the reverse direction as for theory satisfaction: larger types are easier to omit since they contain more formulas.This makes Benda's result somewhat awkward to phrase as he fixes the theory.Our phrasing of compactness for omitting types varies from Benda's formulation in two key ways.First, we also allow the theory to be broken into smaller pieces, which makes the phrasing more natural (at least from the author's perspective).Second, and more crucially, we look at other index sets (or templates) than _κλ.This allows us to capture many more large cardinals than just supercompacts.Other model-theoretic properties have also been used to characterize large cardinals, mainly in the area of reflection properties and the existence of Löwenheim-Skolem-Tarski numbers.Magidor <cit.> characterizes supercompacts as the Löwenheim-Skolem-Tarski numbers for second-order, and Magidor and Väänänen <cit.> explore the possibilities surrounding the Löwenheim-Skolem-Tarski numbers of various fragments of second-order logic.Bagaria and Väänänen <cit.> connect structural reflection properties and Löwenheim-Skolem-Tarski numbers through Väänänen's notion of symbiosis.Of course, Chang's Conjecture has long been known to have large cardinal strength (see <cit.>).Section <ref> fixes our notation and gives some basic results.Section <ref> establishes the main definitions of compactness for type omission and applies it to the logics _κ, κ.The main result in this section is Theorem <ref>.Section <ref> examines type-omitting compactness for higher-order (Theorem <ref>) and sort (Theorem <ref>) logics.This section also deals with compactness characterizations of some other cardinals (e.g., strong) and discusses the notion of elementary substructure in second-order logic.Section <ref> discusses extenders and type omission around the following question: we give characterizations of large cardinals with various logics, from _κ,κ to the all-powerful ^s, Σ_n.However, _κ, κ is already able to work its way up the large cardinal hierarchy, including n-hugeness (Corollary <ref>) and rank-into-rank (Section <ref>).Are these more powerful logics necessary?In other words, can we characterize all model-theoreticaly characaterizable large cardinals (extendible, etc.) by some property of _κ, κ, or is the use of stronger logics necessary to pin down certain cardinals?We focus on strong cardinals, and give a theory and collection of types such that the ability to find a model of the theory omitting the types is equivalent to κ being λ strong.As close as this seems to an _κ, κ (or rather _κ, ω(Q^WF)) characterization of strong cardinals, we still lack a general compactness for type omission for this case.Preliminary results along these lines were first presented at the Workshop on Set-Theoretical Aspects of the Model Theory of Strong Logics hosted by the Centre de Recerca Matemàtica in 2016, and I'd like to thank many of the participants for helpful conversations, especially Jouko Väänänen for discussions about sort logic.I'd also like to thank Gabriel Goldberg for helpful discussions regarding the strength of huge-for-^2_κ, κ cardinals, and Adrian Mathias and Sebastien Vasey for comments on a preliminary draft.I would also like to thank the anonymous referee for helping to improve the paper. § PRELIMINARIESWe begin with an informal introduction to the logics used.The large cardinals notions are standard; consult Kanamori <cit.> or the locally given citation for detail.We introduce some new large cardinal notions, typically naming them and defining them in the statement of a result: see Corollary <ref>, Proposition <ref>,and Theorems <ref> and <ref>._ω, ω is the standard, elementary first-order logic._λ, κ augments _ω, ω by allowing * conjunctions of <λ-many formulas that together contain <κ-many free variables; * <κ-ary functions and relations in the language; and * universal and existential quantification over <κ-many variables at once.We typically restrict to λ≥κ, both regular.^2 = ^2_ω,ω is second-order logic, which extends _ω, ω by allowing quantification over subsets of finite cartesian powers of the universe and has an atomic `membership' relation.The standard interpretation of the second-order quantifiers is quantification over all subsets of (finite cartesian powers of) the universe, but an important concept is the nonstandard Henkin models (M, P, E), where M is a τ-structure, E ⊂ M× P is an extensional relation, and P represents a collection of subsets that the second-order quantifiers can range over.The class of Henkin models of a second-order theory reduces to the models of a sorted _ω, ω-theory, but we will still find use for this definition in Definition <ref> and the characterization of strong cardinals.There is a second-order sentence Ψ that says a Henkin model is standard:∀ X ⊂ M ∃ x ∈ P ∀ y ∈ M ( y ∈ X ↔ y E x)We can also introduce higher-order variants ^n, but these are all codeable in ^2, e.g., a Henkin model for ^3 is (M, P, Q, E), where P represents the subsets of M and Q represents the subsets of M.Then second-order can express the standardness of this model for third-order logic by using Ψ and a copy of Ψ for P and Q.The ^n are coded similarly, although we must use ^2_|α|^+,ω for ^α when α is infinite.Thus, it suffices to talk about second order logic.Additionally, when dealing with second-order logic, we allow the language to include functions and relationswhose domain and range include the second-order part of the model.Given such a second-order language τ, we describe it as consisting of a strictly first-order part (the functions and relations only using first-order inputs) and a second-order part (the functions and relations that use first- and second-order inputs).We want to distinguish a very helpful second-order sentence Φ that will allow us to make a statement about the type of model we are in.The construction of this statement is from Magidor <cit.>, appearing in the proofs of his Theorems 2 and 4.However, it is useful to extract the construction so we can make explicit reference to it. There is a second-order sentence Φ in the language with a single binary relation E such that (A, E) Φ iff (A, E) ≅ (V_α, ∈) for some limit ordinal α.Moreover, Φ is a Π^1_1 sentence. Proof: There is a first-order sentence ϕ(x,y) that is true iff x is an ordinal and y is a copy of V_x (as computed in the model).Then Φ is the conjunction of the statements (note that we use `E' for the relation in the language and `∈' for the logical notion of membership between a first- and second-order variable) * E is well-founded: ∀ X ∃ x ∀ y (y Ex ∧ y ∈ x); * E is extensional: ∀ x,y (x=y↔∀ z (z E x ↔ z E y)); * Every ordinal is in a V_α: ∀ x (On(x) →∃ y ϕ(x,y)); and * Every subset of an element is represented in the model: ∀ x ∀ X ( ∀ y (y ∈ X → y Ex) →∃ z ∀ y (y ∈ X ↔ y E z)). † (Q^WF) is _ω, ω augmented by the quantifier Q^WF that takes in two free variables and so Q^WF xyϕ(x, y, ) is true iff there is no infinite sequence {x_n | n < ω} such that ϕ(x_n+1, x_n, ) holds for all n < ω; that is, ϕ(x, y, ) defines a well-founded relation.Note that, in models of some choice, Q^WF is both _ω_1, ω_1 and ^2 expressible.However, it will be useful to have it, e.g., in Theorem <ref>.Finally, sort logic ^s is a logic introduced by Väänänen <cit.>.This augments second-order logic by adding sort quantifiers ∃^∼, ∀^∼ where ∃^∼ X ϕ(X, ) is true in a structure M iff there is a set X (any set, not just a subset of the universe of M) such that ϕ(X, ) is true.Sort logic is very powerful because it allows one to access a large range of information regardless of the language of the initial structure.For instance, one can easily write down a formula Φ whose truth in any structure implies the existence of an inaccessible cardinal.Väänänen discusses its use as a foundation of mathematics in <cit.>.Since sort logic involves satisfaction of formulas in V, for definability of truth reasons, we must restrict to the logics ^s, Σ_n, where ^s, Σ_n consists only of the formulas of sort logic that are of Σ_n complexity when looking at the quantifiers over sorts.Finally, all of these logics can be combined in the expected way, e.g., ^2_κ, κ.We often take the union of two logics, e.g., ^2 ∪_κ, ω is the logic whose formulas are in ^2 or _κ, ω; however, no second-order quantifier or variable can appear in any formula with an infinite conjunction, which separates it from ^2_κ, ω.We typically use boldfacewhen discussing a particular logic and scriptwhen discussing an abstract logic.For a logicand a language τ, an (τ)-theory T is a collection of sentences (formulas with no free variables) of (τ).An (τ)-type p(x) in x is a collection of formulas from (τ) all of whose free variables are at most x.[Types in single variables suffice for the various characterizations in the paper, but they also extend to types of arity <κ.]A type p(x) is realized in a τ-structure M iff there is an element of the model that satisfies every formula in it and a type is omitted precisely when it is not realized.Note that the “monotonicity of type omission" works the opposite way as theories: if p(x) ⊂ q(x) are both types, then it is easier to omit q than p.We will often refer to filtrations of a theory T.This means there is some ambient partial order (I, ⊂) and a collection of theories {T_s | s ∈ I} such that T = ⋃_s∈ I T_s and s ⊂ t implies T_s ⊂ T_t. In general, we are agnostic about how one codes these logics as sets, except to insist that it is done in a reasonable way, e.g., τ is coded as a set of rank |τ|+ω, _κ, κ(τ) ⊂ V_κ +|τ|, etc.This gives us two nice facts about the interaction between languages τ and elementary embeddings j:V → (or V_α→ V_β, etc.) with j =κ: * if τ is made up of < κ-ary functions and relations, then j"τ and τ are just renamings of each other; and * if ϕ∈^s, Σ_n_κ, κ(τ), then j(ϕ) ∈^s, Σ_n_κ, κ(j"τ).This means that, when searching for a model of T, it will suffice to find a model of j"T, which is a theory in the same logic and an isomorphic language.Given an inner model[In an unfortunate collision of notation, M is commonly used for both inner models in set theory and for τ-structures in model theory.Owing to my model-theoretic roots, this paper uses standard M for τ-structures and scriptfor inner models.](or some V_α), we collect some facts about whenis correct about various logics.That is, the statement “M is a τ-structure" is absolute fromto V and we want to know when the same holds of “ϕ is an (τ)-formula and M_ϕ." *is correct about the logic _ON^, ω(Q^WF). * If ^<κ⊂, thenis correct about the logic _κ, κ. * If (A) ∈, thenis correct about ^2 for structures with universe A. * If ≺_Σ_n V, thenis correct about ^s, Σ_n. As a warm-up, note that any compactness involving an extension of (Q^WF) will entail the existence of large cardinals.Fixing κ, if T:=ED__ω, ω(V_κ+1, ∈, x)_x ∈ V_κ+1∪{c_α < c < c_κ|α < κ}∪{Q^WF xy (x ∈ y)} is satisfiable, then there is a non-surjective elementary embedding j:V_κ+1→ to a well-founded structure with j ≤κ.Standard results imply that j must be measurable.Moreover, T is `locally satisfiable' in the sense that, if T_0 ⊂ T does not contain constants for elements with ranks unbounded in κ, then V_κ+1 can be made a model of T_0 by adding constants.§ TYPE-OMITTING COMPACTNESS IN _∞, ∞We introduce some basic definitions that will be used in each of our characterizations.The notion of containing a strong κ-club becomes very important, and we discuss that concept after the definition.Let κ≤λ and I ⊂(λ). * I is κ-robust iff for every α < λ, [α]_I:={ s ∈ I |α∈ s}≠∅ and I⊂{s : |s ∩κ|<κ}. * C ⊂ I contains a strong κ-club iff there is a function F:[λ]^2 →_κλ such that C(F):= {s ∈ I| sis infinite and, for all x, y, ∈ s, F(x, y) ⊂ s}⊂ C Let U be an ultrafilter on I. (3) U is μ-complete iff for all α < μ and {X_β∈ U |β < α}, we have ⋂_β<α X_β∈ U. (4) U is fine iff for all α∈λ, [α]_I ∈ U. (5) U is normal iff for all F:I →λ such that {s∈ I| F(s) ∈ s}∈ U, there is α_0<λ such that {s ∈ I | F(s) = α_0}∈ U. The conditions of κ-robustness are intended to make sure that I includes enough sets so that the notion of a “κ-complete, normal, fine ultrafilter on I" makes sense and is possible.In particular, any such ultrafilter U will be characterized by an elementary embedding j_U with j_U = κ; this implies that {s ∈ I : |s ∩κ |<κ}∈ U.We define the notion of `contains a strong κ-club' without defining the notion of a strong κ-club.However, thefirst two items of Fact <ref> show that this notion correctly generalizes the notion of containing a club from κ and _κλ.Moreover, the third item shows that a generalization that replaces _κλ with a different set does not work. * If I = _κλ, then containing a strong κ-club is equivalent to containing a club. * Fix a κ-robust I ⊂(λ).If U is a κ-complete, fine, normal ultrafilter on I, then it extends the contains a strong κ-club filter, that is, C(F) ∈ U for all F:[λ]^2→_κλ. * If I = [λ]^κ and U is a κ-complete, fine, normal ultrafilter on I, then there is no s ∈ [λ]^κ such that [s]_I:={t ∈ I | s ⊂ t}∈ U Proof: (1) is a result of Menas, see <cit.>.For (2), fix F: [λ]^2 →_κλ and suppose that C(F) ∉U.For each s ∈ I- C(F), there are α_s < β_s ∈ s such that F(α_s, β_s) ⊄s.By applying normality twice, there is some Z ∈ U and α_* < β_* such that, for all s ∈ Z, α_s = α_* and β_s = β_*.By the κ-completeness and fineness of U, we have that [f(α_*, β_*)]_I∈ U.Thus, there is t ∈ Z ∩ [F(α_*, β_*)]_I; however, this is a contradiction.For (3), given such a U, build the elementary embedding j_U: V →≅∏ V/U.Let s ∈ [λ]^κ.Then, for X ⊂ [λ]^κ, we have that X ∈ U iff j_U”λ∈ j_U(X).However, since |s|=κ =j_U, there is some α∈ j_U(s) - j_U"λ.In particular, this means thatj_U”λ∉{ t ⊂ j(I) | j_U(s) ⊂ t} = j_U([s]_I)† We are interested in model-theoretic conditions that guarantee the existence of a fine, normal, κ-complete ultrafilter on some κ-robust I.Recall that, from Kunen's proof of the inconsistency of Reinhardt cardinals, every countably complete ultrafilter on (λ) must concentrate on _μλ for some μ≤λ.If μ is strictly larger than the completeness of the ultrafilter, we will need the following technical condition.In practice, the set X will always be a theory. Let I ⊂(λ) and X be a set that is filtrated as an increasing union of {X_s | s ∈ I}.Then we say this filtration respects the index iff there is a collection {X^α|α∈λ} such that, for each s ∈ I, X_s = ⋃_α∈ s X^α. This condition says that the filtration at s ∈ I is just determined by the elements of s.Note this condition is trivially satisfied when I ⊂_κλ, but will be important in certain cases, e.g., to characterize huge cardinals (see Corollary <ref>.(<ref>)).There, it will guarantee that if ϕ∈∪_s ∈ [λ]^κ T_s, then there are a large number of s ∈ I such that ϕ∈ T_s.The main concept of this section is the following: Letbe a logic (in the sense of Barwise <cit.> or taken without formal definition), κ≤λ, and I ⊂(λ) be κ-robust.Then we say thatis I-κ-compact for type omission iff the following holds:Suppose that we have * a language τ; * a (τ)-theory T that can be written as an increasing union of {T_s :s ∈ I} that respects the index; and * a collection of types {p^a(x) | a ∈ A} of size λ (for an arbitrary set A), where each type comes with an enumeration p^a(x) = {ϕ^a_α(x) : α < λ} and, for s ⊂λ, we set p^a_s(x):={ϕ^a_α(x):α∈ s}.IfCTO{ s ∈ I | T_shas a model omitting each type in {p^a_s(x) | a ∈ A}}contains a strong κ-club, then there is a model of T omitting each type in { p^a(x) | a ∈ A}. This definition is technical, so we will try to unpack it.A compactness for type omissions result should say that if we want to find a model of a theory T omitting a type p, it should suffice to find a model of all of the small fragments of T omitting the small fragments of p.When there is no p, the monotonicity of satisfying theories makes looser statements of this possible.However, the monotonicity of satisfying theories and omitting types work in opposite directions (smaller theories are easier to satisfy, while smaller types are harder to omit), so much more care has to be taken.The s ∈ I index allows us to associate particular fragments of the theory with particular fragments of the type.In this sense, wether or notthe set (<ref>) contains a strong κ-club depends very strongly on the particular filtration of T and enumerations of the p^a that are chosen.Additionally, because the monotonicities work in opposite directions, given s ⊂ t ∈ I, wether or not s and t are in (<ref>) is independent.As explained in Remark <ref>, we want the set (<ref>) to be in whatever fine, normal, κ-complete ultrafilter our assumptions give us, and the results of Fact <ref> suggest that `contains a strong κ-club' is the right stand in for this notion.The following gives a framework result for linking compactness for type omission for _κ, λ to various large cardinal notions.Let κ≤λ, and I ⊂(λ) be κ-robust.The following are equivalent: * _κ, ω is I-κ-compact for type omission. * _κ, κ is I-κ-compact for type omission. *There is a fine, normal, maximally κ-complete ultrafilter on (λ) concentrating on I; `maximally κ-complete' means κ-complete and not κ^+-complete. *There is an elementary j:V → with j = κ and j"λ∈∩ j(I).Moreover, the first μ such that _μ, ω is I-κ-compact for type omission is the first μ with a fine, normal, μ-complete ultrafilter on I. Proof: The equivalence of (3) and (4) is straightforward from standard methods, and (2) implies (1) is obvious because _κ, ω⊂_κ, κ. (4) → (2): Suppose we have a set-up for I-κ-compact type omission: a theory T in _κ, κ(τ) filtrated as T_s for s ∈ I and a collection {p^a:a ∈ A} of _κ, κ(τ)-types as in Definition <ref>.Set T̅ and p̅^̅a̅ to be the functions that take s ∈ I to T_s and p_s^a, respectively.Similarly, set M̅ to be the partial function that takes s to M_s that models T_s and omits each p^a_s(x) (when defined).Since the set (<ref>) contains a strong κ-club, there is a function F:[λ]^2 →_κλ that witnesses this.Then j(M̅) is a function with domain containing j(C(F)).We know that j comes from an ultrafilter as in (3) and Fact <ref>.(2) implies that C(F) ∈ U.Thus, j"λ∈ j(C(F)).Thenthinks that j(M̅)(j"λ) is a j(τ)-structure that models j(T̅)(j"λ) that omits each j(p̅^a)(j"λ). is correct about this, so j(M̅)(j"λ) models j(T̅)(j"λ) and omits each j(p̅^a)(j"λ) (in V). As discussed in Section <ref>, τ and j"τ are essentially the same, in the sense that there is a canonical bijection between them–taking F ∈τ to j(F)∈τ–that respects arity; such a function is called a renaming.Claim: j"T ⊂ j(T̅)(j"λ) and j"p^a = j(p̅^a)(j"λ) for each a ∈ A.First, let j(ϕ) ∈ j"T and let {T_0^β:β∈λ} witness that the filtration respects the index; this is the crucial use of this property.Then we have that ϕ∈ T_0^α for some α∈λ.Then j(ϕ) ∈ j(T̅_̅0̅)(α) ⊂ j(T̅)(j"λ)The proof that j"p^a ⊂ j(p̅^a)(j"λ) is similar.For the other direction, let ψ∈ j(p̅^a)(j"λ).By elementarity, we have that j(p^a) = {ψ^a_α(x): α < j(λ)} and j(p̅^a)(s) = {ψ^a_α(x):α∈ s}, where ψ^a_j(β) = j(ϕ^a_β).This means that ψ is of the form j(ϕ) for ϕ∈ p^a, which exactly says ψ∈ j"p^a.Here, we have exactly used the condition that j"λ is in the model, and simply using a set that contained this would not work.With the claim proved, we have produced a model j(M̅)(j"λ) of j"T that omits each j"p^a for a ∈ A.After applying the inverse of the canonical naming above, we have the desired model of T that omits each p^a. (1) → (3): We want to write down a theory T and collection of types {p_F:F:I →λ} such that a model of T omitting these types will code the desired ultrafilter.Set τ = {P, Q, E, c_X, d}_X ⊂ I with P and Q unary predicates, E ⊂ P × Q a binary relation, and c_X, d constants.We look at the standard structure M = I, (I), ∈, X_X ⊂ I that has no interpretation for d.Classic results tell us that finding an extension N of M that a new element n (which will be the interpretation of d) codes an ultrafilter U on I byX ∈ UNn E c_XThe completeness of this ultrafilter comes from how strong the elementarity of the extension is, so we need to add conditions that make the fineness and normality hold.Set T_0 = __κ, ω(M) (although much less is necessary).Set T^α := T_0 ∪{“d ∈ c_[α]^I"} for α < λ; T_s := ∪_α∈ s T^α for s ∈ I; and T = ∪_s ∈ I T_s.For each function F:I →λ, define X_F := {s ∈ I| F(s) ∈ s}X_F, α := {s ∈ I | F(s) = α}p_F(x):= { x = d ∧ x E c_X_F∧(x E c_X_F, α) |α < λ} Γ := { p_F | F:I →λ}These types are built so that omitting the type p_F means that the ultrafilter coming from d will be normal with respect to the function F: if F is regressive on a large set, then “dEc_X_F” will hold, so omitting p_F requires that there is an α < λ such that “d E c_X_F, α” holds, and α will then give the constant value of F on a large set.We will use compactness for type omission to find a model of T omitting Γ.Claim 1: If there is a model of T omitting Γ, then there is a fine, normal, κ-complete ultrafilter on I.Let N be this model.Define U on I by X ∈ UN d E c_XIt is straightforward to check that U is a κ-complete ultrafilter on I.For instance, given X_α∈ U |α < μ < κ, we know that the following sentence is in T:∀ x ( ⋀_α < μ x E c_X_α→ x E c_∩_α<μ X_α)Thus, Nd E c_∩_α<μ X_α.Given α < λ, by κ-robustness, there is some s ∈ I such that α∈ s.So T entails that d E c_[α]^I. This means that U is fine.For normality, if F:I →λ is regressive on a U-large set, then Nd E c_X_F.Since N omits p_F, there is α < λ such that X_F, α∈ U, so U is normal.Claim 2: For each s ∈ I, there is a model of T_s omitting {p_F, s| F:I →λ}.Expand M to M_s by interpreting d^M_s = s.This models T_s since s ∈ [α]^I for each α∈ s by definition.Moreover, if there is x ∈ M_s such thatM_s “x = d ∧ x E X_F"for some F:I →λ, then x = s and F(s) ∈ s, so there is α∈ s such that F(s) = α.ThusM_sx E X_F, αSo M_s omits each p_F, s.By the I-κ-compactness for type omission, we are done. The proof of the “moreover" follows similarly.†We chose to show (4)→(2) above because it will more easily generalize to large cardinals characterized by the existence of extenders rather than ultrafilters (this is done in Section <ref>).However, the direct proof of (3)→(2) helps to emphasize the connection between normality of an ultrafilter and type omission, so we outline it here.Suppose that U is an ultrafilter as in Theorem <ref>.(3) and we have a language τ, theory T, and set of types {p^a(x) | a ∈ A} as in the hypothesis of Definition <ref>, and let X be the set in <ref>.For s ∈ X, there is a witnessing model M_s and we can form the ultraproduct[Here we use a more liberal formalism for ultraproducts that allow certain structures to be empty by allowing the choice functions f:I →⋃ M_s to be partial with U-large domain.] ∏ M_s/U.From Łoś' Theorem, the κ-completeness and fineness of U means that ∏ M_s/U models T.To prove type omission, suppose that [f]_U ∈∏ M_s/U and let a ∈ A.Since M_s omits p^a_s(x), there is some α_s ∈ s such thatM_s ϕ^a_α_s(f(s))The function s ↦α_s is regressive on the U-large set f, so normality implies that there is a single α_* such that M_s ϕ^a_α_*(f(s))for a U-large set of s.Then, by Łoś' Theorem, ∏ M_s/U ϕ^a_α_*([f]_U)so [f]_U doesn't realize p^a.Since [f]_U and a were arbitrary, the ultraproduct omits the types {p^a(x) | a ∈ A}.As an example of the moreover, _ω, ω satisfies Benda's supercompactness theorem iff _ωλ carries a fine, normal measure that need not even be countably complete.Note that _ω, ω can never be _ωλ-ω-compact for type omission: the max function shows that no fine ultrafilter on _ωλ can be normal.This general framework directly gives model theoretic characterizations of large cardinals that are characterized by normal ultrafilters.For each numbered item below, all of its subitems are equivalent: * * κ is measurable. * _κ,κ is _κκ-κ-compact for type omission. * * κ is λ-supercompact. * _κ,κ is _κλ-κ-compact for type omission. * * κ is huge at λ; that is, κ is huge and there is j:V→ witnessing this with j(κ)=λ. * _κ,κ is [λ]^κ-κ-compact for type omission. * * κ is n-huge at λ_1, …, λ_n. * _κ,κ is { s ⊂λ : ∀ i < n, |s ∩λ_i+1| = λ_i}-κ-compact for type omission. Proof: The proof follows the standard characterizations of these notions in terms of normal ultrafilters (see <cit.>) and from Theorem <ref>.For instance, in (<ref>), if κ is huge at λ, then there j:V → with j = κ with ^j(κ)⊂ and λ = j(κ).In particular, this means that j"λ∈ and belongs to j([λ]^κ) = ([j(λ)]^j(κ))^.Using Theorem <ref>.(4)→(2), _κ, κ is [λ]^κ-κ-compact for type omission.In the other direction, assume that _κ,κ is [λ]^κ-κ-compact for type omission.By Theorem <ref>.(2)→(3), there is a fine, normal κ-complete ultrafilter on I concentrating on [λ]^κ.By <cit.>, κ is huge at λ.† Item (<ref>) is Benda's supercompactness theorem (Theorem <ref>). Item (<ref>) can be reformulated along the lines of Fact <ref>: If T = ∪_α < κ T_α is an _κ, κ(τ)-theory and p(x) = {ϕ_i(x) | i < κ} is a type such that for every α < κ, there is a model of T_α omitting {ϕ_i (x) | i < α}, then there is a model of T omitting p.This helps highlight the impact that the normality of an ultrafilter has on the resulting ultraproduct: if U is a normal ultrafilter on I ⊂(λ) and {M_s | s ∈ I} are τ-structures, then ∏ M_s/U omits any type p = {ϕ_α(x) |α < λ} such that {s ∈ I | M_somitsp_s}∈ U. As mentioned above, any ultrafilter on (λ) concentrates on some _μλ.We can characterize when an ultrafilter exists on some _μλ with the following large cardinal notion: Fix κ≤μ≤λ.The following are equivalent: * κ is λ-supercompact with μ clearing: there is j:V → with j"λ∈ and j(μ) > λ. * _κ,κ is _μλ-κ-compact for type omission.Proof: This follows from Theorem <ref>: j(μ) > λ ensures that j"λ∈ j(_μλ).† This equivalence even holds for κ=ω, where both conditions fail for all μ≤λ.One feature of the compactness schema are that the theories are not required to have a specific size, but rather should be filtrated by a particular index set.Note this is also true for strongly compact cardinals; that is, rather than characterizing λ-strongly compact cardinals as compactness cardinals for λ-sized theories in _κ, κ, we can give the following. κ is λ-strongly compact iff any _κ, κ-theory T that can be filtrated as an increasing union of satisfiable theories indexed by _κλ is itself satisfiable. This can even be extended to theories of proper class size.Each item of Corollary <ref> remains true if compactness for type omission is generalized to allow for the T and the T_s in Definition <ref> to be definable proper classes and to allow for satisfaction by definable proper class structures.The proof of Theorem <ref> goes through with this generalization. ED_(V, ∈, x)_x ∈ V is used below to denote the -elementary diagram of the structure with universe V, a single binary relation ∈, and a constant for each x that is interpreted as x.Formally, from Tarski's undefinability of truth, this is not a definable class whenextends _ω, ω.Similarly, the statement that j:V → is elementary is not definable.However, e.g., <cit.> shows that, for embeddings between inner models, Σ_1-elementarity implies Σ_n-elementarity for every n < ω.Thus, mentions of elementary embeddings with domain V can be replaced by Σ_1-elementarity.Similarly, the full elementary diagram of V could be replaced by its Σ_1-counterpart.However, following set-theoretic convention, we continue to refer to the full elementary diagram. Armed with a class version of omitting types compactness, we can show equivalences directly between the model-theoretic characterizations and the elementary embedding characterizations without working through an ultrafilter characterization.At each stage, we find a modelof ED__κ, κ(V, ∈, x)_x ∈ V along with the sentences {c_α < c < c_κ|α < κ}, where c is a new constant, and some other sentences.Such a model is well-founded because it models the _ω_1, ω_1-sentence that truthfully asserts well-foundedness, so we can take the Mostowski collapse π: ≅ withtransitive.Then x ↦π(c_x) is an _κ, κ-elementary embedding that necessarily sends α <κ to itself.Moreover, the interpretation of cguarantees that the critical point is at most κ and the use of _κ, ω guarantees the critical point is at least κ.This is enough to show κ is measurable and extra sentences to be satisfied and types to be omitted can be added to characterize the above large cardinal notions. For each numbered item below, all of its subitems are equivalent: * * κ is measurable. * There is a definable class model of ED__κ, κ(V, ∈, x)_x ∈ V∪{c_α < c < c_κ|α < κ}* * κ is λ-strongly compact. * There is a model of ED__κ, κ(V, ∈, x)_x ∈ V∪{c_α < c < c_κ|α < κ}∪{c_α∈ d ∧ |d| < c_κ|α < λ}* * κ is λ-supercompact. * There is a definable class model of ED__κ, κ(V, ∈, x)_x ∈ V∪{c_α < c < c_κ|α < κ}∪{c_α∈ d ∧ |d| < c_κ|α < λ} that omits p(x) = { x E d ∧ x ≠ c_α|α < λ}* * κ is n-huge at λ_1, …, λ_n. * There is a definable class model of ED__κ, κ(V, ∈, x)_x ∈ V∪{c_α < c < c_κ|α < κ}∪{c_α∈ d_i+1∧ |d_i+1| = c_λ_i|α < λ_i+1, i < n} that omits, for i < n, p_i(x) = { x E d_i+1∧ x ≠ c_α|α < λ_i+1}Proof: We prove item (3); the rest of the proofs are similar.First, suppose that κ is λ-supercompact.Then there is j:V → with j=κ, j(κ)>λ, and j"λ∈κ.We claim that (the definable)is our model after expanding the vocabulary: * c_x^ = j(x) for x ∈ V; * c^ = κ; and * d^ = j"λ.The elementarity of j and the closure under κ-sequences ofimply that (V, ∈, x)_x ∈ V≡__κ, κ(∏ V/U, E, [s ↦ x]_U)_x ∈ V≅(, ∈ j(x))_x ∈ VThe other sentences of the theory are true because j(α) < κ < j(κ) for α < κ precisely means j =κ, and we know that j(α) ∈ j"λ for all α < λ, with |j"λ|=λ <j(κ).Second, suppose that we have such a definable class model (, E, a_x, a, b)_x ∈ V.Well-foundedness is _ω_1, ω-expressible[Note that the first-order Axiom of Foundation can hold in ill-founded classes.]:∀x_n:n<ω⋀_n<ω x_n+1 E x_nThus,is well-founded with respect to E.By taking the Mostowski collapse, we can assume thatis transitive and E is ∈.Then define j:V → by j(x) = a_x.On the ordinals, j is increasing.Thus, the second group of sentences tells us that κ≤ a < j(κ), so j ≤κ.On the other hand, for each α < κ, V models the _κ, ω-sentence∀ x (x E c_α→⋁_β<α x = c_β)We can use this to show that c_α = α for every α < κ using induction.Thus, j = κ.Now we claim that b = j"λ.For each α∈λ, we ensure that j(α) E b, so j"λ⊂ b.Given x ∈ b, sinceomits p, we must have x = j(α) for some α < λ.† Note that we didn't directly use compactness for type omission in this proof.However, in each case, the theory has a natural filtration by the appropriate partial order that is easily seen to be locally consistent while omitting the necessary type in ZFC.For instance, in the case of κ being λ-supercompact, for s ∈_κλ, set α_s := otp(s).T_s:=ED__κ, κ(V, ∈, x)_x ∈ (V_≥κ∪ V_α_s)∪{ c_i < c < c_κ| i < α}∪{c_i E d ∧ |d| < c_κ| i ∈ s}p_s(x):= {x E d ∧ x ≠ c_i | i ∈ s}Then V is a model of this theory by interpreting every constant in the language by its index, c as α_s, and d as s.This gives a way to go directly between model-theoretic and elementary embedding characterizations.It also shows that it is enough to omit a single[In the case of n-huge, recall that the omission of finitely many types can be coded by the omission of a single type.] type to obtain the I-κ-type omission for any number of types.The ability to characterize cardinals at the level of huge and above shows that the addition of type omission to attempts to characterize large cardinals is a real necessity.Measurable and strongly compact cardinals have known model-theoretic characaterizations without type omission, so one might wonder if type omission is necessary to characterize huge cardinals.From the following theorem of Makowsky, we can deduce that it is necessary.[<cit.>] The following are equivalent: * Every logichas a strong compactness cardinal; that is, for every logic , there is a cardinal μ_ such that for any language τ and (τ)-theory T, if every T_0 ∈_μ_ T has a model then so does T. * Vopěnka's Principle.Thus, Vopěnka's Principle “rallies at last to force a veritable Götterdammerung" for compactness cardinals for logics[With apologies to Kanamori <cit.>.].Nonetheless, κ being almost huge implies that V_κ satisfies Vopěnka's Principle.Thus, if κ is the first huge cardinal, then V_κ is a model ofVopěnka's Principle + “Every logic is compact, but there are no μ≤λ such that _μ,μ is [λ]^μ-μ-compact for type omission" Indeed, other approaches to model-theoretic characterizations of large cardinals focused solely on compactness or reflection principles have yet to characterize huge cardinals.§ SECOND-ORDER LOGIC AND BEYOND We now turn to characterizations based on logics beyond (or orthogonal to) _∞, ∞.In the spirit of Theorem <ref>, we can characterize compactness for omitting types in second-order logic with a similar theorem.Let κ≤λ and I ⊂(λ) be κ-robust.The following are equivalent: * ^2∪_κ, ω is I-κ-compact for type omission. * ^2_κ, κ is I-κ-compact for type omission. *For every α > λ, there is some j:V_α→ V_β such that j = κ, j(κ)>λ, and j"λ∈ j(I). * For every α > λ, there is some j:V → such that j = κ, j"λ∈ j(I), and V_j(α)⊂.Moreover, the first μ such that ^2 is I-κ-compact for type omission is the first μ that satisfies (<ref>) except with j = μ.Proof: (4) implies (3) and (2) implies (1) are immediate.We show that (1) implies (3) implies (4) implies (2). For (1) implies (3), fix α > λ and consider the ^2 ∪_κ, ω-theory and typeT=ED__κ, ω(V_α, ∈, x)_x∈ V_α∪{c_i < c < c_κ| i < κ}∪{Φ}p(x)= { x E d ∧ x ≠ c_i | i < λ}where d is a new constant symbol and Φ is Magidor's sentence from Fact <ref>.Then, we can filtrate this theory as T_s=ED__κ, ω(V_α, ∈, x)_x∈ V_sup s ∪ [κ,α)∪{c_i < c < c_κ| i < sup s}∪{Φ}p_s(x)= { x E d ∧ x ≠ c_i | i ∈ s}For each s ∈ I, we have that the natural expansion (V_α, ∈, x, s)_x ∈ V_sup s ∪ [κ, α) models T_s and omits p_s.Thus, our compactness principle tells us there is a model of T omitting p, which, after taking the transitive collapse, gives the desired j:V_α→ V_β. For (3) implies (4), fix α≥λ and let α' be the next strong limit cardinal above α.Then there is j:V_α'→ V_β with j =κ and j"λ∈ j(I).Then derive the extender E of length ℶ_j(α) to capture this embedding.Forming the extender power of V and taking the transitive collapse, we get j_E:V →_E with the desired properties. For (4) implies (2), let T̅ = {T_s | s ∈ I} be an increasing filtration of the ^2_κ, κ-theory T that respects the index and {p^a(x) | a ∈ A} be a collection of types indexed as p^a(x) = {ϕ^a_i(x) | i < λ} such that there are a club of s ∈ I with a model M_s that models T_s and omits each p^a_s.Fix strong limit α≥λ to be greater than the rank of these models, their power sets, and the function f that takes each of these s to M_s; form j:V → with j = κ, j"λ∈ j(I) ∩, and V_j(α)⊂.Since the domain of f contains a club, it includes j"λ.Set M_*:= j(f)(j"λ).By the elementarity of j, inside ofwe have that M_*j(T̅)_j"λ and, for each a ∈ A, M_* omits j(p^a)_j"λ = {j(ϕ^a_i) | i < λ} = j"p^a.Since V_j(α)⊂ and M_* < j(α),is correct about this satisfaction.Finally, j"T ⊂ j(T̅)_j"λ because the filtration respects the index.Thus, after a suitable renaming, we have found a model of T omitting {p^a(x) | a ∈ A}.† To aid in the discussion of the implication of this theorem, we introduce the following ad hoc naming convention for large cardinal properties. Suppose a large cardinal property P is characterized by being an I-κ-compactness cardinal for 𝕃_κ,κ.Given a logic , we say that κ is P-for- iffis I-κ-compact for type omission. For instance, Corollary <ref>.(<ref>) characterizes huge as the existence of a λ > κ such that _κ, κ is [λ]^κ-κ-compact for type omission, so saying that κ is huge-for-^2_κ, κ means that there is a λ > κ so ^2_κ, κ is [λ]^κ-κ-compact for type omission.Comparing Theorems <ref> and <ref>, a large difference is that the first-order characterizations are witnessed by a single embedding, while the second-order characterizations require class many embeddings.The reason for this is that a single modelcan be right about _κ, κ everywhere, but cannot be right about ^2 everywhere; otherwise, it would compute the power set of every set correctly and would be V.Similarly, the type omitting compactness does not hold for definable class theories for second-order as it does for first.If it did, one could easily derive an nontrivial embedding j:V → V.The first consequence of Theorem <ref> regards the identity crisis.In the language of Definition <ref>, Magidor has shown that extendible cardinals are exactly those that are strong compact-for-^2_κ,κ <cit.> and additionally shown that the first strongly compact cardinal could be the first measurable or the first supercompact<cit.>.This second result means that various compactness notions for _κ, κ have an imprecise relation to one another: chain compactnes could coincide with compactness, or there could be many chain compact cardinals below the first compactness cardinal.Surprisingly, when moving to ^2, these notions coincide and the identity crisis disappears! The following are equivalent. * κ is measurable-for-^2 ∪_κ,ω. * κ is strongly compact-for-^2_κ,κ. * κ is supercompact-for-^2_κ,κ.In particular, all three of these statements characterize extendible cardinals. Here we take `κ is measurable-for-' as in Fact <ref>.That is, we don't incorporate any type omission; however, the type omission characterization holds as a result of the above.Proof:Clearly, (3) implies (2) implies (1) using (for the first implication) the trivially omitted type {x ≠ x | i < λ}.The condition Theorem <ref>.(3) is clearly stronger than extendibility, so any compactness for ^2_κ, κ (including chain compactness) gives extendibility.In particular, j"κ = κ∈ j(_κκ).So measurable-for-^2 ∪_κ, ω implies extendible.Similarly, the definition of extendibility includes that j(κ) > α.In this case, j"λ has size λ≤α, so j"λ∈ j(_κλ).Thus extendibility implies supercompact-for-^2_κ, κ.†The key to these equivalences is that the condition about j"λ in Theorem <ref>.(<ref>) often had more to do with the closure of the target model (i.e., is j"λ in ?), rather than the nature of the relationship between j(I) and j"λ.When we have extendible-like embeddings, j"λ is always in the target model, so many of the type omitting compactness principles (or even just compactness principles) become trivial.A possible explanation for the collapse of the identity crisis is that type omission in ^n_κ, κ is expressible[This is immediate for ϕ-type omission for fixed ϕ, and any type omission can be coded as ϕ-type omission in an expansion.] in ^n+1_κ, κ, which is again codeable in ^n_κ, κ.Thus, one might expect no difference between strong compact- and supercompact-for-^2_κ, κ.However, this does not explain why measurability coincides with these notions,and the below proposition shows that some notions of type-omitting compactness for ^2 are strictly stronger than extendibility (in consistency strength). * κ is huge at λ-for-^2_κ, κ iff for every α≥λ, there is j:V_α→ V_β such that j = κ and j(κ) = λ. * If κ is almost 2-huge at λ_1, λ_2, then there is a κ-complete, normal ultrafilter on κ containing {α < κ| V_λ_2“α is huge-for-^2_κ, κ"} * If κ is huge-for-^2_κ, κ, then there is a κ-complete, normal ultrafilter on κ containing {α < κ|α is huge} Proof: The first item is just a restatement of Theorem <ref> with I = [λ]^κ.Suppose κ is 2-huge at λ_1, λ_2 and j:V → witnesses this.Fixing α∈ [λ_1, λ_2), j V_α is an embedding from V_α to V_β with j(κ) = λ_1 that is in (V_j(λ_2))^.So(V_j(λ_2))^“∃ j_0:V_α→ V_β such thatj_0 = κ and j_0(κ)=λ_1"Recall that (by arguments of <cit.>), V_λ_2 = (V_λ_2)^≺(V_j(λ_2))^.Thus,V_λ_2“∃ j_0:V_α→ V_β such thatj_0 = κ and j_0(κ)=λ_1"Since α was arbitrary,V_λ_2“∀α≥λ_1, ∃ j_0:V_α→ V_β such thatj_0 = κ and j_0(κ)=λ_1"V_λ_2“κ is huge at λ_1-for-^2_κ, κ"Thus, {α < κ| V_λ_2“α is huge-for-^2_κ, κ"} is in the normal ultrafilter on κ derived from j.Suppose κ is huge at λ-for-^2_κ, κ.Picking α large enough and getting the corresponding j:V_α→ V_β with j(κ) = λ, we can derive a normal, κ-complete, fine ultrafilter U on [λ]^κ.Then U ∈ V_β, so V_β“κ is huge."Thus, {α < κ|α is huge} is in the normal, κ-complete ultrafilter on U generated from j.† Similar results show that n-huge-for-^2_κ, κ lies strictly between n-huge and almost n+1-huge.The preceding argument is due to Gabriel Goldberg, who also reports that he can show that huge-for-^2_κ, κ can be characterized in terms of hyperhugeness. Make a fact, check Gabe's argument.Can we adapt it to get “huge at λ-for-^s, Σ_n_κ, κ implies C^(n)+-hyperhuge?" Recall that κ is λ-hyperhuge iff there is j:V → with j = κ and ^j(λ)⊂ and κ is hyperhuge iff it is λ-hyperhuge for every λ.Hyperhuge cardinals have recently been shown to imply the existence of a minimal inner model of V that can reach V by set-forcing extensions by Usuba <cit.>.Goldberg proves that κ being hyperhuge is equivalent to the existence of a κ_0<κ such that κ_0 is huge at κ-for-^2_κ, κ.Additionally, κ being λ-hyperhuge is equivalent to the existence of μ > λ and a normal, fine, κ-complete ultrafilter on [μ]^λ_*κ:={ s ⊂μ| |s| = λ, |s ∩κ| ∈κ, (s∩λ) < κ}, which is equivalent to _κ, κ being [μ]^λ_*κ-κ-compact for type omission by Theorem <ref>.Examining the proof of Theorem <ref>, we see that a level-by-level characterization of, e.g., α-extendibility is harder due to the tricky nature of the Löwenheim-Skolem number for second-order logics.In first-order, the Löwenheim-Skolem number of _λ, κ for theories of size μ is ((λ+μ)^<κ)^+, which is also its Löwenheim-Skolem-Tarski number.For second-order logic, LS(^2) (for sentences) is the supremum of all Π_2-definable ordinals (Väänänen <cit.>) and LST(^2) is the first supercompact, if one exists <cit.> .However, weak compactness restricts the size of the theory, so admits a more local characterization.Denote the Löwenheim-Skolem number of sentences of ^2_κ,κ by ℓ^2_κ. The following are equivalent for κ. * κ is weakly compact-for-^2 ∪_κ, ω. * κ is weakly compact-for-^2_κ, κ. * Given any κ+1 ⊂⊂ V_ℓ^2_κ of size κ, there is a partial elementary embedding j:V_ℓ^2_κ→ V_β for some β with j = and j = κ.Proof:Clearly, (2) implies (1).We show (1) implies (3) implies (2). Suppose κ is weakly compact-for-^2 ∪_κ, ω and let κ+1 ⊂⊂ V_ℓ^2_κ.Let T be the ^2 theory consisting of * the _κ, ω-elementary diagram ofin V_ℓ^2_κ; * c_i < c < c_κ for i<κ; and * Magidor's Φ from Fact <ref>.Then every <κ-sized subset of T is satisfiable as witnessed by an expansion of V_ℓ^2_κ.By weak compactness, we get a model of T, which must be some V_β.This induces a partial function j:V_ℓ^2_κ→ V_β with j =.Moreover, the elements of T make this a partial elementary embedding with j = κ. Suppose that κ satisfies the embedding property.Let T = {ϕ_i | i < κ} be a ^2_κ, κ(τ)-theory that is <κ-satisfiable with |τ|≤κ.Then, there is a function f with domain κ such that f(α){ϕ_i | i < α} for every α < κ; moreover, by the definition of ℓ^2_κ, we can assume that f(α) ∈ V_ℓ^2_κ.Let ⊂ V_ℓ^2_κ contain all of this information and be of size κ.Then, there is partial elementary j:V_ℓ^2_κ→ V_β with j = and j = κ.In particular, we have that * j(κ) > κ; * V_β“j(f)(κ)j"T" and V_β is correct about this; and * j"T and T are just renamings of the same theory.Thus, the suitably renamed j(f)(κ) witnesses that T is satisfiable.† A key piece in translating weak compactness for second-order into an embedding characterization is the ability to axiomatize well-foundedness.If we look at a fragment of ^2_κ, κ that includes an expression of well-foundedness, then weak compactness for this fragment is characterizable in a similar way, replacing ℓ^2_κ with the Löwenheim-Skolem number of that fragment.However, if the fragment cannot express well-foundedness, then this characterization is harder.Similar results can be proved by restricting the size of the theories under consideration.In the general scheme, the theory T is allowed to be as large as one wants, as are the pieces T_s of the filtration.If one restricts these pieces to be of size ≤μ and wants to characterize ^2_κ, κ being I-κ-compact for type omission, then it suffices to look at an embedding as in Theorem <ref>.(2) for α equal to the Löwenheim-Skolem number of ^2_κ, κ for μ-sized theories.For the characterizations of strong and its variants, we need the concept of a Henkin second-order structure that is full up to some rank.Recall the notion of a Henkin model described in Section <ref>. Let M_* = (M, P, E) be a Henkin structure and A a transitive set. * M_* is full to A iff every X ∈(M) ∩ A is represented in P; this means that there is c_X ∈ P such that, for all y ∈ M, y ∈ XM_*y E c_X * M_* is full up to rank α iff it is full to V_α.While a Henkin structure has a nonstandard interpretation of second-order quantifiers, other additions to the logic must be interpreted standardly.In particular, the next theorem discusses Henkin models of ^2(Q^WF)-theories; while any second-order assertions of well-foundedness–i.e., “∀ X ∃ y ∀ z (y ∈ X ∧ z ∈ X → y R z)"–can be satisfied non-standardly, any Q^WF assertions of well-foundedness–i.e., Q^WFxy(xRY)–must be correctly interpreted (and so R is well-founded). The following are equivalent for κ≤λ. * κ is λ-strong. *If T ⊂^2_κ, ω(Q^WF)(τ) is a theory that can be written as an increasing union T = ∪_α < κ T_α such that every T_α has a (full) model, then T has a Henkin model whose universe is an ordinal and is full up to rank λ. *Same as (<ref>), but there is also a type p={ϕ_i(x) | i <κ} such that T_α has a (full) model omitting p_α, and the resulting model omits p.Note that we add the condition on the universe of the model in (<ref>) to remove the possibility that the “full up to rank λ" condition is vacuous; if the universe of M just consists of elements of rank bigger than λ, then M is trivially full up to rank λ.Proof: First, suppose that κ is λ-strong and let T be a theory and p a type as in (<ref>).We produce a model of T in the standard way: let f be a function with domain κ such that f(α) is a model of T_α.WLOG, f(s) is a full Henkin structure.Then, in , j(f)(κ) is a model of (a theory containing) j"T and “j(f)(κ) is a full Henkin structure that omits j(p)_κ = j"p". is incorrect about second-order satisfaction above rank λ; however, since V_λ⊂, it is correct about second-order satisfaction up to rank λ. Second, suppose we have compactness.Then we wish to build an embedding witnessing strength.By the normal arguments, e.g. <cit.> or see Proposition <ref>, it is enough to derive a (κ, ℶ_λ)-extender from an embedding j: V_κ+2→ with j = κ, V_λ⊂, andwell-founded.We can find such a model by considering the theoryED__κ, ω(V_κ+2, ∈, x)_x ∈ V_κ+2∪{c_i < c < c_κ| i < κ}∪{Φ}∪{Q^WFxy (x E y)}This can be written as an increasing κ-length union of satisfiable theories in the standard way and any model leads to, after taking transitive collapse, the necessary j:V_κ+2→. † We could ask for a variation of (<ref>) that allows for arbitrary κ-satisfiable theories or, equivalently, theories indexed by some _κμ.This would be equivalent to a jointly λ-strong and μ-strongly compact cardinal: there is a j:V → such that j = κ, V_λ⊂, and there is Y ∈ j(_κμ) such that j"μ⊂ Y.If we drop the Q^WF, then we can characterize a weakening of λ-strong.In the following theorem and proof, we break the convention thatalways denotes some transitive model of a fragment of ZFC.In particular, we allow it to be ill-founded.For such models, wfp() denotes the well-founded part of .The following are equivalent for κ≤λ. * κ is non-standardly λ-strong: there is an elementary embedding j: V→ withnot necessarily transitive such that j = κ and V_λ⊂ wfp(). * If τ is a language and T ⊂^2_κ, ω is a theory that can be written as an increasing, continuous union T = ∪_α < κ T_α such that every T_α has a (full) model, then T has a Henkin model whose universe is an ordinal and is full up to rank λ.Proof: The proof is the same as Theorem <ref>, with the changes exactly that we no longer insist on being correct regarding statements about well-foundedness.† An argument of Goldberg shows that the level-by-level notions of non-standardly λ-strong and λ-strong are not equivalent, but full non-standard strong is equivalent to strong. §.§ C^(n) and sort logic Moving to sort logic, we can prove a metatheorem along the lines of Theorems <ref> and <ref> by introducing the notion of a C^(n)-cardinal.The C^(n) variants of large cardinals were introduced by Bagaria <cit.>.Briefly, set C^(n) = {α∈| V_α≺_Σ_n V}, where ≺_Σ_n is elementarity for Σ_n formulas in the Lévy hierarcy (in the language of set theory).For a large cardinal notion P witnessed by a certain type of elementary embedding, κ is C^(n)-P iff there is an elementary embedding j witnessing that κ is P and so j(κ) ∈ C^(n).Recently, Tsaprounis <cit.> and Gitman and Hamkins <cit.> have independently shown that C^(n)-extendibility is equivalent to the a priori stronger notion of C^(n)+-extendibility: κ is C^(n)+-extendible iff for all α > κ in C^(n), there j:V_α→ V_β with j = κ and j(κ), β∈ C^(n).It is the notion of C^(n)+-extendibility that we will use.For some large cardinal notions, there is no increase of strength from moving to the C^(n)-versions (measurable, strong <cit.>, strongly compact <cit.>), but several other notions give an increasing hierarchy of strength. Recall the notions of sort logic described in Section <ref>Let κ≤λ, n < ω, and I ⊂(λ) be κ-robust.The following are equivalent: * ^s, Σ_n∪_κ, ω is I-κ-compact for type omission. * ^s, Σ_n_κ, κ is I-κ-compact for type omission. * For every α≥λ in C^(n), there is some j:V_α→ V_β such that j = κ, j"λ∈ j(I), and β∈ C^(n). * For every α≥λ in C^(n), there is some j:V → such that j = κ, j"λ∈ j(I) ∩,V_j(α)⊂, and j(α) ∈ C^(n).The proof of Theorem <ref> follows the structure of Theorems <ref> and <ref>.To make the necessary changes, we introduce the following notion and lemma.Given a Σ_n formula ϕ() (in the Lévy hierarchy), let ϕ^∼() ∈^s, Σ_n be the same formula where unbounded quantifiers are replaced with the corresponding sort quantifiers.This allows us to characterize C^(n) as follows. Let α be an ordinal.α∈ C^(n) iff V_α models{∀(ϕ() ↔ϕ^∼()) |ϕ is Σ_n}Proof: For ∈ V_α, we always have ϕ() holds in V iff V_αϕ^∼().The above theory makes this equivalent to V_αϕ(). † Proof of <ref>:We sketch the proof and highlight the changes from the proof of Theorem <ref>. Given the compactness, we prove (3) by considering the theory and typeT=ED__κ, ω(V_α, ∈, x)_x∈ V_α∪{c_i < c < c_κ| i < κ}∪{Φ} ∪{∀(ϕ() ↔ϕ^∼()) |ϕ is Σ_n}p(x)= { x E d ∧ x ≠ c_i | i < λ}We filtrate this according to I in the standard way and use expansion of V_α to provide witness models; here it is crucial that we started with α∈ C^(n).The model of T omitting p gives the desired j.We can adjust this proof to get a proof of (4) by finding strong limit α' >α, also in C^(n), and relativizing the appropriate parts of the theory to ensure that j(α) ∈ C^(n).Then, derive the extender E from this model, and j_E: V →_E that retains the desired properties. Given (3) or (4), we prove the compactness by starting with a filtration T̅ = {T_s | s ∈ I} of an ^s, Σ_n_κ, κ-theory and types {p^a(x) = {ϕ^a_i(x) | i < λ}| a ∈ A}, find strong limit α∈ C^(n) above the rank of these objects and the function f that takes s to the model of T_s omitting each p^a_s.V_α reflects these properties since α∈ C^(n), so by elementarity the target model thinks that j(f)(j"λ) models j"T and omits {j"p^a(x) | a ∈ A}.Since V_β or V_j(α) are Σ_n-elementary in V, the target model is correct.† Similar to second-order logic, the identity crisis disappears in sort logic and C^(n)-extendible cardinals witness a wide range of type omitting compactness. First, we show that C^(n)+-extendible cardinals are equivalent to an apparent weakening: the requirement that j(κ) ∈ C^(n) is redundant.The following are equivalent: * κ is C^(n)+-extendible. *For every α > κ in C^(n), there is j:V_α→ V_β such that j = κ and β∈ C^(n). We use the following lemma which will also be useful when examining Löwenheim-Skolem-Tarski numbers.This is similar to Magidor's characterization of supercompacts. Let κ be C^(n)-extendible.Then for all α >κ in C^(n) and R ⊂ V_α, there are cofinally many γ < κ such that there are α̅ <κ in C^(n) and S ⊂ V_α̅ with elementary j:(V_α̅, ∈, S) → (V_α, ∈, R), j = γ, and j(γ) = κ. Proof:Fix α∈ C^(n) above κ, R ⊂ V_α, and β < κ.Find α' >α in C^(n).By assumption, there is j:V_α'→ V_β' with j = κ, j(κ)>α', and β' ∈ C^(n).Given a transitive modelof a fragment of ZFC, write C^(n), for 's version of C^(n).Since α, α' ∈ C^(n), α∈ C^(n), V_α'.By elementarity, j(α) ∈ C^(n), V_β'.Thus, V_β'“∃α̅ < j(κ) andS ⊂ V_α̅, j_0:(V_α̅, ∈, S) →(V_j(α), ∈, j(R)) such thatj_0( j_0) = j(κ),j_0>j(β),and α̅∈ C^(n)"This is witnessed by jV_α.By elementarity,V_α'“∃α̅ < κ andS ⊂ V_α̅, j_0:(V_α̅, ∈, S) →(V_α, ∈, R) such thatj_0( j_0) = κ,j_0>β,and α̅∈ C^(n)"This is the desired result; note that it implies α̅∈ C^(n) because α' ∈ C^(n).† Proof of <ref>:Suppose (2) holds.By Lemma <ref>, the class of C^(n)-ordinals are cofinal in κ.Since this class is club, κ∈ C^(n).Fix α > κ in C^(n) and find j as in (2); it suffices to show j(κ) ∈ C^(n).Since κ, α∈ C^(n), we have κ∈ C^(n), V_α.By elementarity, j(κ) ∈ C^(n),V_β.Since β∈ C^(n), this gives j(κ) ∈ C^(n).† The following are equivalent for every n<ω. * κ is C^(n)-extendible. * κ is measurable-for-^s, Σ_n∪_κ, ω. * κ is strong compact-for-^s, Σ_n_κ, κ. * κ is supercompact-for-^s, Σ_n_κ, κ.Proof:This follows a similar argument as Theorem <ref>, just requiring that the α's be in C^(n).† However, the notion of a huge-for-^s, Σ_n_κ, κ cardinal would be similarly stronger in consistency strength than the notion in Proposition <ref>.While the Löwenheim-Skolem-Tarski number for second order was determined by Magidor in <cit.> and Magidor and Väänänen have explored the Löwenheim-Skolem-Tarski numbers of various fragments of ^2 in <cit.>, the Löwenheim-Skolem-Tarski number of sort logic seems unknown.We give a characterization of these cardinals in terms of a C^(n)+-version of Magidor's characterization of supercompacts.We work with Löwenheim-Skolem-Tarski numbers for strictly first-order languages to avoid the technicalities around trying to develop a notion of elementary substructure for sort logic.See Section <ref> for a definition of elementary substructure in second-order logic. The following are equivalent for κ: * The conclusion of Lemma <ref>. * For all α < κ, if N is a structure in a strictly first-order language τ of size <κ, then there is M ≺__α, α N of size <κ such that M and N have the same ^s, Σ_n_α, α-theory.Proof: (1) implies (2): Let N be a τ-structure with |τ|<κ.Find γ < κ that is above α and |τ| and find α'>κ such that N ∈ V_α'.Code the structure N into a relation R ⊂ V_α'By assumption, there is α̅∈ (γ, κ) in C^(n) and j:(V_α̅, ∈, S) → (V_α, ∈, R) with j > γ.Then S codes a structure M in V_α̅ that, by elementarity, models Th_^s, Σ_n_α, α(N).Moreover, jM is _α, α-elementary, so the range of jM is the desired model. (2) implies (1): Fix α > κ in C^(n), R⊂ V_α, and β < κ.Apply the assumption to the structure (V_α, ∈, R, δ)_δ < β to get M ≺_^β, β N with the same ^s, Σ_n-theory.This includes Magidor's Φ, so after taking the transitive collapse, we get ≺__β, β-elementary j:(V_α̅, ∈, S, δ)_δ<β→ (V_α, ∈, R, δ)_δ < β. The constants for the elements of β force the critical point of j above β.†§.§ Rank-into-rankWe now turn to the strongest large cardinal principles, the rank-into-rank embeddings.For an excellent survey of these, see Dimonte <cit.> or the expanded Dimonte <cit.>.Following <cit.>, this section uses Σ^1_n(_κ,ω) to denote the fragment of infinitary second-order logic ^2_κ, ω consisting of the formulas that are Σ_n in their second-order quantifiers.We can cast I1, I2, and I3 in a uniform way by saying, for n<ω, I2_n(κ, δ) is the assertion There is j:V_δ→ V_δ, κ =j, δ is the supremum of the critical sequence, and j is Σ^1_2n(_κ, ω)-elementary. Laver <cit.> proved[Laver's paper attributes this result to Martin without citation, but other sources attribute it to Laver.] that Σ^1_2n+1-elementarity of such a j implies its Σ^1_2n+2-elementarity, so it suffices to consider the even levels.Then I3 is I2_0, I2 is I2_1[Which is in turn equivalent to being elementary about statements of well-foundedness <cit.>.], and I1 is I2_<ω.Note that second-order elementary embeddings should be understood in the context of Section <ref>.Given j:V_δ→ V_δ, we can naturally extend this to j^+:(V_δ) →(V_δ) by j^+(R) = ⋃_α <δj(R ∩ V_α). Then set A^V_δ = j^+(A).Note that j^+ is the only possible extension of j to V_δ+1 that could be elementary, and its Σ^1_0-elementarity follows from its first-order elementarity.These principles have natural characterizations in terms of extendibility criteria.Recall that κ is weakly compact iff every κ-sized structure has a proper _κ, κ-elementary extension <cit.> and κ is measurable iff every ≥κ-sized structure has a proper _κ, κ-elementary extension. For each n≤ω and δ = ℶ_δ, the following are equivalent. * There is κ < δ such that I2_n(κ, δ). * Every δ-sized structure has a non-identity Σ^1_2n(_ω, ω)-elementary embedding into itself.Proof: Suppose (1) holds, and let j:V_δ→δ with j =κ witness.If M is of size δ, then we can code it as a structure with universe δ.Then jδ is the desired embedding and it inherits the Σ^1_2n(_ω, ω)-elementarity of j.Suppose (2) holds.Then apply it to the structure (V_δ, ∈), which has size ℶ_δ=δ; this gives Σ^1_2n(_ω,ω)-elementary j:V_δ→ V_δ.j must have a critical point below δ, call it κ.As in Section <ref>, this strengthens the elementarity of j to Σ_2n^1(_κ, ω).Since j:V_δ→ V_δ, we must have δ≥sup_n j^n(κ).By Kunen's inconsistency, we must have δ≤sup_n j^n(κ).Since δ is limit, this tells us that δ = sup_n j^n(κ), and I2_n(κ, δ) holds.† I3 and I2 also have standard characterizations in terms of coherent ω-sequences of normal ultrafilters.This allows us to prove the following type omitting compactness from them.Unfortunately, this does not give an equivalence, but does allow us to sandwich these properties between two compactness for omitting types statements.Recall that any type can be trivially extended to an equivalent, larger one by adding instances of “x ≠ x." * If I2_0(κ, δ), then for any theory T= ∪_s ∈_κκT_s ⊂_κ,ω(τ) and set of types {p^β = {ϕ^β_i(x) | i < κ_n_β+1}|β < μ}, if there are club many s ∈_κκ such that T_s has a model M_s with the property for β < μ, {t ∈ [κ_n_β+1]^κ_n_β| M_t ∩κ omitsp^β_t = {ϕ^β_i(x) | i ∈t}} contains a club, then T has a model omitting {p^β|β < μ}. * I2_1(κ, δ) implies the same for the logic _κ, ω(Q^WF).We will use the ultrafilter characterizations of these cardinals, in part because I3 doesn't have a characterization in terms of j:V → and we don't want to restrict to models of size ≤δ.Proof: We prove the second item.The first follows by the same argument, just removing the mentions of well-foundedness.I2_1(κ,δ) is equivalent to the existence of κ-complete, normal ultrafilters U_n on [κ_n+1]^κ_n (where {κ_n| n < ω} is the critical sequence of the witnessing embedding) such that * coherence: For any X ⊂ [κ_n+1]^κ_n and m > n, X ∈ U_n {s ∈ [κ_m+1]^κ_m| s ∩κ_n+1∈ X}∈ U_m * well-founded: For any {n_i < ω| i <ω} and {X_i ∈ U_n_i| i < ω}, there is s ⊂δ such that, for all i < ω, s ∩κ_n_i+1∈ X_i.Let M_s be the desired models.Since they exists for club many s and U_0 contains this club, we can form the direct limit of the ultrapowers as standard: M^*_n = ∏_s ∈ [κ_n+1]^κ_n M_s ∩κ/U and there is a coherent system of _κ, κ-embeddings f_n, m: M^*_n → M^*_m that takes [f]_U_n to [s ↦ f(s ∩κ_n+1)]_U_n+1.Then M^* is the direct limit of these models.Standard arguments (see Proposition <ref> for a more general case) show that Łoś' Theorem holds for formulas of _κ, ω(Q^WF).This guarantees that M^*T.To show the type omission, let β < μ and [n, f]_U̅∈ M^*.Setting n' = maxn, n_β, this impliesX:={t ∈ [κ_n'+1]^κ_n'| M_t ∩κ omitsp^β_t∩κ_n_β+1}∈ U_n'Define a function h on X by setting, for t ∈ X, h(t) ∈ t such that M_t ∩κϕ^β_h(t)(f(t∩κ_n+1)); this is possible exactly because of the type omission.Then, h is regressive on a U_n'-large set, so there is α_0 < κ_n'+1 such thatY:={t ∈ [κ_n'+1]^κ_n'| M_t ∩κϕ^β_α_0(f(t∩κ_n+1))}∈ U_n'Then, by Łoś' Theorem, [n,f]_U̅ = [n', f(s ∩κ_n+1)]_U̅ omits p^β in M^* as M^* ϕ^β_α_0([n, f]_U̅)Since [n, f]_U̅ and β we arbitrary, were are done.† We can also isolate a type omitting compactness stronger than these rank-into-rank axioms.Note that, unlike previous theorems, the types in the following don't shrink in the hypothesis. Fix a cardinal δ=ℶ_δ. * Suppose we have the following for some κ: For any _ω, ω(τ)-theory T with a filtration {T_α|α < κ} and |τ| = δ and any _ω, ω(τ)-type p(x), if every T_α has a model omitting p, then T has a model omitting p. Then there is κ_0 ≤κ such that I2_0(κ_0, δ). * The above implies I2_n(κ, δ) after replacing the logic with Σ^1_2n(_ω, ω). Proof: Fix a bijection f:δ→ V_δ. Consider the theory and typesT =ED__ω, ω(V_δ, ∈, x)_x ∈ V_δ∪{c_i < c < c_κ|i < κ} ∪{d_i E d_j | i, j < δ, f(i) ∈ f(j)}∪{(d_i E d_j) | i, j < δ, f(i) ∉f(j)} p(x)= {x ≠ d_i | i < δ}We claim that a model of T omitting p will give a bijection as required for I2_0(κ_0,δ): the first two parts of T will give an elementary j:V_δ→, where we don't yet know thatis transitive.The third and fourth parts of T ensures thathas a (different) copy of V_δ in it given by x ∈ V_δ↦ d^_f^-1(x)∈.Omitting p means everything inis in this second copy of V_δ, so ≅ V_δ.Thus, we have elementary j:V_δ→ V_δ that is not the surjective because its range does not contain c^.If κ_0 :=j, then this witnesses that I2_0(κ_0, δ) holds; note the second part of T ensures that j ≤κ. To find such a model, we apply our compactness principle.If α < κ, let T_α be T cutting out the constants c_x for elements with rank in the interval [α, κ) as in the discussion following Theorem <ref>, that is,T_α =ED__ω, ω(V_δ, ∈, x)_x ∈ V_α∪ V_≥κ∪{c_i < c < c_κ|i < α} ∪{d_i E d_j | i, j < δ, f(i) ∈ f(j)}∪{(d_i E d_j) | i, j < δ, f(i) ∉f(j)}We can find a model of T_α omitting p by taking the identity from V_δ to itself and interpreting c as α.†§.§ Elementary substructure for ^2In contrast with first-order logics[Here meant as `sublogics of _∞,∞.'], the notion of elementary substructure in second-order logic has not been well-studied.As evidence to this, in Väänänen's paper on second-order logic and set theory <cit.>, the word `structure' appears 210 times, but `substructure,' `sub-structure,' or even `sub structure' never appear.Similarly, Shapiro's book <cit.> never discusses the matter[The notion of “elementary substructure" does appear here, but always in reference to its first-order version.].A guess at the cause for this is that second-order logic is often employed to find categorical theories, whereas first-order logic attempts only to axiomatize classes of structures.In such contexts, there is no reason to talk about one model's relation with others.A first-attempt at second-order elementary substructure would be to work in analogy with first-order and say M is a _2-elementary substructure of N iff every formula holds of parameters from M in M iff it holds in N.However, this notion doesn't allow for any proper elementary extensions as there are definable sets that must grow.Concretely, the formula ϕ(X) := “∀ x (x ∈ X)" must be satisfied by the entire universe, so no extension of M can think ϕ(M) holds.A more modest generalization is used by Magidor and Väänänen in <cit.>.They say that M is an ^2-elementary substructure of N iff the above property holds restricted to formulas with only first-order free variables.This works (in the sense that proper extensions can exist), and they observe that Magidor's theorem on the Löwenheim-Skolem-Tarski number of second-order extends to this notion of elementarity.However, it seems lacking as there's no discussion of free variables in second order.We give a definition of elementary substructure in second-order logic in Definition <ref> below.An important point is that this definition includes a stronger notion of substructure in the second-order context (see Definition <ref>.(3)).Before giving the formal definitions, we give a motivation.Any second-order structure M can be viewed as a Henkin structure M^+ = (M, P[M], E).Then every second-order statement about M can be transferred to a first-order statement about M^+, and the fact that M^+ is a full structure (isomorphic to (M, (M), ∈ )) can be captured by a single second statement Ψ asserting every subset of M is represented by a member of E.Now that we have moved to a more familiar first-order setting, we can ask what relation between the original structures M and N characterizes when M^+ is an elementary substructure of N^+.The key is that, given s ⊂ M and m_s ∈ P[M] representing it, N^+ thinks the same facts about m_s as M does when the parameters come from M, but N also thinks new things about m_s.Crucially, there might be elements of N-M that N thinks are in m_s.Then, setting s^N := {n ∈ N | Nn E m_s}, N thinks all the same facts about s^N that M does about s.This notion of extending subsets is key to defining second-order elementary substructure.This leads to the following definitions[We focus on ≺_^2 for ease, but these definitions could easily be changed to accommodate ≺_^2_α, α for α < κ.]: * Set ^ω(X) = ∪_n < ω(X^n). * For X⊂ Y, an extension function is a function f:^ω(X) →^ω(Y) such that, for all s ⊂ X^n, f(s) ∩ X^n = s and f(s) ⊂ Y^n and f fixes every finite set. * Given τ-structures M and N, we say M is a second-order substructure of N, written M ⊂_2 N iff |M| ⊂ |N| and there is an extension function s ↦ s^N such that for every atomic ϕ(x_1, …, x_n; X_1, …, X_n'), m_i ∈ M, and s_i ⊂ M^n_i, we haveM ϕ(m_1, …, m_n; s_1, …, s_n')N ϕ(m_1, …, m_n; s_1^N, …, s_n'^N) We turn substructure into elementary substructure by letting ϕ range over all formulas. Given τ-structures M and N, we say M is an 𝕃^2-elementary substructure of N, written M ≺_𝕃^2 N iff M ⊂_2 N and there is an extension function s ↦ s^N witnessing this such that for every ϕ(x_1, …, x_n; X_1, …, X_m) ∈𝕃^2, m_i ∈ M, and s_i ⊂ M^n_i, we haveM ϕ(m_1, …, m_n; s_1, …, s_n')N ϕ(m_1, …, m_n; s_1^N, …, s_n'^N) Thus, M ≺_^2 N comes with a choice of extensions for each s ⊂ M.This avoids the issue with the “first-attempt" notion of elementary substructure above.Indeed, for any definable subset A of M, A^N must be the collection of elements satisfying that definition in N if M ≺_^2 N.Note that if τ is a strictly first-order language (as is often the case), then M ⊂ N is equivalent to M⊂_2 N for any collection of extensions.This means that comments about substructures in first-order languages in the context of second-order can be given the normal interpretation.Also, for such a language, ≺_^2 will imply elementarity in the sense of Magidor and Väänänen.For a general notion of ≤, a ≤-embedding is f:M → N such that f is a τ-isomorphism onto its range and f(M) ≤ N (a set theorist might prefer to call this model f"M).Specializing to ≺_^2, f is a map on elements of M and, given s ⊂ M, f"s ⊂ f(M).Then f(M) ≺_^2 N implies there is an extension (f"s)^N ⊂ N that satisfies the definition of ≺_^2.To avoid this unfortunate notation, we say that f:M → N is ^2-elementary means that f is a map from M ∪^ω(M) to N ∪^ω(N) such that, for all ϕ(, ), ∈ M, and ∈^ω (M),Mϕ(, )N ϕ(f(), f())One should be skeptical that this is the “right" notion of ≺_^2 as it adds a strange new condition about global choices of extensions of subsets.In addition to the heuristic with Henkin models above, we offer two “sanity" checks that this is the right notion.The first check is that Magidor's Theorem on the LST number of second order logic holds with this notion. Let κ be supercompact.For any τ of size <κ and α < κ and τ-structure N, there is M ≺_^2_α, α N of size <κ. This follows from the argument given in our heuristic: take a model N, turn it into a full Henkin structure N^+ with Skolem functions, and apply Magidor's version along with the sentence ∀ X ∃ x (X `=' x).This quotational equality is an abbreviation of the statement that X and x have the same elements.The second check is that the natural notion of an elementary diagram characterizes ^2-elementary embedability.Given a τ-structure M, we define it's ^2 elementary diagram by first adding a first-order constant c_a for each a ∈ M and a second-order constant d_s for each s ∈^ω(M) and settingED_^2(M) = {ϕ(c_a_1, …, c_a_n, d_s_1, …, d_s_k) | M ϕ(a_1, …, a_n, s_1, …, s_k) }Let M, N be τ-structures.The following are equivalent. * N has an expansion that models ED_^2(M). * There is a ^2-elementary f:M → N.Proof: The proof follows as the first-order one.Given ^2-elementary f:M→ N, expand N by c^N_a = f(a) and d^N_s = f(s).Similarly, if N^*ED_^2(M), define f:M→ N by the same formula.† Some of the basic results of first-order elementary substructure transfer, while others do not.For instance, the Tarski-Vaught test goes through with the same proof, although a slightly modified statement. Given M ⊂_2 N (with a specified extension map s ↦ s^N), we have that M ≺_^2 N iff both of the following hold: * ∀ m_i ∈ M, s_j ⊂ |M|^n_j and ϕ(x, , ), if N ∃ x ϕ(x, , ^N), then there is m ∈ M such that N ϕ(m, , ^N); and * ∀ m_i ∈ M, s_j ⊂ |M|^n_j and ψ(X, , ), if N ∃ X ϕ(X, , ^N), then there is s ⊂ M^n such that N ϕ(s^N, , ^N).Unfortunately, ≺_^2 does not have properties such as coherence and smoothness under unions of chains fail.These are properties coming from the study of Abstract Elementary Classes, a general framework for nonelementary model theory (see Baldwin <cit.> for a survey): * Coherence: A binary relation ≺ on τ-structures is coherent iff whenever M_0 ≺ M_2 and M_1≺ M_2 and M_0 ⊂ M_1, then M_0 ≺ M_1. * Smoothness: Abinary relation ≺ on τ-structures is smoot under unions of chains iff whenever {M_i | i < α} is a ≺-increasing chain with α limit, then ⋃_i<α M_i is the ≺-least upper bound of the chain.We can show ≺_^2 fails these properties by using Silver's example, the bane of many a nonelementary model theorist's hope.Recall Silver's example[Although typically given as a PC-class (see <cit.>), we give it here as the class of models of a second-order sentence.]: the language consists of two sorts P and Q and the sentence ϕ_Silver is∃ E ⊂ P × Q ( ∀ x,y ∈ Q (x = y ∀ z ∈ P (z E xz Ey )))In other words, a model M of ϕ_Silver consists of two sets P^M and Q^M such that there is an extensional relation on P^M and Q^M.The existence of such a relation is equivalent to there being an injection from P^M to (Q^M), which is equivalent to |Q^M| ≤ 2^|P^M|.Ifis any fragment of ^2 containing ϕ_Silver, then M ≺_ N requires that any extensional relation on P^M × Q^M can be extended to an extensional relation on P^N × Q^N.This is easy unless |Q^M| = 2^|P^M| and |Q^N| > 2^|P^N-P^M|, but it fails in this case.Thus, we can find counter-examples to coherence and smoothness under unions of chains.Note that, since the language is first-order, M ⊂_2 N is equivalent to normal substructure. * Coherence:Set M_0, M_1, M_2 by P^M_0 = P^M_1 = ω; P^M_2 =ω_1; Q^M_0 = (ω); and Q^M_1 = Q^M_2 = (ω+1).Then M_0 ⊂ M_1 and M_0, M_1 ≺_^2 M_2, but M_0 ⊀_^2 M_1. * Smoothness: For α < 2^ω, set M_α by P^M_α = ω and Q^M_α = ω+α.Then this is a ≺_^2-increasing, continuous chain with union M = ( ω, 2^ω).Set N by P^N = ω and Q^N = 2^ω+ω.Then M_α≺_^2 N for each α < 2^ω, but M ⊀_^2 N. We can also define Skolem function for second-order in the same way as first-order.To do so, we must add strictly second-order functions to the language, so that even if τ started out strictly first-order, its Skolemization τ_sk won't be.In the context of first-order, the notions of elementary substructure and that of club sets are very closely intertwined.For instance, given a structure M and cardinal κ < M, {s ∈_κλ : sis the universe of an elementary substructure of M} is club, and, conversely, given a club ⊂_κλ, we can find a structure M with universe λ such that the above set is contained in .This connection is mediated by the fact that both notions can be characterized by being the closure sets of certain functions (Skolem functions, in the case of elementary substructure).With a definition for second-order elementary substructure in hand, we can define a notion of club, which we call superclubs.Recall our focus on ^2.* Fix μ < κ and let F_i:[λ]^<ω× [^ω(λ)]^<ω→^ω(λ) for i < μ.ThenC(F̅):={ s ∈_κλ|∃ extension f:^ω(s) →^ω(λ)such that ∀∈ s, ⊂ s^, i < μ,we haveF_i(, f()) = f(y)for some y ∈^ω(s)} * We call the collectionof all sets containing some C(F̅) the superclub filter on λ.We have calleda filter without proving it is one.This name is justified in Proposition <ref>, although it might be non-proper.Although we defined it with a combinatorial characterization in the spirit of Definition <ref>, our interest in superclubs comes from the following model theoretic characterization.Given a structure M and s ⊂ |M|, set M s to be the τ-substructure of M with universe generated by s (so is the closure of s under the functions of M). The superclub filter is generated by sets of the formD(M):={ s ∈_κλ| M s ≺_^2 M}for M a τ-structure with universe λ and |τ|<κ. Proof:First, suppose we are given {F_α|α < μ}.Set τ ={F_α^n; m_1, …, m_k|α <μ; k, n, m_i < ω} to be a functional language so the domain of F_α^n; m_1, …, m_k is M^n ×(M^m_1) ×…×(M^m_k).Expand λ to a τ-structure by interpreting F_α^n; m_1, …, m_k as F_α restricted to the appropriate arity.Then D(M) ⊂ C(F̅).Second, suppose we are given M with universe λ.WLOG, we can assume that τ has (second-order) Skolem functions.Let F̅ = {F^M | F∈τ} be the functions of M (interpreted as projection on other arities).Then, since we have Skolem functions, C(F̅) ⊂ D(M).†This model-theoretic characterization tends to be more useful to show the things that we want. The superclub filter extends the club filter. Proof:The club filter can be characterized in the same way using first-order elementary substructure.† The next proposition shows that calling this a filter is justified. The superclub filter is a κ-complete, possibly non-proper filter. Proof:The superclub filter is upwards closed by definition, so we only need to show it is closed under the intersection of <κ-many members.Suppose that X_α∈ for α < μ < κ.By Lemma <ref>, there are τ_α-structures M_α with universe λ and |τ_α|<κ such that D(M_α) ⊂ X_α.Set τ_* to be the disjoint union of the τ_α's and M_* to be the τ_*-structure that is simultaneously an expansion of each M_α.Then D(M_*) ⊂ D(M_α), so this witnesses that ⋂_α<μ X_α∈.It is a filter by definition.Given D(M_α) with M_α a τ_α structure, set M_* to be the ∪τ_α-structure that is simultaneously an expansion of each M_α.† Proposition <ref> leaves open the possibility that the superclub filter is non-proper, i.e., contains the empty set.In fact, the properness of the superclub filter characterizes supercompact cardinals. Let κ≤λ. * If κ is not λ-supercompact, then there is an empty superclub.Moreover, there is a uniform definition of the empty superclub. * If κ is λ-supercompact, then every normal, fine, κ-complete ultrafilter on _κλ extends the superclub filter.Proof: If κ is not λ-supercompact, then by Corollary <ref>, there is a structure M of size λ in a language τ of size < κ such that M has no ≺_^2-substructures; looking at Magidor's proof, we can take M to be some (V_β, ∈, α)_α∈μ for some β < λ and μ < β.WLOG |M| is λ by expanding the universe by a trivial sort.Then D(M) = ∅ is in the superclub filter.If κ is λ-supercompact, then let U be a normal, κ-complete ultrafilter on _κλ and derive j_U:V →_U.Let M be a τ-structure with universe λ and, WLOG, it has Skolem functions.By Magidor's result and its extensions, j_U"M ⊂_^2 j_U(M).Since M has Skolem functions, j_U"M≺_^2 j_U(M).Thus, j"λ∈(D(j_U(M)))^_U = j(D(M)).So D(M) ∈ U.†In particular, we get the following: Given κ≤λ, κ is λ-supercompact iff the superclub filter on _κλ is proper. While superclubs give a characterization of supercompact cardinals, they lack a nice characterization in the spirit of “closed unbounded sets" that clubs have.Such a characterization would shed light on properties of the ≺_^2 relations and permit the exploration of superstationary sets.§ EXTENDERS AND OMITTING TYPES A key distinction between Theorem <ref> and Theorems <ref> and <ref>is the lack of an analogue of Theorem <ref>.(<ref>) in the results of Section <ref>.The large cardinals of Section <ref> are typically characterized by the existence of certain kinds of extenders, but the modifier “certain" is typically characterized in a way that nakedly concerns embeddings between models of set theory–e.g., <cit.> characterizes κ being λ-strong by the existence of an extender E such that j_E:V→_E witnesses λ-strength–rather than any combinatorial feature of the extender.We begin with a combinatorial characterization, although it still references the V_α's and may be of limited interest.However, we use this as a starting point to investigate a larger question: to what extent is second-order logic necessary to characterize the large cardinals in Section <ref>?We should say a few words about our definition of extenders.We say that a (κ, λ)-extender is E = {E_|∈ [λ]^<ω}, where each E_ is a κ-complete ultrafilter on ^κ satisfying coherence, normality, and well-foundedness (see <cit.> for a statement of these conditions in a slightly different formalism).Note that this is a compromise between the original definition of Martin-Steel <cit.>–which took ∈ [V_λ]^<ω and required E_ to be on ^ V_κ–and more modern presentations like <cit.>–which takes ∈ [λ]^<ω but requires E_ to be on [κ]^||.Crucially,we also depart from both of these and don't require that {s ∈^κ|∀ a_1, a_2 ∈(a_1 < a_2 ↔ s(a_1) < s(a_2))}∈ E_.Now we are ready to give a combinatorial characterization. We say a bijection h: ℶ_α→ V_α is rank-layering iff for every β < α, h"ℶ_β = V_β. Note this condition is equivalent to x ∈ y ∈ V_α implies h^-1(x) ∈ h^-1(y).The following are equivalent. * κ is λ-strong. * For some rank-layering bijection h:ℶ_λ→ V_λ, there is a (κ, ℶ_λ)-extender E such that for all α, β < λ, we have h(α) ∈ h(β) { s ∈^{α, β}κ| h(s(α)) ∈ h(s(β))}∈ E_{α, β} * For every rank-layering bijection h:ℶ_λ→ V_λ, there is a (κ, ℶ_λ)-extender E such that for all α, β < λ, we have h(α) ∈ h(β) { s ∈^{α, β}κ| h(s(α)) ∈ h(s(β))}∈ E_{α, β} In each of the cases, we have that x ∈ V_λ is the image of [{h^-1(x)}, s ↦ h (s(h^-1(x)))]_E after the transitive collapse.Proof of Proposition <ref>: (3) implies (2) is clear. (2) → (1): Given such an extender, set j_E: V → to come from the extender power of V by E followed by the transitive collapse; then j = κ and j(κ) > λ by standard arguments (see <cit.>).So we are left with showing that V_λ⊂.Let π:∏ V/E → be the Mostowski collapse.Following the above, we will show that, for each x ∈ V_λ, x = π([{h^-1(x)}, s ↦ h (s( h^-1(x)))]_E)We work by induction on the rank of x.For each y ∈ x, the condition on our extender precisely gives that { s ∈^{h^-1(y), h^-1(x)}κ |h(s(h^-1(y))) ∈ h(s(h^-1(x)))}∈ E_{h^-1(y), h^-1(x)} π([{h^-1(y)}, s ↦ h (s( h^-1(y)))]_E)∈ π([{h^-1(x)}, s ↦ h (s( h^-1(x)))]_E)Now suppose that z ∈π([{h^-1(x)}, s ↦ h ∘ s ∘ h^-1(x)]_E).We wish to show that π^-1(z) is one of our terms for some y ∈ x.We know that π^-1(z) is [, f]_E for some ∈ [ℶ_λ]^<ω and f with domain ^κ.WLOG we may assume that h^-1(x) ∈, and coherence implies that [{h^-1(x)}, s ↦ h (s(h^-1(x)))]_E=[, s ↦ h (s(h^-1(x)))]_E.Thus, we have that {s ∈^κ| f(s) ∈ h(s(h^-1(x)))}∈ E_If we set α = h^-1(x), then the rank-layering property tells us that {s ∈^κ| h^-1(f(s)) ∈ s(α)}∈ E_By normality, this means that there is ⊃ and β∈ such that{s ∈^κ| h^-1(f(s)) = s(β) }∈ E_If we set y=h(β) and g(s)=f(s), we can rewrite this as {s ∈^κ| g(s) = h(s(h^-1(y)))}∈ E_Putting all of this together, we have thatπ^-1(z) = [, f]_E = [, g]_E = [{h^-1(y)}, s ↦ h(s(h^-1(y)))]_ESo z must be an element of this form with y ∈ x, as desired.(1) → (3): Fix h:ℶ_λ→ V_λ to be a rank-layering bijection and j:V→ to witness that κ is λ-strong, so j = κ, j(κ)>λ, and V_λ⊂.<cit.> describes the general method of deriving an extender E^* from an elementary embedding by settingX ∈ E^*_𝕀_∈ j(X)The choice of the identity to `seed' the ultrafilters is not necessary; we can (and will) change this function without changing the fact that the derived ultrafilters form an extender.In particular, define E by specifying, for each ∈ [ℶ_λ]^<ω and X ⊂^κ, X ∈ E_ j(h)^-1∘ h ∘ j^-1 j() ∈ j(X)Then E is a (κ, ℶ_λ)-extender and we only need to show the additional property.We verify this with the following chain of equivalences, making crucial use of our choice of seed: for α, β < λ, set = {α, β}{s ∈^κ| h(s(α))∈ h(s(β))} ∈E_ j(h)^-1∘ h ∘ j^-1 j() ∈ j( {s ∈^κ| h(s(α))∈ h(s(β))})j(h)^-1∘ h ∘ j^-1 j() ∈{s ∈^j()j(κ) | j(h)(s(j(α)))∈ j(h)(s(j(β)))}j(h)(j(h)^-1(h(j^-1(j(α))))) =j(h)(j(h)^-1(h(j^-1(j(β)))))h(α) ∈ h(β)as desired. † Now we turn to the issue of how necessary logics beyond _∞, ∞ are to characterize large cardinals and focus on strong cardinals.Theorem <ref> characterizes strong cardinals in terms of the logic ^2; is there a characterization of strong cardinals solely in terms of _∞, ∞?For κ≤λ, consider the following (definable-class) _κ, ω(Q^WF)-theory and types for y ∈ V_λ:τ = {E, c_a, c, d_a'}_a ∈ V, a' ∈ V_λT=ED__κ, ω(Q^WF)(V, ∈, x)_x ∈ V∪{d_i = c_i < c < c_κ| i < κ}∪{d_b ∈ d_a | b ∈ a ∈ V_λ}p_y(x)= { x E d_y ∧ (x = d_z) | z ∈ y}It follows from the methods of the previous sections that T has a model omitting each p_y iff κ is λ-strong.However, what we lack from Proposition <ref> is an appropriate type omitting-compactness scheme and a decomposition of T along that scheme that is satisfiable.The difficulty in constructing this becomes more clear if we look at the extender product construction.Given a (κ,λ)-extender E, for each ∈ [λ]^<ω, we form the ultraproduct ∏ V/E_.The coherence axiom and Łoś' Theorem insures that the restriction function induces a coherent system of _κ, κ-elementary embeddingsfrom ∏ V/E_ to ∏ V/E_ when ⊂.Since [λ]^<ω is directed, we can get a colimit ∏ V/E and _κ, ω-elementary embeddings.Since E is well-founded, so is ∏ V/E, so we can form its transitive collapse _E.We can generalize this to a more general model-theoretic context.Fix some ∈ [λ]^<ω and suppose that we have a collection of τ-structures {M_s | s ∈^κ} and form the ultraproducts ∏_s ∈^κ M_s/E_ for ⊃.Again, the coherence gives a coherent system of embeddings, so we can form the extender product ∏ M_s /E as the colimit of this system.This structure has universe {[, f]_E |⊂∈ [λ]^<ω, f ∈^κ} where [, f]_E = [, g]_E iff {s ∈^κ| f(s ) = g(s ) }∈ E_.Then we have following result for Łoś' Theorem.Let E = {E_|∈ [λ]^<ω} be a system of κ-complete ultrafilters satisfying coherence.The following are equivalent: * E is well-founded. * Łoś' Theorem holds for _κ, ω(Q^WF) formulas.That is, given τ-structures {M_s | s ∈^κ}, ϕ(x_1, …, x_n) ∈_κ, ω(Q^WF)(τ), and [_1, f_1]_E, …, [_n, f_n]_E ∈∏ M_s/E, we have ∏ M_s/E ϕ([_1, f_1]_E, …, [_n, f_n]_E)iff { s ∈^∪_iκ| M_s ϕ( f_1(s_1), …, f_n(s _n))}∈ E_∪_iProof: For one direction, it is known that E is well-founded iff ∏ V/E is well-founded, which follows from Łoś' Theorem applied to Q^WFxy(x∈ y).For the other direction, fix ∈ [λ]^<ω and τ-structures {M_s | s ∈^κ}.We show Łoś' Theorem for _κ, ω(Q^WF) by induction.Standard arguments take care of everything but the Q^WF quantifier.So suppose Łoś' Theorem holds for ϕ(x, y, ) and [, f]_E ∈ M_E := ∏ M_s/E.First, suppose that {s ∈^κ| M_s Q^WF xy ϕ(x, y, f(s))}∉E_.Set X to be the complement of this set.For s ∈ X, there is c^s_r ∈ M_s for r < ω such that M_s ϕ(c^s_r+1, c^2_r, f(s).Then c_r := [, c^s_r]_E ∈ M_E witnesses the illfoundedness of ϕ.Second, suppose that X_0 = {s ∈^κ| M_sQ^WFxyϕ(x,y,f(s))}∈ E_ and M_EQ^WFxyϕ(x, y, [, f]_E).Then there is [_r, f_r]_E ∈ M_E such thatM_E ϕ([_r+1, f_r+1]_E, [_r, f_r]_E, [, f]_E)for all r < ω and, WLOG, ⊂_r ⊂_r+1.ThenX_r+1 := {s ∈^_r+1κ| M_s ϕ(f_r+1(s), f_r(s _r), f(s)}∈ E__r+1The well-foundedness of E gives h:∪→κ such that h_r+1∈ X_r+1 and h ∈ X_0.Then, for r < ω,M_hϕ(f_r+1(h_r+1), f_r(h _r), f(h))Thus, f_r(h _r) ∈ M_h | r < ω witnesses the illfoundedness of ϕ(x, y, f(h )) in M_h, contradicting h ∈ X_0.† So, to decompose T above, one could try to construe ∏ V/ E as an extender product ∏ M_s /E in the appropriate language and for the appropriate , and then see what fragment of T M_s satisfies.However, the problem is that the factors that make up ∏ V/E don't have expansions to τ-structures.Rather, the analysis of ∏ V/E crucially uses that it can be seen as the extender power ∏_s∈^κ V/E for any choice of .Thus, there is no way to analyze which parts of the types each factor omits.However, there is a nice criterion for when an extender product (or just a coherent ultraproduct by a κ-complete, well-founded coherent ultrafilter) omits a _κ, ω(Q^WF)-type based on the behavior of the original models M_s provided that the domain of the type is a subset that appears as an element of _E. Let E be a (κ, ℶ_λ)-extender witnessing that κ is λ-strong.Suppose that ∈ [ℶ_λ]^<ω, {M_s | s ∈^κ} is a collection of τ-structures, A_ℓ⊂∏ M_s/E of rank ≤λ for ℓ = 0, 1, and ϕ(x, y) ∈_κ, ω(Q^WF).Set p(x) = {ϕ(x, a) | a ∈ A_0}∪{ϕ(x, a) | a ∈ A_1} and p_s(x) = {ϕ(x, a) | a ∈ f_0(s)}∪{ϕ(x, a) | a ∈ f_1(s)}, where [, f_ℓ]_E represents A_ℓ in _E.Then, the following are equivalent: * ∏ M_s/E omits p. * {s ∈^κ| M_somitsp_s}∈ E_.Note that we have restricted both to the case of ϕ-types and to the case where E is an extender witnessing strength for simplicity and because those cases suffice for our assumption.We could remove these assumptions, instead requiring that A_ℓ is an element of the subset sort of .Proof: The structure ∏ M_s/E is (isomorphic to) j_E(g)(), where g is the function taking s ∈^κ to M_s and = [, s ↦ s()]_E.Then, since A_ℓ∈ V_λ∈_E, Łoś' Theorem for extenders tells us that _E “j_E(g)()omitsp" _E “j_E(g)()omits the ϕ-type generated byA_0, A_1" {s ∈^κ| g(s )omits the ϕ-type generated by f_0(s), f_1(s)}∈ E_ {s ∈^κ| M_somitsp_s}∈ E_† Still, this doesn't give a syntactic characterization of strength because it deals with type omission for types over sets, whereas Theorem <ref> deals just with type omission over the empty set (in the appropriate language).Thus, we are still left with the following question. Given κ≤λ, is there a syntactic property of logics such that _κ, ω(Q^WF) (or some other sub-logic of _κ, κ) satisfies this property iff κ is λ-strong? amsalpha | http://arxiv.org/abs/1708.07561v3 | {
"authors": [
"Will Boney"
],
"categories": [
"math.LO"
],
"primary_category": "math.LO",
"published": "20170824212957",
"title": "Model-theoretic Characterizations of Large Cardinals"
} |
These authors contributed equally to this work. These authors contributed equally to this work. Electronic address: [email protected] – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Quantum droplets are small clusters of atoms self-bound by the balance of attractive and repulsive forces. Here we report on the observation of a novel type of droplets, solely stabilized by contact interactions in a mixture of two Bose-Einstein condensates. We demonstrate that they are several orders of magnitude more dilute than liquid helium by directly measuring their size and density via in situ imaging. Moreover, by comparison to a single-component condensate, we show that quantum many-body effects stabilize them against collapse. We observe that droplets require a minimum atom number to be stable. Below, quantum pressure drives a liquid-to-gas transition that we map out as a function of interaction strength. These ultra-dilute isotropic liquids remain weakly interacting and constitute an ideal platform to benchmark quantum many-body theories.Quantum liquid droplets in a mixture of Bose-Einstein condensates L. Tarruell December 30, 2023 =================================================================Quantum fluids can be liquids – of fixed volume – or gases, depending on the attractive or repulsive character of the inter-particle interactions and their interplay with quantum pressure. Liquid helium is the prime example of quantum fluid. For small particle numbers it forms self-bound liquid droplets: nanometer-sized, dense and strongly interacting clusters of helium atoms. Understanding their properties, which directly reflect their quantum nature, is challenging and requires a good knowledge of the short-range details of the interatomic potential <cit.>. Very different quantum droplets, more than 2 orders of magnitude larger and 8 orders of magnitude more dilute, have recently been proposed in ultracold atomic gases <cit.>. Interestingly, these ultra-dilute systems enable a much simpler microscopic description, while remaining in the weakly interacting regime. They are thus amenable to well controlled theoretical studies.The formation of quantum droplets requires a balance between attractive forces, which hold them together, and repulsive ones that stabilize them against collapse. In helium droplets, the repulsion is dominated by the electronic Pauli exclusion principle, which arises from quantum statistics. In contrast, in ultracold atomic droplets the repulsion stems from quantum fluctuations, which are a genuine quantum many-body effect. These can be revealed in systems with competing interactions, where mean-field forces of different origins almost completely cancel out and result in a small residual attraction. There, beyond mean-field effects remain sizeable even in the weakly interacting regime. To first order they lead to the Lee-Huang-Yang repulsive energy <cit.>, comparable in strength to the residual mean-field attraction. Recently, ultracold atomic droplets have been realized in magnetic quantum gases with competing attractive dipolar and repulsive contact interactions <cit.>. In this case, the anisotropic character of the magnetic dipole-dipole force leads to the formation of filament-like self-bound droplets with highly anisotropic properties <cit.>. Given the generality of the stabilization mechanism, droplets should in fact also exist in simpler systems with pure isotropic contact interactions. Even though they were originally predicted in this setting <cit.>, their experimental observation has so far remained elusive.In this work, we observe ultracold atomic droplets in a mixture of two Bose-Einstein condensates with competing contact interactions. While a single-component attractive condensate collapses <cit.>, quantum fluctuations stabilize a two-component mixture with inter-component attraction and intra-component repulsion <cit.>. In this case, the repulsion in each component remains large and results in a non-negligible Lee-Huang-Yang energy. The beyond mean-field repulsion and the residual mean-field attraction have different density scalings (in three dimensions the energy densities scale as n^5/2 vs. n^2, respectively). Hence, they can always balance and stabilize droplets. Unlike their dipolar counterparts, these mixture droplets originate exclusively from s-wave contact interactions and are therefore isotropic. We demonstrate the self-bound character of mixture droplets and directly measure their ultra-low densities and micrometer-scaled sizes. Moreover, by comparison to a single-component attractive condensate, we confirm the quantum many-body nature of their stabilization mechanism. Similarly to the dipolar case <cit.>, we observe that for small atom numbers quantum pressure dissociates the droplets and drives a liquid-to-gas transition. We map out the corresponding phase transition line as a function of interaction strength and compare it to a simple theoretical model. Our measurements demonstrate that dipolar and mixture droplets share fundamental features despite the different nature of the underlying interactions. Given the simpler microscopic description of mixture droplets, which includes only well-known contact interactions, they constitute ideal systems to benchmark the validity of complex quantum many-body theories beyond the mean-field approximation.We perform experiments with two ^39K Bose-Einstein condensates in internal states |↑⟩≡|m_F=-1⟩ and |↓⟩≡|m_F=0⟩ of the F=1 hyperfine manifold. An external magnetic field allows to control the interactions parametrized by the intra- and inter-state scattering lengths a_↑↑, a_↓↓ and a_↑↓, see Fig. <ref>(a). These have been computed according to the model of ref. <cit.>. The residual mean-field interaction is proportional to δ a= a_↑↓+√(a_↑↑a_↓↓). The condition δ a=0 separates the repulsive (δ a>0) and attractive (δ a<0) regimes. The experiment starts with a pure condensate in state |↑⟩ loaded in one plane of a vertical blue-detuned lattice potential, see inset of Fig. <ref>(a). We choose a trapping frequency ω_z/2π=635(5) Hz large enough to compensate for gravity, but small enough to be in the three-dimensional regime. Indeed, the vertical harmonic oscillator length a_ho=0.639(3) μm exceeds the characteristic length of the hard Bogoliubov excitation branch by typically a factor of 3 <cit.>. A vertical red-detuned optical dipole trap provides radial confinement in the horizontal plane. In order to prepare a balanced mixture of the two states, we apply a radio-frequency pulse at B≈57.3 G, which lies in the miscible regime (δ a ≈ 7a_0, where a_0 denotes the Bohr radius) <cit.>. Subsequently, we slowly ramp down the magnetic field at a constant rate of 59 G/s and enter the attractive regime δ a<0 <cit.>. We then switch-off the vertical red-detuned optical dipole trap, allowing the atoms to evolve freely in the horizontal plane. The integrated atomic density is imaged in situ at different evolution times. We use a high numerical aperture objective (<1 μm resolution, 1/e Gaussian width) along the vertical direction and a phase-contrast polarization scheme <cit.> which detects both states with almost equal sensitivity (see Fig. <ref>(a) and <cit.>).Typical images of the mixture time evolution in the repulsive and attractive regimes are displayed in Fig. <ref>(b). For δ a=1.2(1) a_0>0 (top row), the cloud expands progressively in the plane, as expected for a repulsive Bose gas in the absence of radial confinement <cit.>. In contrast, in the attractive regime δ a=-3.2(1) a_0<0 (central row), the dynamics of the system is remarkably different and the atoms reorganize in an isotropic self-bound liquid droplet. Its typical sizeremains constant for evolution times up to 25 ms. In an analogous experiment with a single-component attractive condensate |↓⟩ of scattering length a=-2.06(2)a_0<0, the system instead collapses (bottom row). In our experimental geometry, quantum pressure can never stabilize bright solitons due to the presence of a weak anti-confinement in the horizontal plane <cit.>. At the mean-field level, the two-component attractive case has a description equivalent to the single-component one, provided that the scattering length a is replaced by δ a/2 and the density ratio between the two components is fixed to n_↑/ n_↓ = √(g_↓↓/ g_↑↑) <cit.>. However, the role of the first beyond mean-field correction is very different in the two systems, explaining their very different behavior. In the single-component case, the Lee-Huang-Yang energy depends on a and in the weakly interacting regime constitutes a negligible correction to the mean-field term. Its contribution could only be revealed in strongly interacting systems <cit.>. In contrast, in the mixture the mean-field and Lee-Huang-Yang energy densities scale as ℰ_MF∝δ a n^2 and ℰ_LHY∝(√(a_↑↑a_↓↓) n)^5/2, respectively. Since √(a_↑↑a_↓↓)≫|δ a|, for typical experimental parameters they balance at accessible atomic densities and stabilize liquid droplets <cit.>. Therefore, the existence of liquid droplets is a striking manifestation of quantum many-body effects in the weakly interacting regime.To further characterize the mixture, we perform a quantitative analysis of the images fitting the integrated atomic density profiles with a two-dimensional Gaussian <cit.>. We extract the atom number N and radial size σ_r and infer the peakdensity n_0= N/(π^3/2σ_r^2σ_z) by assuming a vertical size σ_z identical to the harmonic oscillator length a_ho. Fig. <ref>(a) (top and central panel) shows the time evolution of N and σ_r measured for the interaction parameters of Fig. <ref>(b). For δ a>0 (red circles) the gas quickly expands while its atom number does not vary. Instead, for δ a <0 (blue circles) the system is in the liquid regime and the radial size of the droplet remains constant at σ_r≈ 6 μm. Initially its atom number is N=24.5(7)× 10^3, corresponding to a peak density of n_0=1.97(8)× 10^14 atoms/cm^3. We attribute the subsequent decay of the droplet atom number shown in the top panel of Fig. <ref>(a) to three-body recombination. The observed timescale is compatible with the measured density and effective three-body loss rate <cit.>. By directly measuring the density of our droplets we confirm that they are more than 8 orders of magnitude more dilute than liquid helium and remain very weakly interacting. Indeed, the interaction parameters of each component are extremely small (n_↑a_↑↑^3, n_↓ a_↓↓^3∼10^-5).A closer view of the droplet size is displayed in the bottom panel of Fig. <ref>(a). At t ∼25 ms, σ_r starts to increase and the system behaves like the δ a>0 gas. Following refs. <cit.>, we attribute the dissociation of the droplet to the effect of quantum pressure, which acts as a repulsive force. As the atom number decreases, the relative weight between kinetic (ℰ_K) and interaction energies (ℰ_MF, ℰ_LHY) changes, for each energy term scales differently with N: ℰ_K∝ N, ℰ_MF∝ N^2 and ℰ_LHY∝ N^5/2. Below a critical atom number, kinetic effects become sufficiently strong to drive a liquid-to-gas transition. To support this scenario, Fig. <ref>(b) depicts the radial size and atomic density as a function of atom number. For δ a<0 (blue circles) we observe that both size (top panel) and density (bottom panel) remain constant at large N. For decreasing atom number, we observe a point where the size diverges and the density drops abruptly. This indicates a liquid-to-gas transition, which takes place at the critical atom number N_c. Below this value, the attractive gas is still stabilized by quantum fluctuations but expands due to kinetic effects, similarly to the repulsive mixture (δ a > 0, red circles).The liquid-to-gas transition is also expected to depend on δ a, as sketched in the inset of Fig. <ref>(b) (top panel). We explore the phase diagram by tuning the interaction strengths with magnetic field, see Fig. <ref>(a). Fig. <ref>(a) displays the measured size as a function of the atom number for magnetic fields corresponding to δ a between -5.5(1) a_0 and -2.4(1) a_0. The critical number N_c shows a strong dependence on the magnetic field. The top panel of Fig. <ref>(b) presents our experimental determination of the phase transition line. We observe that N_c increases when the attraction decreases, confirming that weakly bound droplets are more susceptible to kinetic effects and require a larger atom number to remain self-bound. Fig. <ref>(a) also yields the droplet size as a function of atom number and magnetic field. In the bottom panel of Fig. <ref>(b) we display the measurements obtained at a fixed atom number N=1.5(1)× 10^4, always larger than N_c for our interaction regime. As expected, the droplet size decreases as the attraction increases.We theoretically describe the system using a simple zero-temperature model. It is based on an extended Gross-Pitaevskii equation that includes both the vertical harmonic confinement and an additional repulsive Lee-Huang-Yang term. The latter is obtained assuming the Bogoliubov spectrum of a three-dimensional homogeneous mixture <cit.>. In Fig. <ref>(b) we compare the experimental results to the predicted critical atom number and droplet size (solid lines). We find qualitative agreement for the complete magnetic field range with no adjustable parameters. In the weakly attractive regime the agreement is even quantitative, similarly to the dipolar Erbium experiments of ref. <cit.>. In contrast, when increasing the effective attraction, the droplets are more dilute than expected. In particular, their size exceeds the theoretical predictions by up to a factor of three. This is almost one order of magnitude larger than our imaging resolution, excluding finite-resolution effects. Furthermore, the critical atom number is a factor of two smaller than the theoretical value. Interestingly, a similar discrepancy was reported for dipolar Dysprosium droplets, with a critical atom number one order of magnitude smaller than expected <cit.>. In this case, the deviation was attributed to an insufficient knowledge of the background scattering length. This explanation seems unlikely in the case of potassium <cit.>, where excellent interaction potentials are available <cit.>.Other physical mechanisms might be responsible for the diluteness of the observed droplets. Although our system is three-dimensional, the confinement along the vertical direction might affect the Lee-Huang-Yang energy, modifying its density and interaction dependence or introducing finite-size effects. A description of quantum fluctuations in the dimensional crossover between two and three dimensions is challenging, and goes beyond the scope of this experimental work. Interestingly, the almost perfect cancellation of the mean-field energy could reveal other corrections besides the Lee-Huang-Yang term. Higher-order many-body terms might play a role, as proposed in ref. <cit.> for single-component systems. Taking them into account analytically requires a good knowledge of the three-body interaction parameters of the mixture, which are non-universal and difficult to estimate in our interaction regime. Alternatively, our results could be compared to ab initio quantum Monte Carlo simulations, as recently performed in ref. <cit.>. Given the ultra-dilute character and simple microscopic description of our system, a direct comparison to different theoretical approaches could give new insights on yet unmeasured many-body effects.Future research directions include studying the spectrum of collective modes of the droplets <cit.>. Its unconventional nature not only provides a sensitive testbed for quantum many-body theories, but should also give access to zero-temperature quantum objects <cit.> not present in the dipolar case <cit.>. Our experiments should also enable to explore low-dimensional systems, where the enhanced quantum fluctuations make droplets ubiquitous <cit.>. Finally, a coherent coupling between the two components <cit.> is expected to yield effective three-body interactions <cit.> and provide control over the density dependence of the Lee-Huang-Yang term. We acknowledge insightful discussions with G. Astrakharchik, J. Boronat, A. Celi, I. Ferrier-Barbut, M. A. García-March, M. Lewenstein, P. Massignan, D. Petrov, and A. Recati. We thank A. Simoni and M. Tomza for calculations of the potassium scattering lengths. We are grateful to M. Bosch, V. Brunaud, V. Lienhard, A. Muñoz, L. Saemisch, J. Sastre, and I. Urtiaga for experimental assistance during the construction of the apparatus. We acknowledge funding from Fundació Privada Cellex, EU (MagQUPT-631633 and QUIC-641122), Spanish MINECO (StrongQSIM FIS2014-59546-P and Severo Ochoa SEV-2015-0522), DFG (FOR2414), Generalitat de Catalunya (SGR874 and CERCA program), and Fundación BBVA. CRC acknowledges support from CONACYT (402242/ 384738), JS from FPI (BES-2015-072186), PC from the Marie Skłodowska-Curie actions (TOPDOL-657439) and L. Tarruell from the Ramón y Cajal program (RYC-2015-17890).§ SUPPLEMENTARY MATERIAL§.§ Experimental detailsWe produce a pure ^39K BEC of 8.0(8) × 10^4 atoms in state |↑⟩≡|F=1, m_F=-1⟩ by sympathetic cooling with ^41K. Subsequently, we load the atoms into a single plane of a vertical blue-detuned optical lattice of 10.7 μm spacing. It is created by the interference of two 532 nm laser beams crossing at 1.43. The fraction of atoms loaded into other planes remains below 10%, as verified using a matter-wave focusing technique <cit.>. The lattice yields a trapping frequency ω_z/2π = 635(5) Hz along the vertical direction and a weak anti-confinement in the horizontal plane (estimated to be ∼ 1 Hz in the most anti-confined direction). During the droplet preparation sequence the atoms are radially confined by a red-detuned 1064 nm optical dipole trap. The value of the radial trapping frequency is adjusted for each magnetic field in order to avoid exciting collective modes of the droplets.For the mixture measurements, we create a balanced spin mixture of states |↑⟩ and |↓⟩≡|F=1, m_F=0⟩. To this end, we apply a radio-frequency (rf) pulse at a magnetic field B≈ 57.3 G, which lies in the miscible regime. The same pulse is used to calibrate the magnetic field with an uncertainty of 9 mG, given by the linewidth of the rf transition. The result is corrected for the mean-field interaction shift by comparison to a thermal gas. We then perform an 8 ms linear ramp to B≈ 56.9 G, corresponding to δ a ≈ 0.8a_0. Finally, we ramp down the magnetic field at a constant rate of 59 G/s to its final value, after which we remove the radial confinement.For the single-component experiments, we instead transfer all the atoms to |↓⟩ using a Landau-Zener sweep centered around B≈46.85 G. The magnetic field is subsequently ramped down in 10 ms to its final value. Below B=44.19 G, the scattering length a_↓↓ becomes negative and gives access to a weakly attractive single-component Bose gas. §.§ Inelastic lossesWe attribute the observed decay of the droplet atom number to three-body recombination. To confirm this hypothesis, we have determined the three-body loss rate of the system by studying the time evolution of its atom number and temperature in a deep optical dipole trap <cit.>. The measurements have been performed on single-component thermal clouds of |↑⟩, |↓⟩, and in mixtures of different concentrations. For the magnetic field range of the experiment we find that losses of |↓⟩ dominate over all the other processes and that the effective three-body loss rate of the mixture is proportional to the fraction of atoms in this state. For a pure BEC the measured rate is reduced by 3! <cit.>, yielding K_3^eff=7.5× 10^-28 cm^6/s. This value has a systematic uncertainty up to a factor of 2. The calculated two-body inelastic loss rates, associated to dipolar relaxation, are sufficiently small to neglect two-body processes on the timescale of the experiment <cit.>. §.§ Imaging and atom number calibrationThe main component of our imaging system is an objective consisting of a meniscus-asphere combination with a numerical aperture NA=0.44. We calibrate the magnification of our imaging system M=49.6(9) using Kapitza-Dirac diffraction on a one-dimensional lattice potential. Our imaging resolution is smaller than 0.97(4) μm (Gaussian 1/e width). This upper bound was determined by measuring the size of a two-component bright soliton created at B=54.851(9) G in an optical dipole trap of radial frequency ω_r/2π=110(1) Hz.In order to perform reliable in situ imaging of the droplets, which have column densities as large as n_c∼ 4× 10^10 atoms/cm^2, we employ a dispersive polarization phase-contrast technique <cit.>. We illuminate the atoms for 3 μs with a probe beam linearly polarized along a direction perpendicular to the applied magnetic field. The cloud rotates the polarization by a Faraday angle θ_F=c_F n_c. We record it by inserting a polarizer in a dark-field configuration before the camera.The Faraday coefficients of the two states are calibrated in two steps, using single-component BECs. First, we perform an absolute calibration of the atom number with an independent time-of-flight absorption imaging system, exploiting the atom number dependence on the critical BEC temperature. This measurement has a 25% uncertainty. For each state we then measure directly the polarization phase shift as a function of n_c, which we vary between 1.5×10^10 atoms/cm^2 and 9×10^10 atoms/cm^2by expanding the gas in a single-beam optical dipole trap. We choose an imaging beam frequency detuned by Δ_|↑⟩=-73(1) MHz (Δ_|↓⟩=-105(1) MHz) from the |↑⟩→|m_J'=-3/2, m_I'=-1/2⟩ (|↓⟩→|m_J'=-3/2, m_I'=1/2⟩)transition, with J the total electronic angular momentum and I the nuclear spin. For the magnetic field range of the experiment this yields nearly identical Faraday coefficients for the two components c_F,↑ = 1.1(1) × 10^-11 rad·cm^2 and c_F,↓ = 1.4(1) × 10^-11 rad·cm^2, in agreement with polarizability calculations. Here, the error bars correspond to the statistical error of the fit. However, the error in the atom number is limited by the 25% error of the time-of-flight calibration.We perform an independent crosscheck of our calibration procedure exploiting the cycling transition |F=2,m_F=-2⟩=|m_J=-1/2,m_I=-3/2⟩→|m_J'=-3/2,m_I'=-3/2⟩. To this end, we transfer the atoms from |↑⟩ to |F=2,m_F=-2⟩ with a rf pulse before imaging them on the cycling transition at different detunings. First, this allows us to compare our two-step calibration procedure with the two-level system predictions, finding good agreement. Second, by transferring a variable fraction of atoms to |F=2,m_F=-2⟩, we confirm the linearity of our imaging scheme and rule out the existence of collective effects. As a final check, we image droplets in situ using both imaging schemes. The measured radial sizes and atom numbers agree within the quoted error bars, confirming the validity of our calibration. §.§ Data analysisWe extract the atomic density profiles from the raw images taking into account the intensity of the probe beam and the transfer function of the polarizer <cit.>. In order to obtain the atom number N and radial size σ_r of the system, we fit the images with a two-dimensional Gaussian Ne^-x^2/σ_x^2-y^2/σ_y^2/(πσ_xσ_y). We find σ_x/σ_y≈1 for all our measurements and therefore we define σ_r=√(σ_xσ_y). This fitting function is chosen to simplify the comparison to the theoretical model (see theory section). We have verified that the zeroth and second moments of raw images give compatible results for the atom number and radial size, respectively. Since we do not observe the size along the vertical direction σ_z, we assume it to be identical to the corresponding harmonic oscillator length a_ho=√(ħ/(mω_z)), with ħ the reduced Planck constant and m the mass of ^39K. This assumption is supported by our theoretical model.We extract the critical atom number N_c by fitting the N-dependence of the droplet size with the phenomenological fitting function σ_r(N)= σ_0+A/(N-N_c). We have verified that using a different fitting function leads to changes in N_c well below the atom number calibration error. Therefore, the error bars in Fig. <ref>(b) correspond to the uncertainty of N (25%). §.§ Scattering lengthsWe use the most recent ^39K model interaction potentials for predicting the a_↑↑, a_↓↓ and a_↑↓ scattering lengths <cit.>, which were recently updated to describe the overlapping Feshbach resonances exploited in this work <cit.>. Throughout the paper, the scattering length error bars correspond to the experimental uncertainty of the magnetic field, and do not take into account the systematic errors of the scattering model. We believe that the deviations between the theoretical predictions and our measurements are unlikely to stem from errors in the scattering lengths. Indeed, by modifying them we only observe a shift of the theoretical predictions with δ a, and can never reproduce the critical atom number and droplet sizes observed for the largest attraction, see Fig. <ref>(b). Additionally, we have compared our results to the scattering lengths predicted by the model interaction potentials of ref. <cit.>, obtaining similar results <cit.>. §.§ Theoretical modelThe excitation spectrum of a three-dimensional homogeneous two component Bose gas consists of two branches: a (soft) density mode and a (hard) spin mode. At the mean-field level, the system becomes unstable for δ a=a_↑↓+√(a_↑↑ a_↓↓)=0, where the energy of the soft mode becomes imaginary. However, quantum fluctuations associated to the hard mode prevent the collapse. The characteristic length scales associated to the two Bogoliubov branches are given by ξ_d=ħ/(√(2)m c_d) and ξ_s=ħ/(√(2)m c_s), where c_d and c_s are the sound velocities of the density and spin modes, respectively <cit.>.Following <cit.> we describe the mixture with an effective low-energy theory. It consists in a zero-temperature extended Gross-Pitaevskii equation with an additional repulsive term, which includes the effect of the Lee-Huang-Yang energy. The corresponding energy functional readsℰ= ħ^2/2m(n_↑+n_↓)∇ϕ^2+V_trap(n_↑+n_↓)ϕ^2+n_↑n_↓4πħ^2δ a/mϕ^4 +256√(π)ħ^2/15m(a_↑↑n_↑)^5/2f(a_↑↓^2/a_↑↑a_↓↓,√(a_↓↓/a_↑↑))ϕ^5, with f(x,y)=∑_±(1+y±√((1-y)^2+4xy))^5/2/4√(2). Here, we have assumed identical spatial modes for the two components (Ψ_↑=√(n_↑)ϕ and Ψ_↓=√(n_↓)ϕ), and the density ratio n_↑/n_↓=√(a_↓↓/a_↑↑) which minimizes the energy of the hard mode <cit.>. Furthermore, we have included the experimental confinement along the vertical direction V_trap=1/2mω_z^2z^2.At the mean-field level Eq. (<ref>) is equivalent to a single-component Gross-Pitaevskii equation provided the parameter δ a is replaced by 2a. It includes however the additional beyond mean-field term – negligible for a single-component condensate – which stabilizes the mixture against collapse and is responsible for the formation of quantum droplets. Since the corresponding harmonic oscillator length a_ho typically exceeds ξ_s by a factor of three, the Lee-Huang-Yang term has been calculated assuming the Bogoliubov spectrum of a homogeneous three-dimensional system. Thus, it does not take into account finite-size or dimensional crossover effects due to the presence of the vertical harmonic confinement.We obtain the ground state of the system within a variational approach. We use a Gaussian ansatz ϕ=e^-r^2/2σ_r^2-z^2/2σ_z^2 for the spatial mode of the two components and determine the values of σ_r and σ_z which minimize the energy. For our experimental parameters, numerically solving the extended Gross-Pitaevskii equation yields essentially the same results, with deviations well below the experimental error bars. Therefore, the solid lines of Fig. <ref>(b) show the variational predictions. For the top panel the critical atom number is obtained by applying to the σ_r-N curves the same phenomenological fit used for the experimental data. We have verified that including the residual anti-confinement of the optical lattice does not modify appreciably the theoretical predictions. Furthermore, for deeply bound droplets the role of the magnetic dipole-dipole interactions (computed similarly to ref. <cit.>) can be neglected. We conclude that none of these effects can explain the discrepancies between our model and the experimental measurements.Finally, our theoretical model confirms the absence of mean-field self-bound solutions in our experimental geometry. Indeed, quantum pressure can only stabilize the equivalent of a bright soliton in the presence of a small confinement in the horizontal plane. Experimentally, we create a system without any radial confinement by using a blue-detuned optical lattice potential, which even provides a weak anti-confinement. Under these conditions, a single-component attractive condensate is expected to collapse for any atom number, as observed experimentally. The same behaviour is predicted for a two-component mixture in the absence of quantum fluctuations.99 Dalfovo2001 F. Dalfovo and S. Stringari, Helium nanodroplets and trapped Bose-Einstein condensates as prototypes of finite quantum fluids. http://aip.scitation.org/doi/abs/10.1063/1.1424926J. Chem. Phys. 115, 10078 (2001).Barranco2006 M. Barranco, R. Guardiola, S. Hernández, R. Mayol, J. Navarro, and M. Pi, Helium nanodroplets: An overview. J. Low Temp. Phys. 142 (2006).Petrov2015 D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose-Bose mixture. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.155302Phys. Rev. Lett.115, 155302 (2015).Lee1957 T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. https://journals.aps.org/pr/abstract/10.1103/PhysRev.106.1135Phys. Rev. Lett. 106, 1135 (1957).Kadau2016 H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid. https://www.nature.com/nature/journal/v530/n7589/full/nature16485.htmlNature 530, 194 (2016).FerrierPRL I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of quantum droplets in a strongly dipolar Bose gas. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.215301Phys. Rev. Lett. 116, 215301 (2016).Chomaz2016L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid. https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.041039Phys. Rev. X 6, 041039 (2016).FerrierJPB I. Ferrier-Barbut, M. Schmitt, M. Wenzel, H. Kadau, and T. Pfau, Liquid quantum droplets of ultracold magnetic atoms. http://iopscience.iop.org/article/10.1088/0953-4075/49/21/214004J. Phys. B 49, 214004 (2016).Schmitt2016 M. Schmitt, M. Wenzel, B. Böttcher, I. Ferrier-Barbut, and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid. http://www.nature.com/nature/journal/v539/n7628/full/nature20126.htmlNature 539, 259 (2016).Wenzel2017 M. Wenzel, F. Böttcher, T. Langen, I. Ferrier-Barbut, and T. Pfau, Striped states in a many-body system of tilted dipoles. https://arxiv.org/abs/1706.09388arXiv:1706.09388.Wachtler2016a F. Wächtler and L. Santos, Quantum filaments in dipolar Bose-Einstein condensates. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.93.061603Phys. Rev. A 93, 061603(R) (2016).Baillie2016 D. Baillie, R. M. Wilson, R. N. Bisset, and P. B. Blakie, Self-bound dipolar droplet: A localized matter wave in free space. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.021602Phys. Rev. A 94, 021602(R) (2016).Donley2001 E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Dynamics of collapsing and exploding Bose-Einstein condensates. https://www.nature.com/nature/journal/v412/n6844/full/412295a0.htmlNature 412, 295 (2001).Roy2013 S. Roy, M. Landini, A. Trenkwalder, G. Semeghini, G. Spagnolli, A. Simoni, M. Fattori, M. Inguscio, and G. Modugno, Test of the universality of the three-body Efimov parameter at narrow Feshbach resonances. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.053202Phys. Rev. Lett. 111, 053202 (2013).NoteSupplementary See Supplementary Materials.Hall1998 D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.1539Phys. Rev. Lett. 81, 1539 (1998).Bradley1997 C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein condensation of Lithium: Observation of limited condensate number. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.985Phys. Rev. Lett. 78, 985 (1997).NoteScatteringLength Here and in the following, error bars for the scattering lengths correspond to the 9 mG experimental uncertainty of the magnetic field and do not take into account the systematic uncertainties of the scattering length model.Chevy2016 F. Chevy and C. Salomon, Strongly correlated Bose gases. http://iopscience.iop.org/article/10.1088/0953-4075/49/19/192001/metaJ. Phys. B 49 192001 (2016).Bisset2016 R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuation. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.033619Phys. Rev. A 94, 033619 (2016).Wachtler2016b F. Wächtler and L. Santos, Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einstein condensates. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.043618Phys. Rev. A 94, 043618 (2016).D'Errico2007 C. D'Errico, M. Zaccanti, M. Fattori, G. Roati, M. Inguscio, G. Modugno, and A. Simoni, Feshbach resonances in ultracold ^39K. http://iopscience.iop.org/article/10.1088/1367-2630/9/7/223/metaNew J. Phys. 9, 223 (2007).Falke2008 S. Falke, H. Knöckel, J. Friebe, M. Riedmann, E. Tiemann, and C. Lisdat, Potassium ground-state scattering parameters and Born-Oppenheimer potentials from molecular spectroscopy. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.012503Phys. Rev. A 78, 012503 (2008).Bulgac2002 A. Bulgac, Dilute quantum droplets. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.89.050402Phys. Rev. Lett. 89, 050402 (2002).Cikojevic2017 V. Cikojević, K. Dželalija, P. Stipanović, L. Vranješ Markić, and J. Boronat, Ultradilute quantum liquid drops. https://arxiv.org/abs/1708.01553arXiv:1708.01553.Baillie2017 D. Baillie, R. M. Wilson, and P. B. Blakie, Collective excitations of self-bound droplets of a dipolar quantum fluid. https://arxiv.org/abs/1703.07927arXiv:1703.07927.Petrov2016 D. S. Petrov and G. E. Astrakharchik, Ultradilute low-dimensional liquids. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.100401Phys. Rev. Lett. 117, 100401 (2016).Matthews1999 M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Watching a superfluid untwist itself: recurrence of Rabi oscillations in a Bose-Einstein condensate. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.83.3358Phys. Rev. Lett. 83, 3358 (1999).Petrov2014 D. S. Petrov, Three-body interacting bosons in free space. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.103201Phys. Rev. Lett. 112, 103201 (2014).Hueck2017 K. Hueck, N. Luick, L. Sobirey, J. Siegl, T. Lompe, and H. Moritz, Two-dimensional homogeneous Fermi gases. https://arxiv.org/abs/1704.06315arXiv:1704.06315.Weber2003 T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, Three-body recombination at large scattering lengths in an ultracold atomic gas. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.123201Phys. Rev. Lett. 91, 123201 (2003).Kagan1985 Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Effect of Bose condensation on inelastic processes in gases. http://jetpletters.ac.ru/ps/1420/article_21579.shtmlJETP Lett. 42, 209 (1985).Simoni2016 A. Simoni, private communication.Gajdacz2013 M. Gajdacz, P. L. Pedersen, T. Mørch, A. J. Hilliard, J. Arlt, and J. F. Sherson, Non-destructive Faraday imaging of dynamically controlled ultracold atoms. http://aip.scitation.org/doi/10.1063/1.4818913Rev. Sci. Instrum. 84, 083105 (2013).Tomza2016 M. Tomza, private communication.Goldstein1997 E. V. Goldstein and P. Meystre, Quasiparticle instabilities in multicomponent atomic condensates. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.55.2935Phys. Rev. A 55, 2935 (1997). | http://arxiv.org/abs/1708.07806v1 | {
"authors": [
"C. R. Cabrera",
"L. Tanzi",
"J. Sanz",
"B. Naylor",
"P. Thomas",
"P. Cheiney",
"L. Tarruell"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170825164616",
"title": "Quantum liquid droplets in a mixture of Bose-Einstein condensates"
} |
Interpretation of Mammogram and Chest Radiograph Reports Using Deep Neural Networks Hojjat Salehinejad, Shahrokh Valaee, Senior Member, IEEE, Aren Mnatzakanian, Tim Dowdell,Joseph Barfett, and Errol ColakH. Salehinejad is with the Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto, Toronto, Canada, and Department of Medical Imaging, St. Michael's Hospital, University of Toronto, Toronto, Canada, e-mail: [email protected]. S. Valaee is with the Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto, Toronto, Canada, e-mail: [email protected]. A. Mnatzakanian, T. Dowdell, J. Barfett, and E. Colak are with the Department of Medical Imaging, St. Michael's Hospital, University of Toronto, Toronto, Canada, e-mail: {mnatzakaniana,dowdellt,barfettj,colake}@smh.ca. ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================pageStyleOne Radiology reports are an important means of communication between radiologists and other physicians. These reports express a radiologist's interpretation of a medical imaging examination and are critical in establishing a diagnosis and formulating a treatment plan. In this paper, we propose a Bi-directional convolutional neural network (Bi-CNN) model for the interpretation and classification of mammograms based on breast density and chest radiographic radiology reports based on the basis of chest pathology. The proposed approach is a part of an auditing system that evaluates radiology reports to decrease incorrect diagnoses. Our study revealed that the proposed Bi-CNN outperforms convolutional neural network, random forest and support vector machine methods.Breast density, chest radiograph, convolutional neural networks, mammography, radiology reports classification.§ INTRODUCTIONRadiology reports are an important means of communication between radiologists and other physicians <cit.>. These reports express a radiologist's interpretation of a medical imaging examination and are critical in establishing a diagnosis and formulating a treatment plan. Radiology is among the medical specialties with the highest rate of malpractice claims <cit.>. These claims can arise from a failure to communicate important findings, a failure of perception, lack of knowledge, and misjudgment. A failure to detect an abnormality on a medical imaging examination can lead to significant medical consequences for patients such as a delayed diagnosis. With a delayed or incorrect diagnosis, patients can present later with worsened symptoms and more advanced disease that may require more aggressive treatment or may be untreatable. In this paper, we propose an auditing system for radiologists that has two main components: a natural language processing (NLP) model to process and interpret a radiology report and a machine vision model that interprets the medical imaging examination. This auditing system reviews the radiologist's report and compares it with the interpretation of a machine vision model. The proposed system would notify the radiologist if there is a discrepancy between the two interpretations. Many investigators have aimed to develop machine vision models for the interpretation of medical imaging examinations, such as chest radiographs <cit.>, mammograms <cit.>, and head computerized tomography (CT) scans <cit.>. However, fewer attempts have been made for the NLP of radiology reports <cit.>. The focus of this paper is on the design and performance evaluation of the NLP component.We propose a bi-directional convolutional neural network (Bi-CNN) for the NLP of radiology reports. The Bi-CNN will be trained independently for two distinct report data types: breast mammograms and chest radiographs. In particular, this model will classify the degree of breast density and the type of chest pathology based on the radiology report content. Our proposed NLP model differs from a keyword search algorithm. It is capable of interpreting and classifying a radiology report. The proposed Bi-CNN has two independent input channels, where the order of non-padded input to one channel is the reverse of the non-padded input to the other channel. Performance of the proposed model is compared with a single channel convolutional neural network (CNN), random forest (RF), and support vector machine (SVM) models and a comparative study is conducted. § RELATED WORK The NLP models used for the classification and interpretation of radiology reports can be categorized into rule-based and machine learning models. Rule-based systems are usually built upon a series of “if-then” rules, which require an exhaustive search through documents. Such rules are typically designed by human experts. By contrast machine learning models do not require explicit rules. Instead, they try to train themselves iteratively and extract features from data.§.§ Rule-based ApproachesA radiology report mining system generally has three main components. These include a medical finding extractor, a report and image retriever, and a text-assisted image feature extractor <cit.>. For example, a set of hand-crafted semantic rules are presented in <cit.> for mining brain CT radiology reports. A natural language parser for medical reports is proposed in <cit.> that uses four-gram and higher order systems to assign a stability metric of a reference word within a given sentence. This model is inspired from Field theory and uses a dependency tree that represents the global minimum energy state of the system of words for a given sentence in radiology reports <cit.>. A classifier using extracted keywords from clinical chest radiograph reports is developed in <cit.>. In <cit.>, a method is developed to identify patients with mediastinal findings related to inhalational anthrax. These methods are applied on different case studies with different sets of rules. Such methods are not flexible enough to be applied to case studies for which they were not designed, and hence are not transferable, and their performance comparison requires specific implementation and fine tuning of rules.§.§ Machine Learning ApproachesA NLP study for a large database of radiology reports was conducted in <cit.> based on two classes: positive and negative radiology findings. These classes have different patient attributes such as age groups, gender, and clinical indications <cit.>. A collection of 99 musculoskeletal radiology examinations were studied using a machine learning model versus a naive Bayes classification algorithm, followed by a SVM to detect limb fractures from radiology reports <cit.>. A CNN was used in <cit.> as a supervised tool to map from images to label spaces. This system has an interactive component between supervised and unsupervised machine learning models to extract and mine the semantic interactions of radiology images and reports. Recurrent neural networks (RNN) along with the CNN have been used to interpret chest x-ray reports for a limited number of classes <cit.>. This CNN model learns from the annotated sequences produced by the RNN model. The RNN model with long short term memory (LSTM) is capable of learning long term dependencies between the annotations <cit.>. The model proposed in <cit.> uses a recurrent design to extract features from word representations. Compared with window-based neural networks, which uses a rolling window to extract features, this model generates less noise and is more stable. The max-pooling method distinguishes the words that have a major impact in the report. Machine learning models have also been developed for understanding medical reports in other languages, Korean being one example, using tools such as Naive Bayes classifiers, maximum entropy, and SVMs <cit.>. §.§ Learning Vector Representations of WordsWord2vec is a well-known model used for learning vector space representations of words. It produces a vector space from a large corpus of text with reduced dimensionality. This model assigns a word vector to each unique word in the corpus. Vectors with closer contexts are positioned in closer proximity in this space <cit.>. GloVe approach provides a global vector space representation of words. It uses global matrix factorization and local context windowing methods to construct a global log-bilinear regression model <cit.>. The word-word co-occurrence matrix of the words is a sparse matrix by nature. However, GloVe only uses the non-zero elements for training and does not consider individual context windows in a large corpus <cit.>. CNNs have been shown to work well on n-gram representations of data <cit.>. For example, a convolution layer can perform feature extraction from various n values of a n-gram model and perform personality detection from documents <cit.>. Dynamic k-Max pooling can operate as a global operation over linear sequences. Such dynamic CNNs can perform semantic modelling of sentences <cit.>. The model receives variable dimension size input sentences and induces a feature graph over the sentence that is capable of explicitly capturing short and long-range relations <cit.>.CNN models are also utilized for learning representations, opinion sentiment understanding, and analysis of products. For example, a CNN with a single output softmax layer can classify a vector representation of text with high accuracy <cit.>. The design of output layers in CNNs varies depending on the application of the model. A collaboration of RNNs and CNNs for feature extraction of sentences has been examined, where a CNN learns features from an input sentence and then a gated RNN model discourses the information <cit.>. In such system, a bi-directional long short term memory (Bi-LSTM) RNNs can sequentially learn words from question-answer sentences. The trained network can select an answer sentence for a question and present the corresponding likelihood for correctness <cit.>. § A BRIEF REVIEW ON CONVOLUTIONAL NEURAL NETWORKS As NLP for radiology reports is an interdisciplinary research effort, a brief review on CNN is provided in this section. A CNN is a machine learning model inspired by the visual cortex of cats <cit.>. A CNN has at least one layer of convolution and sub-sampling, in which the number of layers can increase in depth, generally followed by a fully connected multi-layer perceptron (MLP) network. A consecutive arrangement of convolutional layers followed by sub-sampling build a pyramid-shape model, where the number of feature maps increases as the spatial resolution decreases. Instead of hand-designing feature extractors, the convolutional layers extract features from raw data and the MLP network classifies the features <cit.>.A CNN typically has three main pieces which are local receptive fields, shared weights, and spatial and/or temporal sub-sampling. §.§ Local Receptive FieldsLocal receptive fields are made from artificial neurons, which observe and extract features from data such as edges in images. Let us consider an image as input to a CNN such as in Figure <ref>. A rectangular kernel with size M× N scans the input matrix at every single element (i.e., a pixel) and performs convolution such as h_u,v = σ(∑_m=0^M-1∑_n=0^N-1(I_u+m,v+n· w_m,n)+b_u,v)where the output is the state of the neurons at element I_u,v, σ(·) is a non-linear function, W is the shared weights matrix, and b is the bias vector. §.§ Shared WeightsThe term “shared weights" refers to the weight matrix W which appears repeatedly in the convolution operation for every element of I. Weight sharing reduces the number of free parameters of the model and hence improves generalization of the model<cit.>.For a general machine vision task, such as classification of natural images, the convolution kernel uses the correlation among spatial or temporal elements of the image to extract local features. Since the image is stationary (i.e., statistics of one part of the image are similar to any other part), the learned shared weights using one sub-region of the image can also be used at another sub-region to extract different features. §.§ Sub-samplingPassing the whole bag of extracted features after the convolution operation to a classifier is computationally expensive <cit.>. The feature pooling task (i.e. sub-sampling) generalizes the network by reducing the resolution of the dimensionality of intermediate representations (i.e. feature maps) as well as the sensitivity of the output to shifts and distortions <cit.>. The two most popular subsampling methods are mean-pooling and max-pooling. By dividing the feature map into a number of non-overlapping rectangle-shape sub-regions, the mean-pooling computes the average of features sitting inside a sub-region as the pooled feature of that sub-region. Max-pooling performs the same task as mean-pooling but with a maximum operator. A study on these two operations is provided in <cit.>.§ THE PROPOSED METHOD An auditing system for radiology is presented in Figure <ref>. The main components of the system are a machine vision model to interpret the input image and a NLP component to interpret the corresponding radiology report. The system notifies the radiologist if there is a discrepancy between the two components of the system. In this paper, our focus is on the design and evaluation of the NLP component. §.§ PreprocessingIn general, radiology reports include major sections such as “Indication", “Findings", “Impression", andthe name of the reporting radiologist. However, there are variations in report formatting due to “radiologist style" or institutional requirements. Despite brute-force search approaches, the deep learning models have the advantage of requiring the least amount of data cleaning and preprocessing due to their natural adaptation to the data and non-linear feature extraction ability.In the preprocessing step, the model extracts the “Findings" and “Impression" sections of the report and performs tokenization and string cleaning. Some of the major tasks include converting upper-case characters to lower-case, removing unnecessary punctuation and symbols, and separating the remainder from the attached string. We add every unique word to a vocabulary dictionary. Each vocabulary has a unique associated index which represents it in the sentence vector. For example, given the sentence “The breasts show scattered fibroglandular tissue.", the entire vocabulary dictionary would consist of 6 words (breasts, show, scattered, fibroglandular, tissue, the), which is represented as a list of integers such as [1,...,6]. A vector to the length of the longest report in terms of words count in the dataset represents the report. Padding fills up the gap for shorter reports. In the character embedding step, each integer is mapped to a high dimensional (i.e., D) vector with a uniform random distribution.In such a high dimensional space, due to the “curse of dimensionality",the vectors are considered independent <cit.>, <cit.>. Character embedding vector generation for medical vocabularies based on the correlation among the vectors (i.e., similar to Word2Vec) is an interesting topic for further investigation. §.§ Bi-CNN Model ArchitectureAn input report with N words can be represented as a sequence X= [x_1,... ,x_N], where each word is a vector x_n∈ℝ^D, <cit.>. As Figure <ref> shows, the proposed architecture has two input channels followed by two independent convolution layers. Both channels have an identical design, except the order of non-padded input to one channel is the reverse of the non-padded input to the other channel. In the example in Figure <ref>, the dependencies between feature vectors (x_3,x_4,x_5) are visible to the kernels in both channels. However, the dependencies between feature vectors (x_1,x_0,0) are only visible to the kernels in Channel 2, while these dependencies are not visible to the kernels in Channel 1. Other examples are the feature vectors (x_0,0,0) in Channel 2 and feature vectors (x_4,x_5,0) and (x_5,0,0) in Channel 1, which are not visible in the corresponding other channel.Each channel in the proposed model in Figure <ref> has three filters with varying window sizes K∈{3,4,5} that slide across the input layer <cit.>, <cit.>. The filters extract features from the input layer to construct feature maps of size (N-K+1)× F, where F is the number of feature maps for each filter. Each feature map h_(N-K+1) has its own shared weight W_K× D and bias b_ N-K+1. The value of a hidden neuron m is h_m=σ(∑_k=1^K∑_d=1^D x_m+k-1,d· w_k,d +b_m).The window word is x_m:m+K-1 and σ(·) is a rectified linear unit (ReLU) activation function defined as σ(z)=max(0,z) where z∈ℝ. The max pooling <cit.> extracts the most significant feature from the feature map of a filter asĥ_l=max({h_1,..,h_N-K+1}).The output of the max-pooling layer contains the max-pooled features ĥ=[ĥ_1,...,ĥ_L] where the length of the feature vectors is L=C· F and C is the number of classes. For example, for the breast density classification with five classes we have L=5F (see Figure <ref>). The features from the input channels are concatenated and passed to a fully connected perceptron network. The output of the softmax layer is the probability distribution over all the labels P (e.g., for mammograms the breast density classes). The value of output unit p isy_p = ϕ(∑_l=1^L (ĥ_1,l· r_1,l)· w_l,p^ho + b_p^o)where w_l,p^ho is the weight of connection from the hidden unit l in the hidden layer h to the output unit p in the output layer o,b_p^o is the bias of the output unit p, and ϕ(z) is the softmax activation function defined asϕ(z_p)=e^z_p/∑_j=1^Pe^z_jfor p=[1,...,P]. §.§ TrainingAdam optimizer is a first-order gradient-based optimization method with integrated momentum functionality <cit.>. During training, the momentum helps to diminish the fluctuations in weight changes over consecutive iterations. The drop-out regularization method randomly drops units along with their connections from theconcatenated layer (i.e. the input layer to classifier) to the output layer using a binary mask vector r_1× L with Bernoulli random distribution <cit.>. §.§ Evaluation SchemeSince we are dealing with discrete categories, we use cross-entropy between a predicted value y^(i) from the network and the real label t^(i) to measure the loss of networks such asℒ̂(Y, T) = -1/R∑_i=1^Rt^(i)ln(y^(i))+(1-t^(i))ln(1-y^(i))where R is the number of reports, Y is the set of network predictions, and T is the set of targets to predict. Adding the weight decay term (i.e., L_2 regularization) to the loss function helps the network to avoid over-fitting while training such asℒ(Y,T) =ℒ̂(Y, T) + η(𝐖^ho_2)where 𝐖^ho is the weight matrix of connections between the hidden layer h and the output layer o and η is the regularization control parameter. § THE DATASETS Our institutional review board approved this single-center retrospective study with a waiver for informed consent. The Research Ethics Boards (REB) number is 17-167. A search of our Radiology Information System (RIS) (Syngo; Siemens Medical Solutions USA Inc, Malvern, PA) was preformed for mammography and chest radiograph reports using Montage Search and Analytics (Montage Healthcare Solutions, Philadelphia, PA). This search identified 4,080 mammography reports and 1,030 chest radiographic reports (see Table <ref>). These reports were exported from the RIS and removed of any patient identifying information. §.§ Mammogram Reports DatasetThe mammography reports dataset (MRD) contained 4,080 reports. Breast density was classified in each report according to the American College of Radiology Breast Imaging Reporting and Data System (BI-RADS) classification system <cit.>(Figure <ref>). Breast density reflects the relative composition of fat and fibroglandular tissue. Almost entirely fatty refers to breasts with less than 25% areas of fibroglandular tissue. Scattered ares of fibroglandular density is used for breasts composed of 25-50% fibroglandular tissue. Heterogeneously dense breasts are composed of 50-75% fibroglandular tissue and the extremely dense breasts have greater than 75% of fibroglandular tissue density. Descriptive statistics of this dataset are presented in Table <ref> and sample reports are presented in Table <ref>.§.§ Chest Radiographs Reports DatasetThe chest radiograph dataset (CRRD) consisted of 1,030 reports. These reports were classified into 11 categories which includes normal examinations and ten pathologic states (Figure <ref>). Descriptive statistics of this dataset are presented in Table <ref> and sample reports are presented in Table <ref>. This dataset has more categories, fewer reports per category, and is less imbalanced when compared with the MRD. § EXPERIMENTS AND RESULTS ANALYSISIn this section, we compare performance of the proposed Bi-CNN model with the random forest (RF), SVM, and CNN models. The experiments were conducted on both the mammogram and chest radiograph datasets. The CNN and Bi-CNN models are implemented in TensorFlow <cit.> and the RF and SVM models are implemented using the classifiers in scikit-learn <cit.>. The experiments are conducted on a DevBox with an Intel Core i7-5930K 6 Core 3.5GHz desktop processor, 64 GB DDR4 RAM, and two TITAN X GPUs with 12GB of memory per GPU. §.§ Parameters SettingUnless stated, the CNN and Bi-CNN models are trained with a mini-batch size of 64, drop-out probability of 0.5, filter sizes {3, 4, 5} and 120 feature maps per filter size. The number of training iterations is set to 50 and the initial learning rate for Adam optimizer is set to 0.001 <cit.>.An exponential decay adaptive learning rate is applied. The weights at output layers are initializedusing the Xavier method <cit.>, the weights in the convolutional layer are selected based on normal distribution with standard deviation of 0.1, and biases are set to 0.1. The activation function before the max-pooling layer is ReLU <cit.>. The L_2 regularization is set to 1.0 × 10^-4 and early-stopping is applied.For all the experiments, 70%, 15%, and 15% of data is allocated for training, validation, and test, respectively. The data is shuffled before splitting. In each experiment, the model is cross-validated over 30 independent experiments and the results after statistical testing are reported.§.§ Performance Evaluation and AnalysisWe experimented with various configurations of the RF <cit.>, SVM <cit.>, and CNN <cit.> models.* RF: Instead of using a bag-of-the-words technique, which uses the frequency of the words in a query as the features, a n-gram model with a lower boundary of 1 and upper boundary n∈{1,2,3} performs the feature extraction of reports. We use an implementation of RF <cit.> which combines classifiers by averaging their probabilistic predictions. The number of estimators is set to 10.* SVM: Similar to RF, the SVM model uses n-gram feature extraction. The SVM model is deployed for two different kernels, a “sigmoid" and a “polynomial" with degree three.* CNN: The CNN implementation proposed in <cit.> for text classification and later used in <cit.> for the classification of radiology head CT reports is used. This model has a single input channel and the order of input words is similar to the order in the report. The model has a convolution layer followed by a max-pooling layer and a “softmax" classifier. The studies are for kernel sizes k∈{1,2,3}. * Bi-CNN: The proposed Bi-CNN with two input channels. The settings are similar to CNN. The experiments are for breast density classification and chest radiograph classifications based on chest pathology using the MRD and CRRD, respectively.The results of performance comparisons between the RF, SVM, CNN, and Bi-CNN models are presented in Table <ref>. As n increases in n-gram for RF, it considers the dependency between a greater number of words (i.e. n). The RF with 3-gram model has better performance than the 2-gram and 1-gram RF models. The SVM with polynomial of degree three (i.e., “poly") and “sigmoid" kernels almost have the same performance. The number of kernels in the CNN models have a minor impact on accuracy. However, the CNN models with 3-kernel have better performance than models with 1-kernel and 2-kernel. The proposed Bi-CNN with 3-kernel has the best performance compared to the other models. The convergence behaviour of CNN and Bi-CNN over training iterations for the validation dataset is presented in Figure <ref>. For a large learning rate value (i.e., λ=0.1), the models converge rapidly to a local solution. However, for smaller learning rates, the models converge more slowly but with greater stability towards a better solution. As the results in Table <ref> show, the CNN model converges with a lower accuracy for three different learning rates λ∈{0.1,0.01,0.001}. For λ=0.001, the best performance is achieved by Bi-CNN for both MRD and CRRD. One of the advantages of Bi-CNN is adding diversity into the model through the integration of two different representations of feature vectors into the model. This diversity helps the CNN avoid convergence to local solutions with low accuracy. § CONCLUSION Radiology reports contain a radiologist's interpretation of a medical imaging examination. Vast numbers of such reports are digitally stored in health care facilities. Text mining and knowledge extraction from such data may potentially be used for double checking radiologist interpretations as part of an audit system. Furthermore, these methods can be used to audit utilization of limited health care resources and to enhance the study of patient treatment plans over time.In this paper, we have proposed a Bi-CNN for the interpretation of radiology reports. We have collected and anonymized two datasets for our experiments: a mammogram reports dataset and a chest radiograph reports dataset. The Bi-CNN has two input channels where one channel represents the features of the report (including zero-padding) and the other channel represents the reverse non-padded features of the report. The combination of forward and reverse order of feature vectors represents more information about dependencies between feature vectors. Our comparative studies demonstrate that the Bi-CNN model outperforms the CNN, RF, and SVM methods. § ACKNOWLEDGEMENTThe authors thank the support of NVIDIA Corporation with the donation of the Titan X GPUs used for this project.IEEEtran | http://arxiv.org/abs/1708.09254v3 | {
"authors": [
"Hojjat Salehinejad",
"Shahrokh Valaee",
"Aren Mnatzakanian",
"Tim Dowdell",
"Joseph Barfett",
"Errol Colak"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170824122122",
"title": "Interpretation of Mammogram and Chest X-Ray Reports Using Deep Neural Networks - Preliminary Results"
} |
pictexa.texempty (60,40)(-2,-2) 2mm1pt (0,0)(60,0) (0,0)(0,40) (15,0)(35,35) 3(50,0)(50,35) 3(0,35)(50,35) 2(15,0)(15,35) (15,0)194.71245.4978 (17.5,10.5) (1,37)h (51,2)n(h) | http://arxiv.org/abs/1708.07531v2 | {
"authors": [
"Cristian Baldenegro",
"Sylvain Fichet"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170824192537",
"title": "Anomalous gauge interactions in photon collisions at the LHC and the FCC"
} |
[email protected] for Protein Research, Osaka University, 3-2 Yamadaoka, Suita, Osaka, 565-0871, JapanIn the protein sequence space, natural proteins form clusters of families which are characterized by their unique native folds whereas the great majority of random polypeptides are neither clustered nor foldable to unique structures. Since a given polypeptide can be either foldable or unfoldable, a kind of “folding transition” is expected at the boundary of a protein family in the sequence space. By Monte Carlo simulations of a statistical mechanical model of protein sequence alignment that coherently incorporates both short-range and long-range interactions as well as variable-length insertions to reproduce the statistics of the multiple sequence alignment of a given protein family, we demonstrate the existence of such transition between natural-like sequences and random sequences in the sequence subspaces for 15 domain families of various folds. The transition was found to be highly cooperative and two-state-like. Furthermore, enforcing or suppressing consensus residues on a few of the well-conserved sites enhanced or diminished, respectively, the natural-like pattern formation over the entire sequence. In most families, the key sites included ligand binding sites. These results suggest some selective pressure on the key residues, such as ligand binding activity, may cooperatively facilitate the emergence of a protein family during evolution. From a more practical aspect, the present results highlight an essential role of long-range effects in precisely defining protein families, which are absent in conventional sequence models. protein folding molecular evolution protein design sequence analysis Monte Carlo simulation§ INTRODUCTIONNatural proteins can be classified into families based on their sequence similarity<cit.>. This is considered to be primarily a consequence of molecular evolution: proteins evolved from a common ancestral protein share similar sequences. However, evolution alone does not account for the existence of relatively well-defined (domain) families that are distributed rather discretely than continuously in the sequence space <cit.>. A key to understanding the family distribution is protein folding. As observed in protein structure classification databases <cit.>, each protein family corresponds to a unique three-dimensional fold, suggesting the existence of physical constraints imposed on protein sequences during the evolutionary process to maintain the fold <cit.>. While protein structures can tolerate great many mutations to the extent that proteins with little sequence similarity can share the same fold, residue conservation patterns reflect the structural context of protein sequences. This fact has long been exploited in protein structure prediction in the form of position-specific scoring matrices <cit.> and, more recently, direct-coupling analysis and related methods <cit.>.Under a given physiological condition, a polypeptide is either able or unable to fold into some unique structure. This suggests the existence of a “folding transition” at the border between an “island” of a protein family and the “sea” of random polypeptide, that is analogous to the folding transition of a protein molecule in the conformational space <cit.>. It should be noted, however, that there are many families in the sequence space so that a sequence moving in the sequence space may fall into any one of these families. This is in contrast to protein folding in the conformational space where there is usually only one unique native structure for a given protein sequence. Furthermore, a (structural) domain, rather than a whole protein sequence, should be considered as a unit of folding as a particular domain may be found in different proteins in combination with other, different, domains. Therefore, the system in which the analogy of protein folding holds should be limited to the vicinity of each protein domain family rather than the entire sequence space. In the following, we focus on the folding transition in a sequence subspace around a given protein domain family. Although biologically important, intrinsically disordered proteins <cit.> are excluded from the present study for the following two reasons. First, the analogy of the folding transition may not apply to those proteins. Second, it is difficult to obtain reliable and comprehensive multiple sequence alignments for this class of proteins <cit.>, which are required for parameter estimation of the statistical model employed in the present study. There have been a number of theoretical and computational studies on subjects related to the sequence space such as foldability and design <cit.>, molecular evolution within an island <cit.> and between islands <cit.>, or the size and/or distribution of islands in the sequence space <cit.>. On the contrary, relatively little attention has been paid to the transition between an island (a set of sequences belonging to the same family) and the sea (the set of sequences that do not belong to the family) apart from a few exceptions. In the context of protein design, <cit.> theoretically predicted the existence of a “folding transition”.In a study of hierarchical evolution of protein fold families based on a simple model of the evolutionary selection by native stability using a generic contact potential <cit.>, <cit.> observed a sharp transition at a certain design temperature. In neither of these studies, however, the nature of the transition was investigated further. Characterizing the folding transition in the sequence space may help understand essential features that constitute a protein family and possible evolutionary trajectories that may have led to the emergence of a protein family. It also has practical importance in identifying new family members and designing new proteins. In the following, we investigate the folding transition in the sequence subspaces for 15 protein domain families including all-α, all-β, α/β and other folds by performing extensive Monte Carlo (MC) simulations of the modified lattice gas model (LGM) of protein sequence alignment <cit.> that coherently integrates long-range interactions and variable-length insertions. Using the LGM, the existence of a sharp two-state transition between natural-like sequences and random sequences in the sequence subspace is demonstrated. Furthermore, the nature of the transition is examined in detail by analyzing residue distribution of each site along the transition as well as by performing virtual “mutation” experiments.§ THEORYWe briefly summarize the theory to the extent that is necessary for understanding the present study. For more details about the formulation of the LGM as well as the algorithms for MC simulations and parameter optimization, refer to the previous papers <cit.>. The LGM for a given Pfam <cit.> family consists of N “core” sites and N-1 “insert” sites which respectively correspond to the “match” states and “insert” states of the Pfam profile hidden Markov model (HMM) of length N, excluding the N- and C-terminal insert states. Exactly one of the 21 residue types (including the “delete” symbol) can exist on each core site whereas arbitrarily many (including zero) residues out of the standard 20 residue types can reside at each insert site. The array of core and insert sites are connected via “bonds” (solid arrows in Fig. <ref>) that reflect the linear polypeptide structure. A pair of model sites connected via a bond are called a “bonded pair” in the following. Between core sites more than 2 residues apart along the sequence, there may be interactions (dashed lines in Fig. <ref>) based on a representative native structure of the family. Two sites are defined to be interacting if the residues aligned to those sites are in contact in the corresponding representative native structure. Two residues are defined to be in contact if any non-hydrogen atoms in those residues are within 5Å. Interactions are defined only between core sites for simplicity. Interacting core sites are referred to as “non-bonded” pairs in the following. Let 𝐚 = a_1⋯ a_L be an amino acid sequence of L residues and an LGM ℳ of length N consist of core sites O_1,⋯, O_N and insert sites I_1 ,⋯, I_N-1. An alignment between the sequence 𝐚 and the model ℳ is represented as a sequence of pairs of a model site (core or insert) and a residue: 𝐗 = X_1⋯ X_L_𝐗 where L_𝐗 is the length of the alignment and each X_i is a pair such as (S,a) with S∈{O_i}_i=1,⋯,N∪{I_i}_i=1,⋯,N-1 and a ∈{a_1,⋯, a_L}. For example, given an amino acid sequence, say , and a model of length N=6 (Fig. <ref>), one of many possible alignments is represented as 𝐗 = X_1⋯ X_9 = (O_1,𝙺) (O_2,𝙲) (O_3,𝙵) (I_3,𝙿) (I_3,𝙳) (I_3,𝙶) (O_4,-) (O_5,𝚅) (O_6,𝚆). Note there are multiple occurrence of the insert site I_3 whereas other insert sites are completely absent in this particular alignment. Since there may be any number of residues at each insert site, the alignment length is variable.Based on this representation of sequence alignment, the energy function of alignment 𝐗 is defined asE(𝐗)=-∑_k=1^L_𝐗-1J(X_k,X_k+1) -∑_(k,l) ∈𝒯K(X_k,X_l)-∑_k=1^L_𝐗μ(X_k)where J and K are short-range and long-range interaction parameters, respectively, μ's are chemical potentials, and 𝒯 indicates the set of all the interacting non-bonded pairs. The short-range interactions act only between bonded pairs of residues that are consecutive in the alignment (i.e., between X_k and X_k+1 in Eq. <ref>). The long-range interactions act between residues that are aligned to interacting non-bonded pairs. The chemical potentials are so called because they are used to control the residue densities of each site. Only J and K parameters constitute intrinsic energy, and they are to be determined from a given (observed) multiple sequence alignment (MSA) of the family sequences (see below).We assume that the probability P(𝐗) of obtaining an alignment 𝐗 is given by the Boltzmann distribution:P(𝐗) = exp[-E(𝐗)/T]/Ξ[T]where T is the “design (selection) temperature”<cit.> in energy unit and Ξ[T] is the partition function Ξ[T] = ∑_𝐗exp[-E(𝐗)/T].Here, the summation is over all possible alignments (𝐗) of all possible amino acid sequences with the model. Since the alignment length can vary, this ensemble is considered to be a grand canonical ensemble.Given the parameters J, K and μ, we can sample (alignments of) sequences according to the probability distribution Eq. (<ref>) by running grand canonical Monte Carlo (MC) simulations <cit.>.We define the standard condition as the system with μ(X_k) = 0 for all X_k (i.e., no perturbation), and the natural condition as the standard condition with T = 1.The parameters J and K are determined iteratively so that the pair number densities of residues for bonded and non-bonded pairs over the samples produced by MC simulations under the natural condition match those observed in the given MSA <cit.>. This optimization corresponds to minimizing the following free energy function under the natural condition (T=1 in particular):F[T] = -max_J,K(⟨E|_⟩obs - Ω[T])where ⟨E|_⟩obs is the average energy of the observed MSA and Ω[T] = -TlnΞ[T] is the grand potential. It can be easily shown that this is equivalent to the principle of maximum entropy <cit.>.The design temperature in the LGM controls the mutation rate uniformly over the entire sites. The higher the temperature, the higher the mutation rate. The temperature of the natural condition T=1 defines the unit of energy and temperature. In other words, the temperature for each domain family is in a relative unit defined by the natural condition (unlike the absolute scale as in <cit.>). The mutation rate here means the number of accepted mutations per attempted mutation, and mutations are attempted at a constant time interval. The relationship between the present “energy” and physical folding free energy has been recently established by <cit.>.§ RESULTS§.§ Sequence pattern formation is accompanied by a cooperative two-state transitionWe have determined the energy parameters for each of the 15 families of relatively small domains listed in Table <ref> by using MC simulations following the procedure described previously <cit.>. Using the optimized parameters, we run a multicanonical MC simulation <cit.> for each family from which the specific heat,C(T) = ⟨E^2|_⟩T - ⟨E|^⟩2_T/T^2,was computed as a function of temperature (Fig. <ref>A). For all the families but HATPase_c, a single peak in the specific heat was observed, indicating the existence of a transition. An increase in the specific heat of HATPase_c at higher temperatures was due to excessively long insertions. Depending on the family, the peak temperature (T_m; see Table <ref>) ranged between T=1.009 (for DnaB) and T=1.033 (for zf-C2H2).The energy distribution obtained from grand canonical MC simulations at T_m (Fig. <ref>B) showed a clear two-state transition between low energy and high energy states. In order to confirm if the low-energy sequences are indeed similar to the natural sequences, we compared the profile HMM of each family against the set of sequences generated by the MC simulations. Out of 10,000 sequences generated at each temperature ranging from T=0.95 to T=1.05, a profile HMM search was performed (using the hmmsearch program <cit.>) to identify the sequences that significantly matched the Pfam profile HMM with full sequence E-value < 0.01 (Fig. <ref>C). The hit ratio (the fraction of sequences significantly similar to the family's profile HMM) shows the clear two-state transition between natural-like sequences and non-natural-like (≈ random) sequences around the transition temperature T_m for each family, and the sequences generated at lower temperatures were indeed more similar to natural sequences. Since these natural-like sequences are expected to fold into the native fold of the domain family but non-natural-like sequences are not, this transition can be regarded as a “folding transition” in the sequence subspace. The hit ratios at T=1 (the natural condition) ranged from 73.3% (for zf-C2H2) and 92.5% (for HATPase_c), indicating there were a non-negligible fraction of non-natural-like sequences at the natural condition. In particular, the C2H2-type zinc finger domain (zf-C2H2) consisting of only 23 core sites exhibited a large fraction of non-natural-like sequences at lower temperature. For all the families examined, the hit ratio was a decreasing function of temperature. For later reference, we defined a “disordering temperature”, T_D, of each family as the lowest temperature at which the hit ratio was less than 1% (column “T_D” in Table <ref>). §.§ Comparison with a model without long-range interactions: An example The long-range interations incorporated in the LGM are necessary (though not sufficient) for the model to exhibit the cooperative transition. Conventional sequence models, however, do not include such long-range interactions. In other words, they are essentially one-dimensional models. It is a well-known fact that one-dimensional systems do not exhibit cooperative (phase) transitions at finite temperatures <cit.>. Although theoretically trivial, it may be instructive to demonstrate the absence of the transition in such a system. Within the framework of the LGM, we can simply discard the long-range interations and train a model without the K parameters in the energy function (Eq. <ref>). In this “J-only” model, the J parameters for short-range interactions can be obtained exactly <cit.>. We trained a J-only model for the SH3 domain family (Table <ref>) and compared its specific heat, hit ratio and energy distributions with those of the original “J+K” model which includes long-range interactions (Fig. <ref>).The specific heat of the J-only model monotonically increased with temperature (Fig. <ref>A) which is in sharp contrast with the J+K model. The upper limit of the temperature range for the J-only model was set to T=1.17 because it was only at this temperature the hit ratio became less than 1% (Fig. <ref>B), which is significantly greater than the corresponding disordering temperature for the J+K model (T_D = 1.05; Table <ref>). Accordingly, the hit ratio decreases rather slowly as the temperature increases. The energy distributions of the J-only model at varying temperatures are unimodal and the mode energy value continuously increases with increasing temperature (Fig. <ref>C). This indicates that there is no clear boundary between natural-like sequences and random sequences.These results for the J-only model clearly demonstrate the absence of cooperative transition in the model without long-range interactions. It it expected that other conventional sequence models also behave similarly.In the following, we exclusively use the J+K models.§.§ Less conserved sites order at higher temperatures, better conserved sites order at lower temperaturesNext, the transition was characterized in terms of residue distribution at each site over a temperature range. The “naturality” of each core site of simulated sequences at a given temperature was evaluated by the Kullback-Leibler divergence <cit.> D_O_i(T):D_O_i(T) = ∑_a=1^21⟨n_(O_i,a)|_⟩Tlog⟨n_(O_i,a)|_⟩T/⟨n_(O_i,a)|_⟩obswhich measures the difference between the residue distribution ⟨n_(O_i,a)|_⟩T of the site O_i at temperature T from the observed residue distribution ⟨n_(O_i,a)|_⟩obs. An example for the SH3 domain is provided in Fig. <ref>C. Lower divergence values imply more natural-likeliness. By construction, the divergence is nearly 0 for all the sites at T = 1, but the values varied largely among different sites for higher temperatures. In general, better conserved sites (c.f., Fig. <ref>A) have larger divergence at high temperatures. In the following, we use the D_O_i(T_D), i.e., divergence at the disordering temperature T=T_D, as a measure of conservation. As expected, sites with a larger contact number <cit.> (i.e., the number of non-bonded contacts a site makes with other sites; see Fig. <ref>B for example) tend to be more conserved as can be seen in their correlations (Fig. <ref>A). Ordering of each site can be quantified via the derivative of the divergence with respect to temperature (c.f., Fig. <ref>D):C_O_i(T)= dD_O_i(T)/dT= 1/T^2∑_a=1^21[⟨n_(O_i,a)E|_⟩T - ⟨n_(O_i,a)|_⟩T⟨E|_⟩T] ×log[⟨n_(O_i,a)|_⟩T/⟨n_(O_i,a)|_⟩obs]which we call “differential divergence” in the following. Note that this is a weighted sum of covariance between the number density and total energy where the unnormalized and signed weights are log[⟨n_(O_i,a)|_⟩T/⟨n_(O_i,a)|_⟩obs]. Thus, the peaks of this quantity provide the site-specific transition point at which a large change in the residue distribution is accompanied with a large change in total energy. Precisely, the site-wise transition temperature T_m,O_i is defined byT_m,O_i = argmax_T < T_D C_O_i(T)(see Fig. <ref>E for an example). For some sites, this temperature cannot be defined because of the absence of a unique peak for T < T_D (due to lack of non-bonded interactions; this was observed for several sites of the fn3 family). The range of the site-wise transition temperatures are plotted along with T_m and T_D in Fig. <ref>B. The site-wise temperatures all appeared slightly above the global transition temperature T_m, indicating a global (higher order) ordering process between the minimum T_m,O_i and T_m. The site-wise transition temperature was found to be negatively correlated with site conservation D_O_i(T_D) (Fig. <ref>C) although the correlation was not always significant. Thus, as temperature lowers, local ordering starts from less conserved sites, but these sites have little global effects (smaller values of C_O_i(T_m,O_i); see Fig. <ref>E, green line), then it proceeds to increasingly better conserved sites with increasingly greater effects (larger change in total energy). Then, higher-order ordering proceeds down to the global transition temperature, T_m.Interestingly, a nearly perfect correlation (R ≈ 1) was found between {C_O_i(T_m)} and {D_O_i(T_D)}. Thus, the transition of each site appears almost completely determined by the conservation patterns. §.§ Suppressing key residues inhibits sequence pattern formation under the natural condition: “Alanine” scanning Although the above analysis of divergence maps hinted some differential roles of individual sites in determining natural-like sequences, the site-wise transition temperature T_m,O_i did not vary much across the sites.In order to explore the role of each site more directly, we performed virtual site-directed mutation experiments by manipulating the chemical potential μ. First, we performed “alanine” scanning experiments. That is, for each core site O_i, the chemical potential for alanine (“𝙰”) residue (or glycine “𝙶” if the consensus residue was alanine)was set to a large positive value (namely, μ(O_i,𝙰) = +20) while other μ's were set to 0. This effectively fixes alanine residue at the site O_i while allowing other sites to freely adopt residues that are consistent with the perturbation. By performing MC simulations at T=1 and applying the hmmsearch program to the sampled sequences, the hit ratio h_O_i^ALA (the fraction of natural-like sequences) of the “alanine mutant” was determined (c.f., Fig. <ref>C, magenta line). This hit ratio was compared to the hit ratio h_0(T=1) of the natural condition (c.f., Fig. <ref>C). Namely, we evaluated the “mutants” in terms of the following enrichment value:En_O_i^ALA = h_O_i^ALA / h_0(T=1). While most of the sites were not largely affected by alanine mutations (the enrichment ranging between 0.5 and 1.5), a few sites exhibited enrichment of less than 0.5 (Fig. <ref>A), that is, the fraction of natural-like sequences for these few mutants was less than a half of that under the natural condition. The hit ratio of mutants {h_O_i^ALA}_i=1,⋯,N and a measure of conservation of each site {D_O_i(T_D)}_i=1,⋯,N showed negative correlations (Fig. <ref>C; R < -0.50 for most families), indicating better conserved sites are affected more severely by the alanine mutation. This result suggests that fixation of some key residues is essential for the global pattern formation of natural-like sequences.§.§ Enforcing key residues promotes sequence pattern formation under disordering condition: “Enforcedly conserved residue” scanning Conversely to alanine scanning, we can enforce the consensus (the most dominant) residue a_O_i^* of each core site O_i by setting μ(O_i,a_O_i^*) = +20 (and all other μ's were set to 0) at the disordering temperature T = T_D. With this “enforcedly conserved residue (ECR)” mutation at each site, the hit ratio h_O_i^ECR was calculated (c.f., Fig. <ref>C, green line) and the enrichment with respect to the standard hit ratio at T_DEn_O_i^ECR = h_O_i^ECR / h_0(T=T_D).was compared (Fig. <ref>B). While majority of sites exhibited enrichment of less than 10 fold, a few sites showed more than 40-fold enrichment. The hit ratio for ECR mutants are even more correlated with site conservation (Fig. <ref>C; R>0.87 for all the families). Thus, enforcing a few well-conserved residues can enhance the sequence pattern formation over the entire sequence under otherwise disordering conditions.§ DISCUSSIONThe LGM for studying the “folding transition” in the sequence subspace around a given protein family is analogous to Gō-like models for studying the folding transition in the conformational space for a given native structure <cit.>. One major conceptual difference is that while there is a unique native structure as the global energy minimum for a Gō-like model, there is an ensemble of sequences as the global free energy minimum for an LGM (Eq. <ref>). This is because in the latter the parameters are optimized using a set of natural sequences at a finite temperature (T = 1). This may be considered as a consequence of the asymmetric one-to-many relationship between a protein fold and family sequences. The design of the LGM assumes a funnel-like free energy landscape of the sequence subspace analogous to that in the conformational space <cit.>. Based on simple exact models, it has been suggested that the free energy landscape of the sequence subspace within a family of sequences sharing the same ground-state conformation is indeed funnel-like (“superfunnel”) <cit.>. The present LGM extends the superfunnel concept to include sequences slightly beyond the family boundary.The parameters of the LGM are determined so that statistics of simulated sequences match that of an observed MSA. This may imply that the sequences sampled by the LGM are biased towards natural sequences. Nevertheless, based on computational protein design experiments, it has been suggested that the volume of sequence space optimal for a protein structure is restricted to a region around the natural sequence<cit.>. Therefore, the ensemble of sequences sampled by the LGM under the natural condition is expected to be a good approximation for the ensemble of all the sequences, both natural and artificial, compatible with the protein family.The existence of a rather sharp two-state transition implies the existence of a clear boundary between family members and non-family members. This transition is made possible by the inclusion of the long-range (non-bonded) interactions in the LGM. Conventional sequence models such as profile HMMs do not (or cannot) incorporate such long-range effects because they are essentially one-dimensional (1D) systems, and therefore, they cannot exhibit such transition <cit.>. Thus, the discrimination between family and non-family members based on the conventional sequence models is necessarily fuzzy (Fig. <ref>C), inevitably leading to false positives and false negatives. In fact, this situation may have already biased our understanding of protein families. For example, the C-terminal 3_10 helix and β strand (β5 in <cit.>) of the SH3 domain are completely missing in the corresponding Pfam profile HMM (c.f., grey regions in Fig. <ref>), but these regions not only comprise an integral part of the SH3 fold, but also contain a functionally important tyrosine residue (Fig. <ref>) involved in peptide binding <cit.>. More generally, there are many partial domains found in the MSAs of Pfam and other databases that are shorter than 50% of the domain model length, and most of them are considered to be alignment and annotation artifacts <cit.>. The present results suggest that these alignment artifacts are an inherent property of conventional sequence models. A notable exception to the conventional sequence models is MRFalign <cit.> based on essentially the same principle as the LGM <cit.>. Based on the present discussion, the success of MRFalign in remote homology detection is explained by its ability to set a clear boundary for family members owing to the inclusion of long-range correlations. In this study, we have used one of the conventional methods, namely the profile HMM, for quantifying natural-likeliness in terms of the “hit ratio”. However, due to the limitation of profile HMMs just pointed out, such a measure may be inaccurate. In designing artificial WW domain proteins, Socolich et al. <cit.> showed that sequences designed without pairwise correlations were unable to fold whereas all the designed foldable proteins were designed with pairwise correlations (by statistical coupling analysis <cit.>). It has been shown that other statistical models incorporating “direct” long-range correlations can discriminate foldable sequences from unfoldable ones to a very good accuracy <cit.> for the artificially designed WW domains <cit.>. The LGM also incorporates the direct long-range correlations in the form of interacting non-bonded pairs so its energy value is expected to be a good discriminator of foldable proteins. It is of interest and in principle possible to experimentally measure the foldability of the sequences generated by the LGM.In the context of molecular evolution, the present results suggest that if some selective pressure is imposed on one or a few key sites, the entire sequence may evolve rather quickly to form a pattern compatible with the family characterized by a unique fold.How can such selective pressure act on residues that are to be conserved? According to recent studies, it appears that protein families have been naturally (and positively) selected so as to stabilize their native folds <cit.>. In fact, most of the key residues listed in Table <ref> comprise the structural core of native folds, suggesting the structural importance of these residues. However, a specific native structure per se is unlikely to be subject to natural selection unless it accompanies some advantageous phenotype <cit.> through molecular functions. Visual inspection of the native structures in Table <ref> and their homologs revealed that the key residues often included those involved in intermolecular interactions such as binding of small molecules, polypeptides/proteins or polynucleotides, or phosphorylation (Table <ref>, residues marked in boldface). In the case of the SH3 domain (Fig. <ref>), for example, the key residues Y6, W32 and P47 (consensus residues according to Fig. <ref>A) comprise the “xP pockets” essential for the proline-rich peptide binding activity <cit.>. Thus, although it has been suggested that functionally important residues are not necessarily optimal for structural stability <cit.>, some of them play an essential role in cooperatively shaping the overall pattern of family sequences and hence the native fold <cit.>. This may also help explain the observation that most of the structural motifs for ligand binding are confined within single protein families <cit.>. § MATERIALS AND METHODS §.§ Data preparationAll the 15 domain families studied here were selected from the “top 20” list of the largest Pfam families (except for the SH3 domain) that had average sequence length of less than 120 residues. Only those of the “domain” type were used (i.e., “family” and “repeat” were discarded). These domain families consist of various folds. Multiple sequence alignments (MSA) and profile hidden Markov models (HMMs) were downloaded from the Pfam database (version 30.0). The MSAs were based on the “uniprot” set when available or on the largest available representative set, otherwise (Table <ref>). Aligned sequences that contained more than or equal to 10% deleted residues on core sites were discarded. A representative structure of each family was assigned based on resolution and alignment quality using the PDBj Mine2 relational database <cit.> integrated with the SIFTS <cit.> resource (Table <ref>). §.§ Monte Carlo simulationsIn each MC simulation of an LGM model of length N, one sweep consists of N and N-1 attempted mutations for randomly selected core and insert sites, respectively. For the algorithmic details of the MC simulations of the LGM, refer to <cit.>. For the results given in Figs. <ref>B, C, <ref> and <ref>, 20 grand canonical MC simulations (under the conditions described in the text) were performed for 10^6 sweeps after 5× 10^4 sweeps of equilibration, and 10,000 sequences (alignments) were saved every 100 steps for each of the 20 trajectories. For each set of 10,000 sequences resulted from each trajectory, a sequence database was created by removing the “delete” symbols from the sequences, against which the hmmsearch <cit.> program was applied with the Pfam profile HMM with the E-value cutoff of 0.01 in order to identify significantly similar sequences. From the 20 runs, the average hit ratio and the standard deviation were obtained. For the results given in Figs. <ref>A, <ref>, and <ref>, first, the density of states was determined by the Wang-Landau method <cit.> by using the energy bins with bin width of 0.2 energy units. The lower bound of the energy range was fixed at a sufficiently low value, and the upper bound was determined by tentative grand canonical MC simulations at a high temperature T = 1.2. The Wang-Landau iteration was terminated when the density of states was determined to the precision better than 10^-7. Next, using thus determined density of states, a multicanonical MC simulation <cit.> was performed until all the energy bins are filled with at least 2,000 counts (sequences were saved at every sweep). From this trajectory, the specific heat and residue distributions at various temperatures were obtained by the reweighting method <cit.>.§ ACKNOWLEDGEMENTSThe author thanks Ken Nishikawa, Motonori Ota and Gautam Basu for valuable comments, Eugene I. Shakhnovich for providing some references. § REFERENCESelsarticle-harv | http://arxiv.org/abs/1708.07612v3 | {
"authors": [
"Akira R. Kinjo"
],
"categories": [
"q-bio.BM",
"physics.bio-ph"
],
"primary_category": "q-bio.BM",
"published": "20170825043847",
"title": "Cooperative \"folding transition\" in the sequence space facilitates function-driven evolution of protein families"
} |
#1#2#3#4#1 #2, #3 (#4)Phys. Rev. Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Eur. Phys. J. D Eur. Phys. Lett. Science J. Math. Phys. Nature Physica A | http://arxiv.org/abs/1708.07764v1 | {
"authors": [
"Tomas Opatrny",
"Lukas Richterek",
"Martin Opatrny"
],
"categories": [
"quant-ph",
"physics.class-ph"
],
"primary_category": "quant-ph",
"published": "20170825144909",
"title": "Euler top with a rotor: classical analogies of spin squeezing and quantum phase transitions in a generalized Lipkin-Meshkov-Glick model"
} |
This work was supported by NSF grants CCF-1422471, CCF-1223850, CCF-1218344, CCF-1116289, CCF-0954024, IIS-1617157, Air Force grant FA8750-15-2-0075, and a gift from Microsoft Research. Localizing Novice Type Errors with Data-Driven Diagnosis 0000-0002-2529-7790 Department of Computer Science University of California, San Diego La Jolla CA [email protected] Department of Computer Science University of California, San Diego La Jolla CA [email protected] Department of Computer Science University of California, San Diego La Jolla CA [email protected] Department of Computer Science University of Virginia [email protected] Department of Computer Science University of California, San Diego La Jolla CA [email protected] type errors is challenging in languages with global type inference, as the type checker must make assumptions about what the programmer intended to do.We introduce , a data-driven approach to error localization based on supervised learning.analyzes a large corpus of training data — pairs of ill-typed programs and their “fixed” versions — to automatically learn a model of where the error is most likely to be found.Given a new ill-typed program, executes the model to generate a list of potential blame assignments ranked by likelihood.We evaluate by comparing its precision to the state of the art on a set of over 5,000 ill-typed programs drawn from two instances of an introductory programming course.We show that when the top-ranked blame assignment is considered, 's data-driven model is able to correctly predict the exact sub-expression that should be changed % of the time,points higher than and points higher than the state-of-the-art tool.Furthermore, 's accuracy surpasses % when we consider the top two locations and reaches % if we consider the top three. <ccs2012> <concept> <concept_id>10011007.10011006.10011008</concept_id> <concept_desc>Software and its engineering General programming languages</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10003752.10003790.10011740</concept_id> <concept_desc>Theory of computation Type theory</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257</concept_id> <concept_desc>Computing methodologies Machine learning</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> [500]Software and its engineering General programming languages [500]Theory of computation Type theory [300]Computing methodologies Machine learning Learning to Blame Ranjit Jhala December 30, 2023 ===================== § INTRODUCTION Types are awesome. Languages like and make the value-proposition for types even more appealing by using constraints to automatically synthesize the types for all program terms without troubling the programmer for any annotations.Unfortunately, this automation has come at a price. Type annotations signify the programmer's intent and help to correctly blame the erroneous sub-term when the code is ill-typed.In the absence of such signifiers, automatic type inference algorithms are prone to report type errors far from their source <cit.>.While this can seem like a minor annoyance to veteran programmers, <cit.> have found that novices often focus their attention on the location reported and disregard the message.Localizing ErrorsSeveral recent papers have proposed ways to improve feedback via error localization.At a high-level these techniques analyze the set of typing constraints to find the minimum (weighted) subset that, if removed, would make the constraints satisfiable and hence, assertion-safe <cit.> or well-typed <cit.>. The finger of blame is then pointed at the sub-terms that yielded those constraints.This minimum-weight approach suffers from two drawbacks.First, they are not extensible:the constraint languages and algorithms for computing the minimum weighted subset must be designed afresh for different kinds of type systems and constraints <cit.>.Second, and perhaps most importantly, they are not adaptable: the weights are fixed in an ad-hoc fashion, based on the analysis designer's notion of what kinds of errors are morelikely, rather than adapting to the kinds of mistakes programmers actually make in practice.A Data-Driven ApproachIn this paper, we introduce [“Numeric Analysis of Type Errors”; any resemblance to persons living or dead is purely coincidental.] a data-driven approach to error localization based on supervised learning (seefor a survey).analyzes a large corpus of training data — pairs of ill-typed programs and their subsequent fixes — to automatically learn a model of where errors are most likely to be found.Given a new ill-typed program, simply executes the model to generate a list of potential blame assignments ranked by likelihood.We evaluate by comparing its precision against the state-of-the-art on a set of over 5,000 ill-typed programs drawn from two instances of an introductory programming course.We show that, when restricted to a single prediction, 's data-driven model is able to correctly predict the exact sub-expression that should be changed % of the time, points higher than and points higher than the state-of-the-art tool.Furthermore, 's accuracy surpasses % when we consider the top two locations and reaches % if we consider the top three.We achieve these advances by identifying and then solving three key challenges.Challenge 1: Acquiring Labeled ProgramsThe first challenge for supervised learning is to acquire a corpus of training data, in our setting a set of ill-typed programs labeled with the exact sub-terms that are the actual cause of the type error.Prior work has often enlisted expert users to manually judge ill-typed programs and determine the “correct” fix <cit.>, but this approach does not scale well to a dataset large enough to support machine learning.Worse, while expert users have intimate knowledge of the type system, they may have a blind spot with regards to the kinds of mistakes novices make, and cannot know in general what novice users intended.Our first contribution (<ref>) is a set of more than 5,000 labeled programs <cit.>, giving us an accurate ground truth of the kinds of errors and the (locations of the) fixes that novices make in practice.We obtain this set by observing that software development is an iterative process; programmers eventually fix their own ill-typed programs, perhaps after multiple incorrect exploratory attempts.To exploit this observation we instrumented the compiler to collect fine-grained traces of student interactions over two instances of an undergraduate Programming Languages courseat UC San Diego (IRB #140608). at AUTHOR's INSTITUTION (IRB HIDDEN). We then post-process the resulting time-series of programs submitted to the compiler into a set of pairs of ill-typed programs and their subsequent fixes, the first (type-) correct program in the trace suffix.Finally, we compute the blame labels using a tree-diff between the two terms to find the exact sub-terms that changed in the fix.Challenge 2: Modeling Programs as VectorsModern supervised learning algorithms work onfeature vectors: real-valued points in an n-dimensional space. While there are standard techniques for computing such vectors for documents, images, and sound (respectively word-counts, pixel-values, and frequencies),there are no similarly standard representations for programs.Our second contribution (<ref>) solves this problem with a simple, yet expressive, representation called a Bag-of-Abstracted-Terms (BOAT) wherein each program is represented by the bag or multiset of (sub-) terms that appears inside it; and further, each (sub-) term is abstracted as a feature vector comprising the numeric values returned by feature abstraction functions applied to the term. We can even recover contextual information from the parent and child terms by concatenating the feature vectors of each term with those of its parent and children (within a fixed window).We have found this representation to be particularly convenient as it gives us flexibility in modeling the syntactic and semantic structure of programs while retaining compatibility with off-the-shelf classifiers, in contrast to, , <cit.>, who had to develop their own variants of classifiers to obtain their results.Challenge 3: Training Precise ClassifiersFinally, the last and most important challenge is to use our BOAT representation to train classifiers that are capable of precisely pinpointing the errors in real programs.The key here is to find the right set of feature abstractions to model type errors, and classification algorithms that lead to precise blame assignments.Fortunately, our BOAT model allows us a great deal of latitude in our choice of features.We can use abstraction functions to capture different aspects of a term ranging fromsyntactic features (is-a-data-constructor, is-a-literal, is-an-arithmetic-operation, is-a-function-application, ),to semantic features captured by the type system (is-a-list, is-an-integer, is-a-function, ).We can similarly model the blame labels with a simple feature abstraction (is-changed-in-fix).Our third contribution (<ref>) is a systematic evaluation of our data-driven approach using different classes of features like the above, and with four different classification algorithms: logistic regression, decision trees, random forests, and neural networks.We find that 's models generalize well between instances of the same undergraduate course, outperforming the state of the art by at least percentage points at predicting the source of a type error.We also investigate which features and classifiers are most effective at localizing type errors, and empirically characterize the importance of different feature sets. In particular, we find that while machine learning over syntactic features of each term in isolation performs worse than existing purely constraint-based approaches (, ), augmenting the data with a single feature corresponding to the type error slice <cit.> brings our classifiers up to par with the state-of-the-art, and further augmenting the data with contextual features allows our classifiers to outperform the state-of-the-art by percentage points. Thus, by combining modern statistical methods with domain-specific feature engineering, opens the door to a new data-driven path to precise error localization.In the future, we could extend to new languages or forms of correctness checks by swapping in a different set of feature abstraction functions.Furthermore, our data-driven approach allows to adapt to the kinds of errors that programmers (in particular novices, who are in greatest need of precise feedback) actually make rather than hardwiring the biases of compiler authors who, by dint of their training and experience, may suffer from blind spots with regards to such problems.In contrast, our results show that 's data-driven diagnosis can be an effective technique for localizing errors by collectively learning from past mistakes. OverviewLet us start with an overview of 's approach to localizing type errors by collectively learning from the mistakes programmers actually make.The ProblemConsider the |sumList| program in <ref>, written by a student in an undergraduate Programming Languages course.The program is meant to add up the integers in a list, but the student has accidentally given the empty list as the base case, rather than |0|.The compiler collects typing constraints as it traverses the program, and reports an error the moment it finds an inconsistent constraint.In this case it blames the recursive call to |sumList|, complaining that |sumList| returns a |list| while an |int| was expected by the |+| operator.This blame assignment is inconsistent with the programmer's intention and may not help the novice understand the error.It may appear obvious to the reader that |[]| is the correct expression to blame, but how is a type checker to know that?Indeed, recent techniques like and <cit.> fail to distinguish between the |[]| and |+| expressions in <ref>; it would be equally valid to blame either of them alone.The |[]| on line 3 could be changed to |0|, or the |+| on line 4 could be changed to either |@| (list append) or |::|, all of which would give type-correct programs.Thus, these state-of-the-art techniques are forced to either blame both locations, or choose one arbitrarily.Solution: Localization via Supervised ClassificationOur approach is to view error localization as a supervised classification problem <cit.>.A classification problem entails learning a function that maps inputs to a discrete set of output labels (in contrast toregression, where the output is typically a real number).A supervised learning problem is one where we are given a training set where the inputs and labels are known, and the task is to learn a function that accurately maps the inputs to output labels and generalizes to new, yet-unseen inputs.To realize the above approach for error localization as a practical tool, we have to solve four sub-problems.* How can we acquire a training set of blame-labeled ill-typed programs? * How can we represent blame-labeled programs in a format amenable to machine learning? * How can we find features that yield predictive models? * How can we use the models to give localized feedback to the programmer? Step 1: Acquiring a Blame-Labeled Training SetThe first step is to gather a training set of ill-typed programs, where each erroneous sub-term is explicitly labeled.Prior work has often enlisted expert users to curate a set of ill-typed programs and then manually determine the correct fix <cit.>.This method is suitable for evaluating the quality of a localization (or repair) algorithm on a small number (10s–100s) of programs.However, in general it requires a great deal of effort for the expert to divine the original programmer's intentions.Consequently, is difficult to scale the expert-labeling to yield a dataset large enough (1000s of programs) to facilitate machine learning.More importantly, this approach fails to capture the frequency with which errors occur in practice.Solution: Interaction TracesWe solve both the scale and frequency problems by instead extracting blame-labeled data sets from interaction traces.Software development is an iterative process. Programmers, perhaps after a lengthy (and sometimes frustrating) back-and-forth with the type checker, eventually end up fixing their own programs.Thus, we instrumented the compiler to record this conversation, record the sequence of programs submitted by each programmer and whether or not it was deemed type-correct.For each ill-typed program in a particular programmer's trace, we find the first subsequent program in the trace that type checks and declare it to be the fixed version.From this pair of an ill-typed program and its fix, we can extract a diff of the abstract syntax trees, and then assign the blame labels to the smallest sub-tree in the diff.Example Suppose our student fixed the |sumList| program in <ref> by replacing |[]| with |0|, the diff would include only the |[]| expression.Thus we would determine that the |[]| expression (and not the |+| or the recursive call |sumList t|) is to blame.Step 2: Representing Programs as VectorsNext, we must find a way to translate highly structured and variable sized programs into fixed size n-dimensional numeric vectors that are needed for supervised classification.While the Programming Languages literature is full of different program representations, from raw token streams to richly-structured abstract syntax trees (AST) or control-flow graphs, it is unclear how to embed the above into a vector space.Furthermore, it is unclear whether recent program representations that are amenable to one learning task, code completion <cit.> or decompilation <cit.> are suitable for our problem of assigning blame for type errors.Solution: Bags-of-Abstracted-TermsWe present a new representation of programs that draws inspiration from the theory of abstract interpretation <cit.>.Our representation is parameterized by a set of feature abstraction functions, (abbreviated to feature abstractions) f_1, …, f_n, that map terms to a numeric value (or just {0, 1} to encode a boolean property).Given a set of feature abstractions, we can represent a single program's AST as a bag-of-abstracted-terms (BOAT) by:(1) decomposing the AST (term) t into a bag of its constituent sub-trees (terms) {t_1,…,t_m}; and then(2) representing each sub-term t_i with the n-dimensional vector [f_1(t_i),…, f_n(t_i)].Working with ASTs is a natural choice as type-checkers operate on the same representation.Modeling ContextsEach expression occurs in some surrounding context, and we would like the classifier to be able make decisions based on the context as well.The context is particularly important for our task as each expression imposes typing constraints on its neighbors.For example, a |+| operator tells the type checker that both children must have type |int| and that the parent must accept an |int|.Similarly, if the student wrote |h sumList t| forgot the |+|, we might wish to blame the application rather than |h| because |h| does not have a function type.The BOAT representation makes it easy to incorporate contexts: we simply concatenate each term's feature vector with the contextual features of its parent and children.Step 3: Feature DiscoveryNext, we must find a good set of features, that is, a set of features that yields predictive models. Our BOAT representation enables an iterative solution by starting with a simple set of features, and then repeatedly adding more and more to capture important aspects needed to improve precision.Our set of feature abstractionscaptures the syntax, types, and context of each expression.Syntax and Type FeaturesWe start by observing that at the very least, the classifier should be able to distinguish between the |[]| and |+| expressions in <ref> because they represent different syntactic expression forms.We model this by introducing feature abstractions of the form is-|[]|, is-|+|, , for each of a fixed number of expression forms.Modeling the syntactic class of an expression gives the classifier a basic notion of the relative frequency of blame assignment for the various program elements, perhaps |[]| is empirically more likely to be blamed than |+|.Similarly, we can model the type of each sub-expression with features of the form is-|int|, is-|bool|, .We will discuss handling arbitrary, user-defined types in <ref>.Contextual Features: Error SlicesOur contextual features include the syntactic class of the neighboring expressions and their inferred types (when available).However, we have found that the most important contextual signal is whether or not the expression occurs in a minimal type error slice <cit.> which includes a minimal subset of all expressions that are necessary for the error to manifest.(That is, replacing any subterm with |undefined| or |assert false| would yield a well-typed program.)We propose to use type error slices to communicate to the classifier which expressions could potentially be blamed — a change to an expression outside of the minimal slice cannot possibly fix the type error.We empirically demonstrate that the type error slice is so important (<ref>) that it is actually beneficial to automatically discard expressions that are not part of the slice, rather than letting the classifier learn to do so.Indeed, this domain-specific insight is crucial for learning classifiers that significantly outperform the state-of-the-art.ExampleWhen is tasked with localizing the error in the example program of <ref>, the |[]| and |+| sub-terms will each be given their own feature vector, and we will ask the classifier to predict for each independently whether it should be blamed.<ref> lists some of the sub-expressions of the example from <ref>, and their corresponding feature vectors. Step 4: Generating FeedbackFinally, having trained a classifier using the labeled data set, we need to use it to help users localize type errors in their programs.The classifier tells us whether or not a sub-term should be blamed (has the blame label) but this is not yet particularly suitable as user feedback.A recent survey of developers by <cit.> found that developers are unlikely to examine more than around five potentially erroneous locations before falling back to manual debugging.Thus, we should limit our predictions to a select few to be presented to the user.Solution: Rank Locations by ConfidenceFortunately, many machine learning classifiers produce not only a predicted label, but also a metric that can be interpreted as the classifier's confidence in its prediction.Thus, we rank each expression by the classifier's confidence that it should be blamed, and present only the top-k predictions to the user (in practice k=3).The use of ranking to report the results of an analysis is popular in other problem domains <cit.>; we focus explicitly on the use of data-driven machine learning confidence as a ranking source.In <ref> we show that 's ranking approach yields a high-precision localizer: when the top three locations are considered, at least one matches an actual student fix % of the time. Learning to BlameIn this section, we describe our approach to localizing type errors, in the context of (<ref>), a simple lambda calculus with integers, booleans, pairs, and lists.Our goal is to instantiate thefunction of <ref>, which takes as input aof type errors and an ill-typed program e, and returns an ordered list of subexpressions from e paired with the confidence scorethat they should be blamed.Ais produced by , which performs supervised learning on a training set of feature vectorsand (boolean) labels .Once trained, we can uate aon a new input, producing the confidencethat the blame label should be applied.We describe multiple s and their instantiations ofand(<ref>).Of course, theexpects feature vectorsand blame labels , but we are given program pairs.So our first step must be to define a suitable translation from program pairs to feature vectors and labels, we must define thefunction in <ref>.We model features as real-valued functions of terms, and extract a feature vector for each subterm of the ill-typed program (<ref>).Then we define the blame labels for the training set to be the subexpressions that changed between the ill-typed program and its subsequent fix, and modelas a function from a program pair to the set of expressions that changed (<ref>).Thefunction, then, extractsfrom each subexpression and computes the blamed expressions according to .FeaturesThe first issue we must tackle is formulating our learning task in machine learning terms.We are given programs over , but learning algorithms expect to work with feature vectors— vectors of real numbers, where each column describes a particular aspect of the input.Thus, our first task is to convert programs to feature vectors.We choose to model a program as a set of feature vectors, where each element corresponds an expression in the program.Thus, given the |sumList| program in <ref> we would first split it into its constituent sub-expressions and then transform each sub-expression into a single feature vector.We group the features into five categories, using <ref> as a running example of the feature extraction process.Local syntactic features These features describe the syntactic category of each expression e.In other words, for each production of e in <ref> we introduce a feature that is enabled (set to 1) if the expression was built with that production, and disabled (set to 0) otherwise.For example, the feature in <ref> describes whether an expression is the empty list .We distinguish between matching on a list vs. on a pair, as this affects the typing derivation.We also assume that all pattern matches are well-formed — all patterns must match on the same type.Ill-formed match expressions would lead to a type error; however, they are already effectively localized to the match expression itself.We note that this is not a fundamental limitation, and one could easily add features that specify whether a match contains a particular pattern, and thus have a match expression that enables multiple features.Contextual syntactic features These are similar to local syntactic features, but lifted to describe the parent and children of an expression.For example, the feature in <ref> describes whether an expression's parent matches on a list.If a particular e does not have children (a variable x) or a parent (the root expression), we leave the corresponding features disabled.This gives us a notion of the context in which an expression occurs, similar to the n-grams used in linguistic models <cit.>.Expression size We also propose a feature representing the size of each expression, how many sub-expressions does it contain?For example, the feature in <ref> is set to three for the expression |sumList tl| as it contains three expressions: the two variables and the application itself.This allows the model to learn that, , expressions closer to the leaves are more likely to be blamed than expressions closer to the root.Typing features A natural way of summarizing the context in which an expression occurs is with types.Of course, the programs we are given are untypeable, but we can still extract a partial typing derivation from the type checker and use it to provide more information to the model.A difficulty that arises here is that, due to the parametric type constructors ··, ··, and ·, there is an infinite set of possible types — but we must have a finite set of features.Thus, we abstract the type of an expression to the set of type constructors it mentions, and add features for each type constructor that describe whether a given type mentions the type constructor.For example, the typewould only enable thefeature, while the typewould enable the ··, , andfeatures.We add these features for parent and child expressions to summarize the context, but also for the current expression, as the type of an expression is not always clear syntactically.For example, the expressions |tl| and |sumList tl| in <ref> both enable , as they are both inferred to have a type that mentions ·. Note that our use of typing features in an ill-typed program subjects us to traversal bias <cit.>. For example, the |sumList tl| expression might alternatively be assigned the type .Our models will have to learn good localizations in spite of this bias (see <ref>).Type error slice Finally, we wish to distinguish between changes that could fix the error, and changes that cannot possibly fix the error.Thus, we compute a minimal type error slice for the program (the set of expressions that contribute to the error), and add a feature that is enabled for expressions that are part of the slice.The feature in <ref> indicates whether an expression is part of such a minimal slice, and is enabled for all of the sampled expressions except for |tl|, which does not affect the type error.If the program contains multiple type errors, we compute a minimal slice for each error.In practice, we have found that is a particularly important feature, and thus include a post-processing step that discards all expressions where it is disabled.As a result, the |tl| expression would never actually be shown to the classifier. We will demonstrate the importance of empirically in <ref>.LabelsRecall that we make predictions in two stages.First, we useto predict for each subexpression whether it should be blamed, and extract a confidence scorefrom the .Thus, we define the output of theto be a boolean label, where “false” means the expression should not change and “true” means the expression should change.This allows us to predict whether any individual expression should change, but we would actually like to predict the most likely expressions to change.Second, we rank each subexpression by the confidencethat it should be blamed, and return to the user the top k most likely blame assignments (in practice k=3). We identify the fixes for each ill-typed program with an expression-level diff <cit.>.We consider two sources of changes.First, if an expression has been removed wholesale, if fx is rewritten to gx, we will mark the expression f as changed, as it has been replaced by g. Second, if a new expression has been inserted around an existing expression, if fx is rewritten to fx1, we will mark the application expression fx (but not f or x) as changed, as the + operator now occupies the original location of the application. Learning Algorithms| Recall that we formulate type error detection at a single expression as a supervised classification problem.This means that we are given a training data setS : × of labeled examples, and our goal is to use it to build a classifier, a rule that can predict a label b for an input v.Since we apply the classifier on each expression in the program to determine those that are the most likely to be type errors, we also require the classifier to output a confidence score that measures how sure the classifier is about its prediction. There are many learning algorithms to choose from, existing on a spectrum that balances expressiveness with ease of training (and of interpreting the learned model).In this section we consider four standard learning algorithms: (1) logistic regression, (2) decision trees, (3) random forests, and (4) neural networks.A thorough introduction to these techniques can be found in introductory machine learning textbooks <cit.>.Below we briefly introduce each technique by describing the rules it learns, and summarize its advantages and disadvantages.For our application, we are particularly interested in three properties – expressiveness, interpretability and ease of generalization.Expressiveness measures how complex prediction rules are allowed to be, and interpretability measures how easy it is to explain the cause of prediction to a human.Finally ease of generalization measures how easily the rule generalizes to examples that are not in the training set; a rule that is not easily generalizable might perform poorly on an unseen test set even when its training performance is high.Logistic Regression The simplest classifier we investigate is logistic regression: a linear model where the goal is to learn a set of weights W that describe the following model for predicting a label b from a feature vector v: (b = 1 | v) = 1/1 + e^-W^⊤ v The weights W are learnt from training data, and the value of (b | v) naturally leads to a confidence score .Logistic regression is a widely used classification algorithm, preferred for its simplicity, ease of generalization, and interpretability.Its main limitation is that the prediction rule is constrained to be a linear combination of the features, and hence relatively simple.While this can be somewhat mitigated by adding higher order (quadratic or cubic) features, this often requires substantial domain knowledge. Decision Trees Decision tree algorithms learn a tree of binary predicates over the features, recursively partitioning the input space until a final classification can be made.Each node in the tree contains a single predicate of the form v_j ≤ t for some feature v_j and threshold t, which determines whether a given input should proceed down the left or right subtree.Each leaf is labeled with a prediction and the fraction of correctly-labeled training samples that would reach it; the latter quantity can be interpreted as the decision tree's confidence in its prediction.This leads to a prediction rule that can be quite expressive depending on the data used to build it.Training a decision tree entails finding both a set of good partitioning predicates and a good ordering of the predicates based on data.This is usually done in a top-down greedy manner, and there are several standard training algorithms such as C4.5 <cit.> and CART <cit.>.Another advantage of decision trees is their ease of interpretation — the decision rule is a white-box model that can be readily described to a human, especially when the tree is small.However, the main limitation is that these trees often do not generalize well, though this can be somewhat mitigated by pruning the tree. Random ForestsRandom forests improve generalization by training an ensemble of distinct decision trees and using a majority vote to make a prediction.The agreement among the trees forms a natural confidence score.Since each classifier in the ensemble is a decision tree, this still allows for complex and expressive classifiers.The training process involves taking N random subsets of the training data and training a separate decision tree on each subset — the training process for the decision trees is often modified slightly to reduce correlation between trees, by forcing each tree to pick features from a random subset of all features at each node.The diversity of the underlying models tends to make random forests less susceptible to the overfitting, but it also makes the learned model more difficult to interpret.Neural NetworksThe last (and most complex) model we use is a type of neural network called a multi-layer perceptron (see <cit.> for an introduction to neural networks).A multi-layer perceptron can be represented as a directed acyclic graph whose nodes are arranged in layers that are fully connected by weighted edges.The first layer corresponds to the input features, and the final to the output. The output of an internal node v is h_v = g(∑_j ∈ N(v) W_jv h_j ) where N(v) is the set of nodes in the previous layer that are adjacent to v, W_jv is the weight of the (j, v) edge and h_j is the output of node j in the previous layer.Finally g is a non-linear function, called the activation function, which in recent work is commonly chosen to be the rectified linear unit (ReLU), defined as g(x) = 𝗆𝖺𝗑(0,x) <cit.>.The number of layers, the number of neurons per layer, and the connections between layers constitute the architecture of a neural network.In this work, we use relatively simple neural networks which have an input layer, a single hidden layer and an output layer.A major advantage of neural networks is their ability to discover interesting combinations of features through non-linearity, which significantly reduces the need for manual feature engineering, and allows high expressivity.On the other hand, this makes the networks particularly difficult to interpret and also difficult to generalize unless vast amounts of training data are available. | Evaluationcol sep=commadata/sp14-baseline.csv data/sp14-ocaml-results.csv data/sp14-mycroft-results.csv data/sp14-sherrloc-results.csv data/sp14-op+context+type+size-linear-results.csv data/sp14-op+context+type+size-decision-tree-results.csv data/sp14-op+context+type+size-random-forest-results.csv data/sp14-op+context+type+size-hidden-10-results.csv data/sp14-op+context+type+size-hidden-500-results.csvdata/fa15-baseline.csv data/fa15-ocaml-results.csv data/fa15-mycroft-results.csv data/fa15-sherrloc-results.csv data/fa15-op+context+type+size-linear-results.csv data/fa15-op+context+type+size-decision-tree-results.csv data/fa15-op+context+type+size-random-forest-results.csv data/fa15-op+context+type+size-hidden-10-results.csv data/fa15-op+context+type+size-hidden-500-results.csvmodels/linear-op+slice-no-slice.cross.csv models/linear-op+slice.cross.csv models/linear-op+slice-only-slice.cross.csv models/hidden-500-op+slice-no-slice.cross.csv models/hidden-500-op+slice.cross.csv models/hidden-500-op+slice-only-slice.cross.csvmodels/linear-op.cross.csv models/linear-op+size.cross.csv models/linear-op+context.cross.csv models/linear-op+type.cross.csv models/linear-op+context+size.cross.csv models/linear-op+type+size.cross.csv models/linear-op+context+type.cross.csv models/linear-op+context+type+size.cross.csv models/linear-op.cross.csv models/hidden-500-op+size.cross.csv models/hidden-500-op+context.cross.csv models/hidden-500-op+type.cross.csv models/hidden-500-op+context+size.cross.csv models/hidden-500-op+type+size.cross.csv models/hidden-500-op+context+type.cross.csv models/hidden-500-op+context+type+size.cross.csv We have implemented our technique for localizing type errors for a purely functional subset of with polymorphic types and functions.We seek to answer four questions in our evaluation:* Blame AccuracyHow often does blame a correct location for the type error? (<ref>)* Feature UtilityWhich program features are required to localize errors?(<ref>)* Interpretability Are the models interpretable using our intuition about type errors? (<ref>) * Blame Utility Do 's blame assignments help users diagnose type errors? (<ref>) In the sequel we present our experimental methodology <ref> and then drill into how we evaluated each of the questions above.However, for the impatient reader, we begin with a quick summary of our main results: 1. Data Beats Algorithms Our main result is that for (novice) type error localization, data is indeed unreasonably effective <cit.>.When trained on student errors from one instance of an undergraduate course and tested on another instance, 's most sophisticated neural network-based classifier's top-ranked prediction blames the correct sub-term % of the time — a good points higher than the state-of-the-art 's %.However, even 's simple logistic regression-based classifier is correct % of the time, points better than .When the top three predictions are considered, is correct % of the time. 2. Slicing Is CriticalHowever, data is effective only when irrelevant sub-terms have been sliced out of consideration.In fact, perhaps our most surprising result is that type error slicing and local syntax alone yields a classifier that is points better than and on par with .That is, once we focus our classifiers on slices, purely local syntactic features perform as well as the state-of-the-art. 3. Size Doesn't Matter, Types DoWe find that (after slices) typing features provide the biggest improvement in accuracy.Furthermore, we find contextual syntactic features to be mostly (but not entirely) redundant with typing features, which supports the hypothesis that the context's type nicely summarizes the properties of the surrounding expressions.Finally, we found that the size of the sub-expression was not very useful. This was unexpected, as we thought smaller expressions would be simpler, and hence, more likely causes. 4. Models Learn Typing RulesFinally, by investigating a few of the predictions made by the decision tree-based models, we found that the models appear to capture some simple and intuitive rules for predicting well-typedness.For example, if the left child of an application is a function, then the application is likely correct.Methodology We answer our questions on two sets of data gathered from the undergraduate Programming Languages course atUC San Diego (IRB #140608). AUTHOR's INSTITUTION. We recorded each interaction with the top-level system while the students worked on 23 programs from the first three homework assignments, capturing ill-typed programs and, crucially, their subsequent fixes.The first dataset comes from the Spring 2014 class (), with a cohort of 46 students. The second comes from the Fall 2015 class (), with a cohort of 56 students.The extracted programs are relatively small, but they demonstrate a range of functional programming idioms, higher-order functions and (polymorphic) algebraic data types.Feature Selection We extract 282 features from each sub-expression in a program, including:* 45 local syntactic features. In addition to the syntax of , we support the full range of arithmetic operators (integer and floating point), equality and comparison operators, character and string literals, and a user-definedarithmetic expressions. We discuss the challenge of supporting othertypes in <ref>.* 180 contextual syntactic features. For each sub-expression we additionally extract the local syntactic features of its parent and first, second, and third (left-to-right) children. If an expression does not have a parent or children, these features will simply be disabled. If an expression has more than three children, the classifiers will receive no information about the additional children.* 55 typing features. In addition to the types of , we support |int|s, |float|s, |char|s, |string|s, and the user-defined |expr| mentioned above. These features are extracted for each sub-expression and its context. * One feature denoting the size of each sub-expression.* One feature denoting whether each sub-expression is part of the minimal type error slice. We use this feature as a “hard” constraint, sub-expressions that are not part of the minimal slice will be preemptively discarded. We justify this decision in <ref>. Blame Oracle Recall from <ref> that we automatically extract a blame oracle for each ill-typed program from the (AST) diff between it and the student's eventual fix.A disadvantage of using diffs in this manner is that students may have made many, potentially unrelated, changes between compilations; at some point the “fix” becomes a “rewrite”.We do not wish to consider the “rewrites” in our evaluation, so we discard outliers where the fraction of expressions that have changed is more than one standard deviation above the mean, establishing a diff threshold of 40%.This accounts for roughly 14% of each dataset, leaving us with 2,712 program pairs for and 2,365 pairs for .Accuracy Metric All of the tools we compare (with the exception of the standard compiler) can produce a list of potential error locations.However, in a study of fault localization techniques, <cit.> show that most developers will not consider more than around five potential error locations before falling back to manual debugging.Type errors are relatively simple in comparison to general fault localization, thus we limit our evaluation to the top three predictions of each tool.We evaluate each tool on whether a changed expression occurred in its top one, top two, or top three predictions.Blame Utility Finally, to test the explanatory power of our blame assigments, we ran a user study at the University of Virginia (UVA IRB #2014009900).We included three problems in an exam in the Spring 2017 session of UVA's undergraduate Programming Languages course (CS 4501).We presented the 31 students in the course with ill-typed programs and asked them to(1) explain the type error, and(2) fix the type error.For each problem the student was given the ill-typed program and either or 's blame assignment, with no error message.Blame Accuracy First, we compare the accuracy of our predictions to the state of the art in type error localization.Baseline We provide two baselines for the comparison: a random choice of location from the minimized type error slice, and the standard compiler.State of the Art <cit.> localizes type errors by searching for a minimal subset of typing constraints that can be removed, such that the resulting system is satisfiable.When multiple such subsets exist it can enumerate them, though it has no notion of which subsets are more likely to be correct, and thus the order is arbitrary. <cit.> localizes errors by searching the typing constraint graph for constraints that participate in many unsatisfiable paths and comparatively few satisfiable paths.It can also enumerate multiple predictions, in descending order of likelihood.Comparing source locations from multiple tools with their own parsers is not trivial.Our experimental design gives the state of the art tools the “benefit of the doubt” in two ways. First, when evaluating and , we did not consider programs where they predicted locations that our oracle could not match with a program expression: around 6% of programs for and 4% for .Second, we similarly ignored programs where or timed out (after one minute) or where they encountered an unsupported language feature: another 5% for and 12% for . Our Classifiers We evaluate five classifiers, each trained on the full feature set. These include:A logistic regression trained with a learning rate η = 0.001, an L_2 regularization rate λ = 0.001, and a mini-batch size of 200. A decision tree trained with the CART algorithm <cit.> and an impurity threshold of 10^-7 (used to avoid overfitting via early stopping). A random forest <cit.> of 30 estimators, with an impurity threshold of 10^-7.andTwo multi-layer perceptron neural networks, both trained with η = 0.001, λ = 0.001, and a mini-batch size of 200.The first MLP contains a single hidden layer of 10 neurons, and the second contains a hidden layer of 500 neurons.This gives us a measure of the complexity of the MLP's model, if the model requires many compound features, one would expect to outperform .The neurons use rectified linear units (ReLU) as their activation function, a common practice in modern neural networks. All classifiers were trained for 20 epochs on one dataset — they were shown each program 20 times — before being evaluated on the other.The logistic regression and MLPs were trained with the Adam optimizer <cit.>, a variant of stochastic gradient descent that has been found to converge faster.Results <ref> shows the results of our experiment.Localizing the type errors in our benchmarks amounted, on average, to selecting one of 3 correct locations out of a slice of 10.Our classifiers consistently outperform the competition, ranging from % Top-1 accuracy (% Top-3) for the classifier to % Top-1 accuracy (% Top-3) for the .Our baseline of selecting at random achieves % Top-1 accuracy (% Top-3), while achieves a Top-1 accuracy of %.Interestingly, one only needs two random guesses to outperform , with % accuracy.outperforms both baselines, and comes close to our classifier, with % Top-1 accuracy (% Top 3), while underperforms at % Top-1 accuracy.Surprisingly, there is little variation in accuracy between our classifiers.With the exception of the model, they all achieve around 70% Top-1 accuracy and around 90% Top-3 accuracy.This suggests that the model they learn is relatively simple.In particular, notice that although the has 50× fewer hidden neurons than the , it only loses around 4% accuracy.We also note that our classifiers consistently perform better when trained on the programs and tested on the programs than vice versa. Feature UtilityWe have shown that we can train a classifier to effectively localize type errors, but which of the feature classes from <ref> are contributing the most to our accuracy?We focus specifically on feature classes rather than individual features as our 282 features are conceptually grouped into a much smaller number of categorical features.For example, the syntactic class of an expression is conceptually a feature but there are 45 possible values it could take; to encode this feature for learning we split it into 45 distinct binary features.Analyses that focus on individual features, ANOVA, are difficult to interpret in our setting, as they will tell us the importance of the binary features but not the higher-level categorical features.Thus, to answer our question we investigate the performance of classifiers trained on various subsets of the feature classes.Type Error SliceFirst we must justify our decision to automatically exclude expressions outside the minimal type error slice from consideration.Thus, we compare three sets of features:* A baseline with only local syntactic features and no preemptive filtering by .* The features of (1) extended with .* The same features as (1), but we preemptively discard samples where is disabled. The key difference between (2) and (3) is that a classifier for (2) must learn that is a strong predictor.In contrast, a classifier for (3) must only learn about the syntactic features, the decision to discard samples where is disabled has already been made by a human.This has a few additional advantages: it reduces the set of candidate locations by a factor of 7 on average, and it guarantees that any prediction made by the classifier can fix the type error.We expect that (2) will perform better than (1) as it contains more information, and that (3) will perform better than (2) as the classifier does not have to learn the importance of .We tested our hypothesis with the and[A layer of 500 neurons is excessive when we have so few input features — we use for continuity with the surrounding sections.]classifiers, cross-validated (k=10) over the combined SP14/FA15 dataset.We trained for a single epoch on feature sets (1) and (2), and for 8 epochs on (3), so that the total number of training samples would be roughly equal for each feature set.|In addition to accuracy, we report each classifier's recall — “How many true changes can we remember?” — defined as|𝗉𝗋𝖾𝖽𝗂𝖼𝗍𝖾𝖽∩𝗈𝗋𝖺𝖼𝗅𝖾|/|𝗈𝗋𝖺𝖼𝗅𝖾|where 𝗉𝗋𝖾𝖽𝗂𝖼𝗍𝖾𝖽 is limited to the top 3 predictions, and 𝗈𝗋𝖺𝖼𝗅𝖾 is the student's fix, limited to changes that are in the type error slice.We make the latter distinction as:(1) changes that are not part of the type error slice are noise in the data set; and(2) it makes the comparison easier to interpret since 𝗈𝗋𝖺𝖼𝗅𝖾 never changes.| Results <ref> shows the results of our experiment.As expected, the baseline performs the worst, with a mere 25% Top-1 accuracy.Adding improves the results substantially with a 45% Top-1 accuracy, demonstrating the importance of a minimal error slice.However, filtering out expressions that are not part of the slice further improves the results to 54% Top-1 accuracy.Interestingly, while the performs similarly poor with no error slice features, it recovers nearly all of its accuracy after being given the error slice features.Top-1 accuracy jumps from 29% to 53% when we add , and only improves by 1% when we filter out expressions that are not part of the error slice.Still, the accuracy gain comes at zero cost, and given the other benefits of filtering by — shrinking the search space and guaranteeing our predictions are actionable — we choose to filter all programs by .Contextual Features We investigate the relative impact of the other three classes of features discussed in <ref>, assuming we have discarded expressions not in the type error slice.For this experiment we consider again a baseline of only local syntactic features, extended by each combination of(1) expression size; (2) contextual syntactic features; and (3) typing features.As before, we perform a 10-fold cross-validation, but we train for a full 20 epochs to make the differences more apparent.Results <ref> summarizes the results of this experiment.The classifier and the start off competitive when given only local syntactic features, but the quickly outperforms as we add features.is the weakest feature, improving Top-1 accuracy by less than 1% and by only 4%.In contrast, the contextual syntactic features improve Top-1 accuracy by 5% (16%), and the typing features improve Top-1 accuracy by 6% (18%).Furthermore, while does provide some benefit when it is the only additional feature, it does not appear to provide any real increase in accuracy when added alongside the contextual or typing features.This is likely explained by feature overlap, the contextual features of “child” expressions additionally provide some information about the size of the subtree.As one might expect, the typing features are more beneficial than the contextual syntactic features.They improve Top-1 accuracy by an additional 1% (3%), and are much more compact — requiring only 55 typing features compared to 180 contextual syntactic features.This aligns with our intuition that types should be a good summary of the context of an expression.However, typing features do not appear to subsume contextual syntactic features, the gains an additional 4% Top-1 accuracy when both are added.Threats to Validity Although our experiments demonstrate that our technique can pinpoint type errors more accurately than the state of the art and that our features are relevant to blame assignment, our results may not generalize to other problem domains or program sets.One threat to validity associated with supervised machine learning is overfitting (learning a model that is too complex with respect to the data).A similar issue that arises in machine learning is model stability (can small changes to the training set produce large changes in the model?).We mitigate these threats by:(1) using separate training and testing datasets drawn from distinct student populations (<ref>), demonstrating the generality of our models; and(2) via cross-validation on the joint dataset (<ref>), which demonstrates the stability of our models by averaging the accuracy of 10 models trained on distinct subsets of the data.Our benchmarks were drawn from students in an undergraduate course and may not be representative of other student populations.We mitigate this threat by including the largest empirical evaluation of type error localization that we are aware of: over 5,000 pairs of ill-typed programs and fixes from two instances of the course, with programs from 102 different students.We acknowledge, of course, that students are not industrial programmers and our results may not translate to large-scale software development; however, we are particularly interested in aiding novice programmers as they learn to work inside the type system.A related threat to construct validity is our definition of the immedate next well-typed program as the intended ground truth answer (see <ref>, Challenge 2). Students may, in theory, submit intermediate well-typed program “rewrites” between the original ill-typed program and the final intended answer. Our approach to discarding outliers (see <ref>) is designed to mitigate this threat.Our removal of program pairs that changed too much, where our oracle could not identify the blame of the other tools, or where the other tools timed out or encountered unsupported language features is another threat.It is possible that including the programs that changed excessively would hurt our models, or that the other tools would perform better on the programs with unsupported language features.We note however that(1) outlier removal is a standard technique in machine learning; and(2) our Top-1 accuracy margin is large enough that even if we assumed that were perfect on all excluded programs,we would still lead by 9 points.Examining programs written in as opposed to or any other typed functional language poses yet another threat, common type errors may differ in different languages.is, however, a standard target for research in type error localization and thus our choice admits a direct comparison with prior work.Furthermore, the functional core of that we support does not differ significantly from the functional core of or SML, all of which are effectively lambda calculi with a Hindley-Milner-style type system. Finally, our use of student fixes as oraclesassumes that students are able to correctly identify the source of an error.As the students are in the process of learning the language and type system, this assumption may be faulty.It may be that expert users would disagree with many of the student fixes, and that it is harder to learn a model of expert fixes, or that the state of the art would be better at predicting expert fixes.As we have noted before, we believe it is reasonable to use student fixes as oracles because the student is the best judge of what she intended. Interpreting Specific Predictions Next, we present a qualitative evaluation that compares the predictions made by our classifiers with those of .In particular, we demonstrate, with a series of example programs from our student dataset, how our classifiers are able to use past student mistakes to make more accurate predictions of future fixes.We also take this opportunity to examine some of the specific features our classifiers use to assign blame.For each example, we provide(1) the code;(2) 's prediction;(3) our 's prediction; and(4) an explanation of why our classifier made its prediction, in terms of the features used and their values.We choose the classifier for this section as its model is more easily interpreted than the MLP.We also exclude the feature from the model used in this section, as it makes the predictions harder to motivate, and as we saw in <ref> it does not appear to contribute significantly to the model's accuracy.We explain the predictions by analyzing the paths induced in the decision tree by the features of the input expressions.Recall that each node in a decision tree contains a simple predicate of the features, “is feature v_j enabled?”, which determines whether a sample will continue down the left or right subtree.Thus, we can examine the predicates used and the values of the corresponding features to explain why our made its prediction.We will focus particularly on the enabled features, as they generally provide more information than the disabled features.Furthermore, each node is additionally labeled with the ratio of “blamed” vs “not-blamed” training expressions that passed through it.We can use this information to identify particularly important decisions, we consider a decision that changes the ratio to be more interesting than a decision that does not. Failed Predictions We begin with a few programs where our classifier fails to make the correct prediction.For these programs we will additionally highlight the correct blame location.Constructing a List of DuplicatesOur first program is a simple recursive function |clone| that takes an item |x| and a count |n|, and produces a list containing |n| copies of |x|.let rec clone x n = let loop acc n = if n <= 0 then acc else (*@clone ([x] @ acc)@*) (n - 1) in loop [] nThe student has defined a helper function |loop| with an accumulator |acc|, likely meant to call itself tail-recursively.Unfortunately, she has called the top-level function |clone| rather than |loop| in the |else| branch, this induces a cyclic constraint |'a = 'a list| for the |x| argument to |clone|.Our top prediction coincides with (and ), blaming the the first argument to |clone| rather than the occurrence of |clone| itself. We confess that this prediction is difficult to explain by examining the induced paths.In particular, it only references the expression's context, which is surprising.More clear is why we fail to blame the occurrence of |clone|, two of the enabled features on the path are:(1) the parent is an application; and(2) |clone| has a function type.The model appears to have learned that programmers typically call the correct function.Currying Considered Harmful?Our next example is another ill-fated attempt at |clone|. let rec clone x n = let rec loop (*@x n acc@*) = if n < 0 then acc else (*@loop (x, (n - 1), (x :: acc))@*) in loop (x, n, []) The issue here is that functions are curried by default — they take their arguments one at a time — but our student has called the inner |loop| with all three arguments in a tuple.Many experienced functional programmers would choose to keep |loop| curried and rewrite the calls, however our student decides instead to uncurry |loop|, making it take a tuple of arguments.blames the recursive call to |loop| while our classifier blames thetuple of arguments — a reasonable suggestion, but not the answer the student expected.We fail to blame the definition of |loop| because it is defining a function.First, note that we represent |let f x y = e|as|let f = fun x -> fun y -> e|, thus a change to the pattern |x| would be treated as a change to the outer |fun| expression.With this in mind, we can explain our failure to blame the definition of |loop| (the outer |fun|) as follows:(1) it has a function type;(2) its child is a |fun|; and(3) its parent is a |let|.Thus it appears to the model that the outer |fun| is simply part of a function definition, a common and innocuous phenomenon.Correct Predictions Next, we present a few indicative programs where our first prediction is correct, and all of the other tools' top three predictions are incorrect.Extracting the Digits of an IntegerConsider first a simple recursive function |digitsOfInt| that extracts the digits of an |int|. let rec digitsOfInt n = if n <= 0 then [] else [n mod 10] @ (*@[ digitsOfInt (n / 10) ]@*) Unfortunately, the student has decided to wrap the recursive call to |digitsOfInt| with a list literal, even though |digitsOfInt| already returns an |int list|.Thus, the list literal is inferred to have type |int list list|, which is incompatible with the |int list| on the left of the |@| (list append) operator.Both and the compiler blame the recursive call for returning a |int list| rather than |int|, but the recursive call is correct!As our correctly points out (with high confidence), the fault lies with the list literal surrounding the recursive call, remove it and the type error disappears.An examination of the path induced by the list literal reveals that our is basing its decision on the fact that(1) the expression is a list literal;(2) the child expression is an application, whose return type mentions |int|; and(3) the parent expression is also an application.Interestingly, incorrectly predicts that the child application should change as well, but it is less confident of this prediction and ranks it below the correct blame assignment.Padding a listOur next program, |padZero|, is given two |int list|s as input, and must left-pad the shorter one with enough zeros that the two output lists have equal length.The student first defines a helper |clone|. let rec clone x n = if n <= 0 then [] else x :: clone x (n - 1) Then she defines |padZero| with a branch to determine which list is shorter, followed by a |clone| to zero-pad it.firstnumber=lastlet padZero l1 l2 = let n = List.length l1 - List.length l2 in if n < 0 then (clone 0 ((-1) * n) @ l2, l2) else (l1, (*@clone 0 n :: l2@*))firstnumber=1Alas, our student has accidentally used the |::| operator rather than the |@| operator in the |else| branch.and correctly determine that she cannot cons the |int list| returned by |clone| onto |l2|, which is another |int list|, but they decide to blame the call to |clone|, while our correctly blames the |::| constructor.Examining the path induced by the |::|, we can see that our is influenced by the fact that:(1) |::| is a constructor; (2) the parent is a tuple; and(3) the leftmost child is an application.We note that first fact appears to be particularly significant; an examination of the training samples that reach that decision reveals that, before observing the Is-Constructor feature the classifier is slightly in favor of predicting “blame”, but afterwards it is heavily in favor of predicting “blame”.Many of the following decisions change the balance back towards “no blame” if the “true” path is taken, but the |::| constructor always takes the “false” path.It would appear that our has learned that constructors are particularly suspicious, and is looking for exceptions to thisrule. Our correctly predicts that the recursive call blamed by should not be blamed; a similar examination suggests that the crucial observation is that the recursive call's parent is a data constructor application.§.§ Blame UtilityWe have demonstrated in the preceding sections that we can produce more accurate blame assignments by learning from the collective mistakes of prior students; however, users are the final judge of the utility of an error message.Thus, in this final experiment we ask whether 's correct blame assignments aid users in understanding type errors more than incorrect assignments.We assigned three problems to the students in our user study: the |padZero| and |mulByDigit| programs from <ref>, as well as the following |sepConcat| program let rec sepConcat sep sl = match sl with | [] -> "" | h::t -> let f a x = a ^ (sep ^ x) in let base = (*@[]@*) in (*@List.fold_left f base sl@*) where the student has erroneously returned the empty list, rather than the empty string, in the base case of the fold. For each problem the students were additionally given either 's correct blame assignment or 's incorrect blame assignment, with no error message.The full user study is available in true <ref>. Appendix A of the accompanying tech report <cit.>. Due to the nature of an in-class exam, not every student answered every question, but we always received at least 12 (out of a possible 15 or 16) responses for each problem-tool pair. This session of the course was taught in Reason,[<https://reasonml.github.io>] a dialect of with a more C-like syntax, and thus for the study we transcribed the programs to Reason syntax.We then instructed three annotators (one of whom is an author, the others are graduate students at UCSD) to classify the answers as correct or incorrect.We performed an inter-rater reliability (IRR) analysis to determine the degree to which the annotators consistently graded the exams.As we had more than two annotators assigning nominal (“correct” or “incorrect”) ratings we used Fleiss' kappa <cit.> to measure IRR.Fleiss' kappa is measured on a scale from 1, indicating total agreement, to -1, indicating total disagreement, with 0 indicating random agreement.Finally, we used a one-sided Mann-Whitney U test <cit.> to determine the significance of our results.The null hypothesis was that the responses from students given 's blame were drawn from the same distribution as those given 's, had no effect.Since we used a one-sided test, the alternative to the null hypothesis is that had a positive effect on the responses.We reject the null hypothesis in favor of the alternative if the test produces a significance level p < 0.05, a standard threshold for determining statistical significance.ResultsThe measured kappa values were κ = 0.68 for the explanations and κ = 0.77 for the fixes; while there is no formal notion for what consititutes strong agreement <cit.>, kappa values above 0.60 are often called “substantial” agreement <cit.>.Figure <ref> summarizes a single annotator's results, which show that students given 's blame assignment were generally more likely to correctly explain and fix the type error than those given 's.There was no discernible difference between and for |sepConcat|; however, responses for |padZero| and |mulByDigit| were marked correct 5–25% more often than the responses.While the results appear to show a trend in favor of , they do not rise to the level of statistical significance in this experiment; further investigation is merited.Threats to ValidityMeasuring understanding is difficult, and comes with its own set of threats.Construct.We used the correctness of the student's explanation of, and fix for, the type error as a proxy for her understanding, but it is possible that other metrics would produce different results.A further threat arises from our decision to use Reason syntax rather than .Reason and differ only in syntax, the type system is the same; however, the difference in syntax may affect students' understanding of the programs.For example, Reason uses the notation |[h, ...t]| for the list “cons” constructor, in contrast to 's |h::t|.It is quite possible that Reason's syntax could help students remember that |h| is a single element while |t| is a list. Internal.We assigned students randomly to two groups. The first was given 's blame assignment for |sepConcat| and |mulByDigit|, and 's blame for |padZero|; the second was given the opposite assignment. This ensured that each student was given and problems. Students without sufficient knowledge of Reason could affect the results, as could the time-constrained nature of an exam. Thus, we excluded any answers left blank from our analysis. External.Our experiment used students in the process of learning Reason, and thus may not generalize to all developers. The three programs were chosen manually, via a random selection and filtering of the programs from the dataset, where 's top prediction was correct but 's was incorrect. A different selection of programs may lead to different results. Subjects.We collected exams from 31 students, though due to the nature of the study not every student completed every problem.The number of complete submissions was always at least 12 out of a maximum of 15 or 16 per program-tool pair.Limitations We have shown that we can outperform the state of the art in type error localization by learning a model of the errors that programmers make, using a set of features that closely resemble the information the type checker sees.In this section we highlight some limitations of our approach and potential avenues for future work. User-Defined Types Probably the single biggest limitation of our technique is that we have (a finite set of) features for specific data and type constructors.Anything our models learn about errors made with the |::| constructor or the |list| type cannot easily be translated to new, user-defined datatypes the model has never encountered.We can mitigate this, to some extent, by adding generic syntactic features for data constructors and |match| expressions, but it remains to be seen how much these help. Furthermore, there is no obvious analog for transferring knowledge to new type constructors, which we have seen are both more compact and helpful.As an alternative to encoding information about specific constructors, we might use a more abstract representation.For example, instead of modeling |x :: 2| as a |::| constructor with a right child of type |int|, we might model it as some (unknown) constructor whose right child has an incompatible type.We might symmetrically model the |2| as an integer literal whose type is incompatible with its parent.Anything we learn about |::| and |2| can now be transferred to yet unseen types, but we run the risk generalizing too much — perhaps programmers make different errors with |list|s than they do with other types, and are thus likely to choose different fixes.Balancing the trade-off between specificity and generalizability appears to be a challenging task. Additional Features There are a number of other features that could improve the model's ability to localize errors, that would be easier to add than user-defined types.For example, each occurrence of a variable knows only its type and its immediate neighbors, but it may be helpful to know about other occurrences of the same variable.If a variable is generally used as a |float| but has a single use as an |int|, it seems likely that the latter occurrence (or context) is to blame.Similarly, arguments to a function application are not aware of the constraints imposed on them by the function (and vice versa), they only know that they are occurring in the context of an application. Finally, n-grams on the token stream have proven effective for probabilistic modeling of programming languages <cit.>, we may find that they aid in our task as well.For example, if the observed tokens in an expression diverge from the n-gram model's predictions, that indicates that there is something unusual about the program at that point, and it may signal an error.Independent vs Joint Predictions We treat each sub-expression as if it exists in a vacuum, but in reality the program has a rich graphical structure, particularly if one adds edges connecting different occurrences of the same variable.<cit.> have used these richer models to great effect to make interdependent predictions about programs, de-obfuscating variable names or even inferring types.One could even view our task of locating the source of an error as simply another property to be predicted over a graphical model of the program.One of the key advantages of a graphical model is that the predictions made for one node can influence the predictions made for another node, this is known as structured learning.For example, if, given the expression |1 + true|, we predict |true| to be erroneous, we may be much less likely to predict |+| as erroneous.We compensate somewhat for our lack of structure by adding contextual features and by ranking our predictions by “confidence”, but it would be interesting to see how structured learning over graphical models would perform.Related WorkIn this section we describe two relevant aspects of related work:programming languages approaches to diagnosing type errors, andsoftware engineering approaches to fault localization. Localizing Type Errors It is well-known that the original Damas-Milner algorithm 𝒲 produces errors far from their source, that novices percieve as difficult to interpret <cit.>. The type checker reports an error the moment it finds a constraint that contradicts one of the assumptions, blaming the new inconsistent constraint, and thus it is extremely sensitive to the order in which it traverses the source program (the infamous “left-to-right” bias <cit.>).Several alternative traversal have been proposed, top-down rather than bottom-up <cit.>, or a symmetric traversal that checks sub-expressions independently and only reports an error when two inconsistent sets of constraints are merged <cit.>. Type error slicing <cit.> overcomes the constraint-order bias by extracting a complete and minimal subset of terms that contribute to the error, all of the terms that are required for it to manifest and no more.Slicing typically requires rewriting the type checker with a specialized constraint language and solver, though <cit.> shows how to turn any type checker into a slicer by treating it as a black-box.While slicing techniques guarantee enough information to diagnose the error, they can fall into the trap of providing too much information, producing a slice that is not much smaller than the input. Finding Likely Errors Thus, recent work has focused on finding the most likely source of a type error.<cit.> use Bayesian reasoning to search the constraint graph for constraints that participate in many unsatisfiable paths and relatively few satisfiable paths, based on the intuition that the program should be mostly correct.<cit.> translate the localization problem into a MaxSMT problem, asking an off-the-shelf solver to find the smallest set of constraints that can be removed such that the resulting system is satisfiable.<cit.> improve the scalability ofby reusing the existing type checker as a theory solver in the Nelson-Oppen style, and thus require only a MaxSAT solver.All three of these techniques support weighted constraints to incorporate knowledge about the frequency of different errors, but onlyuse the weights, setting them to the size of the term that induced the constraint.In contrast, our classifiers learn a set of heuristics for predicting the source of type errors by observing a set of ill-typed programs and their subsequent fixes, in a sense using only the weights and no constraint solver.It may be profitable to combine both approaches, learn a set of good weights for one of the above techniques from our training data.Explaining Type Errors In this paper we have focused solely on the task of localizing a type error, but a good error report should also explain the error.<cit.>, <cit.>, and <cit.> attempt to explain type errors by collecting the chain of inferences made by the type checkerand presenting them to the user.<cit.> produces a slice enhanced by arrows showing the dataflow from sources with different types to a shared sink, borrowing the insight of dataflows-as-explanations from MrSpidey <cit.>.<cit.> catalog a set of heuristics for improving the quality of error messages by examining errors made by novices.<cit.>, <cit.>, and <cit.> extend the ability to customize error messages to library authors, enabling domain-specific errors. Such static explanations of type errors run the risk of overwhelming the user with too much information, it may be preferable to treat type error diagnosis as an interactive debugging session.<cit.> extend the type inference procedure to handle open expressions (with unbound variables), allowing users to interactively query the type checker for the types of sub-expressions.<cit.> proposes algorithmic debugging of type errors, presenting the user with a sequence of yes-or-no questions about the inferred types of sub-expressions that guide the user to a specific explanation.<cit.> explain type errors by searching for inputs that expose the run-time error that the type system prevented, and present users with an interactive visualization of the erroneous computation.Fixing Type Errors Some techniques go beyond explaining or locating a type error, and actually attempt to fix the error automatically.<cit.> searches for fixes by enumerating a set of local mutations to the program and querying the type checker to see if the error remains.<cit.> use a notion of variation-based typing to track choices made by the type checker and enumerate potential type (and expression) changes that would fix the error.They also extend the algorithmic debugging technique ofby allowing the user to enter the expected type of specific sub-expressions and suggesting fixes based on these desired types .Our classifiers do not attempt to suggest fixes to type errors, but it may be possible to do so by training a classifier to predict the syntactic class of each expression in the fixed program — we believe this is an exciting direction for future work.Fault LocalizationGiven a defect, fault localization is the task of identifying “suspicious” program elements (lines, statements) that are likely implicated in the defect — thus, type error localization can be viewed as an instance of fault localization.The best-known fault localization technique is likely Tarantula, which uses a simple mathematical formula based on measured information from dynamic normal and buggy runs <cit.>.Other similar approaches, including those of <cit.> and <cit.> consider alternate features of information or refined formulae and generally obtain more precise results; see <cit.> for a survey.While some researchers have approached such fault localization with an eye toward optimality (<cit.> determine optimal coefficients), in general such fault localization approaches are limited by their reliance on either running tests or including relevant features.For example, Tarantula-based techniques require a normal and a buggy run of the program.By contrast, we consider incomplete programs with type errors that may not be executed in any standard sense.Similarly, the features available influence the classes of defects that can be localized.For example, a fault localization scheme based purely on control flow features will have difficulty with cross-site scripting or SQL code injection attacks, which follow the same control flow path on normal and buggy runs (differing only in the user-supplied data).Our feature set is comprised entirely of syntactic and typing features, a natural choice for type errors, but it would likely not generalize to other defects. Conclusion We have presented , which combines modern statistical methods with domain-specific feature engineering to open the door to a new data-driven path towards precise error localization, significantly outperforming the state of the art on a new benchmark suite comprising 5,000 student programs.We found that while machine learning over syntactic features of each term in isolation performs worse than existing purely constraint-based approaches,augmenting the data with a single feature corresponding to the type error slice brings our classifiers up to par with the state of the art, and further augmenting the data with features of an expression's parent and children allows our classifiers to outperform the state of the art by percentage points. As with other forms of machine learning, a key concern is that of data-set bias: are 's models specific to our data set, would they fail on other programs?We address this concern in two ways.First, we partition the data by year, and show that models learned from one year generalize to, perform nearly as well on, the programs from the other year.Second, we argue that in our setting this bias is a feature (and not a bug): it allows to adapt to the kinds of errors that programmers (specifically novices, who are in greatest need of precise feedback) actually make, rather than hardwiring the biases of experts who may suffer from blind spots. In this regard, we are particularly pleased that our classifiers can be trained on a modest amount of data, a single course's worth, and envision a future where each course comes equipped with a model of its students' errors.§ ACKNOWLEDGMENTSWe thank Jiani Huang and Yijun Zhang for helping analyze the student interaction traces, Alexander Bakst and Valentin Robert for assisting with the user study, and the anonymous reviewers for their insightful feedback.§ USER STUDY§.§ Version A < g r a p h i c s > < g r a p h i c s > §.§ Version B < g r a p h i c s > < g r a p h i c s > | http://arxiv.org/abs/1708.07583v2 | {
"authors": [
"Eric L. Seidel",
"Huma Sibghat",
"Kamalika Chaudhuri",
"Westley Weimer",
"Ranjit Jhala"
],
"categories": [
"cs.PL"
],
"primary_category": "cs.PL",
"published": "20170825003424",
"title": "Learning to Blame: Localizing Novice Type Errors with Data-Driven Diagnosis"
} |
§ INTRODUCTION POEMMA, the Probe Of Extreme Multi-Messenger Astrophysics, was selected by NASA for an Astrophysics Probe Mission Concept Study (under ROSES-2016) in early 2017. The comprehensive 18-month POEMMA study involves instrument and mission definition at the Integrated Design Center (IDC) of the Goddard Space Flight Center (GSFC) and an independent cost assessment in preparation for the 2020 Decadal Survey. Here we report on the preliminary concept for POEMMA ahead of the POEMMA Study Report to be submitted to NASA by December 31, 2018. POEMMA is being designed to enable charged-particle astronomy with a significant increase in exposure to the highest energy particles ever observed, ultra-high energy cosmic rays (UHECRs), and the capability to discover cosmogenic tau neutrinos (CTNs) through the observation of Cherenkov radiation produced by upward-going tau decays. POEMMA will provide an all-sky survey of UHECRs with an order of magnitude larger exposure compared to ground array measurements and two orders of magnitude higher exposure in fluorescence mode when compared to ground fluorescence observatories (significantly improving the determination of composition above 10s of EeVs). The increase in exposure combined with the full-sky coverage should reveal the sources of these extremely energetic particles that are known to reach Earth from extragalactic sources and that are yet to be identified. These unidentified sources achieve extreme accelerationthrough mechanisms that are not presently understood. As UHECRs propagate from distant extragalactic sources they interact with cosmic background radiation losing energy through the Greisen-Zatsepin-Kuzmin (GZK) effect <cit.> and producing cosmogenic neutrinos <cit.>. Observations from the leading UHECR observatories, the Pierre Auger Observatory <cit.> in Mendoza, Argentina, and the Telescope Array (TA) <cit.> in Utah, USA, show a spectral shape consistent with the GZK effect, but also explainable by the maximum energy of the unidentified astrophysical accelerators, E_ max. Higher statistics measurements of both the flux and the composition of UHECRsabove 10 EeV (1 EeV = 10^18 eV) together with the detection of the flux of cosmogenic neutrinos can settle this long-standing mystery (see e.g., <cit.> for more details). POEMMA is being designed for a significant increase in statistics and the detection of Cherenkov radiation from up-going tau decays produced by cosmogenic neutrinos. The observation of cosmogenic neutrinos will help solve the puzzle of the origin of UHECRs and begin a new field of Astroparticle Physics with the study of neutrino properties at energies orders of magnitude above those reached by human-made accelerators.The POEMMA design combines the concept developed for the Orbiting Wide-field Light-collectors (OWL) <cit.> mission, the experience of the Extreme Universe Space Observatory (EUSO) on the Japanese Experiment Module (JEM-EUSO) <cit.>fluorescence detection camera as recently flown on EUSO-SPB1 by a NASA Super Pressure Balloon (SPB) <cit.> from Wanaka, New Zealand, with the recently proposed CHerenkov from Astrophysical Neutrinos Telescope(CHANT) <cit.> concept to form a multi-messenger probe of the most extreme environments in the Universe. Building on the OWL concept, POEMMA is composed of two identical satellites flying in formation with the ability to observe overlapping regions during moonless nights at angles ranging from Nadir to just above the limb of the Earth. The satellites' altitude is planned to be varied in tandem from about 525 km up to 1,000 km with different separations and pointing strategies. POEMMA satellites detect UHECRs through the observation of particle cascades (or extensive airshowers) produced by the interaction of UHECRs with the Earth's atmosphere. Particles in extensive airshowers excite nitrogen molecules in the atmosphere, which fluoresce in the ultraviolet (UV) and can be observed by ultra-fast UV cameras. The fluorescence technique has been perfected by the leading ground-based UHECR observatories, Auger and TA, while EUSO-Balloon <cit.> and EUSO-SPB1 <cit.>recently pioneered the fluorescence technique from suborbital space.Each POEMMA satellite consists of a large Schmidt telescope with a deployable mirror similar to the OWL concept of a 7-meter diameter deployable optics system. To reach a lower energy threshold, the POEMMA Schmidt telescope is planned to have an f-number of 0.77, which leads to a 6.5 diameter mirror with an optical aperture of about 14 m^2.This is approximately 2 times the aperture for OWL (7.07 m^2), which was an f/1 system. Each POEMMA telescope monitors a 45^ o field of view (FoV) and a 2.3 m diameter optical aperture with a single corrector plate. A lens-cap lid and a “jiffy-pop” cover protect the mirror of stray light and micrometeoroid. The mirrors act as large light collectors with modest imaging requirements.The POEMMA focal surface is composed of a hybrid of two types of cameras: about 90% of the focal surface is dedicated to the POEMMA fluorescence camera (PFC), while POEMMA Cherenkov camera (PCC) occupies the crescent moon shaped edge of the focal surface which images the limb of the Earth. The PFC is composed of EUSO Photo Detector Modules (PDM) based on multi-anode photomultiplier tubes (MAPMTs) as flown in sub-orbital space in EUSO-Balloon <cit.> and EUSO-SPB1 <cit.> and soon to be deployed in the International Space Station (ISS) as mini-EUSO <cit.>. The typical time between images for the PFC is about 1 μsec.The much faster POEMMA Cherenkov camera (PCC) is composed of Silicon photo-multipliers (SiPMs) with sampling time of about 100 nsec. Note that SiPMs flew on EUSO-SPB1 and soon to be tested in space with mini-EUSO. The PFC registers UHECR tracks from Nadir to just below the Earth's limb, while the PCC registers light within the Cherenkov emission cone of up-going showers around the limb of the Earth and also from high energy cosmic rays above the limb of the Earth. § SCIENCE GOALSPOEMMA will provide a new window on the Universe's most energetic environments and events. POEMMA is being designed to significantly increase the statistics of observations of different components of UHECRs and neutrino species over a wide range of energy with greater focus on the highest energies ever observed . The instrument design will focus on answering following science questionWhat objects can accelerate particles to ultra-high energies?To discover the sources of UHECRs, POEMMA will survey from space two orders of magnitude larger volumes of the atmosphere when compared to ground observatories over the full sky with nearly uniform exposure. The related questions of how are the sources distributed in the sky? will be addressed with a full sky map of UHECRs with significantly higher statistics at the highest energies, where pointing to the sources becomes feasible (above ∼10 EeV). POEMMA is designed to reach unprecedented geometrical apertures >10^6 km^2 sr yr, which, after duty cycle corrections, correspond to annual exposures of more than 10^5 km sr yr at the highest energies. POEMMA will also have high angular resolution (∼ 1^ o). POEMMA will enable far more sensitive sky maps leading to the discovery of the brightest sources of UHECRs in the sky, which are likely to be relatively nearby (within ∼100 Mpc). The appearance of nearby sources in the sky is regulated by the GZK effect, which suppress the contribution from very distant accelerators at energies above 10s of EeV. The EeV UHECR sky is isotropic because sources throughout the observable Universe contribute without any damping, while the 100 EeV UHECR sky should only show the nearby sources as the GZK effect obfuscates sources further then 100 Mpc moving closer to 10 Mpc at 100 EeV. A clear source distribution will become apparent when a high statistics map above 60 EeV is produced by POEMMA. In addition, observations above 10s of EeVavoid large deviations (compared to the few degrees angular resolution) from source to arrival directions on Earth due to cosmic magnetic fields. The angular size of the nearby sources in the sky will probe the magnitude and structure of extragalactic and galactic magnetic fields.Above 10s of EeV, Charged-Particle Astronomy is finally attainable.POEMMA will also address what is the composition of the UHECRs above 10 EeV and how does it evolve as energies reach 100 EeV? POEMMA stereo observations of UHECRs will yield significant increase in measurements of the maximum of extensive airshowers, X_ max, with rms resolution of at least ∼ 60 g/cm^2 and a large enough sample of well reconstructed events with better X_ max separation to distinguish light and heavier nuclei above 10 EeV. These composition measurements together with spectrum and sky distribution of anisotropies will determine the source class of UHECRs. What is the flux of cosmogenic neutrinos? By observing the Cherenkov signal from tau decays from the limb of the Earth, POEMMA will determine the flux level of cosmogenic neutrinos for a wide range of UHECR source models. The cosmogenic neutrino flux is very sensitive to the different candidate UHECR sources and add an extra dimension in determining these unidentified sources. In addition, POEMMA can observe tau neutrinos at lower energies addressing the additional question of: What is the flux of astrophysical tau neutrinos?.POEMMA will search for astrophysical and cosmogenic neutrinos with two techniques. With the same system designed to observe UHECRs, the PFCcan detect deeply penetrating horizontal showers initiated by all flavors of EeV neutrinos in the atmosphere. In addition, the PCC based on the CHANT concept can observe the signal produced from tau neutrinos from 10 PeV (where astrophysical IceCube neutrinos are expected) to 10 EeV (where cosmogenic neutrinos can be discovered). The cosmogenic neutrino flux is a by-product of the propagation of UHECR through the GZK interactions via neutron and muon decays. The spectrum of cosmogenic neutrinos depends on the composition and source distribution of UHECRs. A main peak is generically produced around 1 to 10 EeV. A secondary peak, around PeV energies, may occur in models where protons dominate the UHECR composition due to neutron decay. The PeV peak is generally subdominant to the astrophysical neutrinos observed by IceCube, thus the most important evidence of the GZK process is the detection of neutrinos in the 1 to 10 EeV energy range. This range will establish a new Astroparticle Physics field with the study of neutrino properties at EeV energies, well above energies accessible in the laboratory.POEMMA will also be sensitive to ultra-high-energy photons (UHEP) in the most optimistic astrophysical scenarios. The UHEP flux is highly dependenton the model of UHECR sources, being highly sensitive to the location of the closest sources. UHEP are the dominant component of models based on relic decays from the early universe, including super-heavy dark matter. A clear detection of UHEPs would be momentous discovery.Additional science themes include the study of how strong are magnetic fields in the extragalactic medium? Cosmic magnetic fields are traditionally challenging to measure and very little is known about magnetic fields outside galaxies and clusters of galaxies. The pointing pattern to UHECR sources will constrain these extragalactic fields directly. POEMMA will also study atmospheric phenomena in the optical and the UV such as transient luminous events in the upper atmosphere, will observe meteors arriving on Earth, and will search for meteorite (see, e.g., similar studies for JEM-EUSO in <cit.>).§ CALIBRATION AND ATMOSPHERIC MONITORING POEMMA observes UHECRs over a very large area (approximately the size of the state of Utah, USA) moving across the globe atthe orbital speed of ∼7 km/sec. The observed volume of the atmospherewill include variable amounts of clouds with altitudes from sea level to ∼15 km and variable boundary layer aerosols. Consistency between the stereo views of an event provides a powerful tool for understanding scattering through intervening high clouds or aerosol layers. A monitoring and calibration system is being designed for POEMMA, which includes a steerable UV laser, an infrared camera, a LIDAR, and a system of ground-based UV LEDs and lasers forming a worldwide network to continuously calibrate the triggering, intrinsic luminosity, and pointing accuracy of POEMMA's observations. § MISSION OVERVIEWThe POEMMA mission involves two satellites flying in formation in a relatively low-altitude, near-equatorial orbit, with each satellite operating independent. Once in orbit, the satellites will deploy from their stowed, launch configuration.The large mirror, focal surface, and corrector plate will be deployed along with the solar array, sun shield, and the antennas. One of the key items of the POEMMA instrument and mission definition at the IDC is to determine if both satellites can be launched as a dual-manifest on the same launch vehicle, as was considered for OWL. The satellites will be inserted into a circular orbit at an inclination of about 28.5^ o and an initial altitude of 525 km moving further to 1000 km as initial science goals are completed. The satellites will maneuver to the desired separation distance and attitude. To search for Cherenkov signals from Earth-skimming neutrinos the satellites will be separated such that both fall within the same Cherenkov light pool and oriented to view the same area on the limb of the Earth. To focus on extreme energy cosmic rays, the separation willbe larger and both satellites will be oriented to view the same area near the nadir. A sequence of observing formation stages will be developed to address each science goal for the minimum 3 year mission with a 5 year mission goal. § ACKNOWLEDGEMENTThe POEMMA concept study is funded by NASA Award NNX17AJ82 at the University of Chicago and GSFC. 99GZK K. Greisen, Phys. Rev. Lett. 16, 748 (1966). G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 (1966).BZnu V. S. Berezinsky and G. T. Zatsepin, Physics Letters B 28, 423-424 (1969).Auger A. Aab et al., (Pierre Auger Collaboration),The Pierre Auger Cosmic Ray Observatory, Nuclear Instruments and Methods in Physics ResearchA (798) 172-213 (2015).TA T. Abu-Zayyad, et al. (Telescope Array Collaboration), The Surface Detector Array of the Telescope Array Experiment, Nuclear Instruments and Methods in Physics Research A (689) 87-97 (2012).reviewK. Kotera and A. V. Olinto, The Astrophysics of Ultrahigh-Energy Cosmic Rays Annual Review of Astronomy and Astrophysics, vol. 49, issue 1, pp. 119-153 (2011), arXiv:1101.4256OWL1 F.W. Stecker et al., Orbiting Wide-field Light-collectors, Nucl. Phys. B 136C, 433-438 (2004).OWL2 J. F. Krizmanic et. al.,Optimization of the Orbiting Wide-angle Light Collectors (OWL) Mission for Charged-Particle and Neutrino Astronomy, 33rd- Int. Cosmic Ray Conference, Rio De Janeiro, 2013.JE_ExpAst J. H. Adams Jr. et al., (JEM-EUSO Collaboration),The JEM-EUSO mission: An introduction, Experimental Astronomy 40 3-17 (2015).EUSO_SPB1 Lawrence Wiencke for the JEM-EUSO Collaboration, EUSO-SPB1 Mission and Science. These Proceedings.CHANT A. Neronov, D. V. Semikoz, L. A. Anchordoqui, J. H. Adams, A. V. Olinto, Sensitivity of the space-based CHerenkov from Astrophysical Neutrinos Telescope (CHANT), Phys. Rev. D 95, 023004 (2017), arXiv:1606.03629.EUSO-BalloonM. Bertaina et al, (JEM-EUSO Collaboration), Results of the EUSO-Balloon flight. These proceedings.mini_EUSOM. Ricci, Mini-EUSO: a precursor mission to observe and study Atmosphere and Earth UV emission from the International Space Station. These proceedings.TLE J.H. Adams Jr et al, (JEM-EUSO Collaboration), Science of atmospheric phenomena with JEM-EUSO, Experimental Astronomy 40 239-251 (2015).Meteor M. Bertaina et al, (JEM-EUSO Collaboration),JEM-EUSO: Meteor and nuclearite observations, Experimental Astronomy 40 253-279 (2015). | http://arxiv.org/abs/1708.07599v1 | {
"authors": [
"A. V. Olinto",
"J. H. Adams",
"R. Aloisio",
"L. A. Anchordoqui",
"D. R. Bergman",
"M. E. Bertaina",
"P. Bertone",
"M. Bustamante",
"M. J. Christl",
"S. E. Csorna",
"J. B. Eser",
"F. Fenu",
"C. Guépin",
"E. A. Hays",
"S. Hunter",
"E. Judd",
"I. Jun",
"K. Kotera",
"J. F. Krizmanic",
"E. Kuznetsov",
"S. Mackovjak",
"L. M. Martinez-Sierra",
"M. Mastafa",
"J. N. Matthews",
"J. McEnery",
"J. W. Mitchell",
"A. Neronov",
"A. N. Otte",
"E. Parizot",
"T. C. Paul",
"J. S. Perkins",
"G. Prevot",
"P. Reardon",
"M. H. Reno",
"F. Sarazin",
"K. Shinozaki",
"F. Stecker",
"R. Streitmatter",
"T. Venters",
"L. Wiencke",
"R. M. Young"
],
"categories": [
"astro-ph.IM",
"astro-ph.HE"
],
"primary_category": "astro-ph.IM",
"published": "20170825021335",
"title": "POEMMA: Probe Of Extreme Multi-Messenger Astrophysics"
} |
[email protected] ^1 Instituto de Fsica, Universidade Federal de Gois, 74.001-970, Goinia-GO, Brazil We perform the calculation of the dc resistivity as a function of temperature of the “strange-metal” state that emerges in the vicinity of a spin-density-wave phase transition in the presence of weak disorder. This scenario is relevant to the phenomenology of many important correlated materials, such as, e.g., the pnictides, the heavy-fermion compounds and the cuprates. To accomplish this task, we implement the memory-matrix approach that allows the calculation of the transport coefficients of the model beyond the quasiparticle paradigm. Our computation is also inspired by the ϵ=3-d expansion in a hot-spot model embedded in d-space dimensions recently put forth by Sur and Lee [Phys. Rev. B 91, 125136 (2015)], in which they find a new low-energy non-Fermi liquid fixed point that is perturbatively accessible near three dimensions. As a consequence, we are able to establish here the temperature and doping dependence of the electrical resistivity at intermediate temperatures of a two-dimensional disordered antiferromagnetic metallic model with a composite operator that couples the order-parameter fluctuations to the entire Fermi surface. We argue that our present theory provides a good basis in order to unify the experimental transport data, e.g., in the cuprates and the pnictide superconductors, within a wide range of doping regimes. Keywords: Transport properties; electrical resistivity; spin-fermion model; high-T_c superconductors 74.20.Mn, 74.20.-z, 71.10.HfMemory matrix theory of the dc resistivity of a disordered antiferromagnetic metal with an effective composite operator Hermann Freire^1 December 30, 2023 =======================================================================================================================§ INTRODUCTION The ubiquitous “strange-metal” phase that appears in the cuprate superconductors near optimal doping <cit.>, the iron-based superconductors <cit.>, and heavy fermions compounds <cit.>, to name a few, is probably the most outstanding challenge in the field of strongly correlated systems and quantum criticality.This metallic phase displays a universal linear-in-T resistivity at intermediate temperatures. Since this phase clearly possesses no well-defined quasiparticle excitations at low energies, it represents a universal example of a non-Fermi liquid state that emerges in those correlated materials. Notwithstanding this fact, the underlying microscopic mechanism of such a state remains largely unknown up to this date <cit.>. One problem associated with this conundrum is related to the issue that it is very difficult, from a theoretical viewpoint, to write down a fairly “realistic" model that describes a highly resistive metallic state with no quasiparticles whose momentum relaxation mechanism yields ρ∼ T^α, such that α<2. By contrast, conventional metals are normally described by the paradigmatic Landau's Fermi liquid theory, in which the resistivity follows the well-known scaling relation ρ∼ T^2 due to both umklapp scattering and disorder.In this respect, a prominent theoretical proposal to describe a “strange-metal” phase consists of assuming the existence of a quantum critical point<cit.> (QCP) at T=0 that is responsible for generating such a state at finite temperatures <cit.>. Experimental results in many compounds are by now quite numerous and provide a strong support to this perspective. Interesting possibilities of quantum critical points include spin-density-wave<cit.> (SDW), charge-density-wave<cit.> (CDW), pair-density-wave <cit.> (PDW), onset of various nematic orders <cit.>, loop-current orders <cit.>, fractionalized phases <cit.>, preformed excitonic pairs <cit.>, among many others (see, e.g., <cit.>). We note in passing that recently an evidence of a novel PDW phase in the cuprate superconductor Bi_2Sr_2CaCu_2O_8+x has been given by Hamidian et al. <cit.> using scanned Josephson tunneling microscopy, which was in fact anticipated by the microscopic theories in the Refs. <cit.>. In the present work, we shall focus only on the question of a SDW quantum phase transition underlying the phase diagram of a given correlated material in order to assess specifically what is the corresponding effect on the dc electrical resistivity as a function of doping of the adjacent metallic phases of the system at intermediate temperatures. The study of the SDW quantum criticality has quite a long history in the field (see, e.g., <cit.>), but we will not detail all those works here. In this description, the microscopic mechanism of the Cooper pair formation is associated with the exchange of short-range antiferromagnetic spin-density-wave (SDW) fluctuations <cit.>, which can be enhanced in the vicinity of a SDW quantum critical point. From a numerical viewpoint, it has been recently established<cit.> via a sign-problem-free Quantum Monte Carlo approach that the paradigmatic spin-fermion model, originally proposed by Abanov and Chubukov <cit.>, indeed describes a high-T_c dome-shaped superconducting phase with the correct d-wave symmetry, in agreement with the experimental situation. Despite this statement, we note that in the underdoped regime of the hole-doped cuprates it has been argued <cit.> that the spin-fermion model must be altered to account for the non-double occupancy constraint that should be enforced for this case. From a weak-to-moderate coupling perspective, many analytical works have pointed out that the Abanov-Chubukov spin-fermion model essentially flows to strong-coupling at low energies <cit.>, and one has to be very careful in devising new approximate perturbative schemes to calculate its physical properties. In this sense, an ingenious proposal consists of the ϵ=3-d expansion within a hot-spot model embedded in d-space dimensions recently developed by Sur and Lee <cit.>. Interestingly, they obtain in their work a stable non-Fermi liquid fixed point at low energies that is perturbatively controlled near three spatial dimensions. Transport calculations of the two-dimensional spin-fermion model are also of interest and have been performed by several researchers in the field <cit.>. Since it has become increasingly clear in the last years that there are no long-lived quasiparticles near the so-called “hot spots” (i.e., the points in reciprocal space where the antiferromagnetic boundary intersects the underlying Fermi surface) in the spin-fermion model <cit.>, a strong emphasis has been put on analytical approaches that allow the calculation of transport coefficients of this model without making any reference to the quasiparticle picture. One such approach is the Mori-Zwanzig memory matrix formalism <cit.>. The memory matrix is a generalization of the concept of scattering rate in Boltzmann theory and, for this reason, it can also be applied to strongly-correlated models describing non-Fermi liquid states in which this quantity is clearly not well-defined. In this respect, we point out that two recent papers <cit.> have applied this approach in order to investigate only the contribution to the resistivity due to the “hot spots” on the Fermi surface of the model. Consequently, they have shown that the resistivity contribution associated with those isolated points on the Fermi surface leads naturally to a constant term and a correction that is linear in temperature as a result of disorder. However, given the strong-coupling nature of the problem of SDW quantum criticality in two dimensions <cit.>, it is conceivable that the order-parameter (bosonic) fluctuations of the spin-fermion model might ultimately couple not only to the aforementioned “hot spots”, but crucially also to the remaining parts of the underlying Fermi surface of the model. Technically speaking, this proposal was first suggested by Hartnoll et al. in Ref. <cit.>. In that work, they have shown that this can be achieved via the addition of a composite operator to the original model, which has the important effect of making the rest of the Fermi surface at least “lukewarm” (i.e., strongly renormalized), rather than simply “cold” (i.e., weakly renormalized) as is conventionally assumed in many works <cit.>. Recently, Wei et al. <cit.> were able to demonstrate microscopically the existence of such a composite operator explicitly from the spin-fermion model via an intermediate off-shell state. For this reason, we shall not repeat their derivation here. In other words, we assume, from the outset, the existence of such an effective term in the Lagrangian of the model. This higher-order interaction can be described in terms of a scattering process off a composite mode that includes, e.g., two spin fluctuations, such that the momentum transfer of fermions in the vicinity of the Fermi surface can be effectively small.However, since this interaction involving the “lukewarm” fermions does not contain explicitly umklapp processes, it turns out that such a composite operator alone cannot render the dc resistivity of the model finite. Therefore, it is crucial to include other sources of momentum relaxation (e.g., disorder, phonons, etc) in the model on top of the composite interaction for the corresponding resistivity to become finite. Physically speaking, at high temperatures, the contribution from phonons is of course expected to play an important role in the temperature dependence of the resistivity of the present system. However, as the temperature is lowered, the impurity contribution will surely become the dominant one. For this reason, we shall focus here, as a first step, on impurity scattering alone as the main mechanism for momentum relaxation in the spin-fermion model with composite scattering, and leave the interesting analysis of the phonon contribution at higher temperatures for a future work. Thus, two significant questions, which one may legitimately ask at this point, are the following: What is the interplay of such a quantum critical composite interaction term with the effects of weak disorder on the transport properties (e.g., electrical resistivity) of the spin-fermion model at intermediate temperatures? And, secondly, can it possibly lead to an additional (perhaps, stronger) contribution to the resistivity of the model exhibiting the hallmarks of non-Fermi liquid scaling at finite temperatures with the correct doping dependence as observed experimentally? These are precisely the questions that we intend to address in the present work.Our findings in the present work can be summarized pictorially (e.g., in the context of the cuprate superconductors) in Fig. 1. The two-dimensional model analyzed here describes a “strange-metal” phase at intermediate temperatures, insofar as the dc resistivity is described by ρ_xx(T)∼ AT+BT^2 (for A≫ B at optimal doping). Moreover, on a more phenomenological level, in case there is pseudogap formation in the model at low doping (various proposals for this mechanism exist in the literature – see, e.g., <cit.>), the hot spot regions would become clearly gapped out. In this scenario, the dc resistivity of the model revealed from our calculation would recover a traditional Fermi-liquid-like scaling given by ρ_xx(T)∼ BT^2. This result is surprisingly consistent with many recent transport experiments performed, e.g., in the cuprate superconductors<cit.>.Our paper is organized as follows. First, we define the spin-fermion model with an effective composite mode in the presence of weak disorder. Secondly, we explain the formalism of the Mori-Zwanzig memory matrix approach in order to calculate transport properties of the model beyond the quasiparticle paradigm. Then, we move on to discuss our main results in this work and to compare them with previous results obtained in the literature. Lastly, we present our final conclusions.§ SDW CRITICAL THEORY WITH COMPOSITE MODES As previously explained for the SDW quantum criticality problem, our starting point will be the spin-fermion model<cit.> with the inclusion of a composite operator<cit.> that effectively transfers small momenta and couples the order-parameter (bosonic) fluctuations to the whole Fermi surface of the model. In this way, the SDW critical field theory becomes described by the following Lagrangian ℒ = ∑_σψ̅_σ(∂_τ+ε̅_𝐤)ψ_σ+1/2χ^-1_q(ϕ⃗.ϕ⃗)+u/4!(ϕ⃗.ϕ⃗)^2+ λ∑_σσ'ψ̅_σ(ϕ⃗.τ⃗_σσ')ψ_σ'+λ'∑_σψ̅_σψ_σ(ϕ⃗.ϕ⃗), where χ_q^-1=(q_0^2+c^2𝐪^2+m) with the bosonic momentum 𝐪 centered around the commensurate antiferromagnetic ordering wavevector 𝐐=(π,π), ψ̅_σ and ψ_σ are the fermionic Grassmann fields with spin projection σ, ε̅_𝐤=ε_𝐤-μ is the fermionic energy dispersion relative to the chemical potential μ, ϕ⃗=(ϕ_x,ϕ_y, ϕ_z) is the three-dimensional order-parameter collective field, τ⃗=(τ_x,τ_y, τ_z) are the Pauli matrices, c is the spin-wave velocity, m is the bosonic mass that measures the distance of the theory to the antiferromagnetic QCP, λ is the spin-fermion coupling constant and λ' is the composite operator coupling constant. Regarding the last term in the Lagrangian (i.e., the composite interaction), it is worth mentioning that, by expressing it in momentum space, one can see that while the first spin fluctuation ϕ indeed carries the momentum 𝐐+𝐪_1 (where |𝐪_1|≪ |𝐐|), the second order-parameter field ϕ must necessarily carry the momentum -𝐐+𝐪_2 (where |𝐪_2|≪ |𝐐|), such that the transferred momentum 𝐪_1+𝐪_2 to the low-energy fermions is effectively small in the present theory.In the next step, we proceed to introduce weak disorder in the model. We will add two types of disorder, which couple to different degrees of freedom in the present system, i.e., the short-wavelength disorder and the long-wavelength disorder. The former will couple directly to the fermionic fields in the model, while the latter will correspond to random shifts of the precise location of the SDW quantum critical point. Therefore, we must add to Eq. (<ref>) the following termsℒ_imp = ∑_σV(r⃗)ψ̅_σ(r⃗)ψ_σ(r⃗)+∑_σm(r⃗)[ϕ⃗(r⃗).ϕ⃗(r⃗)], which should satisfy the Gaussian white-noise disorder averaging relations: ⟨⟨ V(r⃗) ⟩⟩=⟨⟨ m(r⃗) ⟩⟩=0, ⟨⟨ V(r⃗)V(r⃗'⃗) ⟩⟩=V_0^2δ^2(r⃗-r⃗'⃗), and ⟨⟨ m(r⃗)m(r⃗'⃗) ⟩⟩=m_0^2δ^2(r⃗-r⃗'⃗). Henceforth, we will refer to the parameter V_0 as a random potential for the fermionic field and the parameter m_0 as the random mass term for the bosonic field.At this point, it is important to emphasize that the renormalization group scaling that is put forward by Sur and Lee in Ref. <cit.> for a hot-spot model embedded in d-space dimensions within a ϵ=d-3 expansion implies the following power counting analysis: [ω]=Λ^z_b, [k_i]=Λ, [u]=Λ^3z_b-3+ϵ, [λ]=Λ^(3z_b-3+ϵ)/2, [λ']=Λ^(2z_b-3+ϵ), [c]=Λ^z_b-1,where Λ is a high-energy (ultraviolet) cutoff of the field theory, z_b is the bosonic dynamical critical exponent, andi=x,y are the components of the momenta. As a result of this scaling, one finds straightforwardly a new stable non-Fermi liquid fixed point at low energies given by: λ^*→ 20πϵ/3, (λ')^*→ 0, u^*→ 0 and z_b^*→ 1+5ϵ/6. We will assume here, as a first approximation, that at intermediate temperatures the effects of weak disorder will not change qualitatively the above fixed-point structure of the model in the clean limit. Another important remark here is related to the fact that this low-energy non-Fermi liquid fixed point is perturbatively controlled in the limit ϵ→ 0, but, strictly speaking, not in the limit ϵ→ 1. Fortunately, this is not a big limitation, in view of the fact that we will restrict our present analysis to an intermediate temperature regime such that T>E_s (where E_s∼ c^2γ, with the parameter γ∝λ^2 being the Landau damping constant of the model), i.e., before the fixed point of Sur and Lee has been reached.We point out that this latter choice is also made on physical grounds, since at lower temperatures the above non-Fermi liquid fixed point is expected to be preempted by a d-wave superconducting phase, whose existence was recently confirmed via sign-problem-free Quantum Monte Carlo simulations of the spin-fermion model<cit.>. Therefore, in the intermediate temperature regime that we will consider here, we may assume, to a good approximation, that the spin-fermion coupling λ lies within a perturbative regime, the composite operator coupling λ' is small but finite, and the bosonic dynamical critical exponent is described by z_b ≃ 1 (i.e., the perturbative effects of Landau damping in the theory may be neglected). Finally, we also note the very recent works by Lee and collaborators <cit.>, in which these authors were able to extend the above renormalization group results in the context of a non-perturbative calculation, and confirmed that z_b=1 at the low-energy non-perturbative fixed point of the spin-fermion model in d=2.§ MEMORY MATRIX APPROACH Since the entire Fermi surface of the spin-fermion model described by the Lagrangian in Eq. (<ref>) can be potentially affected as a consequence of the aforementioned stable non-Fermi-liquid fixed point, it is absolutely necessary to have a methodology at our disposal that calculates transport properties without making any assumption with regard to the existence of quasiparticle excitations in our theory. Inspired by recent holographic methods applied to many fundamental problems in condensed matter systems (see, e.g., Ref. <cit.>), a crucial realization emerged in the community that this can be achieved by using, e.g., the so-called Mori-Zwanzig memory matrix approach (for excellent explanations of this method in different contexts, see, e.g., Refs. <cit.>). In this formalism, only the operators that are either conserved or nearly conserved, which have a finite overlap with the appropriate current of interest, are expected to play an important role in the computation of the transport coefficients. This is because such nearly-conserved operators have naturally the longest relaxation times in the theory. Therefore, these conservation laws turn out to be central to the memory-matrix approach and follow straightforwardly from the underlying symmetries of the Lagrangian. We start here with the matrix of generalized conductivities σ(ω,T) as a function of both frequency ω and temperature T, which can be written in this method asσ(ω,T)=χ^R(T)/-iω+M(ω,T)[χ^R(T)]^-1, where χ^R_AB(T)=(A|B)=χ^R_AB(ω=0) is the matrix of the static retarded susceptibilities of some conserved (or nearly conserved) operators A and B in the system. This latter matrix is defined asχ_AB(iω,T)=∫_0^1/Tdτ e^iωτ⟨ T_τA^†(τ)B(0)⟩, where χ^R_AB(ω)=χ_AB(iω→ω +i0^+), ⟨ ...⟩ is the grand-canonical statistical average, T_τ is the time-ordering operator, and the volume V of the system has been set to unity. As for the memory matrix conventionally denoted by M(ω,T), it can be calculated in this formalism as followsM_AB(ω,T)=∫_0^1/Tdτ⟨Ȧ^†(0)Qi/ω-QLQQḂ(iτ)⟩, where L is the Liouville super-operator, which is defined as LA=[H,A]=-iȦ, with H being the Hamiltonian of the system and Q is a projection operator that projects out of a space of operators spanned by all the conserved and nearly-conserved operators denoted by {A,B,...}. To simplify the memory matrix calculation, we have taken into account the fact that the nearly-conserved operators of the spin-fermion model have the same signature under time-reversal symmetry in Eq. (<ref>), i.e., (Ȧ|B)=0. The memory matrix is fundamentally related to the mechanism of relaxation of all nearly-conserved operators in the system. In addition to this fact, due to the presence of various projection operations in Eq. (<ref>), this matrix is expected to be a smooth function of all coupling constants in the theory. Consequently, within a weak-to-moderate coupling regime, this quantity can be computed via perturbative means.The Lagrangian defined by Eq. (<ref>) is invariant under both global U(1) symmetry and spatial translation. As a result, using Noether's theorem, one finds that both electrical current 𝐉 and the momentum operator 𝐏 of the model are conserved at the classical level. In the present model, they are given by𝐏 = ∫ d^2 x[i∑_σ∇ψ̅_σψ_σ + (∂_t ϕ) ∇ϕ], 𝐉 = -i/m∑_σ∫ d^2 x ∇ψ̅_σψ_σ. At the quantum level, one expects that the corresponding operators will have the longest relaxation times and, for this reason, we will argue that they should dominate the transport properties in the present model. With the subsequent addition of weak disorder [Eq. (<ref>)] to the Lagrangian describing the field theory, it is important to emphasize that the momentum operator does not conserve any longer due to the resulting breaking of translation symmetry. Thus, we obtaini𝐏̇ = ∫d^2𝐪/(2π)^2∫d^2𝐤/(2π)^2𝐤[V(𝐤)∑_σψ̅_σ(𝐤+𝐪)ψ_σ(𝐤)+ m(𝐤)ϕ(𝐪)ϕ(-𝐪-𝐤)]. In the following, we shall use the propagator of z_b= 1 bosons at intermediate temperatures in the spin-fermion model at a critical doping (we set c=1)χ(q_0,𝐪) = 1/q_0^2+𝐪^2+R(T), where q_0 denotes the Matsubara bosonic frequency, 𝐪 is the wavevector and we have neglected the spin-wave mass term from this point on. Additionally, following previous works<cit.>, we have included a low-energy cutoff in the theory described by R(T). This term was exactly calculated in Ref. <cit.>, and it was shown to be given by R(T)=4ln^2[(√(5)+1)/2]T^2 for the case of the dynamical critical exponent z_b=1. As for the fermions, we shall restrict our analysis to the immediate vicinity of the underlying Fermi surface and linearize the energy dispersion as follows: ε̅_𝐤=v⃗_𝐤.𝐤+u'𝐤^2, where v⃗_𝐤 is the Fermi velocity and u' is related to the curvature of the Fermi surface.Therefore, the dc resistivity ρ_xx(T)=lim_ω→ 0[1/σ_xx] of the present model becomes given byρ^-1_xx(T) = Γ^-1(T)χ_P_x J_x(T), where we have introduced the matrix of relaxation rates Γ^-1(T)=χ_J_x P_x(T)M^-1_P_xP_x(T). To leading order, the susceptibility χ_J_x P_x(T) is taken to be the noninteracting susceptibility that naturally evaluates toχ_J_x P_x(T) = 1/m∫d^2𝐤/(2π)^2 k_x^2 [n_F(ε̅_𝐤)-n_F(ε̅_𝐤+𝐪)]/(ε̅_𝐤-ε̅_𝐤+𝐪)≈ m/πμ(T=0)+O(e^-βμ), where n_F(ε)=1/(e^βε+1) is the Fermi-Dirac distribution, μ is the chemical potential and β=1/T is the inverse temperature. From the above expression, one can clearly see that χ_J_x P_x(T) has no temperature dependence. Consequently, the T-dependence of the resistivity will come solely from the memory matrix M_P_x P_x(T). For this reason, we now move on to this calculation.We now consider the effects of weak disorder to the memory matrix. For the reasons already explained before, we may assume that, at intermediate temperatures, the spin-fermion coupling λ lies within a perturbative regime, the composite operator coupling λ' is small but finite, and both the random potential V_0 and the random mass term m_0 are also small. Since the equation of motion of the momentum [Eq. (<ref>)] is of order linear in the parameters V_0 and m_0, the leading contribution of M turns out to be quadratic in the same parameters. Since we would like to keep only the dominant contribution to the Liouville operator, we shall also replace this operator by its noninteracting value (L≈ L_0). In addition to this, for the same reason, instead of using the full Hamiltonian in all grand-canonical averages, we will replace it by the noninteracting Hamiltonian. As a result, the leading contribution of the memory matrix in the spin-fermion model becomesM_P_x P_x(ω→ 0,T) = lim_ω→ 0Im G^R_Ṗ_̇ẋṖ_̇ẋ(ω,T)/ω, where G^R_Ṗ_̇ẋṖ_̇ẋ(ω,T)=⟨Ṗ_̇ẋ(ω)Ṗ_̇ẋ(-ω)⟩ is the corresponding retarded Green's function within the Matsubara formalism at finite temperature T. The corresponding Feynman diagrams are displayed in Fig. 2.§ RESULTS Here, we compute in an explicit way the memory matrix of the present model. In what follows, we stress that we will not take into account the Cooperon channel in the calculation. This contribution is generally expected to lead to the phenomenon of weak localization at low temperatures, which we will ignore in this paper for simplicity. Therefore, the perturbative contributions to the memory matrix in the present model turn out to be the ones depicted in Fig. 2. Henceforth, these terms will be referred to in the exact same order as they appear in this latter figure. Thus, M_P_xP_x(T)=∑_i=0^10M^(i)(T)+⋯ . The zeroth order Feynman diagram stands for the lowest order coupling of short-range disorder to the fermion fields. Its leading contribution will come from pairs of hot spots (i,j), such that i≠ j, that are connected by a large momentum transfer denoted by a vector Q⃗_ij in momentum space. The diagram then evaluates toM^(0)=V_0^2 Im{∑_i,j,i≠ jQ^ij 2_x/ω∫_𝐤,𝐪[n_F(ε̅_𝐪)-n_F(ε̅_𝐤+𝐪)]/ω+ε̅_𝐪-ε̅_𝐤+𝐪+i0^+}, where ∫_𝐤=∫ d^2 𝐤/(2π)^2 and the limit ω→ 0 should be taken. Using the identity 1/(x+i0^+)=𝒫(1/x)-iπδ(x), the above inter-hot-spot scattering term essentially yields a temperature-independent contribution, which is given byM^(0)=-∑_i,j,i≠ jQ^ij 2_x V_0^2Λ^2/4π^3 |v⃗_i×v⃗_j|, where the ultraviolet cutoff Λ must be imposed on the integrations over all the energies ε̅_iα relative to the chemical potential in the present theory. Since the local curvature parameter u' in the fermionic energy dispersion turns out to be irrelevant in the low-energy stable fixed point<cit.> discussed in Sec. II, we neglected, for simplicity, this curvature term in the above computation. We point out that we will use the same approximation in all contributions to the memory matrix that follow in this work. Finally, the above anisotropic equation for M^(0) will naturally contribute to the residual resistivity ρ_0 that should appear in the model at low temperatures (e.g., if the superconducting phase that exists in the model is suppressed by the application of an external field).The next diagram in Fig. 2 refers to the lowest order coupling of long-range disorder to the bosonic modes. It eventually becomesM^(1)(T) = m_0^2 Im{∫_𝐤,𝐪k_x^2∫_E_1,E_2π^2sign(E_1)sign(E_2)× (E_1^2-R(T))θ(E_1^2-R(T))θ(E_2^2-R(T))× (1/2ω)[n_B(E_2)-n_B(E_1)]/ω+E_2-E_1+i0^+} where n_B(ε)=1/(e^βε-1) is the Bose-Einstein distribution and θ(x) is the Heaviside step function. In the above derivation (and also in the calculations that follow in the present work), we use the spectral function A(E,𝐤) of the bosonic propagator, i.e.χ(k_0,𝐤)=∫_-∞^∞dE A(E,𝐤)/ik_0 -E where A(E,𝐤)=1/2α_𝐤[δ(E+α_𝐤)-δ(E-α_𝐤)] with α_𝐤=√(𝐤^2+ R(T)). Additionally, we use also the following identities ∫d^2𝐤/α_𝐤[δ(E+α_𝐤) - δ(E-α_𝐤)]=-πsgn(E)× θ(E^2-R(T)), and∫d^2𝐤/α_𝐤 k_x^2 [δ(E+α_𝐤)-δ(E-α_𝐤)]=-π/2sgn(E)× (E^2-R(T))θ(E^2-R(T)). By calculating Eq. (<ref>), M^(1)(T) yields a Fermi-liquid-like T^2-contribution to the memory matrix calculation, i.e.M^(1)(T)≈(1.77 m_0^2/16π^2)T^2. A quick examination of the above result also reveals that the prefactor is isotropic and independent of doping in the model. This should be contrasted with the other contributions to the memory matrix calculated in this work [such as, e.g., Eq. (<ref>)], whose prefactors are anisotropic and dependent on doping, in view of the appearance of cross products of Fermi velocities in those expressions that are clearly related to the band structure. As for the Feynman diagrams given by M^(2) and M^(8) in Fig. 2 that represent corrections due to the renormalization of the bosonic propagator at one-loop and two-loop orders respectively, it is straightforward to show using Eqs. (<ref>) and (<ref>) that these terms evaluate to zero. The same result also holds to M^(6), M^(7), M^(9) and M^(10), which represent various corrections to the fermionic propagators in the model. In the calculation of the latter terms, since the fermionic propagators have independent energies given by ε̅_iα, expressions of the type ∫ dε̅/(iω - ε̅)^n=0 for n≥ 2 naturally appear, and the corresponding Feynman diagrams vanish as well. The fourth diagram in Fig. 2 is one of the leading vertex corrections to the resistivity with composite interaction λ'. As we have explained before, it couples to the whole Fermi surface of the model. Computing this term analytically for the present model is quite cumbersome, but it can be of course evaluated numerically. We obtain that it yields a leading contribution given by M^(3)(T)=-Im{V_0^2λ'^2/ω∫_𝐤,𝐪,𝐤',𝐪'k_x^21/β^3∑_k_0,q_0,k'_01/[(k'_0+q'_0)^2+(𝐤'+𝐪')^2+R(T)]1/[k_0^'2+𝐤'^2+R(T)]1/(iω+iq_0-ε̅_𝐤+𝐪) ×1/(iω+iq_0+iq'_0-ε̅_𝐤+𝐪+𝐪')1/(iq_0-ε̅_𝐪)1/(iq_0+iq'_0-ε̅_𝐪+𝐪')}≈∑_i,j,i≠ j∑_m,n,m≠ n(7.93 V_0^2 λ'^2/256π^3 | v⃗_i×v⃗_j| | v⃗_m×v⃗_n|)T^4.where k_0 and q_0 stand for fermionic Matsubara frequencies, whereas k'_0 refers to a bosonic frequency and the limit ω→ 0 should be taken. The pairs of indices (i,j) and (m,n) run over all points of the Fermi surface connected by a small wavevector 𝐪'.As can be seen from Eq. (<ref>), the composite interaction λ' gives a non-Fermi liquid contribution to the resistivity that will be analyzed more carefully below.Like the previous calculation, the solution of the fifth Feynman diagram in Fig. 2, which is an important vertex correction to the model with composite interaction λ' as well, turns out to be also somewhat lengthy. For this term, we obtain that the leading contribution isM^(4)=-Im{m_0^2λ'^2/ω∫_𝐤,𝐪,𝐤',𝐪'k_x'^2 1/β^3∑_k_0,q_0,k'_01/(ik_0+iq_0-ε̅_𝐤+𝐪)1/(ik_0-ε̅_𝐤)1/[(ω+q'_0)^2+(𝐤'+𝐪')^2+R(T)]1/[q_0^'2+𝐪'^2+R(T)] ×1/[(q'_0-q_0)^2+(𝐪'-𝐪)^2+R(T)]1/[(ω+q'_0-q_0)^2+(𝐤'+𝐪'-𝐪)^2+R(T)]}≈∑_i,j,i≠ j(0.96 m_0^2 λ'^2/256π^2|v⃗_i×v⃗_j|)Λ^2, where the limit ω→ 0 should be taken and the indices i and j run over all points of the Fermi surface connected by a small wavevector 𝐪. Therefore, Eq. (<ref>) yields a temperature-independent contribution to the resistivity. Because of this property, this term will also contribute to the residual resistivity ρ_0 of the model. Lastly, the sixth Feynman diagram is the so-called Altshuler-Aronov-type vertex correction with inter-hot-spot interaction λ to the memory matrix. We mention here that this contribution has been explicitly calculated previously in Ref. <cit.>. We agree here with their result for the present case, in which the bosonic dynamical critical exponent z≃ 1. In this way, the corresponding diagram evaluates toM^(5)(T)=-lim_ω→ 0Im{V_0^2λ^2/ω∫_𝐤,𝐪,𝐤' Q^ij 2_x1/β^2∑_k_0,q_01/[k_0^'2+𝐤^'2+R(T)]1/(iω+iq_0-ε̅_𝐤+𝐪)1/(iω+iq_0+ik'_0-ε̅_𝐤+𝐪+𝐤') ×1/(ik'_0+iq_0-ε̅_𝐤'+𝐪)1/(iq_0-ε̅_𝐪)}≈∑_i,j,i≠ j∑_α,β(0.001 V_0^2 λ^2 Q^ij 2_x / | v⃗_iα×v⃗_iβ| | v⃗_jα×v⃗_jβ|)T, where we have slightly changed our previous notation by including new indices α, β that now run over only the hot spots of the Fermi surface, which are connected by the large momentum transfer denoted by the wavevector Q⃗_ij. Therefore, the above result gives a non-Fermi-liquid linear-in-T contribution to the resistivity. Using Eq. (<ref>), and collecting all the above expressions, we obtain that the dc electrical resistivity of the present model finally becomes ρ_xx(T)=ρ_0+AT+BT^2+CT^4, where ρ_0∝ [M^(0)(T)+M^(4)(T)]/χ_J_x P_x^2(T), A∝ M^(5)(T)/(Tχ_J_x P_x^2(T)), B∝ M^(1)(T)/(T^2χ_J_x P_x^2), and C∝ M^(3)(T)/(T^4χ_J_x P_x^2). As can be seen, the residual resistivity of the model has therefore two independent contributions associated with different scattering mechanisms in the present model. As a consequence, our transport theory implies that even though the magnitude of the residual resistivity ρ_0 is not entirely related to the coefficient of the linear resistivity A, there are some important correlations among each other in the present effective model.It is also interesting to note here that the non-Fermi-liquid T^4-contribution to the resistivity of the model turns out to be subleading with respect to the short-wavelength disorder contribution calculated in Eq. (<ref>). For this reason, Eq. (<ref>) may be further approximated toρ_xx(T)≈ρ_0+AT+BT^2. A crucial point that we wish to stress here is that the BT^2-contribution to the resistivity is not necessarily subleading to the AT-term. The reason for this is that the two contributions come from different sources of momentum relaxation: the AT-term is associated with a hot-spot scattering mechanism off short-wavelength disorder, whereas the BT^2-contribution in turn requires long-wavelength disorder coupled to the bosonic order-parameter fluctuations in the model. In other words, our results add support to the point of view that the spin-fermion model may indeed describe a “strange-metal” phase at intermediate temperatures, but with a resistivity described by ρ_xx∼ AT+BT^2 (the relative magnitude of the prefactors A and B are such that A≫ B at optimal doping).In addition to this fact, we also would like to point out that, while the scattering rate associated with the coefficient A in the resistivity is highly anisotropic and doping-dependent [see Eq. (<ref>)], the scattering rate related to the coefficient B turns out to be isotropic and independent of doping [see Eq. (<ref>)]. This feature of our present transport theory is remarkably consistent with recent experimental measurements performed in the cuprate compounds, such as, e.g., HgBa_2CuO_4+δ (Ref. <cit.>) and Tl_2Ba_2CuO_6+δ (Ref. <cit.>). It is also worthwhile to comment on that our theoretical prediction clearly agrees with a proposed phenomenology put forward by Hussey and collaborators in order to describe various aspects of the transport data of the cuprates in the literature <cit.>. In addition, we mention that some aspects of our present work also agree qualitatively with other transport theories, which include the cold-spot-model proposed by Ioffe and Millis <cit.> and the nearly antiferromagnetic Fermi liquid model put forward by Stojkovic and Pines <cit.>.Another important consequence of our theory is that, for the physical situation in which there is pseudogap opening in the antinodal directions of the Fermi surface as revealed by angle-resolved photoemission experiments (we mention here that several proposals for this mechanism exist in the literature – see, e.g., <cit.>), the corresponding hot spots become gapped out and disappear altogether. Therefore, the resulting dc resistivity of the present model should recover a Fermi-liquid-like scaling described by ρ_xx∼ T^2 within this range of temperatures. This analysis is surprisingly consistent with recent state-of-the-art experiments performed by Mirzaeia et al. <cit.> and Barisic et al. <cit.> for the single-layer HgBa_2CuO_4+δ compound inside the pseudogap state, in which they provided strong evidence that both optical conductivity and electrical conductivity of this phase conform to Fermi-liquid-like scalings. An additional outcome of the present result is that, for doping regimes that lie beyond the situation of van Hove band-filling in the single-band model, the hot spots cease to exist. The reason is that, in this case, there is no intersection of the underlying Fermi surface of the model with the antiferromagnetic zone boundary any longer. Therefore, in such a highly overdoped regime, the resistivity of the model according to our theory should again recover a Fermi-liquid-like result ρ_xx∼ T^2 as a function of temperature. This is of course consistent with transport experiments performed in all cuprate materials, where it is observed that the normal state in the overdoped regime is indeed described by the Fermi liquid theory. To make evident this point and to emphasize other similarities with the physics of the high-T_c cuprates, we depict pictorially a phase diagram from the point of view of transport of the two-dimensional spin-fermion model obtained in the present work together with recent theoretical results<cit.> related to this model in Fig. 1.Lastly, we point out that a similar evolution of doping and temperature dependence of the resistivity has also been observed in the metallic phase of many iron-based superconductors [see, e.g., the compounds BaFe_2(As_1-xP_x)_2 in the Refs. <cit.> and Ba(Fe_1-xCo_x)_2As_2 in the Refs. <cit.>]. Even though these are in some sense complicated materials that exhibit many bands at the Fermi energy level, our present result could also suggest that the spin-fermion model with an effective composite interaction may capture universal aspects of correlated metallic systems in the presence of strong antiferromagnetic fluctuations. § CONCLUSIONS We have presented the computation of the dc resistivity as a function of doping and temperature of the metallic states that emerge due to the existence of a putative SDW quantum critical point in the presence of weak disorder. This analysis is relevant to the phenomenology of many important correlated materials. For this calculation, we have implemented the Mori-Zwanzig memory-matrix approach that importantly does not rely on the existence of well-defined quasiparticle excitations in this model at low energies. As an application of the present transport theory, we have compared our predictions to the experimental situation in the cuprate superconductors. In the “strange-metal” phase, the dc resistivity evaluates to ρ(T )∼ AT + BT^2 at intermediate temperatures (for A≫ B, close to optimal doping), where the scattering rate related to the prefactor A is non-universal and strongly doping-dependent, while the scattering rate related to B is universal. We have also correctly obtained the temperature dependence of the dc resistivity of the system as measured experimentally in recent works <cit.>, both inside the pseudogap phase and in the overdoped metallic regime, whose transport coefficients indeed conform to Fermi-liquid-like scalings. It will be clearly very important to analyze also the magneto-transport of the two-dimensional spin-fermion model with an effective composite operator using the present theory. For this reason, we plan to perform in a subsequent work a similar analysis for the Hall angle and the magnetoresistance of the model within the memory matrix formalism. From an experimental point of view, the cotangent of the Hall angle θ_H in the strange metal phase of the cuprates at optimal doping is characterized by a scaling law given by (θ_H)∼ T^2, which is seemingly Fermi-liquid-like, despite the fact that the resistivity is given by ρ(T )∼ AT + BT^2 (for A≫ B) within the same temperature regime. Therefore, only an approach that allows the calculation of the transport properties of a model, which does not rely on the quasiparticle picture is capable of reproducing such an important result. Indeed, given our encouraging results obtained in this work concerning the doping dependence of the dc resistivity of a disordered nearly antiferromagnetic two-dimensional metal, we believe that the present framework may provide a good basis in order to unify all the available experimental transport data, e.g., in the cuprate superconductors and also in the closely related iron-based superconductors, within a wide range of doping and temperature regimes. I would like to thank V. S. de Carvalho for his help with the figures in the present manuscript. Financial support from CNPq under Grant Number 405584/2016-4 is also acknowledged. 99 Ong T. R. Chien, Z. Z. Wang, and N. P. Ong, Phys. Rev. Lett. 67, 2088 (1991). Yoshida K. Yoshida, A. I. Rykov, S. Tajima and I. Terasaki, Phys. Rev. B 60, R15035 (1999). Tyler A.W. Tyler and A.P. Mackenzie, Physica C 282, 1185 (1997). Matsuda S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Phys. Rev. B 81, 184519 (2010). Analytis J. G. Analytis, H-H. Kuo, R. D. McDonald, M. Wartenbe, P. M. C. Rourke, N. E. Hussey, and I. R. Fisher, Nature Physics 10, 194 (2014). Coleman J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y. Tokiwa, O. Trovarelli, C. Geibel, F. Steglich, C. Ppin and P. Coleman, Nature 424, 524 (2003). McGreevy J. McGreevy, Physics 3, 83 (2010). Hussey N. E. Hussey, J. Phys: Condens. Matter 20, 123201 (2008), and references therein. Hertz J. A. Hertz, Phys. Rev. B 14, 1165 (1976). Millis A. J. Millis, Phys. Rev. B 48, 7183 (1993). Varma C. M. Varma, Phys. Rev. B 55, 14554 (1997). Sachdev S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). Chubukov1 Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 (2000). Chubukov2 A. Abanov, A. V. Chubukov, and J. Schmalian, Adv. Phys. 52, 119 (2003). Metlitski M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010). Efetov K. B. Efetov, H. Meier, and C. Ppin, Nat. Phys. 9, 442 (2013). Freire1 V. S. de Carvalho and H. Freire, Annals of Physics 348, 32 (2014); V. S. de Carvalho and H. Freire, Nucl. Phys. B 875, 738 (2013). Castellani C. Castellani, C. DiCastro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995); C. Castellani, C. DiCastro, and M. Grilli, Z. Phys. B: Condens. Matter 103, 137 (1997). Wang2 Y. Wang and A. Chubukov, Phys. Rev. B 90, 035149 (2014). PALee P. A. Lee, Phys. Rev. X 4, 031017 (2014). Agterberg D. F. Agterberg, D. S. Melchert, and M. K. Kashyap, Phys. Rev. B 91, 054502 (2015). Fradkin E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015). Freire2 H. Freire, V. S. de Carvalho, and C. Pepin, Phys. Rev. B 92, 045132 (2015). Wang Y. Wang, D. F. Agterberg, and A. Chubukov, Phys. Rev. B 91, 115103 (2015); Y. Wang, D. F. Agterberg, and A. Chubukov, Phys. Rev. Lett. 114, 197001 (2015). Kivelson V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev. B 64, 195109 (2001). Varma2 C. M. Varma, Phys. Rev. B 73, 155113 (2006). Freire3 V. S. de Carvalho, T. Kloss, X. Montiel, H. Freire, and C. Pepin, Phys. Rev. B 92, 075123 (2015); V. S. de Carvalho, C. Pepin, and H. Freire, Phys. Rev. B 93, 115144 (2016). Chowdhury D. Chowdhury and S. Sachdev, Phys. Rev. B 90, 245136 (2014); D. Chowdhury and S. Sachdev, arXiv:1501.00002 (2015). Pepin X. Montiel, T. Kloss, and C. Ppin, EPL 115, 57001 (2016); X. Montiel, T. Kloss, and C. Ppin, arXiv:1510.03038 (2015). Eberlein A. Eberlein, W. Metzner, S. Sachdev, and H. Yamase, Phys. Rev. Lett. 117, 187001 (2016). Hamidian M. H. Hamidian, S. D. Edkins, Sang Hyun Joo, A. Kostin, H. Eisaki, S. Uchida, M. J. Lawler, E. -A. Kim, A. P. Mackenzie, K. Fujita, Jinho Lee, and J. C. Samus Davis, Nature 532, 343 (2016). Pines P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).Freire4 For a recent review, see, e.g., T. Kloss, X. Montiel, V. S. de Carvalho, H. Freire, and C. Ppin, Rep. Prog. Phys. 79, 084507 (2016). Schattner Y. Schattner, M. H. Gerlach, S. Trebst, and E. Berg, Phys. Rev. Lett. 117, 097002 (2016); M. H. Gerlach, Y. Schattner, E. Berg, and S. Trebst, Phys. Rev. B 95, 035124 (2017). Kochetov A. Ferraz and E. Kochetov, EPL 109, 37003 (2015). Sachdev2 S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, Phys. Rev. B 94, 115147 (2016). SSLee S. Sur and S. S. Lee, Phys. Rev. B 91, 125136 (2015). Hartnoll S. A. Hartnoll, D. M. Hofman, M. A. Metlitski, and S. Sachdev, Phys. Rev. B 84, 125115 (2011). Patel A. A. Patel and S. Sachdev, Phys. Rev. B 90, 165146 (2014). Patel2 A. A. Patel, P. Strack, and S. Sachdev, Phys. Rev. B 92, 165105 (2015). Maslov2 A. V. Chubukov, D. L. Maslov, and V. I. Yudson, Phys. Rev. B 89, 155126 (2014). Efetov2 K. B. Efetov, Phys. Rev. B 91, 045110 (2015). Maslov D. L. Maslov and A. V. Chubukov, arXiv:1608.02514 (2016). Pepin2 X. Montiel, T. Kloss, and C. Ppin, arXiv:1611.06597 (2016). Forster D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Cambridge, 1975). Rosch A. Rosch and N. Andrei, Phys. Rev. Lett. 85, 1092 (2000). Hartnoll3 S. A. Hartnoll, R. Mahajan, M. Punk, and S. Sachdev, Phys. Rev. B 89, 155130 (2014). Freire5 H. Freire, Annals of Physics 349, 357 (2014). Hartnoll2 R. Mahajan, M. Barkeshli, and S. A. Hartnoll, Phys. Rev. B 88, 125107 (2013). Lucas A. Lucas and S. Sachdev, Phys. Rev. B 91, 195122 (2015). Hlubina R. Hlubina and T. M. Rice, Phys. Rev. B 51, 9253 (1995). Schmalian P. S. Weiss, B. N. Narozhny, J. Schmalian, and P. Wolfle, Phys. Rev. B 93, 045128 (2016). Harrison N. Harrison and S. E. Sebastian, New Journal of Physics 14, 095023 (2012). Greven N. Barisic, M. K. Chan, M. J. Veit, C. J. Dorow, Y. Ge, Y. Tang, W. Tabis, G. Yu, X. Zhao, and M. Greven, arXiv:1507.07885 (2015). Mirzaeia S. I. Mirzaeia, D. Strickera, J. N. Hancock, C. Berthoda, A. Georges, E. v. Heumen, M. K. Chan, X. Zhao, Y. Li, M. Greven, N. Barisic, and D. van der Marel, Proc. Natl. Acad. Sci. 110, 5774 (2013). Abdel-Jawad M. Abdel-Jawad, J. G. Analytis, L. Balicas, A. Carrington, J. P. H. Charmant, M. M. J. French, and N. E. Hussey, Phys. Rev. Lett. 99, 107002 (2007). Abrahams E. Abrahams, J. Schmalian, and P. Wolfle, Phys. Rev. B 90, 045105 (2014). SSLee2 A. Schlief, P. Lunts, and S. S. Lee, Phys. Rev. X 7, 021010 (2017); P. Lunts, A. Schlief, and S. S. Lee, Phys. Rev. B 95, 245109 (2017). Holography A. Lucas, S. Sachdev, and K. Schalm, Phys. Rev. D 89, 066018 (2014) and references therein. Chubukov3 A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994). Hussey2 N. E. Hussey, K. Takenaka, and H. Takagi, Phil. Mag. 84, 2847 (2004).Ioffe L. B. Ioffe and A. J. Millis, Phys. Rev. B 58, 11631 (1998). Pines2 B. P. Stojkovic and D. Pines, Phys. Rev. Lett. 76, 811 (1996). Canfield N. Ni, M. E. Tillman, J.-Q. Yan, A. Kracher, S. T. Hannahs, S. L. Bud'ko, and P. C. Canfield, Phys. Rev. B 78, 214515 (2008). Efremov K. Iida, V. Grinenko, F. Kurth, A. Ichinose, I. Tsukada, E. Ahrens, A. Pukenas, P. Chekhonin, W. Skrotzki, A. Teresiak, R. Huehne, S. Aswartham, S. Wurmehl, I. Moench, M. Erbe, J. Haenisch, B. Holzapfel, S. L. Drechsler, and D. V. Efremov, Scientific Reports 6, 28390 (2016). | http://arxiv.org/abs/1708.07537v1 | {
"authors": [
"Hermann Freire"
],
"categories": [
"cond-mat.str-el",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.str-el",
"published": "20170824195338",
"title": "Memory matrix theory of the dc resistivity of a disordered antiferromagnetic metal with an effective composite operator"
} |
APL Department of Applied Sciences and Mechatronics, Munich University of Applied Sciences, Lothstr. 34, 80335 Munich, Germany These two authors contributed equally to this work.Department of Applied Sciences and Mechatronics, Munich University of Applied Sciences, Lothstr. 34, 80335 Munich, Germany These two authors contributed equally to this work. NaMLab gGmbH, Noethnitzer Strasse 64, 01187 Dresden, Germany Chair of Nanoelectronic Materials, Technische Universität Dresden, Noethnitzer Strasse 64, 01187 Dresden, Germany [email protected] Department of Applied Sciences and Mechatronics, Munich University of Applied Sciences, Lothstr. 34, 80335 Munich, GermanyDifferent dopants with their specific dopant concentration can be utilized to produce ferroelectric HfO_2 thin films. In this work it is explored for the example of Sr in a comprehensive first-principles study. Density functional calculations reveal structure, formation energy and total energy of the Sr related defects in HfO_2. We found the charge compensated defect including an associated oxygen vacancy Sr_HfV_O to strongly favour the non-ferroelectric, tetragonal P4_2/mnc phase energetically. In contrast, the uncompensated defect without oxygen vacancy Sr_Hf favours the ferroelectric, orthorhombic Pca2_1 phase. According to the formation energy the uncompensated defect can form easily under oxygen rich conditions in the production process. Low oxygen partial pressure existing over the lifetime promotes the loss of oxygen leading to V_O and, thus, the destabilization of the ferroelectric, orthorhombic Pca2_1 phase accompanied by an increase of the leakage current. This study attempts to fundamentally explain the stabilization of the ferroelectric, orthorhombic Pca2_1 phase by doping. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in Appl. Phys. Lett., vol. 111, no. 8, p. 82902, 2017 and may be found at http://dx.doi.org/10.1063/1.4993110. The impact of charge compensated and uncompensated strontium defects on the stabilization of the ferroelectric phase in HfO2 Alfred Kersch December 30, 2023 ============================================================================================================================Polycrystalline HfO_2 thin films produced by Atomic Layer Deposition (ALD) or Chemical Solution Deposition (CSD) can exhibit ferroelectric properties if they are appropriately doped <cit.>. An orthorhombic, non-centrosymmetric phase (Pca2_1) has been proposed as the source of these properties which has since been confirmed by electron diffraction study <cit.>. Furthermore, another theoretically proposed ferroelectric Pmn2_1 phase has been ruled out by the same study and is therefore not included in this work. Pure HfO_2 occurs naturally in a monoclinic (P2_1/c) phase. With increasing temperature a transformation into the tetragonal (P4_2/mnc) and then the cubic (Fm3m) phase occurs <cit.> avoiding the orthorhombic phase. Different Density Functional Theory (DFT) studies consistently calculate the total energy of the orthorhombic phase as second most stable after the monoclinic phase and are able to reproduce the thermally driven phase transformation <cit.> giving credibility to the used density functionals.To explain the occurrence of the ferroelectric phenomena, factors favouring the orthorhombic phase have been proposed <cit.> including entropy contribution, surface or interface energy, stress, and doping. Surface or interface energy stems from the large surface to volume ratio of the individual crystals in the polycristalline HfO_2 thin films <cit.> with grain sizes typically in the range of the film thickness (530nm) <cit.>. It explains the generally observed decrease or disappearance of the ferroelectric properties with increasing film thickness<cit.>. For the case of Hf_1-xZr_xO_2, at x = 0.5, surface energy or interface energy has been found to be sufficient to explain stability of the orthorhombic phase <cit.>. For thin films based on pure HfO_2 surface or interface energy is insufficient except for the case of very small grains <cit.>.In such thin films further stabilization by appropriate doping is required <cit.>. For the case of Sr doping, ferroelectricity was observed in a 10nm film between 1.7 and 7.9mol% SrO content with the maximum polarization observed at around 3.4mol% SrO <cit.>. The effect of doping on HfO_2 phases has been investigated in earlier works <cit.> but the Pca2_1 as well as II-valent dopants were not included in the study. The authors found stabilization of the tetragonal phase by IV-valent and stabilization of the cubic phase by III-valent dopants. Due to its II-valent nature, it is expected that each Sr dopant atom is accompanied by an oxygen vacancy for charge compensation. Furthermore, due to opposite charges, the and defect should strongly attract each other leading to similar to the case of Mg_Hf^-2 or Ba_Hf^-2 doping investigated in <cit.>. However, the defect concentration created during the manufacturing process is not explicit known and strongly depends on the chemical potential of the defects. In this work the defect notation of Freysoldt et al. is used <cit.>.To propose a consistent scenario for the ferroelectric stability of a Sr doped HfO_2 thin film, we determined total energy and defect formation energy for various defects in monoclinic, orthorhombic, tetragonal and cubic HfO_2 from first principle calculations. These defects include single oxygen vacancies V_O^q with the charges q = 0,+1,+2, Sr substituted for Hf with Sr_Hf^q (q = 0, -1, -2) as well as the compensated defect [Sr_Hf V_O]^q(q= 0, -1, -2). Oxygen vacancies were placed on the eight next neighboring oxygen sites of a given Sr or Hf atom excluding structural equivalent positions. All shown results always depict the energetically most favourable position. Placing one defect in a 96 or 48 atomic super cell corresponds to a concentration of 3.125% (= 1 defect/ 32 formula units) and 6.25% (= 1 defect/ 16 formula units) respectively.DFT calculations were performed using the Local Density Approximation (LDA) and Projector Augmented Wave (PAW) <cit.> Pseudo Potentials (PP) from the GBRV library <cit.> with the ABINIT code <cit.>. Several LDA calculations were repeated with the all electron code FHI-AIMS <cit.> based upon numeric, atom-centered orbitals of type tight with first and second tier enabled. In the remainder of this work we will refer to those two methods as plane waves (PW) and numerical orbitals (NO), respectively. The stopping criteria for the electronic convergence was a force criteria of 10^-6 Hartree/Bohr (PW) and 10^-4 eV/Å(NO). The stopping criteria for the structural convergence was a force criteria of 10^-5 Hartree/Bohr (PW) and 10^-3 eV/Å(NO). Charged and neutral defect calculations in monoclinic, tetragonal, cubic, and orthorhombic HfO_2 were performed with 96 atomic super cells using a 2 × 2 × 2 Monkhorst-Pack k-point set, a plane wave cut off of 18 Ha and a PAW cut off of 22 Ha in accordance with a convergence study. Charge neutral 48 atomic super cells with a 2 × 4 × 2 k-point grid were used to determine the phase stability at 6.25% defect concentration.The defect formation energies E_f were calculated as E_f(X, q) = U(X, q)-U(pure)-∑_i n_iμ_i +q(ϵ_F+ϵ_VB(pure) + Δ V(X, 0) )+E_Corr(X, q)using the DFT total energies U of both HfO_2 without and with a defect X∈{Sr_Hf^q,[Sr_HfV_O]^q, V_O^q} and charge q. The chemical potential and number of defect atoms of each species is given by μ_i and n_i respectively. The Fermi energy is ϵ_F and the valence band edge is ϵ_VB. A charge correction E_Corr with the scaling law<cit.> E_f∼a/L + c using a 324 atomic super cell and a potential alignment Δ V was applied. a and c are fit parameters and L is the size of the super cell. The chemical potential of Hf was set to the total energy of hcp Hf and of Sr was calculated by the equilibrium condition μ_Sr = μ_SrO - μ_O. For the chemical potential of oxygen two cases are considered: oxygen rich and oxygen deficient <cit.>. In the oxygen rich caseμ_O is set to μ_ O_2/2. Ferroelectric HfO_2 is often deposited on TiN electrodes <cit.> which can exist in a partially oxidized state. The oxygen chemical potential μ_O for the deficient conditions uses oxygen precipitation into anatase TiO_2. In similar studies <cit.>, precipitation into SiO_2 has been used adapting to a Si substrate. Both assumptions, however, lead to very similar formation enthalpies. We therefore calculate μ_O=(μ_TiO_2-μ_Ti)/2 for the oxygen deficient case.The main result from this paper is the connection between the phase stability of defective HfO_2 and the conditions under which the defective material can form. FIG. <ref> (a) shows the total energy difference Δ U to the monoclinic phase for the Sr_Hf defect as a function of the Sr concentration. Both the orthorhombic and tetragonal phase are depicted calculated with PW and NO. The cubic phase turned out to be unstable and is therefore not shown here.The defect free orthorhombic phase has a Δ U of 53meV (PW) and 49meV (NO) while the tetragonal phase has 115meV (PW) or 114meV (NO). The Sr_Hf defects lead to a decrease of Δ U of about 20meV for 6Sr-%which is roughly the same for both the tetragonal and orthorhombic phase. Therefore, the defect contributes to the stabilization but not sufficiently to fully stabilize the orthorhombic phase on its own. However, according to previous works, in Hf_1-xZr_xO_2 <cit.> the surface or interface energy of grains can decrease the energy of the tetragonal and orthorhombic phase below the monoclinic phase and, thus, suppress the formation of the monoclinic phase. The surface or interface energy for the tetragonal and orthorhombic phase are expected to be very similar. Proposing a surface or interface energy penalty for the monoclinic phase is difficult in this case since the issue has not been investigated for doped HfO_2 so far. In Hf_1-xZr_xO_2 considering Zr as a dopant, a typical energy penalty of about 20meV (for typical grains of 10nm diameter in a 10nm film) was found for HfO_2 linearly increasing to about 60meV for ZrO_2. At the same time the interface energies increased from 174490mJ/m^2 <cit.>. There is another argument in favour of a significant increase of the energy penalty for the monoclinic phase with doping. The authors <cit.> indentified the energy penalty with the energy of the tetragonal/monoclinic interface observed by Grimley <cit.>. An interface energy, however, is expected to depend sensitively on doping. Altogether, we expect a surface or interface related energy penalty for the monoclinic phase starting at around 30meV for pure HfO_2 and increasing significantly with doping. We therefore expect Δ U of the orthorhombic phase to become negative for some Sr concentrations and the film to become ferroelectric. Important for the ferroelectric stabilization is that the orthorhombic phase turns always out to be more favourable than the tetragonal phase.This is not the case for the compensated defect as shown in FIG. <ref> (b). The change in Δ U is much larger for the tetragonal phase than for the orthorhombic phase. Above a threshold of 23% (NO) or 5% (PW) the material looses ferroelectricity and the tetragonal phase replaces the orthorhombic phase as the most favorable. This would severely limit the dopant concentration range in which ferroelectric properties can be observed and is therefore in conflict with the experimentally observed range for ferroelectricity of 1.77.9 mol% dopant concentration<cit.>.This leads to the question, whether the Sr_Hf is indeed always compensated with an oxygen vacancy V_O as stoichiometry suggests. An estimation of the vacancy concentration results from an electrical measurement of the leakage current in Sr doped Hf by Pešić et al. <cit.>, who extracted a vacancy concentration of 5e19cm^-3 which is significantly less than required to pair every Sr atom (1.4e21cm^-3 for 5%) with a vacancy. Crucial for the question, whether Sr_Hf or Sr_HfV_O should be expected is the formation energy as a function of the oxygen chemical potential and a kinetic process creating the defect <cit.>.FIG. <ref> (a) shows the formation energies under oxygen rich conditions for the orthorhombic phase and for oxygen in the III-valent and IV-valent position. The formation energy does not differ very much from the monoclinic phase (not shown here). The LDA band gap for the orthorhombic phase was found to be 4.41eV (3.98eV for the monoclinic and 4.56eV for the tetragonal). The individual formation energy of charged and defects is lower than the formation energy of the combined charge neutral defect. This might lead to a separated creation of and . However, since vacancies are very mobile, the positively charged vacancies combine with the negatively charged creating with an energy release of 2.36eV. Under oxygen rich conditions, few Sr_HfV_O are expected in the end except close to the interface where some oxygen loss towards the electrode has to be expected. As a result, a film with substitutional Sr_Hf defects and few compensated defects is expected, but at the electrode interface a significant amount of compensated Sr_HfV_O defects is possible which may stabilize a tetragonal interlayer <cit.> and may be a prerequisite of the energy penalty to suppress the monoclinic phase. This would support the assumptions made by Pešić <cit.>. The acceptor doping without charge compensation achieved under oxygen rich conditions is often desired to improve electric isolation since the negative space charge increases the band offset to the electrode.During the life time of a ferroelectric HfO_2 stack, the external oxygen partial pressure is defined by the oxidized electrodes. FIG. <ref> (b) shows the formation energy under such oxygen deficient conditions. As there is no new Sr-source, only vacancies can be created possibly due to field cycling. Since the energy of is lower than the sum of and , these vacancies will recombine quickly with the already present substitutional Sr defects leading to a charge compensation. The concentration of will decrease and that of will increase. The implication on the phase stability is a gradual degradation of the orthorhombic phase content accompanied by a decrease of the remanent polarization. A further implication concerning the electron transport is that the charge transition level of a deep defect state promotes trap assisted tunneling (TAT). The related charge transition levels ϵ(0/-1)= 3.63eV and ϵ(-1/-2) = 3.92eV close to the conduction band release electrons which modify the space charge and contribute to TAT. Therefore, a moderate increase of leakage current with time would be expected indicating an increase of charge compensated defects.As the creation of Sr_HfV_O under oxygen deficient conditions is preferred, the concentration of V_O will stay on a relatively low level and constant over time. However, the V_O defects with charge transition levels at ϵ(+2/+1)= 2.41eV and ϵ(+1/0)= 2.81eV are about 2eV below the conduction band and, therefore, can be occupied by tunneling electrons promoting leakage current. A last argument explains why the Sr_Hf defect favors the orthorhombic and Sr_HfV_O defect the tetragonal phase in total energy. The cause for the stabilization of the orthorhombic and tetragonal phase by Sr_Hf defects can be found in the bond length of the Sr atom to its neighboring oxygen atoms. Calculations of SrO and SrO_2 show a bond length between 2.53 and 2.60Å, respectively. In undoped HfO_2 the average bond length is 2.12Å for the monoclinic and orthorhombic phase and 2.17Å for the tetragonal phase. Substituting a Sr atom on a Hf site, the bond length increases to only 2.35Å for the monoclinic phase but to 2.37 Å for the orthorhombic and tetragonal phase. Sr in monoclinic HfO_2 is therefore energetically more unfavourable than in the orthorhombic or tetragonal phase, therefore the energy difference to the monoclinic phase decreases with doping. Introducing vacancies, the monoclinic average bond length increases to 2.38Å, but the tetragonal value of 2.47Å almost matches the value of SrO and is accompanied by the significant decrease in total energy difference, see FIG. <ref>.In summary a mechanism is proposed, based on first-principles DFT calculations, to explain the influence of Sr doping on the phase stability in HfO_2. The tetragonal phase is strongly preferred by the incorporation of the Sr_HfV_O defects while the Sr_Hf allows for the stabilization of the ferroelectric orthorhombic phase. The uncompensated defect can form in sufficiently oxygen rich environments, which might exist during the production process. The loss of oxygen during field cycling may increase the charge compensation which promotes the phase transformation into other HfO_2 polymorphs. This contributes to the fatigue behavior. The proposed mechanism has the potential to describe the action of other dopants on the ferroelectric phase in HfO_2 if appropriately adapted and expanded. The author wants to thank U. Schröder, T.Schenk, Min Hyuk Park from NamLab/SNU, and U. Böttger and S. Starschich from RWTH Aachen for discussions. The German Research Foundation (Deutsche Forschungsgemeinschaft) is acknowledged for funding this research in the frame of the project “Inferox” (Project No. MI 1247/11–1). The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.(www.gauss-centre.eu)forfundingthisprojectby providingcomputingtimeontheGCSSupercomputer SuperMUCatLeibniz SupercomputingCenter(LRZ, www.lrz.de). | http://arxiv.org/abs/1708.07778v1 | {
"authors": [
"Robin Materlik",
"Christopher Künneth",
"Thomas Mikolajick",
"Alfred Kersch"
],
"categories": [
"cond-mat.mtrl-sci",
"physics.app-ph"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170825153158",
"title": "The impact of charge compensated and uncompensated strontium defects on the stabilization of the ferroelectric phase in HfO$_2$"
} |
Department of Physics & Astronomy, University of Kentucky, Lexington, KY-40506-0055Department of Physics & Astronomy, University of Kentucky, Lexington, KY-40506-0055 Department of Physics & Astronomy, University of Kentucky, Lexington, KY-40506-0055We study the ground stateordering of quadrupolar ordered S=1 magnets as a function of spin dilution probability p on the triangular lattice. In sharp contrast to the ordering of S=1/2 dipolar Néel magnets on percolating clusters, we find that the quadrupolar magnets are quantum disordered at the percolation threshold, p=p^*. Further we find that long-range quadrupolar order is present for all p<p^* and vanishes first exactly at p^*. Strong evidence for scaling behavior close to p^* points to an unusual quantum criticality without fine tuning that arises from an interplay of quantum fluctuations and randomness.Quadrupolar quantum criticality on a fractal Ribhu K. Kaul December 30, 2023 ============================================§ INTRODUCTION Quantum spin models with random vacancies provide a rich playground for the study of the combined effects of strong interactions and quenched disorder that are relevant to experimental measurements on a number of doped magnetic alloys <cit.>. Magnetic order in higher dimensions is generally stable to a small concentration of vacancies. On the otherhand, if its moments are diluted beyond the percolationthreshold, the system breaks up into zero-dimensional clusters and hence magnetic order must be lost. What is the nature of themagnetic quantum phase transition as the dilution is varied? The answer to this question is most thoroughly understood for the randomly diluted transverse field Ising model. The expected ground state phase diagram <cit.> as a function of the transverse field g and site dilution probability p, has three kinds of quantum phase transitions, Fig. <ref>. For g>g_ C the transition takes place at a value of p smaller than the percolation threshold p_c, the critical point is then described by the random transverse field Ising model which possesses an infinite randomness fixed point <cit.>; the physics of percolation however plays no role (“R” in Fig. <ref>). For g<g_ C on the other hand percolating clusters are magnetically ordered and the singularities at the transition are determined by those of classical percolation (“P” in Fig. <ref>) <cit.>.Finally exactly at g=g_ C a critical point is obtained: Crucial to our discussion, quantum criticality at the percolation threshold(“C” in Fig. <ref>) requires fine tuning of the quantum fluctuations and hence “C” is not the generic transition for Ising magnets on dilution.Another family of quantum spin models where this question has been addressed in detail are bipartite Néel ordered Heisenberg models. For random depletion of the square lattice it was found that percolating clusters have long range order <cit.>, the destruction is thus akin to Fig. <ref> “P”. The surprising stability of the Néel order has been traced back to the effect of uncompensated Berry phases which result in “orphan” moments <cit.>.In the bilayer geometry, with bond dilution that circumvents random Berry phases, a phase diagram similar to Fig. <ref> was found <cit.>, with quantum phase transitions corresponding to “P” <cit.> in some regions and corresponding to “R” in others, in contrast to the Ising case the critical phenomena here is controlled by a finite disorder fixed point with conventional scaling <cit.>. The critical point “C” which requires fine tuning has also been studied <cit.>. A variety of phase diagrams and critical phenomena can be accessed by dilution of Néel magnets in geometries different from the single and bilayer <cit.>.The two examples of dipolar ordered magnets (Ising and Heisenberg) make it clear that the details such as symmetry of the order parameter and Berry phases play a crucial role in determining how quantum magnetism is destroyed by random dilution. In this work we address the dilution transition in a different kind of system: quadrupolar ordered (also referred to as spin nematic) magnets in their most common S=1 realization <cit.>. The study of the quadrupolar phase of S=1 magnets has become increasingly popular motivated by their possible sighting in some triangular lattice Ni based magnets (see e.g. <cit.>). Experimental studies of Zn replacement of Ni in NiGa_2S_4 provide a direct experimental motivation for the site diluted S=1 magnets we study here <cit.>. As we shall describe below, in contrast to what has been observed for Néel order in S=1/2 magnets and transverse field Ising models, we find here that quadrupolar order vanishes exactly at the percolation threshold without any fine tuning, resulting in new quantum critical behavior.§ MODEL Just as the bilinear Heisenberg model is the archetype for realizing Néel order, the biquadratic Heisenberg interaction is the archetypical model system that realizes quadrupolar order in S=1 magnets. We shall consider the following Hamiltonian,H = - ∑_⟨ ij⟩ J_ij (S⃗_i ·S⃗_j )^2, where ⟨ ij ⟩ denote nearest neighbors in the triangularlattice, and S⃗_i are the S=1 Pauli matrices on the i^th site. The model possesses an explicit physical SO(3) internal spinsymmetry.We use standard uncorrelated random site and bonddilution: For site dilution J_ij=|J| r_i r_j where r_i=0 withprobability p and r_i=1 with probability 1-p. For bond dilutionJ_ij=|J| χ_ij where χ_ij=0 with probability p andχ_ij=1 with probability 1-p. While site dilution is closerto experimental realizations such as Zn doped NiGa_2S_4,bond dilution provides an alternate access to the percolationthreshold. Our numerical results were obtainedusing stochastic series expansion <cit.> with an efficient sign-problem free algorithmthat has been described and testedpreviously <cit.>. We calculate averages on finite sizeclusters at finite-temperature, T.To study the quadrupolar order in the magnet we define the quadrupolar order parameter through the equal time two point correlation function of the quadrupolar order parameter. O_Q^2= ⟨1/N_c^2∑_a,i,j⟨Q̂^aa_i Q̂^aa_j⟩⟩_p,where the quadrupolar order parameter is defined in terms of the local S=1 Pauli matrices as Q̂^ab= Ŝ^a Ŝ^b+ Ŝ^bŜ^a/2- 2/3δ^ab, N_c is the size of the cluster, ⟨…⟩ is the quantum mechanical average over the thermal density matrix and ⟨…⟩_p is a disorder average over realization of the ensemble of clusters.§NUMERICAL RESULTS It is now well established that the model Eq. (<ref>) without any depletion (p=0) has quadrupolar order with ferromagnetically aligned directors <cit.> (i.e. O_Q^2 in Eq. <ref> is finite in the ground state in the thermodynamic limit). We begin our numerical study by asking the following question: As p is increased does the quadrupolar order parameter eventually vanish like “R”, “C” or “P” (referring to Fig. <ref>).As we have discussed in the introduction, generic expectations would be either “R” or “P”. In Fig. <ref> we show extrapolations of the quadrupolar order parameter to the thermodynamic limit at the percolation threshold. We define two different ensembles for finite-size scaling which allow us to approach the thermodynamic limit in two different ways, with fixed cluster size N_c (on an infinite underlying lattice) and the largest cluster on a finite L× L lattices, following <cit.>. We have also studied both bond and site dilution at the percolation threshold. Consistently across all finite-size scaling schemes we find that the order parameter vanishes at the percolation threshold. For a similar scaling analysis for the Néel order, see Fig. 10 in Ref. <cit.> which clearly shows the ordering of S=1/2 Heisenberg model on the percolating cluster in two dimensions. In contrast here we find the quadrupolar order vanishes in the thermodynamic limit. This data clearly eliminates the “P” possibility since this requires the percolating clusters to be magnetically ordered. We have taken great care to make sure our data is equilibrated and in the limit of T=0. The ground state limit requires extraordinarily low temperatures becasue of the weak links that connect percolating clusters. Details are provided in the Appendix. Now that we have shown that the percolating cluster are magnetically disordered, the generic expectation is that the quadrupolar order vanishes before the percolation threshold is reached at some p<p^* as illustrated for “R”in Fig. <ref>.To address this question quantitively we study a quadrupolar “Binder” ratio R_Q≡⟨ O_Q^4 ⟩/⟨ O_Q^2 ⟩^2. R_Q is expected to be monotonically decreasing with L in a quadrupolar ordered phase. When graphed as a function of the tuning parameter p, R_Q data for different L are expected to cross at the phase transition to a non-magnetic phase. Our zero temperature data for the case of site dilution in Fig. <ref> clearly shows that the R_Q crosses and as the thermodynamic limit is reachedthe crossing point approaches the percolation threshold p^*=0.5 with high accuracy. This establishes that the quadrupolar order vanishes continuously as the percolation threshold is approached. This allows us to eliminate the possibility “R”, in which the order parameter vanishes before the percolation threshold is reached. We hence conclude that the order parameter vanishes first precisely at the percolation threshold p^*, as illustrated in the cartoon “C” shown in Fig. <ref>.The vanishing of quadrupolar order right at the percolation threshold raises the interesting possibility that at p=p^* the system is quantum critical and the quadrupolar correlation possess scale invariance.Random systems are well known to display a range of novel scaling behavior. We present evidence however thatour model possesses conventional power law scaling. In order to test the scaling hypothesis, we first study the dynamic scaling in imaginary time of the order parameter at p=p^*. In Fig. <ref> we show scaling collapse of the order parameter data at the percolation threshold p=p^* for the maximum sized cluster on an L× L lattice as a function of the putative scaling variable L^zT. For a quantum critical system, we expect the scaling form O_Q^2 = 1/L^1+η F ( L ^z T). Note the absence of a tuning parameter (we only require the system to be at the percolation threshold) that would normally be expected for instance from previously studied phase diagrams of quantum rotor models in Fig. <ref> (the tuning parameter there is the choice of g=g_ C). We find an excellent collapse with small finite size corrections to the exponents. An alternate test of scaling can be made by varying p away from the percolation threshold. In Fig. <ref> we study the scaling of the order parameter with the deviation from the percolation theshold p-p^*. We find excellent scaling behavior assuming the simple scaling form O_Q^2 = 1/L^1+η G ( (p-p^*)L^1/ν) . The value of η so obtained is in excellent agreement with the estimate obtained from the previous scaling analysisat the percolation threshold Fig. <ref>. § DISCUSSION We have presented extensive evidence that in two dimensions quadrupolar order vanishes continuously as a function of dilution right at the percolation threshold. At the percolation threshold the systems shows conventional scaling behavior symptomatic of a finite disorder fixed point. Our study is an unusual example of quantum criticality at the percolation threshold without any fine tuning. We contrast this with the generic phase diagram Fig. <ref> where at the percolation threshold, tuning of quantum fluctuations to a special value if required to observe quantum criticality.An interesting open question is whether the critical exponents we have found are universal or can vary continuously as non-universal properties of the magnet are varied at the percolation threshold. Such varying exponents were reported in a study of specific diluted dimerized S=1/2 magnet <cit.>. Interestingly, the dynamic critical exponent z we find from our scaling analysis is numerically somewhat close to d_f=91/48≈ 1.8958… (the fractal dimension of a percolating cluster). A number of previous works have found or predicted such scaling at the percolation threshold and our finding could be consistent with such behavior <cit.>, though numerically z=2 would also be consistent with our value within errors.A complete theory of the unusual behavior and scaling we have found here is an interesting direction for future work.We acknowledge financial support fron NSF DMR-1611161. The numerical simulations reported here were carried out on the DLX cluster at University of Kentucky and by resources allocated by XSEDE.§ EQUILIBRATION OF DISORDERED CLUSTERSIn order to ensure the proper equilibration of our disordered clusters, we have adopted the same equilibration and measurement protocol as in Ref. sandvik2002:s12perc.For each disorder realization we begin by performing N_e equilibration sweeps at some large initial temperature (β≈ 1, setting |J|=1), followed by N_m more measurements sweeps, then again N_e equilibration sweeps, and finally N_m measurement sweeps.Once this process is complete we perform β doubling on our configurations (see appendix <ref>) and start the process again.Thus for each disorder realization we have two separate measurement cycles at many different values of β.Separating out two distinct measurement segments allows us to check the equilibration of our disordered clusters.In order to achieve the best QMC averages with minimal computational time, we have set N_m=2 N_e.We can then study the percent difference of our two measurement segments (each averaged over disorder realizations) as a function of N_m to determine its optimal value, which is given in Fig. <ref>.We observe that at the percolation threshold (where the clusters are most fragmented), the difference between our first and second measurement cycles becomes statistically insignificant when N_m ≈ 64.We therefore cautiously set N_m=200 throughout the course of our numerical studies.§ ZERO TEMPERATURE CONVERGENCEThe most computationally expensive component of our numerical studies is to converge our configurations to the ground state on disordered clusters.In order to achieve this, extremely large values of the inverse temperature are needed (relative to that of a clean lattice). Furthermore, the free energy landscape of configurations is rugged, and thus quenching a randomly initialized starting configuration to low temperature abruptly may leave it stuck in a local minimum.In order to circumvent this issue, and to efficiently reach the low temperatures required with minimal equilibration, we implement the β doubling procedure <cit.>.The procedure works as follows: given a disorder realization, we begin the equilibration and measurement cycles at some initial high temperature (β_0 on the order of unity).Configurations at high temperatures are relatively easy to equilibrate, since there are few operators acting in the operator string.After the equilibration and measurement cycles, the value of β and the QMC configuration are both doubled.In terms of the configuration, this corresponds to repeating the operator sequence twice.This procedure gives another valid partition function configuration that is close to being equilibrated at an inverse temperature 2β_0.From here the equilibration and measurement sequence is again carried out.The process is continued until the final target value β_max is reached.In order to illustrate the convergence of our zero temperature data, in Fig. <ref> we show the binder ratio for max clusters on an L=32 lattice near the site percolation threshold for all of our β doubled values.We see that very large values of β are required to converge to the ground state.This is a reflection of the fact that z is close to 2 at this disordered quantum critical point, meaning that doubling the size of the lattice would require quadrupling β to remain near the ground state. | http://arxiv.org/abs/1708.07924v1 | {
"authors": [
"Jonathan D'Emidio",
"Simon Lovell",
"Ribhu K. Kaul"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170826031134",
"title": "Quadrupolar quantum criticality on a fractal"
} |
1in -0.5in 1.8in -1inthmTheorem[section]lemmaLemma[section]corCorollary[section] propProposition[section] definitionDefinition[section] condCondition[section] | http://arxiv.org/abs/1708.07852v3 | {
"authors": [
"Jiashun Jin",
"Zheng Tracy Ke",
"Shengming Luo"
],
"categories": [
"stat.ME",
"62H30, 91C20, 62P25"
],
"primary_category": "stat.ME",
"published": "20170825181613",
"title": "Mixed Membership Estimation for Social Networks"
} |
alphaStacks and Higgs bundles]An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves Casalaina-Martin]Sebastian Casalaina-MartinUniversity of Colorado at Boulder, Department of Mathematics, Boulder, CO,USA [email protected] Wise]Jonathan Wise University of Colorado at Boulder, Department of Mathematics, Boulder, CO,USA [email protected] first author was partially supported by NSF grant DMS-1101333, a Simons Foundation Collaboration Grant for Mathematicians (317572), and NSA Grant H98230-16-1-005.The second author was partially supported by an NSA Young Investigator's GrantH98230-14-1-0107.This article is based in part on lecture notes prepared forthe summer school “The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles”at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014.The aim is to provide a brief introduction to algebraic stacks, and thento giveseveral constructions of the moduli stack of Higgs bundles on algebraic curves.The first construction is via a “bootstrap” methodfrom the algebraic stack of vector bundles on an algebraic curve.This construction is motivated in part byNitsure's GIT construction ofa projective moduli space of semi-stable Higgs bundles, andwe describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here.The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.[ [ December 30, 2023 =====================§ INTRODUCTIONStacks have been usedwidely by algebraic geometers since the 1960s for studying parameter spaces for algebraic and geometric objects <cit.>. Their popularity is growing in other areas as well (e.g., <cit.>).Nevertheless, despite their utility, and their having been around for many years, stacks still do not seem to be aspopular as might be expected. Of course, echoing theintroduction of <cit.>,this mayhave something to do with the technical nature of the topic, which may dissuade the uninitiated, particularly when there may be less technical methods available that can be used instead. To capture a common sentiment, it is hard to improve on the following excerpt fromHarris–Morrison <cit.>,which serves as an introduction to their section on stacks:`Of course, here I'm working with the moduli stack rather than with the moduli space.For those of you who aren't familiar with stacks, don't worry: basically, all it means is that I'm allowed to pretend that the moduli space is smooth and there's a universal family over it.' .1 inWho hasn't heard these words, or their equivalent spoken at a talk?And who hasn't fantasized about grabbing the speaker by the lapels and shaking him until he says what – exactly – he means by them? At the same time, the reason stacks are used as widely as they areis that theyreally are a natural language for talking about parameterization. While there is no doubt a great deal of technical background that goes into the set-up, the pay-off is that once this foundation has been laid,stacksprovide a clean, unifiedlanguage for discussing what otherwise may require many caveats and special cases.Our goal here is to give a brief introduction to algebraic stacks, with a view towardsdefining the moduli stack of Higgs bundles. We have tried to provide enough motivation for stacks that the reader is inclined to proceed to the definitions, and a sufficiently streamlinedpresentation of the definitions that the reader does not immediately stop at that point!In the end, we hope the reader has a good sense of what the moduli stack of Higgs bundles is, why it is an algebraic stack, and how the stack relates to the quasi-projective variety of Higgs bundles constructed by Nitsure <cit.>. .1 inDue to the authors' backgrounds, for precise statements, the topic will be treated in the language algebraic geometry, i.e., schemes. However, the aim is to have a presentation that is accessible to those in other fields, as well, particularly complex geometers,and most of the presentationcan be made replacing the word “scheme” with the words “complex analytic space” (or even “manifold”) and “étale cover” with “open cover”.This is certainly the case up though, and including, the definition of a stack in <ref>. The one possible exception to this rule is the topic of algebraic stacks <ref>, for which definitions in the literature are reallygeared towards the category of schemes. In order to make our presentation as accessible as possible, in <ref> and<ref> we providedefinitions ofalgebraic stacksthat make sense for any presite; in particular, the definitiongives a notion of an “algebraic” stack in the category of complex analytic spaces. There are other notions of analytic stacks in the literature, but for concision,we do not pursue the connection between the definitions..1 in The final sections of this survey (<ref> and <ref>) study algebraic stacks infinitesimally, with the double purpose of giving modular meaning to infinitesimal motion in an algebraic stack and introducing Artin's criterion as a means to prove stacks are algebraic.Higgs bundles are treated as an extended example, and along they way we give cohomological interpretations of the tangent bundles of the moduli stacks of curves, vector bundles, and morphisms between them.In the end we obtain a direct proof of the algebraicity of the stack of Higgs bundles, complementing the one based on established general algebraicity results given earlier. .1 in While these notes are meant to be somewhat self contained, in the sense that essentially all relevant definitions are included, and all results are referenced to the literature, these notes are by no means comprehensive.There are now a number of introductions to the topic of algebraic stacks that have appeared recently (and even more are in preparation), and these notes areinspired byvarious parts of thosetreatments.While the following list is not exhaustive, it provides a brief summary of some of the other resources available.To begin, we direct the reader to thenow classic, concise introduction by Fantechi <cit.>. The reader may very well want to read that ten page introduction before this one.A more detailed introduction toDeligne–Mumford stacks is given inthe book <cit.>. The book <cit.> provides another detailed introduction, with less emphasis on the algebraic property of algebraic stacks.The book<cit.> provides a concise, detailed,but perhaps less widely accessible introduction to the topic. There is also the comprehensive treatise<cit.>, which has a complete treatment of all of the details. The reader who would like to follow the early development of the subject might want to consult <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>. §.§ NotationWe denote by 𝖲the category of schemes over a fixed base, which the reader may feel free to assume is Specℂ, where ℂ is the field of complex numbers.All schemes will be members of this category.Until we discuss algebraic stacks, one could even take 𝖲 to be the category of complex analytic spaces, or any other reasonable category of “spaces” with which one would like to work. Our convention is generally to use roman letters for schemes (e.g., M), script letters for functors to sets (e.g., ℳ), and calligraphic letters for categories fibered in groupoids; (e.g.,ℳ). We will typically use sans serif letters for various standardcategories (e.g., 𝖬).§ MODULI PROBLEMSAS FUNCTORS When one wants to parameterize some kind of algebraic or geometric objects, one says one has a moduli problem.The goal is to find another geometric object, called a moduli space,that parameterizes the objects of interest. One of the most natural ways to phrase a moduli problem is in terms of the corresponding moduli functor.From this perspective, the hope is that the moduli functor will be representable by ageometric object, which will be the moduli space. In fact, it is possible to say a great deal about the moduli space purely in terms of the moduli functor, without even knowing the moduli space exists!For example, the tangent space can be computed by evaluating the moduli functor on the spectrum of the ring of dual numbers (<ref>). We make this precise in what follows. The best way to get a feel for this is through examples.Forthis reason, in this section we start by considering the familiar problems of parameterizing linear subspaces of a fixed vector space, andof parameterizing Riemann surfaces of a fixed genus.§.§ GrassmanniansWe expect this example is familiar to most of the readers, and the aim will be to motivate an approach to these types of problems, which we will continue to use throughout. We direct the reader to <cit.> for details.In this section we explicitly take 𝖲 to be the category of ℂ-schemes, or the category of analytic spaces, for simplicity; a “scheme” refers to a member of this category.For an object S of 𝖲, we adopt the notation ℂ^n_S:=ℂ^n× S for the trivial vector bundle of rank n over S.Recall that Grassmannians arise from the moduli problem that consistsof parameterizingr-dimensional complex subspaces ofℂ^n.There is clearly a set of such spaces G(r,n):={W⊆_linℂ^n: W=r}.In fact, one can easily put a “natural” complex structure on G(r,n) that makes this set into a smooth complex projective variety of dimension r(n-r), which we call the Grassmannian.Of key importancefrom our perspective is the fact thatthere is a rank r vector bundle 𝕌 over G(r,n), and an inclusion of vector bundles: 0→→ℂ^n_G(r,n)such that for any ℂ-scheme S and any inclusion of vector bundles 𝔽↪ℂ^n_S over S with rank𝔽=r, there is a unique ℂ-morphismf:S→ G(r,n)such that the inclusion 𝔽↪ℂ^n_S is isomorphicto the pullback of the inclusion ↪ℂ^n_G(r,n) by f.Diagramatically: @C=.2cm @R=.2cm𝔽@->[rdd]@^(->@<-2pt>[rrd] @[rrr]^(0.1)="a" @<2pt> "a";[rrr]@[rrd]^(0.1)="b" @^(->@<-2pt> "b";[rrd] @->[rdd]ℂ^n_S @->[rrr] @->[ld] ℂ^n_G(r,n)@->[ld] S @->[rrr]^f_∃ ! G(r,n)Set theoretically, we defineby setting the fiber ofover a point [W]∈ G(r,n) corresponding to a subspace W⊆ℂ^n to be the subspace W itself; i.e., we have_[W]=W↪ℂ^n=( ℂ^n_G(r,n))_[W].The bundleis called the universal subbundle. In fact, the topology and geometry of G(r,n), as well asthe vector bundle structure on , are all encoded in the universal property.To see the topology, consider a linear projection p : ℂ^n →ℂ^r.If S is any scheme and 𝔽↪ℂ^n_S is a rank r vector subbundle, then for a general projection p,denoting byp_S the induced morphism p_S:ℂ^n_S→ℂ^r_S, then the composition p_𝔽 : 𝔽→ℂ^n_Sp_S→ℂ^r_S is an isomorphism over a nonemptyopen subset S_p⊆ S.Choosing appropriate projections, these open subsets cover S.This applies in particular to the Grassmannian itself, so that we obtain an open cover of G(r,n) by subsets U_p where p_𝕌|_U_p:𝕌|_U_p→ℂ^r_U_p is an isomorphism.Moreover, we can easily see that locally, after trivializing 𝕌,a point of U_p may be identified with a section of the projection p : ℂ^n →ℂ^r; or more precisely, a map S → U_p may be identified with a section of p _S: ℂ_S^n →ℂ_S^r.Consequently, one can check from the universal property that U_p must be isomorphic to the space of sections of the projection, which is isomorphic toHom_ℂ(ℂ^r,ℂ^n-r)= ℂ^(n-r) × r, the spaceof (n-r) × r complex matrices. This of course agrees with the standard method of constructing charts for the Grassmanian by choosing bases for the subvector space, and performing row reduction.We also have the identification p_𝕌|_U_p : _U_p→ℂ^r_U_p, which gives the vector bundle structure on .The take-away from this discussion is that there is an identification of families (up toisomorphism)of r-dimensional subspaces of ℂ^n parameterized by S, withmorphisms of ℂ-schemes S into G(r,n) and, moreover, that we can understand everything about G(r,n) in terms of this identification.As perverse as it might seem, we could throw away the topological space and sheaf of rings and considerinstead the Grassmannian functor:𝒢(r,n):𝖲^op→(𝖲𝖾𝗍) 𝒢(r,n)(S):= { (𝔽,i)|-8pt𝔽 is a vector bundle on Si : 𝔽↪ℂ_S^n is a linear inclusion}/ ∼where two inclusions of vector bundles 𝔽↪ S×ℂ^n and 𝔽'↪ S×ℂ^n over S are equivalent if there is a commutative diagram of the form@R=10pt𝔽@^(->@<-1pt>[r] @->[d]_-90∼ ℂ^n_S @=[d] 𝔽'@^(->@<-1pt>[r] ℂ^n_S.The functor 𝒢(r,n) acts on morphisms in the following way. Associated to a morphism f:S'→ S, we havef^*:𝒢(r,n)(S)→𝒢(r,n)(S')taking the class of theinclusion 𝔽↪ S×ℂ^n to the class of the inclusion 𝔽'↪ S'×ℂ^ndefined via pullback diagrams:@C=.3cm @R=.3cm𝔽': @=[r] f^∗𝔽@->[rdd]@^(->@<-1pt>[rrd] @->@<1pt>[rrr] 𝔽@^(->@<-1pt>[rrd] @->[rdd] ℂ^n_S'@->[rrr]_<>(0.25)ℂ^n_f @->[ld] ℂ^n_S. @->[ld]S' @->[rrr]_f SHere f^∗𝔽 is the pullback of 𝔽 to S' via f and ℂ^n_f is the canonical morphism ℂ^n_S' = ℂ^n × S' ℂ^n × S = ℂ^n_S.The assertion that there is an identification of families (up toisomorphism)of r-dimensional subspaces of ℂ^n parameterized by S, withmorphisms of schemes S into G(r,n), can be formulated precisely as an isomorphism of functors@R=4ptHom_𝖲(-,G(r,n)) [r]𝒢(r,n)(f:S→ G(r,n)) @|->[r] f^*(↪ℂ^n_G(r,n)).In other words, the Grassmannian functor is representable by the Grassmannian scheme.We also say that the Grassmannianis a fine moduli space for the Grassmannian functor.Our previous extraction of the geometric structure of G(r,n) from the universal property, in other words, from the functor 𝒢(r,n), is a manifestation of the general principle encoded in Yoneda's lemma (stated below).Indeed, if there is another space G(r,n)' such that Hom_𝖲(-,G(r,n)')≅𝒢(r,n),then Hom_𝖲(-,G(r,n)')≅Hom_𝖲(-,G(r,n)),and one may use Yoneda's lemma to conclude that G(r,n)≅ G(r,n)'.Let 𝖲 be a category. There is a fully faithful functorh:𝖲→𝖥𝗎𝗇(𝖲^𝗈𝗉,(𝖲𝖾𝗍)) S↦ h_S:=Hom_𝖲(-,S) (Sϕ→ S')↦ (h_S h_S'),which identifies 𝖲 with a full subcategory of the category of functors𝖥𝗎𝗇(𝖲^𝗈𝗉,(𝖲𝖾𝗍)).For each S∈Ob(𝖲), the map Hom_𝖥𝗎𝗇(𝖲^𝗈𝗉,(𝖲𝖾𝗍))(h_S, ℱ)→ℱ(S) (η:h_S→ℱ)↦η(S)(id_S)is a bijection.This is best worked through on one's own, preferably in private.Nevertheless, in the spirit of completeness that inspires these notes, we refer the reader to <cit.> or <cit.> for more details.[Representable functor] A functor isomorphic to one of the formHom_𝖲(-,S) for some S in 𝖲 is called a representable functor.Due to the Yoneda Lemma, we will often write S in place of h_S orHom_𝖲(-,S).Note that under the identification of (<ref>), the identity mapid_G(r,n)∈Hom_𝖲(G(r,n),G(r,n)) is sent to the class of the inclusion of the universal subbundle↪ℂ^n_G(r,n) in 𝒢(r,n)(G(r,n)). Summarizing the discussion above, the fact that the Grassmannian functor is representable is equivalent to the factthat there is a universal family, , of dimension r-subspaces of ℂ^n, parameterized by the Grassmannian G(r,n), such thatany other family ofdimension r-subspaces of ℂ^n is obtained from this universal family via pullback along a (unique) map to the Grassmannian. This whole construction could have been dualized, considering instead r-dimensional quotients of the dual vector space (ℂ^n)^∨.The problem is clearly equivalent, but this dualformulation is more obviously related to the Quot scheme we will discuss later.We leave it to the reader to make this comparison when the Quot scheme is introduced.§.§ Moduli of Riemann surfaces The example of the Grassmannian gives amoduli space that is representable by a projective variety.Anotheraccessible example of a moduli problem that on the other handcaptures the complications leading to the development of stacks is that of parameterizing Riemann surfaces.We will work in this section with the algebraic analogues ofcompactRiemann surfaces of genus g; i.e.,smooth proper complex algebraic curves of genus g. Our moduli problem is that of parameterizing all such curves up to isomorphism.Motivated by the discussion of the Grassmannian, we make this more precise by defining a moduli functor.To do this, we define a relative curve of genus g to bea surjective morphism of schemesπ:X→ S such that* π is smooth,* π is proper,* every geometric fiber is a connected curve of genus g. Recall that if X and S are smooth, then π:X→ S is smooth if and only the differential is everywhere surjective.In other words, this includes the analogues ofsurjectivemorphisms of smooth complex manifolds, with surjective differential, where every (set theoretic)fiber is a compact Riemann surface of genus g.We now define the moduli functor of genus g curves as:ℳ_g:𝖲^𝗈𝗉→ (𝖲𝖾𝗍)ℳ_g(S):={π:X→ S , a relative curve of genusg }/∼where two relative curves π:X→ S and π':X'→ S are equivalent if there is a commutativediagram (<ref>), in which the upper horizontal arrow is an isomorphism:X' [r]^∼[d]_π' X [d]^πS @=[r] S.The functor ℳ_g acts on morphisms in the following way.Associated to a morphism f:S'→ S, we havef^*:ℳ_g(S)→ℳ_g(S')defined via pullback diagrams; in other words, given [π:X→ S]∈ℳ_g(S), we define f^*[π:X→ S] as the class of the curve π':X'→ S', definedby the fiber product diagram:X' [r] [d]_π'X [d]^πS' [r]^f S.[Isotrivial family]Consider the relative curve π:X→ S=ℂ^* of genus 1 given by the equationy^2=x^3+t,t 0.Here X is the projective completion of {(x,y,t): y^2-x^3-t=0} in ℙ^2_ℂ, and the map π:X→ S is inducedby (x,y,t)↦ t. It turns out this family is isotrivial (i.e., all of the fibers are abstractly isomorphic); this can be seen quickly by confirming that the j-invariant of each fiber is 0, or simply writing down the isomorphism (x,y,t) ↦ (λ^-2 x, λ^-3 y, 1), where λ^6 = t, between X_t and X_1.One can also show that this family is not equivalent to a trivial family; i.e., not isomorphic to S× X_1. One way to see this is to check that themonodromy action of π_1(ℂ^*,1)onH^1(X_1,ℂ) is nontrivial (see e.g., <cit.>), and therefore, that X/S is not a trivial family. Anotherway to see this is to observe that the explicit isomorphism given above trivializes X over the pullback via ℂ^∗→ℂ^∗ : λ↦λ^6.We can therefore characterize X as the quotient of X_1 ×ℂ^∗ by the action ζ . (x,y,λ) = (ζ^-2 x, ζ^-3 y, ζλ) of a 6-th root of unity ζ∈μ_6. The étalesheaf on S of isomorphisms between X and the X_1 × S is a torsor under the automorphism group of X_1, which is μ_6.It is therefore classified by an element of H^1(S, μ_6) = Hom(μ_6,μ_6)=ℤ/6ℤ,which one can check is1.Since this corresponds to a nontrivialtorsor, X/S is not a trivial family.One can generalize this exampleto show that for every g, there exist isotrivial, but nontrivial families of curves(see Example <ref>).The functor ℳ_g is not representable. See e.g. <cit.> for more details.The main point is thatif ℳ_g were representable, then every isotrivial family of curves would be equivalent to atrivial family, which we have just seen is not the case.To this end, suppose that ℳ_g=Hom_𝖲(-,M_g) for some scheme M_g.Let C_g→ M_g be the family of curves corresponding to the identity morphism id_M_g. Now let π:X→ S be an isotrivial family of curves over a connected base S, that is not equivalentto S× X_s for anys∈ S, with s=Specℂ (here X_s is the fiber of X/S; see e.g., Examples <ref> and <ref>).The curve X_s→ s corresponds to an element ofℳ_g(Specℂ)=Hom_𝖲(Specℂ,M_g); i.e., the isomorphism class of the curvecorresponds to a point, say [X_s]∈ M_g.Similarly, the family π:X→ S corresponds to a morphism f:S→ M_g.Since every fiber of π:X→ S is isomorphic to [X_s], the image of S under f must be the point [X_s].By definition of the representability of the functor, we must then have that the pullback of the universal family C_g→ M_g along f, i.e., X_s× S, is equivalent to π:X→ S, which we assumed was not the case. While the functor ℳ_g is not representable, it does admit a coarse moduli space.In other words,there is a quasi-projective variety M_g and a morphismΦ:ℳ_g→ M_g (here we are using the convention mentioned in Remark <ref> of denoting a scheme and its associated functor with the same letter) such that:* for any scheme S and anymorphism Ψ:ℳ_g→ S, there is a unique morphism η :M_g→ S such that the following diagram commutes:ℳ_g @->[r]^Φ@->[d]_ΨM_g@–>[ld]^∃ !_ηS* Φ is a bijection when evaluated on any algebraically closed field.A scheme representing a functor, i.e., a fine moduli space,is clearly a coarse moduli space.For many moduli functors that we consider, we will be able to find a scheme satisfying the first condition above; however, when the automorphisms of our objects are positive dimensional, it will not in general be possible to find a scheme satisfying the second condition, as well. For brevity (or at least for lesser verbosity), we have not included a discussion of coarse moduli spaces.The interested reader may consult <cit.>, or <cit.>. There was not anything special about curves in Proposition <ref>:all we needed was an isotrivial family that was nontrivial. In fact,nontrivial automorphism groups almost always give rise to nontrivial isotrivial families (see Corollary <ref>) so we conclude that moduli problems involving objects with nontrivial automorphisms will almost never be representable by genuine spaces.Since most moduli problems of interest involve objects with nontrivial automorphisms, this meansfine moduli spaces almost never exist; i.e.,moduli problems are almost never representable by schemes.There are a few ways one might try to go around this problem:one could rigidify the problem by imposing additional structure in the hopes of eliminating nontrivial automorphisms; one could give up on the idea of a fine moduli space and settle for a coarse moduli spacethat at least gets the points right, even if it botches the universal property (see Remark <ref>).Stacks do not go around the problem of isomorphisms so much as they go through it.By remembering how objects are equivalent, as opposed to merely that they are equivalent, we can eliminate the issue that is responsible for Proposition <ref>.The price we must pay is to enlarge the class of objects we are willing to call spaces andsacrifice some of our geometric intuition. What we hope to illustrate in these notes is that the cost in geometric intuition is less than one might first expect, and that the payoff in newly available moduli spaces more than compensates for it.Indeed, stacks will allow us to bring our intuition about geometric families to bear on the geometry of their moduli spaces, giving us a new—and, we would argue, more powerful—sort of intuition to replace what we have sacrified. §.§ The tangent space of a moduli functor Because of Yoneda's lemma, a fine moduli space is uniquely characterized by the functor it represents.In particular, the tangent space to a moduli space at a point can be determined directly from the moduli functor, without even knowing that the moduli space is representable!This is not merely a formal convenience:the tangent space is an essential tool in proving that moduli functors are representable (see <ref> and <ref>).As an example of the definition, we compute the tangent space of the Grassmannian.The same calculation actually computes the tangent space of the Quot scheme, as well (see also <cit.>). Notice that we do not make use anywhere of the representability of the Grassmannian in this calculation.Further examples will appear in <ref>.We have already seen on p.p:GrCovhow the cover of G(r,n) by open subsets U_p associated to projections p : ℂ^n →ℂ^r is visible from the functor 𝒢(r,n).From this, near a point of the GrassmannianW↪ℂ^n,we get a local identification of 𝒢(r,n) with the functor represented by Hom_ℂ(ℂ^r, ℂ^n-r) = Hom(W, ℂ^n / W).Since this is a vector space, this gives a local identification of the tangent space T_𝒢(r,n),W with Hom(W, ℂ^n / W).As is well known, this identification can be made globally.Rather than verify the compatibility of this identification with gluing, which would require us to use the fact that 𝒢(r,n) is a sheaf, we will arrive at it via a global construction working directly fromthe definition of 𝒢(r,n).[Ring of dual numbers] The ring of dual numbers (over ℂ) is 𝔻 = ℂ[ϵ] / (ϵ^2).For any scheme S over ℂ, we write S[ϵ] for S ×Spec𝔻.[Tangent space to a moduli functor] If ℱ : 𝖲^𝗈𝗉→ (𝖲𝖾𝗍) is a functor, the tangent bundle to ℱ is the following functor:T_ℱ : 𝖲^𝗈𝗉→ (𝖲𝖾𝗍) T_ℱ(S) = ℱ(S[ϵ])Reduction modulo ϵ gives a map 𝔻→ℂ and therefore a closed embedding S → S[ϵ] for any scheme S.This induces natural functionsT_ℱ(S) = ℱ(S[ϵ]) →ℱ(S)and therefore a natural transformationT_ℱ→ℱ .We allow ourselves the following notational shortcut, which some may feel is abusive:When ξ∈ℱ(S), in other words, when (S, ξ) is an S-point of ℱ, we write T_ℱ(S, ξ) or T_ℱ(ξ) for the fiber of T_ℱ(S) over ξ∈ℱ(S).For instance, if we haveℱ=Hom_𝖲(-,M) for some complex manifold M, and ξ:Specℂ→ M is a point of M, then T_ℱ(ξ)=T_M(ξ), the holomorphic tangent space to M at x. The tangent space to an arbitrary functor ℱ may not be at all well-behaved!For example T_ℱ(ξ) might not even be a vector space.When trying to characterize the functors that are representable by schemes (or algebraic spaces or algebraic stacks) one of the first axioms we impose is that ℱ should behave infinitesimally like a scheme, and in particular that its tangent spaces should be vector spaces (see <ref>).§.§.§ Tangent space to the GrassmannianSuppose that S is a scheme and 𝔽↪ℂ_S^n is an S-point of 𝒢(r,n).Denote by F the sheaf of regularsections of 𝔽. Note that F is aquasicoherent locally free sheaf of rank r, and recall that 𝔽 can be recovered from F, either from transition functions, or directly as 𝔽=0pt[0pt][-.5pt]Spec_S(Sym^∙ F^∨),where F^∨=Hom_𝒪_S(F,𝒪_S) is the sheaf of sections of the dual vector bundle𝔽^∨=Hom_S(𝔽,ℂ_S).By definition, T_𝒢(r,n)(S,𝔽) is the set of isomorphism classes of extensions of 𝔽 to a vector subbundle 𝔽_1 ⊆ℂ^n_S[ϵ].Because vector bundles are determined by their sheaves of sections, deforming 𝔽 is the same as finding a locally freedeformation F to S[ϵ]. One can easily check by dualizing that this is equivalent to finding a locally free deformation of F^∨ to S[ϵ]. Since 𝔽↪ℂ^n_S, we have a quotient 𝒪_S^n →F^∨ by duality. We write E^∨ for the kernel of this quotient.To find a locally freedeformation of F^∨ to S[ϵ]is the same as to complete the diagram below with a locally free 𝒪_S[ϵ]-module F_1^∨:𝒪_S[ϵ]^n @–>[r] [d]F_1^∨@–>[d] @–>[r] 0 0 [r]E^∨[r]𝒪_S^n [r]F^∨[r] 0.Since F_1^∨ is locally free, tensoring the exact sequence0→ϵ𝒪_S→𝒪_S[ϵ]→𝒪_S→ 0with F_1^∨ we see that the kernel of F_1^∨→F^∨ is ϵF_1^∨≃F^∨.Note, moreover, that a short computation shows that any quasicoherent sheaf F_1^∨ fitting into the diagram (<ref>) above with ϵF_1^∨≃F^∨will be locally free, because a local basis for F^∨ can be lifted via the projection F_1^∨→F^∨ to a basis for F_1^∨.This can also be seen using the infinitesimal criterion for flatness <cit.>.Therefore our problem is, equivalently, to complete the diagram below with 𝒪_S[ϵ]-modules E^∨_1 and F^∨_1, so that the middle row is a short exact sequence, and the vertical arrows are those induced from (<ref>): @R=1.5em 0[d] 0[d] 0[d]0 [r] ϵE^∨[r] @–>[d] ϵ𝒪_S^n [r] [d]ϵF^∨[r]@–>[d]00@–>[r] E_1^∨@–>[r] @–>[d] 𝒪_S[ϵ]^n @–>[r] [d]F_1^∨@–>[d] @–>[r] 0 0 [r]E^∨[r]𝒪_S^n [r]F^∨[r] 0.0@<-[u] 0@<-[u] 0@<-[u]In particular, all rows and columns above are short exact sequences. Since ϵE^∨ necessarily maps to zero in F_1^∨, to produce a quotient F_1^∨ of 𝒪_S[ϵ]^n lifting the quotient F^∨ of 𝒪_S^n, it is equivalent to produce a quotient of 𝒪_S[ϵ]^n / ϵE^∨.We are therefore trying to complete the diagram below: 0 [r]E_1^∨ / ϵE^∨@=[d] @–>[r]𝒪_S[ϵ]^n / ϵE^∨@–>[r] [d]F_1^∨@–>[d] @–>[r] 0 0 [r]E^∨[r]𝒪_S^n [r]F^∨[r] 0.But note that E_1^∨ / ϵE^∨ projects isomorphically to E^∨. Therefore T_𝒢(r,n)(S,F) is isomorphic to the set of lifts of the following diagram:𝒪_S[ϵ]^n / ϵE^∨[d]E^∨@–>[ur] [r]𝒪_S^n.We have one canonical lift by composing the inclusion E^∨↪𝒪_S^n with the section 𝒪_S^n →𝒪_S[ϵ]^n.(Note that this is not an 𝒪_S[ϵ] homomorphism until it is restricted to E^∨.)An easy diagram chase in (<ref>) shows that the difference between anytwo lifts is a homomorphism E^∨→F^∨, so we get a canonical bijection:T_𝒢(r,n)(S,𝔽) = Hom_𝒪_S( E^∨, F^∨) = Hom_𝒪_S( F, E)=Hom_S(𝔽,𝔼)where 𝔼=0pt[0pt][-.5pt]Spec_S(Sym^∙E^∨) is the quotient of the trivial vector bundle ℂ_S^n by the subbundle 𝔽.§ MODULI FUNCTORS AS CATEGORIES FIBERED IN GROUPOIDS As nontrivial automorphisms tend to preclude the representability of a moduli problem (by a scheme), one plausible way to proceed is to account for these automorphisms by retaining them in thedefinition of the moduli functor.We are led to consider functors valued in groupoids (categories in which all morphisms are isomorphisms), rather than in sets, and immediately find ourselves in a morass of technicalities (see <ref>). In our judgment, a more elegant solutioncan be found in the notion of a category fibered in groupoids (see Remark <ref> for the reasoning that motivates our point of view). In <ref> we briefly discuss functors valued in groupoids, with the primary objective of convincing the reader that another approach would be preferable.We then introduce categories fibered in groupoids in <ref>. §.§ (Lax 2-)Functors to groupoidsInstead of studying a moduli problem by defining a functor whose value on a scheme is the set of isomorphism classes of families over that scheme, we try to define one whose value is the category of families parameterized by that scheme, with morphisms being isomorphisms of families.Contrary to the set of isomorphism classes, the category explicitly allows two families to be isomorphic in more than one way.[Groupoid]A groupoid is a category in which all morphisms are isomorphisms. In the literature, in the definition of a groupoid, it is common to require the additional condition that the category be small (the class of objects is a set).This is not required for our treatment, and so we drop the condition since almost every category we will consider will not be small.For instance, the category of sets with one element is obviously not small (for every set E there is a one element set {E}), and one can immediately generalize this to examples we consider here. However, the groupoids we study will usually be essentially small, meaning they are equivalent to small categories, so none of the pathologies that compel one to include a hypothesis of smallness will trouble us. One way to avoid worrying about small categories (or at least, to transfer the worry to somewhere else) is to introduce Grothendieck universes.Essentially, one assumes axiomatically that there are very large sets, called universes, that are large enough to `do set theory' within.All objects of interest occur within the universe, but one can still use set-theoretic language to speak about the universe itself.For example, the collection of all 1-element sets within the universe does form a set, namely one in bijection with the universe itself. Ultimately, these considerations are technical from our perspective, and we will remain silent about them in the sequel.One may consult <cit.> for yet another way around these technical issues. Continuing on, at first glance, it seems thatwe wanta “functor” to the category of groupoids:ℳ:𝖲^𝗈𝗉→ (𝖦𝗋𝗈𝗎𝗉𝗈𝗂𝖽).Making this precise leads to the morass of technicalities alluded to above. The issue here is the assignment for morphisms.In our examples, we pulled back families along morphisms.The fact that pullbacks are only defined up to isomorphism, albeit a canonical one, means that one does not have an equality ofg^∗ f^∗ with(fg)^∗ but only an isomorphism between them.One then has not one but two ways of identifying h^∗ g^∗ f^∗≃ (fgh)^∗ and one must require these be the same.Here is how this plays out for themoduli of curves: Letℳ^L2_g:𝖲^op→(𝖦𝗋𝗈𝗎𝗉𝗈𝗂𝖽)be the moduli functor in groupoidsfor curves of genus g; by definition ℳ^L2_g(S) is the category of families of curves over S, with S-isomorphisms as the morphisms.If T → S is a morphism of schemes we obtain a functor ℳ^L2_g(S) →ℳ^L2_g(T) sending C to C ×_S T.If we have a pair of morphisms U → T → S then we obtain two maps ℳ^L2_g(S) →ℳ^L2_g(U), one sending C to (C ×_S T) ×_T U and the other sending C to C ×_S U.These are canonically isomorphic, but they are not equal!Do we have to keep track of this canonical isomorphism, as well as the compatibilities it must satisfy when we encounter a sequence V → U → T → S? Pursuing this line of reasoning, one ultimately arrives at the definition of a pseudo-functor or lax 2-functor.However, just to define lax 2-functors is an unpleasant task, with little to do with the geometry that ultimately motivates us.(The reader who desires one may find a definition in <cit.>.)Fortunately, we will not have to think too hard about lax 2-functors because Grothendieck has supplied a more elegant solution:categories fibered in groupoids.The fundamental observation is that the pullbacks we needare canonically isomorphic because they satisfy universal properties that are literally the same.If we keep track of universal properties rather than the objects possessing them, we arrive at a more efficient definition.According to the philosphy behind categories fibered in groupoids, the mistake in Example <ref> was to choose a fiber product C ×_S T.We were then forced to carry it around and keep track of the compatibilities that it obviously satisfies.The category fibered in groupoids posits only that an object satisfying the universal property of the fiber product exists, i.e., that there is some cartesian diagram (<ref>),D [r] [d] C [d] T [r] Swithout actually specifying a construction of one. §.§ Categories fibered in groupoids From our perspective, the motivation for acategory fibered in groupoids is to avoid the technical complications of the definition of a lax 2-functor by essentially clumping all of the groupiods of interestinto one large categoryover the categoryof schemes; the issues we ran into defining the lax 2-functor on morphisms are avoided by using the universal properties offibered products in our category. We now make this precise. Temporarily, we let 𝖲 denote any category. A category over 𝖲 is a pair(ℳ,π) consisting of a category ℳ together with a covariant functor π : ℳ→𝖲. If S is an object of𝖲, the fiber of ℳ over 𝖲, denoted ℳ_S or ℳ(S) is defined to be the subcategory consisting of allobjects over S, and all morphisms over the identity of S. An object X in ℳ is said to lie over S if it is in ℳ_S.A morphism f̃:X'→ X in ℳ is said to lie overa morphism f:S'→ S if π(f̃)=f. A morphism between categories (ℳ, π) and (ℳ', π') over 𝖲 is a functor F : ℳ→ℳ' such that π' ∘ F = π.We indicate objects and morphisms lying above other objects and morphisms in diagrams like this:ℳ@->[d]^π X'@->[r]^f̃@|->[d]X @|->[d] 𝖲 S' @->[r]^fS. A fibered category is essentially one for which “pullback diagrams” exist. Keeping in mind the definition of a fibered product, a quick glance at (<ref>) should make clear the meaning of a pullback diagram in this setting. [Cartesian morphism]Let (ℳ,π) be a category over 𝖲.A morphism f̃:X'→ X in ℳ is cartesian if the following condition holds. Denote byf : S' → S the morphism π(f̃) in 𝖲 (as in(<ref>)).Given any morphism g:S”→ S' in 𝖲, and any morphism f∘ g:X”→ X in ℳ lying over f∘ g, there is a uniquemorphism g̃:X”→ X' in ℳ lying over g such that f∘ g =f̃∘g̃.Pictorially, every diagram (<ref>) has a unique completion:X”@->@/^.5pc/[rrd]^f∘ g@|->@/_.0pc/[d] @–>[rd]_∃ !^g̃S”@->@/_.3pc/[rd]_g@.>@/^.5pc/[rrd]^<>(0.7)f∘ gX'@->[r]^<>(0.4)f̃@|->[d]X @|->[d]S' @->[r]_fS. [Fibered category] A category (ℳ,π) over 𝖲 is said to befibered over 𝖲 if for anymorphism f : S' → S in 𝖲 and any object X of ℳ lying overS, there exists a cartesianmorphismf̃: X' →X in ℳ lying overf. [Category fibered in groupoids (CFG)]A category (ℳ, π) over 𝖲 is said to be fibered in groupoids if it is fibered over 𝖲, and for every S in 𝖲, we have that ℳ(S) is a groupoid.Categories fibered in groupoids are typically introduced as fibered categories in which all morphisms are cartesian <cit.>, <cit.>, or equivalently, a fibered category where thefibers are allgroupoids. In examples, as we have seen here, where we define all morphisms via fibered product diagrams, and pullbacks, one is led naturally to this definition. The added generality of fibered categories (not neccesarily fibered in groupoids) is important in order to formulate faithfully flat descent efficiently <cit.>, <cit.>, <cit.>.However, it is not particularly relevant to the study of moduli problems that is our focus here.Fibered categories were originally defined in <cit.> to be what we would call lax 2-functors (albeit valued in categories rather than in groupoids), and what others call pseudo-functors <cit.>, <cit.>.The definition of a fibered category given in <cit.>, is equivalent to the slightly different formulation in <cit.>.Lax 2-functors are equivalent (we do not attempt to say precisely in what sense) to fibered categories with cleavage <cit.>, <cit.>.The notation ℳ(S) for π^-1 (S) is meant to be suggestive of the relationship between categories fibered in groupoids and functors valued in groupoids.Indeed, one may construct an equivalence between the notions such that the groupoid-valued functor associated to ℳ has value ℳ(S) on S ∈𝖲.[CFG associated to a presheaf]Let ℱ:𝖲^𝗈𝗉→ (𝖲𝖾𝗍) be a functor in sets.One obtains a CFG π:ℱ→𝖲 in the following way.For each S∈Ob(𝖲), let ℱ_S=ℱ(S).Let S,S'∈Ob(𝖲) and suppose that X_S∈ℱ_S and X_S'∈ℱ_S'.Then we assign a morphism X_S'→ X_S in ℱ if there is a morphism f:S'→ S in 𝖲, and ℱ(f)(X_S)=X_S'.The functor π:ℱ→𝖲 is given by sending X_S to S, and similarly for morphisms. It is not difficult to generalize this construction to yield a CFG associated to a lax 2-functor <cit.>, provided that one has first given a precise definition of the latter.[CFG associated to an object of 𝖲] Let S be in 𝖲. Theslice category 𝖲/S is the CFG defined to have objects that are pairs (S', f) where S' ∈𝖲 and f : S' → S is a morphism.A morphism (S', f) → (S”, g) is a morphism h : S' → S” such that gh = f:@R=1em S' [rr]^h [dr]_fS”[dl]^gSThe projection π : 𝖲/S →𝖲 is π(S',f) = S'.A CFG equivalent to one of the form 𝖲/S for some S in 𝖲 is said to berepresentable by S. For any S∈Ob(𝖲), we can assign the functor h_S, and the category fibered in groupoids associated to h_S. This agrees with the CFG 𝖲/S. [The CFG (sieve) associated to a family of maps]Given a collection of morphisms ℛ={S_i→ S} in 𝖲, we define an associated full sub-CFG of 𝖲/S, which will also be denoted by ℛ.The objects of ℛ are the objectsS'→ S of 𝖲/S that factor through one of the S_i→ S. Sub-CFGs of representable CFGs (i.e., a sub-CFGs of 𝖲/S for some S in 𝖲) are known as sieves.To give the full statement of the Yoneda lemma for CFGs we need another definition.The correct language for discussing this is that of the 2-category of CFGs over 𝖲. We postpone this more technical discussion until later (<ref>).Here we give a working definition that suffices for our purposes.Let p_ℳ:ℳ→𝖲 and p_𝒩:𝒩→𝖲 be CFGs over 𝖲. There is a category Hom_𝖢𝖥𝖦/𝖲(ℳ,𝒩) with objects being morphisms ℳ→𝒩 of fibered categories over 𝖲 and morphisms beingnatural isomorphisms. Let 𝖲 be a category, and let S∈Ob(𝖲).For any fibered category π:ℳ→𝖲 the natural transformationHom_𝖢𝖥𝖦/𝖲(𝖲/S,ℳ)→ℳ(S) F ↦ F(S S)(defined similarly for morphisms) is an equivalence of categories.In view of the 2-Yoneda lemma, we may introduce the following notation:Suppose that S ∈𝖲 and that ℳ is a CFG over 𝖲.We write X : S →ℳ to mean X ∈Hom_𝖢𝖥𝖦/𝖲(𝖲/S, ℳ).By the Yoneda lemma, this is the same as specifying an object of ℳ(S), and we frequently do not distinguish notationally between X : S →ℳ and X ∈ℳ(S). Let 𝖲 be asmall category, and let S,S'∈Ob(𝖲). The mapHom_𝖲(S',S)→Ob(Hom_𝖢𝖥𝖦/𝖲(𝖲/S',𝖲/S))obtained by post-composition of arrows (e.g., a morphism f:S'→ S is sent to the functor that assigns to an arrow (g:S”→ S')∈Ob(𝖲/S'), the composition (f∘ g:S”→ S)∈Ob(𝖲/S)) is a bijection. Let ℳ,𝒩:𝖲^𝗈𝗉→ (𝖲𝖾𝗍) be two functors.Let ℳ,𝒩 be the associated categories fibered in groupoids over 𝖲.In a similar way, there is a bijection Hom_𝖥𝗎𝗇(𝖲^𝗈𝗉,(𝖲𝖾𝗍))(ℳ,𝒩)→Ob( Hom_𝖢𝖥𝖦/𝖲(ℳ,𝒩)).We delay introducing the notion of 2-categories until later (<ref>), but the consequence of the 2-Yoneda Lemma, and this observation, is that the category 𝖲, and the category of functors 𝖥𝗎𝗇(𝖲^𝗈𝗉,(𝖲𝖾𝗍)) can be viewed as full 2-subcategories of the 2-category of CFGs over 𝖲.Consequently, wewill frequently identify objects S of 𝖲, and functors to sets ℳ:𝖲^𝗈𝗉→ (𝖲𝖾𝗍) with their associated CFGs, 𝖲/S, and ℳ, respectively.§.§.§ The Grassmannian as a category fibered in groupoids For this section, let 𝖲 be the category of schemes over ℂ.The Grassmannian CFGπ:𝒢(r,n)→𝖲is defined as follows.Objects of 𝒢(r,n) are pairs (S, 𝔽) where S is a scheme and 𝔽↪ℂ^n_S is a vector subbundle of rank r.Morphisms in 𝒢(r,n) are defined via pullback.More precisely, a morphism (S',𝔽') → (S,𝔽) is a diagram@C=.3cm @R=.3cm𝔽' @->[rdd]@^(->@<-1pt>[rrd] @<2pt>@->[rrr] 𝔽@^(->[rrd] @->[rdd]ℂ^n_S'@->[rrr]_<>(0.25)ℂ^n_f@->[ld] ℂ^n_S @->[ld] S' @->[rrr]_f Swhere f:S'→ S is a morphism in 𝖲 and the rectangles are all cartesian. The mapπ:𝒢(r,n)→𝖲 is the forgetful functor π(S,𝔽) = S. To dispel any confusion that may arise from the notation in Example <ref>, we emphasize that the CFG associated to the functor 𝒢(r,n) is equivalent to𝒢(r,n). The former is fibered in sets whereas the latter is fibered in groupoids that are equivalent, but not isomorphic, to sets.We use the two notations to emphasize that one is a CFG and one is a functor. §.§.§ The CFG of curves of genus gAgain we take 𝖲 to be the category of schemes over ℂ.We define the CFG of genus g curvesπ:ℳ_g→𝖲in the following way.Objects of ℳ_g are pairs (S, X) where X→ S is a relative curve of genus g (see <ref>).Morphisms in ℳ_g are defined via pullback.More precisely, a morphism from (S',X') → (S,X) is a cartesiandiagramX' [r] [d] X [d] S' [r]^f Swhere f:S'→ S is a morphism in 𝖲.The map π:ℳ_g→𝖲 is π(S,X) = S.We emphasizethat the CFG associated to the functor ℳ_g is not equivalent to ℳ_g.The former is fibered in sets whereas the latter is fibered in groupoids that are not equivalent to sets. § STACKS Stacks are the categories fibered in groupoids that respect topology, in the sense that compatible locally defined morphisms into a stack can be glued together into global morphisms.This is the most basic requirement a category fibered in groupoids must satisfy in order to be studied geometrically.As usual, algebraic geometry introduces a troublesome technicality here:the Zariski topology is much too coarse to do much interesting gluing.Indeed, suppose that ℱ is a CFG that `deserves to be studied geometrically' and consider a scheme S with a free action of G = ℤ / 2 ℤ and a ξ∈ℱ(S) that is equivariant with respect to the G-action.If ℱ were a stack in the complex analytic topology, we could descend ξ to an element of ℱ(S/G) because S → S/G is a covering space, an in particular a local homeomorphism, hence a cover in the analytic topology.There is no such luck in the Zariski topology, where S → S/G may fail to be a local homeomorphism and may therefore also fail to be a cover in the Zariski topology. It is important to be able to do this kind of descent, so one must introduce an abstract replacement for the concept of a topology, called a Grothendieck topology.The essence of the definition is to isolate exactly the aspects of topology that are necessary to speak about gluing.It is possible to express the sheaf conditions without ever making reference to points, or even to open sets.All that is needed is the concept of a cover.Grothendieck's definition actually goes further, replacing even the concept of a cover with the abstract notion of a sieve.Even though topologies afford a few pleasant properties that pretopologies do not (see <ref> for more details), we will primarily limit our discussion to pretopologies in this introduction. Lest it appear genuinely pointless to remove the points from topology, consider the vast expansion of settings that can be considered topologically by way of Grothendieck topologies.To take just one, Quillen <cit.> and Rim <cit.> were able to understand extensions of commutative rings—that is, deformations of affine schemes—by putting a Grothendieck topology on the category of commutative rings (see <cit.> for further developments of this idea). We begin this section by reviewing the definition of a Grothendieck pretopology, using sheaves on a topological space as our motivation. We then define the isomorphism presheaf, and descent data, and finally, we give the definition of a stack.§.§ Sheaves and pretopologiesWe take asmotivation for Grothendieck pretopologies the definition of a sheaf on a topological space. §.§.§ Sheaves on a topological space Let X be a topological space.For any open subsets U' ⊆ U of X, let ι_U',U denote the inclusion of U' inside of U. Define thecategory of open sets on X,𝖮_X,with the following objects and morphisms:Obj𝖮_X={U⊆ X: Uopen} Hom_𝖮_X(U',U)= {ι_U',U}U'⊆ U ∅ else A presheaf (of sets) is a functorℱ:𝖮_X^𝗈𝗉→ (𝖲𝖾𝗍).Given a∈ℱ(U), and a subset ι_U',U:U'⊆ U, we denote by a|_U', or ι_U',U^*(a),the image of a under the map ℱ(ι_U',U):ℱ(U)→ℱ(U'). A presheaf is:* separated if, given an opencover {U_i→ U} and two sections a and b in ℱ(U) such thata|_U_i=b|_U_i inℱ(U_i)for all i, one hasa=b. * a sheaf if, given an open cover { U_i → U } with intersections U_ij = U_i ∩ U_j, and elements a_i ∈ℱ(U_i) for all i, satisfying a_i |_U_ij = a_j |_U_ij for all i and j, there is a unique a ∈ℱ(U) such that a |_U_i = a_i for all i. A morphism of presheaves, separated presheaves, or sheaves is a natural transformation of functors.Now, to motivate the definition of a pretopology, a presite,and a sheaf on a presite, we rephrase this definitionin the language of equalizers.Recall that if U' and U” are open subsets of U, then U'∩ U”=U'×_U U”. Given any open cover 𝒰={U_i→ U}, denote by ℱ(𝒰) the equalizer of diagram (<ref>): ∏_i ℱ(U_i)@->@<-2pt>[r]_-pr_2^*@->@<2pt>[r]^<>(0.5)pr_1^* ∏_i,jℱ(U_i×_U U_j).Recall that the equalizer is the categorical limit for morphisms into the diagram; in other words,we obtain a diagramℱ(𝒰) @->[r] ∏_i ℱ(U_i)@->@<-2pt>[r]_-pr_2^*@->@<2pt>[r]^-pr_1^* ∏_ijℱ(U_i×_U U_j),and the arrow on the left is universal (terminal) for morphisms into (<ref>).The natural map on the left in the diagram below isinduced by the restriction maps:ℱ(U) @->[r] ∏_i ℱ(U_i)@->@<-2pt>[r]_-pr_2^*@->@<2pt>[r]^-pr_1^* ∏_ijℱ(U_i×_U U_j).By the universal property of the equalizer, this induces a map:ℱ(U)→ℱ(𝒰)The sheaf conditions have the following translations into the language of equalizers. Let ℱ:𝖮_X^𝗈𝗉→ (𝖲𝖾𝗍) be a presheaf. * ℱ is separated if and only ifℱ(U)→ℱ(𝒰) is injective for all open U in X and all open covers 𝒰 of U. * ℱ is a sheaf if and only if ℱ(U)→ℱ(𝒰) is a bijection for all open U in X and all open covers 𝒰 of U.The main takeaway from this discussion is that we can repackage the sheaf condition in terms of fibered products and equalizers.This provides the motivation for a Grothendieck pretopology, and a sheaf on a presite. §.§.§ PretopologiesTemporarily denote by 𝖲 any category. A Grothendieck pretopology 𝒯 on𝖲 consists of the following data: for each object S in 𝖲, a collection of families of maps {S_α→ S}, called covers of S in 𝒯 (or covering families in 𝒯), such that: (PT 0)For all objects S in 𝖲, and for all morphisms S_α→ S which appear in some covering family of S, and for all morphisms S' →S, the fibered product S_α×_ S S' exists.(PT 1)For all objects S in 𝖲, all morphisms S' →S, and allcovering families {S_α→ S}, the family {S_α×_S S' → S'}is a covering family.(PT 2)If {S_α→ S} is a covering family, and if for all α, {S_βα→ S_α}is a covering family, then the family of composites {S_βα→ S_α→ S}is a covering family.(PT 3) If S' → S is an isomorphism, then itis a covering family. Let X be a topological space.The category 𝖮_X together with open covers is a Grothendieck pretopology. Let X be a topological space, define a category 𝖲 to have as objects 𝒫(X), the set of all subsets of X, and as morphisms the inclusions.We give every subset S⊆ X the induced topology.Then the collection of allopen covers of subsets of X gives a Grothendieck pretopology on 𝖲.Indeed, (PT0) is satisfied, since the fibered product is given by intersection.(PT1) holds since we are giving every subset the induced topology, so an opencover of a superset gives an open cover ofa subset.(PT2) holds since refinements of open covers are open covers.(PT3) holds since isomorphisms are equalities.Let π : 𝒳→𝖲 be a CFG over a category 𝖲 equipped with a Grothendieck pretopology.Call a family of maps { X_i → X } in 𝒳 covering if {π(X_i) →π(X) } is covering in 𝖲.Then this determines a Grothendieck pretopology on 𝒳. To verify this, it may be helpful to observe that the induced morphismof CFGs 𝒳 / X→𝖲/π(X) is anequivalence of categories; indeed, by the definition, 𝒳 being a CFG implies the morphism is essentially surjective, and fully faithful. [Presite] A pair (𝖲,𝒯) consisting of a category 𝖲 together with a Grothendieck pretopology 𝒯is called a presite.Often 𝒯 is left tacit and one uses 𝖲 to stand for both the presite and its underlying category of objects.[Covers in the étale pretopology]The primary example of a Grothendieck pretopology that we will use is the étale pretopologyon the category of schemes.We denote the associated presite by 𝖲_et. Covers in 𝖲_et are collections ofjointly surjective étale morphisms. Recall that étale morphisms are the algebro-geometric analogue of local isomorphisms in the complex analytic category.Given a pretopology on 𝖲 and a scheme S in 𝖲, we obtain the category 𝖲/S, and an induced pretopology defined in the obvious way. For instance, (𝖲/S)_et has covers given by jointly surjective étale morphisms in that category.Readers who prefer working in the analytic category,should feel free totake 𝖲 to be the category of complex analytic spaces, and to work with the pretopology 𝒯 generated by theusual open covers of complex analytic spaces,in the analytic topology. In fact, one could as easilytake 𝖲 to be the category of complex manifolds, with smooth morphisms (so that fibered products remain in the category), and work with the pretopology 𝒯 generated by theusual open covers of complex manifolds.[Standard pretopologies on schemes] The most commonly used Grothendieck pretopologies on the category of schemes are the: * Zariski pretopology,* étale pretopology,* faithfully flat finite presentation (fppf) pretopology,* faithfully flat quasicompact (fpqc) pretopology.Each of these pretopologies is a refinement of the one preceding it.We will write 𝖲__Zar, 𝖲__et, etc., for the respective presites.The Zariski pretopology (covers by Zariski open sets) is too coarse for most of the applications we have in mind.For simplicity, we will work almost exclusively with the étale pretopology. §.§.§ Sheaves on a presiteThe definition of a pretopology is exactly set up to allow us to make a definition ofa sheaf following our discussion of sheaves on topological spaces. A presheaf (of sets) on 𝖲 is just a functorℱ:𝖲^𝗈𝗉→ (𝖲𝖾𝗍). Given any cover ℛ={S_i→ S}, denote by ℱ(ℛ)=ℱ({S_i→ S}) the equalizer of the diagram∏_i ℱ(S_i)@->@<-2pt>[r]_-pr_2^*@->@<2pt>[r]^-pr_1^* ∏_ijℱ(S_i×_S S_j).As before, thereis a natural mapℱ(S)→ℱ(ℛ).Following Lemma <ref>, we make the following definition:[Sheaf on a site] Let ℱ be apresheafon apresite (𝖲,𝒯). * ℱ is separated ifℱ(S)→ℱ(ℛ) is an injection for every covering family ℛ of every object S of 𝖲. * ℱ is a sheafif ℱ(S)→ℱ(ℛ) is a bijection for every covering family ℛ of every object S of 𝖲. On occasion we will be more precise about how fine apretopology can be used to obtain a given statement.For instance, we may specify that a presheaf is a sheaf with respect to the fpqc pretopology, which implies it is also a sheaf with respect to all of the other pretopologies mentioned above. [Subcanonical presite] A pretopology 𝒯 on a category 𝖲 is called subcanonical if every representable functor on 𝖲 is a sheaf with respect to 𝒯. A presite (𝖲,𝒯) is called subcanonical if 𝒯 is subcanonical.[Grothendieck <cit.>] Let S be a scheme. The presite(𝖲/S)__fpqc is subcanonical; in particular(𝖲/S)__et is subcanonical. §.§ The isomorphism presheaf Let π:ℳ→𝖲 be a CFG. If we view ℳ as a space, as is our intention, and view an object X ∈ℳ(S) as a map S →ℳ then we expect that morphisms into ℳ that agree locally should agree globally.The only proviso is that because two maps into ℳ can agree with one another in more than one way, we must interpret local agreement of X and Y to include not only choices of local isomorphisms between X and Y over a cover, but also compatibility of these choices on the overlaps in the cover.In order to state this condition precisely, we introduce the presheaf of witnesses to the agreement of objects of ℳ.The condition we want to impose is that this presheaf be a sheaf. [Isomorphism presheaf] Let π:ℳ→𝖲 be a CFG. Given S in 𝖲, and X,Y∈ℳ(S), we obtain a presheaf ℐsom_ℳ(X,Y):(𝖲/S)^𝗈𝗉→ (𝖲𝖾𝗍)in the following way.For every object(S'→ S) in 𝖲/S, we setℐsom_ℳ(X,Y)(S'→ S):= Isom_ℳ(S')(X |_S', Y |_S')Thus ℐsom_ℳ(X,Y)(S'→ S) consists of all isomorphisms α : X |_S'→ Y |_S' in ℳ that lie over the identity id_S' in 𝖲. The assignment for morphisms is left to the reader (see <cit.>).An observant reader will note that the restrictions in the definition of ℐsom_ℳ(X,Y) depend, albeit only up to a canonical isomorphism, on a choice of inverse to the functor Hom(S, ℳ) →ℳ(S), as guaranteed by the Yoneda lemma. Concretely, to say that isomorphisms form a sheaf means the following.Given a cover {S_i→ S} in the pretopology on 𝖲, and any collection of isomorphisms α_i:X|_S_i→ Y|_S_i over the identity on S_i such that α_i|_S_ij=α_j|_S_ij, there is a unique isomorphism α :X→ Y such that α|_S_i=α_i.Here we are using the shorthand S_ij:=S_i×_S S_j. [Prestack] A CFG ℳ→𝖲 such that forevery S in 𝖲, and every X,Y in ℳ(S),the presheafℐsom_ℳ(X,Y) is a sheaf, is called a prestack.We will also say that isomorphisms are a sheaf.The notation would be more consistent with sheaf notation(see Proposition <ref>) if we called categories fibered in groupoids (or fibered categories) `prestacks' and called prestacks `separated prestacks'.But we will keep to tradition. It is typically very easy to prove that a category fibered in groupoids arising from a moduli problem is a prestack in a subcanonical topology.This is because an object Y ∈ℳ(S) is usually representable by a scheme, perhaps with some extra structure or properties, and descending local isomorphisms between X ∈ℳ(S) and Y amounts to descending locally defined morphisms from X to Y, which is automatic because Y represents a sheaf in any subcanonical topology!Let us see how this works concretely in the example of ℳ_g.Let S be a scheme in 𝖲, and let X→ S and Y→ S be relative curves of genus g.Suppose that there exists an étale cover {S_i→ S} so that for each S_i there are S_i-isomorphisms α_i:X_S_i→ Y_S_i such that α_i|_S_ij=α_j|_S_ij (using the shorthand S_ij = S_i ×_S S_j).These correspond by the universal property of fiber product to morphisms X_S_i→ Y and X_S_ij→ Y satisfying the same compatibility condition.As Y represents a sheaf and {X_S_i→ X} isa cover of X in the pretopology, these glue to a morphism α : X → Y such that α|_S_i = α_S_i.To check that this is in fact a morphism over S, note that the commutativity of the diagrams (<ref>) X_S_i[r] [d] Y [d] S_i [r] Sfor all i implies the commutativity of diagram (<ref>),@R=1.5em X [rr]^α[dr]Y [dl]Sthis time because morphisms into S are a sheaf. In order to make a precise general statement along these lines, we make a very general definition:[Stable class of arrows <cit.>] A class of arrows 𝐏 in a category 𝖲 is stable (under base change) if morphisms in 𝐏 can be pulled back via arbitrary morphisms in 𝖲, and the result of any such pullback is also in 𝐏. [CFG associated to a stable class of arrows]Given any stable class ofarrows 𝐏 in a category 𝖲, one may make 𝐏 into a category 𝒫 by settingobjects to be arrows in 𝐏 and setting morphisms to be cartesian squares. There is a morphism𝒫→𝖲 given by sending an object X→ S to the target S (and similarly for morphisms). Then one can check that 𝒫→𝖲 is a CFG if and only if 𝐏 is a stable class of arrows. [<cit.>] Let (𝖲,𝒯) be a subcanonical presite, and 𝐏 a stable class of arrows.Let π : 𝒫→𝖲 be the associated CFG (Example <ref>).Then (𝒫, π) is a prestack.The discussion in Example <ref> adapts in a straightforward way to a proof of Theorem <ref>. The CFGs 𝒢(r,n) and ℳ_g are prestacks in the étale topology on schemes.The case of ℳ_g follows directly from the theorem with (𝖲,𝒯)=𝖲_et and 𝒫 the class of relative curves of genus g.The case of 𝒢(r,n) requires a slight modification (since there is more than one arrow in the definition of the objects), but is essentially the same.§.§ Descent for categories fibered in groupoidsRecall that we can view vector bundles either as global geometric objects admitting local trivializations, or alternatively, as collections of locally trivial objects together withtransition functions, which satisfy the cocycle condition. We now use this motivation to define the notion of descent datum in the setting of CFGs. This isthe last definition we will need before defining a stack. §.§.§ Vector bundles on open subsets of a complex manifold Let X be a complex manifold.Let 𝖮_X be the category of open sets.Give 𝖮_Xthe structure of a presite in the usual way, by taking covering families to beopen covers.Define a CFG of vector bundles π:𝒱^r_X→𝖮_Xin the following way.For each U⊆ X open, the objects in thefiber 𝒱_U^r are therank r, holomorphic vector bundles on U.Morphisms in 𝒱^r are given by pullback diagrams. Consider an open covering {U_i ⊆U}.We use the notation U_ij = U_i ∩ U_j and U_ijk = U_i ∩ U_j ∩ U_k for the double and triple overlaps. Suppose we are given a vector bundle E_ioverU_i for every i,and an isomorphism α_ij : E_i | _U_ij→E_j | _U_ij for every i, j,all of which satisfy the cocycle condition α_ik = α_jk∘α_ij over U_ijk.Then there exists a vector bundle E lying overU,together with isomorphisms α_i: E |_U_i→E_isuch that α_ij = α_j |_U_ij∘ (α_i |_U_ij)^-1. This means precisely that the category fibered in groupoids of vector bundles on a topological space satisfies descent and therefore is a stack in the usual topology.The formulation of descent and the definition of a stack axiomatize this familiar gluing process.§.§.§ Intuitive definition of descentIn this section we will give a direct translation of the gluing condition for vector bundles encountered in <ref> in the context of categories fibered in groupoids.While intuitive, this formulation has both technical and practical definiencies.We correct these in <ref> and <ref>, but a reader looking to develop intuition about stacks may safely ignore these matters, especially in a first reading.[Descent using gluing data]Let ℳ be a category fibered in groupoids over a presite (𝖲,𝒯) with a cleavage (Definition <ref>).A descent datum for ℳ over a space S is the following: a covering {S_i →S}; for every i, an object X_ioverS_i; for every i, j an isomorphism α_ij : X_i | _S_ij→X_j | _S_ijin the fiber ℳ(S_ij), which satisfies the cocycle condition α_ik = α_jk∘α_ij over S_ijk. The descent datum is said to beeffective if there exists an X lying overS,together with isomorphisms α_i: X |_S_i→X_i in the fiber such that α_ij = α_j |_S_ij∘ (α_i |_S_ij)^-1.From the example in <ref>, we see that every descent datum for the CFG 𝒱^r_X→𝖮_X is effective.We now discuss the category of descent data, and the meaning of effective descent data in this context.[The category of descent data]Let (𝖲,𝒯) be a presite, and let π:ℳ→𝖲 be a CFG. Let ℛ={S_i→ S} be a covering in 𝖲.An object with descent data on ℛ, or descent datum on ℛ, is a collection ({X_i},{α_ij}) of objects X_i∈ℳ(S_i), together with isomorphisms α_ij:pr_2^*X_j ≅pr_1^* X_i in ℳ(S_i×_S S_j), such that the following cocycle condition is satisfied: For any triple of indices i,j,k, we have the equalitypr_13^*α_ik=pr_12^*α_ij∘pr_23^*α_jk:pr_3^*X_k→pr_1^* X_i.An arrow between objects with descent data {ϕ_i}:({X_i},{α_ij})→ ({X_i'},{α_ij'})is a collection of arrows ϕ_i:X_i→ X_i' with the property that for each pair of indices, i,j, the diagram pr_2^*X_j@>pr_2^*ϕ_j>> pr_2^*X_j'@Vα_ijVV @Vα_ij'VV pr_1^*X_i@>pr_1^*ϕ_i>> pr_1^*X_i'commutes.The objects and morphisms above determine a category of descent dataℳ({ S_i → S }).We have deliberately omitted a number of canonical isomorphism from the notation here.This obscures some technical issues (which can, of course, be resolved: see <ref>).For example, different choices of fiber products S_ij and S_ijk will lead to different but equivalent categories ℳ({ S_i → S }). If ℛ is a covering, as in Definition <ref>, it is technically convenient to indicate the category of descent data with respect to ℛ as a map ℛ→ℳ.This notation is consistent with the Yoneda identification between ℳ(S) and the category of morphisms S →ℳ, which is the special case where ℛ is the trivial covering { S → S }.In general, the notation is justified by replacing the covering with the associated sieve or simplicial object (see Section <ref>).Note that whenever a cover ℛ' refines a cover ℛ there is an induced morphisms ℳ(ℛ) →ℳ(ℛ'), which we notate as a composition:ℛ' →ℛ→ℳWe identify the object S with the trivial cover of itself, so that it makes sense to write ℛ→ S for any cover ℛ of S.Given X∈ℳ(S), we can construct an object with descent data on {σ_i:S_i→ S} as follows.The objects are the pullbacks σ_i^*X; the isomorphisms α_ij:pr_2^*σ_j^*X ≅pr_1^* σ_i^*X are the isomorphisms that come from the fact that bothpr_2^*σ_j^*X andpr_1^* σ_i^*X are pullbacks of X to S_i×_S S_j (they are both equipped with canonical isomorphisms with (σ_i∘pr_1)^*X=(σ_j∘pr_2)^*X).(If we identifypr_2^*σ_j^*=(σ_j∘pr_2)^*, etc., as is common, then the α_ij are identity morphisms.) If ϕ:X→ X' is an arrow in ℳ(S), then we get arrows σ_i^*X σ_i^*X' yielding an arrow from the object with descent data associated to X with the object with descent data associated to X'.In short, we have defined a functor ℳ(S)→ℳ({ S_i → S }).[Effectivity of descent data] A descent datum ( { X_i }, {α_ij}) ∈ℳ({ S_i → S }) is said to be effective if it is isomorphic to the image of an object of ℳ(S). §.§ The long-awaited definition of a stack We are finally ready for the definition of a stack: [Stack] A stackis a category fibered in groupoids (Definition <ref>) over a presite(𝖲,𝒯) (Definition <ref>) such that isomorphisms are a sheaf (Definition <ref>) and every descent datum is effective (Definition <ref>).A prestack is a category fibered in groupoids over (𝖲,𝒯) such that isomorphisms are a sheaf. A morphism of (pre)stacks over (𝖲,𝒯)is a morphism of the underlying CFGs over 𝖲. The definition can be rephrasedin the following way, emphasizing the connection with sheaves.[<cit.>] Let ℳ→𝖲 be a category fibered in groupoids over a presite (𝖲,𝒯). * ℳ is a prestack over 𝖲 if and only iffor each covering ℛ={S_i→ S}, the functor ℳ(S)→ℳ(ℛ) is fully faithful. * ℳ is a stack over 𝖲 if and only if for each covering ℛ={S_i→ S}, the functor ℳ(S)→ℳ(ℛ) is an equivalence of categories.This essentially leads driectly to the following proposition.[<cit.>] Let (𝖲,𝒯) be a presite, and let ℳ:𝖲^𝗈𝗉→ (𝖲𝖾𝗍𝗌) be a presheaf.Denote by ℳ→𝖲 the associated category fibered in groupoids.* ℳ is a prestack if and only if ℳis a separated presheaf. * ℳ is a stack if and only if ℳ is a sheaf.If (𝖲,𝒯) is subcanonical, then every object in 𝖲 represents a stack. More precisely, given an object S, weassociate to it the category fibered in groupoids𝖲/S, whichis a stack over 𝖲. In fact,using the language of 2-categories, for a subcanonical presite (𝖲,𝒯), the category 𝖲, and the category of sheaveson (𝖲, 𝒯) can be viewed as full 2-subcategories of the 2-category of stacksover 𝖲. [Local class of arrows] A class 𝐏 of arrows in a presite (𝖲,𝒯) is local (on the target) if it is stableand has the “converse” property that for any cover {S_i→ S} and any arrow X→ S, ifthe pullbacks S_i×_ SX→ S_i are in 𝐏 for all i, then X→ S is also in 𝐏.Fix a morphism X → Y in 𝖲 and consider the collection of all S → Y such that X ×_Y S → S has property 𝐏.This is a category fibered in groupoids, in fact a sieve of Y, that we denote 𝒫(X → Y).Then 𝐏 is a local property if and only if 𝒫(X → Y) is a stack for all morphisms X → Y. This observation can be used to prove the following theorem:[<cit.>]Let S be a scheme.Let 𝐏 be a class of flat projectivecanonically polarized morphisms of finite presentation in (𝖲/S)__fpqc, which is local. Then the associated CFG𝒫→ (𝖲/S)__fpqc(from Definition <ref>) is a stack.Canonically polarized morphisms include smooth morphisms suchthat the determinant of the relative cotangent bundle is relatively ample; for instance, families of smooth curves of genus g≥ 2.The theorem in fact holds in more generality forpolarized morphisms, but then one must add a compatibility conditionfor the polarizations, which lengthens the statement (see <cit.>).The above will be sufficient for many of our applications.The CFG ℳ_g isa stack in the fpqc topology (and therefore in the étale topology) for g ≥ 2.Let us make the descent condition concrete in the case of ℳ_g.Suppose we are given an étale cover {S_i→ S} and for each S_i a relative curve X_i→ S_i.Suppose moreover that for each i,j, we are given an S_ij-isomorphism α_ij : X_i | _S_ij→X_j | _S_ij, which satisfies the cocycle condition α_ik = α_jk∘α_ij over S_ijk. Then there exists a relative curve X→ S,together with S_i-isomorphisms α_i: X |_S_i→X_isuch that α_ij = α_j |_S_ij∘ (α_i |_S_ij)^-1. A similar argument can be used to show that 𝒢(r,n) is a stack.For brevity, we omit the detailsas we will also give references for Quot stacks, of which 𝒢(r,n) is a special case.One can also show that ℳ_g is a stack for g=0; it does not follow immediately from the theorem above since the canonical bundle of such curves is not ample.However, for g=0, the anti-canonical bundle is ample, and this gives a polarization that can be used in a more general formulation of the theorem <cit.>.However, ℳ_1 is not a stack!See Section <ref> for a demonstration and a discussion of the fix.§ FIBERED PRODUCTS OF STACKSBecause of the ubiquity of base change in algebraic geometry, it is essential to know that one can take fiber products of stacks.In this section we present a construction of the fiber product of categories fibered in groupoids, which yields a fiber product of stacks when applied to CFGs that are stacks. A reader who wants to get to algebraic stacks as quickly as possible may prefer to look briefly at <ref> and then skip the remainder of <ref>, referring back as necessary. A more detailed discussion of 2-categories and universal properties within them can be found in the Stacks Project <cit.>.§.§ A working definitionof 2-fibered products Here we give a working definition of a 2-fibered product of CFGs.This should suffice for understanding the definitions of an algebraic stack.Again, on a first pass, the reader is encouraged tolook at this section, and then skip the remainder of <ref>.Our presentation here is taken largely from <cit.>.Recall that a fiber product of morphisms of sheaves f : 𝒳→𝒵 and g : 𝒴→𝒵 is defined by setting (𝒳×_𝒵𝒴)(S) = 𝒳(S) ×_𝒵(S)𝒴(S) for all S.The same definition could be used for lax 2-functors valued in groupoids, except one must first define a fiber product of groupoids.Here there is a subtle, but crucial point, since objects of 𝒵(S) can be `equal' to each other in more than one way.The fiber product must therefore keep track of all of the different ways f(X) and g(Y) are equal to each other.One defines 𝒳(S) ×_𝒵(S)𝒴(S) to be the category of triples (X,Y,α) where X ∈𝒳(S), Y ∈𝒴(S), and α : f(X) ≃ g(Y) is an isomorphism.Morphisms (X,Y,α) → (X',Y',α') in this groupoid are pairs (u,v) with u : X → X' and v : Y → Y' are morphisms such that α' f(u) = f(v) α as morphisms f(X) → g(Y'). This construction is analogous to one construction of the homotopy fiber product in algebraic topology, with isomorphism playing the role of homotopy.This is not an accident, as homotopy fiber products are intended to be invariant under replacement of the spaces involved with homotopy equivalent spaces; the fiber product of groupoids is intended to be invariant under equivalence of categories.Even more directly, groupoids can be realized as topological spaces by way of the geometric representation of the simplicial nerve,under which the fiber product of groupoids is transformed into the homotopy fiber product. Since we are not defining stacks in terms of lax 2-functors here, we make the straightforward translation of the above idea to categories fibered in groupoids:[<cit.>]Let f : 𝒳→𝒵 and g : 𝒴→𝒵 be morphisms of CFGs over 𝖲.Then the fiber product of 𝒳 and 𝒴 over 𝒵,denoted 𝒳×_𝒵𝒴, is the category over 𝖲 whose objects are quadruples (S, X, Y, α) where S ∈𝖲, X ∈𝒳(S), Y ∈𝒴(S), and α : f(X) ≃ g(Y) is an isomorphism in 𝒵(S).A morphism (S,X,Y,α) → (S',X',Y',α') is triple (φ, u, v) where φ : S → S' is a morphism in 𝖲, u : X → X' is a morphism in 𝒳 lying above φ, and v : Y → Y' is a morphism in 𝒴 lying above φ, and α' f(u) = g(v) α. The fiber product of CFGs is a CFG.The fibered product of stacks is a stack. The 2-fibered product of CFGs 𝒳×_𝒵𝒴 has a universal property similar to that satisfied by a fiber product of sheaves.Indeed, there are forgetfulmorphisms p:𝒳×_𝒵𝒴→𝒳 and q:𝒳×_𝒵𝒴→𝒴, respectively sending (S,X,Y,α) to X and to Y.Then α gives an isomorphism f p(S,X,Y,α) = f(X) ≃ g(Y) = g q(S,X,Y,α).By the definition of morphisms in 𝒳×_𝒵𝒴, this isomorphism is natural in (S,X,Y,α).We denote this natural isomorphism ψ : f p ≃ g q.In standard terminology, the following diagram is 2-commutative:𝒳×_𝒵𝒴[r]^p[d]_q𝒳[d]^f@=>[ld]_ψ𝒴[r]_g 𝒵.The universal property of the 2-fibered product is that (𝒳×_𝒵𝒴, p, q, ψ) is the universal completion of the diagram of solid arrows below to a 2-commutative diagram𝒲@–>[r]^p@–>[d]_q𝒳[d]^f@==>[ld]_ψ𝒴[r]_g 𝒵.In other words, given a 2-commutative diagram (<ref>), there is a 2-commutative diagram:𝒲@->@/^1 pc/ [rrd]^a @–>[rd]_γ@/_1 pc/ [rdd]_b 𝒳×_𝒵𝒴[r]_-p [d]_q𝒳[d]^f𝒴[r]^g 𝒵Note that the 2-commutativity of this diagram includes the tacit specification of natural isomorphisms p γ≃ a and q γ≃ b.The functor γ is determined by these data uniquely up to a unique natural transformation.Using this one can showthat if 𝒲 also satisfies the universal property of 𝒳×_𝒵𝒴 then γ : 𝒲→𝒳×_𝒵𝒴 is an equivalence. Suppose that X,Y,Z∈𝖲 and 𝖲 has fibered products.Then it follows from the 2-Yoneda Lemma that𝖲/X×_𝖲/Z𝖲/Y is equivalent to𝖲/(X×_ZY). Similarly, suppose that 𝒳,𝒴,𝒵 are pre-sheaves on 𝖲 with associated CFGs 𝒳,𝒴,𝒵.Then 𝒳×_𝒵𝒴 is equivalentto the CFG associated to 𝒳×_𝒵𝒴.§.§ The diagonalThe following example is used repeatedly.[The diagonal and the sheaf of isomorphisms]Let ℳ be an S-stack over 𝖲, let S be in 𝖲, and let X,Y in ℳ(S) be two objects corresponding under the 2-Yoneda Lemmato S-morphisms X,Y:S→ℳ.Then, using the notation from <ref>,there is 2-cartesian diagramℐ𝑠𝑜𝑚_ℳ(X,Y) [r][d]^S [d]^(X, Y) ℳ[r]^<>(0.5)Δ ℳ×ℳ.We will verify this by way of the universal property (<ref>) (see also <cit.>).Suppose that we have a map f : T → S, an object Z ∈ℳ(T), and an isomorphism(φ, ψ) : (Z,Z) = Δ(Z) ≃ f^∗ (X,Y) = (f^∗ X, f^∗ Y),in (M×ℳ)(Z).Then the composition ψ∘φ^-1 is an isomorphism f^∗ X ≃ f^∗ Y, hence yields a section of ℐ𝑠𝑜𝑚_ℳ(X,Y) over T.Conversely, given such an isomorphism α : f^∗ X ≃ f^∗ Y, we obtain 2-commutative diagram by taking Z = (f^∗ X, f^∗ X) and (φ, ψ) = (id_f^∗ X, α).Intuitively, ℐ𝑠𝑜𝑚_ℳ(X,Y) is the sheaf of witnesses to the equality of X and Y, in the same way that the diagonal is the moduli space of pairs of objects of ℳ that are equal to one another.Notably, ℐ𝑠𝑜𝑚_ℳ(X,Y) is not a subobject of S, reflecting the fact that Δ is not an embedding.This is because a pair of objects of a groupoid can be equal—that is, isomorphic—to each other in more than one way. From the perspective of moduli, the isomorphisms of the objects of study were of central importance.On the other hand,the diagonal map is central in the definition of many properties of schemes. The diagram aboverelates the two notions. [Injective morphism ofstacks]A morphism f:𝒳→𝒴 ofstacks over 𝖲 is called injective (resp. an isomorphism) if for each S∈𝖲, the functor f(S):𝒳(S)→𝒴(S) is fully faithful (resp. an equivalence of categories).A substack is an injective morphism of stacks. A stack 𝒳 has injectivediagonal if and only if 𝒳 is representable by a sheaf (i.e., equivalent to the stack associated to a sheaf). Injective diagonal means that ℐ𝑠𝑜𝑚_𝒳(x,y) is a subobject of S, or, equivalently, that for any x,y ∈𝒳(S) there is at most one isomorphism between x and y.Thus 𝒳(S) is equivalent to a set. §.§ Fibered products and the stack condition As an application of the formalism of fiber products introduced above, we give a simple but often useful criterion for a CFG over a presite to be stack, which is simply a translation of Proposition <ref> (see also Definition <ref>). Let π:ℳ→𝖲 be a CFG over a presite (𝖲,𝒯). * ℳ is a prestackif and only for every covering family ℛ = {S_i→ S},every morphism f:ℛ→ℳ (see Remark <ref>), and every pair of morphisms f_1,f_2:S→ℳ making diagram (<ref>) 2-commutative,there is a unique 2-isomorphism f_1⇒ f_2. ℛ[r]^f [d]ℳ[d]^π 𝖲/S @–>@/^.5 pc/[ru]^<>(0.5)^f_1@–>@/_.5 pc/[ru]^<>(0.5)_f_2[r]𝖲 * ℳ is a stackif and only for every sieve ℛ associated to a covering family {S_i→ S}, with natural map ℛ→ S,any morphism f:ℛ→ℳ, there exists a morphism f_1:S→ℳ making the diagram below 2-commutative, which is unique up to 2-isomorphism (as explained in (1)).ℛ[r]^f [d]ℳ[d]^π 𝖲/S @–>[ru]^f_1_∃ ![r]𝖲The lemma can be summarized with the following efficient characterization of stacks: Let ℳ be a category fibered in groupoids over 𝖲.Then ℳ is a stack over 𝖲 if and only if for every object S of 𝖲 and every cover ℛ of S, the functorHom(S, ℳ) →Hom(ℛ, ℳ)is an equivalence of categories. The first part of Lemma <ref> is the full faithfulness, and the second part is the essential surjectivity.Recall that the notation in the corollary is based on the abuse explained in Remark <ref>.The statement requires no such abuse when formulated with sieves (see <ref>), the efficiency of which is one reason we like using sieves to think about Grothendieck topologies.For example, Corollary <ref> generalizes immediately to give a definition of higher stacks, while the other formulations of descent from <ref> become even more combinatorially complicated.The following proposition is quite useful for making boot-strap arguments.It allows us to show that a CFG is a stack by showing it is a stack relative to a CFG already known to be a stack. Suppose that p : 𝒳→𝒴 is a morphism of CFGs over 𝖲 and that 𝒴 is a stack over 𝖲.Then 𝒳 is a stack over 𝖲 if and only if it is a stack over 𝒴, where 𝒴 is given the pretopology of Example <ref>. We will verify the second part of the criterion in Lemma <ref>; the first part is very similar, so we omit it.Assume first that 𝒳 is a stack over 𝒴.Consider the following diagram:@C=4em@R=1.3emℛ[r]^f [dd]𝒳[d]^p 𝒴[d]S @–>[uur]^f_1@–>[ru]_h[r] 𝖲We would like to find a morphism f_1 rendering the outer square 2-commutative.Since 𝒴 is a stack over 𝖲, we can findan appropriatelift h as in the diagram.But then the assumption that 𝒳 is a stack over 𝒴 guarantees the existence of the desired liftingf_1 ofh. Thus 𝒳 is a stack over 𝖲. The converse is similar.Suppose that p : 𝒳→𝒴 is a morphism of CFGs over a presite 𝖲.Assume that 𝒴 is a stack over 𝖲.Then 𝒳 is a stack over 𝖲 if and only if for S ∈𝖲 and every y: S →𝒴, the fiber product 𝒳_S = 𝒳×_𝒴 S is a stack on 𝖲/S. We show the harder direction,that 𝒳 is a stack over 𝖲.From Proposition <ref>, it is enough to show that 𝒳 is a stack over 𝒴.From Lemma <ref> and the definition of the fibered product, one can easily deducethat𝒳 is a stack over 𝒴 if and only if 𝒳×_𝒴 (𝒴/Y) is a stack over 𝒴/Y for all Y ∈𝒴.Now using theequivalence of categories 𝒴/Y≅𝖲/S (Example <ref>), we are done.As an example of the utility of Proposition <ref> and Corollary <ref>, consider the CFG 𝒳 whose S-points are triples (C, C', f) where C and C' are families of smooth curves over S of genera g and h, both ≥ 2, and f : C → C' an S-morphism.Then there is a projection p:𝒳→ℳ_g ×ℳ_h, where ℳ_g denotes the CFG of families of smooth curves of genus g.We know that ℳ_g and ℳ_hare stacks in the étale topology by Corollary <ref>, and therefore the product ℳ_g ×ℳ_h is a stack (Lemma <ref>). Applying Corollary <ref> to p, to verify that 𝒳 is a stack it is therefore sufficient to show that, for any fixed pair of smooth curves C and C' over a scheme S, the functor ℳor(C,C')(-):(𝖲/S)^𝗈𝗉→ (𝖲𝖾𝗍) that assigns to an S-scheme T the set of T-morphisms C_T → C'_T, is a sheaf.This is easily verified using the fact that the étale topology is subcanonical (see Example <ref>).§ STACKS ADAPTED TO A PRESITE In this section, we take the most naive approach to defining an algebraic stack on a presite, namely, we define what we call a stack adapted to a presite (Definition <ref>).Thisis simply a stack with a representable coverby an object in the presite (in the sense of Definition <ref>).While there are technical reasons (see Example <ref>)that in total generalitythis is not really the `right' definition, it neverthelessprovides aquick definition, that immediately suffices for many of the standard examples one sees (e.g., algebraic spaces,FantechiDeligne–Mumford stacks, and Laument–Moret-Bailly Deligne–Mumford stacks). In fact, by iteratively enlarging the presite, and then reducing back down to the original presite, one can obtain all of the definitions of algebraic stacks we discuss in this paper (see <ref>). We hope that in this generality,and brevity, the salient points of an algebraic stack will be apparent to readers who prefer to work in categories other than schemes. The reader interested in moving quickly to the definition of the algebraic stack of smooth curves, or the algebraic stack ofHiggs bundles, may prefer to read this section, and then skip directly to <ref>.Technically, the stack of Higgs bundlesisadapted to the smooth presite of algebraic spaces, rather thanto the étale presite of schemes; nevertheless, the main aspects of the formalism should already be apparent to the reader after this section. §.§ Definiton of stacks adapted to a presiteIn order to define a stack adapted to a presite, we want the definition of a representable morphism:[𝖲-representable morphism]Let𝖲 be a subcanonicalpresite that admits fibered products.We saya morphism 𝒳→𝒴 of CFGsis 𝖲-representableiffor every S in 𝖲, the fiber product 𝒳×_𝒴 S is in 𝖲 (i.e., equivalent to a stack 𝖲/S'for some S' in 𝖲).@R=1em𝒳×_𝒴 S[r] [d]S [d] 𝒳[r]𝒴. With this definition, we definean 𝖲-adapted stackby requiring that the stack admitan `𝖲-representable cover' in 𝖲.More precisely:[𝖲-adapted stack]Let𝖲 be a subcanonical presite that admits fibered products. Then an 𝖲-adapted stack (or a stack adapted to the presite 𝖲) is a stack ℳ over 𝖲 admitting an𝖲-representable cover in 𝖲of the following form: there exists a U in 𝖲 and an 𝖲-representablemorphism U@>p>>ℳ suchthat for every S in 𝖲 and every S→ℳ, the morphism U×_ℳS→ S in 𝖲 obtained from base change is a cover in the pretopology on 𝖲.Such a morphismp is called a presentation ofℳ.A morphism of𝖲-adapted stacks isa morphism of stacks. We will define more generally a covering morphism of CFGs in Definition <ref>.The cover defined in Definition <ref> above is a cover in that sense as well (Remark <ref>). [Algebraic spaces and adapted Deligne–Mumfordstacks]If S is a scheme and 𝖲 is the étale presite on schemes over S, then the 𝖲-adapted stacks arecalled Fantechi Deligne–Mumford (F DM) stacks (over S). The 𝖲-adapted stacks that are representable by sheaves are called algebraicspaces. If 𝖲 is the presite of complex analytic spaces with the pretopology induced fromusual open covers, then one obtains a notion of an adapted complex analytic stack. Similarly, if 𝖲 is the presite of topological spaces with the presite induced from usual open covers, then one obtains a notion of an adapted topologicalstack.There are other definitions of complex analytic and topologicalstacks appearing in the literature; we do not pursue the relationship among the definitions.Other notions of algebraic stacks are determined by requiring the diagonal Δ :ℳ→ℳ×_𝖲ℳ to be 𝖲-representable, and putting further geometric conditions on the diagonal. For instance, an Laument–Moret-Bailly Deligne–Mumford (LMB DM) stack over the étale presite𝖲 of schemes over a fixed scheme S isan 𝖲-adapted stack, with 𝖲-representable, separated, and quasicompact diagonal.To see how all the other algebraic stacks discussed in this paper can be defined using stacks adapted to a presite,see <ref>.§.§ Bootstrapping stacks adapted to a presite It can often be useful to show that a stack is 𝖲-adapted usingbootstrap methods; in other words, it will often be the case that one can exhibit a morphism from astack of interest to a well-known 𝖲-adapted stack, so that it is easy to check the 𝖲-adaptedconditionon the fibers.The following propositionstates that in this situation the stack is an 𝖲-adapted stack.Let f:𝒳→𝒴 be a morphism of stacks over a subcanonical presite 𝖲 that admits fibered products.Assume that 𝒴 is an𝖲-adapted stack and that for all S in 𝖲 and all morphisms S→𝒴 we have that 𝒳×_𝒴S is𝖲-adapted.Then 𝒳 is an 𝖲-adaptedstack. Later we will prove Proposition <ref>, whose proof can be readily modifiedto fit this situation.We sketch the argument here. Choose a presentationY →𝒴.Thenset 𝒵 =𝒳×_𝒴 Y. One can check that 𝖲-representable morphisms are stable under base change (see e.g., Lemma <ref>; in that notation, take 𝖢=𝖲). Thus the projection 𝒵→𝒳 is𝖲-representable.One can then check directly from the definition that for every S in 𝖲 and every morphism S→𝒳, the morphism 𝒵×_𝒳S→ S is a cover in 𝖲 (use Y× _𝒴S=Y×_𝒴𝒳×_𝒳S=𝒵×_𝒳S). Furthermore, 𝒵 is 𝖲-adapted, by assumption, so there is a presentation Z →𝒵. One can check that 𝖲-representable morphisms are stable under composition (see e.g., Lemma <ref>; in that notation, take 𝖢=𝖲).Thus the composition Z→𝒵→𝒳 is 𝖲-representable.One can then check directly from the definition that this morphism is a cover in the sense of Definition <ref>, and thus provides the desired presentation for 𝒳.A result similar to Proposition <ref> holds when additional conditions are placed on the diagonal of the stacks (see Corollary<ref>). § ALGEBRAIC STACKSIn the previous section we introduced the notion of a stack adapted to a presite. This provided aquick definition, that suffices in many cases.However, in general, that approach is a little too naive, particularly if one does not enlarge the presite iteratively.In this section, we take a slightly more lengthy approach, which considers stacks that are adapted to larger classes of morphisms in the the presite.After iterating this process, we arrive at the definition of an algebraic stack (adapted to a class of morphisms in the presite); stacks adapted to the presiteare algebraic stacks..1 cmWe now also provide a lengthier motivation to the study of algebraic stacks than we provided in the previous section. Suppose that 𝖲 is a subcanonical presite, meaning that every S ∈𝖲 represents a sheaf, or equivalently that 𝖲/S is a stack on 𝖲.A stack on 𝖲 is fundamentally a topological object, and the category of stacks on 𝖲 is therefore too inclusive a milieu for our geometric purposes.Even when 𝖲 is a category of geometric objects, the stacks (and even the sheaves) on 𝖲 need not behave at all geometrically.To take just one example, a stack on the category of schemes always has a tangent space at a point, but this tangent space may not have the structure of a vector space (see <ref>).Nevertheless, some sheaves and stacks that are not representable do behave geometrically, and our goal will be to identify those that do: algebraic spaces and algebraic stacks. Granting our post hoc reasoning for the definition of algebraic stacks, it might have been more sensible to call them `geometric stacks'.This terminology has caught on in some places <cit.>, but we stick to the traditional nomenclature here.The essential idea in the definition of algebraic stacks is that a stack that resembles geometric objects locally can itself be studied geometrically, provided that the meaning of `local' is sufficiently geometric.In the algebraic category, `locally' is interpreted `over a smooth cover' and the `geometric objects' are taken to be schemes.Here a technicality arises:there are étale sheaves, known as algebraic spaces, that resemble schemes étale-locally, but are not themselves schemes.More worryingly, the class of algebraic stacks modelled locally by algebraic spaces is strictly larger than the class of those modelled locally by schemes.In order to make the whole theory satisfyingly formal, one takes the smallest class of stacks that includes affine schemes and is stable under disjoint unions and groupoid quotients.Thankfully, this turns out to be the same as the class of algebraic stacks with smooth covers by algebraic spaces.This section will be quite formal, and applies to essentially any presite.The main idea is that of a 𝐏-adapted stack in Section <ref> over a presite 𝖲, which is precisely a stack modelled locally, according to a suitable property 𝐏, on morphisms in𝖲.In Section <ref>, we iterate the construction to arrive at the class of algebraic stacks. In fact, most stacks we encounter in practice (e.g., quasiseparated Deligne–Mumford stacks) are adapted to the class of morphisms of schemes (or even to affine schemes), and the iteration in Section <ref> is useful only to have a category with good formal properties.A reader interested in seeing stacks in geometric action may prefer to skip Section <ref>.The key point is an elementary notion of a cover, and to this end, we must first discuss representable morphisms (<ref>).We also presentProposition <ref>, which is a useful tool for bootstrapping from one adapted stack to another, analogous to Corollary <ref> for stacks.§.§ CoversWe explain what it means for a morphism of stacks over a presite 𝖲 to be a cover:[Covering morphism of CFGs] A morphism of CFGs 𝒳→𝒴 on a presite 𝖲 is said to be covering if, for any morphism S →𝒴, there is a cover { S_i → S } such that the induced maps S_i →𝒴 lift to 𝒳: @R=1em S_i[d] @/_10pt/@–>[ld] 𝒳×_𝒴S [d] [r]S[d]𝒳[r] 𝒴. Note that a morphism of objects of 𝖲 is covering according to Definition <ref> if and only if it is covering in the topology associated to the pretopology of 𝖲 (Definition <ref>).The presentation P:U→ℳ in Definition <ref> is covering the sense ofDefinition <ref>. Indeed, choose an S in 𝖲 and a morphism S→ℳ,and consider the fibered product @R=1em U×_ℳ S [r] [d] S [d] U [r]^Pℳ.In Definition <ref>werequire that there is a cover {S_i→ S}so that for each i, the map S_i→ S factors through U×_ℳS.Of course, sinceU×_ℳS→ S is a cover by Definition <ref>,this condition is automatically satisfied taking {S_i→ S}={U×_ℳS→ S}.Coverings are stable under composition, base change, and fiberedproducts of morphisms. We prove stability under composition: Suppose that 𝒳→𝒴→𝒵 is a composition of covering morphism and W is a scheme over 𝒵.Since 𝒴 covers 𝒵,there is a cover { W_i → W } such that the maps W_i →𝒵 lift to 𝒴.Now, since 𝒳 covers 𝒴 there are covers { W_ij→ W_i } such that W_ij→𝒴 lifts to 𝒳.The family { W_ij→ W } is a cover of a cover, hence is covering. We prove stability under base change:Suppose that 𝒳→𝒴 is covering and 𝒵→𝒴 is an arbitrary morphism.Let W be a scheme over Z.Then there is a cover { W_i → W } such that the maps W_i →𝒴 lift to 𝒳, as 𝒳 covers 𝒴.But then by definition of the fiber product, the maps W_i →𝒵 lift to 𝒵×_𝒴𝒳. Stability under fiber product now follows formally from the parts already demonstrated (e.g., the proof of Lemma <ref> can easily be adapted to this purpose).§.§ Representable morphismsA representable morphism of stacks on 𝖲 is roughly one whose fibers are representable by some prescribed class of stacks: [Representable morphism]Let 𝖢 be a class of CFGs over a subcanonical presite 𝖲 such that 𝖢 is stable under isomorphism and fiber product and every CFG over 𝖲 can be covered (Defintion <ref>) by objects of 𝖢.We say that a morphism 𝒳→𝒴 of CFGs is representable by objects of 𝖢 or 𝖢-representable (or representable, when 𝖢 is clear from context) if, for every 𝒵∈𝖢, the fiber product 𝒳×_𝒴𝒵 is in 𝖢.@R=1em𝒳×_𝒴𝒵[r] [d] 𝒵[d] 𝒳[r]𝒴.When 𝖲 is the category of schemes and 𝖢 is also the category of schemes, i.e., the collection of stacks on the étale site of schemes representable by schemes, we call a 𝖢-representable morphism schematic (i.e., we recover Definition <ref>).For obvious reasons, schematic morphisms are sometimes also called representable morphisms, without qualification.However, it is also common to use the term representable morphism for morphisms representable by algebraic spaces, so for clarity we use `schematic' to refer to morphisms representable by schemes.For 𝖢a class of CFGs over a category 𝖲 that is stable under isomorphism and fiber product and such that every CFG over 𝖲 can be covered (Definition <ref>) by objects of 𝖢: * The composition of 𝖢-representable morphisms is 𝖢-representable. * The base change of a 𝖢-representable morphism is 𝖢-representable. * The fibered product of 𝖢-representable morphisms is 𝖢-representable. * If f : 𝒳→𝒴 and g : 𝒴→𝒵 are morphisms such that gf is 𝖢-representable and the diagonal of g is 𝖢-representable then f is 𝖢-representable. The verification is formal, so the proof in <cit.> for schematic morphisms applies here.Variants appear in <cit.>, <cit.>.For completeness, we give a proof. (<ref>) Suppose 𝒳→𝒴 and 𝒴→𝒵 are 𝖢-representable.To see that the composition is 𝖢-representable, considera 𝒲∈𝖢 and a morphism 𝒲→𝒵. Then 𝒲×_𝒵𝒳 = (𝒲×_𝒵𝒴) ×_𝒴𝒳 is 𝖢-representableusing first the 𝖢-representability of 𝒴→𝒵, and then the 𝖢-representability of𝒳→𝒴. (<ref>) If 𝒳→𝒴 is a 𝖢-representable morphism, 𝒲→𝒵→𝒴 are morphisms of CFGs with 𝒲∈𝖢, then 𝒲×_𝒵 (𝒵×_𝒴𝒳) = 𝒲×_𝒴𝒳, hence is in 𝖢.(<ref>) is essentially <cit.>, which observes that the conclusion follows from (1) and (2), together withthe fact that given morphisms f:𝒳→𝒳' and g:𝒴→𝒴' over a stack 𝒵, the fiberedproduct 𝒳×_𝒵𝒴𝒳'×_𝒵𝒴' is given by the composition of morphisms obtained from fibered product diagrams:𝒳×_𝒵𝒴 @>f×_id_𝒵id_𝒴>> 𝒳'×_𝒵𝒴 @>id_𝒳'×_id_𝒵 g>> 𝒳'×_𝒵𝒴'. (<ref>)For any 𝒲→𝒴 with 𝒲 in 𝖢, we have a cartesian diagram@R=1em𝒲×_𝒴𝒳[r] [d] 𝒲×_𝒵𝒳[d] 𝒴[r]^<>(0.5)Δ_g 𝒴×_𝒵𝒴. We know 𝒲×_𝒵𝒳 is in 𝖢 since gf is representable, so 𝒲×_𝒴𝒳 is in 𝖢 by representability of the diagonal of g. [Locality to the target for representable morphisms]Let 𝐏 be a property of morphisms between CFGs in𝖢 that is stable under base change (Definition <ref>).We say 𝐏 is local to the target if, for any morphism 𝒳→ Y and any cover 𝒵→𝒴, the morphism 𝒳×_𝒵𝒴→𝒵 has property 𝐏 if and only if 𝒳→𝒴 does. Now that we have these definitions, we can translateproperties of morphisms in 𝖲 into properties of representable morphisms of stacks:[𝐏-representable morphism]Let 𝖢 be a class of CFGs over a subcanonical presite 𝖲 such that 𝖢 is stable under isomorphism and fiber product and every CFG over 𝖲 can be covered by objects of 𝖢. Let 𝐏 be a property of morphisms between CFGs in𝖢 that is stable under base change (Definition <ref>), stable under composition,and local on the target (Definition <ref>).A 𝖢-representable morphism 𝒳→𝒴 of stacks over𝖲 (Definition <ref>)is said to have property 𝐏 if for every 𝒵 in 𝖢 and every 𝒵→𝒴, the morphism 𝒳×_𝒴𝒵→𝒵 of CFGs between objects of 𝖢 has property 𝐏. We also call this a 𝐏-representable morphism.Here are some examples of classes of morphisms of schemes that are stable under base change and local to the target for the étale topology on schemes: * quasiseparated <cit.>;* quasicompact <cit.>;* flat <cit.>;* smooth <cit.>;* étale <cit.>;* unramified <cit.>;* separated <cit.>;* proper <cit.>;* finite type <cit.>;* locally of finite type <cit.>;* finite presentation <cit.>;* locally of finite presentation <cit.>;* locally of finite type and pure relative dimension d <cit.>;* surjective <cit.> or <cit.>.For more references, see <cit.> or <cit.>.§.§ Locality to the source[Local to the source] Let 𝐏 be a property of morphisms in 𝖲 that is local (on the target) and stable under base change and composition.We call 𝐏 local to the source if a morphism X → Yin 𝖲 has the property 𝐏 if and only if, for any covering family U_i → X in the pretopology 𝖲, all of the composed morphisms U_i → Y have property 𝐏.Let 𝐏 be the class of étale morphisms of schemes.Then 𝐏 is local to the source in the étale topology.Let 𝐏 be the class of local isomorphisms of complex analytic spaces.Then 𝐏 is local to the source in theanalytictopology.Recall that smooth surjections are covering (in the sense of Definition <ref>)in the étale topology (Example <ref>). The class of smooth morphisms is local to the source in the étale topology <cit.>.We will prove a more general version of this statement in Lemma <ref>. The same proof shows that smooth morphisms in the category of complex analytic spaces, and submersions in the category of C^∞-manifolds, are local to the source in the usual topologies. §.§ Adapted stacks Let 𝖢 be a class of CFGs over a subcanonical presite 𝖲 such that 𝖢 is stable under isomorphism and fiber product and every CFG over 𝖲 can be covered by objects of 𝖢.Let 𝐏 be a property of morphisms between CFGs in𝖢 that is stable under base change (Definition <ref>),stable under composition, local to the source(Definition <ref>), and local to the target (Definition <ref>).In this section we will introduce the class of stacks that admit 𝐏-representable covers by objects of 𝖢, calling these (Definition <ref>) stacks 𝐏-adapted (to 𝖢), or just adapted stacks, if the context is clear (in particular, if it will not be confused with a stack adapted to the presite).In the next section, we will show that once one has the class of 𝐏-adapted stacks,the property 𝐏 can always be definedfor morphisms between 𝐏-adapted stacks, which will allow us to iterate this procedure in the next section and arrive at the definition of an algebraic stack (with respect to a class of morphisms 𝐏 in 𝖲).Perhaps remarkably, many of the algebraic stacks we consider in this paper are smooth-adapted to schemes (the class 𝖢 is taken to be 𝖲 and the class of morphisms 𝐏 is taken to be the class of smooth morphisms), so the reader who so desires may safely ignore the question of iteration and proceed after this section to Section <ref>. In fact, for the purposes of this paper, we will only need to iterate once, to obtain the class of algebraic spaces, and smooth morphisms between algebraic spaces, as all of the stacks we will work with are smooth-adapted to algebraic spaces. [Stacks 𝐏-adapted to 𝖢] Let 𝖢⊇𝖲 be a class of CFGs over a subcanonical presite 𝖲 such that 𝖢 is stable under isomorphism and fiber product and every CFG over 𝖲 can be covered by objects of 𝖢.Let 𝐏 be a property of morphisms between CFGs in𝖢 that is stable under base change (Definition <ref>), stable under composition, local to the source(Definition <ref>), and local to the target (Definition <ref>).Then a stack on 𝖲 is 𝐏-adapted to 𝖢 if it admits a 𝐏-representable (Definition <ref>) cover (Definition <ref>) by an object of 𝖢.In other words, ℳ is 𝐏-adapted to 𝖢 if there exists a U in 𝖢 and amorphism p :U ⟶ℳ suchthat for every 𝒵 in 𝖢 and every 𝒵→ℳ, we have that U×_ℳ𝒵 is in 𝖢,and the morphism U×_ℳ𝒵→𝒵 has property𝐏, and is covering. Such a morphismp is called a presentation ofℳ.When 𝐏, 𝖲, and 𝖢 are clear, we abbreviate the terminology to adapted stacks.A morphism of adapted stacks isa morphism of stacks. The presence of both 𝖲 and 𝖢 in the definition is for technical reasons.We will later wish to iterate this construction, and we do not wish to undertake a definition of Grothendieck topologies on 2-categories.Had we given such a definition, we would have simply taken 𝖢 to be a 2-site and defined 𝐏-adapted stacks to 𝖢 without any reference to 𝖲.This would be similar to the approach taken in <ref>; note that in that approach one can avoid 2-cagetories by simply working withpresites on the category of algebraic spaces (rather than with all the stacks that arise from the adaption process to the presite). If 𝖢=𝖲 (as categories)and 𝐏 is the class of coverings in the presite (S,𝒯), then then 𝐏-adapted stacks are the same as the 𝖲-adapted stacks of Definition <ref>. In particular,if S is a scheme and 𝖲 is the étale presite on schemes over S(i.e., (𝖲/S)_et), we take 𝖢=𝖲 and 𝐏 to be the class of étale covers, then the 𝐏-adapted stacks arecalled adapted Deligne–Mumford (F DM) stacks. The 𝐏-adapted stacks that are representable by sheaves are called algebraicspaces (see <ref>). If 𝖲 = 𝖢 is the presite of complex analytic spaces, and 𝐏 is the class of smooth morphisms, with the pretopology induced fromusual open covers, then one obtains a notion of an adapted complex analytic stack. Similarly, if 𝖲 = 𝖢 is the presite of topological spaces with the presite induced from usual open covers, then one obtains a notion of an adapted topologicalstack. There are other definitions of complex analytic and topologicalstacks appearing in the literature; we do not pursue the relationship among the definitions. Suppose that 𝒳→𝒵 and 𝒴→𝒵 are morphisms of 𝐏-adapted stacks.Then 𝒳×_𝒵𝒴 is a 𝐏-adapted stack. Choose 𝐏 covers X_0 →𝒳, Y_0 →𝒴, and Z_0 →𝒵.Then the map X_0 ×_𝒵 Y_0 ×_𝒵 Z_0 →𝒳×_𝒵𝒴×_𝒵𝒵 = 𝒳×_𝒵𝒴is a composition of changes of base of representable 𝐏 covers, hence is a representable 𝐏 cover.§.§ Iterated adaptationIn order to iterate the definition of 𝐏-adapted stacks, we must extend the property 𝐏 to morphisms representable by adapted stacks.The main content of this section is that there is a unique way to do this so that 𝐏 remains stable under base change and composition, and local to the source and target. Note that Lemma <ref> gives a canonical way of extending the definition of smooth morphisms to all morphisms of CFGs over the category of schemes in a way that is still stable under composition and base change and local to source and target.This definition necessarily agrees with Definition <ref>, below, which is valid for an arbitrary class of morphisms that is stable under composition to base change and local to source and target.The reader should feel free to skip to Definition <ref> if he or she is only interested in algebraic stacks over schemes and is willing to rely on Lemma <ref>.[Bootstrapping property 𝐏]Let 𝖢⊇𝖲 be a class of CFGs over a subcanonical presite 𝖲 such that 𝖢 is stable under isomorphism and fiber product and every CFG over 𝖲 can be covered by objects of 𝖢.Let 𝐏 be a property of morphisms in 𝖢 that is stable under base change (Definition <ref>), stable under composition, local to the source(Definition <ref>), and local to the target (Definition <ref>). Suppose that 𝒳 and 𝒴 are 𝐏-adapted stacks, so that there arepresentations X_0 →𝒳 and Y_0 →𝒴; i.e., 𝐏-representable covers from objects in 𝖢.We say that 𝒳→𝒴 is in class 𝐏 if the map X_0 ×_𝒴 Y_0 → Y_0 is in class 𝐏.For brevity in what follows, we will sometimes write 𝒳→𝒴 is 𝐏 if it is in class 𝐏.The definition of the class 𝐏 of morphisms between 𝐏-adapted stacks given above does not depend on the choices of presentationsX_0 →𝒳 and Y_0 →𝒴. Suppose that X'_0 →𝒳 is a different presentation.Then let X”_0 = X'_0 ×_𝒳 X_0.The projection X”_0 → X'_0 is a 𝐏 cover, since 𝐏 covers are stable under base change, and X_0 →𝒳 is a 𝐏 cover.Pulling back to Y_0 over 𝒴, we have adiagram:X”_0 ×_𝒴 Y_0 [r]^f [d]_g X'_0 ×_𝒴 Y_0 [d]^p X_0 ×_𝒴 Y_0 [r]^-q Y_0By assumption q is 𝐏, and g is the base change of a 𝐏 morphism, so it is 𝐏.Therefore qg = pf is 𝐏.But 𝐏 is local to the source, and f is the base change of the 𝐏 cover, X”_0 → X'_0, so p is 𝐏, as required.This proves the independence of the choice of X_0.Now suppose that Y'_0 →𝒴 is another presentation.Again we form Y”_0 = Y'_0 ×_𝒴 Y_0 and note that the two projections are 𝐏 covers, as is the map Y”_0 →𝒴.Now consider the diagram with cartesian squares:X_0 ×_𝒴 Y_0 [r]^-f Y_0 X_0 ×_𝒴 Y”_0 [r]^-h [u] [d]_p Y”_0 [d]^q [u] X_0 ×_𝒴 Y'_0 [r]^-g Y'_0The map f is 𝐏, by assumption, so h is 𝐏 as well.As q is 𝐏, so is qh = gp.But p is a 𝐏 cover (being the base change of q), so it follows that g is 𝐏, as required.The property 𝐏 of morphisms of stacks admitting 𝐏-representable covers is stable under composition, stable under base change, local to the target, and local to the source. Consider a composition 𝒳→𝒴→𝒵 where X_0 → X, Y_0 → Y, and Z_0 → Z are all 𝐏-representable covering maps, where 𝒳→𝒴 is 𝐏.Form the following diagram:@R=20pt@!C=4pc X_0 ×_𝒴 Y_0 ×_𝒵 Z_0 [dr]^(.6)f_(.3)𝐏_(.6)(𝐂𝐨𝐯)[dl]_(.5)𝐏^𝐂𝐨𝐯 X_0 ×_𝒴 Y_0 [dr]^𝐏_(𝐂𝐨𝐯)[dl]^𝐂𝐨𝐯_𝐏 Y_0 ×_𝒵 Z_0 [dl]_(.6)𝐂𝐨𝐯^𝐏[dr]^g X_0 [d]_𝐏^𝐂𝐨𝐯 Y_0 [d]_𝐏^𝐂𝐨𝐯 Z_0 [d]_𝐏^𝐂𝐨𝐯 𝒳[rr]^(𝐂𝐨𝐯)𝒴[rr] 𝒵Morphisms that are 𝐏 or covering without any further assumptions are labelled with a 𝐏 or 𝐂𝐨𝐯.All of these morphisms are 𝖢-representable.If 𝒳→𝒴 is covering then the other arrows labelled (𝐂𝐨𝐯) are covering as well.To prove stability under composition, suppose that 𝒴→𝒵 is 𝐏.Then g is 𝐏, so gf is 𝐏. On the other hand, we have a cartesian diagram:@!C=3pc@R=1pc X_0 ×_𝒴 Y_0 ×_𝒵 Z_0 [rr]_(.65)𝐂𝐨𝐯_(.4)𝐏^(.525)p[d]X_0 ×_𝒵 Z_0 [d] [r]_-𝐂𝐨𝐯^-q Z_0 Y_0 [rr]^(.35)𝐏^(.65)𝐂𝐨𝐯𝒴Since Y_0 is 𝐏 over 𝒴, we deduce that p is 𝐏 and therefore that qp = gf is 𝐏.But p is covering, so by locality to the source, q is 𝐏.This means that 𝒳→𝒵 is 𝐏, by definition.This proves stability under composition.To prove locality to the source, we use the same diagrams.Suppose that 𝒳→𝒵 is 𝐏.Then q is 𝐏, so qp = gf is 𝐏.But then f is covering and 𝐏, so we deduce that g is 𝐏, as required.Now we prove stability under base change.Consider a cartesian diagram𝒳' [r] [d]𝒳[d]𝒴' [r]𝒴where 𝒳 is 𝐏 over 𝒴.Let 𝒳' →𝒳 be the base change.Suppose that X_0 →𝒳 and Y_0 →𝒴 are 𝐏 covers.Then p : X_0 ×_𝒴 Y_0 → Y_0 is 𝐏, by assumption.Changing base to 𝒴', we get 𝐏 covers X'_0 →𝒳 and Y'_0 →𝒴.The map q : X'_0 ×_𝒴' Y'_0 → Y'_0 is the base change of p.But both X_0 ×_𝒴 Y_0 and Y_0 are in 𝖢, by assumption, and 𝐏 is stable under base change for morphisms in 𝖢.Therefore q is 𝐏, from which it follows that 𝒳' →𝒴' is 𝐏, by definition.Finally, we prove locality on the target.Suppose that we have a cartesian diagram (<ref>).Assume that 𝒴' is a cover of 𝒴 and that 𝒳' is 𝐏 over 𝒴.Let X_0 →𝒳 and Y_0 →𝒴 be 𝐏 covers.Assume that 𝒳' is 𝐏 over 𝒴'.Set Y'_0 = Y_0 ×_𝒴𝒴' and X'_0 = X_0 ×_𝒳𝒳'.Then X'_0 ×_𝒴' Y'_0 → Y'_0 is the base change of X_0 ×_𝒴 Y_0.By assumption Y'_0 → Y_0 is a cover, so by locality to the target for morphisms between objects of 𝖢, we deduce that X_0 ×_𝒴 Y_0 is 𝐏 over Y_0.This means that 𝒳 is 𝐏 over 𝒴, by definition. Given a class of morphisms 𝐏' of 𝐏-adapted stacks that extends the class of morphisms 𝐏 to the class of 𝐏-adapted stacks, and which is stable under base change and composition, and local to the source and target,𝐏'=𝐏. We leave the proof to the reader.§.§ Algebraic stacks The lemmas in the previous section imply that the property 𝐏 makes sense for stacks that are 𝐏-adapted to 𝖢.Replacing 𝖢 with the category of stacks on 𝖲 that are 𝐏-adapted to 𝖢, we may iterate the procedure, enlarging the class of stacks under consideration and ensuring that the property 𝐏 makes sense for them at each step.Taking the union of all of these classes, we arrive at the class of algebraic stacks:[Algebraic stack]Let 𝐏 be a property of morphisms of 𝖲 that isstable under base change, stable under composition, local to the target, and local to the source.An algebraic stack (with respect to the property 𝐏) is a member of the smallest class 𝖢 of stacks on 𝖲such that 𝖢⊇𝖲, 𝖢 hasproperty 𝐏 defined on it,𝐏 is stable under base change and composition, and local to the source and target,and the class of stacks on 𝖲 that are 𝐏-adapted to 𝖢(Definition <ref>) is 𝖢. [Deligne–Mumford stacks and algebraic spaces] A stack on the category of schemes is called an algebraic stack if it is algebraic with respect to the étale topology and the class of smooth maps.It is called a Deligne–Mumford stack if it is algebraic with respect to the étale topology and the class of étale maps.It is called a algebraic space if it is an algebraic stack and it is representable by a sheaf.One can also use the usual topology on complex analytic spaces and take 𝐏 again to be smooth morphisms.Much of what we do here in the algebraic category works analytically as well, but we will not address the analytic category directly in what follows. Likewise, there is a class of `algebraic stacks' on the category of manifolds, obtained by taking the usual topology on the category of manifolds and taking 𝐏 to be the class of submersions.§.§ Fiber products and bootstrapping Fiber products of algebraic stacks are algebraic. This is immediate by iteration of Lemma <ref>. It can often be useful to show that a stack is 𝐏-adapted to 𝖢 usingbootstrap methods; in other words, it will often be the case that one can exhibit a morphism from astack of interest to a well-known adapted stack, so that it is easy to check the 𝐏-adaptedconditionon the fibers.The following propositionstates that in this situation the stack is a 𝐏-adapted stack.We make this precise in the following remark and proposition.The class of 𝐏-adapted stacks is stable under isomorphism and fibered products (Lemma <ref>).Therefore, from Definition <ref>, we have the notion of a morphism of stacks representable by adapted stacks. Let f:𝒳→𝒴 be a morphism of stacks over a subcanonical presite 𝖲.Assume that 𝒴 is a 𝐏-adapted stack and that f isrepresentable by 𝐏-adapted stacks.Then 𝒳 is a 𝐏-adaptedstack. Choose a 𝐏-representable cover Y_0 →𝒴, with Y_0 ∈𝖢.Then then set 𝒵 = Y_0 ×_𝒴𝒳.The projection 𝒵→𝒳 is a 𝐏-representable cover.Furthermore, 𝒵 is 𝐏-adapted, by assumption, so there is a 𝐏-representable cover X_0 →𝒵, with X_0 ∈𝖢.Then X_0 →𝒳 is a composition of 𝐏-representable covers, hence is a 𝐏-representable cover, as required. Suppose that 𝒴 is an algebraic stack and f : 𝒳→𝒴 is a morphism of stacks that is representable by algebraic stacks.Then 𝒳 is an algebraic stack. This is immediate by iteration ofProposition <ref>.§ MODULI STACKS OF HIGGS BUNDLESIn this section we construct the moduli stack of Higgs bundles on a smooth complex projective curve.We also construct several related moduli stacks of interest. In what follows, the reader should always feel free to assume that the morphism π:X→ S is a projective morphism between schemes, and that all sheaves are coherent (or even vector bundles).For instance, one case of special interest that will always satisfy the given hypotheseswill be the case where S=Specℂ, X isa smooth complex projective curve (or compact Riemann surface), and thesheaves are takento be the sheaves of sections of holomorphic vector bundles on X.In order not to avoid repetition below when defining various categories fibered in groupoids, we fix the following notation. Suppose that f : T → S is an S-scheme.If ξ is an object of some CFG on 𝖲/S, we denote by ξ_T a pullback of ξ to T.For example, if X is an S-scheme, we denote by X_T a T-scheme making the the following diagram cartesian:X_T [r] [d] X [d] T [r]^f S.If L is a line bundle on X, we denote by L_T = f^∗ L a pullback of that line bundle to X_T.Note that we have somewhat abusively used f^*L to denote the pullback of L via the morphism X_T → X induced from f.We shall make this abuse repeatedly in what follows.When T is denoted by decorating S, we abbreviate our notation further and denote X_T by decorating X the same way as S; i.e., if T = S' then we allow ourselves to write X' instead of X_S'. §.§ The moduli space of curvesWe will show that the stack of smooth curves is algebraicusing the representability of the Hilbert scheme.We give another proof in Section <ref>, using Artin's criteria (see Theorem <ref>).The following is a useful criterion to check properties of the diagonal of an algebraicmoduli stack of schemes. [<cit.>]Let S be a noetherian scheme, let X→ S be a flat, projective scheme over S, and let Y→ S be a quasi-projective scheme over S.Then the functor ℐ𝑠𝑜𝑚_𝖲/S(X,Y) on schemes over S is representable by a scheme. In fact ℐ𝑠𝑜𝑚_𝖲/S(X,Y) is representableby an open subscheme of the Hilbert schemeHilb_X×_SY/S.A special case of a theorem of Olsson(<cit.>) allows one to drop the noetherian and projective hypotheses, in exchange for requiring finite presentation and settling for an algebraic space rather than scheme. Let S be ascheme, let X be a flat, proper scheme of finite presentation over S, and let Y beaseparated scheme of finite presentation over S.Then the functor ℐ𝑠𝑜𝑚_𝖲/S(X,Y) is representable by an algebraic space over S.[<cit.>] For g≥ 2, the stackℳ_g→ (𝖲/ℂ)_𝖾𝗍 is a Deligne–Mumford stack. The previous theorem asserts that Δ:ℳ_g→ℳ_g×_ℂℳ_g is schematic.More detailed analysis of the isomorphisms of algebraic curves establishes that the the diagonal is unramified.Since we are in characterstic 0, this essentially follows from the fact thatthe automorphism groupof a smooth curve of genus g≥ 2is finite, and group schemes in characteristic 0 are smooth.Now let H_g be the open subset of the Hilbert scheme parameterizing smooth, ν-canonically embedded curves for ν≫ 0. The universal family C_g→ H_g determines a morphism P:H_g→ℳ_g, that is schematic by virtue of the fact that the diagonal is schematic (see Lemma <ref>). One can check that P is smooth, and therefore provides a presentation of the stack. Therefore ℳ_g is DM, since the diagonal is unramified (see Lemma <ref>).§.§ The 𝖰𝗎𝗈𝗍 stack and the Quot schemeLet π:X→ S be a propermorphism of finite presentation between schemes. Let E be a quasicoherent sheaf of finite presentation onX.We define a category fibered in groupoids 𝖰𝗎𝗈𝗍_E/X/S→𝖲/S as follows. For each S-schemef:S'→ S, we set the objects of 𝖰𝗎𝗈𝗍_E/X/S(S') to be surjections E'@>q'>> F'@>>> 0where, as was introduced at the beginning of Section <ref>, E' denotes the pullback of E to X' = X×_S S'.The sheaf F' is required to be an S'-flat quasicoherent sheaf of finite presentation on X'.We will denote such an object by the pair (F',q').Given an S-morphismg:S”→ S'and object (F”,q”) ∈𝖰𝗎𝗈𝗍_E/X/S(S”), a morphism over g from (F”,q”) to (F',q') is a commutative diagram of quasicoherent sheaves on X”:@R=15pt E”[r]^q”@=[d] F”[r] [d]^<>(0.5)≀0 g^∗ E' [r]^g^∗ q'g^∗ F' [r] 0(The equality on the left indicates the canonical isomorphism induced by pullback.) As quotients of a coherent sheaf cannot have any nontrivial automorphisms respecting the quotient map, the 𝖰𝗎𝗈𝗍_E/X/S CFG is equivalent to the CFG associated to the presheaf 𝒬uot_E/X/S:(𝖲/S)^𝗈𝗉→ (𝖲𝖾𝗍)wherein 𝒬uot_E/X/S(T) is defined to be the set of isomorphism classes of objects of 𝖰𝗎𝗈𝗍_E/X/S(T).Of course, 𝖰𝗎𝗈𝗍_E/X/S is not literally equal to 𝒬uot_E/X/S, since one is a CFG and the other is a presheaf.By dualizing, one can check that theGrassmannian 𝒢(r,n) is a substackof𝖰𝗎𝗈𝗍_𝒪_Specℂ^⊕ n/Specℂ/Specℂ. The following is a special case of a result of Lieblich: [<cit.>, <cit.>]Let π:X→ S be a propermorphism of finite presentation between schemes.The CFG 𝖰𝗎𝗈𝗍_E/X/S is an algebraic space that is locally of finite presentation over S. More can be said for projective morphisms π:X→ S. Let L be a relatively very ample line bundle on X/S.If S'=Speck for a field k, and F' is a coherent sheaf on X_S', then the Hilbert function of F' with respect to L' is defined to be Φ(m) : = χ(X',F'(m))=∑_i=0^ X(-1)^i_k H^i(X',F'⊗ (L')^⊗ m).This is in fact a polynomial in ℚ[m] (see <cit.> for more details).There is a decomposition 𝖰𝗎𝗈𝗍_E/X/S =∐_Φ∈ℚ[m]𝖰𝗎𝗈𝗍^Φ,L_E/X/Swhere 𝖰𝗎𝗈𝗍_E/X/S^Φ,L(S') is the set of equivalence classes ⟨ F',q' ⟩ over S' such that for each s'∈ S', the Hilbert polynomial of F'_t with respect to L'_t is equal to Φ.Crucially, cohomology and base change implies that the inclusions of the subfunctors 𝖰𝗎𝗈𝗍^Φ,L_E/X/S⊆𝖰𝗎𝗈𝗍_E/X/S are representable by open and closed subfunctors of 𝖰𝗎𝗈𝗍_E/X/S.This reduces the study of 𝖰𝗎𝗈𝗍_E/X/S to that of the subfunctors 𝖰𝗎𝗈𝗍_E/X/S^Φ,L.The main theorem is due to Grothendieck <cit.> (see also <cit.>, and <cit.>). [<cit.>]Let S be anoetherian scheme, π:X→ S a projective morphism, and L a relatively very ample line bundle on X/S.Then for any coherent sheaf E and any polynomial Φ∈ℚ[m], the CFG 𝖰𝗎𝗈𝗍_E/X/S^Φ,L is representable by a projective S-scheme Quot_E/X/S^Φ,L. By dualizing one can checkthat the Grassman CFG 𝒢(r,n) is isomorphic to the CFG 𝖰𝗎𝗈𝗍_𝒪_Specℂ^⊕ n/Specℂ/Specℂ^r,𝒪_Specℂ, andthe Grassmannian G(r,n) is isomorphic to the Quot schemeQuot_𝒪_Specℂ^⊕ n/Specℂ/Specℂ^r,𝒪_Specℂ.Of course, one cannot use this observation to construct the Grassmannian, as Grothendieck's proof of Theorem <ref> relies on the representability of the Grassmannian by a projective scheme. §.§ Stacks of quasicoherent sheaves Let S be a scheme. Let π:X→ Sbe a proper morphism offinite presentation between schemes. We define a category fibered in groupoids𝖰𝖢𝗈𝗁_X/S→𝖲/Sin the following way.For an S-scheme f:S'→ S in 𝖲/S, we take 𝖰𝖢𝗈𝗁_X/S(S') to consist (in the notation introduced at the beginning of Section <ref>) of the S'-flat quasicoherentsheaves <cit.>on X_S'. Morphisms in 𝖰𝖢𝗈𝗁_X/S are defined by pullback; i.e., if g:S”→ S' is a morphism, F” is an object of 𝖰𝖢𝗈𝗁_X/S(S”) and F' is an object of 𝖰𝖢𝗈𝗁_X/S(S'), we define a morhpism F”→ F' to be an isomorphism F”→ g^*F'. Following <cit.>, we use 𝖢𝗈𝗁_X/S to denote the substack of 𝖰𝖢𝗈𝗁_X/S consisting of quasicoherent sheaves of finite presentation<cit.>(when the structure sheaf is coherent, finitely presented sheaves coincide with coherent sheaves <cit.>).We similarly define 𝖥𝗂𝖻_X/S and 𝖥𝗂𝖻(r)_X/S by restricting to locally free sheaves of finite rank and locally free sheaves of rank r, respectively.The first statement we want is descent for quasicoherent sheaves.This is <cit.>.See also <cit.>, and <cit.>.The sameargument shows thatthestatements hold for coherent sheaves, and for vector bundles.[<cit.>]Let S be a scheme, and let X be a scheme over S.The CFG 𝖰𝖢𝗈𝗁_X/S→𝖲/S is a stack with respect to the fpqc topology.The same is true for𝖢𝗈𝗁_X/S, 𝖥𝗂𝖻_X/S and 𝖥𝗂𝖻(r)_X/S. Moreover, these are algebraic stacks.The following isa special case of a result of Lieblich:[<cit.>]Let π:X→ Sbe a proper morphism offinite presentation between schemes, with S an excellent scheme (see e.g., <cit.>; for instance S can be a scheme of finite type over a field). The stack 𝖢𝗈𝗁_X/S is an algebraicstack, locally of finite presentation over S.The same is true for 𝖥𝗂𝖻_X/S and 𝖥𝗂𝖻(r)_X/S. For a projective morphism of noetherian schemes, the Quot scheme can be used to establish the presentation of the Artin stack (see Remark <ref>). For our purposes, the benefit of this perspective will be in comparing the moduli stack of Higgs bundles to the moduli scheme of semistable Higgs bundles constructed via GIT. [<cit.>] Let S be a noetherian scheme, and let π:X→ S be a projective morphism.Assume that π_*𝒪_X=𝒪_S universally (i.e., π'_*𝒪_X'=𝒪_S' for all S'→ S). Then the S-stack 𝖢𝗈𝗁_X/S is an algebraic S-stack locallyof finite type. The same is true for 𝖥𝗂𝖻_X/S and 𝖥𝗂𝖻(r)_X/S. The hypothesis that π_∗𝒪_X = 𝒪_S universally is satisfied whenever X is projective and flat over S with reduced, connected geometric fibers <cit.>.For each coherent sheaf E on X,a universal quotient over the schemeQuot_E/X/S induces a morphism Quot_E/X/S→𝖢𝗈𝗁_X/S.Taking appropriate open subsets of these Quot schemes gives a presentation of the algebraicstack (see <cit.> for more details).Suppose that S=Speck for an algebraically closed field k, and that X isa smooth projective curve.We can define the CFG 𝖥𝗂𝖻_X/S(r,d) by restricting to vector bundles of degree d.All of the statements above hold in this setting; i.e., the CFG 𝖥𝗂𝖻_X/S(r,d) of vector bundles of rank r and degree d on X is an algebraic stack, of finite type over k, that admits a presentation from open subsets of Quot schemes.§.§ The stack of Higgs bundles over a smooth projective curveLet X be a smooth, projective curve over ℂ, of genus g. Fix r≥ 1 and d∈ℤ.A Higgs bundle on X of rank r and degree d consists of a pair(E,ϕ)whereE is a vector bundle (locally free sheaf of finite rank) on X with rankE=r and E=d, andϕ∈Hom_𝒪_X(E,E⊗ K_X),where K_X=Ω^1_X=ω_X is the canonical bundle on X. The aim of this section is to constructanalgebraic stack ℋ_X/Specℂ(r,d) of Higgs bundles on X of rank r and degree d,over the étale site 𝖲/Specℂ. §.§.§ Higgs bundles as a category fibered in groupoidsWe begin by defining the CFG underlying the stackℋ_X/Specℂ(r,d). Given a ℂ-scheme f:S'→Specℂ,we start by defining ℋ_X/Specℂ(r,d)(S') to have objectsconsistingofpairs(E',ϕ')where E' ∈𝖥𝗂𝖻_X/Specℂ(r,d)(S') is a relativevector bundle of rank r and degree d on X'/S',and ϕ' ∈Hom_𝒪_X'(E', E'⊗ f^*K_X). Given a ℂ-morphismg:S”→ S', and (E',ϕ') in ℋ_X/Specℂ(r,d)(S'),we obtain a pulled-back family ( E”,ϕ”):=g^*( E',ϕ')in ℋ_X/Specℂ(r,d)(S”),where (in the notation from the beginning of Section <ref>) E”= g^* E' andϕ”= g^*ϕ'. We now define the morphisms of ℋ_X/Specℂ(r,d) in the following way.Given ( E”,ϕ”) in ℋ_X/Specℂ( F)(S”), and ( E',ϕ') in ℋ_X/Specℂ( F)(S'), then a morphism ( E”,ϕ”)→ ( E',ϕ') over g:S”→ S' consists of an isomorphism α : E”→ g^∗ E'such that the following diagram commutes:@R=15pt E”[r]^-ϕ”[dd]_αE”⊗ (fg)^∗ K_X [d]^α⊗id g^∗ E' ⊗ (fg)^∗ K_X @=[d] g^∗ E' [r]^-g^∗ϕ'g^∗(E' ⊗ f^∗ K_X). We now have a category fibered in groupoids:ℋ_X/Specℂ(r,d)→𝖲/Specℂ. §.§.§ Higgs bundles as an algebraic stackThe key point is the following well-known lemma: Let π:X→ S be a flat, proper morphism of finite presentation between schemes. Let G be an S-flat quasicoherent sheaf of 𝒪_X-modules that is of finite presentation.There exists a quasicoherent 𝒪_S-module M of finite presentation, such that the linear schemeV=0pt[0pt][-.5pt]Spec_S(Sym_𝒪_S^∙ M)defined by Mrepresents the functor(S'S) ↦Γ(X',f^*G),(see the beginning of Section <ref> for notation). The formation of V commutes with all base changes S'→ S. This functor willbe denoted by ℋom_X/S(𝒪_X,G). Lemma <ref> isessentially contained in <cit.>.This can also be found in <cit.>, <cit.>, and detailed proofs are given in <cit.> and <cit.>.The forgetful morphism of CFGsℋ_X/Specℂ(r,d)→𝖥𝗂𝖻_X/Specℂ(r,d)is schematic.An S-morphism S'→𝖥𝗂𝖻_X/Specℂ(r,d)corresponds to a vector bundle E' over X'.Using the construction of the fibered product ℋ_X/Specℂ(r,d)×_𝖥𝗂𝖻_X/S(r,d) S' in Definition<ref>, one can establish that this is equivalent to thefunctor on 𝖲/S'ℋom_X'/S'(𝒪_X',E'^∨⊗ E'⊗ f^*K_X). Thisis representable by a scheme, by virtue of the lemma above.The CFGℋ_X/Specℂ(r,d)→𝖲/Specℂ is a stack. We have seen in Theorem <ref> that 𝖥𝗂𝖻_X/Specℂ(r,d) is a stack.Therefore, the corollary follows from Corollary <ref> and Corollary <ref>. The CFG of Higgs bundlesℋ_X/Specℂ(r,d)→𝖲/Specℂ is an algebraic stack, locally of finite type over Specℂ. We have seen in Corollary <ref>,that ℋ_X/Specℂ(r,d) is a stack.Under the hypotheses here, it follows from Theorem <ref> (see also Remark <ref>) that 𝖥𝗂𝖻_X/Specℂ(r,d) is an algebraic stack locally of finite type over ℂ.Thus fromCorollary <ref> and Corollary <ref> we have that ℋ_X/Specℂ(r,d) is an algebraic stack, locallyof finite type over ℂ.§.§ Meromorphic Higgs bundles, and the stack of sheavesand endomorphisms There is alsointerest in considering so-called meromorphic Higgs bundles on a smooth projective curve X; that is pairs (E,ϕ) where E is a vector bundle and ϕ:E→ E⊗ K_X(D) is a morphism of sheaves, for some fixed divisor D on X. The construction of such a stack can be made in essentially the same way as for Higgs bundles.Since it is not much more work, we provide here a more general construction. In this setting, the input data are the following: * π:X→ Sa proper morphism of finite presentation between schemes, such that either* S is excellent, or,* S is noetherian, π is projective, and π_*𝒪_X=𝒪_S universally. * Fan S-flatquasicoherentsheaf of 𝒪_X-modules of finite presentation.The output from this data will be analgebraic stack ℰ_X/S( F) of endomorphisms of coherent sheaves with values in F,over the site (𝖲/S)_et. Taking X/S to be a smooth complex projective curve over S=Specℂ, F=K_X(D), and restricting to vector bundles, one obtains the stack of meromorphic Higgs bundles associated to the divisor D. While the construction above generalizes the construction of Higgs bundles for curves, it seems that for many applications thesestacks arenot the correct generalizations. First in higher dimension (even in the smooth case over S=Specℂ andwith F=Ω_X^1), one should at least include the integrability condition ϕ∧ϕ =0 (in End(E)⊗Ω_X^2). Second,in any dimension, it is oftenbetter to putderived structures on X and S so that the relative cotangent complex becomes perfect, and then consider the moduli problem on the derived scheme X.The result will be a derived stack (a notion which we will not introduce here).One case where the generalized construction we take here does suffice is for families of stable curves, where one can consider endomorphismswith values in the relative dualizing sheaf ω_X/S. This will be discussed further below.§.§.§ The CFG ℰ_X/S( F)We begin by defining the category fibered in groupoids underlying the stackℰ_X/S( F). Given a morphism f:S'→ S,we start by defining ℰ_X/S( F)(S') to have objectsthosepairs(E',ϕ')whereE' is an S'-flat coherent sheafon X' and ϕ' ∈Hom_𝒪_X'(E', E'⊗ F'), where F'= f^*F and X' = X ×_S S' (as always, we use the notation introduced at the beginning of Section <ref>).Given an S-morphismg:S”→ S', we againdenote the composition f∘ g=f', and willuse the notation introduced at the beginning of Section <ref>. Given(E',ϕ') in ℰ_X/S( F)(S'),we obtain a pulled-back family g^*( E',ϕ') = (g^∗ E', g^∗ϕ')in ℰ_X/S( F)(S”), We now define morphisms in the following way.Given ( E”,ϕ”) in ℰ_X/S( F)(S”), and ( E',ϕ') in ℰ_X/S( F)(S'), a morphism ( E”,ϕ”)→ ( E',ϕ') over g:S”→ S' consists of an isomorphism α : E”→ g^∗ E'such that the following diagram commutes:@R=15pt E”[r]^-ϕ”[dd]_αE”⊗ F”[d]^α⊗id g^∗ E' ⊗ F”@=[d] g^∗ E' [r]^-g^∗ϕ'g^∗(E' ⊗ f^∗ F') Now we have the category fibered in groupoidsℰ_X/S( F)→𝖲/S. §.§.§ ℰ_X/S( F) is a stack In this setting, we replace Lemma <ref> with a special case of a result due to Lieblich.Let X → Sbe a proper morphism of finite presentation between schemes.Let E and G be finitely presented, quasicoherent sheaves on X such that G is S-flat.We define a CFGℋom_X/S(E , G ) →𝖲/Sby assigningto each morphism f:S'→ Sthe set of homomorphismsHom_𝒪_X'(E_S' , G_S' ).Morphisms are defined by pullback.[Leiblich <cit.>]Let X → Sbe a proper morphism of finite presentation between schemes.Let E and G be finitely presented, quasicoherent sheaves on X such that G is S-flat. The CFG ℋom_X/S(E , G )→𝖲/S is representable by an algebraic space over S, locally of finite type. The forgetful morphism of CFGsℰ_X/S(F)→𝖢𝗈𝗁_X/Sis representable by algebraic spaces,locally of finite type over S.An S-morphism S'→𝖢𝗈𝗁_X/Scorresponds to E' in 𝖢𝗈𝗁_X/S(S').Therefore the fibered product ℰ_X/S(F)×_𝖢𝗈𝗁_X/S S' is the CFGℋom_X/S(E' , E'⊗ F'), which is representable by virtue of Lieblich's result above.The CFGℰ_X/S(F) is a stack. We have seen in Theorem <ref> that 𝖢𝗈𝗁_X/S is a stack.Therefore, the corollary follows from Corollary <ref> and Corollary<ref>.§.§.§ ℰ_X/S( F) is an algebraicstackℰ_X/S(F) is an algebraic stack,locally of finite presentation over S. We have seen in Corollary <ref>,that ℰ_X/S(F) is a stack.Under the hypotheses here, it follows from Theorem <ref> that 𝖢𝗈𝗁_X/S is an algebraic stack,locally of finite presentation over S.Thus fromCorollary <ref> and Corollary <ref> we have that ℰ_X/S(F) is an algebraic stack, locally of finite presentationover S.§.§ The stack of Higgs bundles over the moduli of stable curvesWe now construct a moduli stack of Higgs bundles over the moduli stack of stable curves:ℋ𝒮h_ℳ_g→ℳ_g.We start by fixing a base S in 𝖲 (for instance S=Specℂ, or even S=Specℤ).[Stable curve]A stable curve of genus g≥ 2 over an algebraically closed field k is a connected curve of arithmetic genus g,with no singularities apart from nodes, and with finite automorphism group (every component isomorphic to ℙ^1_k meets the rest of the curve in at least 3 points). For an S-scheme S'→ S,a relative stablecurve of genus g over S'is a surjective morphism of schemesπ:X'→ S'that is flat, proper and whose every geometric fiber is a stable curve of genus g. We define the CFG ℳ_g over 𝖲/S.For every S-scheme S'→ S we take the objects of ℳ_g(S') to be the relative stable curves over S'.The morphisms in ℳ_g are given by pullback diagrams, exactly as in the definition of ℳ_g. Using the factthat for every relative stable curve of genus g≥ 2 the relative dualizing sheaf ω_X/S is relatively ample, one can showthat ℳ_g is an algebraic stack (in fact DM)exactly as was done for ℳ_g.In fact ℳ_g is an open substack of ℳ_g, since smoothness is an open condition.Now we define the CFG 𝖢𝗈𝗁_ℳ_g/S.Over an S-scheme S'→ S, the objects of 𝖢𝗈𝗁_ℳ_g/S(S') are pairs (X'/S',E') where X'→ S' is a relative stable curve, and E' is an S'-flat finitely, presented quasicoherent sheaf on X'.Morphism are defined by pullback in both entries.There is a natural morphism of CFGs 𝖢𝗈𝗁_ℳ_g/S→ℳ_ggiven by forgetting the sheaf. For every S-scheme S'→ S, and every morphism S'→ℳ_g induced by a relative stable curve X'→ S', the fibered product 𝖢𝗈𝗁_ℳ_g/S×_ℳ_gS' is equivalent to 𝖢𝗈𝗁_X'/S'.Since this is aalgebraic stack, we have immediately from Corollary <ref> and Corollary <ref> that 𝖢𝗈𝗁_ℳ_g/S is aalgebraic stack (this is also a special case of <cit.>). Similarly, we have stacks 𝖢𝗈𝗁_ℳ_g/S(r,d) (resp. 𝖥𝗂𝖻_ℳ_g/S(r,d)) where we restrict to sheaves (resp. bundles) of relative rank r and degree d. Recall that the rank and degree for sheaves on a reducible curve are defined via the Hilbert polynomial with respect tothe relative dualizing sheaf.Finally we define the CFG ℋ𝒮h_ℳ_g/S. Over an S-scheme S'→ S, let the objects of ℋ𝒮h_ℳ_g/S(S') be triples (X'/S',E',ϕ') where X'→ S' is a relative stable curve, and E' is an S'-flat finitely, presented quasicoherent sheaf on X', and ϕ'∈Hom_𝒪_X'(E',E'⊗ω_X/S). Morphism are defined by pullback in the first two entries, and in the same way as in the definition of Higgs bundles in the last entry.There is a natural morphism of CFGs ℋ𝒮h_ℳ_g/S→𝖢𝗈𝗁_ℳ_g/Sgiven by forgetting ϕ'. For every S-scheme S'→ S, and every morphism S'→𝖢𝗈𝗁_ℳ_g/S induced by a pair (X'/S',E'), the fibered product ℋ𝒮h_ℳ_g/S×_𝖢𝗈𝗁_ℳ_g/SS' is equivalent to ℰ_X'/S'(ω_X'/S').Since this is an algebraic stack, we have immediately from Corollary <ref> and Corollary<ref> that ℋ𝒮h_ℳ_g/S is an algebraic stack.Similarly, we have stacks ℋ𝒮h_ℳ_g/S(r,d) (resp. ℋ_ℳ_g/S(r,d)) where we restrict to sheaves (resp. bundles) of relative rank r and degree d. In the discussion above we omitted some noetherian hypotheses on the test S-schemes S'. This is possible via the standard noetherian reduction arguments of <cit.>.Similar arguments are made in <ref>, and we direct the reader there for more details on these types of arguments.§.§ Semi-stable Higgs bundles and the quotient stack In <cit.> Nitsure constructs a moduli scheme of semi-stable Higgs bundles on a smooth projective curve taking values in a line bundle L.The construction of the moduli stack of Higgs bundles on an algebraic curve above essentially follows Nitsure's construction in the setting of stacks.Here we introduce Nitsure's space, and compare it to the moduli stack. Nitsure's construction uses geometric invariant theory, so our stack-oriented perspective will rely on quotient stacks (Section <ref>).§.§.§ Nitsure's construction Recall that the slope of a bundle E on a smooth curve X is defined to be μ(E):= (E)/rank(E).We will say that (E',ϕ') is a sub-Higgs bundle of (E,ϕ) if E'ι↪ E and there is a commutative diagramE@->[r]^-ϕE⊗ LE' @->[r]^-ϕ'@^(->[u]^ιE'⊗ L @^(->[u]_ι⊗id_L.A Higgs bundle (E,ϕ) is said to be slope stable (resp. semi-stable) iffor every sub-Higgs bundle (E',ϕ')⊆ (E,ϕ), one has μ(E')< μ(E) (resp. μ(E') ≤μ(E)).Fix positive integers r and d.By <cit.>, semistability is an open condition on a flat family of coherent sheaves.Therefore there is an open substack ℋ_X/Specℂ^ss(L)(r,d) ⊆ℋ_X/SpecC(r,d) of semi-stable Higgs bundles over X of rank r and degree d, taking values in L.Here we aim to relate this stack to a quasi-projective variety constructed by Nitsure. We review Nitsure's construction briefly. Let 𝒪_X(1) be an ample line bundle on X.Let N∈ℤ be the minimal positive integer such that d/r+ N𝒪_X(1)>max{2g-1,2g-1+(r-1)^2/r L}.Set p=(d+rN𝒪_X(1))+r(g-1).Let Q be the component of the Quot scheme containingquotients 𝒪_X(-N)^⊕ p→ E→ 0 where E is a rank r, degree d vector bundle on X.Let Q^∘ be the locus in Q where E is locally free, the quotient 𝒪_X^p→ E(N)→ 0 obtained by twisting by 𝒪_X(N) induces an isomorphism on the space of global sections, and H^1(X,E(N))=0.It is known that Q is projective, and that Q^∘ is reduced and open in Q(see <cit.>). Let ℰ be the universalbundle on X× Q^∘ obtained from the universal quotient.Lemma <ref> implies there is a linear schemeF→ Q^∘ parameterizing pairs (𝒪_X(-N)^⊕ p→ E→ 0,ϕ), where ϕ:E→ E⊗ L. <cit.> implies that every slope semi-stable Higgs bundle of rank r and degree d (together with a presentation as a quotient) shows up in this family. Let F^ss be the locus of pairs where the bundle and the endomorphism form a slope semi-stable Higgs bundle.The universal family over F induces a morphismF^ss→ℋ^ss_X/Specℂ(L)(r,d).It is essentially the content of <cit.> that this morphism is smooth. Moreover, the obvious action of ℙGL_p on Q given by changing coordinates for the choice of generators of thebundle lifts to an action of ℙGL_p on F <cit.>. The closed orbits in F^ss are given by S-equivalence classes of slope semi-stable Higgs bundles <cit.>. Nitsure constructs a quotient of F in the following way.There is a quasi-projective variety H equipped with a ℙGL_p-linearizedample line bundle ℒ, and aℙGL_p-equivariantmorphism τ̂:F→ H such thatslope (semi-)stability on F corresponds to GIT (semi-)stability on H<cit.>. Via τ̂, the quasiprojective GIT quotient H//_ℒℙGL_p induces a quasiprojective scheme structure on the set F^ss/ℙGL_p of S-equivalence classes of slope semi-stable Higgs bundles <cit.>, whichwe will denote by H_X/Specℂ^ss(L)(r,d).(We expect one can also obtain the quotient directly via GIT on F with respect to the semiample line bundle obtained from ℒ by pullback via τ̂.) This is the moduli scheme of Higgs bundles constructed by Nitsure.It is shown in <cit.> that H_X/Specℂ^ss(L)(r,d) is a good quotient for F^ss by the action of ℙGL_p, and is a categorical moduli schemefor the associated moduli functor ℋ^ss_X/Specℂ(L)(r,d) of Higgs bundles (defined in the obvious way). Consequently there is a diagram@C=2em@R=.5em ℋ^ss_X/Specℂ(L)(r,d)[r] ℋ^ss_X/Specℂ(L)(r,d) [dd]F^ss[ru] [rd] [F^ss/ℙGL_p] [r] H_X/Specℂ^ss(L)(r,d).The compositionℋ^ss_X/Specℂ(L)(r,d)→ H^ss_X/Specℂ(L)(r,d)is a categorical moduli scheme for the stack ℋ^ss_X/Specℂ(L)(r,d), as well, in the sense that it is initial among all morphisms to schemes. §.§ The stack of principal G-Higgs bundles The focus of our presentation has been on Higgs vector bundles.For completeness we include abrief section of principal Higgs bundles on smooth complex projective curves. We direct the reader to<cit.> for more on the topic.We will give a deformation-theoretic perspective on G-Higgs bundles in Section <ref>. Let X be a smooth complex projective curve, and letG be a complex semisimple Lie group.Let 𝔤 be the complex Lie algebra associated to G, and let Ad:G→Aut(𝔤)be the adjoint representation of G. Given an algebraicprincipal G-bundle P on X, the adjoint bundle of P is the associated algebraic vector bundleadP:=P× _G𝔤:=(P×𝔤)/Gwhere the action of G is the product action via the natural action of G on P and the adjoint action of G on 𝔤.[G-Higgs bundle]Let X be a smooth complex projective curve, and letG be a complex semisimple Lie group.A G-Higgs bundle on X is a pair (P,Φ) where P is a principal G-bundle over Xand Φ∈ H^0(X,adP⊗ K_X).If G↪GL_n, then the data of a G-Higgs bundle (P,Φ) gives rise to a Higgs bundle (E,ϕ), where E := P ×_Gℂ^n, and ϕ:E→ E⊗ K_C is described as follows.First observe that End(E)=P×_GEnd(ℂ^n).The embedding G↪GL_n induces on tangent spaces a G-equivariant map 𝔤→End(ℂ^n), where by definition G acts by the adjoint representationon 𝔤, and by conjugation on End(ℂ^n) via the embeddingG↪GL_n.Thus we obtain a morphism ad P=P×_G𝔤⟶ P×_GEnd(ℂ^n)=End(E).Tensoring by K_X and taking global sections, one obtains the morphism ϕ induced by Φ. Following the constructionin <ref>, one can define the CFG ℋ^G_X/Specℂ(r,d) over the étale site𝖲/Specℂconsisting of principal G-Higgs bundles whose associated vector bundle is of rank r and degree d.The category fibered in groupoidsof principal G-Higgs bundles, ℋ^G_X/Specℂ(r,d)→𝖲/Specℂ, is analgebraic stack locally of finite type over Specℂ. The CFG𝖯𝗋𝗂𝗇^G_X/Specℂ(r,d), of principal G-bundles over X with associated vector bundle of rank r and degree d, is analgebraic stack locally of finite type over ℂ (see e.g., <cit.>).The forgetful functor ℋ^G_X/Specℂ(r,d) →𝖯𝗋𝗂𝗇^G_X/Specℂ(r,d) is schematic(this is similar to Corollary <ref>). It follows fromCorollary <ref> that ℋ^G_X/Specℂ(r,d) is an algebraic stack locally of finite type over ℂ.As in <ref>, one can easily generalize this construction to the case of a proper morphism π:X→ S of finite presentation between schemes over ℂ, and pairs (P,Φ), where P is againa principal G-bundle, but Φ is a global section of adP⊗ F, where F is some fixed S-flat quasicoherent sheaf of 𝒪_X-modules of finite presentation.As in the case of Higgs bundles, for many applications this is not the correct generalization. In the smooth case, the following is standard: Let X be a smooth projective manifold. A G-Higgs bundle is a pair (P,Φ) where P is a principal G-bundle over Xand Φ∈ H^0(X,adP⊗Ω_X^1)is such that [Φ, Φ]=0∈ H^0(X,ad P⊗⋀^2Ω_X^1). § THE HITCHIN FIBRATION In this section we describe the Hitchin fibration at the level of stacks. §.§ Characteristic polynomials and the Hitchin morphism In this section we introduce characteristic polynomials and the Hitchin morphism.§.§.§ Characteristic polynomials Let X be a scheme, let E be a locally free sheaf of rank n on X, let L be a locally free sheaf of rank 1 on X, and let ϕ:E→ E⊗ Lbe a morphism of sheaves. Let𝕃=0pt[0pt][-.5pt]Spec_X(Sym^∙ L^∨) be the geometric line bundle associated to L, and letp:𝕃→ Xbe the structure map.Let 𝒪_𝕃@>T>> p^* Lbe the tautological section (this is the section corresponding to the tautological map of geometric line bundles𝕃×_X𝔸^1_X→𝕃×_X𝕃, given heuristicallyby“(v_x,λ_x)↦ (v_x,λ_x v_x) for v_x∈𝕃_x and λ_x ∈𝔸^1_X,x”; see <ref> below).Tensoring by p^*E, we obtain p^*E@>T>> p^*E⊗ p^*L.Consequently we obtain the endomorphismp^*E@>T-p^*ϕ>> p^*E⊗ p^*L.Taking the determinant we obtainp^* E@> (T-p^*ϕ)>> p^* E⊗ p^*L^⊗ n.We can view this as a global section 𝒪_𝕃@> (T-p^*ϕ)>>p^*L^⊗ n.Therefore we have(T-p^*ϕ)∈Γ(𝕃,p^*L^⊗ n)=Γ(X,p_*𝒪_𝕃⊗ L^⊗ n)=⊕_m≥ 0T^mΓ(X, L^⊗ (n-m)).On the right, T is a formal variable, which we introduce to make the bookkeeping and some local computations easier to follow. We call (T-p^*ϕ) the characteristic polynomial of ϕ (it is an element of the graded ring of globalsections of tensor powers of L, which we formally view as a polynomial by introducing the formal variable T).The component of (T-p^*ϕ) in Γ(X,L^⊗ i) can be obtained in the following way.The morphismϕ:E→ E⊗ Ldetermines a global section of L via the composition 𝒪_X→ E^∨⊗ E→ L,where the first map takes 1 to id_E and the second is induced by ϕ.We define the trace Tr(ϕ)∈ H^0(X,L) to bethis global section.The component of (T-p^*ϕ) in H^0(X,L^⊗ i) is given by (-1)^iTr(∧^iϕ). In particular, (T-p^*ϕ)∈⊕_i= 0^n T^n-iΓ(X, L^⊗ i).Moreover, the component in Γ(X,𝒪_X) is always equal to 1, and so we will drop this term in what follows.§.§.§ The Hitchin morphismFrom the discussion abovewe can definethe Hitchin morphism h:ℋ_X/S(L)(n) → ⊕_i = 1^n T^n-iΓ(X, L^⊗ i) (E,ϕ) ↦ (T-p^*ϕ).Here we are denoting by ℋ_X/S(L)(n) the sub-algebraic stack of ℰ_X/S(L)(n)consisting ofendomorphisms of locally free coherent sheaves of rank n with values in L,over the site (𝖲/S)_et. On the right, we suggestively indicate the functor that on an S-scheme S'→ S takes values ⊕_i = 1^n T^n-iΓ(X', L'^⊗ i), where X'=S'×_SX and L' is the pullback to X'.In other words it is the functor ℋ𝑜𝑚_X/S(𝒪_X,⊕_i=1^nT^n-iL^⊗ i) of Lemma <ref>.By virtue of Lemma <ref>,this functor is representable by a scheme𝔸_X/S(L)(n) over S, .We call 𝔸_X/S(L)(n) the Hitchin base for ℋ_X/S(L)(n).We obtain the Hitchin morphismh:ℋ_X/S(L)(n) → 𝔸_X/S(L)(n). When X/S is a smooth projective curve over S=Specℂ (and we consider ℂ-points of the moduli problem) the Hitchin maptakes a rank n Higgs bundle (E,ϕ)on X with values in a line bundle L, and sends it to the corresponding n-tuple of “coefficients” of the characteristic polynomial of ϕ in the complexvector space ⊕_i= 1^n Γ(X, L^⊗ i).§.§.§ The tautological section of p^*LThe main goal of this subsection is to define the tautological section (<ref>), and give a local description of the map. There are several equivalent ways to define it.Onecan use adjunction to identify the groupsHom_𝒪_𝕃(𝒪_𝕃,p^*L) =Hom_𝒪_X(L^∨,Sym^∙ L^∨) and then use the tautological morphism of sheaves on the right (see Remark <ref>).Alternatively, one could consider the global section of the geometric line bundle p^*𝕃 on 𝕃 given pointwise by assigning to v∈𝕃 the point v in the fiber of p^*𝕃 over v (see Remark <ref>). Finally, one can describe it from a morphism of geometric line bundles T:𝔸^1_X×_X𝕃→𝕃×_X𝕃 over 𝕃; since this is the how we will use the tautological section, we consider this approach in detail.The 𝒪_X-module structure map for the rank 1, locally free sheaf L𝒪_X× L→ Linduces a multiplication map𝔸^1_X×_X𝕃 @>μ >> 𝕃,as is seen easily from the following diagram, where U is an open set, and the dashed arrow indicates a section of 𝒪_X× L over U:@C=.5cm @R=.5cm 𝔸^1_X×_X 𝕃@->[r]^<>(0.5)μ@->[d] 𝕃@->[d]U@–>[ru] @^(->[r]X@=[r]X.This in turn induces a diagram@C=.5cm @R=.5cm𝔸^1_X×_X𝕃@–>[rd]^T @->@/^1pc/[rrd]^μ@->@/_1pc/[rdd]_pr_2𝕃×_X𝕃@->[r] @->[d] 𝕃@->[d]^p𝕃@->[r]_p XThe map T is the geometric version of the tautological global section.More precisely, 𝔸^1_X×_X𝕃 is the pullback of thetrivial geometric line bundle on X (i.e., p^*𝔸^1_X), and is hence the trivial geometric line bundle on 𝕃 (i.e., 𝔸^1_𝕃), and 𝕃×_X𝕃 is p^*𝕃. The associated morphism of sheaves is a morphism𝒪_𝕃 @>T >> p^*L,whichcorresponds to a global section T∈ H^0(𝕃,p^*L). .2 cmIt can be useful to describe the tautological section locally.LetU=SpecR⊆ X be an affine open subset.Assume that L is trivialized over U, corresponding to the trivial R-module R.We canidentify the R-algebra Sym^∙ L^∨(U) with R[T], so that p:𝕃→ X is identified over U as p:L|_U=𝔸^1_U=SpecR[T]→ U=SpecR.The multiplication map μ:𝔸^1_U×_U𝕃|_U→𝕃|_U is then identified with the R-algebra mapR[T,T']=R[T]⊗_R R[T']@<μ<< R[T']given by T'↦ TT'. The diagram (<ref>) defining the geometric tautological section, i.e., the map T:𝔸^1_U×_U 𝕃|_U→𝕃|_U×_U𝕃|_U, is givenat the level of R-algebras by the daigram:@C=.5cm @R=.5cm R[T]⊗_R R[ T'] @<–[rd]^T @<-@/^1pc/[rrd]^μ@<-@/_1pc/[rdd]_R[T']⊗_R R[ T']@<-^)[r] @<-_)[d] R[ T'] @<-_)[d] R[T']@<-^)[r] RThe tautological map T is given here byT'↦ TT' and T'↦ T'.Now p^-1(U)=𝕃|_U=SpecR[T]. We have that 𝔸^1_U×_U 𝕃|_U and𝕃|_U×_U𝕃|_U are geometric line bundles, which on 𝕃|_U, where they are trivialized, correspond to the trivial rank 1, locally free module R[T]. The tautological section T:𝒪_𝕃→ p^*L is given under these identificationsby the multiplication map R[T]@>· T>> R[T]. Alternatively, one can describe the tautological section as follows.We have identificationsHom_𝒪_𝕃(𝒪_𝕃,p^*L)=Hom_𝒪_𝕃(p^*𝒪_X,p^*L)=Hom_𝒪_X(𝒪_X,p_*(𝒪_𝕃⊗ p^*L)) =Hom_𝒪_X(𝒪_X,(Sym^∙ L^∨)⊗ L) =Hom_𝒪_X(L^∨,Sym^∙ L^∨).There is a natural inclusion L^∨↪𝒪_X⊕ L^∨⊕ (L^∨)^⊗ 2⊕⋯ onto the second factor, and this corresponds to the tautological section.Locally, if we set U=SpecR⊆ X to be an open affine over which 𝕃 is trivial, then Sym^∙ L^∨is identified with R[T], andthe natural map L^∨→Sym^∙ L^∨ is associated to the mapR→ R[T] by multiplication by T. One can also describe the tautological section geometrically as follows.The sheaf p^*L is the sheaf of sections of the line bundle p^*𝕃:=𝕃×_X𝕃→𝕃, where the structure morphism to 𝕃 is the first projection. There is a global section of the structure morphism given by the diagonal map 𝕃→𝕃×_X𝕃.This is the tautological global section. §.§.§ The characteristic polynomial locally Let (E,ϕ) be a Higgs bundle on X with values inL. LetU=SpecR⊆ X be an affine open subset.Assume that E is trivial over U, corresponding to R^n, and thatL is also trivialized, corresponding to R.Then ϕ:E→ E⊗ L can be identified as a map R^n→ R^n, and thus with an n× n matrix (ϕ_ij) over R.We can also identify Sym^∙ L^∨ with R[T], so that p:𝕃→ X is identified over U as p:𝔸^1_U→ U, and the tautological section T:𝒪_𝕃→ p^*L is given under these identificationsby the multiplication map R[T]@>· T>> R[T].The map (T-p^*ϕ):p^*E→ p^*E⊗ L can then be identified over 𝕃|_U with the map R[T]^n→ R[T]^ngiven by the matrix T-p^*ϕ=( [ T-ϕ_11⋯-ϕ_1n;⋮⋱⋮;-ϕ_n1⋯ T-ϕ_nn;]).We then have (T-p^*ϕ)|_U=T^n-Tr(ϕ)T^n-1+⋯ +(-1)^nϕ∈ R[T].Recall that globally we had(T-p^*ϕ)∈Γ(𝕃,p^*L^⊗ n)=Γ(X,p_*𝒪_𝕃⊗ L^⊗ n)=⊕_m≥ 0 T^mΓ(X, L^⊗ (n-m)); the coefficients of the powers of T in this description andin (<ref>) agree.§.§ Spectral covers and fibers of the Hitchin morphism Here we describe spectral covers, and the connection with the fibers of the Hitchin morphism.The main point for Higgs bundles on smooth curves are the results of <cit.> reviewed in Remark <ref>. We give a weaker statement that holds in more generality in Proposition <ref> and Lemma<ref>.§.§.§ Spectral coversEvery σ:𝒪_X→⊕_i=1^nL^⊗ i (corresponding to a map σ:S→𝔸_X/S(L)(n)) determines via (<ref>) a global section σ:𝒪_𝕃→ p^*L^⊗ n of the line bundle p^*L^⊗ n on 𝕃, and consequently a zero set of the section:X(σ):=V(σ)⊆𝕃.The map X(σ)→ X (obtained by composition from the map p:𝕃→ X) is called the spectral cover associated to σ. One can check (see the local computation below) that the spectral cover is a finite morphism of degree n. Associated to the scheme X(σ)=V(σ) is an ideal sheaf ℐ_σ, defined via the short exact sequence:0→ℐ_σ→Sym^∙ L^∨→𝒪_V(σ)→ 0. More generally, the computation showsthat for any σ':S'→𝔸_X/S(L)(n), corresponding to σ' :𝒪_X'→⊕_i=1^nL^⊗ i|_X', where X'=X×_SS', there is a corresponding spectral coverp':X'(σ')→ X'. In particular, there is a universal spectral cover X(σ_id_𝔸_X/S(L)(n))→ X×_S𝔸_X/S(L)(n)determined by the identity morphism on 𝔸_X/S(L)(n), fromwhich all the spectral covers are obtained by pullback. §.§.§ Local description of the spectral coverTo describe X(σ), it can be useful to consider the construction locally on X. LetU=SpecR⊆ X be an affine open subset.Assume that L istrivialized over U, corresponding to the R-module R. Given σ∈⊕_i=1^n T^n-iΓ(X,L^⊗ i), with components σ_i∈Γ(X,L^⊗ i), then we can write σ|_U= T^n-σ_1|_UT^n-1+⋯ +(-1)^nσ_n|_U.Since L, and hence L^⊗ i is trivalized over U, we may view the σ_i|_U as elements of R.The short exact sequence (<ref>) is then written locally as 0@>>> R[T] @>σ|_U>> R[T] @>>> R[T]/ℐ_σ|_U @>>>0 .In other words, ℐ_σ|_U is given by (σ|_U)⊆ R[T], and X(σ)|_U=SpecR[T]/(σ|_U).§.§.§ Minimal ideals for endomorphisms Let X be a scheme, L a rank 1 locally free sheaf on X, and E a quasicoherent sheaf on X.A morphism of 𝒪_X-modulesϕ:L^∨→ End(E)is equivalent to a morphism of 𝒪_X-algebrasϕ^∙ :Sym^∙ L^∨→End(E).The minimal ideal of ϕ, denoted ℐ_ϕ,is defined to be the kernel of ϕ^∙. The minimal cover X(ϕ)→ X associated toϕ is defined to be the subscheme of 𝕃 defined by ℐ_ϕ. Note that E induces a quasicoherent sheaf M on X(ϕ) such that thepushforward of M to X is equal toE. If E is locally finitelygenerated, the support of M is exactly X(ϕ) (i.e., locally, the support isthe set ofprimes containing the annihilator of the finitely generated module). More generally, for any ideal sheaf ℐ⊆Sym^∙ L^∨, a quasicoherent sheaf M on the schemeV(ℐ)⊆𝕃 is equivalent to a quasicoherent sheaf Eon X together with a morphism ϕ:L^∨→End(E) such that ℐ⊆ℐ_ϕ; theidentification is made via pushforward; i.e., E is the pushforward ofM.Locally on X, for locally free sheaves this is described as follows.Suppose that E is locally free of rank n, and is trivial over U=SpecR⊆ X.Then ϕ:L^∨→End(E)induces an evaluationmorphism ϕ^∙|_U:R[T]→End_R(R^n) sending T to ϕ|_U.The kernel ϕ^∙|_U is the restriction of theminimal ideal ℐ_ϕ|_U.We have that R^n is an R[T]/ℐ_ϕ|_U-module, with support SpecR[T]/ℐ_ϕ|_U.In addition, so long as σ|_U∈ℐ_ϕ|_U, we can view R^n as an R[T]/(σ|_U)-module, as well.In other words, E is obtained by push forward from a sheaf on the spectral cover associated to σ (although it may only be supported on the possibly smaller minimal cover associated to ϕ). This discussion allows us to identify Higgs bundles with given minimal ideal with certain sheaves on the spectral cover. [<cit.>] Let X→ S be a proper morphism of finite presentation between schemes with S excellent, and let L be a locally free sheaf of rank 1 onX. Given an S-morphismσ':S'→𝔸_X/S(L)(n),to give a pair(E',ϕ') in ℋ_X/S(L)(n)(S') such that ϕ' has minimal ideal ℐ_ϕ' equal toℐ_σ', it is equivalent to give a coherent sheaf M'on X'(σ') (where X' = X ×_S S') such thatthe pushforward of M' to X' is a rank n locally free sheaf and the support of M' is X'(σ').As we have seen above, (E', ϕ') ∈ℋ_X/S(L)(n)(S') corresponds to a quasicoherent sheaf M' on 𝕃' (the total space of L') whose support is X'(ϕ').But X'(ϕ') coincides with X'(σ') if and only if ℐ_σ' = ℐ_ϕ'.It is easy to see that every locally free sheaf M' of rank 1 on X'(σ) satisfies the conditions in Proposition <ref>.Note that one caneasily construct examples (even with X'(σ') a union of smooth complex projective curves) where there are sheaves M' on X'(σ') that push forward to rank n locally free sheaves E' on X such that the induced endomorphism ϕ':E'→ E'⊗ L' does not have minimal polynomial equal to σ'; this is the reason for the hypothesis on the support of M'. §.§.§ Fibers of the Hitchin map The following lemma asserts that the category of sheaves in Proposition <ref> above is contained in the fiber of the Hitchin map. A pair (E',ϕ') inℋ_X/S(L)(n)(S') such that the minimal ideal ℐ_ϕ' is equal to ℐ_σ' has characteristic polynomial given by σ'.Moreover, if X' is integral and the fiber of X'(σ') over the generic point of X is geometrically reduced (e.g., X' is a variety over ℂ and X'(σ') is reduced), then the converse holds.In other words, under these hypotheses, Proposition <ref> describes the fiber of the Hitchin morphism in terms of sheaves on the spectral cover.Let (E',ϕ') be a Higgs bundle as in the statement of thelemma, and let p_ϕ'(T) be the characteristic polynomial of ϕ'.Assume first that ℐ_ϕ'=ℐ_σ'. Viewing the characteristic polynomial as a global section p_ϕ'(T):𝒪_X'→⊕_i=1^nL^⊗ i|_X'defines a locally principalideal ℐ_p_ϕ'(T) of Sym^∙ L^∨ (<ref>).The Cayley–Hamilton theorem says that ℐ_p_ϕ'(T)⊆ℐ_ϕ'. But we are assuming that ℐ_ϕ'=ℐ_σ'. So we have ℐ_p_ϕ'(T)⊆ℐ_σ', with both being locally principal generated by monic polynomials of degree n.So we have reduced to the local statement:Given a ring R, and two principal ideals (f(T)) and (g(T)) in R[T] with both f(T) and g(T) monic of degree n, if (f(T))⊆ (g(T)), then f(T)=g(T).Thus the characteristic polynomial p_ϕ'(T) is given by σ.Conversely, suppose X is integral, the fiber of X'(σ') over the generic point of X is geometricallyreduced, and that the characteristic polynomialp_ϕ'(T) is given by σ'.We want to show thatℐ_ϕ'=ℐ_σ'.It is enough to do this locally.In other words, we assume that R is an integral domain with field of fractions K contained in an algebraic closure K, that σ(T)∈ R[T] is a monic polynomial of degree n, that K[T]/(σ(T)) is reduced, and that ϕ:R^n→ R^n is an endomorphism with characteristic polynomial σ(T).We want to show the containment ℐ_ϕ⊇ (σ(T)) is an equality.So consider0 f(T)∈ℐ_ϕ. The claim is that f(T) is divisible by σ(T) in R[T].To see this, note that with ϕ_K:K^n→ K^n the induced morphism, we have that f(ϕ_K)=0. Thus f(T) is divisible in K[T] by the minimal polynomial for ϕ_K.Note also that from the local definition of the characteristic polynomial, it is clear that the the characteristic polynomial of ϕ_K is the same as for ϕ, namely σ(T). Our assumption that K[T]/(σ(T)) is reduced, i.e., that the characteristic polynomial of ϕ_K has distinct roots, implies that the characteristic polynomial for ϕ_K agrees with the minimal polynomial.Thus f(T) is divisible by σ(T) in K[T].But since σ(T) is monic, one can then conclude (see the remark below) that f(T) is divisible by σ(T) in R[T]. Let R be an integral domain and let K be its field of fractions.Suppose that f(T),g(T)∈ R[T] with g(T) monic, and there exists h(T)∈ K[T] such that f(T)=g(T)h(T)∈ K[T].Then h(T)∈ R[T].Indeed, let us write f(T)=∑_i=0^n+ma_iT^i,g(T)=∑_i=0^nb_iT^i, h(T)=∑_i=0^mc_iT^i.Then we havea_n+m =b_nc_ma_n+m-1 =b_nc_m-1+b_n-1c_m⋮a_n = b_nc_0+b_n-1c_1+⋯ +b_0c_n.(Here c_j is taken to be 0 if j>m.) Since b_n is assumed to be 1, the first equality shows that c_m∈ R.The second then shows that c_m-1∈ R, and so on until we have established that c_0∈ R.It can be instructive to consider Proposition <ref> and Lemma <ref> the case where X=S=Specℂ and L=𝒪_X. In the case where π:X→ S is a smooth relative curve; i.e., a smooth proper morphism such that every geometric fiber is connected, and dimension 1, more can be said.If σ':S'=Spec (k)→𝔸_X/S(L)(n) is a geometric point inducing a reduced spectral curveX'(σ')over Speck, then h^-1(σ') is the the compactified Picard stack, parameterizingrank 1, torsion-free sheaves on X'(σ') <cit.>. If X'(σ') is assumed further to be anintegral curve, which is the case considered in<cit.>, then the key point is that in the notation of Proposition <ref>,for π'_*M to be locally free, and thus torsion-free, it must be that M is torsion-free.Then,since X' is a smooth curve over k, any torsion-free sheaf is locally free, and thus for any torsion-free sheaf on X', the push-forward is locally free.The case of a general spectral curve X'(σ') is considered in <cit.>; the connection between slope stability of Higgs bundles on X'and slope stability of rank 1 torsion free sheaves on X'(σ')in the sense of Oda–Seshadri is also considered there. See also <cit.>. § MORPHISMS OFSTACKS IN ALGEBRAIC GEOMETRY As explained in Definition <ref>, many reasonable properties of morphisms in our presite 𝖲 may be extended to 𝖲-representable morphisms of stacks.However, not every morphism of algebraic stacks that deserves to be called quasicompact or étale or smooth (to name just a few) is necessarily schematic.In Section <ref>, we saw how to extend the notion of smoothness to morphisms between algebraic stacks, but this definition required the choice of a presentation of the stack in question.Since stacks representing moduli problems rarely come with an easily described presentation, it would be better to have an intrinsic definition of smoothness.To give such a definition, as well as definitions of other geometric properties or morphisms between algebraic stacks, will be the purpose of this section. We caution that although many of the definitions given here make sense for arbitrary morphisms of CFGs, or for arbitrary morphisms of stacks, they should not necessarily be regarded as reasonable generalizations from schemes without a further algebraicity assumption.The characterization of smoothness we give in Section <ref>, for example, is only reasonable for morphisms of algebraic stacks (or at least morphisms representable by algebraic stacks) and not necessarily for all stacks.§.§ Injections, isomorphisms, and substacks In Definition <ref> we gave the definition of an injection and of anisomorphism of stacks. We have not given a special name to morphisms of algebraic stacks that are only faithful (rather than being fully faithful or equivalences, respectively)when viewed as functors,because we already have one: A morphism of algebraic stacks f : 𝒳→𝒴 that is objectwise a faithful functor is representable by algebraic spaces. By saying f : 𝒳→𝒴is objectwise a faithful functor we mean that for each scheme S the induced functor of groupoids f(X):𝒳(S)→𝒴(S) isfaithful. Suppose that Z is an algebraic space and Z →𝒴 is a morphism.Let 𝒳_Z be the base change, which is algebraic by Corollary <ref>.The projection 𝒳_Z → Z is also faithful.But if Z is viewed as a stack then, for any scheme S, the fiber Z(S) is equivalent to a set.Since𝒳_Z(S) is equivalent to a subcategory of Z(S), this means that 𝒳_Z(S) is equivalent to a set, which means that 𝒳_Z is an algebraic space. [Open and closed substacks] Let 𝒳 be a stack over 𝖲.A substack 𝒰⊆𝒳 (resp. 𝒵⊆𝒳) is called an open substack (resp. closed substack) if for every scheme W, the fiber product 𝒰×_𝒳 W(resp. 𝒵×_𝒳 W)is representable by an open (resp. closed)subscheme of W. §.§ The underlying topological space [Topological space of a stack] Let 𝒳 be a stack on the category 𝖲 of schemes.For each scheme S, write |S| for the underlying topological space of S.The underlying topological space of 𝒳 is the universal topological space |𝒳| that receives a continuous map |S| → |𝒳| for each map S →𝒳. In other words, |𝒳|=_S→𝒳|S| is the colimit of the spaces |S|, taken over all maps from schemes S to 𝒳. In <cit.>, the underlying topological space was defined only for algebraic stacks, but <cit.> makes sense for arbitrary categories fibered in groupoids and agrees with the definition given here because the underlying set of a colimit of topological spaces is the colimit of the underlying sets. The topology on the underlying set was only defined in <cit.> for algebraic stacks.However, the topology of loc. cit. agrees with Definition <ref>.Recall that a subset of |𝒳| is called open if its preimage under a fixed flat, finite presentation, surjective map U →𝒳 is open.This topology is clearly at least as fine than the topology we have defined, so we verify that every subset of |𝒳| that is open in the sense of <cit.> is open in the sense of Definition <ref>. Indeed, if U →𝒳 is flat and locally of finite presentation then for any V →𝒳, the map U ×_𝒳 V → V is also flat and of finite presentation.The preimage in |V| of the image of |U| in | 𝒳| is the same as the image of |U ×_𝒳 V| in |V|.Since U ×_𝒳 V is flat of finite presentation over V, its image in |V| is open, so the image of |U| in |𝒳| pulls back to an open subset of |V|.This holds for any V →𝒳 so the image of |U| in |𝒳| is open, by definition of the colimit topology.As the topology in loc. cit. is uniquely characterized by this property, it must agree with |𝒳|, as defined in Definition <ref>. §.§ Quasicompact and quasiseparated morphisms [Quasicompact morphisms] We call a stack 𝒳over 𝖲 quasicompact if every covering of 𝒳(Definition <ref>) by open substacks has a finite subcover. A morphism of stacks 𝒳→𝒴 is quasicompact if 𝒳×_𝒴 Z is quasicompact for all quasicompact schemes Z and all morphisms Z →𝒴. A morphism of stacks 𝒳→𝒴 is quasiseparated if the diagonal morphism 𝒳→𝒳×_𝒴𝒳 is quasicompact.§.§ Separation and properness We will not actually discuss the separatedness or properness of morphisms of algebraic stacks in any examples in this survey, but we include the definitions for the sake of completeness.Since smooth morphisms of schemes are always locally of finite type, Section <ref> shows that there is a unique way to make sense of locally finite type morphisms of algebraic stacks that is stable under base change and composition and local to the source and target.Technically, this breaks our promise to give only intrinsic definitions in this section.However, in the noetherian situation, finite type coincides with finite presentation, which is characterized intrinsically in Section <ref>.[Proper and separated morphisms of algebraic stacks] A morphism of algebraic stacks 𝒳→𝒴 is said to be proper if it is an isomorphism or it is separated, of finite type, and universally closed.It is said to be separated if its diagonal 𝒳→𝒳×_𝒴𝒳 is proper.Definition <ref> is not as circular as it appears.The diagonal morphism of a morphism of algebraic stacks is representable by algebraic spaces (Lemma <ref>, <cit.>),so the definition of separatedness for algebraic stacks depends only on the definition of properness for algebraic spaces.Iterating Definition <ref>, we see that definition of properness for algebraic spaces depends on separatedness for morphisms of algebraic spaces,and therefore,by iterating again, the definition depends on the definition of properness for diagonals of algebraic spaces.Continuing further, we see thatwe must define separatedness for diagonals of algebraic spaces. But the diagonal of a morphism of algebraic spaces is injective, so the definition ultimately depends on the definition of separatedness for injections of algebraic spaces.But the diagonal of an injection is an isomorphism, so is automatically proper. §.§ Formal infinitesimal properties [Infinitesimal extension <cit.>] An infinitesimal extension or nilpotent extension of a scheme S is a closed embedding S ⊆ S' such that the ideal of S in S' is nilpotent. Suppose that S ⊆ S' is an infinitesimal extension and that T → S is étale.Then there is an infinitesimal extension T ⊆ T' and étale map T' → S' inducing T as the fiber product T' ×_S' S.Moreover, T' is unique up to unique isomorphism. Since T' will be unique when it is constructed, this is a local problem in the Zariski topology on S':if T' has been constructed over a suitable open cover, the uniquenes will imply that the various T's can be glued together.We can therefore work Zariski-locally in S', or equivalently in S, since S and S' have the same Zariski topology.The same reasoning shows that we can work Zariski-locally in T as well.This permits us to assume that S, S', and T are all affine.By the Jacobian criterion (e.g., <cit.>), we can assume that S = Spec A and that T = Spec B where B = A[x_1, …, x_n] / (f_1, …, f_n) and the determinant ∂ f_i/∂ x_j is a unit of B.If S' = Spec A', we take T' = Spec B' where B' = A'[x_1, …, x_n] / (g_1, …, g_n) where g_i is an arbitrary lift of f_i to a polynomial with coefficients in A'.Then ∂ g_i/∂ x_j reduces to ∂ f_i/∂ x_j in B.Since B' is an infinitesimal extension of B, this means that ∂ g_i/∂ x_j is a unit in B', so by the Jacobian criterion, B' is étale over A'.This establishes that T' exists locally in the Zariski topology of S'.It remains to prove the uniqueness of T'.Suppose that T' and T” are two such extensions.Then by the infinitesimal lifting criterion for étale maps, applied to the diagramT [r] [d] T' [d] @–>@/_5pt/[dl]_-u T”[r] @/_5pt/@–>[ur]_-v S'and there are unique lifts u : T' → T” and v : T”→ T'.These must compose to the identity maps, again by the infinitesimal criterion for étale maps, this time applied to the diagrams below:T [r] [d] T' [d] T”[r] @/^5pt/[ur]^vu@/_5pt/[ur]_idS'T [r] [d] T' @/^5pt/[dl]^-uv@/_5pt/[dl]_-id[d] T”[r]S'.Suppose that S ⊆ S' is an infintisimal extension of schemes.Then the étale sites of S and S' are equivalent.A morphism of CFGs f : 𝒳→𝒴 is said, respectively, to be formally unramified, formally étale, or formally smooth if every commutative diagram (<ref>) below, in which S ⊆ S' is an infinitesimal extension of schemes, admits at most one (up to unique isomorphism), exactly one (up to unique isomorphism), or at least one lift étale-locally in S. S [r] [d]𝒳[d] S' [r] @–>[ur]𝒴.We explicate a bit the meaning of`étale-locally'in the definition.To say that 𝒳→𝒴 is formally smooth means that, given any lifting problem (<ref>), there is an étale cover { S_α→ S } such that, denoting by S'_α the unique infintesimal extension of S_α lifting S (by Lemma <ref>), the diagramS_α[r] [d] S [r] [d]𝒳[d] S'_α@–>[urr] [r] S' [r] 𝒴admits a lift for every α.To say that 𝒳→𝒴 is formally unramified means, first, that that given any two lifts of (<ref>), there is a cover of S by S_α such that the induced lifts of (<ref>) are isomorphic, and, second, that any two isomorphisms between lifts of (<ref>) agree after passage to a suitable étale cover of S.To be formally étale is the conjunction of these properties. §.§ Local finite presentationIt was observed in <cit.> that a scheme X is locally of finite presentation if and only if whenever A =A_i is a filtered colimit of commutative rings, the natural mapX(A_i) → X(A)is a bijection.The same formula characterizes algebraic stacks that are locally of finite presentation, provided one interprets a filtered colimit of groupoids correctly.It is therefore reasonable to use (<ref>) as the definition of local finite presentation for stacks that are not known to be algebraic.To make sense of the filtered colimit of groupoids in equation (<ref>), one can take𝖮𝖻𝗃𝒳(A_i) = ⋃_i 𝖮𝖻𝗃𝒳(A_i)and, for any objects ξ∈𝒳(A_i) and η∈𝒳(A_j),Hom(ξ, η) = _k ≥ i,jHom_𝒳(A_k)(ξ_k, η_k)where ξ_k and η_k denote pullbacks of ξ and η, respectively, to 𝒳(A_k).In order to formulate local finite presentation for morphisms of stacks, it is useful to introduce the pro-object “" Spec A_i associated to a filtered system of commutative rings A_i.By definition, “" Spec A_i is the covariant functor on schemes (and stacks) obtained by taking the filtered colimit of the functors represented by the Spec A_i.What this actually means is that one should interpret a morphism“"Spec A_i →Xto a scheme (or stack)as an object ofHom(Spec A_i,X) =X(A_i);i.e., as compatible systems of morphisms SpecA_i→ X.In more technical terms, the quotation marks indicate that one is taking a colimit in the category of covariant functors valued in sets (or groupoids).We absolve ourselves of responsibility for the notation <cit.>. A morphism 𝒳→𝒴 of stacks in the étale topology on schemes is said to be locally of finite presentation if whenever A =A_i is a filtered colimit of commutative rings, then every commutative diagram of solid lines (<ref>) can be completed uniquely by a dashed arrow.Spec A [r] [d]𝒳[d]“" Spec A_i [r] @–>[ur]𝒴 Observe the resemblance between the lifting diagram (<ref>) and the lifting diagram (<ref>).This allows us to reason formally about étale maps and local finite presentation maps at the same time.Under this analogy, Lemma <ref> below is the analogue of Lemma <ref>. Suppose that a ring A is the filtered colimit of rings A_ℓ.Set S = Spec A and S_ℓ = Spec A_ℓ.Then for any affine[In fact, quasicompact and quasiseparated would suffice.] étale map T → S there is for some ℓ an affine scheme T_ℓ admitting anétale map T_ℓ→ S_ℓ, and this map is unique up to unique isomorphism and enlargement of ℓ. Suppose that T = Spec B is affine and étale over S.By the Jacobian criterion, there is an open cover of T by finitely many subsets U = Spec C where C = A[x_1, …, x_n] / (f_1, …, f_n) and ∂ f_i/∂ x_j is a unit of C.Since we are proving a uniqueness statement and only finitely many such subsets are involved, we may treat the subsets individually.We may therefore assume that T = U.For k sufficiently large, the coefficients of the f_j all appear in the image of A_k in A, so that we may form the ring B_k = A[x_1, …, x_n] / (g_1, …, g_n) for some lifts g_1, …, g_n of f_1, …, f_n to A_k.The determinant ∂ g_i/∂ x_j maps to the unit ∂ f_i/∂ x_j in B.Pick t ∈ B inverse to ∂ f_i/∂ x_j.As B = _ℓ≥ k B_ℓ, the element t must be in the image of some B_ℓ with ℓ≥ k.Furthermore t ∂ g_i/∂ x_j must equal 1 in all sufficiently large B_ℓ.This implies that by taking ℓ sufficiently large, the image of ∂ g_i/∂ x_j is a unit.Thus B_ℓ is étale over A_ℓ for all sufficiently large ℓ, and B_ℓ induces B over A.We must still prove that B_ℓ is unique up to enlargement of ℓ.Suppose that C_k were étale over A_k, also inducing B over A.Since B_ℓ is of finite presentation over A_ℓ and C_k is of finite presentation over A_k, the isomorphism B_ℓ≃ B ≃ C_k must come from an isomorphism B_m ≃ C_m defined over some m that is ≥ k and ≥ℓ.This map is necessarily unique up to further enlargement of m (see <cit.> for more details).By an étale map “" U_i →“" Spec A_i, we mean a family of maps U_i →Spec A_i that are étale for all sufficiently large i. In terms of this definition, Lemma <ref> saysany étale map U →“" Spec A_i is induced from an étale map “" U_i →“" Spec A_i, and that this map is unique up to a unique, suitably defined, isomorphism. Lemma <ref> says, in a sense that we do not attempt to make precise, that the étale site of Spec A is the colimit of the étale sites of the Spec A_i, whenever A is the filtered colimit of the A_i.See <cit.>.§.§ Smooth, étale, and unramified morphisms[Unramified, étale, smooth]<cit.>] A morphismofalgebraic stacks f : 𝒳→𝒴 is said, respectively, to be unramified, étale, or smooth if it is locally of finite presentation and formally unramified, formally étale, or formally smooth. We have now defined smooth morphisms between algebraic stacks in two ways.On one hand, we have defined smooth morphisms in Definition <ref> as the unique extension of smoothness from morphisms of schemes in a way that is stable under composition and base change and local to the source and target.On the other hand, we have defined smoothness intrinsically for morphisms of algebraic stacks in Definition <ref>.We are obliged to verify that they are equivalent.It is sufficient to show that local finite presentation and smoothness are stable under composition and base change and local to the source and target, as these definitions clearly agree with the usual definitions in the category of schemes and the extension in Definition <ref> was uniquely characterized by these properties.Let 𝐏 be one of the following properties of morphisms of stacks in the étale topology on schemes: * formal smoothness,* formal unramifiedness,* formal étaleness, or* local finite presentation.Then 𝐏 is stable under composition and base change and is local to the source and target. We omit the verification for composition and base change, since these are formal and straightforward.All of these properties may be phrased in terms of existence or uniqueness (or both) of lifts of a diagramS [r] [d]𝒳[d] S' @–>[ur] [r]𝒴In the case of formal unramifiedness, formal étaleness, or formal smoothness, S' is an infinitesimal extension of S.In the case of local finite presentation, we have a ring A that is the filtered colimit of rings A_i, and S = Spec A and S' = “" Spec A_i.The only fact we will use here is that if S → S' is the left side of one of these diagrams and T → S is an étale map, then there is a unique (up to unique isomorphism) extension of T to an étale map T' → S'.In the case where S → S' is an infinitesimal extension, this is Lemma <ref>; in the case where S = Spec A and S' = “" Spec A_i with the A_i filtered and A_i = A, this is Lemma <ref>.We prove locality to the target.Consider a lifting problem (<ref>), and assume that 𝒴' →𝒴 is a covering map such that the base change 𝒳' →𝒴' is 𝐏.We may freely replace S by an étale cover, so we may assume that the map S' →𝒴 factors through 𝒴'.This induces a factorization of S →𝒳 through 𝒳'.Property 𝐏 for 𝒳' over 𝒴' gives a lift f in diagram (<ref>), which yields g by composition.S [r] [d]𝒳' [r] [d]𝒳[d] S' [r] @–>[ur]^-f @–>[urr]_(0.7)g𝒴' [r]𝒴 Now we prove locality to the source.Again, consider a lifting problem (<ref>) and suppose that X_0 → X is a cover that is formally smooth over 𝒴.Since we can replace S by an étale cover, we can assume that S →𝒳 factors through X_0.Formal smoothness of X_0 over 𝒴 gives a lift f in diagram (<ref>), which gives us g by composition.S [r] [dd] X_0 [d] 𝒳[d] S' @–>[uur]^(0.6)f @–>[ur]_-g [r]𝒴 § INFINITESIMAL DEFORMATION THEORYIn this section we introduce some deformation theory, with a view towards Artin's criterion for a stack to be algebraic. Roughly speaking, deformation theory is the study of families of objects over Artin rings. To get a concrete idea, considerthe Kodaira–Spencer approach to deforming a complex manifold Xover the unit disk by extendingthe given transition functions τ_ij of the manifold to transition functions τ_ij=τ_ij+τ_ij^(1)t+τ_ij^(2)t^2+⋯, where t is the parameter on the disk (for each fixed t, one obtains a manifold with the given transition functions). Deformation theory would then bethe problem ofiteratively accomplishing this formally, by first extending up to t (e.g., τ_ij+τ_ij^(1)tt^2), and then extending up to t^2, and so on.In the process, one would observe that the first order deformations are governed by H^1(X,T_X) (called the tangent space), and that once one has extended to first order, the ability to extend to second order would be goverened by H^2(X,T_X) (called an obstruction space).In general, deformation theory aims to abstract this to the setting of any stack over schemes (and to deforming over bases other than the unit disk). The heart of the matter turns out to be defining the tangent space to a stack, and an obstruction theory to a stack.We take an abstract point of view and identify precisely the condition, homogeneity, on a CFG that gives it a well-behaved tangent space.The advantage of this abstraction is that the existence of a well-behaved tangent space, and obstruction theory,can be used to prove that a stack is algebraic, as we will discuss in Section <ref>. §.§ Homogeneous categories fibered in groupoidsAfter introducing homogeneity and demonstrating its basic properties, our goal will be to show that the stack of Higgs bundles is homogeneous, without relying on its algebraicity.In this first section we stick to the abstract setting, and also introduce the tangent space to a stack.We discuss the pertinent deformation theory in the following section, <ref>, where we establish that the stack of Higgs bundles in homogeneous. §.§.§ The tangent bundle of a stackThe following definition reprises Definition <ref>:[Square-zero extension]An infinitesimal extension of schemes is a closed embedding S ⊆ S' such that the ideal I_S/S' is nilpotent.A square-zero extension is an infinitesimal extension S ⊆ S' such that I_S/S'^2 = 0. The utility of square-zero extensions is twofold:every infinitesimal extension can be factored as a sequence of square-zero extensions, and square-zero extensions behave `linearly', in the sense that deformations and obstructions over square-zero extensions are classified by linear-algebraic data. These observations may be viewed as a functorial perspective on Taylor series.The most important examples of infinitesimal extensions are the trivial ones:[Dual numbers]The ring of dual numbers is 𝔻 = ℤ[ϵ] / (ϵ^2). For any scheme S, we write S[ϵ] = S ×_SpecℤSpec𝔻.More generally, if V is a quasicoherent sheaf of 𝒪_S-modules, we write 𝔻(V) for the sheaf of 𝒪_S-algebras,𝒪_S + ϵ V = Sym^∙ (ϵ V) / (ϵ^2 Sym^2 V),whose underlying sheaf of abelian groups consists locally of symbols f + ϵ v with f ∈𝒪_S and v ∈ V and has multiplication law(f + ϵ v)(g + ϵ w) = fg + ϵ (fw + gv).We write S[ϵ V] for the scheme whose underlying topological space is S and whose sheaf of rings is 𝔻(V); i.e., S[ϵ V]=Spec_S 𝔻(V). There is a canonical morphism S[ϵ V] → S corresponding to the following homomorphism of sheaves of rings:𝒪_S →𝒪_S + ϵ V f ↦ f + 0 ϵUnless otherwise specified, when it is necessary to equip S[ϵ V] with the structure of a scheme over S, we do so with this morphism. The construction of S[ϵ V] → S commutes with base change in S.It can therefore be extended to apply to any CFG, as follows.If 𝒳 is a CFG and V is a quasicoherent sheaf on 𝒳, then we define 𝒳[ϵ V](S) to the category of pairs (ξ, δ) where ξ∈𝒳(S) and δ is a section of S[ϵξ^∗ V] over S. When V = 𝒪_𝒳, we write 𝒳[ϵ] rather than 𝒳[ϵ𝒪_𝒳]; in this case𝒳[ϵ] = 𝒳×_SpecℤSpec𝔻.[Tangent bundle]Let𝒳 bea CFG over 𝖲/S.We give R[ϵ] the structure of an S-scheme via the canonical projection R[ϵ] → R → S, andthe relative tangent bundle of 𝒳 over Sisthe category fibered in groupoids T_𝒳/S over 𝖲/S with T_𝒳 / S(R) = 𝒳(R[ϵ])for each S-scheme R.If 𝒳 is a presheaf, then T_𝒳/S is also a presheaf.In the case of S=Specℤwe write T_𝒳=T_𝒳/Specℤ. The idea to study the tangent space this way, and to think of the spectrum of the ring of dual numbers as a pair of infinitesimally nearby points, goes back at least to Weil <cit.>. For any morphism of schemes X → S, we haveT_X/S = Spec_X ( Sym^∙Ω_X/S )This reduces to the affine situation, where it comes downto the following identities: given a ring k and algebrask→ Aϕ→ B, then Hom_k-alg,ϕ(A,B[ϵ])=Der_k(A,B)=Hom_A-mod(Ω_A/k,B) =Hom_A-alg(Sym^∙Ω_A/k,B),where the first group of homomorphisms consists ofthe k-algebra homomorphisms that reduce to ϕ modulo ϵ.When X is a smooth scheme over ℂ, then T_X/Specℂ is a vector bundle over X and coincides with any familiar definition of the tangent bundle.When 𝒳 is an algebraic stack over S, so is T_𝒳 /S.It is almost immediate that T_𝒳/S is a stack in the étale topology:If we have an étale cover of R by U_i then the U_i[ϵ] form an étale cover of R[ϵ].Étale descent for the maps U_i[ϵ] →𝒳 to R[ϵ] →𝒳 yields étale descent for U_i → T_𝒳/S to R → T_𝒳/S.To see that T_𝒳/S is an algebraic stack, note that if X_0 →𝒳 is a smooth cover of 𝒳 by a scheme then then T_X_0/S→ T_𝒳/S is also a smooth cover.We will see in a moment (item (<ref>) on p. I:proj) that there is a projection T_𝒳 /S→𝒳, as one expects, but that it is not necessarily representable by algebraic spaces!This is because the objects parameterized by 𝒳 may possess infintisimal automorphisms.Nevertheless we will see that the fibers of T_𝒳/S over 𝒳 behave like `groupoids with vector space structure'.This is why we insist on calling T_𝒳/S the tangent bundle:it has all the features of a vector bundle except for being aset! §.§.§ Homogeneity [<cit.>]Let S be a scheme.Suppose that Q ⊆ Q' is an infinitesimal extension of S-schemes and f : Q → R is an affine S-morphism.Then there is a universal (initial) S-scheme R' completing diagram (<ref>):Q @^(->[r]^inf.[d]^f_affineQ' @–>[d] R @–>[r] R'Furthermore, R' is also universal (initial) among algebraic S-stacks completing the diagram.The underlying topological space of R' is the same as that of R and viewing 𝒪_Q' and 𝒪_R' as sheaves on Q and R respectively, 𝒪_R' =𝒪_R ×_f_∗𝒪_Q f_∗𝒪_Q'.In particular, if R = Spec A (hence Q and Q' are also affine, say Q = Spec B and Q' = Spec B') thenR' = Spec (A ×_B B') .The universality here means that for any algebraic stack 𝒳 the natural map displayed in equation (<ref>) is an equivalence of groupoids:𝒳(R') →𝒳(R) ×_𝒳(Q)𝒳(Q') We will prove this in the case where R is also an infinitesimal extension of Q, which implies that the étale sites of Q, Q', R, and R' are all equivalent (Corollary <ref>).First, note that schemes are homogeneous.Indeed, suppose we have a commutative diagramQ @^(->[r] [d] Q' [d] R [r] Xwhere X is a scheme.All of Q, Q', and R have the same underlying topological space, so we denote by f the continuous map from that space to the underlying topological space of X.The maps of schemes give homomorphisms of sheaves of rings:f^-1𝒪_X [r] [d]𝒪_Q'[d]𝒪_R [r]𝒪_QBy the universal property of 𝒪_R' = 𝒪_R ×_𝒪_Q𝒪_Q', there is a unique factorization f^-1𝒪_X →𝒪_R', which gives the desired, uniquely determined, map R' → X.To extend this to algebraic stacks, it is sufficient by Definition <ref> to show that whenever 𝒳 is a stack admitting a smooth cover by a homogeneous stack, 𝒳 is homogeneous.Suppose that we have a commutative diagramQ @^(->[r] [d] Q' [d] R [r]𝒳where 𝒳 has a smooth cover X_0 →𝒳 and X_0 is homogeneous.Since we are trying to prove a uniqueness assertion, it is sufficient to work étale-locally on Q.Replace Q by an étale cover such that the map Q →𝒳 factors through X_0.Since X_0 is formally smooth over 𝒳, we can assume, after refining the cover further, that the maps Q' →𝒳 and R →𝒳 also lift to X_0, extending the lift already constructed over Q.By the homogeneity of X_0, there is a unique map R' → X_0 compatible with the maps from Q, Q', and R.Composing with the projection to 𝒳 gives the desired map R' →𝒳.[Pushout]Suppose that Q ⊆ Q' is an infinitesimal extension of S-schemes and f : Q → R is an affineS-morphism. We refer to the scheme R' in Theorem<ref> as the pushout of Q ⊆ Q' along the map f:Q → R.A CFG is said to be homogeneous if it behaves like an algebraic stack with respect to (<ref>): [Homogeneous CFG]A CFG (not necessarily an algebraic stack) is said to be homogeneous if for every pushout as in Definition <ref>,we have (<ref>) is an equivalence of categories.This definition is a natural extension of <cit.>, which was stated only in the context of artinian algebras.For much of what we have to say, we will only be interested in homogeneity with respect to morphisms Q → R that are isomorphisms on topological spaces.However, in practice, it is little more difficult to verify homogeneity in the generality we have formulated it.As we have seen in Theorem <ref>, homogeneity is a necessary condition for a stack over schemes to be algebraic.It is not sufficient, but in the presence of a reasonable finiteness condition, it does guarantee formal representability by an algebraic stack (see Theorem <ref>). .2 cmThe following theorem is very helpful for checking homogeneity. It was asserted by Schlessinger in <cit.> but proved there only for free modules; Milnor gave a proof for projective modules <cit.>; a proof in the general case can be found in <cit.>.[Schlessinger, Milnor, Ferrand] Let 𝒬 be the CFG whose fiber over a scheme S is the category of flat, quasicoherent 𝒪_S-modules.Then 𝒬 is homogeneous. In language closer to the original statements, this says that if one has a cartesian diagram of commutative ringsBB'[l]_inf. A[u]A' [u] [l]in which B' → B (and therefore also A' → A) is a nilpotent extension then it is equivalent to specify either of the following data: * a flat A'-module M', or * a flat A-module M, a flat B'-module N', a flat B-module N, and an A-module map M → N and a B'-module map N' → N inducing by adjunction isomorphisms M⊗_AB → N and N'⊗_B'B → N of B-modules.One direction of the correspondence sends M' to M = M' ⊗_A' A, N' = M' ⊗_A' B', N = M' ⊗_A' B with the canonical maps M' ⊗_A' A → M' ⊗_A' B and M' ⊗_A' B' → M' ⊗_A' B.The other direction has M' = M ×_N N'. §.§.§ Tangent bundle to a homogeneous stackTo begin to appreciate the significance of homogeneity, consider that in general, the tangent bundle of a stack, and even the tangent bundle of a presheaf, need not be a very well behaved object:it might not even be a vector space.However, the tangent space of a homogeneous stack has all of the familiar structure one expects: *(projection to base) The map ℤ[ϵ] / (ϵ^2) →ℤ sending ϵ to 0 induces closed embeddings R → R[ϵ] for all schemes R.For any S-scheme R, we use this to obtain a morphismT_𝒳/S(R) = 𝒳(R[ϵ]) →𝒳(R), whence a projection T_𝒳/S→𝒳. (This does not require homogeneity.) * (zero section) The map ℤ→ℤ[ϵ] / (ϵ^2) induces R[ϵ] → R.This gives a map 𝒳(R) →𝒳(R[ϵ]) = T_𝒳/S(R), whence a section 𝒳→ T_𝒳/S of the aforementioned projection.This is the zero section in the vector bundle when T_𝒳/S is a vector bundle. (This does not require homogeneity.)* (addition law) We have ℤ[ϵ_1,ϵ_2]/(ϵ_1^2, ϵ_1 ϵ_2, ϵ_2^2) ∼⟶𝔻×_ℤ𝔻 given byx+yϵ_1+zϵ_2↦ (x+yϵ_1,x+zϵ_2). In addition to the two projections 𝔻×_ℤ𝔻→𝔻, there is a third mapsending both ϵ_1 and ϵ_2 to ϵ. We obtain a commutative diagram whose outer square is cartesian (in fact the upper right and lower left are cartesian as well):@!C=5em@R=1emℤℤ[ϵ_1] [ll]ℤ[ϵ] [lu]ℤ[ϵ_2] [uu] ℤ[ϵ_1,ϵ_2]/(ϵ_1,ϵ_2)^2 [ll]_<>(0.5)ϵ_1↦ 0[uu]_<>(0.65)ϵ_2 ↦ 0[lu]_ϵ_1,ϵ_2↦ϵApplying this to any S-scheme R, we obtain a pushout diagram with an extra morphismσ: R[ϵ] → R[ϵ_1, ϵ_2],whereR[ϵ_1, ϵ_2] = R ×Spec (𝔻×_ℤ𝔻) (it is also the vanishing locus of ϵ_1 ϵ_2 in R[ϵ_1][ϵ_2]):@!C=5em@R=1em RR[ϵ_1] @<-[ll]R[ϵ]@<-[lu] R[ϵ_2] @<-[uu] R[ϵ_1,ϵ_2] @<-[ll] @<-[uu]@<-[lu].Now consider maps from the diagram above into our homogeneous stack 𝒳; the map 𝒳(R[ϵ_1, ϵ_2]) →𝒳(R[ϵ_1]) ×_𝒳(R)𝒳(R[ϵ_2]) is an equivalence of categories.Combining these facts, and abbreviating T_𝒳/S to T, we have a diagram: .9T(R) ×_𝒳(R)T(R) = 𝒳(R[ϵ_1]) ×_𝒳(R)𝒳(R[ϵ_2]) 𝒳(R[ϵ_1, ϵ_2]) 𝒳(R[ϵ]) = T(R)This gives the fibers of T_𝒳/S(R) →𝒳(R) an addition law; i.e., given an R-point R→𝒳, the collection of all extensions R[ϵ]→𝒳 has an induced addition law.This coincides with the addition law of the vector space structure in the familiar situation where 𝒳 is a smooth scheme and R is the spectrum of a field.* (scalar multiplication) If λ∈Γ(R, 𝒪_R), there is an induced map:@R=0pt𝒪_R + ϵ𝒪_R [r]𝒪_R + ϵ𝒪_R f + ϵ v @|->[r]f + ϵλ vgiving a map R[ϵ]→ R[ϵ]. Applying 𝒳,this induces a mapT_𝒳/S(R) → T_𝒳/S(R)commuting with the projection to 𝒳(R).This is the action of scalars in the vector space structure when R is a point and 𝒳 is smooth.* (vector space structure) The axioms of a vector space can be verified in similar ways, occasionally making use of 𝔻×_ℤ𝔻×_ℤ𝔻 = ℤ[ϵ_1, ϵ_2, ϵ_3] / (ϵ_1, ϵ_2, ϵ_3)^2.We leave these verifications to the reader.* (Lie bracket) The Lie bracket will not be used in the rest of the paper, and may be skipped. There is a map𝔻⊗_ℤ𝔻 = ℤ[ϵ_1, ϵ_2] / (ϵ_1^2, ϵ_2^2) →ℤ[ϵ_1, ϵ_2] / (ϵ_1, ϵ_2)^2 = 𝔻×_ℤ𝔻sending ϵ_1 ϵ_2 to 0. There is also an isomorphism(𝔻⊗_ℤ𝔻) ×_ℤ[ϵ_1, ϵ_2] / (ϵ_1, ϵ_2)^2 (𝔻⊗_ℤ𝔻) ≅ (𝔻⊗_ℤ𝔻) ×_ℤℤ[ϵ_1 ϵ_2] / (ϵ_1 ϵ_2)^2sending (a,b) on the left to (a,(a (ϵ_1, ϵ_2)) + b-a) on the right (this choice of isomorphism will give rise to the choice of a sign for the Lie bracket).This all leads to a commutative diagram, in which both the small square and the outer rectangle are cartesian: 1exℤ[ϵ_1, ϵ_2]ℤ[ϵ_1][ϵ_2] [l]ℤ[ϵ_1 ϵ_2] @/_20pt/[ll]ℤ[ϵ_1][ϵ_2] [u]ℤ[ϵ_1][ϵ_2] ×_ℤ[ϵ_1, ϵ_2]ℤ[ϵ_1][ϵ_2] @-^-∼[r] [l] [u]ℤ[ϵ_1][ϵ_2] ×_ℤℤ[ϵ_1 ϵ_2] [u] Suppose v, w ∈ T_𝒳(𝒳) are vector fields on 𝒳.We view them as maps (see Definition <ref> and Remark <ref> for notation):v : 𝒳[ϵ_1] →𝒳w : 𝒳[ϵ_2] →𝒳 .We obtain two morphismsid× v : 𝒳[ϵ_1]×_𝒳𝒳[ϵ_2] →𝒳[ϵ_2] w×id : 𝒳[ϵ_1]×_𝒳𝒳[ϵ_2] →𝒳[ϵ_1];note that the fibered product is over the standard projections on both factors. Note that𝒳[ϵ_1] ×_𝒳𝒳[ϵ_2] = 𝒳[ϵ_1][ϵ_2] = 𝒳×Spec (𝔻⊗𝔻)=𝒳×Specℤ[ϵ_1,ϵ_2]. The two compositions v(id× w), w(v ×id): 𝒳[ϵ_1] ×_𝒳𝒳[ϵ_2] →𝒳 are retractions(they agree with the identity when restricted to the canonical inclusion of 𝒳 in 𝒳[ϵ_1] ×_𝒳𝒳[ϵ_2]) that agree with one another when restricted to 𝒳[ϵ_1, ϵ_2] ⊆𝒳[ϵ_1] ×_𝒳𝒳[ϵ_2]. In other words, they give us a commutative diagram, dual to (<ref>):𝒳[ϵ_1, ϵ_2] [r] [d]𝒳[ϵ_1][ϵ_2] [d] @/^25pt/[ddr]𝒳[ϵ_1][ϵ_2] [r] @/_15pt/[drr]𝒳[ϵ_1][ϵ_2] ⨿_𝒳[ϵ_1, ϵ_2]𝒳[ϵ_1][ϵ_2]@–>[dr] 𝒳All of the morphisms restrict to the identity on 𝒳.Now we restrict under the inclusion 𝒳[ϵ_1 ϵ_2] ⊂𝒳[ϵ_1][ ϵ_2] ×_𝒳[ϵ_1, ϵ_2]𝒳[ϵ_1][ϵ_2] to get[v,w] : 𝒳[ϵ_1 ϵ_2] ⊂𝒳[ϵ_1][ϵ_2] ⨿_𝒳𝒳[ϵ_1 ϵ_2] 𝒳[ϵ_1][ ϵ_2] ⨿_𝒳[ϵ_1, ϵ_2]𝒳[ϵ_1][ϵ_2] →𝒳The point is that the homogeneity of 𝒳 ensures the existence of the dashed arrow above, and the composition of the dashed arrow with the canonical inclusion of 𝒳[ϵ_1 ϵ_2]. For the reader's convenience, we include a verification that this does indeed compute the Lie bracket when 𝒳 is representable by an affine scheme Spec A.In that case, v and w correspond to derivations δ_1 and δ_2 from A to itself.We consider these as ring homomorphisms:id + ϵ_i δ_i : A → A + ϵ_i AThe map v(id× w) is dual to the compositionAA + ϵ_1 AA + ϵ_1 A + ϵ_2 A + ϵ_1 ϵ_2 Asending f ∈ A to f + ϵ_1 δ_1(f) + ϵ_2 δ_2(f) + ϵ_1 ϵ_2 δ_2 ∘δ_1(f).To be clear, the second map is obtained from id + ϵ_2 δ_2 by application of A[ϵ_1] ⊗_A (-) = ℤ[ϵ_1] ⊗_ℤ (-) = (-)[ϵ_1] and carries f_0 + ϵ_1 f_1 to (f_0 + ϵ_2 δ_2(f_0)) + ϵ_1(f_1 + ϵ_2 δ_2(f_1)). The map w(v ×id) sends f to f + ϵ_1 δ_1(f) + ϵ_2 δ_2(f) + ϵ_1 ϵ_2 δ_1 ∘δ_2(f).Taking the difference of these recovers the derivation δ_2 δ_1 - δ_1 δ_2. Suppose ξ is a k-point of 𝒳. The discussion above shows that T_𝒳(ξ) has all the trappings of a k-vector space structure, except that T_𝒳(ξ) is a groupoid that may not be equivalent to a set.Nevertheless, it is useful to think of T_𝒳(ξ) as a `2-vector space'.In fact, given a scheme R and a morphism ξ:R→𝒳, we can extract two important invariants from this groupoid:T_𝒳^-1(ξ) := Aut_T_𝒳(ξ)(0)T_𝒳^0(ξ) := T_𝒳(ξ) / isomHere 0, the zero section 0:R→ T_𝒳(ξ),corresponds to an object of the groupoidT_𝒳(ξ), and Aut_T_𝒳(ξ)(0) is the automorphism group of that object.Likewise, T_𝒳(ξ) / isom is the set of isomorphism classes of objects of the groupoid. The following lemma is a formal consequence of the discussion above: When k is a field and ξ:Spec k →𝒳, the sets T_𝒳^-1(ξ) and T_𝒳^0(ξ) are k-vector spaces. It will be important to have a relative variant of the tangent bundle:[Relative tangent bundle] Let f : 𝒳→𝒴 be a morphism of CFGs over 𝖲.We define the relative tangent bundle T_f = T_𝒳/𝒴 to be the kernel of the morphism of stacks T_𝒳/S→ T_𝒴/S.In other words, the relative tangent bundle is the fiber product of stacksT_𝒳/𝒴 = T_𝒳/S×_T_𝒴/S𝒴 = T_𝒳/S×_f^-1 T_𝒴/S𝒳where the morphisms 𝒴→ T_𝒴/S and 𝒳→ f^-1 T_𝒴/S are the zero sections. In slightly more concrete terms, a section of T_𝒳 / 𝒴 is a section of T_𝒳/S, together with an isomorphism between its image in T_𝒴 and the zero section.§.§.§ Relative homogeneity [Relative homogeneity] Let f : 𝒳→𝒴 be a morphism of CFGs over 𝖲.We say that f is homogeneous, or that 𝒳 is homogeneous over 𝒴, if for any scheme S and any morphism S →𝒴, the CFG 𝒳_S=𝒳×_𝒴S is homogeneous. Another way of formulating the definition is that for any cocartesian diagram (<ref>) and any compatible objects (we leave it to the reader to formulate compatibility precisely)η' ∈𝒴(R') η∈𝒳(R) ξ' ∈𝒳(Q') ξ∈𝒳(Q)there is a η∈𝒳(R') inducing all of them, and this η is unique up to unique isomorphism. The following lemma is proved by a standard formal argument: * Let f : 𝒳→𝒴 and g : 𝒴→𝒵 be morphisms of CFGs over 𝖲.If g is homogeneous then f is homogeneous if and only if gf is.* The base change of a homogeneous morphism of CFGs is homogeneous.§.§ Deformation theory When we speak of the deformation theory of a stack 𝒳 we mean extending morphisms R→𝒳 to morphisms R'→𝒳, where R' is an infinitesimal extension of R.The definition of the tangent space of a stack in <ref> connects the tangent space of a stack 𝒳 to the deformation theory of the objects it parameterizes.Indeed, if ξ∈𝒳(k) then T_𝒳(ξ) over ξ is precisely the groupoid of extensions of ξ to ξ' ∈𝒳(k[ϵ] / (ϵ^2)).We now work out several examples that will be used in the construction of the moduli of Higgs bundles.Our main focus is on using the deformation theory to show that various stacks are homogeneous. §.§.§ Deformations of morphisms of vector bundlesHere we discuss the deformation theory for morphisms of vector bundles. From our perspective, the main point isLemma <ref> (see also Remark <ref>) on homogeneity.Let π : X → S be a flat family of schemes over S and let E and F be two vector bundles on X.Let ℋ = ℋom_X/S(E, F)be the S-sheaf of morphisms from E to F (see <ref>).We compute the tangent space T_ℋ/S(ξ)of ℋ over S at an R-point ξ : E_R → F_R, where R→ S is an S-scheme, and E_R and F_R are the pullbacks of E and F to R.An element of T_ℋ/S(ξ) is an extension of the S-map R →ℋ to R[ϵ] such that the composition R[ϵ] →ℋ→ S is the same as the composition R[ϵ] → R → S of the canonical retraction R[ϵ] → R and the fixed map R → S.In other words, it is a morphism of vector bundles E_R[ϵ]→ F_R[ϵ] that reduces to u modulo ϵ.The vector bundle E_R[ϵ] is identified with E_R+ ϵ E_R, and similarly for F_R[ϵ]. Notice that ξ(e_0 + ϵ e_1) = ξ(e_0) + ϵξ(e_1) gives one such morphismE_R[ϵ]→ F_R[ϵ] that reduces to ξ modulo ϵ.If ξ' is any other then the difference ξ' - ξ is a map E_R →ϵ F_R ≅ F_R,so we get an identification T_ℋ/S(ξ) ≃ℋ(R).As this identification is natural, we getT_ℋ/S = ℋ×_S ℋ .Of course, it is no surprise that the tangent space of ℋ is ℋ itself, since ℋ already has a linear structure.We have T^-1_ℋ/S(ξ)= 0 T^0_ℋ/S(ξ)= Hom(E_R, F_R)This gives the tangent space to ℋ over S, but does not explain the higher order infinitesimal structure.For that we must pose a more general lifting problem:given ξ∈ℋ(R) and an arbitrary square-zero extension R' of R as an S-scheme, can ξ be lifted to ξ' ∈ℋ(R')? To answer this question, we consider it locally in X_R, that is, we cover X by open subsets U with corresponding extensions U' over R' and ask for extensions of E_R |_U → F_R |_U to E_R'|_U'→ F_R'|_U'.We make two observations:* there is a cover of X_R' by open sets U' such that ξ|_X_R∩ U' extends to a morphism E_U'→ F_U'. * if ξ' and ξ” are any two extensions, then ξ' - ξ” may be viewed as a morphism E_R → F_R ⊗π^∗ Jwhere J is the ideal of R in R' and, by abuse of notation,π:X_R→ R is the morphism obtained from π:X→ S by pullback.These observations combine to imply that there is a Hom(E_R,F_R ⊗π^∗ J)-torsor P on X (in the Zariski topology) whose sections are in bijection with the lifts of ξ to H(R') (those who would rather avoid the language of torsors may obtain the following lemma by a Čech cohomology calculation).This yields a deformation-obstruction theory:Let X be an S-scheme, let E and F be vector bundles on X, and let R ⊆ R' be a square-zero extension of S-schemes with ideal J.Associated to any homomorphism ξ : E_R → F_R there is an obstruction ω∈ H^1( X_R, Hom(E_R,F_R ⊗π^∗ J)) whose vanishing is equivalent to the existence of an extension of ξ to some ξ' : E_R'→ F_R'.If there is at least one extension then the set of all extensions possesses a simply transitive action of H^0(X_R, Hom(E_R,F_R ⊗π^∗ J)) = Hom(E_R,F_R ⊗π^∗ J).Beyond its finite dimensionality, the particular deformation-obstruction theory is not actually necessary for the proof of algebraicity.What is important is homogeneity: The functor ℋom_X/S(E,F) is homogeneous. If f : Q → R is an affine morphism of S-schemes and Q ⊆ Q' is a square-zero extension of S-schemes, let R' be the pushout (as in Theorem <ref>), with f' : Q' → R' denoting the tautological morphism.(Since Q and Q' have the same underlying topological space, we do not bother to introduce notation for the inclusion Q ⊆ Q'.)Since Q and Q', as well as R and R' have the same underlying topological space, and we have a natural morphismF_Q'→ F_Q, we can push forward to obtain a morphism f'_*F_Q'→ f_*F_Q.Using adjunction we have have that the identityf^*F_R=F_Q induces a morphism F_R→ f_*F_Q.Thus we obtain a fibered product f'_*F_Q'×_f_*F_QF_R. Assume we have v ∈ H(R) and u' ∈ H(Q'), both extending u = f^∗ v.Then u' gives usf'^∗ E_R' = E_Q' F_Q' = f'^∗ F_R',whence E_R'→ f'_∗f'^∗ F_R' = f'_∗ F_Q' by adjunction.We also have a map E_R'→ E_RF_R.These induce the same map E_R'→ f_∗ F_Qso we get a mapη' : E_R'→ f'_∗ F_Q'×_f_∗ F_Q F_Rby the universal property of the fiber product.On the other hand, the canonical map γ : F_R'→ f'_∗ F_Q'×_f_∗ F_Q F_R is an isomorphism, by Theorem <ref>, or directly: γ is an isomorphism modulo J, since it reduces to the identity F_R = f_∗ F_Q ×_f_∗ F_Q F_R, and the kernel of reduction is the fiber product of the kernels, namely (f_∗ F_Q ⊗π^∗ J) ×_0 0 = F_R ⊗π^∗ J.Therefore γ is an isomorphism by the 5-lemma.To conclude we can view η' as a map E_R'→ F_R' by composition with γ^-1. As was pointed out above (see Proposition <ref>), under mild hypotheses,a result of Lieblich implies that ℋom_X/S(E,F) is an algebraic space over S, locally of finite type.In particular, thisalso showsthat ℋom_X/S(E,F) is homogeneous.However, Lieblich's proof relies on the homogeneity, so this reasoning is actually circular. §.§.§ Deformations of vector bundles Here we discuss the deformation theory for vector bundles; see also <cit.>.The main points are Corollary <ref> and Lemma <ref> (see also Remark <ref>).Let π : X→ S be a family of schemes over S.Suppose that R is an S-scheme and E ∈𝖥𝗂𝖻_X/S(R) is a vector bundle over X_R. Let R' be a square-zero S-scheme extension with ideal J.We askwhether E can be extended to E' ∈𝖥𝗂𝖻_X/S(R'), and if so, in how many ways.Again we make several observations: * there is a cover of X_R' by open subsets U' such that E |_X_R ∩ U' can be extended in at least one way to a vector bundle on U';* if E' and E” are two extensions of E |_X_R ∩ U' to U' then there is a cover of U' by open subsets V' such that E' |_V'≃ E”|_V' as extensions of E |_X_R ∩ V';* if u, v : E' |_V'→ E”|_V' are two isomorphisms of extensions of E |_X_R ∩ V' then u and v differ by a homomorphism E |_X_R ∩ V'→E |_X_R ∩ V'⊗π^∗ J. Choosing a suitable cover, we therefore obtain a Čech 2-cocycle for the sheaf of groups Hom(E, E⊗π^∗ J).This is a coboundary if and only if a deformation exists.A more careful analysis shows that the isomorphism classes of all deformations then correspond to 1-cocycles modulo coboundaries.Here is the statement in its usual form:Fix a flat family of schemes π : X → S and a square-zero S-extension R ⊆ R' with ideal J.A vector bundle E on X_R induces an obstruction ω∈ H^2(X_R, Hom(E, E ⊗π^* J)) whose vanishing is equivalent to the existence of an extension of E to X_R'.If ω = 0, the set of isomorphism classes of extensions is a principal homogeneous set under a natural action of H^1(X_R, Hom(E, E ⊗π^* J)).The automorphisms (as an extension) of any given extension are canonically H^0(X_R, Hom(E, E ⊗π^* J)).In particular, this lemma gives the tangent space of 𝖥𝗂𝖻_X/S: Suppose that R is an S-scheme and E is a vector bundle over X_R.We have T_𝖥𝗂𝖻_X/S(E) = BHom(E, E)(X_R)where E is the universal vector bundle on 𝖥𝗂𝖻_X/Sand the prefix B denotes the classifying stack (Section <ref>). In particular,T^-1_𝖥𝗂𝖻_X/S(E) = Hom(E,E) T^0_𝖥𝗂𝖻_X/S(E) = Ext^1(E,E) . We have just seen that sections of T_𝖥𝗂𝖻_X/S correspond to 1-cocycles for Hom(E,E) modulo coboundaries.But 1-cocycles modulo coboundaries also classify torsors. The Čech calculations can be abstracted into the observation that properties (i) – (iii) above imply there is a gerbe 𝒢 over X_R(see e.g., <cit.> for the definition of a gerbe) whose sections correspond to extensions of E to E' ∈𝖥𝗂𝖻_X/R(R'), and that this gerbe is banded by the sheaf of abelian groups Hom(E_R, E_R ⊗π^∗ J).Giraud classifies banded gerbes cohomologically: [<cit.>] Let 𝒢 be a gerbe on X, banded by an abelian group A.There is an obstruction ω∈ H^2(X, A) to the existence of a global section of 𝒢.Should this obstruction vanish, global sections up to isomorphism form a principal homogeneous set under the action of H^1(X,A).Automorphisms of any given section are in canonical bijection with H^0(X,A). We also record the homogeneity of 𝖥𝗂𝖻_X/S:For any scheme X, the CFG 𝖥𝗂𝖻_X/S is homogeneous. Since S is representable, it is homogeneous by Theorem <ref>.Therefore by Lemma <ref>, it is sufficient to show that 𝖥𝗂𝖻_X/S is homogeneous over S.We will content ourselves to sketch the construction of the inverse to the functor𝖥𝗂𝖻_X/S(R') →𝖥𝗂𝖻_X/S(Q') ×_𝖥𝗂𝖻_X/S(Q)𝖥𝗂𝖻_X/S(R)associated to a cocartesian diagramQ @^(->[r] [d] Q' [d] R [r] R'of S-schemes, where Q → R is affine and Q ↪ Q' is an infinitesimal extension.An object of the right side of (<ref>) consists of vector bundles E' on X_Q', F on X_R, and E on X_Q, along with identifications E' |_X_Q = E = F |_X_Q.By applying <cit.> to a cover of X_R' by open affines and then to covers of the intersections by open affines, we obtain a vector bundle F' on X_R' restricting to E' on X_Q', to E on X_Q, and to F on X_R, as required.This also follows from the fact that 𝖥𝗂𝖻_X/S is an algebraic stack, but, as in Remark <ref>, this reasoning is circular. Let ℱ be the stack of triples (E,F,σ) where E and F are vector bundles on X and σ : E → F is a morphism of vector bundles.Then ℱ is homogeneous. We have a morphism ℱ→𝖥𝗂𝖻_X/S×_S 𝖥𝗂𝖻_X/S.We have just seen that 𝖥𝗂𝖻_X/S is homogeneous, so by two applications of Lemma <ref>, it follows that 𝖥𝗂𝖻_X/S×_S 𝖥𝗂𝖻_X/S is homogeneous.Lemma <ref> says that ℱ is relatively homogeneous over 𝖥𝗂𝖻_X/S×_S 𝖥𝗂𝖻_X/S, so we mayconclude by another application of Lemma <ref>. §.§.§ Deformations of nodal curvesWe now discuss the deformation theory of nodal curves.[Nodal curve] Let S be a scheme.A nodal curve over S is an algebraic space C and a projection π : C → S that is étale-locally isomorphic to Spec𝒪_S[x,y] / (xy - t) for a local section t of 𝒪_S.That is, there is an étale cover of S by affines U = Spec A and an étale cover of π^-1 U by schemes V, each of which admits an étale map to Spec A[x,y] / (xy - t) for some t ∈ A. We shall write 𝒩 for the stack over schemes whose S-points are the families of nodal curves over S. A more conventional definition of a nodal curve over S is as a flat family π : C → S whose fibers are 1-dimensional, reduced schemes whose only singularities are ordinary double points.It follows from <cit.> that the two notions are equivalent. We will show that nodal curves form a homogeneous stack (Lemma <ref>), compute the tangent space of this stack (Lemmma <ref>), and show that it is formally smooth (Corollary<ref>).This whole section could easily be adapted to curves with locally planar singularities (or to more general schemes of finite type with hypersurface singularities), but for concreteness, we stick to nodal curves. The method used in the last section to study deformations of vector bundles works quite well for deformations of smoothcurves, and even families of smooth schemes π : X → S.If S' is a square-zero extension of S with ideal J, one discovers that extensions of X to S' always exist locally, that any two deformations are locally isomorphic, and that any two deformations are related by a section of T_X/S⊗π^*J, from which it follows by a Čech calculation (as in the last section) that (see <cit.>):* there is obstruction to the existence of a flat extension X' over S extending X lying in H^2(X, T_X/S⊗π^∗ J);* should a deformation exist, the isomorphism classes of flat extensions form a torsor under H^1(X, T_X/S⊗π^∗ J);* automorphisms of a fixed flat extension are canonically H^0(X, T_X/S⊗π^∗ J). The deformation theory of nodal curves introduces a new complication:It is still the case that deformations exist locally, and one can easily compute that automorphisms of deformations can be identified with sections of T_X/S⊗ J.However, it is no longer the case any two deformations are locally isomorphic, since the node xy = 0 has a 1-parameter family of first order deformations xy = λϵ.A naive Čech calculation will therefore not suffice, but with a little more effort, we will see that the deformations and obstructions for nodal curves can be classified in much the same way as for smooth curves.Suppose that C is an S-point of 𝒩. An S-point of T_𝒩 lying above C is a cartesian diagram (<ref>), in which C' is an S[ϵ]-point (see Definition <ref> for notation) of 𝒩: C [r] [d]_πC' [d] S [r] S[ϵ].We consider the more general problem of extending an S-point C of 𝒩 to a S[ϵ J]-point: C [r] [d]_πC' [d] S [r]@–>@/_1pc/[l] S[ϵ J].Here J is a quasicoherent sheaf on S and C' is an S[ϵ J]-point of 𝒩.Since there is a canonical retraction S[ϵ J] → S, we can consider C' as an S-scheme.As C is the closed sub-scheme of C' determined by π^∗ J (with π^*J^2=0), we mayform the exact sequence (<ref>): 0 →π^∗ J →𝒪_C ⊗_𝒪_C'Ω_C'/S→Ω_C/S→ 0The right exactness of this sequence follows from general principles and does not depend on the fact that C is a nodal curve.To see the left exactness, we may work étale-locally and assume:S = Spec A S' = Spec A' C = Spec BC' = Spec B' B = A[x,y] / I B' = A'[x,y] / I' I = (xy - t) I' = (xy - t')In fact, the local structure of nodes implies we can arrange for C and C' to be étale over A[x,y] / I and A'[x,y] / I', respectively. Then the sequence (<ref>) on C is pulled back via the étale (and in particular flat) map C →Spec B from the corresponding sequence over Spec B.For the sake of proving (<ref>) is exact, we can therefore replace C with Spec B and C' withSpec B' without loss of generality.Our diagram of rings isB=A[x,y]/(xy-t) B'=A'[x,y]/(xy-t') [l] A [u] @–>@/^1pc/[r] [l] [u] A'=A[ϵ J].We are trying to show the exactness of0→ B⊗_A J → B ⊗_B'Ω_B'/A→Ω_B/A→ 0. The left exactness of (<ref>) is equivalent to showing that, for every injective [Recall that a morphism M'→ M of modules over a ring A is injective if and only if for every injective A-module I ,Hom_A(M,I)→Hom_A(M',I) is surjective.To see this, choose an inclusionM'↪ I for some injective module I.Then we get a map M → I extending the inclusion M' ⊆ I, so M' → M must be injective.] B-module M,the mapHom_B(B⊗_B'Ω_B'/A,M)→Hom_B(B⊗_A J,M) is surjective; i.e.,every B-module homomorphism B⊗_A J→ M extends to a morphism B⊗_B'Ω_B'/A→ M.Using tensor-hom adjunction, and the universal property of the module of Kähler differentials, this translates into the statement that, every A-module homomorphism J → M should extend to an A-derivation B' → M:B'=A[ϵ J][x,y]/(xy-t')@–>[rd]^∂ J [r]^φ@^(->[u]^·ϵM. First we will extendφ to a derivationδ' : A[ϵ J][x,y] → M.To do this, we will invoke the identificationsHom_A(J,M)=Hom_A-alg,Id_A(A[ϵ J],A[ϵ M]) = Der_A(A[ϵ J],M).On the left, we take a homomorphism φ:J→ M and send it to the ring homomorphism Id_A⊕φ:A⊕ϵ J→ A⊕ϵ M.The group in the middle consists of the A-algebra homomorphisms that are the identity on the first term.On the right, we just project to get a derivation. Similarly, we have indentifications Hom_A-alg,Id_A(A[ϵ J][x,y],A[ϵ M]) = Der_A(A[ϵ J][x,y],M).In summary, the homomorphism φ:J→ M induces a ring homomorphism φ̃:A[ϵ J]→ A[ϵ M].We can extend this toφ̂:A[ϵ J][x,y]→ A[ϵ M] by sending x and y to any element of M.This then gives a derivation δ' as desired.The choices of such extensions δ' are a torsor under Der_A(A[x,y], M), and we will adjust δ' by such a derivation so that it descends to a derivation B' → M.The obstruction to descending δ' to B' is the homomorphism δ' |_I', so we want to find a derivation δ∈Der(A[x,y], M) such that the restriction of δ to I' (via the composition I' ⊆ A'[x,y] → A[x,y]) agrees with δ'. To execute this, let us take a fixed extension δ'. Thenδ'(I'J) ⊆ I' δ(J) + J δ(I') = 0 because M is a B-module, and therefore I' M = JM = 0.This implies δ|_I' descends to I'/JI' = I.We also have δ'(I'^2) = I' δ'(I') = 0, for the same reason, so δ' gives us a homomorphism u : I/I^2 → M.Now, we have an exact sequence0 → I/I^2 → B ⊗_A[x,y]Ω_A[x,y] / A→Ω_B/A→ 0,either by <cit.> or a direct verification.Since M is injective, the homomorphism u : I/I^2 → M extends to a derivation δ : A[x,y] → M (again we are using tensor-hom adjunction and the universal property of the sheaf of Kähler differentials).Trivially extendingδ to A'[x,y] we obtain a derivationδ'-δ on A'[x,y]. We have that δ'-δ =0 on I' essentially by construction.Moreover, δ' - δ agrees with δ' |_J = φ on J because δ vanishes there.Thus δ' - δ descends to a derivation in Der_A(B,M) that agrees with φ on J, as required.We conclude that (<ref>) is exact..2cmConversely, given any extension of 𝒪_C-modules0 →π^∗ J →Ω' →Ω_C/S→ 0we can define an extension of C/S to S[ϵ J].Namely,one may define𝒪_C' = 𝒪_C×_Ω_C/SΩ', where the map from 𝒪_C to Ω_C/S is the universal differential, d. This is clearly an 𝒪_C-module, but it is not a priori obvious that 𝒪_C' is equipped with a ring structure.The ring structure can be constructed easilyby factoring d as𝒪_C 𝒪_C +ϵΩ_C/S⟶Ω_C/Sand recognizing that𝒪_C'≃𝒪_C ×_𝒪_C+ϵΩ_C/S (𝒪_C +ϵΩ').Here the map on the right is𝒪_C+ϵΩ'→𝒪_C+ϵΩ_C/S a+ω'↦ a+ω̃'where we are using the given map Ω'→Ω_C/S, ω'↦ω̃'; this is the standard map takingHom_𝒪_C(Ω',Ω_C/S)→Hom_𝒪_C-alg,Id(𝒪_C[ϵΩ'],𝒪_C[ϵΩ_C/S]). This above description of 𝒪_C' is as afiber product of rings, hence is naturally equipped with a ring structure.The map C'→ S[ϵ J]is given topologically by C→ S, and at the level of sheaves by a map𝒪_S[ϵ]=𝒪_S⊕ϵ J →𝒪_C'= 𝒪_C ×_𝒪_C⊕ϵΩ_C/S (𝒪_C ⊕ϵΩ').We are given maps 𝒪_S→𝒪_C, and J→Ω'. The map 𝒪_S ⊕ϵ J→𝒪_C is definedby the given map on the first term, plus the zero map on the second term. The map 𝒪_S[ϵ J]→𝒪_C[ϵΩ'] is given by the map 𝒪_S[ϵ J]→𝒪_C[ϵπ^*J], and then the natural map 𝒪_C[ϵπ^*J]→𝒪_C[ϵΩ'].These maps agree on composition to 𝒪_C[ϵΩ_C/S], and so define a morphism to the fibered product.One can check that these processes are inverses of one another, and sothis yields an equivalence of categories between the tangent space of 𝒩at C/S and the category of extensions of Ω_C/S by π^*J. Let π : C → S be an S-point of the stack 𝒩 of all nodal curves.Then T_𝒩(C/S) = 𝐄𝐱𝐭(Ω_C/S, 𝒪_C) ,where we write 𝐄𝐱𝐭(A,B) for the groupoid of extensions of B by A. In particular,T^-1_𝒩(C/S)= Hom_𝒪_C(Ω_C/S,𝒪_C)=Γ(C, T_C/S) T^0_𝒩(C/S)= Ext^1_𝒪_C(Ω_C/S, 𝒪_C) .Observe that the discussion above never made use of the assumption that C be proper over S.Thus Ext^1(Ω_C/S, 𝒪_C) may be interpreted as the associated sheaf of the presheaf of isomorphism classes of local deformations of C.This allows us to interpret the 5-term exact sequence of the local-to-global spectral sequence for Ext(Ω_C/S, 𝒪_C) deformation theoretically:0 → H^1(C, T_C/S) →Ext^1(Ω_C/S, 𝒪_C) →Γ(C, Ext^1(Ω_C/S, 𝒪_C)) → H^2(C, T_C/S).If S is affine then H^2(C, T_C/S) = 0 because C has relative dimension 1 over S.The first term of the sequence may be interpreted, via a Čech calculation, as the set of isomorphism classes of locally trivial deformations of C.Somewhat imprecisely, we have:0 → (loc. triv. defs. of C) → (defs. of C) → (defs. of nodes of C) → 0.When S is not affine, the final term H^2(C, T_C/S) may be interpreted as the obstruction to finding a deformation of the curve realizing specified deformations of the nodes. .2 cmNow we turn to the more general problem of extending a curve over a general square-zero extension S ⊆ S' with ideal J:C [d]_π@–>[r]C' @–>[d] S [r] S'We will make use of homogeneity: The CFG of all nodal curves (not necessarily proper!) is homogeneous, as are the substack of all proper nodal curves, the substack of curves of fixed genus, and the substack of canonically polarized curves. LetQ ⊆ Q' be an infinitesimal extension of schemes and Q → R an affine morphism.Form the pushout R' of Q' and R under Q using Theorem <ref>.Letting 𝒩 denote the stack of all nodal curves, we construct the inverse to the map (<ref>). 𝒩(R') →𝒩(Q') ×_𝒩(Q)𝒩(R) Suppose that C is a nodal curve over Q, that C' is an extension of C to Q', and that C = D ×_R Q for nodal curve D over R and observe that C ⊆ C' is an infinitesimal extension and C → D is affine, so we can also form the pushout D' of C' and D under C (Theorem <ref>). By Theorem <ref>, D' is flat over R' and D' ×_R' R = D, D' ×_R' Q' = C', and D' ×_R' Q = C.We argue that D' is a nodal curve over R'.Since the fibers of D' over R' are the same as the fibers of D over R, the fibers of D' over R' are nodal curves.We have already seen that D' is flat over R', so we therefore only need to check that D' is locally of finite presentation over R'. One can verify easily that local generators and relations for 𝒪_D as an algebra over 𝒪_R lift to local generators and relations for 𝒪_D' as an algebra over 𝒪_R', implying it is locally of finite presentation as well. This proves the homogeneity of the stack of all nodal curves.For the remaining statements, note that they are stable under infinitesimal deformation.Now we return to the problem of completing (<ref>): * The problem of completing (<ref>) can be solved locally in C.Indeed, one may find a cover of S by affine open subschemes U = Spec A and elements t ∈ A such that π^-1 U has a cover by open affines V, each of which is étale over Spec B, with B = A[x,y] / (xy - t).If U' = Spec A' is the open subset of S' whose preimage in S is U then one may form B' = A'[x,y] / (xy - t') where t' is any lift of t to A'.Then Spec B' is a deformation of Spec B to Spec A' and this lifts uniquely to a deformation of V by <cit.>.We may thus select local deformations V' for each V in an open cover of C and attempt to glue.* Observe that 𝒪_S'×_𝒪_S𝒪_S'≃𝒪_S' + ϵ J,via the map (f,g) ↦ f + ϵ(g-f), so that S' ⨿_S S' ≃ S' ⨿_S S[ϵ J] and therefore by homogeneity, we have𝒩(S') ×_𝒩(S)𝒩(S') = 𝒩(S' ⨿_S S') ≃𝒩(S' ⨿_S S[ϵ J]) = 𝒩(S') ×_𝒩(S)𝒩(S[ϵ J]) .In other words, any two extensions of an open subset of C to a nodal curve over S' differ by a uniquely determined element of 𝒩(S[ϵ J]).* We have already seen that 𝒩(S[ϵ J]) may be identified with the category of extensions 𝐄𝐱𝐭(Ω_C/S, π^∗ J).Thus the first obstruction to gluing the deformations U' comes from making sure that the isomorphism classes of the deformations U' agree on overlaps.Since the isomorphism classes form a torsor under Ext^1(Ω_C/S, π^∗ J), this obstruction lies in H^1(C, Ext^1(Ω_C/S, π^∗ J)).* One can arrange very easily for this obstruction to vanish by working locally in S.Indeed, Ext^1(Ω_C/S, π^∗ J) is quasicoherent and is supported on the nodes of C, which is finite (and in particular affine) over S.Therefore when S is affine the first obstruction vanishes.* Now assuming that the first obstruction vanishes we can fix a compatible system of isomorphism classes of local deformations.We must determine whether one can find genuine local deformations within those isomorphism classes in a compatible way.For each open U ⊆ C in a suitable cover we select a deformation U' in the specified isomorphism class.If U_i and U_j are two such open sets then write U'_ij for the restriction of U'_i to U_i ∩ U_j.Because we have chosen the isomorphism classes compatibly, U'_ij≃ U'_ji, and we may select such an isomorphism φ_ij.Over triple overlaps the cocycle condition φ_ki∘φ_jk∘φ_ij = id obstructs the gluing.As the element φ_ki∘φ_jk∘φ_ij lies in the automorphism group of U'_ijk, which is canonically identified with Γ(U_ijk, Hom(Ω_C/S,π^∗ J)) we obtain a Čech 2-cocycle in H^2(C, Hom(Ω_C/S, π^∗ J)).Once again, this obstruction vanishes when S is affine, this time because C has relative dimension 1 over S and the coefficients are taken in a quasicoherent sheaf. The stack of all nodal curves satisfies the formal criterion for smoothness. We have just seen that all obstructions to infinitesimal deformation vanish over an affine base, which is precisely the formal criterion for smoothness. Any algebraic substack of the stack of all nodal curves that is locally of finite presentation and stable under infinitesimal extensions is smooth.In particular, the stack of all proper nodal curves is smooth and the open substack of all canonically polarized nodal curves is smooth. An observant reader will have noticed that the obstructions to (<ref>) constructed above lie in the graded pieces of Ext^2(Ω_C/S, π^∗ J) induced by the local-to-global spectral sequence.This is not an accident:There is a single obstruction in Ext^2(Ω_C/S, π^∗ J) whose vanishing is equivalent to the existence of a solution to (<ref>).We will discuss one way to obtain this obstruction in <ref>.In fact, one can dispense with the assumption that C be a curve over S and obtain an obstruction to deformation in Ext^2(𝐋_C/S, π^∗ J) where 𝐋_C/S is the relative cotangent complex of C over S, constructed by Illusie <cit.>.For a nodal curve, one has 𝐋_C/S = Ω_C/S.In general, 𝐋_C/S is concentrated in nonpositive degrees and Ω_C/S is the 0-th homology group. §.§.§ Simultaneous deformation of curves and vector bundles Let S be a scheme, C a curve over S, and E a vector bundle on C.We consider the problem of extending C and E to the scheme S[ϵ J] where J is a quasicoherent sheaf of 𝒪_S-modules.We begin by defining a sheaf of modules Υ_C(E) on C to play a role analogous to the one played by the module of differentials when we studied deformations of curves.For each 𝒪_C-module F, define Φ(F) to be the set of pairs (δ, φ) where δ : 𝒪_C → F is an 𝒪_S-derivation and φ : E → F ⊗ E is what we will call a δ-connection.That is, for any local sections f ∈𝒪_C and x ∈ E, we haveφ(fx) = δ(f) ⊗ x + f φ(x) .Then Φ(F) is naturally a covariant functor of F.There is a natural Γ(C, 𝒪_C)-module structure on Φ(F), in which λ∈Γ(C, 𝒪_C) acts byλ . (δ, φ) = (λδ, λφ) .There is also an evident exact sequence (<ref>).0 →Hom(E, F ⊗ E) →Φ(F) →Der_𝒪_S(𝒪_C, F)Here the map Hom(E, F ⊗ E) →Φ(F) sends φ to (0, φ) and the map Φ(F) →Der_𝒪_S(𝒪_C, F) sends (δ, φ) to δ.For each open set U⊆ C,setting Φ(F)(U):=Γ(U, Φ(F)) = Φ(F |_U), we obtain a covariant functorΦ:(𝒪_C)→ (𝒪_C)with Φ(F)=Φ(F)(C), and an exact sequence 0→Hom_𝒪_C(E^∨⊗ E,-)→Φ(-)→Der_𝒪_S(𝒪_C,-)=Hom_𝒪_C(Ω_C/S,-),in the sense that it is exact when applied to any 𝒪_C-module.In fact, we claim that this is surjective on the right, and that Φ (and hence Φ) is representable; i.e., there is asheaf of 𝒪_C-modules Υ_C/S(E)such that Φ(-)=Hom_𝒪_C(Υ_C/S(E),-), and Φ(-)= Hom_𝒪_C(Υ_C/S(E),-).The functors Φ and Φ defined above are both representable by the same sheaf of 𝒪_C-modules Υ_C/S(E) fitting into a short exact sequence (<ref>) inducing (<ref>):0 →Ω_C/S→Υ_C/S(E) →End(E) → 0 For the representability we give an explicit construction, although it is possible to obtain the same result more quickly by an application ofthe adjoint functor theorem. As mentioned above, to prove that Φ is representable it is equivalent to prove that Φ is representable, for which we can work locally in C.We can therefore assume E = 𝒪_C^⊕ n.But then if δ : 𝒪_C → F is any derivation, it itself gives a δ-connection by δ^× n(x_1, …, x_n) = (δ(x_1) ⊗ 1, …, δ(x_n) ⊗ 1).This gives a natural bijection between Φ(F) and Der_𝒪_S(𝒪_C, F) ×Hom(E, F ⊗ E) sending (δ, φ) to (δ, φ - δ^× n). But Der_𝒪_S(𝒪_C, -) is representable by Ω_C/S and Hom(E, (-) ⊗ E) is representable by E ⊗ E^∨.Thus Φ is representable.To check that the sequence(<ref>) in the lemma is exact, it is enough to check locally, which we have just done. We can view Der_𝒪_S(𝒪_C, π^*J) = Hom(Ω_C/S, π^*J) as the group of automorphisms of C[ϵπ^*J] that act as the identity on C and on π^*J (the group of `infinitesimal automorphisms').Similarly, we can view the groupΦ(π^*J) = Hom(Υ_C/S(E), π^*J) as the group of automorphisms of the pair (C[ϵπ^*J], E + ϵ J ⊗π^*E), the trivial square-zero extension of (C,E) by J, that act trivially on C, J, and E. In many deformation problems, deformations and obstructions are classified by analyzing the ways to glue, and the obstruction to gluing, deformations along infinitesimal automorphisms.Thus computing the infinitesimal automorphism group of the objects under consideration goes a long way toward understand how the object can deform.Indeed, it was by calculating the infinitesimal automorphisms of a deformation of (C,E) that we arrived at the definition of Φ in the first place. Now consider an extension (C', E') of (C, E) to S[ϵ J].Let π : C → S be the projection.We have a canonical projectionΥ_C'/S(E') →Υ_C/S(E)and it is easy to see that this is surjective.This induces an exact sequence0 →π^∗ J →𝒪_C ⊗_𝒪_C'Υ_C'/S(E') →Υ_C/S(E) → 0,where the morphism π^∗ J →𝒪_C ⊗_𝒪_C'Υ_C'/S(E') is the compositionπ^∗ J →𝒪_C ⊗_𝒪_C'Ω_C'/S→𝒪_C ⊗_𝒪_C'Υ_C'/S(E').As in the proof of the exactness of (<ref>), the right exactness and exactness in the middle of (<ref>) is formal using the universal properties.The left exactness follows easily from the exactness of (<ref>) and (<ref>).Indeed, End(E') is a flat 𝒪_C'-module, so tensoring the short exactsequence (<ref>) for E' on C'/S with 𝒪_C⊗_𝒪_C'(-), we obtain theshort exact sequence0 →𝒪_C ⊗_𝒪_C'Ω_C'/S→𝒪_C ⊗_𝒪_C'Υ_C'/S(E') →𝒪_C ⊗_𝒪_C'End(E') → 0. Therefore using also the exactness of (<ref>), the morphisms in (<ref>) are both injections. The discussion shows that each extension (C',E') of (C,E) induces an extension of Υ_C/S(E) by π^∗ J..2cmConversely, given such an extension,0 →π^∗ J →Υ' →Υ_C/S(E) → 0we can recover 𝒪_C' as δ^-1Υ' = 𝒪_C'×_ΥΥ', where δ : 𝒪_C →Υ_C/S(E) is the universal derivation; i.e., derivation δ in the universal pair (δ,φ) obtained from the identity mapunder the identification Hom(Υ_C/S(E),Υ_C/S(E))=Φ(Υ_C/S(E)).Likewise, we can recover E' by first tensoring (<ref>) by E and then pulling back via the universal δ-connection φ:@R=15pt 0 [r]π^∗ J ⊗ E @–>[r] @=[d] E' @–>[r] @–>[d] E [r] [d]^φ0 0 [r]π^∗ J ⊗ E [r]Υ' ⊗ E [r]Υ_C/S(E) ⊗ E [r] 0Summarizing the discussion above: Let 𝒢be the stack of pairs (C,E) where C is a nodal curve and E is a vector bundle on C.Suppose that (C,E) is an S-point of ℱ.ThenT_ℱ(C,E) = 𝐄𝐱𝐭(Υ_C/S(E), 𝒪_C) .In particular,T^-1_ℱ(C,E)= Hom(Υ_C/S(E), 𝒪_C) T^0_ℱ(C,E)= Ext^1(Υ_C/S(E), 𝒪_C) .Note that for a nodal curve C/S (or more generally a curve with locally planar singularities), applying Hom(-,𝒪_C) to Lemma <ref>, and utilizing Lemmas <ref>, <ref>, and Corollary <ref>, weobtain a long exact sequence@R=4em @C=1em0 [r] T^-1_𝖥𝗂𝖻_C/S(E) [r] T^-1_ℱ(C,E) [r] T^-1_𝒩(C/S)@-> `r/8pt[d] `/10pt[l] `^dl[ll]| `^r/3pt[dll] [dll]T^0_𝖥𝗂𝖻_C/S(E) [r] T^0_ℱ(C,E) [r] T^0_𝒩(C/S) @-> `r/8pt[d] `/10pt[l] `^dl[ll]| `^r/3pt[dll] [dll] Ext^2(End(E),𝒪_C) [r]Ext^2(Υ_C/S(E),𝒪_C) [r]Ext^2(Ω_C/S,𝒪_C)@->[r] ⋯Note that if S is affine, then the last row above is 0. There is an explicit treatment of infinitesimal deformations of pairs (C,E) where C is a curve and E is a line bundle in <cit.>, with infinitesimal automorphisms, deformations, and obstructions lying respectivelyin Ext^i(𝒫^1_X/S(E), E), i=0,1,2, where 𝒫^1(E) is the sheaf of principal parts of E, and fits in an exact sequence 0→Ω_C/S(E)→𝒫^1_X/S(E) → E→ 0. The same formulas do not give a deformation–obstruction theory for the pair (C,E) when the rank of E is larger than 1. Illusie provides a 2-step obstruction theory for deformations of the pair (C,E).The primary obstruction is to deforming C, which lies in this case in Ext^1(Ω_C/S, π^∗ J), and in general in Ext^2(𝐋_C/S, π^∗ J), as we have discussed (see Remark <ref>).A secondary obstruction in Ext^2(𝒫^1(E), E) obstructs the existence of a deformation of E that is compatible with a fixed deformation C' of C <cit.>.This latter obstruction may be constructed as the cup product of the class [C'] ∈Ext^1(𝐋_C/S, π^∗ J) and the Atiyah class <cit.>. One can arrive at this 2-step obstruction theory[We have only verified the obstruction groups coincide, not the obstruction classes.] from the extension (<ref>), which induces an exact sequence: Ext^2(End(E), π^∗ J) →Ext^2(Υ_C/S(E), π^∗ J) →Ext^2(Ω_C/S, π^∗ J) The first term may of course be identified canonically with Ext^2(E, E ⊗π^∗ J) since E is flat.For the sake of completeness, we also observe homogeneity: The stack of pairs (C,E) where C is a curve and E is a vector bundle on E is homogeneous. Let 𝒳 denote the stack of pairs (C,E) as in the statement of the lemma and let 𝒩 be the stack of nodal curves.The forgetful map 𝒳→𝒩 is relatively homogeneous by Lemma <ref> and 𝒩 is homogeneous by Lemma <ref> so 𝒳 is homogeneous by Lemma <ref>. §.§.§ Simultaneous deformation of curves, vector bundles, and morphisms of vector bundles We will consider a curve C over S, vector bundles E and F on C, and a homomorphism σ : E → F.We ask in how many ways these data can be extended to C', E', F', and σ' over S[ϵ J] where J is a quasicoherent sheaf on S.This time, deformations will be controlled by a complex rather than by a module.Imitating the last section we can construct a quasicoherent sheaf Υ_C/S(E,F)controlling simultaneous deformations of the two vector bundles E and F.This will be the universal example of a quasicoherent 𝒪_C-module, equipped with a derivationδ : 𝒪_C →Υ_C/S(E,F)and δ-connections,φ_E : E →Υ_C/S(E,F) ⊗ Eφ_F : F →Υ_C/S(E,F) ⊗ F.As it would be similar to the proof of Lemma <ref>, we will omit an explicit verification of the existence of such a universal object, as well as the construction of the natural exact sequence:0 →Ω_C/S→Υ_C/S(E, F) →End(E) ×End(F) → 0Before studying the question of deforming a homomorphism σ : E → F, we analyze how we can recover the deformation Hom(E',F') of Hom(E,F) from deformations of E and F, encoded as an extension of Υ_C/S(E,F) by π^∗ J. If σ : E → F is an homomorphism of vector bundles we obtain an element[σ, φ] ∈Υ_C/S(E,F) ⊗Hom(E,F) [σ, φ] := (id_Υ_C/S(E,F)⊗σ) ∘φ_E - φ_F ∘σ : E →Υ_C/S(E,F) ⊗ F .As [σ, φ] depends linearly on σ, this induces a homomorphism of 𝒪_C-modules:φ_Hom(E,F) : Hom(E,F) →Υ_C/S(E,F) ⊗Hom(E,F)Suppose that Υ' ∈𝐄𝐱𝐭(Υ_C/S(E,F), π^∗ J) corresponds to extensions C' of C, E' of E, and F' of F.Then there is a cartesian square of sheaves of abelian groups: Hom(E',F') [r] [d]_φ_Hom(E',F') Hom(E,F) [d]^φ_Hom(E,F) Υ' ⊗Hom(E,F) [r]Υ⊗Hom(E,F) For the sake of readability, we will write Υ= Υ_C/S(E,F) H = Hom_𝒪_C(E,F) H' = Hom_𝒪_C'(E',F')φ= φ_H We also observe that we can identify Υ' = 𝒪_C⊗_𝒪_C'Υ_C'/S(E',F') by Lemma <ref> (which allows us to identify the exact sequences (<ref>) and (<ref>)) and then write φ' for the composition of φ_H' and reduction modulo π^∗ J: φ' : H' Υ_C'/S(E',F') ⊗_𝒪_C' H' →Υ' ⊗_𝒪_C H Consider the following commutative squares: 𝒪_C'[r] [d]_ρ + ϵδ' 𝒪_C [d]^id + ϵδ 𝒪_C + ϵΥ' [r]𝒪_C + ϵΥ1cm H' [r] [d]_ρ + ϵφH [d]^id + ϵφ H + ϵ (Υ' ⊗ H) [r] H + ϵ (Υ⊗ H)We have written ρ for reduction modulo π^∗ J and δ' for the composition𝒪_C'→Υ_C'/S(E',F') →𝒪_C ⊗_𝒪_C'Υ_C'/S(E',F') = Υ' .We have already seen that the square on the left is cartesian inthe discussion of (<ref>) and Lemma <ref>.The square on the right is cartesian as well, since its horizontal arrows are surjective and we can identify the kernels of both horizontal arrows with π^∗ J ⊗ H:on the top we have applied ⊗_𝒪_C' H' to the exact sequence 0 →π^∗ J →𝒪_C'→𝒪_C → 0 noting that H' is flat over 𝒪_C', and on the bottom we have applied ⊗_𝒪_C H to the exact sequence (<ref>), this time noting that H is flat over 𝒪_C.Each entry in the square on the right is a module under the ring in the corresponding entry of the square on the left and the arrows in the square on the right are homomorphisms with respect to the arrows in the square on the left.It follows that the module structure on H' is induced from the fiber product.The lemma now follows, since the square H + ϵ(Υ' ⊗ H) [r] [d] H + ϵ (Υ⊗ H) [d]Υ' ⊗ H [r]Υ⊗ H is cartesian. If we fix σ : E → F, then φ_Hom(E,F)(σ) ∈Υ_C/S(E,F) ⊗Hom(E,F) induces a linear map:Hom(E,F)^∨→Υ_C/S(E,F):Υ^∙_C/S(E,F,σ)via evaluation against Hom(E,F). As indicatedwrite Υ^∙_C/S(E,F,σ) for the complex (<ref>), concentrated in degrees [-1,0]. As with Υ_C/S(E), one may arrive at the definition of Υ^∙_C/S(E,F,σ) by contemplating, with a bit of care, the automorphism group of the trivial extension of (C,E,F,σ) to S[ϵ J] (cf. Remark <ref>). In order to state the next lemma, we recall the category𝐄𝐱𝐭(K^∙, L) of extensions of a complex K^-1→ K^0 by a module L is, by definition, the category of commutative diagramsK^-1[dl] [d] 0 [r] L [r] M [r] K^0[r] 0in which the bottom row is exact <cit.>. Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles on C, and σ : E → F is a morphism of vector bundles.At an S-point (C,E,F,σ), we haveT_ℱ(C,E,F,σ) = 𝐄𝐱𝐭(Υ^∙_C/S(E,F,σ), 𝒪_C)and thereforeT_ℱ^-1(C,E,F,σ) = Ext^0(Υ^∙_C/S(E,F,σ), 𝒪_C) T_ℱ^0(C,E,F,σ) = Ext^1(Υ^∙_C/S(E,F,σ), 𝒪_C) .As in the previous sections we will actually prove the analogous statement about extensions to S[ϵ J].Suppose that (C',E',F',σ') is an extension of (C,E,F,σ) to S[ϵ J].We have a commutative diagram:𝒪_C ⊗_𝒪_C'Hom(E',F')^∨[r]^<>(0.5)∼[d]Hom(E,F)^∨[d] 0 [r]π^∗ J [r]𝒪_C ⊗_𝒪_C'Υ_C'/S(E',F') [r]Υ_C/S(E,F) [r] 0The bottom row is exact, as in the last section.As the upper horizontal arrow is an isomorphism, we obtain an element of 𝐄𝐱𝐭(Υ^∙_C/S(E,F,σ), π^∗ J). Conversely, suppose we are given an extension:Hom(E,F)^∨[dl]_β[d] 0 [r]π^∗ J [r]Υ' [r]Υ_C/S(E,F) [r] 0We may construct 𝒪_C'= δ^-1Υ' = 𝒪_C ×_ΥΥ' E' = φ_E^-1 (Υ' ⊗ E) = E ×_Υ⊗ E (Υ' ⊗ E) F' = φ_F^-1 (Υ' ⊗ F)= F ×_Υ⊗ F (Υ' ⊗ F)To get σ' ∈Hom(E',F'), regard β as an element of Υ' ⊗Hom(E,F).Then the image of β in Υ_C/S(E,F) ⊗Hom(E,F) is [σ, φ], by the commutativity of (<ref>).Thereforeσ' = (σ, β) defines an element ofHom(E',F') = Hom(E,F) ×_Υ⊗Hom(E,F)Υ' ⊗Hom(E,F).We leave the verification that these constructions are mutually inverse to the reader. Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles on C, and σ : E → F is a homomorphism of vector bundles.Then ℱ is homogeneous. Let 𝒩 be the stack of nodal curves.The forgetful map ℱ→𝒩 is relatively homogeneous by Corollary <ref> and 𝒩 is homogeneous by Lemma <ref>.Therefore ℱ is homogeneous by Lemma <ref>. §.§.§ Deformations of Higgs bundles We may apply the method of the previous section to study deformations of a Higgs bundle ϕ : E → E ⊗ω_C/S on a nodal curve C over S.We find that the deformation theory of (C,E,ϕ) is controlled by the complex Υ_C/S(E,σ), in degrees [-1,0]:Hom(E, E ⊗ω_C/S)^∨→Υ_C/S(E)Let (C,E,ϕ) be a nodal curve of genus g over S equipped with a Higgs bundle.ThenT_ℋ𝒮h_M_g(C,E,ϕ) = 𝐄𝐱𝐭(Υ_C/S(E,σ), 𝒪_C)andT^-1_ℋ𝒮h_M_g(C,E,ϕ) = Ext^0(Υ_C/S(E,σ), 𝒪_C)T^0_ℋ𝒮h_M_g(C,E,ϕ) = Ext^1(Υ_C/S(E,σ), 𝒪_C) The stack of Higgs bundles on nodal (resp. stable) curves is homogeneous. Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve over a tacit base S, E and F are vector bundles on C, and σ : E → F is a homomorphism.Let ℰ be the stack of pairs (C,E) where C is a nodal curve and E is a vector bundle on C.Let ℋ be the stack of Higgs bundles on nodal curves. We have seen in Lemma <ref> that ℰ is homogeneous and in Lemma <ref> that ℱ is homogeneous.Then we have two projections ℱ→ℰ, one sending (C,E,F,σ) to F and the latter sending (C,E,F,σ) to ω_C/S⊗ E.We can identify ℋ with the equalizer of these two maps, that is, with the fiber product ℱ×_ℰ×ℰℰ, hence is homogeneous by Lemma <ref>. Again by Lemma <ref>, we deduce that ℋ→𝒩 is relatively homogeneous, where 𝒩 denotes the stack of nodal curves.By base change, the stack ℋ𝒮h_ℳ_g of Higgs bundles on stable curves of genus g is homogeneous over ℳ_g (again by Lemma <ref>).But we know ℳ_g is homogeneous (Lemma <ref>) so we deduce that ℋ𝒮h_ℳ_g is homogeneous as well. §.§.§ Deformations of principal bundlesSuppose that X ⊆ X' is a square-zero extension of schemes by the ideal J and G' is a smooth algebraic group over X'.Denote by G the preimage of X in G'.Let P be an étale G-torsor over X, meaning that the map G ×_X P → P ×_X P is an isomorphism, and that P covers X in the étale topology (Definition <ref>).We would like to classify the extensions of P to G'-torsors over X'. An algebraic group can have étale torsors that are not torsors in the Zariski topology (see <cit.>), so it is important that we have specified étale torsors here.Since G is smooth, working in the fppf or fpqc topology would not yield any more torsors:flat descent implies that any G-torsor would be representable by a smooth algebraic space over X, hence would have a section étale-locally.This discussion of G-torsors generalizes the discussion of <ref>:given a vector bundle E over X, we may take G = GL_n and let P to be the G-torsor of isomorphisms between E and 𝒪_X^n. Before analyzing the deformation problem, we introduce the (underlying vector bundle of the) Lie algebra of G.Let e : X → G and e' : X' → G' denote the identity sections. We set 𝔤 = e^∗ T_G/Xto bethe Lie algebra of G. We will also be interested in a closely related object.To introduce it,note that the small étale sites of X and X' are equivalent (Lemma <ref>), and that if U is an étale scheme over X we denote by U' the unique (up to unique isomorphism) extension of U to an étale scheme over X'.We write G'_ét and G_ét for the restrictions of G' and G to this common étale site, and we introduce 𝔤' for the kernel of the projection from G'_ét to G_ét:0→𝔤'→ G'_ét→ G_ét→ 0In general 𝔤' does not coincide with 𝔤, but the following lemma relates them: Let 𝔤 and 𝔤' be as above. Then 𝔤' ≃𝔤⊗ J as sheaves of groups on the étale site of X.In particular, 𝔤' is a sheaf of commutative groups. We observe that 𝔤' is canonically a torsor under 𝔤⊗ J.Indeed, for an open subset U⊆ X, denoting by G'_e(U) the morphisms U' → G' whose restriction to U are the identity section, then we have:𝔤'(U) ×𝔤'(U)= G'_e(U') × G'_e(U')= G'_e(U' ⨿_U U') by homogeneity of G = G'_e(U' ⨿_U U[ϵ J_U]) 𝒪_U'×_𝒪_U𝒪_U'≃𝒪_U'×_𝒪_U𝒪_U[ϵ J], p. B:coprodExamp = G_e(U') × G_e(U[ϵ J])by homogeneity again = G_e(U') × (𝔤⊗ J)(U)since 𝔤 is the tangent space to G at the origin.But 𝔤' also has a canonical section over X coming from e'.Therefore 𝔤' ≃𝔤⊗ J as a 𝔤⊗ J-torsor.In particular, 𝔤' inherits a group structure from 𝔤⊗ J.There is another group structure on 𝔤' by virtue of its construction as a kernel.These two group structures commute with one another, in the sense that the multiplication map of either group structure is a homomorphism with respect to the other, and the identity elements are the same.By a standard argument, this means the two group structures coincide (and that both are commutative). As in the earlier sections, we make some observations about the local triviality of this problem: * There is an étale cover of X' by maps U' → X' such that, setting U = U' ×_X' X, each G_U-torsor P_U extends to a G'_U'-torsor on U'.* Any two extensions of P to X' are isomorphic on a suitable étale cover of X'.*If P' and P” are two extensions of P and u, v : P' → P” are two isomorphisms between them, then there is a unique map h : P' → G' over X' such that h.u = v (we are denoting the groupaction map for the torsors with a dot, and the map h.u is the composition of h× u : P' × P' → G' × P” with the action map G' × P”→ P”).As u and v both reduce to the identity map on P, the map h must reduce to the constant map P → e.Therefore h factors uniquely through 𝔤':@R=1em@C=1em P' [rr]^h @–>[rd]G'𝔤'@^^(->[ru] Furthermore, the induced map h:P'→𝔤' is equivariant with respect to the natural action of G' on P' and on 𝔤' (by conjugation).Indeed, we haveh(g.x) . u(g.x) = v(g.x) h(g.x) g . u(x) = g . v(x) = g h(x) . u(x)for all x in P'.It follows that h(g.x) = g h(x) g^-1 for all g ∈ G'.Finally, h:P'→𝔤' factors through the projection P' → P=P'/𝔤'.This is because 𝔤' is commutative, so h(g.x) = g h(x) g^-1 = h(x) whenever g ∈𝔤'.Therefore h factors through P' / 𝔤' = P.Thus h factors uniquely as a G-equivariant mapP ⟶𝔤'=𝔤⊗ J.Putting all of this together using the same Čech or gerbe argument from Section <ref>, we obtain a theory of deformations and obstructions:Suppose that G is a smooth algebraic group over a scheme X, that X' is a square-zero extension of X, and that G' is an extension of G to X' by the ideal J.Let P be a G-torsor on X.There is an obstruction to extending P to a G'-torsor on X' lying in H^2(X, Hom_G(P, 𝔤⊗ J)).Should this obstruction vanish, deformations are canonically a torsor on X under H^1(X, Hom_G(P, 𝔤⊗ J)), and automorphisms of any given deformation are canonically isomorphic to H^0(X, Hom_G(P, 𝔤⊗ J) ). In the case of the trivial extension X[ϵ], so that J=𝒪_X, and 𝔤=𝔤⊗ J=𝔤', the bundle𝔭:= Hom_G(P, 𝔤)is known as the adjoint bundle of P. The adjoint bundle can also be constructed as the quotient of P ×𝔤 by the diagonal action of G.To see that these are equivalent, note that a section of P ×_G 𝔤 corresponds to a G-orbit in P ×𝔤, which is the graph of an equivariant map P →𝔤.Thus the definition of the adjoint bundle given above is equivalent to the one given in Section <ref>.The adjoint bundle 𝔭 of a G-torsor P over X is isomorphic to the space of G-invariant vector fields on the fibers of P over X. Recall that a vector field on the fibers of P over X is an X-morphism V : P[ϵ] → P that lifts the identity on P ⊂ P[ϵ].To be invariant means that the diagramG × P[ϵ] [r]^-id× V[d]_p_2G × P [d]^αP[ϵ] [r]^V Pcommutes, α being the action map and p_2 the second projection.Now, the zero vector field also gives a map P[ϵ] → P, and the pair (0, V) gives an X-morphismP[ϵ] (0,V)⟶ P ×_X P ∼⟵ G ×_X P .The first map is equivariant with respect to the diagonal G-action on P ×_X P.Since the map G ×_X P → P ×_X P sends (g,y) to (gy,y), the second arrow is also equivariant, provided we let G act by conjugation on itself.The original condition that P[ϵ] → P restrict to the identity on P ⊂ P[ϵ] means that P ⊂ P[ϵ] → P ×_X P factors through the diagonal, and therefore that P ⊂ P[ϵ] → G ×_X P → G factors through the identity section of G.Now composing(<ref>) with the projection G ×_X P → G, we get an equivariant mapP[ϵ] = P × X[ϵ] → G,that factors through the identity section of G when restricted to P.This is the same as to give an equivariant mapP →Hom_X(X[ϵ], G) ×_Hom_X(X, G){ e } = T_G/X×_G{ e } = 𝔤 ,which, by definition is the same as to give a section of the adjoint bundle of P. The adjoint bundle 𝔭 of a G-torsor P has the structure of a sheaf of Lie algebras.If G is semisimple then the Killing form furnishes an isomorphism between 𝔭 and its dual. The point is to verify that the bracket of equivariant vector fields on P is equivariant.This can be done with a diagram chase, using the definition of the Lie bracket in Section <ref>, but we omit it.Then P is locally isomorphic to G so 𝔭 is locally isomorphic to 𝔤, and this isomorphism preserves the Killing form.If G is semisimple then 𝔤 is self dual with respect to the Killing form. For the next statement, let X be an S-scheme and let G be a smooth group scheme over X.Let 𝖥𝗂𝖻^G_X/S denote the S-stack whose sections over T are the G-torsors on X_T. We have T_𝖥𝗂𝖻^G_X/S(P) = BHom_G(P, 𝔤) = B1pt𝔭 where 𝔤 denotes the Lie algebra of G.In particular,T^-1_𝖥𝗂𝖻^G_X/S(P)= H^0(X, Hom_G(P, 𝔤)) = H^0(X, 𝔭) T^0_𝖥𝗂𝖻^G_X/S(P)= H^1(X, Hom_G(P, 𝔤)) = H^1(X, 𝔭).Let X' = X[ϵ].Let P' be the trivial extension of P to X'.Suppose that P” is another extension.Then Isom_G'^P(P”, P') (isomorphisms respecting the G'-action and the maps to P) is a torsor on X' under Hom_G(P, 𝔤) as explained in (<ref>) above (X' has the same étale site as X).The same argument shows that Aut_G'^P(P') = 𝔭, so that, given any torsor Q under 𝔭=Hom_G(P, 𝔤), we can form the sheaf of 𝔭-equivariant maps from Q to P'P” = Hom_𝔭(Q, P') .These constructions are inverse isomorphisms between T_𝖥𝗂𝖻^G_X/S(P) and the stack of torsors under 𝔭 = Hom_G(P, 𝔤). Now suppose that X is a reduced (and therefore Cohen–Macaulay) projective curve over a field k (i.e., S=Speck).Then by Serre duality, we haveT^0_𝖥𝗂𝖻^G_X/S(P)=H^1(X, 𝔭) ≃ H^0(X, Hom(𝔭, ω_X))^∨ . Generalizing Definition <ref>:[G-Higgs bundle]Let X be areduced projective curve over a field k, and letG be a smooth algebraic group over k.A G-Higgs bundle on X is a pair (P,Φ) where P is a G-torsorover Xand Φ∈Hom(𝔭,ω_X), where 𝔭:= Hom_G(P, 𝔤). If G is semisimple then 𝔭 is self-dual (Corollary <ref>), so a G-Higgs bundle on X can also be viewed as an element of H^0(X, 𝔭⊗ω_X), or, in another popular notation, of H^0(X, ad P ⊗ K_X).In other words, Definition <ref> agrees withDefinition <ref>.In the case where P is the GL_n-torsor associated to a vector bundle E, we have 𝔭 = End(E), so thatHom(𝔭, ω_X) = Hom(E, E ⊗ω_X),and T^0_𝖥𝗂𝖻^G_X/k(P)= H^1(X, 𝔭)≃Hom(𝔭,ω_X)^∨=Hom(E,E⊗ω_X)^∨. More generally, extending the definition of G-Higgs bundles to families of curves, and using Grothendieck duality,the discussion aboveproves the following corollary: Let X be a family of smooth curves over S.Higgs bundles on X/S correspond to isomorphism classes of relative cotangent vectors for 𝖥𝗂𝖻_X/S over S.Likewise, G-Higgs bundles on X/S correspond to relative cotangent vectors over S for the stack 𝖥𝗂𝖻^G_X/S of principal G-bundles on X. §.§ Obstruction theoryIn this section, we will consider a stack ℱ(not a priori algebraic) and a lifting problemS [r]^ξ[d]ℱ S' @–>[ur]in which S' is a square-zero extension of S' with ideal J.We are looking for an obstruction theory that can detect whether this lifting problem has a solution.The obstruction theory will be an 𝒪_S-module T^1_ℱ(ξ, J) that depends only on J and ξ∈ℱ(S), not on the particular extension S'.The above lifting problem will then produce a natural obstruction ω∈ T^1_ℱ(ξ, J) whose vanishing is equivalent to the existence of a lift.We will see that for moduli problems that are locally unobstructed in a suitable sense (that includes all of the examples considered here) there is a natural choice for T^1_ℱ(ξ,J) arising from the infinitesimal deformation theory.This phenomenon could be better explained in the language of torsors under abelian group stacks, but to introduce such objects would take us too far afield.We will rely instead on injective resolutions and Čech methods. Various definitions of obstruction theories have appeared in the literature <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.From the perspective of derived algebraic geometry, an obstruction theory arises from the promotion of a moduli problem to a derived moduli problem <cit.>.The definition we adopt here is closest to <cit.>. Let ℱ be a stack on the étale site of schemes.An obstruction theory for ℱ is a system of abelian groups T^1(ξ, J) depending contravariantly on a scheme S and an element ξ∈ℱ(S), and covariantly on a quasicoherent 𝒪_S-module J, together with an obstruction map ω : Exal(𝒪_S, J) → T^1(ξ, J)that is natural in S and J and has the following property:ω(S') = 0 if and only if diagram (<ref>) admits a lift. Above we use the notation Exal(𝒪_S, J) for the 𝒪_S-module of algebras that are extensions of 𝒪_S by Jwith J^2=0 (see e.g., <cit.> for more details). We spell out precisely the meaning of naturality in Definition <ref>.Suppose there is a commutative (not necessarily cartesian) diagramof square-zero extensions (<ref>) where R ⊆ R' has ideal I and S ⊆ S' has ideal J.R [r] [d]_f R' [d] S [r] S'Then we get a homomorphism f^∗ J → I and therefore a morphism:T^1(ξ, J) → T^1(f^∗ξ, f^∗ J) → T^1(f^∗ξ, I).We also have obstructions ω(S') ∈ T^1(ξ, J) and ω(R') ∈ T^1(f^∗ξ, I).The naturality alluded to in Definition <ref> requires that ω(S') is carried under the morphism (<ref>) to ω(R').Our goal in this section will be to illustrate a technique for constructing an obstruction theory for a stack.For concreteness, we will consider the case where ℱ is the stack of quadruples (C,E,F,σ) in which π:C→ S is a nodal curve, E and F are vector bundles on C, and σ : E → F is a morphism of vector bundles.However, apart from the following observations, analogues of which frequently hold for other stacks, the particular choice of ℱ will not enter into the rest of the discussion: *Fix a scheme S, an S-point ξ = (C,E,F,σ) of ℱ, and a square-zero extension S ⊆ S' with ideal J.We obtain a square-zero extension of sheaves of commutative rings on the Zariski site, π^-1𝒪_S'→π^-1𝒪_S, with ideal π^-1 J.Our problem can be phrased as the search for a square-zero extension 𝒪_C' of 𝒪_C by π^∗ J compatible with the extension π^-1𝒪_S'→π^-1𝒪_S and the homomorphism π^-1 J →π^∗ J, together with locally free 𝒪_C'-modules E' and F' extending E and F and a morphism σ' : E' → F' extending σ. We could replace π^-1𝒪_S' by the extension 𝒜 of π^-1𝒪_S by π^∗ J obtained by pushout:0 [r]π^-1 J [r] [d]π^-1𝒪_S'[r] [d]π^-1𝒪_S [r] @=[d] 0 0 [r]π^∗ J [r]𝒜[r]π^-1𝒪_S [r] 0To find an extension of 𝒪_C by π^∗ J compatible with 𝒜 is the same as to find a π^-1𝒪_S'-algebra extension compatible with the homomorphism π^-1 J →π^∗ J. Indeed, an 𝒜-algebra extension induces a π^-1𝒪_S'-algebra extension by composition with π^-1𝒪_S'→𝒜; conversely, if ℬ is a π^-1𝒪_S'-algebra extension of 𝒪_C by π^∗ J then the map π^-1 J →ℬ factors through π^∗ J so the map π^-1𝒪_S'→ℬ factors through 𝒜.Moreover, we may consider an arbitrary square-zero extension 𝒜 of π^-1𝒪_S by an 𝒪_C-module 𝒥 and define ℱ_ξ(𝒜) to be the category of quadruples (𝒪_C', E', F', σ') where 𝒪_C' is an extension of 𝒪_C by 𝒥, compatible with 𝒜; both E' and F' are locally free 𝒪_C'-modules extending E and F, respectively; and σ' : E' → F' is a morphism of 𝒪_C'-modules extending σ.We observe that ℱ_ξ is a homogeneous functor.This allows us to define T^-1_ℱ(ξ, 𝒥) and T^0_ℱ(ξ, 𝒥). *There is a complex Υ^∙of 𝒪_C-modules with quasicoherent cohomology such that T_ℱ(ξ, 𝒥) = 𝐄𝐱𝐭(Υ^∙, 𝒥) .In other words, the infinitesimal deformations and automorphisms of ξ should be representable by a complex.In the case of interest, Υ^∙ is the complex Υ^∙_C/S(E, F, σ), constructed in <ref>.For the representability of the obstruction theory by a complex on S, we will also need the observation that Υ^∙ is perfect as an object of the derived category.That is, locally in C it can be represented by a bounded complex of locally free modules.This is even true globally for Υ_C/S(E, F, σ).*Whenever 𝒥 is an injective 𝒪_C-module and 𝒜 is a square-zero extension of π^-1𝒪_S by 𝒥, there is some ξ' ∈ℱ_ξ(𝒜).In our situation of interest, we can see that this is the case by considering the successive obstructions introduced in earlier sections.There is no local obstruction to deforming curves (Corollary <ref>).The first global obstruction (gluing isomorphism classes of local deformations) lies in H^1(C, Ext^1(Ω_C/S, 𝒥)) = 0 .The obstruction to finding a global deformation inside compatible local isomorphism classes lies inH^2(C, Hom(Ω_C/S, 𝒥)) = 0 .Therefore we can find 𝒪_C'.Now the obstructions to extending E and F lie inExt^2(End(E)^∨, 𝒥) = 0Ext^2(End(F)^∨, 𝒥) = 0 .Finally, the obstruction to extending σ, once 𝒪_C', E', and F' have been chosen, lies inExt^1(Hom(E,F)^∨, 𝒥) = 0.One could also verify the existence of ξ' directly by choosing local deformations arbitrarily and using the fact that 𝒥 is a flasque sheaf to assemble them into a global deformation.Thus, if one wants to axiomatize the discussion of this section, one should require that ℱ_ξ(𝒜) be locally nonempty, when regarded as a CFG over the Zariski site of C. Now we will build a global obstruction to the existence of ξ' ∈ℱ(S') lifting ξ∈ℱ(S).The obstruction will lie in Ext^2(Υ^∙, π^∗ J), which we will take as the definition of T^1_ℱ(ξ, J).We begin by choosing a resolution 0 →π^∗ J →𝒥^0 →𝒥^1 → 0where 𝒥^0 and 𝒥^1 are sheaves of 𝒪_C-modules and 𝒥^0 is injective.We leave it to the reader to verify that the construction of the obstruction given here is independent of the choice of resolution.We can push out the extension:0 [r]π^-1 J [r] [d]π^-1𝒪_S'[r] [d]π^-1𝒪_S [r] @=[d] 0 0 [r]𝒥^0[r] [d]𝒜[r] [d]π^-1𝒪_S [r] @=[d] 0 0 [r]𝒥^1 [r]ℬ[r]π^-1𝒪_S [r] 0Note ℬ is canonically isomorphic to π^-1𝒪_S + ϵ𝒥^1 because the map π^-1 J →𝒥^1 is zero.The commutative diagram above induces a map:ℱ_ξ(𝒜) →ℱ_ξ(ℬ) = T_ℱ(ξ, 𝒥^1).The identification with T_ℱ(ξ, 𝒥^1) comes by virtue of the canonical isomorphism ℬ≃π^-1𝒪_S + ϵ𝒥^1.Note that ℱ_ξ(ℬ) contains a canonical zero element (corresponding to the zero element of T_ℱ(ξ, 𝒥^1)), namely the image of ξ under the map ℱ_ξ(π^-1𝒪_S) →ℱ_ξ(ℬ) induced from the homomorphism π^-1𝒪_S →π^-1𝒪_S + ϵ𝒥^1 = ℬ, so we can speak of the kernel of (<ref>).By definition, this is the fiber product of groupoids 0 ×_ℱ_ξ(ℬ)ℱ_ξ(𝒜). There is a canonical identification between ℱ_ξ(π^-1𝒪_S') and the kernel of (<ref>). The diagramπ^-1𝒪_S'[r] [d]𝒜[d]π^-1𝒪_S [r]ℬis cartesain (the bottom arrow is the canonical splitting of the projection ℬ→π^-1𝒪_S).Therefore the homogeneity of ℱ_ξ implies thatℱ_ξ(π^-1𝒪_S') [r] [d]ℱ_ξ(𝒜) [d]ℱ_ξ(π^-1𝒪_S) [r]ℱ_ξ(ℬ)is also cartesian.But ℱ_ξ(π^-1𝒪_S) = 0, by definition, so this identifies ℱ_ξ(π^-1𝒪_S') with the kernel of (<ref>). As 𝒥^0 is injective, ℱ_ξ(𝒜) ≠∅ (observation <ref> on p. obs:3),so that Ext^1(Υ^∙, 𝒥^0) acts simply transitively on the set | ℱ_ξ(𝒜) | of isomorphism classes in ℱ_ξ(𝒜) (by observation (<ref>) on p. obs:2).As ℬ is a (canonically) split extension of π^-1𝒪_S, we have a (canonical) identification Ext^1(Υ^∙, 𝒥^1) = | ℱ_ξ(ℬ) |.Now we have an exact sequence in the top row below, with compatible actions illustrated in the bottom row:@R=10ptExt^1(Υ^∙, π^-1 J) [r]Ext^1(Υ^∙, 𝒥^0) [r]Ext^1(Υ^∙, 𝒥^1) [r]Ext^2(Υ^∙, π^-1 J) |ℱ_ξ(π^-1𝒪_S')| [r] @(ul,ur)|ℱ_ξ(𝒜)| [r] @(ul,ur)|ℱ_ξ(ℬ)| @=[u]Note that ℱ_ξ(π^-1𝒪_S') may be empty.As the action of Ext^1(Υ^∙, 𝒥^0) on | ℱ_ξ(𝒜) | is faithful and transitive, we have thatthe image of | ℱ_ξ(𝒜) | is an Ext^1(Υ^∙, 𝒥^0) coset in Ext^1(Υ^∙, 𝒥^1).In other words, it gives a well-defined element of Ext^2(Υ^∙, π^-1 J) obstructing the existence of an element of ℱ_ξ(π^-1𝒪_S').We therefore defineT^1_ℱ(S,J) := Ext^2(Υ^∙, π^-1 J). By construction, T^1_ℱ(S,J) is functorial with respect to S (contravariant) and J (covariant) and the obstruction class is natural.[Representable deformation-obstruction theory] Let ℱ be a stack in the étale topology on schemes.We will say that a deformation-obstruction theory T^i_ℱ, i = -1, 0, 1, is representable at an S-point ξ if there is a complex of locally free sheaves 𝐄^∙ on S, such that for any f : T → S and any quasicoherent sheaf J on T, we have a natural (in T and in J) identificationT^i_ℱ(f^∗ξ, J) = Ext^i(f^∗𝐄^∙, J)for i = -1, 0, 1.We will say it is finitely presentable if the vector bundles in the complex 𝐄^∙ may be chosen to have finite rank.We say that the deformation theory is locally representable if it is representable at all points valued in affine schemes. We say it is locally finitely presentable if it is finitely presentable at all points valued in affine schemes.Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles over C, and σ : E → F is a morphism of vector bundles.Then the obstruction theory for ℱ introduced above is locally finitely presentable. Let A be a commutative ring and let ξ = (C,E,F,σ) be an A-point of ℱ.We want to show that there is a complex of locally free A-modules 𝐄^∙ representing T^i_ℱ(ξ, J) for i = -1, 0, 1 and all A-modules J.We assume first that A is noetherian.Let Ξ^∙ be a bounded above complexof locally free sheaves on C that is dual to Υ^∙.ThenT^i_ℱ(ξ, J) = Ext^i(Υ^∙, π^∗ J) = H^i(C, Ξ^∙⊗π^∗ J) .Therefore, by <cit.>, there is a complex L^∙ of finite rank vector bundles on S such that[Loc. cit. requires that Ξ^∙ be a quasicoherent sheaf, but the proof works for a complex as long as one takes the total complex of the Čech double complex at the bottom of p. 182.]T^i_ℱ(ξ, J) = h^i(L^∙⊗_A J) .But now we may take 𝐄^∙ to be a complex dual to L^∙ and obtainExt^i(𝐄^∙, J) = h^i(L^∙⊗_A J)as required.Now we pass to the general case.We can always write A as a filtered colimit of commutative rings of finite type A =A_i.As ℱ is locally of finite presentation (Corollary <ref>) there is some index j and some ξ_j = (C_j,E_j,F_j,σ_j) ∈ℱ(A_j) inducing (C,E,F,σ).Now, T^i_ℱ(ξ, J) = T^i_ℱ(ξ_j, J), with J regarded as an A_j-module via the map A_j → A.ThereforeT^i_ℱ(ξ, J) = T^i_ℱ(ξ_j, J) = Ext^i_A_j(𝐄^∙, J) = Ext^i_A(A ⊗_A_j𝐄^∙, J)so T_ℱ is locally representable at ξ. Suppose that 𝒳 is a stack with a locally finitely presentable obstruction theory.Then, for any field-valued point ξ of 𝒳, the vector spaces T^-1_𝒳(ξ), T^0_𝒳(ξ), and T^1_𝒳(ξ) are finite dimensional. This is immediate, because the vector spaces in question may be identified with cohomology groups of the dual complex of a complex of finite rank vector bundles.§ ARTIN'S CRITERION FOR ALGEBRAICITYIntuition from analytic moduli spaces suggests that moduli spaces should locally be embedded as closed subspaces of their tangent spaces.For this to apply in an algebraic context, locally must be interpreted to mean étale-locally for schemes and algebraic spaces, and smooth-locally for algebraic stacks.Artin gives criteria under which a stack is locally cut out by polynomial equations inside its tangent space, thereby ensuring the stack is algebraic. Since Artin's original formulation, there have been a number of improvements <cit.>.The statement we give here is close to the form given by Hall <cit.>, but with some hypotheses strengthened for the sake of transparency:[Artin's criterion <cit.>]Let S be an excellent scheme and let 𝒳 be a CFG over the category 𝖲/S of S-schemes.Then 𝒳 is analgebraic stack over (𝖲/S)_et that is locally of finite presentation over S if and only if it has the following properties: *𝒳 is a stack in the étale topology (Definition <ref>).*𝒳 is homogeneous (Definition <ref>).*𝒳 has finite dimensional tangent and automorphism spaces (Section <ref>).*𝒳 is integrable (Definition <ref>).*𝒳 is locally of finite presentation (Sections <ref> and <ref>). *𝒳 has a locally finitely presentable obstruction theory (Definition <ref>).The assumption (<ref>) actually follows from (<ref>) (see Lemma <ref>); however, we include the hypothesis (<ref>)since it is useful from an expository perspective.In particular, a theorem of Schlessinger–Rim (see Theorem <ref>) uses the hypotheses (<ref>) and (<ref>) on a CFG. Hall only requires the existence of a multistep obstruction theory, which is an a priori weaker hypothesis than (<ref>).A posteriori, every algebraic stack has a cotangent complex, whence a single step obstruction theory. In our case, we actually constructed the obstruction theory for Higgs bundles in pieces in Section <ref>, but we were able to assemble it into a single-step obstruction theory in Section <ref>.Property (<ref>) is sometimes phrased, `𝒳 is limit preserving'. For Property (<ref>), some would say `𝒳 is effective' or `formal objects of 𝒳 may be effectivized' or `formal objects of 𝒳 can be algebraized' (in the literature, the term `algebraized' is sometimes reserved for algebraization over a scheme of finite type; see Definition <ref>).We picked up the term `integrable' from <cit.>, the intuition being that infinitesimal arcs can be integrated into formal arcs, in the manner that tangent vectors are integrated to curves on smooth manifolds.The rest of this section will be devoted to explaining these properties and verifying them for the stack of Higgs bundles and related moduli problems; at the end, we give a brief explanation of how these properties combine to imply a stack is algebraic.§.§ The Schlessinger–Rim criterionWe have seen that homogeneity is a necessary condition for representability by an algebraic stack.The Schlessinger–Rim criterion implies that it is sufficient for prorepresentability.By abstract nonsense, a covariant functor is prorepresentable if and only if it preserves finite limits.Generally, it is not very practical to check that functors on Artin rings preserve arbitrary finite limits, ultimately due to the restrictions on descent for flat modules along non-flat morphisms in Theorem <ref>.Fortunately, Schlessinger was able to prove that homogeneity, that is, respect for a restricted class of limits, along with a finite dimensional tangent space, are sufficient to imply a functor is prorepresentable <cit.>.In this section, we will discuss Rim's generalization of Schlessinger's result to groupoids <cit.>. .2 cmSuppose that 𝒳 is a CFGon Λ-schemes, where Λ is a complete noetherianlocal ring with residue field k, and ξ:Speck→𝒳 is a k-point.Let 𝒞_Λ be the category of local artinian Λ-algebras with residue field k.One may interpret 𝒞_Λ^𝗈𝗉 as the category of infinitesimal extensions of Spec k.Let 𝒳_ξ be the fibered category on 𝒞_Λ^𝗈𝗉 whose fiber over a ring A in 𝒞_Λ^𝗈𝗉 consists of all η∈𝒳(A) whose image via the projection A → k is ξ.Th CFG 𝒳_ξgives a formal picture of 𝒳 near the point ξ.Note that the restriction of the étale topology to 𝒞^𝗈𝗉_Λ is trivial, as every morphism in 𝒞^𝗈𝗉_Λ has a section over the residue field, and sections of étale maps extend infintisimally.Therefore all covers in 𝒞^𝗈𝗉_Λ have sections, every presheaf is a sheaf, and every CFG is a stack.Therefore we can use the terms `stack' and `CFG' interchangeably over 𝒞^𝗈𝗉_Λ. Typically, we will start with a CFG 𝒳 over 𝖲.This induces a CFG 𝒳_𝒞_Λ^𝗈𝗉 over 𝒞_Λ^𝗈𝗉 by restricting to the full subcategory obtained by takingobjects over the spectrum of such a ring. Given an object ξ∈𝒳(Speck), the category 𝒳_ξ over 𝒞_Λ^𝗈𝗉 has objects the pairs (η, ϕ) where ϕ : ξ→η is a morphism of 𝒳 (necessarily cartesian) lying above an infintiesimal extension Spec k →Spec A, with A in 𝒞_Λ^𝗈𝗉:ξ@->[r]^@|->[d] η@|->[d] Speck @->[r]^ Spec AMorphisms in 𝒳_ξ are defined in the obvious way.We write 𝒞_Λ for the category of complete local Λ-algebras with residue field k that are formally of finite type over Λ; i.e., completions of rings of finite type over Λ.Recall the notion of a groupoid object of 𝒞^𝗈𝗉_Λ from Definition <ref> and its associated CFG <ref>.Note that this coincides with the associated stack from Definition <ref>, since the topology of 𝒞^𝗈𝗉_Λ is trivial.[<cit.>] We saya category fibered in groupoids over 𝒞_Λ^𝗈𝗉 is prorepresentable if it is representable by (i.e., equivalent to) the CFG associated to a groupoid object @C=10ptU_1 @<1.5pt>[r]^s @<-1.5pt>[r]_t U_0in 𝒞_Λ^𝗈𝗉 (Example <ref>).The groupoid is said to be smooth if the morphisms s and t satisfythe formal criterion for smoothness (Definition <ref>).In the above discussion, it is important to note that we are considering stacks over 𝒞^𝗈𝗉_Λ, not 𝒞^𝗈𝗉_Λ.In particular, if 𝒳 is representable by a groupoid object @C=10ptU_1 @<1.5pt>[r]^s @<-1.5pt>[r]_t U_0in 𝒞_Λ^𝗈𝗉, the morphism U_0→𝒳 (Example <ref>)is not in general defined by an object over U_0 (as U_0 is not in general in 𝒞^𝗈𝗉_Λ).However, if 𝔪 is the maximal ideal of U_0, and V_k is the vanishing locus of 𝔪^j in U_0, we can write U_0=_k V_k with each V_k in 𝒞^𝗈𝗉_Λ and there is a family of compatible morphisms in V_k →𝒳 defined by objects of 𝒳 over V_k—in other words, an element of _k 𝒳(U_0/𝔪^j). Such an element iscalled a formal element of 𝒳 over U_0 (see Definition <ref>). We will return to this topicagain in <ref>. Note finally that if @C=10ptU_1 @<1.5pt>[r]^s @<-1.5pt>[r]_t U_0 is a smooth groupoid object, then the morphism U_0→𝒳 is formally smooth (Proposition <ref>).The following theorem says, essentially, that a stack looks like an algebraic stack in a formal neighborhood of a point if and only if it is homogeneous:[Schlessinger <cit.>, Rim <cit.>]Let 𝒳 be a fibered category over 𝒞_Λ^𝗈𝗉 such that 𝒳(k) is a single point.Then 𝒳 is prorepresentable by a smooth groupoid object of 𝒞_Λ^𝗈𝗉 if and only if 𝒳 is homogeneous and T^-1_𝒳 and T^0_𝒳 are finite dimensional k-vector spaces. One direction of the implications in the theorem is clear: it is quite easy to see that the stack associated to aa smooth groupoid object @C=10ptU_1 @<1.5pt>[r]^s @<-1.5pt>[r]_t U_0 of 𝒞_Λ^𝗈𝗉is homogeneous, sincethe stacks associated to the U_iare homogeneous.The finite dimensionality of T^-1_𝒳(ξ) and T^0_𝒳(ξ) follows from the finite dimensionality of the tangent spaces of stacks represented by objects of 𝒞_Λ^𝗈𝗉, which consists of objects formally of finite type. §.§ Local finite presentationThe definition of morphisms locally of finite presentation was given in <ref>.The proof of the following lemma is formal: * Suppose that g : 𝒴→𝒵 is locally of finite presentation.Then f : 𝒳→𝒴 is locally of finite presentation if and only if gf : 𝒳→𝒵 is locally of finite presentation. * The base change of a morphism that is locally of finite presentation is also locally of finite presentation. There is a repertoire of techniques for proving moduli problems are locally of finite presentation to be found in <cit.>.We combine these with Lemma <ref> to prove that the stack of Higgs bundles is locally of finite presentation.The stack of proper nodal curves is locally of finite presentation. Let 𝒩 denote the stack of proper nodal curves.Suppose that a commutative ring A is the filtered colimit of commutative rings A_i.Put S = Spec A and S_i = Spec A_i.Let C be an element of 𝒩(A).That is, C is a flat family of nodal curves over A.We want to show that C is induced by base change from a nodal curve C_i over A_i for some i, and that (up to increasing the index i) this curve is unique up unique isomorphism.First of all, C is of finite presentation over S.By <cit.>, there is an index i and a scheme C_i of finite presentation over A_i such that C = C_i ×_S_i S.It follows from <cit.> that C is unique up to unique isomorphism and increase of the index i.By <cit.>, we can arrange for C_i to be proper over S_i by replacing i with a larger index. Now C has a cover by open subsets U, each of which admits an étale map U → V, where V = Spec A[x,y] / (xy - t_U) for some t_U ∈ A.Refining the cover, we can assume that the open subsets U are affine, and hence of finite presentation over A.As C is quasicompact, this cover can be assumed finite, so by increasing i, we can assume that t_U appears in A_i for all i.By increasing i still further, we can assume each U is the preimage in C of an open subset U_i ⊆ C_i <cit.> and that these open subsets cover C_i <cit.>.Now V = V_i ×_S_i S so by <cit.>, the map U → V is induced from a map U_i → V_i over S_i, at least after increasing i still further.By <cit.>, we can ensure that the map U_i → V_i is étale, at least after increasing i.Then C_i is a family of nodal curves over S_i, and the proof is complete. Let ℰ be the stack of pairs (C, E) where C is a proper nodal curve and E is a vector bundle on C and let 𝒩 be the stack of proper nodal curves.The projection ℰ→𝒩 is locally of finite presentation. As before, A is the filtered colimit of commutative rings A_i.We suppose that (C, E) is an A-point of ℰ and that C is induced from a nodal curve C_i over A_i.We want to show that, up to increasing i, the vector bundle E is induced from a unique (up to unique isomorphism) vector bundle E_i over C_i.As E is of finite presentation, we may increase i to obtain a quasicoherent sheaf of finite presentation E_i over C_i inducing E by pullback <cit.>.The uniqueness of E_i follows from <cit.>.Increasing i still further, we can ensure that E_i is a vector bundle <cit.>. Let ℱ be the stack of tuples (C, E, F, σ) where C is a nodal curve, E and F are vector bundles on C, and σ : E → F is a morphism of vector bundles.Let 𝒢 be the stack of triples (C,E,F) as above, and ℱ→𝒢 the projection forgetting σ.Then ℱ is locally of finite presentation over 𝒢. We assume A =A_i is a filtered colimit of commutative rings andthat (C,E,F,σ) ∈ℱ(A) and (C_i,E_i,F_i) ∈𝒢(A_i) induces (C,E,F) ∈𝒢(A).By an immediate application of <cit.>, we discover that, after increasing i, we can find σ_i : E_i → F_i inducing σ and that σ_i is unique up to further increasing i. Combining Lemmas <ref>, <ref>, <ref>, and <ref>, we obtainThe stack of tuples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles on C, and σ : E → F is a morphism of vector bundles is locally of finite presentation.The stack of Higgs bundles is locally of finite presentation. This is deduced from the previous corollary by the same argument as in Lemma <ref>.§.§ Integration of formal objectsLet A be a complete noetherianlocal ring with maximal ideal 𝔪.A formal A-point of X is an object of the inverse limit _k X(A/𝔪^j).It can be checked easily that for any scheme X, the function X(A) → X(A/𝔪^j)is a bijection.It is only slightly more difficult to verify this gives an equivalence when X is an algebraic stack, provided that one interprets the limit of groupoids correctly.One efficient way of describing the limit is as the category of sections of 𝒳 over the subcategorySpec(A/𝔪) →Spec(A/𝔪^2) →Spec(A/𝔪^3) →⋯of 𝒞^𝗈𝗉_Λ.Lifting a formal A-point of X to an A-point of X may be seen as an analogue of integrating a tangent vector to a curve.We must require a formal point, as opposed to merely a tangent vector, because not every tangent vector can be integrated on a singular space.[Many formulations of Artin's criterion only require (<ref>), when applied to a stack 𝒳,to have dense image.This strengthens the analogy to integrating tangent vectors, since there is not a unique curve with a given tangent vector.However, it is generally no more difficult to prove (<ref>) is an equivalence than it is to prove it has dense image.][Integrating formal points]Let 𝒳 be a fibered category over the category of schemes and let A be a complete noetherianlocal ring with maximal ideal 𝔪.By a formal A-point of 𝒳 we mean an object of the category_j 𝒳(A/ 𝔪^j) .We say that a formal A-point of 𝒳 can be algebraized, or that it can be effectivized, or that it is integrable if it lies in the essential image of the functor𝒳(A) →_j 𝒳(A/𝔪^j) .If every formal A-pointof 𝒳 can be algebraized, for every complete noetherianlocal ring A, then we say formal objects of 𝒳 can be algebraized, or can be effectivized, or are integrable. For a long time, the main algebraization theorem was Grothendieck's existence theorem, which asserts that formal objects of the stack of coherent sheaves on a proper scheme can be algebraized:[Groth. existence <cit.>, <cit.>]Let X be a proper scheme over S = Spec A with A a complete noetherianlocal ring with maximal ideal 𝔪.For each j, let S_j = Spec A / 𝔪^j and let X_j = X ×_S S_j.ThenCoh(X) →_j Coh(X_j)is an equivalence of categories. Very recently, Bhatt <cit.> and Hall and Rydh <cit.> have proved strong new integration theorems extending Grothendieck's.Since Grothendieck's existence theorem will suffice for the stack of Higgs bundles on curves, we will not need to state these new results.Formal families of proper nodal curves can be algebraized. Let 𝒩 denote the stack of proper nodal curves. Let A be a complete noetherianlocal ring with maximal ideal 𝔪.Set A_j = A/ 𝔪^j+1 and S_j = Spec A_j and suppose that C_j ∈𝒩(S_j) are the components of a formal A-point of 𝒩.Then C_0 is a curve over the field A_0.Pick a very ample line bundle L_0 on C_0 with H^1(C_0, L_0) = 0.Extend L_0 inductively to a compatible system of line bundles L_j on each C_j:By Lemma <ref>, the obstruction to extending L_j to L_k+1 lies inH^2 (C, π^∗ (𝔪^j / 𝔪^j+1) ) = 𝔪^j /𝔪^j+1⊗ H^2(C_0, 𝒪_C_0) = 0.Then V_j = π_∗ L_j is a (trivial) vector bundle on S_j.Moreover, by Lemma <ref>, the obstruction to extending a section of L_j to a section of L_k+1 lies inH^1 (C, π^∗ (𝔪^j / 𝔪^j+1) ⊗ L_0 ) = 𝔪^j / 𝔪^j+1⊗ H^1(C_0, L_0) = 0so V_j |_S_ℓ = V_ℓ for ℓ≤ k.There is therefore a (trivial) vector bundle V on S whose restriction to S_j is V_j for all k.The complete linear series of the L_j give a system of closed embeddings C_j →𝐏(V_j).We may regard the structure sheaves 𝒪_C_j as a compatible system of quotients of the structure sheaf of 𝐏_A_j(V_j), so by Grothendieck's existence theorem they can be algebraized to a quotient 𝒪_C of the structure sheaf of 𝐏_A(V).Let C be the corresponding closed subscheme of 𝐏_A(V).By construction C is proper over Spec A.It is also flat by the infinitesimal criterion for flatness <cit.>.Therefore it is a family of nodal curves (see Rem. <ref>).The above argument can be used more generally to show that formal families of proper schemes can integrated if the central fiber X has H^2(X, 𝒪_X) = 0.See <cit.> or <cit.> for different ways of organizing the ideas. Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles on C, and σ : E → F is a morphism of vector bundles.Then formal objects of ℱ can be integrated. Suppose A is a complete noetherianlocal ring with maximal ideal 𝔪, set A_j = A / 𝔪^j+1.Given a formal family (C_j,E_j,F_j,σ_j) ∈ℱ(A_j), we seek an element(C,E,F,σ) ∈ℱ(A) inducing it.We may find C by the Lemma <ref>.Note that C is projective over A and the E_j and F_j are each formal families of coherent sheaves over C, so by Grothendieck's existence theorem (Theorem <ref>) they can be algebraized to coherent sheaves E and F over C.Moreover, both E and F are flat, by the infinitesimal criterion for flatness, so they are vector bundles.One more application of Grothendieck's existence theorem algebraizes the family of homomorphisms of coherent sheaves σ_j : E_j → F_j to a homomorphism σ : E → F and the lemma is complete. §.§ Artin's theorems on algebraization and approximation The question of integrability concerns objects ofa stack 𝒳 lying over the spectrum of a complete local algebra over a field. With the Schlessinger–Rim theorem and integration, we can factor any morphism Spec k →𝒳 through a map Spec A →𝒳 such that A is a complete noetherian local ring and is formally smooth over 𝒳.This is tantalizingly close to showing 𝒳 is an algebraic stack:we need too find a factorization that is genuinely smooth over 𝒳.The distinction between smoothness and formal smoothness is finiteness of presentation, so we need to find a finite type ring B that is still formally smooth over 𝒳 at the k-point, and induces A by completion at a point (this will give smoothness of the map B→𝒳 at the given k-point of B; to extend to smoothness on an open neighborhood of the k-point, see <ref>).This may be the subtlest part of the proof of Artin's criterion.It is resolved by Artin's approximation theorem, proved originally by Artin <cit.> with some technical hypotheses, and in its current form by B. Conrad and J. de Jong <cit.> using a spectacular theorem of Popescu <cit.>. The algebraizationtheorem (Theorem <ref>) asserts that under our assumption that 𝒳 is locally of finite presentation (Theorem <ref>(<ref>)),given ξ∈𝒳(A) in an appropriatecomplete local ring with residue field k, one can find a finite type k-algebraB with a marked point (maximal ideal 𝔫), whose completionatthe marked point is A, and an element η∈𝒳(B) that agrees to a specifiable finite order with ξ.That is one may select j beforehand and then find η such that the restriction of η to B / 𝔫^j ≃ A / 𝔪^j agrees with the restriction ξ_j of ξ.Even though η∈𝒳(B) does not necessarily restrict to ξ∈𝒳(A), it will differ from ξ only up to an automorphism of A that is the identity modulo 𝔪^j.In particular, it will still be formally smooth at the closed point (see the proof of Theorem <ref>). .2 cmWe begin by introducing notation. Let k be a field and let Λ be a complete noetherian local ring with residue field k.Recall, we denote by 𝒞_Λ the category of local artinian Λ-algebras with residue field k, and by𝒞_Λthe category of complete local Λ-algebras with residue field k that are formally of finite type over Λ. [Algebraization over a scheme of finite type] Let 𝒳 be a CFG over 𝖲 andlet (A,𝔪) ∈𝒞_Λ.We say a formal element η̂ of 𝒳 over SpecA, i.e., an objectof _j𝒳(A/𝔪^j), can be algebraized over a scheme of finite type if there exist: * a finitely generated Λ-algebra B,* a k-point s:Speck→SpecB corresponding to a maximal ideal 𝔫⊆ B, and, * an object θ∈𝒳(B), such that * A=B_𝔫, the completion of B at 𝔫, and,* the formal element η̂ is isomorphic to the image of θ under the map 𝒳(B)→_j𝒳(A/𝔪^j) induced by the morphism B→ B_𝔫→ B_𝔫/𝔫^j=A/𝔪^j. Artin's algebraization theorem asserts the following: [Artin's algebraization theorem] Let 𝒳 be a CFG that is locally of finite presentation over anexcellent DVR or field Λ. Let (A,𝔪) be a complete local Λ-algebra with residue field k that is formally of finite type over Λ.If η̅∈𝒳(A) then there is a finite type Λ-algebra B with a maximal ideal 𝔫 at whose completion B is isomorphic to A, along with η∈𝒳(B) inducing η̅∈𝒳(A).The proof is almost an immediate consequence of Artin's approximation theorem:[Artin's approx. <cit.>, <cit.>]Let 𝒳 be a CFG that is locally of finite presentation over anexcellent DVR or field Λ. Let (A,𝔪) be a complete local Λ-algebra with residue field k that is formally of finite type over Λ. Given η̅∈𝒳(A), there is a finitely generated Λ-algebra B with k-point s:Spec k→Spec B corresponding to a maximal ideal 𝔫 and B_𝔫=A, such that for any positive integer j there is an element η∈𝒳(B) (depending on j) such that the images of η and η̅ are isomorphic in 𝒳(A/𝔪^j).Take B and η as in the Approximation theorem, with j=2. In other words, η_2 and η̂_2 are isomorphic in 𝒳(A/𝔪^2). Now, inductively, using formal smoothness, we can use diagram (<ref>)Spec A/𝔪^j [d] [r]^ψ_j-1 Spec A [dd]^η̅ Spec A/𝔪^j+1@–>[ur]_ψ_j[d]Spec B [r]^η 𝒳to construct for all j morphisms ψ_j:SpecA/𝔪^j+1→Spec A so that η̅ψ_j≅η_j.This inducesa morphismψ:SpecA →Spec A so that η̅ψ≅η.But ψ is an isomorphism modulo 𝔪^2, so it must be surjective, and a surjective endomorphism of a noetherian ring is an isomorphism (e.g., <cit.>). §.§ Algebraicity of the stack of Higgs bundlesInLemma <ref>,Lemma <ref>, Lemma <ref>, Corollary <ref>, andLemma <ref>,we have verified the conditions of Theorem <ref> for the stack ℱ that parameterizes proper nodal curves equipped with a homomorphism of vector bundles: Let ℱ be the stack of quadruples (C,E,F,σ) where C is a nodal curve, E and F are vector bundles, and σ : E → F is a morphism of vector bundles.Then ℱ is an algebraic stack. Analogous arguments show that the stack of Higgs bundles satisfies the axioms, hence is algebraic.Alternately, one may consider the stacks ℰ_1 of pairs (C,E) where C is a nodal curve and E is a vector bundle on C and ℰ_2 of triples (C,E,F) where C is a nodal curve and E and F are vector bundles on C.(Note that ℰ_2 = ℰ_1 ×_ℳℰ_1.)Then we can construct the stack of Higgs bundles as the fiber product ℰ_1 ×_ℰ_2ℱ where the map ℰ_1 →ℰ_2 sends (C,E) to (C,E,E ⊗ω_C).We therefore have The stack of Higgs bundles on proper nodal curves is an algebraic stack.§.§ Outline of the proof of Artin's criterion We will briefly summarize the proof of Theorem <ref>. Since assumption Theorem <ref>(<ref>)is that 𝒳 isa stack, the key point is to establish the existence of a smoothrepresentable covering of 𝒳 by a scheme.The basic idea of the proof is to begin with an arbitrary point ξ_0 ∈𝒳(k), valued in a field k, and find a smooth neighborhood U →𝒳 of this point by enlarging Spec k until it is smooth over 𝒳.In more concrete terms, U will be a versal deformation of ξ_0.Repeating this for every point of 𝒳 and taking a disjoint union of the different U gives a smooth cover of 𝒳 by a scheme. We now explain in more detail how the arguments in the previous sections imply the existence of the schemes U..2 cmVersality at a point .2 cm By the Schlessinger–Rim theorem (see Theorem <ref>), the homogeneity of 𝒳 (Theorem <ref>(<ref>)) and the finite dimensionality of T_𝒳^-1(ξ_0) and T_𝒳^0(ξ_0) (Theorem <ref>(<ref>)) guarantee that 𝒳 is prorepresentable at ξ_0.That is, there is a formal groupoid V_1 ⇉V_0, with V_i = SpecR_i for complete noetherianlocal rings R_i, whose associated CFG (Definition <ref>) agreeswith 𝒳 on infinitesimal extensions of ξ_0.This gives a formal morphism V_0 →𝒳 that is formally smooth (Example <ref>). In other words, we have compatible elements ξ_j ∈𝒳(SpecR_0 / 𝔪^j+1), where 𝔪 is the maximal ideal of R_0.The assumption that formal objects of 𝒳 integrateuniquely (Theorem <ref>(<ref>)) guarantees that this formal morphism comes from a genuine morphism V_0 →𝒳, i.e., from an element of ξ∈𝒳(SpecR_0).Now the map V_0 →𝒳 is formally smooth at ξ_0, but it is not of finite type.To remedy this we can use the local finite presentation of 𝒳 (Theorem <ref>(<ref>)), which ensures that V_0 →𝒳 must factor through some scheme V of finite type.Unfortunately, it is not clear we can exert any control over V, even formally, to guarantee it is formally smooth over 𝒳.Fortunately, we may rely on Artin's algebraization theorem (Theorem <ref>) to ensure that V is still formally smooth over 𝒳 at the central point.This does not yet guarantee that the map V →𝒳 is smooth:we only have formal smoothness at one point.The next step will be to show formal smoothness at a point implies formal smoothness nearby..2 cmVersality in a neighborhood .2 cm Write V = Spec R.We know that R is of finite type and we now have a map η : V →𝒳 that we know to be formally smooth at a point lifting ξ.All that is left is to find an open neighborhood of this point at which the map is actually smooth.For this we use the obstruction theory T^1_𝒳 (Theorem <ref>(<ref>)).Hall shows that the existence of a locally presentable obstruction theory representing the automorphisms, deformations, and obstructions of 𝒳 implies that there is a relative obstruction theory T^1_V/𝒳 for V over 𝒳 and that T^1_V/𝒳 is a coherent functor.As V is formally smooth over 𝒳 at ξ_0, we know that T^1_V/𝒳(η, J) = 0 for any quasicoherent sheaf J on V supported at ξ_0.By a theorem of Ogus and Bergman <cit.>, coherent functors over noetherianrings satisfy an analogue of Nakayama's lemma, which guarantees that T^1_W/𝒳(η|_W, J) = 0 for all quasicoherent sheaves on W, where W is the localization of Vat the point ξ_0.Making use of the local finite presentation of 𝒳 (Theorem <ref>(<ref>)), Hall shows that this implies T^1_U/𝒳(η|_U, J) = 0 for all quasicoherent J on an open subset U ⊆ V containing W.Then U →𝒳 is locally of finite presentation and satisfies the formal criterion for smoothness..2cm Bootstrapping to representability .2cmTo conclude that U →𝒳 is smooth, we only need to show that it is representable by algebraic spaces.This is proved by observing that the hypotheses of Theorem <ref> on 𝒳 imply relative versions of themselves for the map U →𝒳 (cf. Lemma <ref>, Lemma <ref>, and the relative obstruction theory T^1_U/𝒳 mentioned above).The same hypotheses then hold for any base change U_Z → Z via any map Z →𝒳.Taking Z to be a scheme and viewing U_Z as a sheaf in the étale topology on schemes over Z, this reduces the problem to showing that U_Z is an algebraic space.Now we can try to prove the theorem for 𝒳 = U_Z:In effect, Theorem <ref> is reduced to the case where 𝒳 is a sheaf as opposed to a stack. Now we have a scheme U and a map U →𝒳 that is formally smooth and locally of finite presentation, and we are faced with the same problem:to show U →𝒳 is representable by algebraic spaces.But this time, the relative diagonal of U over 𝒳 is injective, so that if we iterate the process one more time, we discover once again that our task is to prove U_Z is an algebraic space.But U_Z = U ×_𝒳 Z is a subsheaf of U × Z, since the diagonal of 𝒳 is injective.Therefore taking U × Z to be the new base scheme and replacing 𝒳 with U × Z, we discover we may assume further that 𝒳 is even a subsheaf of the base scheme S.But now, if W →𝒳 is any map, we have W ×_𝒳 U = W ×_S Usince 𝒳 is a subsheaf of S.Since U → S is schematic—it is a morphism of schemes, after all—so must be U →𝒳. Alph@̧section§ SHEAVES, TOPOLOGIES, AND DESCENTWe collect a few more technical topics surrounding the subject of descent.In particular, we show that the presence of nontrivial automorphisms always prevents an algebraic stack from having a representable sheaf of isomorphism classes (Corollary <ref>), we give a few more technically efficient ways of thinking about descent (<ref>), we discuss a bit more about saturations of pretopologies (<ref>) and we recollect and example of Raynaud showing that genus 1 curves do not form a stack unless one admits algebraic spaces into the moduli problem (<ref>). §.§ Torsors and twists The point of this section is to explain how nontrivial automorphisms give rise to nontrivial families, and prevent a stack from having a representable sheaf of isomorphism classes.In other words, moduli functors parameterizing isomorphism classes ofobjects with nontrivial automorphism groups are never representable. §.§.§ Locally trivial families and cohomologyIn general, one can build a locally trivial family with fiber X over a base S from cohomology classes in H^1(S, Aut(X)) (<cit.>).If one represents the cohomology class with a Čech cocycle then it is a recipe for assembling the family from trivialized families on open subsets.The cocycle condition is precisely the one necessary to ensure that the family can be glued together, while the coboundaries act via isomorphisms.The cocyle condition appears in the definition of the gluing condition for a stack in <ref> for precisely this reason. §.§.§ Locally trivial families from torsors over the base Example <ref> describes a special case of a standardmethod for constructing locally trivial families is via automorphisms, and covers of the base.Namely, givenschemes X_1 and S, and an étale principal Γ-bundle S→ S for some discrete group Γ, then for eachϕ:Γ→Aut(X_1)one can construct a (étale) locally trivial family X/S with fibers isomorphic to X_1 by setting X=(X_1×S)/Γ, where the quotient is via the (free)diagonal action induced by ϕ.Alternately, this can be constructed as the space of equivariant morphisms from S to X_1.Using the previous remark, thisdefines a mapHom(Γ,Aut(X_1))→ H^1(S,Aut(X_1))(of course, the cohomology should be taken in a topology in which S is a torsor). When S→ S is a universal cover, we obtaina map Hom(π_1(S),Aut(X_1))→ H^1(S,Aut(X_1)).In the discussion above (<ref>),it is possible for the isotrivial family X to be trivial, even if the homomorphism ϕ is nontrivial.We will highlight one situation in which the family X can be assured to be nontrivial. Let X_1 and S→ S be as in the discussion above (<ref>). LetG=Aut(X_1), and let G'≤ G be a discrete subgroup.From <ref> and <ref> above, we obtaina commutative diagramHom(π_1(S),G') @^(->[r] @=[d]Hom(π_1(S),G) [r] [d]Hom(π_1(S),π_0(G)) @=[d] H^1(S,G') [r] H^1(S,G) [r] H^1(S,π_0(G))where the horizontal maps are the natural maps, andthe verticalequalities on the left and right come from the fact that G' and π_0(G) are totally disconnected. In summary, we can conclude that a locally trivialfamily X/S obtained from a homomorphism ϕ:π_1(S)→ G' will be a nontrivial family so long as the image of ϕ in Hom(π_1(S),π_0(G)) is nontrivial.Note that the kernel of thenatural mapH^1(S,G') →H^1(S,G) corresponds toprincipal G' bundles that can be equivariantly embedded in G× S; i.e., the kernel is given bysections over S of the coset space (G/G')× S.[Disconnected automorphism groups]If Aut(X_1) is disconnected, then there exist nontrivial locally trivial families X/S with fibers isomorphic to X_1. This follows immediately from the previous remark, so long as one can find a space S with π_1(S)=ℤ, or at least that has a nontrivial principal ℤ-bundle. Indeed, for anyα∈Aut(X_1) notin the connected component of the identity, one would take G'=⟨α⟩ and set X/S to be the locally trivial family corresponding to the map ℤ→⟨α⟩ by 1↦α. In the complex analytic setting, we can simply take S=ℂ^*. However, we can arrange for this even in the category of schemes by joining a pair of rational curves at two points;this has fundamental group ℤ in the sense that it has a simply connected covering space (even in the Zariski topology!) with a simply transitive action action of ℤ.[Finite automorphism groups] If Aut(X_1) is finite and nontrivial (or more generally, has a finite subgroup not contained in the connected component of the identity), one can easilyconstruct similar examples using finite covers.Indeed, then G=Aut(X_1) contains a nontrivial finite cyclic subgroup G'=μ_n (not contained in the connected component of the identity).Let S=ℂ^* and S→ S be the cyclic cover z↦ z^n, which is an étale principal μ_n-bundle.Then, as in(<ref>) of Remark <ref> above,we obtain a diagram @C=1em@R=1emHom(μ_n,G') @=[r] [rd] Hom(π_1(S),G') @^(->[r] @=[d]Hom(π_1(S),G) @->[r] [d]Hom(π_1(S),π_0(G)) @=[d] H^1(S,G') [r] H^1(S,G) [r] H^1(S,π_0(G)).Therefore, a generator of μ_n=Hom(μ_n,G')determines a nontrivial locally trivial family given explicitly by X=(X_1×ℂ^*)/μ_n→ S=ℂ^*/μ_n=ℂ^*, where the quotient is by the diagonal action under the identifications μ_n≤Aut(X_1), and μ_n≤Aut(ℂ^*) acting by a primitive n-th root of unity. [Isotrivial families of curves] For every g there exists a relative curve π:X→ S of genus g that is isotrivial, but not isomorphic to a trivial family.For g=0, any nontrivial ruled surface X→ S provides an example.The previous Example <ref>, and Example <ref>, provide nontrivial isotrivial families for g≥ 1. For instance, for g≥ 2, one could begin with a hyperelliptic curve { y^2 = f(x) } carrying the nontrivial action of μ_2 sending y to -y.The construction in Example <ref>yields the family { t y^2 = f(x) }, where t is the coordinate on S=ℂ^*.§.§.§ Twisting by a torsor All of the previous examples are special cases of the general process of twisting by a torsor.[Twist by a torsor] Suppose that Z is an S-point of a stack 𝒳 and that the automorphisms group of Z is a smooth group scheme G over S (in fact, a flat group scheme is enough if 𝒳 is an algebraic stack).Let P be a G-torsor.As G is smooth, P is smooth over S, and hence has sections over some étale cover of S.Therefore P is covering in the étale topology.(Note that P→ S it is not necessarily an étale morphism,but this is not an obstacle to using it for descent, since we may use smooth descent; alternatively, see Appendix <ref>.)By descent, 𝒳(S) may be identified with the G-equivariant objects of 𝒳(P).In particular, let Z_P in 𝒳(P) be the pullback of Z to P.Then Z ∈𝒳(S) corresponds to Z_P with the trivial action of G.However, we can also ask G to act on Z_P=P×_SZ by the given action of G on P and by automorphisms on Z, giving an object Y:=P ×_G Z:=(P×_S Z)/Gin𝒳(S) by descent. We call Y the twist of Z by the torsor P.We note that the torsor P can be recovered from a twist Y of Z as the sheaf ℐsom_𝒳(Z, Y).Thus the twist is nontrivial if and only if the torsor P was.Moreover, one has an equivalence of categories between the full subcategory of 𝒳(S) consisting of twists of Z and the category BG(S) consisting of G-torsors on S.The following theorem was communicated to us by Jason Starr <cit.>. It implies that if 𝒳 is a stack in the fppf topology in which there are objects with nontrivial automorphisms, and X is its associated sheaf of isomorphism classes, then X cannot be representable by a scheme (Corollary <ref>).Let G be an algebraic group of finite type over an algebraically closed field k.Then there is a scheme S of finite typeover k and a nontrivial G-torsor over S. We suppose that G is an algebraic group of finite type and that H^1(S, G) = 0 for every scheme S.We wish to show that G=0. We will break the proof into several steps. .2 cm Step 1: G is connected. We will show that G is connected by showing that any homomorphism ℤ→ G has image in the connected component of the identity.To this end, as in Example <ref>,let S be a k-scheme with a nontrivial ℤ-torsor P.For concreteness, let us take S to be an irreducible rational curve with a single node (otherwise smooth) and take P to be an infinite chain of copies of the normalization of S, attached at nodes. Then using the right hand side of the commutative diagram (<ref>), we may conclude every homomorphism ℤ→ Gcomposes to a morphism ℤ→ G→π_0(G), with trivial image, and we are done..2 cm We now assume always that G is connected, andproceed to considerG_red⊆ G, the maximal reduced closed subscheme (necessarily a subgroup since we work over an algebraically closed field, e.g.,<cit.>). ByChevalley's theorem (see for instance <cit.> for a modern treatment), there is an exact sequence1 → G_aff→ G_red→ A → 0where G_aff is smooth connected and affine, and A is an abelian variety. Our next goal will be to show that G_red=0; we will do this in several steps. .2 cm Step 2: H^1(S,G_red)=0 for all S. For any scheme S, we have an exact sequence:Hom(S, G) →Hom(S, G/G_red) → H^1(S, G_red) → H^1(S, G)We have assumed that H^1(S, G) = 0.Furthermore, G → G/G_red is smooth (since G_red is) and G/G_red is artinian, so G admits a (not necessarily homomorphic) section over G/G_red.This implies Hom(S,G) →Hom(S, G/G_red) is surjective, so H^1(S, G_red) = 0..2 cm Step 3: G_aff is unipotent. Consider the exact sequenceHom(ℙ^1, G_red) →Hom(ℙ^1, A) → H^1(ℙ^1, G_aff) →Hom(ℙ^1, G_red) = 0.As A is an abelian variety, every map from ℙ^1 to A is constant and therefore lifts to G_red.It follows that H^1(ℙ^1, G_aff) = 0.Let U ⊆ G_aff be the unipotent radical <cit.>.Consider the exact sequence: H^1(ℙ^1, G_aff) → H^1(ℙ^1, G_aff/U) → H^2(ℙ^1, U) Since U is an iterated extension of 𝔾_as <cit.>, we have that H^2(ℙ^1, U) = 0, whence H^1(ℙ^1, G_aff/U) = 0.Let T ⊆ G_aff/U be a maximal torus, and let W be its Weyl group.As G_aff/U is reductive by definition <cit.>, there is an injection (in fact a bijection!)H^1(ℙ^1, T) / W → H^1(ℙ^1, G_aff/U)=0by a theorem of Grothendieck <cit.> (note that Grothendieck works analytically, but his proof of the injectivity is valid algebraically <cit.>).Thus H^1(ℙ^1,T) is finite; although we do not need it, note thatsince the Weyl group coinvariants of H^1(ℙ^1, T) are trivial if and only if H^1(ℙ^1, T) is, we can actually conclude immediately that H^1(ℙ^1, T) = 0.But T ≃𝔾_m^r for some r, and H^1(ℙ^1, 𝔾_m) = ℤ.Therefore r = 0.That is, the maximal torus T of G_aff/U is trivial, so that G_aff is unipotent <cit.>. .2 cm Step 4: G_red is affine. Now that we know G_aff is unipotent, we argue that A = 0.Let S = 𝔸^1 ∖{ 0 }.Consider the exact sequenceH^1(S, G_red) → H^1(S, A) → H^2(S, G_aff) .Since S is affine, and G_aff is unipotent and thereforean iterated extension ofquasicoherent sheaves (associated to 𝔾_as), we haveH^2(S, G_aff) = 0.This implies H^1(S, A) = 0.Choose an integer n relatively prime to the characteristic of k.Consider the exact sequenceHom(S, A) Hom(S, A) → H^1(S, A[n]) → H^1(S, A)where [n] denotes multiplication by n and A[n] is the n-torsion subgroup.Since S is rational, every map S → A is constant.In particular, Hom(S,A) Hom(S,A) is surjective.Therefore H^1(S, A[n]) injects into H^1(S,A) = 0, and so is also 0.Now, A[n] ≃ (ℤ / n ℤ)^2g where g is the dimension of A.We know that H^1(S, ℤ/n ℤ) = ℤ/ n ℤ, so we deduce that g = 0, and therefore A = 0. .2 cm Step 4: G_red=0. Now we know that G_red = G_aff is affine and unipotent.We can choose an injective homomorphism G_red⊆ G' where G' is smooth affine and reductive (i.e., embed it in an appropriate GL_n).Consider the G_red-torsor G' over G'/G_red.This must be trivial, since H^1(G'/G_red, G_red) = 0 from Step 2, so G' ≃ G'/G_red× G_red as a scheme.But G' is affine, so this implies G'/G_red is affine.Therefore G_red is reductive, by Matsushima's criterion <cit.>.As it is also unipotent, this means G_red = 0. .2 cm Step 5: G=0. Since G_red=0, thismeans that G = G/G_red is the spectrum of an artinian local ring.But we can once again choose a closed embedding G ⊆ G' where G' is smooth and affine (for example, let G act on its ring of regular functions).The quotient G'/G is also reduced and affine.But H^1(G'/G, G) = 0 by assumption, so that G' ≃ G'/G × G as a scheme.As G' is reduced, this means G is reduced, and therefore G = 0.Suppose that X is the presheaf of isomorphism classes in an algebraic stack 𝒳 such that all objects over algebraically closed fields have automorphism groups that are algebraic groups of finite type over the field.If X is a sheaf in the fppf topology then X ≃𝒳.In particular, if 𝒳 admits an object over an algebraically closed field kwith a nontrivial algebraic automorphism group of finite type over k, then X is not representable by a scheme. Let x be a k-point of 𝒳, where k is algebraically closed.Let G be the stabilizer group of x.We obtain a monomorphism BG →𝒳_k.If P is any G-torsor over a schemeS then we obtain a twist S →BG →𝒳 (Example <ref>).By definition, the twists agree locally, so that the induced maps to X coincide locally.But we have assumed X is a sheaf, so that the maps agree globally as well.But the torsor can be recovered, up to isomorphism, from the twist, so all G-torsors over all k-schemes are trivial.Therefore by Theorem <ref>, G is trivial, so no point of 𝒳 has a nontrivial stabilizer group and 𝒳≃ X. §.§ More on descent We highlight a few alternate formulations of the descent properties (Section <ref>).The definitions we present here are more efficient than those given Section <ref>, and often lead to more streamlined proofs, but they come at a cost of abstraction. §.§.§ Descent using gluing dataIn Definition <ref> we needed a cleavage in order to be able to write things like X_i |_S_ij.To do this properly requires keeping track of a number of canonical isomorphisms, which were intentionally elided in Definition <ref>.Reliance on a cleavage can be avoided by explicitly choosing a restriction at each step, instead of insisting on a canonical choice from the beginning.Although this definition avoids the technical deficiencies of Definition <ref>, it only exacerbates the proliferation of indices.Nevertheless, we will see in <ref> that it points the way towards a definition that is both technically correct and pleasantly efficient.[Descent datum via gluing data]Let (𝖲, P 𝒯) be a presite and let { S_i → S } be a cover of S in 𝖲.A descent datum with respect to this cover consists of the following data: * objects X_i ∈ℳ(S_i), X_ij∈ℳ(S_ij), and X_ijk∈ℳ(S_ijk) for all indices i,j,k of the cover;* morphismsX_ij→ X_iX_ij→ X_jX_ijk→ X_ij X_ijk→ X_ik X_ijk→ X_jkin ℳ respectively covering the canonical projectionsS_ij→ S_iS_ij→ S_jS_ijk→ S_ij S_ijk→ S_ik S_ijk→ S_jk . The category of descent data with respect to { S_i → S }, denoted ℳ(S_∙), has as objects the descent data as defined above.A morphism (X_∙) → (Y_∙) consists of morphisms X_i → Y_i X_ij→ Y_ij X_ijk→ X_ijkfor all indices i,j,k, commuting with the sructural morphisms (a more precise formulation of this condition will appear below in Remark <ref>).An observant reader may be wondering where the cocycle condition (<ref>) is hiding in Definition <ref>.It is built into the commutativity diagram (<ref>),@C=10pt X_iX_ij[ur] [dl][u] [l] [r] X_ijk[dll] [d] [drr] X_ik[ul] [dr] X_j[ll] X_jk[rr]X_kwhich itself is forced by the commutativity of the corresponding diagram with each X replaced by an S, and the fact that every arrow in ℳ is cartesian.The morphism α_ij of Definition <ref> is the compositionX_i |_S_ij X_ij X_j |_S_ijwith α_ik and α_jk defined similarly.The cocycle condition is the identity of the compositionsX_i |_S_ijk X_ij|_S_ijk X_j |_S_ijk X_jk|_S_ijk X_k |_S_ijkX_i |_S_ijk X_jk|_S_ijk X_k |_S_ijkby virtue of the restriction of Diagram (<ref>) to S_ijk. If X is an object of ℳ(S) then using Axiom (i) of a category fibered in groupoids (Definition <ref>) we can find induced objects X_i ∈ℳ(S_i), X_ij∈ℳ(S_ij), and X_ijk∈ℳ(S_ijk) for all indices i,j,k.We also obtain the required morphisms among these by many applications of Axiom (ii) of a category fibered in groupoids.By an application of the axiom of choice, we can do this for all objects of ℳ(S_∙) and obtain a functor ℳ(S) →ℳ(S_∙).[Effective descent datum via gluing data] A descent datum X ∈ℳ(S_∙) for ℳ with respect to a cover { S_i → S } is said to be effective if it lies in the essential image of ℳ(S) →ℳ(S_∙). We give a technical reformulation of Definition <ref>, motivated by the idea that the data we are keeping track of can be organized conciselyby the nerve of subcoversconsisting of three open sets; i.e., 2-simplices. Suppose that the cover { S_i → S } is indexed by a set I.Let Δ be the category of nonempty, totally ordered, finite sets of cardinality ≤ 3, equipped with a morphism to I.That is, an object of Δ is a pair (T, f) where T is a totally ordered finite set with 1 ≤ |T| ≤ 3 (i.e., T is either { 0 }, { 0 < 1 }, or { 0 < 1 < 2 } up to isomorphism) and f : T → I is a function.A morphism (T, f) → (T', f') is an order preserving function g : T → T' such that f' ∘ g = f.We introduce abbreviations for certain objects of Δ.Every object of Δ is isomorphic to one of the following: * for i ∈ I, we write i for the object ({ 0 }, (0 ↦ i)) of Δ;* for i,j ∈ I, we write ij for the object ({ 0 < 1 }, (0 ↦ i, 1 ↦ j));* for i,j,k ∈ I, we write ijk for the object ({ 0 < 1 < 2 }, (0 ↦ i, 1 ↦ j, 2 ↦ k)). The cover { S_i → S } induces a functor S_∙ : Δ^𝗈𝗉→𝖲 sending x to S_x.In other words, it sends i to S_i and ij to S_ij and ijk to S_ijk.The morphisms are all the canonical projections among fiber products.Now a descent datum, in the sense of Definition <ref> is a functor X_∙ : Δ^𝗈𝗉→ℳ such that π X_∙ = S_∙:ℳ[d]^π Δ[r]_S_∙[ur]^X_∙ 𝖲A morphism of descent data X_∙→ Y_∙ is a natural transformation of functors that projects to the identity natural transformation of S_∙. §.§.§ Descent using sieves Sieves on topological spacesIn fact, there is an even more efficient formulation of the definition of a sheaf.Recall that the if V is an object of 𝖮_X then h_V is the functor represented by V.It is convenient to work with the related functor, which we again denote by h_V, defined as h_V(W) = {ι_W,V} if W ⊆ V and h_V(W) = ∅ otherwise. Suppose that U ∈𝖮_X.For any open cover 𝒰 of U, define a presheaf h_𝒰:h_𝒰(W) = ⋃_V ∈𝒰 h_V(W) = {ι_W,X} ∃ V ∈𝒰 such thatW ⊆ V∅ elseIf ℱ : 𝖮_X^𝗈𝗉→ (𝖲𝖾𝗍) is a presheaf then, as h_𝒰⊆ h_X (e.g., if W⊆ U for U∈𝒰, then we useW⊆ U⊆ X), we get a morphismℱ(X) Hom(h_X, ℱ) →Hom(h_𝒰, ℱ) .The bijectivity of the arrow on the left is Yoneda's lemma.The interested reader may prove the following proposition:Let ℱ be a presheaf. * ℱ is separated if and only if (<ref>) is an injection for all open covers 𝒰 of all U ∈𝖮_X.* ℱ is a sheaf if and only if (<ref>) is a bijection for all open covers 𝒰 of all U ∈𝖮_X. Sieves in generalLet 𝖲 be a category and S an object of 𝖲.A sieve of 𝖲 is a subcategory of 𝖲/S that is fibered in groupoids over 𝖲/S. In other words, a sieve of S is a subcategory ℛ⊆𝖲/S such that whenever S”→ S' is a morphism in 𝖲/S and S' is in ℛ then S” is also in ℛ.In Example <ref> we saw how to associate a sieve ℛ to any family of maps { S_i → S }.By a covering sieve we mean a sieve associated to a covering family in the pretopology.In fact, all of the axioms of a Grothendieck topology can be formulated purely in terms ofsieves <cit.>, but that will not concern us here.[Descent datum via sieves] Let 𝖲 be a presite, let π : ℳ→𝖲 be a category fibered in groupoids over 𝖲, and let ℛ be a covering sieve of S ∈𝖲.A descent datum for ℳ with respect to ℛ is a functor X : ℛ→ℳ lifting the canonical projection ℛ⊆𝖲/S →𝖲 (sending an object S' → S of 𝖲/S to S'):ℳ[d]ℛ @^(->@<-1pt>[r] @/^10pt/[urr]^X𝖲/S [r]𝖲.Descent data over the sieve ℛ are the objects of a category ℳ(ℛ) where a morphism X → Y is a natural transformation projecting to the identity natural transformation of the projection ℛ→𝖲.To construct the functor ℳ(S) →ℳ(ℛ) for a sieve ℛ of S, observe that ℳ(S) ≃ℳ(𝖲/S) by the 2-Yoneda lemma (on the left we mean the fiber of ℳ over S and on the right we mean the category of morphisms from 𝖲/S to ℳ).Composing this equivalence with the restriction ℳ(𝖲/S) →ℳ(ℛ) induced from the inclusion ℛ⊆𝖲/S inducesℳ(S) ≃ℳ(𝖲/S) →ℳ(ℛ) [Effective descent datum via sieves]A descent datum X ∈ℳ(ℛ) for ℳ with respect to a sieve ℛ is said to be effective if it lies in the essential image of ℳ(S) →ℳ(ℛ). §.§ Grothendieck topologiesGrothendieck pretopologies seem quite natural from the definition of a sheaf, but have the deficiency that many different pretopologies can give rise to the same category of sheaves.This is not unlike the way different bases of a topological space should be considered equivalent.The topology associated to a pretopology is the finest pretopology that gives the same category of sheaves (see <cit.>).In this section, we give an idea of how this works, providing a review of <cit.>, although here for brevity we define the topology directly, without a discussion of refinements of pretopologies. Note that typically a Grothendieck topology is defined in terms of sieves, not coverings; the point is that the sieves associated to a pretopology are the same as the sieves associatedto the topology obtained from the pretopology <cit.>, and therefore both induce the same Grothendieck topology in the sense of sieves..2 cmFor clarity of the discussion in this section, we will refer to coverings {S_α→ S} in a pretopology 𝒯 on a category 𝖲 as basic coverings.We start with a definition (cf. <cit.>):[Coverings with respect to 𝒯]Let 𝒯 be a pretopology on a category 𝖲.We call a family of morphisms {S_α→ S} covering (with respect to 𝒯) if there is a basic covering family {T_β→ S} such that, for each β, there is an α such that the morphism T_β→ S factors through S_α.The covering family { T_β→ S } is called a basic covering refinement of { S_α→ S }.Note that the covering{S_α→ S} in Definition <ref>is a covering in the sense ofDefinition <ref>.Assume that 𝖲 has all fiber products.Let 𝒯 be a pretopology on 𝖲 and let 𝒯' be the collection of all covering families with respect to 𝒯.Then 𝒯' is a pretopology on 𝖲. Property (PT0) is automatic, since 𝖲 has all fiber products.Likewise (PT3) is immediate.If { S_α→ S } has a basic covering refinement { T_β→ S } and S' → S is any morphism then { S_α×_S S' → S' } has the basic covering refinement { T_β×_S S' → S' }, hence is in 𝒯'.This proves (PT1).Suppose that { S_α→ S } has a basic covering refinement { T_β→ S }, and each S_α has a family { S_αγ→ S_α} with a basic covering refinement { T_αδ→ S_α}.For each β, choose an α(β) and a factorization of T_β→ S through S_α(β).Then the maps T_α(β) δ×_S_α(β) T_β→ T_β are covering.As the T_β cover S, the property (PT2) implies that the T_α(β) δ cover S.Therefore the T_α(β) δ give a basic covering refinement of the family S_αγ→ S. [Grothendieck topology] A pretopology 𝒯 is calleda (Grothendieck) topology if 𝒯' = 𝒯, in the notation of Lemma <ref>.If 𝒯 is a pretopology then 𝒯' is called the associated topology to 𝒯.In <cit.> what we call atopology is calleda saturated pretopology.Let 𝒯 be the pretopology on topological spaces where the basic covering families are open covers.Then every surjective local isomorphism has a section over a suitable open cover, so surjective local isomorphisms are covering in the associated saturated topology. Consider the étale pretopology on schemes, defined in Example <ref>.Every smooth surjection admits a section over some étale cover, so every smooth surjection is covering in the associated topology to the étale pretopology. The following lemma shows that a pretopology and its associated topology have the same sheaves.Passage to the associated topology may therefore significantly expand the class of morphisms with respect to which one can use descent. If 𝒯 is a pretopology on 𝖲 then a presheaf on 𝖲 is a sheaf with respect to 𝒯 if and only if it is a sheaf with respect to 𝒯'. Since 𝒯⊆𝒯', it is immediate that sheaves in 𝒯' are sheaves in 𝒯.For the converse, suppose that ℱis a sheaf with respect to 𝒯 and let { S_α→ S } be a covering family (with respect to 𝒯').Construct a presheaf ℱ' over S by the following formula:ℱ'(R) = eq( ∏_αℱ(U_α×_S R) ⇉∏_α,βℱ(U_α×_S U_β×_S R) ).There is a natural map φ : ℱ→ℱ', which we would like to show is an isomorphism.Both ℱ and ℱ' are sheaves in the pretopology 𝒯, so this is a local problem in 𝒯.We can therefore replace S by a basic cover from 𝒯.Since { U_α→ U } has a refinement by a basic cover, we can assume that there is a section σ : S → U_α for some α.We can use σ to construct an inverse ψ to the map φ : ℱ→ℱ'.Indeed, if ξ∈ℱ'(S) then let ξ_α be its projection on the α component.Then σ^∗(ξ_α) ∈ℱ(S) and we set ψ(ξ) = σ^∗(ξ_α).It is immediate that ψφ(η) = η for all η∈ℱ(S).We check that φψ(ξ) = ξ.What we need to check is that, for all indices β,ξ_β = σ^∗(ξ_α) |_U_β .By assumption, ξ is equalized by the maps to ∏_α,βℱ(U_αβ), where U_αβ = U_α×_S U_β.Therefore ξ_α|_U_αβ = ξ_β|_U_αβ.But (σ, id_U_β) determines a section of U_αβ over U_β, so we determine that ξ_β = σ^∗ (ξ_β|_U_αβ) = σ^∗ (ξ_α|_U_αβ) = σ^∗(ξ_α) |_U_β,as required.§.§ An example of ineffective descentThe category fibered in groupoids ℳ_1 is not an étale stack!As we will see, it is possible to create a descent datum for genus 1 curves that is not effective.This is really a deficiency of our definition of ℳ_1 in <ref>, by which ℳ_1 parameterizes schematic families of smooth, proper curves of genus 1.The proper thing to do would be to include in our moduli problem smooth proper families of algebraic spaces (Definition <ref>) whose fibers are curves of genus 1. To construct an ineffective descent datum for ℳ_1, it will help to notice that every descent datum for S can at least be descended to a sheaf on the big étale site of S that is locally on S representable by a family of genus 1 curves.This sheaf is precisely the algebraic space we should have admitted into the moduli problem for ℳ_1.Our task in this section is to construct such a sheaf X that is locally in S representable by genus 1 curves, but is not globally representable by a scheme.To begin, note that if X→ S is any family of smooth curves of genus 1 over a base S, then the relativeJacobian J→ S of X→ S is a family of abelian schemes of dimension 1 over S.This construction is local on S, so that even if X is merely a sheaf over S that is locally representable by a family of smooth curves of genus 1, one obtains a descent datum for a family of elliptic curves over S.But the descent datum comes with a compatible family of ample line bundles (coming from the origin of the group structure) so by Theorem <ref>, it can be descended to a family of elliptic curves over S.Furthermore, J acts on X making X into a J-torsor.Therefore X is classified up to isomorphism by an element [X]∈ H^1(S, J) (this can be étale or flat cohomology).Raynaud shows that, provided S is quasicompact, this element [X] is torsion if and only if X is projective over S <cit.>, and that, provided S is normal, X is projective over S if and only if it is representable by a scheme <cit.>.In fact, Raynaud proves these statements more generally about torsors under abelian varieties: [<cit.>]Assume that S is a quasicompact scheme and let J be a projective abelian variety over S.Then a J-torsor X is projective if and only if its class in H^1(S, J) is torsion. [<cit.>]Let S be a quasicompact, normal scheme and J an abelian variety.Then a J-torsor X is representable by a scheme if and only if it is projective.First, suppose the class represented by X is torsion, say n [X] = 0.As n[X] is represented by [X / J[n]], where J[n] is the n-torsion of J, we have a finite mapX → X / J[n] ≃ J.This implies X is projective over S, as J is. Conversely, if X is projectiveover S, then a relativelyample line bundle L on X over S induces a relativelyample line bundle L' on J <cit.>.To see this, note that the relative Néron–Severi group NS_X/S of X over S is isomorphic to NS_J/S, so that L determines a class in NS_J/S.Any line bundle M on J determines a line bundle μ^∗ M ⊗ p_1^∗ M^∨⊗ p_2^∗ M^∨⊗ e^∗ M(where μ : J × J → J is the addition map, p_i are the projections, and e is the composition of the projection J → S and the zero section) on J ×_S J that depends only on the Néron–Severi class of M.Restricting this to the diagonal of J × J recovers a line bundle on J whose image in the Néron–Severi group is twice that of M.Altogether, this gives a map:NS_X/S≃NS_J/S→Pic_J/S.Applying this to Lyields a line bundle L' on J.Moreover, under a local isomorphism between J and X, the Néron–Severi class of L' is double that of L.Thus L' is relatively ample on J over S.In general, a relatively ample line bundle M on J induces an isogeny J →Ĵ of abelian schemes over S (where Ĵ is the dual abelian variety) by way of (<ref>), and hence a morphism H^1(S,J) → H^1(S,Ĵ).The image of [X] is the class [X̂], where X̂ is the Ĵ-torsor consisting of line bundles on X that lie in the same Néron–Severi class as L' <cit.>.By assumption this torsor is trivial (the line bundle L^⊗ 2 provides a section), so [X̂] = 0.On the other hand, we have an exact sequenceH^1(S,K) → H^1(S,J) → H^1(S,Ĵ) ,where K is the kernel of J →Ĵ, so [X] lies in the image of H^1(S,K).Since L is ample, K is finite, so H^1(S,K) is torsion, and therefore so is [X].We will prove a special case of Theorem <ref>, following the proof of <cit.>, that will suffice to construct our example.We assume that S is the spectrum of a noetherian local ring with closed point s and generic point η.Let X be a J-torsor over S.Choose an effective Cartier divisor D ⊆ X whose complement U ⊆ X is quasi-affine and meets the closed fiber of X.We will argue that the line bundle L = 𝒪_X(D) must be ample.We make a few observations: *The J-orbit of U is all of X.This is because U meets X_s and J_s acts transitively on X_s, so JU contains X_s.But every point of X specializes to a point of X_s, since X is proper over S, and JU is open, hence contains all of X.* Let M be the line bundleM = (p_1 + p_2 + p_3)^∗ L ⊗ (p_1 + p_3)^∗ L^∨⊗ (p_2 + p_3)^∗ L^∨⊗ p_3^∗ Lon J ×_S J ×_S X.There is some positive integer n such that M^⊗ n is the pullback of a line bundle on J ×_S J.As J ×_S J is normal, it is sufficient to verify this over the generic point of S <cit.>. We may therefore assume that S = η is the spectrum of a field.If X(η) ≠∅ then X ≃ J and this is a version of the theorem of the cube (see <cit.>); in this case n = 1.If X does not have a section over η then it will certainly have one over some finite extension p : η' →η, so p^∗ M is the pullback of a line bundle M' on J_η'×_η' J_η'.If η' has degree n over η then the norm of p^∗ M is M^⊗ n and this is the pullback of the line bundle Norm_η'/η(M') on J_η×_η J_η.We replace D with nD so that (<ref>) is the pullback of a line bundle on J ×_S J without passing to a tensor power.*Restricting (<ref>) to a point (g,-g) of J yields an isomorphism of line bundles on X:T_g^∗ L ⊗ T_-g^∗ L ≃ L^⊗ 2where T_g denotes translation by g. To prove the ampleness of L, we must show that, for any point x of X, and any open neighborhood V of x in X, there is some index n and some f ∈Γ(X, L^⊗ n) such that the open set X_f ⊆ X defined by the nonvanishing of f is contained in V <cit.>.Suppose first that x ∈ U.Let z be the tautological section of 𝒪_X(D) that vanishes exactly along D.As U is quasi-affine, there is some affine open neighborhood W of x in V ∩ U, defined by the nonvanishing of a function h on U.Then there is some n ≥ 1 such that z^n h extends to all of X.That is, we may regard z^n h as a section without poles of 𝒪_X(nD) and W_h = W.If x is not in U, we can at least find a g ∈ J such that T_g(x) ∈ U (by observation (<ref>), above).Applying the argument above to T_g(x) and T_g(V), we can find a section f ∈Γ(X, L^⊗ n) such that x ∈ W_f ⊆ T_g(V) by the argument above.We have some freedom in the choice of g, and by avoiding a closed set of possibilities, we can ensure that T_-g(x) ∈ U as well.Then we can consider f' = T_g^∗ f ⊗ T_-g^∗ z^n as a section of T_g^∗ L^⊗ n⊗ T_-g^∗ L^⊗ n≃ L^⊗ 2n (by (<ref>), above).Now, W_f' = T_g^-1(W_f) ∩ T_-g^-1(W_z) = T_g^-1(W_f) ∩ T_-g^-1(U).By our choice of g, we have x ∈ W_f' and W_f'⊆ T_g^-1 T_g(V) = V, as required. Finally, <cit.> proves that there is a genus 1 curve over a normal noetherian local ring of dimension 2 whose image in H^1(S,J) is non-torsion, establishing the existence of the desired family X→ S.Another version of the construction (and a slightly stronger conclusion) can be found in <cit.>.We will summarize Raynaud's construction.We begin with a discrete valuation ring R with algebraically closed residue field.Let T = Spec R and assume that T has a connected, étale double cover T' → T.Let π be a uniformizer for T.For any object Y over T, we will write Y' for its base change to T'.We write τ for the generic point of T and t for the special point; τ' is the generic point of T' and t_1 and t_2 are the two points in the fiber of T' over t.Let E be an elliptic curve over T and let V ⊆ E be the complement of the zero section in E.Let W be the quotient of V by the inversion in the group law of E, so W ≃𝔸^1_T.Let γ : W → W be multiplication by π and let Z be the normalization in V of the compositionV → WW . All we will use about Z is the following lemma, whose proof is straightforward:The map f : V → Z is an isomorphism over τ∈ T and constant over t ∈ T.Furthermore, Z is normal. Let s be the unique closed point of Z and let S = Spec𝒪_Z,s.Let η' be the generic point of S' and let s_1 and s_2 the two closed points lying above s.Let U_i be the complement in S' of s_i, and let U_12 be their intersection.Then U_1 ∪ U_2 = S', so we use the Mayer–Vietoris sequence to find a class in H^1(S', E'):H^0(U_1, E') × H^0(U_2, E') → H^0(U_12, E') → H^1(S', E')Now, U_12 = {η' } so H^0(U_12, E') = E'(η').Recall that we have a map E' → Z' that is an isomorphism over the generic points.Therefore we have a canonical element ξ of E'(η') corresponding to the inclusion of the generic point.An element of H^0(U_i, E') can be seen as a rational map from E' to itself over T' that restricts to a constant map over t_i.Any such map must in fact factor through a section of E' over T'.Therefore we haveH^0(U_i, E') = E'(T') . Consider the image of ξ in H^1(S', E').This cannot possibly be torsion, for if it were then it would have a multiple in the image of H^0(U_1, E') × H^0(U_2, E').That is impossible, because no multiple of the identity map on an elliptic curve is a difference of constant maps.Therefore we have found a non-torsion element in H^1(S', E').Since S' is étale over the normal scheme S, it is normal, and therefore by Theorem <ref>, the corresponding sheaf is a descent datum for an elliptic curve over S' that is not projective, hence not effective by Theorem <ref>.Raynaud goes a bit further and shows that the base for the descent may be chosen to be local.Indeed, letting q denote the projection from S' to S, we haveH^1(S', E') = H^1(S, q_∗ E') .Now, q_∗ E' = q_∗ q^∗ E' is the Weil restriction of scalars of E' via the finite, étale map q, hence is an abelian scheme of dimension 2 over S.It comes with a canonical inclusion E ⊆ q_∗ E' whose quotient is another elliptic curve F (in fact a quadratic twist of E).Then we have an exact sequenceH^1(S, E) → H^1(S, q_∗ E') → H^1(S, F)so the non-torsion class ξ∈ H^1(S', E') = H^1(S, q_∗ E) determines a non-torsion class either in H^1(S, F) or in H^1(S, E).Either way, we obtain a non-effective descent datum for genus 1 curves over S. § THE MANY MEANINGS OF ALGEBRAICITY We workover the presite 𝖲 of étale covers of schemes (over some fixed base scheme).Many authors have given different definitions of algebraicity.Deligne and Mumford required a schematic diagonal and an étale cover by a scheme <cit.>, but insisted their definition was not the right one except for quasiseparated stacks.Knutson required all algebraic spaces to be quasiseparated <cit.>.Artin gave his definition only for stacks that are locally of finite presentation and required a diagonal representable by algebraic spaces (in the sense of Knutson) and a smooth cover by a scheme <cit.>.Laumon and Moret-Bailly defined an `algebraic stack (understood quasiseparated)' by adding quasiseparation to Artin's conditions <cit.>.The Stacks Project uses Artin's conditions, but without requiring the algebraic spaces to be quasiseparated <cit.>.Recall that in Definition <ref> we defined algebraic stacks as the smallest class of stacks on the category of schemes that includes all schemes and includes all stacks that admit smooth covers by stacks in the class.Our definition is equivalent to the Stacks Project's, but usage in the literature varies widely.§.§ Stacks with flat covers by schemes and stacks over the fppf siteThe étale topology is not the only natural topology on schemes.From the perspective of descent, at least for quasicoherent sheaves, the flat topologies (fppf and fpqc) might be even more natural.The fpqc topology presents certain technical issues, owing to the absence of a sheafification functor, so we will not discuss it.There are two ways one might try to replace the étale topology with the fppf topology in our discussion of algebraic stacks.We might try to limit the class of stacks by insisting they be stacks in the fppf, as opposed to just the étale, topology.Or, we might enlarge the class of stacks under consideration by permitting them to have fppf, as opposed to necessarily smooth, covers by schemes.It turns out that either modification yields the same class of algebraic stacks.We will sketch the main ideas behind this result.We start by stating the following lemma whose proof we omitas it is well-known and not difficult. The class of flat morphisms of schemes is stable under composition and base change and is local to the source and target in the fppf topology.The lemma implies that one could develop a theory of flat-adapted algebraic stacks in the étale topology.The following theorem of Artin explains that to do so would yield nothing new:[<cit.>, <cit.>, <cit.>] Let 𝒳 be a stack in the fppf topology on the category of schemes. If there exists morphismU@>P>> 𝒳from analgebraic space U that is representable by algebraic spaces, faithfully flat, and of finite presentation, then there is such a morphism P that is smooth and surjective.In particular,𝒳 is anSP algebraic stack in the sense of Definition <ref>.We give a rough sketch of the proof, following Artin.By an `induction on stackiness', it is sufficient to assume that the diagonal of 𝒳 is representable by algebraic spaces.We consider the following moduli problem 𝒱.Choose a cover of U by a disjoint union U_0 of affine schemes (which is certainly possible, since U is a scheme).Since U_0 ×_𝒳 U_0 is an algebraic space, we can choose a disjoint union of affine schemes U_1 and a smooth cover U_1 → U_0 ×_𝒳 U_0.Let s and t denote the two projections from U_1 to U_0.Let U_2 be a smooth cover of the space of triples (α, β, γ) ∈ U_1 × U_1 × U_1 such that s(α) = s(γ), t(α) = s(β), and t(β) = t(γ) and β∘α = γ as isomorphisms between objects of 𝒳.For any scheme S, we define an S-point of 𝒲 to be * the choice of a finite union of components V_i ⊆ U_i for each i such that V_∙ forms a subgroupoid of U_∙,* a finite, locally free, surjective S-scheme Z with a distinguished basis 𝒪_Z ≃𝒪_S^d, and * a morphism of groupoids Z_∙→ V_∙:Z_2 [d] @<2pt>[r] [r] @<-2pt>[r] Z_1 [d] @<1.5pt>[r] @<-1.5pt>[r] Z_0 [d] [r] S V_2 @<2pt>[r] [r] @<-2pt>[r] V_1 @<1.5pt>[r] @<-1.5pt>[r] V_0 [r]𝒳 * where we have set Z_0 = Z and Z_i = Z_i-1×_S Z for i ≥ 1. We argue that 𝒱 is representable by a disjoint union of affine schemes, indexed by the choice of V_∙.The algebra structure on 𝒪_Z is determined by its structure constants and various identities among them, hence is parameterized by an affine scheme.The maps Z_i → V_i are determined by various elements of 𝒪_Z and relations among them (since the V_i are affine schemes).For each commutativity condition we have a pair of maps Z_i → V_i-1 that we wish to coincide.That is a closed condition (since affine schemes are separated).Any S-point of 𝒱 determines a descent datum for a morphism S→𝒳 in the fppf topology.Since 𝒳 is a stack in the fppf topology, this descends to a morphism S →𝒳 and we obtain a morphism of groupoids 𝒱→𝒳.We have just seen that 𝒱 is representable by a disjoint union of affine schemes, so it remains to verify this map is smooth. Now let 𝒲⊆𝒱 be the open substack where Z_0 → V_0 ×_𝒳 S is a local complete intersection morphism.To see that 𝒲 is indeed open in 𝒱, note that there are open subsets W_i ⊆ Z_i where the maps W_i → V_i are local complete intersection morphisms.Since the Z_i are proper over S, the image in S of the complement of W_i is closed, so the condition that the fiber of Z_i → V_i be a closed immersion and a local complete intersection morphisms is open on S.Now we verify that 𝒲 is locally of finite presentation, formally smooth, and surjective over 𝒳.To see that 𝒲→𝒳 is locally of finite presentation, one observes that once a morphism S →𝒳 is specified, a lift to 𝒲 involves only a finite amount of additional data.Next we verify the smoothness, for which we can use the infinitesimal criterion.Given a lifting problemS [r] [d]𝒲[d] S' @–>[ur] [r]𝒳in which S is affine and S' is an infinitesimal extension of S, we have, by definition a morphism of groupoids Z_∙→ V_∙, with the Z_i finite and locally free over S, that we would like to extend to Z'_∙→ V_∙ with Z'_i finite and locally free over S'.In this case, the map Z_0 → U_0 is a local complete intersection morphism, and Z_0 is affine, so there is no obstruction to extending it to a morphism Z'_0 → U_0 with Z'_0 finite and locally free over S'.This induces a pair of morphisms Z'_1 ⇉ V_0, hence a map Z'_1 → V_0 ×_𝒳 V_0.Now, V_1 → V_0 ×_𝒳 V_0 is smooth, so the map Z_1 → V_0 ×_𝒳 V_0 lifts to V_1.Now let R ⊆ V_1 × V_1 × V_1 be the set of triples (α, β, γ) such that the equation β∘α = γ makes sense and holds in 𝒳.Then we obtain Z'_2 → R and V_2 is smooth over R, so Z'_2 → R lifts to V_2.An infinitesimal deformation of a local complete intersection morphism is still a local complete intersection morphism, and an infinitesimal deformation of a closed embedding is still a closed embedding, so we have produced the required extension.Finally, we have to check 𝒲→𝒳 is surjective.Let k be the spectrum of an algebraically closed field and let S = Spec k.Let ξ be a k-point of 𝒳.The fiber of U_0 over S is a nonempty algebraic space T.Choose a smooth cover P of T by a scheme.This scheme is certainly flat over k, so it has a dense open subset where it is Cohen–Macaulay <cit.>.Pick a point p of P where P is Cohen–Macaulay, and let Z_0 be the vanishing locus of a regular sequence at p.Now let V_0 be a component of U_0 that contains the image of p under the composition Z_0 → P → T → U_0.Then Z_0 → V_0 ×_𝒳 S is a local complete intersection morphism by construction.Furthermore, we obtain a map Z_1 → V_0 ×_𝒳 V_0 ⊆ U_1.But recall that Z_1 has just one point, by construction, so we choose a component V_1 of U_1 whose image in U_0 ×_𝒳 U_0 contains the image of Z_0.Since Z_1 is artinian, there is a lift of Z_1 → V_0 ×_𝒳 V_0 to V_1.Then we repeat the same process to get Z_2 → V_2 and we conclude. Note that the condition that 𝒳have separated and quasicompact diagonal does not depend on the presentation. Suppose that G is a flat group scheme over S, acting on an S-scheme X.Then the stack [X/G] (see <ref>) isalgebraic.If G is quasiseparated over S(e.g., quasiprojective) then [X/G] is a quasiseparated algebraic stack. The cover X → [X/G] is a G-torsor, hence is an fppf cover.Therefore from the theorem, [X/G]is an algebraic stack.We may identify X ×_[X/G] X with X ×_S G, under which identification the diagonal map becomes the inclusion (id_X, e) : X → X ×_S G, with e denoting the identity section of G over S.This morphism is certainly representable and separated (it is an injective morphism of schemes).It is quasicompact if G is quasiseparated over S, since a section of a quasiseparated morphism is quasicompact <cit.>. Next we consider the question of stacks in the fppf topology.Note first that an algebraic stack in the fppf topology is clearly an algebraic stack in the étale topology, by restriction.Now we show the converse: Algebraic stacks are stacks in the fppf toplogy. Suppose that 𝒳 is an algebraic stack.Let 𝒳' be the fppf stackification.By induction, we can assume that we have already shown the diagonal of 𝒳 is a relative fppf sheaf, which means that 𝒳→𝒳' is injective.Now let U →𝒳 be a smooth cover.We argue that U →𝒳' is representable by algebraic spaces.Indeed, if S →𝒳' is any morphism, we can find an fppf cover T → S such that T →𝒳' lifts to 𝒳.Then T ×_𝒳' U = T ×_𝒳 U since 𝒳⊆𝒳', and T ×_𝒳 U is an algebraic space.But the mapT ×_𝒳' U → S ×_𝒳 Uis the base change of the fppf cover T → U, so S ×_𝒳 U has an fppf cover by an algebraic space.It is therefore an algebraic space, as required.Furthermore, S ×_𝒳' U → S is a smooth cover, since smoothness can be verified locally in the fppf topology and T ×_𝒳' U = T ×_𝒳 U → T is a smooth cover.Therefore U →𝒳' is a smooth cover, and as this factors through 𝒳, the map 𝒳→𝒳' is a smooth cover.Now both 𝒳 and 𝒳' are stacks in the étale topology, and 𝒳→𝒳' was already seen to be injective, so 𝒳→𝒳' is an isomorphism.§.§ Other definitions of algebraicity The following definition collects some of the most common meanings attributed to algebraicity of a stack on the étale site of schemes, in roughly chronological order.After giving the definition, we analyze the relationships among them, as well as to our Definition<ref>. [Algebraic stack] Let 𝒳 be a category fibered in groupoids over 𝖲=𝖲_et.[Much of the literature works over the étale site of affine schemes.Since every scheme has an étale (even Zariski) cover by affine schemes, the notions of stacks agree.] We define various notions of algebraic stack using the table below. Namely, we call 𝒳 a Deligne–Mumford algebraic stack (resp. Knutson algebraic space, resp. Artin algebraic stack, etc.)if the diagonal Δ:𝒳⟶𝒳×𝒳 satisfies the condition specified in the second column of the table on page F:AlgStackDef, and there is a scheme U and a surjection p:U⟶𝒳 that satisfies the conditions in the third column. Themorphism p : U →𝒳is called a presentation of 𝒳. A morphism between any such stacks is a morphism of the underlying CFGs. tempfoot Notethat our definition of representability is different, but equivalent to some of the notions used in the literature (see Lemma <ref>). The notions of quasicompact and separated morphisms of algebraic spaces defined in <ref> carry over directly to all of the notions of algebraic spaces discussed here. Also, in the sources, the various notions of algebraic spaces are defined for sheaves, rather than stacks. Here, for uniformity,we have simply added the injectivity hypothesis on the diagonal (see Lemma <ref>). The morphism p:U→𝒳 is either assumed to be schematic, or isrepresentable by the same class of algebraic spaces for which the diagonal is representable by virtue of Lemma <ref>.In <ref>, we definedsurjective, étale, andsmooth morphisms of algebraic spaces; these definitions carry over directly to all of the notions of algebraic spaces discussed here. Deligne and Mumford caution that their definition is correct only for quasiseparated stacks <cit.>. Quasicompactness of thediagonal immediately implies quasicompactness of the map in <cit.>.We leave the converse to the reader.See also <cit.>, cf. Stacks Project Algebraic Spaces. Artin gives his definition only under an additional assumption of local finite presentation. Laumon and Moret-Bailly ask only for a Laumon–Moret-Bailly algebraic space U and a smooth surjection onto 𝒳 <cit.>, but then U has an étale cover by a scheme, so the definition is equivalent. The Stacks Project requires its algebraic spaces and algebraic stacks to be sheaves in the fppf topology <cit.>.This yields an equivalent definition by <cit.> in the case of algebraic spaces and by <cit.> in the case of algebraic stacks (see <ref>). There is an unfortunate, confusing point in the nomenclature introduced in Definition <ref>.Deligne and Mumford defined an algebraic stack to be what we have, for the sake of historical verisimilitude, called a `Deligne–Mumford algebraic stack' above.Artin defined algebraic stacks more inclusively, and the modern terminology is more inclusive still.Meanwhile, the term Deligne–Mumford stack has come to refer to algebraic stacks with unramified diagonal.As the term `algebraic' has become ever more inclusive, so has `Deligne–Mumford', so that now the class of `Deligne–Mumford stacks', whilecontained in the class of algebraic stacks,unfortunately includes some stacks that are not`Deligne–Mumford algebraic stacks' in the sense we defined them here. The relationship among the definitions is clarified in Figure <ref>.A smooth (resp. étale) morphism of algebraic stacks π : 𝒴→𝒳 that is representable byalgebraic spaces is surjective if and only if it is covering in the étale topology.Indeed, π is surjective if and only if its base change to any scheme is surjective, if and only if its base change to any scheme is covering, if and only if it is covering.In short, for Definition <ref>, in the table on page F:AlgStackDefwe could replace theheading`... and there is a scheme U and a surjective morphism p:U→𝒳 ...' with the heading`... and there is a scheme U and a cover p:U→𝒳 ...'. Indeed, in the definition, we have either stipulated that p is at leastrepresentable by algebraic spaces, or we obtain this from the condition on the diagonal (see Lemma <ref>). §.§ Remarks on representabilityNote that in order to be able to speak about morphisms representable by the classes of algebraic stacks inDefinition <ref>, we need to know that these categories admit fiber products. All of the classes of stacks in Definition <ref> admit fiber products and these coincide with fiber products taken on the underlying CFGs. To deal with the entries in Definition <ref> that involve a schematic cover (DM stack, K algebraic space, LMB algebraic space, FDM stack, F algebraic stack, SP algebraic space), suppose that 𝒳→𝒵 and 𝒴→𝒵 are morphisms of stacks of the appropriate type.Choose smooth schematic covers (or étale schematic covers, as the case warrants) X →𝒳, Y →𝒴, and Z →𝒵 by schemes X, Y, and Z.Set X_Z = X ×_𝒵 Z and Y_Z = Y ×_𝒵 Z, as in the diagram below:@!C=2pc@R=2ex X_Z [d] [rrdd]X_Z×_𝒵Y_Z [rr] [ll] [d]Y_Z [lldd] [d]X [rd] 𝒳×_𝒵𝒴[dl] [dr]Y [ld]𝒳[rd] Z[d] 𝒴[ld]𝒵The projections X_Z →𝒳 and Y_Z →𝒴 are both smooth (or étale, according to the case) and covering (by base change and composition; Lemma <ref>).Now, X_Z ×_𝒵 Y_Z = (X_Z × Y_Z) ×_Z × Z (Z ×_𝒵 Z) so X_Z ×_𝒵 Y_Z is a fiber product of schemes, hence is a scheme. Moreover, the map X_Z ×_𝒵 Y_Z →𝒳×_𝒵𝒴 is a fibered product of smooth (or étale, as the case warrants) coverings hence is a smooth (or étale) covering.Finally,via composition and base change, Lemma <ref> implies the maps X_Z →𝒳 and Y_Z →𝒴 are schematic, so the same applies to X_Z ×_𝒵 Y_Z →𝒳×_𝒵𝒴 from Lemma <ref>(<ref>). Now this implies that fiber products of SP algebraic spaces are SP algebraic spaces so that the same argument can be repeated, with `representable by algebraic spaces' substituted for `schematic' and `algebraic space' substituted for `scheme'.This proves the lemma for SP algebraic stacks and SP DM stacks. All of the remaining classes in Definition <ref> can be characterized as algebraic stacks with additional conditions on the diagonal.The conditions imposed on the diagonal are all properties that are preserved by fibered products of morphisms (note that properties preserved by composition and fibered product are preserved by fibered products of morphisms; see the proof of Lemma <ref>(<ref>)), and the diagonal morphism of a fibered product is the fibered product of the diagonal morphisms. Using Definition <ref>, we now may speak of morphisms representable by any of the classes of stacks in Definition <ref>. Although we will not need to do so, it is sometimes convenient to test representability by various classes of stacks using smaller classes.The following lemma shows that the class of affine schemes suffices to address most representability questions. A morphism of stacks f:𝒳→𝒴 is representable bya class of algebraic spaces in Definition <ref> (resp. schematic) if and only if for every affine scheme S=SpecA, the fibered product 𝒳×_𝒴S is representably by an algebraic space in that class (resp. a scheme).All of the properties involved in the various definitions of algebraic space satisfy étale descent, so they can be verified étale locally.Since algebraic spaces have étale covers by affine schemes (Theorem <ref>) we can check if a morphism is representable by algebraic spaces in any sense by testing with affine schemes.The same argument with Zariski descent and schemes takes care of the respected case. If 𝒳 is a CFG then conditions on the diagonal correspond by base change to conditions on the fiber product S ×_𝒳×𝒳 S for all schemes S and all pairs of morphisms x,y : S →𝒳.Recall from Example <ref> that this fiber product may be identified with ℐ𝑠𝑜𝑚_𝒳(x,y) and from Lemma <ref> that a CFG has injective diagonal if and only if it is equivalent to a sheaf.Let ℳ be a CFG over over 𝖲.The following conditions are equivalent:* The diagonal morphism ℳΔ⟶ℳ×ℳ is representable by SP algebraic spaces (resp. representable by K algebraic spaces, resp. schematic); * For all S in 𝖲, and all x,y in ℳ(S), the presheaf ℐ𝑠𝑜𝑚(x,y) on 𝖲 is representable by an SP algebraic S-space (resp. K algebraic S-space, resp. S-scheme); * For all S in 𝖲 and all x in ℳ(S), the morphism x : S →ℳ (guaranteed by the Yoneda lemma) is representable by SP algebraic spaces (resp. representable by K algebraic spaces, resp. schematic); * For every SP algebraic space (resp. K algebraic space, resp. scheme) S, we have that every morphism S→ℳ is SP-representable (resp. K representable, resp. schematic). §.§ Overview of the relationships among the definitions of algebraicityOur definition of an algebraic stack can also be characterized in a similar manner to Definition <ref>. Each type of stack 𝒳 in the first column below is characterized by the condition on the diagonal in the second column and the existence of a surjection p : U →𝒳 satisfying the condition in the third column. A stack 𝒳 is... if the diagonalΔ : 𝒳→𝒳×𝒳 is…and there is a scheme U and a surjective morphism p : U ⟶𝒳that is … an algebraic space(Definition <ref>)injective schematic andétale. a Deligne–Mumford stack (Definition <ref>) unramified representable by algebraic spaces and smooth. a Deligne–Mumford stack (Definition <ref>)representable by algebraic spaces and étale. an algebraic stack(Definition <ref>)representable by algebraic spaces and smooth. Suppose𝒳 is an algebraic stack. Considering the iterative nature of Definition <ref>, it is clear that 𝒳 has a smooth cover P:U→𝒳 by a scheme U. We need to show that P is representable by algebraic spaces. This morphism must have injective relative diagonal (since U ×_𝒳 U is a sheaf of sets and U → U ×_𝒳 U → U × U is injective),so it is representable by algebraic spaces, as required.This argument applies also to show that Deligne–Mumford stacks have étale covers by algebraic spaces. Suppose now that 𝒳 is an algebraic space.Then the diagonal of 𝒳 is injective and representable by algebraic spaces.In particular, it is locally quasifinite and separated, so by separated, locally quasifinite descent <cit.>, it is schematic. To complete the proof, we need to show that algebraic spaces and algebraic stacks with unramified diagonals have étale covers by schemes.We will show these statements simultaneously (using an `induction on stackiness'), mostly following <cit.>. We argue that every finite-type point of 𝒳 has an étale neighborhood that is a scheme.This will suffice, since if x is a geometric point of 𝒳, we can find a smooth map U →𝒳 where U is an affine scheme and x lifts to U.Then this lift has a specialization to a closed point of U, which induces a point y of 𝒳 of finite type.Any étale neighborhood of y will also contain x. Let x = Spec k be a point of 𝒳 of finite type.We argue first that there is a factorization x → y →𝒳 where y is unramified over 𝒳.Let R = x ×_𝒳 x.Then R is flat over x (since x is the spectrum of a field) and of finite type (since the composition u ×_𝒳 u → u ×_𝒳𝒳 is the base change of a morphism u →𝒳 of finite type). Since R is of finite type over x via the first projection, its geometric fiber has finitely many components.Replacing k with a finite extension, we can therefore assume that the connected components of R are geometrically connected.Let R_0 ⊂ R be the connected component of the diagonal section.Note that R_0 → x × x is unramified and its fiber over the diagonal x → x × x is injective on geometric points.But every geometric fiber is either empty or is a torsor under the fiber over the diagonal, so that every fiber is injective on geometric points.It follows that R_0 → x × x is a injection, so that it defines an equivalence relation on x. Let y = x / R_0 be the quotient of x by this equivalence relation.Since R_0 ⊂ R is open, R_0 is flat over x and therefore this is an algebraic space (Theorem <ref>) equipped with a map y →𝒵.In fact, y must be the spectrum of a field:choose a smooth cover Spec A → R_0 by a scheme, and let ℓ be the equalizer of the two maps k → A.Then ℓ is a field (since an element of k is equalized by two homomorphisms if and only if its inverse is).Moreover, we obtain a map y →Specℓ that is covering and injective in the fppf topology.Therefore it is an isomorphism (see <cit.>). Now we argue that y →𝒳 is unramified.Indeed, the diagonal map y → y ×_𝒳 y pulls back via the fppf cover R_0 = x ×_𝒳 x → y ×_𝒳 y to x ×_y x = R_0, which is open in R.Therefore y → y ×_𝒳 y is an open embedding, so y is unramified over 𝒳. Now choose a smooth morphism U →𝒳 containing y in its image, with U = Spec A affine.Let V = U ×_𝒳 y.Then V is a smooth algebraic space over y and V → U is unramified.If 𝒳 is an algebraic space then we have seen that U →𝒳 is schematic, so that V is a scheme; in that case we write W = V.In general, then we know the diagonal of 𝒳 is at least representable by algebraic spaces, so V is an algebraic space.By `induction on stackiness' we may assume that there is a scheme W and an étale map W → V whose image in 𝒳 contains the image of y. Up to an étale extension of ℓ, we can assume that y is the image of an ℓ-point of W, which we denote w.We take u to be its image in U.The map W → U is unramified, so the maximal ideal 𝔪 of w in 𝒪_W is generated by the image of the maximal ideal 𝔫 of u in U.We can therefore choose functions f_1, …, f_d ∈𝒪_U,u whose images in 𝒪_W,w form a basis for 𝔪/𝔪^2.Replacing W by an open neighborhood of w, the vanishing locus of f_1, …, f_d in W will be { w }.In particular, the vanishing locus is unramified over y.But the locus where U →𝒳 is unramified is open, so that there is an open neighborhood U' of u ∈ U where U' →𝒳 is unramified.Since it is also smooth, it is étale, as required. The implications among all of the definitions are described in Figure <ref>.They can all essentially be explained by putting various conditions on the diagonal of an algebraic stack.In this sense, from an expository perspective,the definition of an algebraic stack is the basic definition, and the rest can easily be obtained from this. In practice thisis somewhat misleading, however, since one must first define an algebraic space to define an algebraic stack. §.§ Relationships among definitions of algebraic spaces The following implications hold for algebraic spaces: @C=.9cm @R=.5cmSch.@=>[r]@<=>@/_.8 pc/[dd]_<>(0.5) Δ q.c. SP Alg. Sp.@<=>@/_.8 pc/[dd]_Δ q.c. Alg. Sp.@<=>[l] q.s. Sch.@=>[r] @=>[uu] LMB Alg. Sp.@=>[uu] K Alg. Sp.@<=>[l]An arrow with a label indicates that the implication holds under the additional assumption indicated on the diagonal.A two headed arrow implies that the definitions are equivalent under the given assumption on the diagonal.The only arrows that require justification are the equivalence between LMB algebraic spaces and K algebraic spaces, and the equivalence betweenalgebraic spaces and SP algebraic spaces. LMB algebraic spaces are the same as K algebraic spaces andSP algebraic spaces are the same asalgebraic spaces.It is clear that LMB algebraic spaces are K algebraic spaces and that SP algebraic spaces are algebraic spaces.For the converse, we only need to show that the diagonal is schematic.This is <cit.>.In fact, the diagonal of an algebraic space is injective, hence separated and locally quasifinite.Therefore the diagonal is schematic, by separated, locally quasifinite descent <cit.>. §.§ Relationships among the definitions of algebraic stacksAlg.@<=>[d] LMB Alg.@=>[r]@<=>@/^15pt/[r]^Δ q.c.+sep. SP Alg.@<=[r] @<=>@/^15pt/[rr]^Δsch. F Alg.@<=[r]SP Alg. Δ sch.Most of the implications are immediate, so we only make a few comments.The diagonal morphism of an algebraic stackis representable by algebraic spaces. One can check this using Theorem <ref> and a slight modification of the proof of<cit.>. Algebraic stacks are the same asSP algebraic stacks. From Theorem <ref>, we only need to show that the diagonal morphism is representable by SP algebraic spaces.Since we have shown already that algebraic spaces are SP algebraic spaces,we conclude using the previous lemma.An SP algebraic stack with quasicompact and separated diagonal is an LMB algebraic stack. Suppose that 𝒳 isSP algebraic, with quasicompact and separated diagonal.By Lemma <ref>, thediagonal morphism for 𝒳 is representable by algebraic spaces.An algebraic space is an LMB algebraic space if and only if its diagonal is quasicompact (see Section <ref>).Since the diagonal of 𝒳 is separated, the double diagonal is a closed embedding and, a fortiori, quasicompact.Hence the diagonal of 𝒳 is representable by LMB algebraic spaces, as required.We expectthere are F algebraic stacks that do not have schematic diagonal, but we do not know of any example.We also expect that there are SP algebraic stacks that do not admit a smooth schematic morphism from a scheme (and therefore are not F algebraic stacks), but we do not know of any example.See Example <ref> foran SP algebraic stack that does not have schematic diagonal.§.§ Relationships among the definitions of Deligne–Mumford stacksFor Deligne–Mumford stacks, we have implications: DM@<=>[d] LMB DM@=>[r]@<=>@/^15pt/[r]^Δ q.c.+sep.@<=>@/_15pt/[rrr]_Δ q.c.+sep. SP DM@<=[r] @<=>@/^15pt/[rr]^Δsch. F DM@<=[r]DM Alg.Most of the implications are immediate, but we focus onthe main points:DM stacks are the same as SP DM stacks. From Theorem <ref> we only need to show that a DM stack has diagonal representable by algebraic spaces (as these are the same as SP algebraic spaces). However, by Theorem <ref> it is immediate that DM stacks are algebraic stacks, and we have seen in Lemma <ref> that the diagonal of an algebraic stack is representable by algebraic spaces.All LMB DM stacks have schematic diagonal. Any locally quasifinite, separated morphism that is representable by algebraic spaces is schematic <cit.>.The rest of the implications are obvious from Theorem <ref> and the two lemmas above. We will construct an SP Deligne–Mumford stack withoutschematic diagonal (i.e., an SP DM stack that is not a DM algebraic stack).Let G be 𝔸^1 with a doubled origin.This can be regarded as a group scheme over 𝔸^1 by distinguishing one of the two origins as the identity element.If X →𝔸^1 is a morphism of schemes, with X_0 the fiber over the origin, then a G-torsor on X is a ℤ/2ℤ-torsor on X_0.We will show that the map p : 𝔸^1 →BG is not schematic.Indeed, if Z →BG is any morphism then the base change of p is the total space of the corresponding torsor.Therefore we have to find a scheme Z over 𝔸^1 and a G-torsor over Z that is not representable by a scheme.To find such a torsor, choose a scheme W and an étale double cover W' → W that does not have a section Zariski-locally.Let Z = 𝔸^1_W (so that Z_0 = W) and let P be the G-torsor over Z corresponding to W' → Z_0. In regards to the example above, we expect that there are also F DM stacks that do not have schematic diagonal (and are therefore not DM algebraic stacks), but we do not know of any example.We also expect that there are SP DM stacks that do not admit a smooth schematic morphism from a scheme (and therefore are not F DM stacks), but we do not know of any example. Example <ref> shows there are SP algebraic stacks that do not have schematic diagonal, and are therefore not DM algebraic stacks.§.§ Stacks with unramified diagonal The implications in Figure <ref> regarding unramified diagonal all follow immediately from the following lemma: An algebraic stack with unramified diagonal is a DM stack. Now that we have the identification between algebraic stacks and SP algebraic stacks, and DM stacks and SP DM stacks, this is<cit.>. §.§ The adapted perspective We show how all of the stacks that have appeared in this paper can be described asstacks adapted to a given presite.§.§.§ Stacks adapted to the étale presite of schemesWe have already seen the following.A stack adapted to the étale presite with injective diagonal is the same thing as an algebraic space. A stack adapted to the étale presite with injective and quasicompact diagonal is the same thing as an LMBalgebraic space. A stack adapted to the étale presite is the same thing as an F DM stack. A stack adapted to the étale presite with quasicompact and separated diagonal is the same thing as an LMB DM stack (Lemma <ref>).§.§.§ Stacks adapted to the étale presite of algebraic spacesStacks adapted to the étale presite of algebraic spacesinduceDeligne–Mumford stacks on the étale presite of schemes.Indeed, given astack in the étale topology on algebraic spaces one obtainsa stack on the étale presite of schemes by restriction.Conversely, a stack on the étale presite of schemes extends uniquely to the étale presite of algebraic spaces since every algebraic space has an étale cover by schemes.Using again that every algebraic space has an étale cover by schemes,the definition of stack adapted to the étale presite of algebraic spaces agrees with the characterization of Deligne–Mumford stacks in Theorem <ref>.§.§.§ Stacks adapted to the smooth presite of algebraic spacesStacks adapted to the smooth presite of algebraic spaces induce algebraic stacks by restricting from the category of algebraic spaces to schemes.§.§.§ Stacks adapted to the fppf presite of algebraic spacesA stack adapted to the fppf presite ofalgebraic spaces induces an algebraicstack on the étale presite of schemes, by restricting from the category of algebraic spaces to the category of schemes (see Theorem <ref>). §.§.§ An example of a stack that is not adapted to a presiteConsider the logarithmic abelian varieties of Kajiwara, Kato, and Nakayama <cit.>, which are sheaves in the étale topology and possess logarithmically étale covers by logarithmic schemes but have no such étale covers (logarithmically étale maps are a more general class of morphisms including all étale maps but also some blowups and other non-flat morphisms). However, the logarithmic étale topology is not subcanonical and there is no subcanonical topology in which logarithmic abelian varieties are covered by logarithmic schemes. §.§ Conditions on the relative diagonal of a stack and bootstrappingAs we have seen, most of the variations on the definition of analgebraic stacks outlined in Definition <ref>can be obtained from the definition of an algebraic stack by imposing conditions on the diagonal.Here we aim to extend the bootstrapping result of Proposition <ref> to these other cases. In other words, our goal here is to show that a stack that is relatively algebraic over an algebraic stack (according to any of the definitions in Definition <ref>) is itself algebraic.For the next lemma, recall that for a morphism f:𝒳→𝒴 of algebraic stacks,the diagonal Δ_f is representable byalgebraicspaces (<cit.>).Let 𝐏 be a property of morphisms ofalgebraic spaces that is stable under composition and base change. Let 𝒳,𝒴,𝒵 bealgebraic stacks over 𝖲.* If 𝒳f→𝒴has property 𝐏 for Δ_f and 𝒴'→𝒴 is any morphism, then the morphism 𝒳×_𝒴𝒴'f'→𝒴' obtained from the fibered product has property 𝐏 for Δ_f'. * If 𝒳f→𝒴 and 𝒴g→𝒵are morphisms such that Δ_f and Δ_g have property 𝐏, then for the composition 𝒳g∘ f→𝒴, the diagonal morphism Δ_g∘ f has property 𝐏.* If 𝒳f→𝒳' and 𝒴g→𝒴' are morphisms over a stack 𝒵 such that Δ_f and Δ_g have property 𝐏, then for the fibered product morphism𝒳×_𝒵𝒴f×_id_𝒵 g⟶𝒳'×_𝒵𝒴' the diagonalmorphism Δ_f×_id_𝒵 g has property 𝐏.* A morphism f : 𝒳→𝒴 has property 𝐏 for Δ_f if and only if for every scheme S and every morphism S →𝒴, the base change f':𝒳×_𝒴 S → S has property 𝐏 for Δ_f'.* Suppose that property 𝐏 satisfies the following condition:if If Xf→ Y and Yg→ Z are morphisms of algebraic spaces such that g and f∘ g satisfy property 𝐏, then f satisfies property 𝐏.ThenIf 𝒳f→𝒴 and 𝒴g→𝒵are morphisms such that Δ_g and Δ_g∘ f have property 𝐏, then Δ_f has property 𝐏. This is essentiallycontained in <cit.>. (1)follows from the 2-cartesian diagrams:𝒳' @->[r]^f'@->[d] 𝒴' @->[d] 𝒳@->[r]^f 𝒴.5 in𝒳' @->[r]^Δ_f' @->[d] 𝒳'×_𝒴'𝒳' @->[d] 𝒳@->[r]^Δ_f𝒳×_𝒴𝒳The diagram on the left induces the diagram on the right, and it then follows from base change that Δ_f' has property 𝐏. (2) follows from (1) using the 2-commutative diagram below forthe morphisms f:𝒳→𝒴 and g:𝒴→𝒵. The square is a 2-fibered product.𝒳@->[r]^<>(0.7)Δ_f@->[rd]@->@/^2pc/[rr]^Δ_g∘ f 𝒳×_𝒴𝒳@->[r]@->[d] 𝒳×_𝒵𝒳@->[d]𝒴@->[r]^<>(0.5)Δ_g 𝒴×_𝒵𝒴. (3) is essentially <cit.>, which observes that the conclusion follows from (1) and (2), together withthe fact that given morphisms f:𝒳→𝒳' and g:𝒴→𝒴' over a stack 𝒵, the product 𝒳×_𝒵𝒴f×_id_𝒵g⟶𝒳'×_𝒵𝒴' is given by the composition of morphisms obtained from fibered product diagrams:𝒳×_𝒵𝒴 @>f×_id_𝒵id_𝒴>> 𝒳'×_𝒵𝒴 @>id_𝒳'×_id_𝒵 g>> 𝒳'×_𝒵𝒴'. (4)If Δ_f has property 𝐏, this follows from (1).Conversely, assume that for every scheme S and every morphism S →𝒴, the base change 𝒳×_𝒴 S → S has property 𝐏 for its diagonal. By definition, for Δ_f : 𝒳→𝒳×_𝒴𝒳 to haveproperty 𝐏 means that the base change S ×_𝒳×_𝒴𝒳𝒳→ S has property 𝐏.But S ×_𝒳×_𝒴𝒳𝒳 = S ×_𝒳_S ×_S 𝒳_S𝒳_S where 𝒳_S = 𝒳×_𝒴 S and by assumption, the diagonal of 𝒳_S → S has property 𝐏. (5) This follows from diagram (<ref>), and the previous parts of the lemma. For any of the classes 𝖢 of objects introduced in Definition <ref>, if 𝒴 is of class 𝖢 and f : 𝒳→𝒴 is representable by objects of 𝖢 then 𝒳 is of class 𝖢. Lemma <ref> covers the case of algebraic stacks. ForF algebraic stacks one can easily adapt the proof of Lemma <ref>.The remainingclasses 𝖢 of stacks introduced in Definition <ref> can can be obtained by imposing various conditionson the diagonal of analgebraic stack, all of which are stable under composition and base change. Fix a class of such stacks, and call the necessary conditions on the diagonal condition 𝐏. In particular,𝒴 is of class 𝖢 means that the diagonal Δ_π of the structure map π:𝒴→𝖲 has property 𝐏. If f : 𝒳→𝒴 is representable by objects of 𝖢, then from Lemma <ref>(4), we have that Δ_f has property 𝐏.Then by Lemma <ref>(2), we have that Δ_π∘ f has property 𝐏.In other words, 𝒳 is of class 𝖢.The arguments above show the following, as well (<cit.>).Let f:𝒳→𝒴 be a morphism of algebraic stacks. The morphismf is representable by LMB algebraic stacks(resp. LMB DM stacks, resp. LMB algebraic spaces)if and only ifΔ_f is quasicompact and separated (resp. quasicompact, separated, and unramified, resp. quasicompact, separated, and injective). § GROUPOIDS AND STACKS Stacks are often studied via groupoid objects.In this section we discuss torsors, groupoid objects, and stacks arising as quotients of groupoid objects.In the end weshow that groupoid objects adapted to a presite, i.e., those where the source and target maps are coverings in the presite, are essentially the same thing as stacks adapted to the presite.§.§ Torsors and group quotients Let X be in 𝖲/S, and let G be a sheaf of groups over S acting on the righton X:X×_SG @>σ >> X We define a CFG over 𝖲/S, [X/G]in the following way.The objects over an S-scheme f:S'→ S arediagrams@C=1.5em@R=1.5em P'@->[d]@->[r] X_S'S'where P' is a G_S'-torsor (principal bundle) over S', and P'→ X_S' is a G_S'-equivariant morphism.Morphisms are defined by pullback.There is a morphism [X/G]→ S given by forgetting everything except the S-scheme f:S'→ S, and there is an S-morphism q:X→ [X/G] given by the trivial G_X-bundle X×_SG@->[d]_pr_1@->[r]^id×σ X×_SX X.This induces a 2-cartesian diagramX×_SG@->[d]_pr_1@->[r]^σX @->[d]^q X@->[r]^q [X/G]that is a co-equalizer forX×_SG @->@<-2pt>[r]_<>(0.5)pr_1@->@<2pt>[r]^<>(0.5)σ X (i.e., initial in the category of stacks for the diagram (<ref>)); in other words,[X/G] is a quotient in the category of stacksfor the action of G on X, in the sense that any G-equivariant map out of X factors through it.Note that if there exists a schemeX/G that is a quotient in the category of schemesfor the action of G on X (i.e., a co-equalizer in the category of schemes), then there is a morphism[X/G]→ X/G. Of particular importance is the trivial action of G on X = S.The quotient [S/G] is denoted BG.As a CFG, BG consists of the pairs (S', P') where S' ∈𝖲/S and P' is a G-torsor over S. There isa similar construction for left group actions; in the notation above, we would have the stack [G\ X].§.§ Groupoid objects and groupoid quotients §.§.§ Groupoid objects Let 𝖦𝗉𝖽 denote the category (not 2-category) of small groupoids.There are functors 𝖮𝖻𝗃 and 𝖬𝗈𝗋 from (𝖦𝗉𝖽) to (𝖲𝖾𝗍) sending a groupoid X, respectively, to its set X_0 of objects and its set X_1 of morphisms.There are two canonical morphisms s, t : X_1 → X_0 sending a morphism to its source and target.More data are required to specify a groupoid, but these are often left tacit and the groupoid is usually denoted by a pair of morphisms of sets@C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0. A groupoid object in a category 𝖲 is defined by taking X_1, X_0, and all of the morphisms involved in specifying the groupoid to lie in 𝖲 (as opposed to 𝖲𝖾𝗍).If this is the case, thenwe obtainfunctors Hom_𝖲(-,X_i) from 𝖲^op to (𝖲𝖾𝗍). This can all be said moreconcisely as follows:[Groupoid object]A groupoid object of a category 𝖲 is a functor 𝒳 : 𝖲^ op→ (𝖦𝗉𝖽), along with objects X_0 and X_1 in 𝖲, respectively representing the composition offunctors 𝖮𝖻𝗃∘𝒳 and 𝖬𝗈𝗋∘𝒳 from 𝖲^ op to (𝖲𝖾𝗍).The groupoid object 𝒳 is often denoted @C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0. A morphism of groupoids objects is a morphism (natural transformation) of functors. There are morphisms s,t:X_1→ X_0 associated to athe groupoid object 𝒳in 𝖲 obtained from the source and target maps associated to a groupoid.Moreover, associated to a morphism 𝒳→𝒴 of groupoid objects in 𝖲 are morphisms X_0→ Y_0 and X_1→ Y_1. [Constant groupoid object] Let X be any object of 𝖲.Define X_0 = X_1 = X.Then @C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)Id@->@<2pt>[r]^<>(0.5)Id X_0is a groupoid object of 𝖲.We call such a groupoid constant and denote it, abusively, by the same letter X. [Action groupoid] Let G be a group object of 𝖲 acting on the left on an object X_0.Define 𝖮𝖻𝗃𝒳(U) = Hom(U, X_0) and let 𝖬𝗈𝗋𝒳(U) be the set of all triples (g, x, y) where x, y ∈𝖮𝖻𝗃 X(U) and g ∈Hom(U, G) is a U-point of G such that gx = y.The composition of (g, x, y) and (g', y, z) is the triple (g'g, x, z). This is known as the action groupoid. It is also typically denoted byG× X_0 @->@<-2pt>[r]_<>(0.5)σ@->@<2pt>[r]^<>(0.5)pr_2 X_0. One can immediately associate a category fibered in groupoids to any groupoid object.Indeed, suppose that 𝒳 is a groupoid object of 𝖲.Construct a category 𝒳 whose objects are pairs (U, ξ) where U is an object of 𝖲 and ξ∈𝖮𝖻𝗃𝒳(U).A morphism (U, ξ) → (V, η) consists of a morphism f : U → V of 𝖲 and a morphism ϕ : ξ→ f^∗η of 𝒳(U).The composition of (f, ϕ) : (U, ξ) → (V, η) and (g, ψ) : (V, η) → (W, ζ) is (gf, ϕ∘ f^∗ψ).It is easy to verify that this category is fibered in groupoids over 𝖲 with the projection sending (U, ξ) to U. When 𝖲 has a topology, this groupoid is rarely a stack, although it is a prestack if the topology is subcanonical. If 𝖲 is subcanonical and @C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0 is a groupoid object, then it is common to denote the associated prestack by [@C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0 ]^pre. There is an abstract process of stackification of prestacks, analogous to sheafification of presheaves, by which a CFG 𝒳 is replaced by the initial stack receiving a map from 𝒳.In the situation of groupoid objects, this stack is typically denoted by [@C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0 ] (see e.g., <cit.> for more details on this approach).Moreover, there is a morphism X_0→ [@C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0 ] that makes the following diagram@C=.2em X_1 [rrrr]^t[d]_s X_0 [d(0.7)] X_0 [rrr][X_1 @->@<-2pt>[rr]_<>(0.5)s@->@<2pt>[rr]^<>(0.5)tX_0]2-cartesian (<cit.>) and essentially a 2-coequalizer for@C=1.5em X_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t X_0 (see <cit.> for more details on the precise meaning of this).In other words, the stack provides a “quotient” for the groupoid. Rather than undertake an explanation of this construction and the attendant 2-universal property, we will give a direct construction of the stack associated to a groupoid object.In the case of left group action, the stack [ G× X_0 @->@<-2pt>[r]_<>(0.5)σ@->@<2pt>[r]^<>(0.5)pr_2 X_0] is equivalent to the stack [G\ X_0].§.§.§ Augmented groupoids [Augmented groupoid object] A groupoid object 𝒳 of a category 𝖲 is said to be augmented toward an object X of 𝖲 when it is equipped with a morphism 𝒳→ X.A groupoid 𝒳 augmented toward X is often denoted X_1 ⇉ X_0 → X.If 𝒳→ X and 𝒴→ Y are augmented groupoids, a morphism of augmented groupoids from 𝒳→ X to 𝒴→ Y is a commutative diagram of groupoid objects as in (<ref>):𝒳[r] [d]𝒴[d] X [r]^f Y.To augment a groupoid 𝒳 torwards X, it is equivalent to give a morphism f : X_0 → X such that fs = ft.[Cartesian morphism of augmented groupoid objects] If 𝖲 admits fibered products,a morphism of augmented groupoid objects as in (<ref>) is called cartesian if X_0 → X ×_Y Y_0 and X_1 → X ×_Y Y_1 are isomorphisms.Suppose that q : X_0 → X is a morphism in a category admitting fiber products.We define a groupoid as follows:𝖮𝖻𝗃𝒳(U) = Hom(U, X_0) and 𝖬𝗈𝗋𝒳(U) is the set of pairs (f, g) ∈Hom(U, X_0) such that qf = qg.In other words, 𝖬𝗈𝗋𝒳 is represented by X_1 = X_0 ×_X X_0.The composition of the pair (f,g) and (g,h) is, by definition, the pair (f,h), and the identity of f ∈𝖮𝖻𝗃𝒳(U) is the pair (f,f). The groupoid X_1 ⇉ X_0 is augmented toward X by construction. Let 𝖦𝗉𝖽^+_𝖲 denote the category of augmented groupoid objects of 𝖲, with cartesian morphisms.The projection sending an augmented groupoid object (𝒳→ X) to X makes 𝖦𝗉𝖽^+_𝖲 into a CFG over 𝖲.§.§.§ Stacks associated to groupoid objects Now suppose that 𝖲 is equipped with a pretopology.[Presentation of an augmented groupoid obejct]Let 𝖲 be a presite admitting fibered products.Let X_1 ⇉ X_0 → X be a groupoid of 𝖲 augmented toward X.We call it a presentation of X if X_0 → X is covering (Definition <ref>) and the canonical map X_1 → X_0 ×_X X_0 is an isomorphism. [Category associated to a groupoid object]Let 𝖲 be a presite admitting fibered products.Let 𝒳 = (X_1 ⇉ X_0) be a groupoid object of 𝖲.We construct a CFG, 𝒳, called the CFG associated to a groupoid object.The objects of 𝒳 are triples (U, 𝒰, ξ) where U is an object of 𝖲, where 𝒰→ U is a presentation of U, and where ξ : 𝒰→𝒳 is a morphism of groupoid objects.A morphism in 𝒳 from (U, 𝒰, ξ) to (V, 𝒱, η) is a cartesian morphism (f, φ) of augmented groupoids from (𝒰→ U) to (𝒱→ V) such that η∘φ = ξ as morphisms of groupoids 𝒰→𝒳.The morphism 𝒳→𝖲 is given by sending (U, 𝒰, ξ) to U. The following lemma asserts that the category 𝒳 is a CFG; the proof is straightforward, so it is omitted.If 𝖲 admits fiber products, then the category 𝒳 over 𝖲 constructed above is a CFG. For the following lemma, let 𝖢𝗈𝗏 denote the category whose objects are covering morphisms X → U in 𝖲 and whose morphisms are cartesian squares.If 𝖲 has fiber products then the projection 𝖢𝗈𝗏→𝖲 sending (X → U) to U makes 𝖢𝗈𝗏 into a CFG over 𝖲 (Example <ref>). Assume that 𝖲 is a subcanonical presite with fiber products.Let𝒳 be the CFG constructed as above in Definition <ref>.If 𝖢𝗈𝗏 is a stack over 𝖲 then so is 𝒳. We give just a sketch. The idea is to use Lemma <ref>. We start with an object S of 𝖲 and the canonical morphism𝖲/S→𝖲. Given a cover ℛ = { U_i → S } of S and a morphism ℛ→𝒳 we need to show how to obtain the lift 𝖲/S→𝒳.So, givengroupoid objects 𝒰_i over each U_i, along with compatible data over the double and triple fiber products U_ij and U_ijk, these descend to a groupoid object 𝒮 over S by descending the objects 𝖮𝖻𝗃𝒰_i and 𝖬𝗈𝗋𝒰_i of 𝖢𝗈𝗏 and the morphisms between them (using that 𝖢𝗈𝗏 forms a stack and that morphisms between representable objects form sheaves).Then the maps 𝒰_i →𝒳 descend to 𝒮→𝒳 by descending the maps on objects and morphisms, again using the subcanonicity of the site.This gives the desired morphism 𝖲/S→𝒳.If 𝒳 is the groupoid object @C=10ptU_1 @<1.5pt>[r]^s @<-1.5pt>[r]_t U_0 and 𝒳 is its associated stack,there is a canonical map U_0 →𝒳, which is covering. We construct the triple (U_0,𝒰_0,𝒰_0→𝒳) giving this morphism as follows. The presentation 𝒰_0→ U_0 is given by @C=10pt U_1×_U_0U_1 @<1.5pt>[rr]^<>(0.5)pr_1@<-1.5pt>[rr]_<>(0.5)comp U_1 [rr]^s U_0;here the maps for the fibered product are the source and target maps repsectively, and the bottom arrow comp is the composition morphism taking a pair (α,β) in U_1×_U_0U_1(over some S) to the composition β∘α in U_1 (over S).This is a presentation of U_0 sinces is covering (the projections s,t:U_1 → U_0 are always covering in a groupoid object because of the `identity map' section U_0→ U_1).To describe 𝒰_0→𝒳, it is convenient to describe 𝒰_0 as follows. For any scheme S,𝒰_0(S) is the groupoid in which the objects are the morphisms of 𝒳(S).The morphisms from (ξ→ζ) to (η→ω) in 𝒰_0(S) are the commutative squaresξ[r] [d]ζ@=[d]η[r]ω .That is, there are no morphisms unless ζ = ω. Clearly𝖮𝖻𝗃𝒰_0(S) = Hom(S, U_1) and 𝖬𝗈𝗋𝒰_0(S) = Hom(S, U_1×_U_0U_1).There is a canonical map 𝒰_0 →𝒳 sending an object (ξ→ζ) to ξ. Finally, we show that U_0→𝒳 is covering.For a scheme S, a morphism S→𝒳 is a triple (S,𝒮,ξ:𝒮→𝒳), where 𝒮=@C=10ptS_1 @<1.5pt>[r]^s'@<-1.5pt>[r]_t' S_0→ S is a presentation.The morphism 𝒮→𝒳 induces a morphism S_0→ U_0.The fact that S_0→ S is covering means that there is a cover {T_α→ S} of S that lifts to S_0, and therefore composinggivesmorphisms T_α→ U_0.This gives the cover {T_α→ S} of S whose compositions T_α→ S→𝒳 lift to U_0. In fact, there is a cocartesian diagram in the 2-category of stacks:U_1 [r]^s[d]_tU_0 [d]U_0 [r] 𝒳We will not use this property, so we do not give a proof (see e.g., <cit.>).§.§ Adapted groupoid objects and adapted stacks Here we show that a stack associated to a groupoid object having source and target that arecoverings in the presite is the same as a stack adapted to the presite; i.e., it is an algebraic stack.Moreover, the groupoid object induces a presentation of the stack. [<cit.>] Let 𝖲 be a subcanonical presite admitting fibered products. Let@C=1emU_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)tU_0 be a groupoid object, and set 𝒳=[@C=1emU_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)tU_0].If 𝐏 is any property of morphisms in 𝖲 that is stable under base change and local (on the target), then: * Assuming the diagonal 𝒳→𝒳×𝒳 is 𝖲-representable, it has property 𝐏 if and only if (s,t):U_1→ U_0× U_0has property 𝐏. * Assuming themorphism U_0 →𝒳 (corresponding to the identity of U_0) is 𝖲-representable, it has property 𝐏 if and only if s (or t) has property 𝐏.For the first claim, U_1 → U_0 × U_0 is the base change of the diagonal via U_0 × U_0 →𝒳×𝒳, so the former inherits property 𝐏 from the latter.Conversely, if S →𝒳×𝒳 is any morphism, letV [r] [d] W [d] T [r] Sbe the base change of the cartesian diagramU_1 [r] [d] U_0 × U_0 [d]𝒳[r]^<>(0.5)Δ 𝒳×𝒳 .Then V → W has property 𝐏 by base change.But W → S is a cover, since it is the base change of U_0 × U_0 →𝒳×𝒳, so property 𝐏 descends to T → S.This applies to any morphism S →𝒳×𝒳, so Δ has property 𝐏.For the second claim, there is a cartesian diagramU_1 [r]^t [d]_s U_0 [d] U_0 [r]𝒳so s and t inherit property 𝐏 from U_0 →𝒳.Conversely, suppose S →𝒳 is any morphism.Let V [r] [d] W [d]T [r] Sbe the base change of (<ref>).Then V → W is 𝐏 by base change.But W → S is covering (by base change; see Example<ref>),so 𝐏 descends to T → S.This applies to any S →𝒳, so U_0 →𝒳 is 𝐏, as required.[<cit.>] Let 𝖲 be a subcanonical presite admitting fibered products.Let 𝒳 be a stack, let P:U→𝒳 be an𝖲-representable morphism from an object U of 𝖲. Then there is an associatedgroupoid object U×_𝒳U @->@<-2pt>[r]_<>(0.5)pr_1@->@<2pt>[r]^<>(0.5)pr_2U in 𝖲 (see e.g., <cit.>). If moreover the morphism P:U→𝒳 is a cover in the sense of Definition <ref>, then 𝒳 is equivalent to [U×_𝒳U @->@<-2pt>[r]_<>(0.5)pr_1@->@<2pt>[r]^<>(0.5)pr_2U], and the projections pr_1,pr_2 are covers in the presite. Set U_0 = U and U_1 = U ×_𝒳 U.Let 𝒰 be the stack associated to the groupoid object𝒰 = U_1 ⇉ U_0.We construct a map 𝒰→𝒳 by descent.Let Z →𝒰 be any morphism, where Z is a scheme.By definition, this corresponds to a groupoid presentation of 𝒵 of Z and a cartesian morphism 𝒵→𝒰.By composition, this gives a map to the constant groupoid object 𝒳, and this descends uniquely to a map Z →𝒳.This is easily shown to be functorial in Z, hence gives a morphism 𝒰→𝒳.Now we argue that 𝒰→𝒳 is an isomorphism if U_0 covers 𝒳.Indeed, the map U →𝒳 factors through 𝒰, so 𝒰→𝒳 is surjective.On the other hand, U_0 ×_𝒰 U_0 → U_0 ×_𝒳 U_0 is an isomorphism.This is the pullback under the cover U_0 ×_𝒳 U_0 →𝒰×_𝒳𝒰 of the diagonal 𝒰→𝒰×_𝒳𝒰.Therefore the relative diagonal of 𝒰→𝒳 is an isomorphism, which is to say that 𝒰→𝒳 is injective.Combined with the surjectivity, this means 𝒰→𝒳 is an isomorphism.The statement that pr_1 and pr_2 are covers can be obtained from the 2-cartesian diagram (<ref>).[Groupoid object adapted to a presite] Let 𝖲 be a subcanonical presite admitting fibered products.We say a groupoid object @C=1emU_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)tU_0 is adapted to the presite if s and t are covers in the presite and the natural morphism U_0→𝒳=[@C=1em U_1 @->@<-2pt>[r]_<>(0.5)s@->@<2pt>[r]^<>(0.5)t U_0 ] is 𝖲-representable.The stack associated to a groupoid object adapted to a presite is a stack adapted to the presite; in particular, it is algebraic.Conversely, a stack adapted to a presite is the stack associated to a groupoid object adapted to the presite.§ ACKNOWLEDGMENTS These notes provide an elaboration on lecturesthe first author gave at the summer school The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundlesat the Institute for Mathematical Sciences at the National University of Singapore in July of 2014.He would like to thank the organizers for their invitation, and the IMS at NUS for their hospitality.He would also like to thankTony Pantev for discussions on the modulistack of Higgs bundles.Both authors would like to thank the referees and Sebastian Bozlee for their comments on earlier versions of the paper. | http://arxiv.org/abs/1708.08124v1 | {
"authors": [
"Sebastian Casalaina-Martin",
"Jonathan Wise"
],
"categories": [
"math.AG",
"14D20, 14D23, 14H60"
],
"primary_category": "math.AG",
"published": "20170827185218",
"title": "An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves"
} |
lemmaLemma corollaryCorollaryDimensional Reduction of DSSA. Allawala et al1 Department of Physics, Brown University, Providence, Rhode Island 02912-1893, USA 2 JPMorgan Chase & Co., 237 Park Ave., New York, New York 10016 3 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK Dimensional Reduction of Direct Statistical Simulation Altan Allawala1,2, S. M. Tobias3 J. B. [email protected] ================================================================================= Direct Statistical Simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by Direct Numerical Simulation (DNS).That low-order statistics usually evolve slowly compared withinstantaneous dynamics is one important advantage of DSS.Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low-order statistics typically have higher dimension than the underlying fields. Here we show that it is possible to go much further by using a form of unsupervised learning, Proper Orthogonal Decomposition (POD), to address the “curse of dimensionality.” We apply POD directly to DSS in the form of expansions in the equal-time cumulants to second order (CE2).We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models of fluid on a rotating sphere (a jet that relaxes deterministically towards an unstable profile, and a stochastically-driven flow that spontaneously organizes into jets).Order-of-magnitude savings in computational cost are obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS. § INTRODUCTIONStatistical descriptions are appropriate for turbulent flows.In nature such flows are rarely homogeneous and isotropic; instead they typically exhibit rich correlations that reflect the presence of coherent structures, with statistics that evolve only slowly in time, or not at all.The large range of spatial and temporal scales spanned by turbulent flows often makes their Direct Numerical Simulation (DNS) computationally prohibitive <cit.>.Fifty years ago Lorenz pointed to an alternative approach that “consists of deriving a new system of equations whose unknowns are the statistics themselves” <cit.>.Such Direct Statistical Simulation (DSS) has seen, in recent years, a number of successful applications <cit.>.Many well-established approaches can be considered instances of DSS. For example the probability distribution function can be obtained from the Fokker-Planck equationby numerical methods, but this is limited to dynamical systems of at most a few dimensions<cit.>. Another form of DSS, Large Deviation Theory <cit.>, focuses on extreme or rare events.Other methods such as those developed by Kolmogorov <cit.>, Kraichnan <cit.>, and others <cit.> provide an approximate description of some statistical properties of turbulent flows but assume homogeneity and usually isotropy. Many flows in geophysics and astrophysicsspontaneously develop features such as coherent vortices and zonal banding <cit.>. Furthermore, in engineering applications turbulence often interacts with non-trivial mean flows <cit.>. Therefore, any statistical method that can appropriately treat such systems needs to respect such asymmetries. One such scheme of DSS that meets these requirements is that of low-order expansions in equal-time (but spatially nonlocal) cumulants.Since low-order statistics are spatially smoother than the corresponding dynamical fields (or instantaneous flow), the approach can capture the macroscopic features of turbulent flows using fewer degrees of freedom. An added benefit of such a cumulant expansion scheme is that the detailed time evolution of the flow is replaced by a description of the statistics of most interest.The modes associated with the low-order statistics may be described by a fixed point or a slow manifold that can be quickly accessed. It is important to note that naive implementations of expansions in cumulants may be much more expensive computationally than full DNS because the second cumulant may have higher dimension than the underlying fields (depending on the symmetry of the problem and choice of averaging operation).In this paper we investigate a reduced dimensionality method for DSS, based on a Proper Orthogonal Decomposition (POD) of the eigenvectors of the second moment.This is a form of unsupervised learning, with training based upon full resolution simulations.The equations of motion (EOMs) for the cumulants are rotated into a sub-basis formed by the eigenvectors of the second zonally-averaged moment after removal of eigenvectors with small eigenvalues.We implement POD directly on the simplest non-trivial closure, one that goes to second order in an expansion of cumulants (CE2), of two model problems in fluid dynamics.The rest of the paper is organized as follows.In Section <ref> we introduce the two different types of cumulant expansions that are explored.Although CE2 is often performed using a zonal average <cit.> (Section <ref>), this has the drawback that scattering of eddies off non-zonal coherent structures such as vortices are neglected <cit.>.We therefore also explore a variant of CE2 that is based upon an ensemble average <cit.> (Section <ref>). Although more accurate, ensemble-averaged cumulants have higher dimensionality compared with those based upon the zonal average; this is partly overcome with our POD method as discussed in Section <ref>.In Section <ref> both types of CE2 are evaluated against DNS which serves as the reference truth. We test the approaches on two different highly idealized barotropic models of planetary atmospheres on a spherical geodesic grid: A deterministic point jet relaxed toward an unstable profile (Section <ref>), and a stochastically-forced jet (Section <ref>). The order of magnitude computational savings of DSS in a reduced basis with little loss to accuracy promises a fast and accurate alternative to accessing directly the low-order statistics of turbulent flows, and offers the possibility that flow regimes inaccessible to DNS will come within reach.This speed-up is illustrated by a continuation in parameter space, keeping the POD basis fixed, in Section <ref>.Section <ref> concludes with some discussion. § CUMULANT EXPANSIONSWe carry out a non-equilibrium statistical closure of the low order equal-time statistics of the flow <cit.>. The approach can be more easily understood by application to a simple toy model. We do so here by considering a barotropic (two-dimensional) fluid on a rotating sphere of unit radius, where relative vorticity evolves under the action of a bilinear Jacobian operator, J[A, B]=r̂·(∇ A×∇ B), a linear operator that contains frictional and hyperviscous termsL[A] ≡ -[κ-ν_3(∇^2+2)∇^4] A, and either a deterministic forcing term F, stochastic forcing η, or both. The EOM of the barotropic model is then given by:ζ̇ = J[ζ+f, ψ] + L[ζ] + F + η,where ψ is the stream function, ζ=∇^2ψ is the relative vorticity, f=2Ωcos(θ) is the Coriolis parameter, θ is the co-latitude and ϕ is the azimuth angle.We set Ω = 2 π and thus the unit of time is the period of rotation, a day.The cumulant expansion may then be implemented by Reynolds decomposing the relative vorticity into the sum of an average vorticity field ζ and a fluctuation about that average ζ^' so thatζ(θ,ϕ)=ζ(θ,ϕ)+ζ^'(θ,ϕ). The choice of this averaging operation will be postponed until later, but it will be required to satisfy the Reynolds averaging rules:ζ^'(θ,ϕ)=0, ζ(θ,ϕ)=ζ(θ,ϕ), ζζ =ζ ζ.The first three cumulants are centered moments, and on the surface of a sphere the first two read:c(θ_1,ϕ_1)≡ ζ(θ_1,ϕ_1), c(θ_1,ϕ_1;θ_2,ϕ_2) ≡ ζ^'(θ_1,ϕ_1)ζ^'(θ_2,ϕ_2). Since the Jacobian couples the relative vorticity and the stream function, it is useful to define their correlations as auxiliary cumulants,p(θ_1,ϕ_1) ≡ ψ(θ_1,ϕ_1), p(θ_1,ϕ_1;θ_2,ϕ_2) ≡ ζ^'(θ_1,ϕ_1)ψ^'(θ_2,ϕ_2).The EOMs of the cumulants are derived by applying the averaging operation to Equation (<ref>). Refer to <cit.> for a detailed derivation. The first cumulant evolves as:∂/∂ tc(Ω⃗_1)= ∫ J_1[p(Ω⃗_1, Ω⃗_2),δ(Ω⃗_1 - Ω⃗_2)] dΩ_2 +J_1[c(Ω⃗_1) + f(θ_1), p(Ω⃗_1)] +L_1[c(Ω⃗_1)] + F(Ω⃗_1)where the subscript 1 on J and L indicates that these operators act upon the vector coordinate Ω⃗_1 ≡ (θ_1, ϕ_1) and δ(Ω⃗_1 - Ω⃗_2) is the two-dimensional Dirac functional.As averages of the product of two fields do not generally equal the product of their separate averages,the quadratic nonlinearity leads to the well-known closure problem with the equation of motion of the first cumulant depending on the second cumulant. Likewise, the equation of motion for the second cumulant involves the first, second and third cumulant.A closure should be performed at the lowest order possible; this may be achieved by decoupling the third cumulant from the EOM of the second cumulant, known as the CE2 approximation:ζ^'(Ω⃗_1) ζ^'(Ω⃗_2) ψ(Ω⃗_⃗3⃗)≃c(Ω⃗_1, Ω⃗_2) p(Ω⃗_3)(CE2).With this approximation the EOM for the second cumulant closes to give∂/∂ t c(Ω⃗_1, Ω⃗_2) =2 { J_1[ c(Ω⃗_1) + f(θ_1), p(Ω⃗_2, Ω⃗_1) ]+2J_1[c(Ω⃗_1, Ω⃗_2), p(Ω⃗_1)]+L_1[c(Ω⃗_1,Ω⃗_2)] } + 2 Γ(Ω⃗_1,Ω⃗_2),where the symmetrization operator {⋯} performs an average over all interchanges of the field points. Here Γ(Ω⃗_1,Ω⃗_2) is the covariance ofthe Gaussian stochastic forcing η(Ω⃗,t) that is assumed to be δ-correlated in time. CE2 neglects interaction between two fluctuations to produce another fluctuation because of the decoupling of the third and higher cumulants <cit.>. Thus the fluctuation-fluctuation scattering process is neglected, making it formally equivalent to a quasi-linear approximation <cit.>. This confers upon the CE2 approximation two attractive properties: conservation up to quadratic order <cit.> (angular momentum, energy and enstrophy) in the limit of no forcing or dissipation, and physical realizability <cit.> with a positive-definite second cumulant. These properties ensure numerical stability.Both the DNS and DSS performed hereare carried out in a basis of spherical harmonics (see <cit.>) with the spectral decomposition:ζ(θ, ϕ) = ∑_ℓ = 1^L ∑_m^|m| ≤min(ℓ, M)ζ_ℓ m Y_ℓ^m(θ, ϕ).Spectral cutoffs L and M are specified below.[A program that implements the computations is available on the Apple Mac App Store at URL https://apps.apple.com/us/app/gcm/id592404494?mt=12.] §.§ Zonal AverageCommon choices for the DSS averaging operation, ⋯, are temporal, spatial and ensemble means. For models with zonal symmetry such as idealized geophysical or astrophysical flows the zonal spatial average, defined as q(θ) ≡1/2π∫_0^2π q(θ,ϕ) dϕ. is often the simplest and most natural. Henceforth ⋯ will be used to only denote such a zonal average. Since the fluctuations q^' with respect to a zonal average now represent eddies that may be classified by zonal wavenumber, zonal CE2 allows for the interaction between a mean flow and an eddy to produce an eddy, and also for the interaction between two eddies to produce a mean flow as shown in Figure <ref>(a).An important virtue of the zonal average is that it reduces the dimension of each cumulant by one in the zonal direction, reducing computational complexity. The first cumulant depends only on the co-latitude, c(Ω⃗_1) = c(θ_1), and the second cumulant depends only on the two co-latitudes of either points and the difference in their longitudes, c(Ω⃗_1,Ω⃗_2)=c(θ_1,θ_2,ϕ_2-ϕ_1). Since c(Ω⃗_⃗1⃗), p(Ω⃗_⃗1⃗) and f(θ_1) do not vary with longitude, the second term in Equation (<ref>) vanishes. §.§ Ensemble Average In systems where the eddy-eddy scattering processes dominate, or where non-zonal coherent structures dominate, zonal CE2 can give an inaccurate reproduction of the statistics <cit.>.An alternative version of the cumulant expansion that does not entirely neglect these processes can be formulated by replacing the zonal average with an ensemble average, to be denoted in the following with ⟨⋯⟩. Formally the triadic interactions that are retained in ensemble CE2 are identical to those in zonal CE2 as shown in Figure <ref>(b) except for the addition of a diagram with three mean fields. This addition reflects the fact that, for ensemble averaging, the mean field may contain non-zonal coherent structures <cit.> instead of only the zonal mean; the fluctuations about this average represent incoherent perturbations, instead of eddies, that may interact. The coherent part of the flow, c(θ, ϕ) = ⟨ζ(θ, ϕ) ⟩, may for instance consist of long-lived vortices rather than zonal jets <cit.> in which case departures from the zonal mean do not necessarily constitute a fluctuation, and some of the eddy + eddy → eddyscattering processes (where eddies are defined as deviations from the zonal mean) are retained. On the other hand, because only the first two cumulants are retained, ensemble-averaged CE2 is equivalent to the requirement that the probability distribution functionbe purely Gaussian – an approximation known to be poor for some flows such as isotropic and homogeneous turbulence.It is well known that for ergodic flows, the average over an infinite ensemble of realizations should equal a long-time average (in the limit of the averaging time going to infinity). For such flows one therefore expects that the statistics in ensemble CE2 typically flow to a single fixed point — this would be guaranteed if there were no closure approximation —and those for zonal CE2 to match this behaviour. However in flows where long-lived coherent structures such as vortices and jets play an important role <cit.>, reducing the chaoticity of the flow, it is not true that the two versions of CE2 described here should yield the same results. If ensemble CE2 is initialised with non-zonally symmetric initial conditions then we find that it oftn flows to a fixed point. By contrast zonal CE2 often does not flow to a fixed point; instead the statistics may oscillate in time <cit.>. § PROPER ORTHOGONAL DECOMPOSITIONThe zonal average second cumulant is a three-dimensional object, one dimension higherthan that of the underlying dynamical fields;forensemble averages the second cumulant is of dimension four. This “curse of dimensionality”<cit.> can be tamed by application of Proper Orthogonal Decomposition (POD) <cit.> directly to the low order statistics. Traditionally, PODhas been applied directly to the instantaneous master PDEs; instead, here we apply these methods directly to the statistical formulation and find improved performance.Because the two operations of formulating DSS, and reducing dimensionality, do not commute, a given number of POD modes may better represent the DSS statistics than they would the full instantaneous dynamics. In the reduced basis the two-point function may be efficiently evolved forward in time without encountering the instabilities that plague DNS in POD bases <cit.>.Here we illustrate how the procedure keeps dimensionality in check without significant loss of accuracy.A new basis of lower dimensionality that represents the first and secondcumulants may be found by Schmidt decomposition of the zonally-averaged second moment:ζ(Ω⃗_1) ζ(Ω⃗_2) = c(Ω⃗_1, Ω⃗_2) + c(θ_1) c(θ_2) = ∑_i λ_i ϕ_i(Ω⃗_1) ϕ_i(Ω⃗_2) .Here ϕ_i(Ω⃗_1) is an eigenvector of the second moment with eigenvalue λ_ithat is both real and non-negative <cit.>.Eq. <ref> can equivalently be expressed in the space of spherical harmonics where the second moment m_ℓℓ^' m≡ζ_ℓ mζ_ℓ^' m^* is block-diagonal in the zonal wavenumber m andm_ℓℓ^' m = c_ℓℓ^' m + c_ℓ c_ℓ^'δ_m,0 = ∑_i U_ℓ i^(m)λ_i^(m) U^† (m)_i ℓ^' .Here U_ℓ i^(m) are unitary matrices (∑_ℓ U^† (m)_i ℓ U_ℓ j^(m) = δ_i j)that are composed of the eigenvectors of the second moments:∑_ℓ^' m_ℓℓ^' m U^(m)_ℓ^' j = λ_j U^(m)_ℓ j. The dimension of the EOMs for the cumulants may now be reduced by setting acutoff for the eigenvalues, λ_c, and discarding all eigenvectors with eigenvalues below this value.The truncation generally breaks conservation of angular momentum, energy, and enstrophy, but for driven and damped systems this will generally not cause divergences as long as the truncation is not too severe.Positivity of the second cumulant is maintained by periodically projecting out eigenvectors with negative eigenvalues.The zonally-averaged second moment is accumulated after spin-up to a statistical steady state.For each type of simulation (DNS, zonal CE2, or ensemble CE2) we first compute the second moment in the full basis, perform the POD decomposition, and implement the corresponding reduced-dimensionality version of CE2.Figure <ref>, for instance, shows the second cumulant for the stochastically-forced jet (defined below) as calculated by zonal CE2 at different levels of truncation.A severe truncation that retains only the six eigenvectors with largest eigenvalues is still able to reproduce most features of the cumulant. § TESTSWe first implement unreduced ensemble CE2 and zonal CE2 on two idealized barotropic models on a rotating unit sphere. After comparing these two variants of CE2 against the statistics gathered by DNS that are used as the reference truth, westudy the efficacy of dimensional reduction of the two forms of DSS using the POD procedureoutlined in Section <ref>.§.§ Point JetWe study a deterministic point jet that relaxes on a timescale τ toward an unstable retrograde jet with meridional profile ζ_jet(θ) = - Ξtanh((π / 2 - θ)/Δθ).Parameters Ξ = 0.6 Ω and Δθ = 0.05 are chosen to correspond with the jetpreviously examined in <cit.>.The flow is damped and driven by the terms that appear in Equation (<ref>), i.e.L[ζ] + F =ζ_jet - ζ/τwith η = 0. Spectral simulations are performed at a modest resolution of 0 ≤ℓ≤ L and |m| ≤ min{ℓ, M} with L = 20 and M = 12.For a short relaxation time of τ=2 days the flow is dominated by critical-layer waves and eddies are suppressed by the strong coupling to the fixed jet.For a longer relaxation time of τ=20 days the flow is turbulent and well-mixed near the equator.Since zonal CE2 neglects eddy-eddy interactions it increasingly differs from DNS as τ grows. Here we set τ=20 days to demonstrate that ensemble-averaged CE2 captures features of the non-zonal coherent structures that zonal average CE2 misses.Figure <ref>(a) shows that ensembleCE2 accurately reproduces the zonal mean absolute vorticity, whereas zonal CE2 overmixes the absolute vorticity at low latitudes <cit.>. Figure <ref> shows that whereas zonal CE2 has power only in the zonal mean m=0 mode, and a single eddy of wavenumber m=3, the power spectrum of ensemble CE2 matches that found by DNS.DNS statistics are accumulated from time 500 to 1000 days after spin-up of500 days.Ensemble CE2 reaches a statistical steady-state by 400 days and no time-averaging is required. Whereas the second cumulant of the vorticity field as determined by zonal CE2exhibits artificial reflection symmetryabout the equator <cit.>, ensemble CE2 does not have thisdefect; see Figure <ref>(b, d, f).We turn now to model reduction via POD truncation. Figure <ref>(b)-(d) show the first cumulants, or zonal mean absolute vorticity, at different levels of truncation. While POD offers a substantial dimensional reduction for DNS and ensemble CE2 with slight loss of accuracy, extreme truncation of zonal CE2 down to just a few modes is possible. This is consistent with the spectrum of the eigenvalues of the second moment shown in Figure <ref>(a). The steep decay of the eigenvalues of zonal CE2 reflects the fact thatonly the m = 3 eddy has power. §.§ Stochastic JetJets that form spontaneously in the presence of rotation and small scale random forcingprovide another, perhaps more stringent, test of DSS.An idealized barotropic model that has been much studied <cit.> is governed by:L[ζ] = - [κ-ν_3(∇^2 + 2)∇^4]ζ,with F = 0.We examine this model for the set of parameters used in <cit.>; friction κ=0.02 with hyperviscosity ν_3 set such that the mode at the smallest length scale decays at a rate of 1. The covariance of the Gaussian white noise of the modes that are forced stochastically is Γ_ℓℓ^' m = 0.1 δ_ℓℓ^' for 8 ≤ℓ≤ 12 and 8 ≤ |m| ≤ℓ, and set the stochastic renewal time to be τ_r = 0.1. For this experiment, the forcing is concentrated at low latitudes, enabling an evaluation of the ability of differentDSS approximations to convey angular momentum towards the poles <cit.>. Spectral simulations are performed at a resolution of 0≤ l≤ L and |m|≤ min{l,M} with L = 30 for M = 20.After a spin-up time of 300 days, statistics are accumulated for a further 700 days.Figure <ref>(a) compares the zonal mean zonal velocity as a function of latitude for ensemble CE2, zonal CE2, and DNS.Owing to the neglect of eddy-eddy scattering in zonal CE2,there is no mechanism to transport eddy angular momentum from the equator, where the forcing is concentrated, towards the poles, and the method underestimates the mean zonal velocity at high latitudes.Ensemble CE2 does somewhat better, but thematch at high latitudes remains poor.(Higher-order closures do much better – see <cit.> – but for simplicity we do not study them here.)The second vorticity cumulant is shown in Figure <ref>(b,d,f).Vorticity correlations are non-local in space, and zonal CE2 exaggerates the range of the correlations because the waves are coherent, again owing to the lack of eddy-eddy scattering.Ensemble CE2 more closely matches DNS.We now examine the reduction in dimensionality of DSS by POD. Figure <ref>(b)-(d) shows the zonal mean zonal velocity as a function of latitude at different levels of truncation for DNS, ensemble CE2 and zonal CE2.It is apparent that both types of CE2 are better suited to the POD method than DNS. Zonal CE2 in particular allows for a more severe truncation compared with ensemble CE2 and DNS, a fact that can again be explained by the spectrum of eigenvalues shown in Figure <ref>(b). The eigenvalues decay more slowly for the stochastically-forced jet than for the point jet. It can be seen that statistics accumulated from DNS do notconverge monotonically in truncation level toward those of the full simulation;those obtained from the cumulant expansions are better behaved. Convergence of the second cumulant with increasing number of retained modes is also evident; see Figures <ref> and <ref>.The number of operations required for a POD reduced CE2 time step scales with the size of the retained basis N_ret as 𝒪(N_ret^3). Therefore a dimensional reduction from N_ret=441 to N_ret=13 would naively equate to a speed up of order 10^4.In the case of unreduced zonal CE2, however, zonal symmetry may be exploited to decrease the number of operations per time step <cit.> to𝒪(L^3M) << 𝒪((LM)^3).Nevertheless, the speed up isconsiderable. A machine with a 2.6 GHz quad-core Intel Core i7 processor runs 28 × faster for reduced zonal CE2with N_ret=13 in comparison to full unreduced zonal CE2, with little loss in accuracy.§ CONTINUATION IN PARAMETER SPACEThe test problems studied in the preceding section show that it is possible to implement DSS in a subspace of reduced dimensionality, supporting the idea that relatively few modes are required for a description of the low-order statistics.The reduced statistical simulations still, however, require a full resolution training run for POD.Thus, from a practical point of view, the reduced simulations do not offer a speed-up.In order to address this,we show here that it is possible to fix the reduced basis obtained from a single training run, alter model parameters, and still obtain good agreement with both DNS and CE2 in a reduced basis. Thus, the reduced basis obtained from one training run can be used to perform DSS for a range of parameters.We illustrate the continuation by changing the relaxation time of the point jet. This parameter changes the relative degree of turbulence to mean flow in the jet and is related to the Kubo number, which is a measure of the degree of applicability of the quasilinear approximation <cit.>. We utilize the τ = 20 days jet as a training run for POD on the ensemble CE2 solution, and calculate the reduced basis for this parameter set.Figure <ref> shows the (zonally averaged) first cumulant for ensemble CE2 in this reduced basis of 40 retained modes for τ = 10 and 5 days.Note that the dynamics (and indeed the statistics) does change significantly when this parameter is altered. For τ=10 days the agreement is exceptionally good. However, as the parameter τ is moved further away from the training run to τ = 5 days the agreement between reduced ensemble CE2 with full ensemble CE2 and DNS worsens as expected. Even here, qualitative agreement is retained. Comparison of power spectra in Figure <ref> shows that second-order statistics also continue to show qualitative agreement at τ=10 days. The reduction in dimensionality simplifies the spectrum of the reduced ensemble CE2 spectra in comparison with the full-resolution simulation as expected. This initial exploration of parameter continuation shows a promising direction for future research.§ DISCUSSION AND CONCLUSIONProper Orthogonal Decomposition (POD) can be used to reduce the dimensionality of second order cumulant expansions (CE2) by discarding modes that are unimportant for the low-order statistics.For the idealized models that we studied, the first and second cumulants can be accurately reproducedwith relatively few modes, permitting a substantial reduction in dimensionality, and an increase in computational speed. This is particularly true for zonal CE2.The degree of truncation and hence computational saving that can be made while maintaining accuracy depends on the specifics of the system, as demonstrated by our results for the two different test problems.It would be interesting to explore dimensional reduction of Direct Statistical Simulation (DSS) for more realistic models such as those explored in <cit.>.Dimensional reduction by POD has been tested in a quasilinear model of the ocean boundary layer <cit.>. It would also be interesting to explore whether or not a dimensional reduction algorithm could beconstructed that acts dynamically on DSS, bypassing the step of first acquiring statistics for the full non-truncated problem, as in the current work.With such an advance it may be possible for DSS to access regimes that cannot be reached by DNS.We note that CE2 by itself has already been used for a dynamo problem to access lower magnetic Prandtl number than is possible by DNS <cit.>. As reduced-dimensionality DSS is able to quickly describe some fluid dynamical systems via parameter continuation of the low-order statistics, the combination of POD with cumulant expansions offers a good prospect for other simulations to reach beyond DNS. § DECLARATION OF INTERESTSThe authors report no conflict of interest.§ ACKNOWLEDGEMENTSSupported in part by NSF DMR-1306806 and by a grant from the Simons Foundation (Grant number 662962, GF).SMT would also like to acknowledge support offunding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement no. D5S-DLV-786780).We thank Greg Chini and Joe Skitka for useful discussions.49 natexlab#1#1#1#1 #1#1 #1#1#1#1#1#1 #1#1#1#1 #1#1 #1#1 #1#1#1#1#1#1#1#1[Ait-Chaalal et al.(2016)Ait-Chaalal, Schneider, Meyer & Marston]ait2016cumulant Ait-Chaalal, Farid, Schneider, Tapio, Meyer, Bettina & Marston, JB 2016Cumulant expansions for atmospheric flows.New Journal of Physics18 (2),025019.[Allawala & Marston(2016)]Allawala:2016hx Allawala, Altan & Marston, J B 2016Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions.Physical Review E94,052218(9).[Bakas & Ioannou(2011)]bakas2011structural Bakas, Nikolaos A & Ioannou, Petros J 2011 Structural stability theory of two-dimensional fluid flow under stochastic forcing.Journal of Fluid Mechanics682, 332–361.[Bakas & Ioannou(2013)]bakas2013emergence Bakas, Nikolaos A & Ioannou, Petros J 2013 Emergence of large scale structure in barotropic β-plane turbulence.Physical review letters110 (22),224501.[Bakas & Ioannou(2014)]bakas2014theory Bakas, Nikolaos A & Ioannou, Petros J 2014A theory for the emergence of coherent structures in beta-plane turbulence. Journal of Fluid Mechanics740,312–341.[Barkley(2016)]barkley2016 Barkley, D. 2016Theoretical perspective on the route to turbulence in a pipe.Journal of Fluid Mechanics803, P1.[Batchelor(1947)]batchelor1947kolmogoroff Batchelor, GK 1947 Kolmogoroff's theory of locally isotropic turbulence.In Mathematical Proceedings of the Cambridge Philosophical Society, ,vol. 43,pp. 533–559. Cambridge Univ Press.[Bauer et al.(2015)Bauer, Thorpe & Brunet]bauer2015quiet Bauer, Peter, Thorpe, Alan & Brunet, Gilbert 2015 The quiet revolution of numerical weather prediction.Nature 525 (7567),47–55.[Bellman(1961)]bellman1961 Bellman, R.E. 1961 Adaptive Control Processes: A Guided Tour.Princeton University Press.[Bellman(1957)]bellman1957 Bellman, R. E. 1957 Dynamic Programming. Princeton University Press.[Bergman & Spencer Jr(1992)]bergman1992robust Bergman, LA & Spencer Jr, BF 1992Robust numerical solution of the transient Fokker-Planck equation for nonlinear dynamical systems.In Nonlinear Stochastic Mechanics,pp. 49–60.Springer.[Bouchet et al.(2018)Bouchet, Marston & Tangarife]Bouchet:2018er Bouchet, F, Marston, J B & Tangarife, T 2018 Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics.Physics of Fluids30 (1),015110–20.[Bouchet & Simonnet(2009)]Bouchet:2009cl Bouchet, Freddy & Simonnet, Eric 2009Random Changes of Flow Topology in Two-Dimensional and Geophysical Turbulence. Physical Review Letters102 (9),094504–4.[Constantinou et al.(2014)Constantinou, Farrell & Ioannou]Constantinou:2013fh Constantinou, Navid C, Farrell, Brian F & Ioannou, Petros J 2014Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. Journal of the Atmospheric Sciences71,1818 – 1842.[Farrell & Ioannou(2007)]Farrell:2007fq Farrell, Brian F & Ioannou, Petros J 2007Structure and Spacing of Jets in Barotropic Turbulence.Journal of the Atmospheric Sciences64 (10),3652–3665.[Frisch(1995)]frisch1995turbulence Frisch, Uriel 1995 Turbulence: the legacy of AN Kolmogorov.Cambridge university press.[Frishman et al.(2017)Frishman, Laurie & Falkovich]frishman2017 Frishman, A., Laurie, J. & Falkovich, G. 2017 Jets or vortices — What flows are generated by an inverse turbulent cascade?Physical Review Fluids2 (3),032602, arXiv: 1608.04628.[Hänggi & Talkner(1980)]hanggi1980remark Hänggi, P & Talkner, P 1980A remark on truncation schemes of cumulant hierarchies.Journal of Statistical Physics22 (1),65–67.[Herring(1963)]herring1963investigation Herring, JR 1963Investigation of problems in thermal convection.Journal of the Atmospheric Sciences20 (4), 325–338.[Holloway & Hendershott(1977)]holloway1977stochastic Holloway, Greg & Hendershott, Myrl C 1977 Stochastic closure for nonlinear Rossby waves.Journal of Fluid Mechanics82 (04),747–765.[Holmes et al.(1998)Holmes, Lumley & Berkooz]holmes1998turbulence Holmes, Philip, Lumley, John L & Berkooz, Gal 1998 Turbulence, coherent structures, dynamical systems and symmetry. Cambridge university press.[Huang et al.(2001)Huang, Galperin & Sukoriansky]huang2001anisotropic Huang, Huei-Ping, Galperin, Boris & Sukoriansky, Semion 2001Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere.Physics of Fluids (1994-present) 13 (1),225–240.[Kraichnan(1980)]kraichnan1980realizability Kraichnan, Robert H 1980Realizability Inequalities and Closed Moment Equations.Annals of the New York Academy of Sciences 357 (1),37–46.[Kumar & Narayanan(2006)]kumar2006solution Kumar, Pankaj & Narayanan, S 2006Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems.Sadhana31 (4),445–461.[Laurie et al.(2014)Laurie, Boffetta, Falkovich, Kolokolov & Lebedev]Laurie:2014dn Laurie, Jason, Boffetta, Guido, Falkovich, Gregory, Kolokolov, Igor & Lebedev, Vladimir 2014Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence. Physical Review Letters113 (25),254503–5.[Laurie & Bouchet(2015)]Laurie:2015co Laurie, Jason & Bouchet, Freddy 2015Computation of rare transitions in the barotropic quasi-geostrophic equations.New Journal of Physics17 (1),1–25.[Legras(1980)]legras1980turbulent Legras, Bernard 1980Turbulent phase shift of Rossby waves.Geophysical & Astrophysical Fluid Dynamics15 (1), 253–281.[Lorenz(1967)]lorenz1967nature Lorenz, Edward N 1967 The nature and theory of the general circulation of the atmosphere, ,vol. 218.World Meteorological Organization Geneva.[Marston et al.(2008)Marston, Conover & Schneider]marston65conover Marston, JB, Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. Journal of the Atmospheric Sciences65,1955–1966.[Marston(2010)]marston2010statistics Marston, J B 2010Statistics of the general circulation from cumulant expansions.Chaos20,041107.[Marston(2012)]Marston:2012co Marston, J B 2012Planetary Atmospheres as Nonequilibrium Condensed Matter.Annual Review of Condensed Matter Physics 3 (1),285–310.[Marston et al.(2016)Marston, Chini & Tobias]Marston:2016ff Marston, J B, Chini, G P & Tobias, S M 2016 Generalized Quasilinear Approximation: Application to Zonal Jets. Physical Review Letters116 (21),214501.[Marston et al.(2019)Marston, Qi & Tobias]marston2014direct Marston, J. B., Qi, Wanming & Tobias, S. M. 2019 Direct Statistical Simulation of a Jet in Zonal Jets: Phenomenology, Genesis, Physics (arXiv:1412.0381).Cambridge University Press.[Muld et al.(2012)Muld, Efraimsson & Henningson]muld2012flow Muld, Tomas W, Efraimsson, Gunilla & Henningson, Dan S 2012Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Computers & Fluids57,87–97.[Naess & Hegstad(1994)]naess1994response Naess, A & Hegstad, BK 1994Response statistics of van der Pol oscillators excited by white noise.Nonlinear Dynamics 5 (3),287–297.[O'Gorman & Schneider(2007)]o2007recovery O'Gorman, Paul A & Schneider, Tapio 2007Recovery of atmospheric flow statistics in a general circulation model without nonlinear eddy-eddy interactions.Geophysical Research Letters 34 (22).[Pichler et al.(2013)Pichler, Masud & Bergman]pichler2013numerical Pichler, L, Masud, A & Bergman, LA 2013 Numerical Solution of the Fokker-Planck Equation by Finite Difference and Finite Element Methods - A Comparative Study.In Computational Methods in Stochastic Dynamics,pp. 69–85. Springer.[Resseguier et al.(2015)Resseguier, Mémin & Chapron]Resseguier:2015tr Resseguier, Valentin, Mémin, Etienne & Chapron, Bertrand 2015 Stochastic Fluid Dynamic Model and Dimensional Reduction.In International Symposium on Turbulence and Shear Flow Phenomena (TSFP-9),pp. 1–6. Melbourne.[Salmon(1998)]salmon1998lectures Salmon, Rick 1998 Lectures on geophysical fluid dynamics.Oxford University Press.[Schoeberl & Lindzen(1984)]schoeberl1984numerical Schoeberl, Mark R & Lindzen, Richard S 1984A numerical simulation of barotropic instability. Part I: Wave-mean flow interaction.Journal of the atmospheric sciences41 (8), 1368–1379.[Skitka et al.(2020)Skitka, Marston & Fox-Kemper]Skitka:2020 Skitka, Joseph, Marston, J. B. & Fox-Kemper, Baylor 2020Reduced-order quasilinear model of ocean boundary-layer turbulence.Journal of Physical Oceanograph (in press) arXiv:1906.11671 .[Squire & Bhattacharjee(2015)]Squire:2015ia Squire, J & Bhattacharjee, A 2015Generation of Large-Scale Magnetic Fields by Small-Scale Dynamo in Shear Flows. Physical Review Letters115 (17),175003.[Tobias(2019)]tobias2019 Tobias, Steven 2019The Turbulent Dynamo.arXiv e-printsp. arXiv:1907.03685,arXiv: 1907.03685.[Tobias et al.(2011)Tobias, Dagon & Marston]tobias2011astrophysical Tobias, SM, Dagon, K & Marston, JB 2011 Astrophysical fluid dynamics via direct statistical simulation. The Astrophysical Journal727 (2),127.[Tobias & Marston(2013)]tobias2013direct Tobias, SM & Marston, JB 2013Direct statistical simulation of out-of-equilibrium jets.Physical review letters 110 (10),104502.[Tobias & Marston(2017a)]kolmogorovForcing2017 Tobias, SM & Marston, JB 2017a Direct statistical simulation of jets and vortices in 2d flows. Physics of Fluids .[Tobias & Marston(2017b)]Tobias:2017im Tobias, S M & Marston, J B 2017b Direct statistical simulation of jets and vortices in 2D flows. Physics of Fluids (1994-present)29 (11),111111–9.[Varadhan(1966)]largedeviationtheory Varadhan, SR Srinivasa 1966Asymptotic probabilities and differential equations.Communications on Pure and Applied Mathematics19 (3),261–286.[von Wagner & Wedig(2000)]von2000calculation von Wagner, Utz & Wedig, Walter V 2000On the calculation of stationary solutions of multi-dimensional Fokker–Planck equations by orthogonal functions.Nonlinear Dynamics21 (3),289–306. | http://arxiv.org/abs/1708.07805v3 | {
"authors": [
"Altan Allawala",
"S. M. Tobias",
"J. B. Marston"
],
"categories": [
"physics.flu-dyn",
"astro-ph.EP",
"cond-mat.stat-mech",
"physics.ao-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20170825164228",
"title": "Dimensional Reduction of Direct Statistical Simulation"
} |
Brendan P. Bowler [email protected]]Brendan P. Bowler Hubble Fellow Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaiiunder contract NNH14CK55B with the National Aeronautics and Space Administration McDonald Observatory and the Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA McDonald Observatory and the Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA Hubble Fellow Center for Astrophysics and Space Astronomy, University of Colorado at Boulder, Boulder, CO 80309, USA McDonald Observatory and the Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA McDonald Observatory and the Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA NSF Graduate Research Fellow Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA Institute for Astronomy, University of Hawai`i at Mānoa; 2680 Woodlawn Drive, Honolulu, HI 96822, USA California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, ChileROXs 12 (2MASS J16262803–2526477)is a young star hosting a directly imagedcompanion near the deuterium-burning limit. We present a suite of spectroscopic, imaging, and time-series observations tocharacterize the physical and environmental properties of this system. Moderate-resolution near-infrared spectroscopy of ROXs 12 B from Gemini-North/NIFS and Keck/OSIRISreveals signatures of low surface gravity including weak alkali absorption lines and a triangular H-band pseudo-continuum shape. No signs of Paβ emission are evident. As a population, however, we find thatabout half (46 ± 14%) of young (≲15 Myr) companions with masses≲20 possess actively accreting subdisks detected via Paβ line emission, which represents a lower limit on the prevalence of circumplanetary disks in generalas some are expected to be in a quiescent phase of accretion.The bolometric luminosity of the companion and age of the host star (6^+4_-2 Myr) imply a mass of17.5 ± 1.5for ROXs 12 B based on hot-start evolutionary models. We identify a wide (5100 AU) tertiary companion to this system, 2MASS J16262774–2527247,which is heavily accreting and exhibits stochastic variability in its K2 light curve. By combining vsini_* measurements with rotation periods from K2, we constrain the line-of-sight inclinations of ROXs 12 A and 2MASS J16262774–2527247 and find that they are misaligned by 60^+7_-11^∘. In addition, the orbital axis of ROXs 12 B is likely misalignedfrom the spin axis of its host star ROXs 12 A, suggesting that ROXs 12 Bformed akin to fragmenting binary stars or in anequatorial disk that was torqued by the wide stellar tertiary. § INTRODUCTION A growing number of directly imaged planetary-mass companions (PMCs; ≲13 ) spanning orbital distances of hundreds to thousands of AUhave been discovered over the past decade through adaptive optics imaging and seeing-limited common proper motion searches (e.g., ;; ; ; ; ). The origin of this remarkable population is under debate but probably differs fromgiant planets located at smaller separations within ∼10 AU. Planet scattering to large separations (; ),a binary-like formation scenario via turbulent fragmentation(; ; ), and instabilities in massive protoplanetary disks have all been proposed to explain their existence. Unfortunately, robust tests of these scenarios are difficult with the relatively small sample of known objects().In principle, clues about their origin can be inferred from a comparative abundance analysis with their host stars(e.g., , ), measurements of their occurrence rate over time, long-term monitoring of their orbits(e.g., ; ; ),the physical properties of their circumplanetary disks (; ; ),and constraints on their mass and semi-major axis distributions (; ; ). Indeed, <cit.> recently concluded that dynamical scattering is probably not the dominant origin of wide PMCs based on the lack of close-in scatterers, the low rate of close-in giant planets in the field, and early orbital constraints for wide PMCs.Regardless of their formation route, the favorable angular separations and contrasts of wide PMCsmake them attractive targets for detailed spectroscopic characterization in the near-infrared(e.g., ; ). As such, they represent excellent targets to study the atmospheres of younggas giant planets and serve as empirical templates for discoveries with second-generationplanet-finding instruments like the Gemini Planet Imager, SPHERE, and SCExAO.<cit.> presented the discovery offaint companions comoving withthe pre-main sequence stars FW Tau AB, ROXs 42B, and ROXs 12 at projected separations between 100 to 400 AU. At the young ages of these systems (1–10 Myr), the implied masses of the companions span about 5–20 assuming hot-start cooling models. Follow-up spectroscopy by <cit.> showed that ROXs 42B b is a young early L dwarf with clear spectroscopic signs of low surface gravity. On the other hand, the companion to FW Tau exhibits a mostly featureless near-infrared pseudo-continuum spectrum with strong veilingand emission lines indicating ongoing accretion and outflow activity.The implication is that the faint companion to FW Tau may be a brown dwarf withan edge-on disk rather than a widely-separated planet, as suggested from photometry alone. Follow-up spectroscopic confirmation is evidently a critical step inconfirming the low temperatures and masses of young planet candidates found with direct imaging.Here we present near-infrared integral-field spectroscopy of the substellar companion to ROXs 12 (2MASS J16262803–2526477), an M0-type pre-main sequence star[Therehas been some confusion regarding the coordinates of ROXs 12 in the literature. <cit.> originally identified 47 “Rho Oph X-ray” (ROX) detections in observations with the Einstein Observatory X-ray telescope. ROX 12 is listed at α_J2000.0 = 16:26:24.8, δ_J2000.0 =–25:27:28 with a 40” positional uncertainty. In a follow-up study to identify the sources of the X-ray emission,<cit.> list ROXs 12 (“ROX star 12”), the candidate optical counterpart to the ROX 12 detection,as the star at α_J2000.0 = 16:26:28.0, δ_J2000.0 =–25:26:47(also known as 2MASS J16262803–2526477).Later, <cit.> used speckle imaging to identifya candidate companion to ROXs 12 ata position angle of 10.3^∘, a separation of 175, and a flux ratio of Δ K=5.7 mag, but the coordinates they list are for a different star at α_J2000.0 = 16:26:27.75, δ_J2000.0 =–25:27:24.7(also known as 2MASS J16262774–2527247). obtained followup adaptive optics imaging of ROXs 12 (2MASS J16263803–2526477); while constructing the target list, they used coordinates based on the 2MASS source closest to the position in <cit.>, observing the correct source. However, in the manuscript that confirmed comovement for the companion identified by Ratzka et al., they adopted the same (incorrect) coordinates for the star 37” south of ROXs 12 that were listed by Ratzka et al.Most of the observations we collected between 2011 and 2015 targeted the erroneous published coordinates for ROXs 12,but have nevertheless provided useful data for the third companion 2MASS J16262774–2527247.] near the boundary between the Ophiuchus and Upper Scorpius star-forming regions. ROXs 12 B[Here we adopt “B” rather than “b” as used in the discovery paper because the mass we infer for this companion is securely above the deuterium-burning limit.]was first identified as a candidate companion by <cit.> at 178 (240 AU in projected separation)and confirmed to be comoving with its host star by <cit.>. More recently, <cit.> obtained follow-up high-contrast imaging of this system and found that ROXs 12 B has undergonemeasurable orbital motion based on astrometry spanning about 15 years. As one of only a handful of young (<10 Myr) very low-mass companions at wide orbital distances beyond 100 AU, ROXs 12 B offers a valuable opportunity to study the atmosphere of a young object spanning thebrown dwarf-planetary mass boundary.In addition to characterizing the atmosphere of ROXs 12 B, we also show that the young disk-bearing star 2MASS J16262774–2527247 (hereinafter 2M1626–2527) located 37” south of ROXs 12 islikely a wide binary companion based on a consistent radial velocity and proper motion with ROXs 12. Figure <ref> shows an overview of the system. The combination of rotation periods from K2 together with projected rotational velocities from our high-resolution spectra allow us to assess the mutual inclinations of both stars. Combining these results with orbital inclination constraints for ROXs 12 B allows us to then examine whether the ROXs 12 system is in spin-orbit alignment, as expected for a planet forming in a disk. To place ROXs 12 B in context, we conclude with an overview of accreting subdisks around young planetary-mass companions and derive the frequency of accreting circumplanetary disks fromcompanions with existing moderate-resolution J-band spectra.lcccccccc 0pt 9 Spectroscopic ObservationsObjectDateTelescope/ Filter Slit WidthPlate ScaleTot. Exp. Resolution Standarda (UT) Instrument(”) (mas pix^-1)(min) (=λ/δλ)ROXs 12 B 2016 April 18+22 Gemini-North/NIFS J⋯ 100×401106000 HD 145127ROXs 12 B 2016 May 22 Keck/OSIRIS Kbb⋯ 50503800 HIP 69021ROXs 12 B 2016 May 22 Keck/OSIRIS Hbb⋯ 50 60 3800 HIP 93691ROXs 12 B 2016 May 22 Keck/OSIRIS Jbb⋯ 50903800 HIP 93691ROXs 12 A 2011 June 28Keck/HIRES KV4180.861 ⋯5 48000 ⋯ROXs 12 A 2016 July 26McDonald 2.7 m/IGRINS ⋯0.98 ⋯30 45000 HD 1553792M1626–2527 2011 Apr 29 IRTF/SpeX ⋯ 0.3 ⋯ 82000 HD 1449252M1626–2527 2014 May 21 Mayall/RC-Spec GG 495 1.5 ⋯ 202600 HZ 442M1626–2527 2015 May 03Keck/HIRES GG4750.861 ⋯10 48000 ⋯2M1626–25272016 July 26McDonald 2.7 m/IGRINS ⋯0.98 ⋯30 45000 HD 155379aTelluric or radial velocity standard.§ OBSERVATIONS§.§ Gemini North/Near-Infrared Integral Field Spectrometer J-Band Spectroscopy of ROXs 12 B We acquired moderate-resolution (R≡λ/δλ≈6000) J-band spectroscopy of ROXs 12 B spanning 1.15–1.36 μm with theNear-Infrared Integral Field Spectrometer (NIFS; ) at Gemini-North 8.1 m telescope on UT 2016 April 18 and 22 (Gemini Program ID: GN-2016A-Q-37). The facility AO system ALTAIR () provided diffraction-limitedcorrection using the Gemini laser guide star system with ROXs 12 as a bright (R = 13.5 mag) tip tilt reference. NIFS has a 3”×3” field of view with 01×004 rectangular spaxels. To better sample the PSF wing of the host star, we rotated the instrument so that theshort side of the spaxels was oriented in the direction of the ROXs 12 A and B position angle (P.A.) with the host star located immediately off the detector (see Figure <ref>). Our observations of ROXs 12 B were then carried out in an ABBA pattern by nodding 15 orthogonal to the binary P.A.We acquired a total of 40 min (eight exposures of 300 s each) with the J-band grating centered at 1.25 μmand ZJ filter on UT 2016 April 18 in queue mode.On UT 2016 April 22 an additional 70 min (14 exposures of 300 s each)were taken with the same configuration.On both nights the A0V standard HD 145127 was targeted at a similar airmass for telluric correction. Our observations are summarized in Table <ref>.Basic data reduction was carried out with NIFS reduction packages in IRAF which include flat fielding, bad pixel interpolation, sky subtraction, and image rectification to data cubes. ROXs 12 B is clearly visible in each cube but overlaps with the PSF wing from the host star. To remove this low-level contaminating flux, we performed PSF subtraction in the same fashion as in <cit.> for NIFS data of ROXs 42B b. For each column of each spectral channel, we masked out the companion and then fit and subtracted various parameterized PSF models to the 2D wing of ROXs 12 A using the Levenberg-Marquardt least-squares curve fitting package (). Gaussian, Lorentzian, and Moffat PSF models were tested to examine their influence on the extracted spectra. All three profiles resulted in similar spectra; ultimately we adopted the Gaussian model as this produced the lowest systematic over- or under-subtraction in the region surrounding ROXs 12 B (Figure <ref>). After PSF subtraction we extracted the spectra from each cube using aperture photometryand median-combined them after scaling individual spectra to their median values. Telluric correction wasperformed using theroutine in the Spextool data reduction package for IRTF/SpeX (; ). These steps were carried out separately for each night and the resulting spectra were subsequently combinedby calculating the weighted mean and uncertainty of both spectra. Note that standards were taken immediately before and after the science observations and the S/N levels were low for individual spectra, so we performed telluric correction on the coadded spectra from each night instead of carrying this out for each separate exposure. The final NIFS spectrum has an effective on-source integration time of 110 min and is shown in Figure <ref> after having been dereddened by A_V=1.8 mag— the extinction to ROXs 12 A measured by<cit.> based on moderate-resolution optical spectroscopy— following the extinction curve from <cit.>.§.§ Keck/OSIRIS Near-Infrared Spectroscopy of ROXs 12 B We targeted ROXs 12 B with the OH-Suppressing InfraRed Imaging Spectrograph (OSIRIS; ; ) using natural guide star adaptive optics (NSG AO; ) at the Keck I telescope on 2016 May 22 UT. Broad band filters were used with the 50 mas per spaxel plate scale, producing a 16×64 spaxel (08×32) rectangular field of view with a spectral resolution of R≈3800. The sky was clear with seeing between 05–08 throughout the night. To minimize contamination from the host star we oriented the long axis of the array to be orthogonal to the binary position angle. Our observations consisted of nodded ABBA sequences with about 1” offsets for pairwise sky subtraction. A total of 50 min of integration time was acquired in Kbb band (10 300-s exposures), 60 min in Hbb band (12 300-s exposures),and 90 min in J band (18 300-s exposures). Multiple A0V standards were targeted between filter changes at a similar airmass as the science observations (see Table <ref>).Basic data reduction was carried out using the OSIRIS data reduction pipeline.Images were flat-fielded, corrected for bad pixels, and assembled into data cubes using the latest rectification matrices provided by Keck Observatory.We then extracted the spectra from the data cubes using aperture photometry at each wavelength at the centroided location of the target in median-collapsed cubes. Spectra were median-combined after scaling them to their individual median levels. Telluric correction was then carried outwith theroutine in Spextool (; ). Each spectral band was then flux calibrated usingphotometry of ROXs 12 B (Figure <ref>), whichwas derived by transforming the 2MASS photometry of ROXs 12 A to the MKO system using relations from <cit.> and then using the relative photometry of ROXs 12 AB from <cit.>. The resulting photometry is reported in Table <ref>. §.§ Harlan J. Smith Telescope/IGRINS High-Resolution Near-Infrared Spectroscopy We obtained high-resolution (R=45000) 1.45–2.45 μm spectra ofROXs 12 A and 2M1626–2527 on UT 2016 July 26 with the Immersion Grating Infrared Spectrometer(IGRINS; ; ) at the McDonald Observatory's 2.7-m Harlan J. Smith Telescope to measureradial velocities for both components of this common proper motion pair in order to test whether they share common space velocities. Six exposures of 300 s each were acquired while nodding in an ABBA pattern along the slit, totaling 30 min of integration for each target. The A0V star HD 155379 was observed on the same night. Spectra were extracted and reduced using version 2.1 of the IGRINS reduction pipeline[https://github.com/igrins/plp] following the description in <cit.>. The S/N per pixel of both spectra range from 80–100. In summary, after bias subtraction and flat fielding, the spectra were optimallyextracted and cross correlated with over one hundred other IGRINS spectra of comparable spectral type taken over the past two years to measure radial velocities. The final radial velocities for ROXs 12 A and 2M1626–2527 are –6.09 ± 0.16 km s^-1and –6.03 ± 0.16 km s^-1, respectively, where the uncertainties are dominated by the transformation to the external reference frame (as opposed to relative uncertainties, which are 0.04 km s^-1).§.§ Mayall/Ritchey-Chretien Spectrograph Optical Spectroscopy of 2M1626–2527 We observed 2M1626–2527 with the Ritchey-Chretien Spectrograph (RC-Spec) usingthe T2KA CCD at the Kitt Peak National Observatory's 4-m Mayall telescope on UT 2014 May 21. The 15×98” slit was used with the BL420 grating resulting in a resolving power of R≈2600 spanning 6300–9200 Å. The slit was oriented in a fixed North-South direction throughout the night.We targeted 2M1626–2527 near transit at an airmass of 1.86 to minimize effects fromdifferential atmospheric refraction. However, because the slit was not oriented exactly at parallactic angle, some slit losses may occur which may have affected the slope of our RC-Spec spectrum.These data are therefore useful for spectral classification but not for detailed reddening measurements. A single exposure was acquired with the GG495 filter and a total integration time of 1200 s.The raw data was first bias-subtracted and flat-fielded to remove pixel-to-pixel sensitivity variations.Night sky lines were removed using median sky values in the spatial direction on either side of the spectrum. We then extracted the spectrum of the science target by summing flux in the spatial direction. The overall throughput response was then corrected using the spectrophotometric standard HZ 44. Finally, wavelength calibration was achieved using HeNeAr lamp observations taken throughout the night. The optical spectrum of 2M1626–2527 is discussed in more detail in Section <ref>. §.§ IRTF/SpeX Moderate-Resolution Near-Infrared Spectroscopy of 2M1626–2527 2M1626–2527 was targeted with the InfraRed Telescope Facility's SpeX instrument ()in short cross-dispersed (SXD) mode on UT 2011 April 29. We used the 03 slit rotated to the parallactic angle, which yielded a mean resolving power of ≈2000 across the 0.8–2.4 μm spectrum. Four nodded pairs were obtained with 60 s per exposure in an ABBA pattern. The A0V standard HD 144925 was observed prior to our science target. Standard data reduction including spectral extraction, wavelength calibration, and telluric correction was carried outusing the Spextool package (; ). The near-infrared spectrum of 2M1626–2527 is discussed in more detail in Section <ref>. §.§ Keck/HIRES High-Resolution Optical Spectroscopy We obtained high-resolution (R≈48000) optical spectra of ROXs 12 A and 2M1626–2527 with HIRES () at Keck Observatory on UT 2011 June 28 and UT 2015 May 3, respectively. Both spectra were optimized for long wavelengths with the red cross disperser.A single 300 s exposure of ROXs 12 A was taken with the C1 decker using the KV418 order blocking filter, resulting in a slit size of 70×0861on the sky and a wavelength range of 4310–8770 Å. For 2M1626–2527, we obtained one 600 s exposure spanning a wavelength range of 4800–9220 Åusing the C1 decker, the GG475 order blocking filter, and a slit size of 70×0861.Basic data reduction and spectral extraction were carried out following the description in <cit.>. The extraction pipeline MAKEE was used for bias subtraction, flat fielding, cosmic ray rejection, spectral extraction, andwavelength calibration. Zero-point corrections to the wavelength solution were derived by cross correlating the telluric band of the O7V star S Mon. Radial velocities and projected rotational velocities (vsini_*) were then measured for both targets using early-M dwarf RV standards from <cit.> after removingorders with strong telluric features or low S/N. A broadening function () was measured relative to these standards to find theabsolute RV and line broadening, which was then translated into projected rotational velocities using rotationally broadened template spectra to empirically correlate line broadening andprojected rotational velocities following <cit.>. Our final RV values from HIRES are –5.67 ± 0.12 km ^-1 for ROXs 12and –6.97 ± 0.15 km s^-1 for 2M1626–2527, comparable to the radial velocities we found from our IGIRNS spectra. The slight differences (<1 km s^-1) between the HIRES and IGRINS RVs are likely caused by jitter in these young and active stars. Our vsini_* measurements from this analysis are 8.2 ± 0.7 km s^-1 and 4.5 ± 1.0 km s^-1 for ROXs 12 and 2M1626–2527, respectively. lcccc Physical, Kinematic, and Photometric Properties of the ROXs 12 Triple SystemProperty ROXs 12 A ROXs 12 B2M1626–2527 Ref2MASS ID J16262803–2526477 ⋯ J16262774–25272471B (mag) 16.172 ± 0.141 ⋯ 17.1 ± 0.7 2V (mag) 14.346 ± 0.066 ⋯15.8 ± 0.3 2R (mag) 13.5 ⋯ 15.3 3r' (mag) 13.5 ⋯ 15.84I (mag) 11.870 ± 0.04 ⋯12.91 ± 03 5J_2MASS (mag) 10.282 ± 0.02415.82 ± 0.03a11.021 ± 0.024 1, 6H_2MASS (mag) 9.386± 0.02614.83 ± 0.03a9.930 ± 0.0261, 6K_s, 2MASS (mag)9.099 ± 0.025 14.14 ± 0.03a9.211 ± 0.025 1, 6 L'(mag) [9.1 ± 0.1]b 13.2 ± 0.1[9.2 ± 0.1]b 6 W1(mag) 8.805 ± 0.023 ⋯ 8.360 ± 0.023 7 W2(mag)8.712 ± 0.021⋯ 7.831 ± 0.021 7 W3(mag)8.393 ± 0.042⋯ 5.983 ± 0.017 7 W4(mag) 6.558 ± 0.089⋯ 3.784 ± 0.025 7 μ_αcosδ (mas yr^-1)–12.7 ± 2.3⋯–11.9 ± 2.3 8μ_δ(mas yr^-1)–29.0 ± 2.3⋯ –30.5 ± 2.3 8v_rad(km s^-1)c–6.09 ± 0.16⋯–6.03 ± 0.16 9v_rad(km s^-1)d–5.67 ± 0.12⋯–6.97 ± 0.15 9vsin i(km s^-1)d 8.2 ± 0.7⋯4.5 ± 1.0 9 log(L_bol/L_⊙)e–0.57 ± 0.06 –2.87 ± 0.06 –0.81 ± 0.06 9Mass (M_⊙) 0.65^+0.05_-0.090.0167 ± 0.0014 0.5^+0.1_-0.19T_eff (K) 3900 ± 100 3100^+400_-5003700 ± 1509P_rot (d) 9.1 ± 0.4 ⋯3.30 ± 0.059Age (Myr)6^+4_-2 6^+4_-28^+7_-4 Myr 9Spectral TypeM0 ± 0.5 L0 ± 2 M1 ± 1 9, 10 i_* (^∘)f77^+7_-9⋯ 17^+5_-4 9 R_* (R_⊙)g 1.14 ± 0.07⋯ 0.96 ± 0.08 9aJ and H band photometry for ROXs 12 B is on the MKO filter system.The Ks band photometry assumes K_s≈K' to convert the contrast measurement from <cit.> to an apparent magnitude. bEstimated L'-band magnitudes assuming K–L'=0.0 ± 0.1 mag for M0 and M1 spectral types (). cFrom our IGRINS high-resolution near-infrared spectra. dFrom our HIRES high-resolution optical spectra. eAssumes a distance of 137 ± 10 pc. fLine-of-sight stellar inclination. gRadius from bolometric luminosity and effective temperature. (1) 2MASS (); (2) APASS DR9 (); (3) USNO-B1.0 (); (4) CMC15 (); (5) DENIS; (6) <cit.>; (7) AllWISE Data Release ();(8) HSOY (), which includes data from Gaia DR1 (); (9) this work; (10)<cit.>. §.§ K2 Time Series Photometry ROXs 12 (EPIC 203640875) and 2MASS J16262774–2527247 (EPIC 203637940)were both observed with K2 (),the extended mission of the Kepler spacecraft, during Campaign 2 between2014 August 23 and 2014 November 10 (GO 2052, PI: K. Covey; GO 2063, PI: A. Kraus). The raw long cadence (30-min) photometry for ROXs 12 were corrected forsystematic features caused by pixel-to-pixel drift andspacecraft thruster firings using the reduction pipeline described in <cit.>. For 2MASS J16262774–2527247, we used the raw light curve instead of applyingcorrections because this star exhibits very large photometric variations— up to ∼50% in amplitude—which are far larger than systematics present in the data. Light curves for both targets are presented in Section <ref>. §.§ Keck/NIRC2 Aperture Masking Interferometry ROXs 12 was observed on UT 2011 April 24 usingNIRC2's 9-hole aperture mask together with natural guide star adaptive optics at Keck Observatoryas part of the multiplicity survey of Ophiuchus members by <cit.>. The observations consist of four 20 s images (5 s × 4 coadds) in the Kp filter using NIRC2's narrow camera mode. After basic reduction (bias subtraction, flat fielding, and bad pixel correction), we re-examined the images and found that the interferogram from ROXs 12 B is clearly visible in all four frames. This serendipitous detection opens the opportunity to search for a potential binary companion to ROXS 12 B at the Keck diffraction limit— analogous to brown dwarf-brown dwarf binary companions to stars likeHD 130948 BC () and ϵ Indi Bab (; )—albeit at modest flux ratios because of flux loss inherent to this technique and ROXs 12 B's intrinsic faintness.Figure <ref> displays the coadded image of ROXs 12 AB system after applying the distortion solution and north orientation from <cit.>. Using the aperture masking pipeline described in <cit.>,we did not find evidence for an additional companion to ROXs 12 B downto angular scales of about 30 mas.The corresponding 99%-level contrast curve in Kp band over which we can exclude companions is {0.08, 1.77, 2.02, 2.03, 1.97, 1.98} mag at {15, 30, 60, 120, 200, 280} mas. This corresponds to mass limits of ≈10 at about 01 based on the evolutionary models of <cit.>for an age of ≈6 Myr (see Section <ref>). It therefore appears that ROXs 12 B is single down to physical separations of ≈4 AU, which is well within its Hill radius of about 50 AU. Contrast limits for the host star from these same data are about 3 magnitudes deeper;these are shown in Figure <ref> and presented in <cit.>. § RESULTSlcc 0pt 3 ROXs 12 B NIFS J-Band Equivalent Widths λ_0a LineEW (μm) ID(Å) 1.1693K11.10 ±0.09 1.1773 K12.60 ± 0.10 1.1887 Fe1 0.89± 0.04 1.1976 Fe1 0.73± 0.05 1.2436 K1 0.88 ±0.04 1.2526 K11.26 ± 0.051.2903Mn1 0.42 ± 0.12 1.3127Al10.54 ±0.06 1.3154 Al1 0.47 ±0.03 aNominal air wavelength.lcccccc 0pt 5 K1 Equivalent Widths of Young Planetary-Mass CompanionsObject SpTK1 1.169 μm K1 1.177 μmK1 1.243 μmK1 1.253 μm ReferenceEW (Å)EW (Å)EW (Å)EW (Å)GSC 6214-210 B M9.5 ± 1 ⋯ ⋯ 3.0 ±0.5 3.2 ± 0.2 1 ROXs 12 B L0 ± 21.10 ± 0.09 2.60 ± 0.100.88 ±0.04 1.26 ± 0.05 2 ROXs42B bL1 ± 1 2.8 ± 0.5 3.12 ± 0.19 1.9 ±0.3 1.76 ± 0.16 1 2M0441+2301 Bb L1 ± 1⋯ ⋯ 5.4 ±0.6 4.0 ± 0.63 1RXS J1609–2105 B L2 ± 1⋯⋯ 5.33 ±1.0 3.8 ± 0.4 4, 5, 6 (1) <cit.>; (2) This work; (3) <cit.>; (4) ; (5) ; (6) .§.§ Spectral Properties of ROXs 12 B Our moderate-resolution spectra of ROXs 12 B from Gemini-N/NIFS (J band) andKeck/OSIRIS (J, H, and K bands) are shown inFigures <ref> and <ref>. The NIFS spectrum shows prominent atomic and molecular absorption features fromK1, Fe1, Mn1, Al1, H_2O, and FeH, as well as hints of broad VO absorption at ≈1.2 μm.Notably absent is Paβ emission at 1.282 μm, a signature of active accretion from a circumsubstellar disk; such emission has been detected in several other young brown dwarf companionsover the past few years (e.g., ; ).We compare our NIFS spectrum to brown dwarfs from the Brown Dwarf Spectroscopic Survey (BDSS; ) in Figure <ref>. Relative to field objects, ROXs 12 B exhibits shallower K1 doublets at 1.169/1.178 μm and 1.244/1.253 μm as well as diminished FeH molecular absorption features at ≈1.195–1.205 and 1.239 μm—all signs of low surface gravity (e.g., ; ; ; ). The pseudocontinuum is a good match to the L2 template spectrum. Equivalent widths of absorption lines are reported in Table <ref>.Our OSIRIS spectrum broadly resembles a late-M and early-L dwarf with deep H_2O bands at ≈1.4 and ≈1.9 μm, FeH, ^12CO, Na1, andweak K1.The triangular H band shape is typical of low-gravity brown dwarfs and giant planets (e.g., ; ; ), offering independent evidence that the system is young, although note thatthe triangular H-band shape does not strictly track low surface gravity ().In Figure <ref> we compare our OSIRIS and NIFS spectra to “very low gravity” (“VL-G”) templates from <cit.>. The full 1.2–2.4 μm spectrum is most similar to the young L0 template, but individually each bandpass shows inconsistent matches: the J band resembles L1–L2 objects, the H band is closest to the M8 template, and the K band appears similar to M8–M9 objects.Altogether we adopt a near-infrared spectral type of L0 ± 2 for ROXs 12 B.Note thatthe OSIRIS broadband filterbandpasses do not cover spectral regions used in the <cit.> classification system so weadopt visual-based classifications for ROXs 12 B.Equivalent widths of the J-band potassium doublets from our NIFS spectrumare shown relative to ultracool objects in Figure <ref>. High-gravity field objects possess deep lines which trace out an upper envelope of equivalent widths, whereas young objects exhibit weaker alkali features as a result of their low-pressure atmospheres. Compared to low-mass stars and brown dwarfs spanning a range of ages and surface gravities, ROXs 12 B has shallow potassium lines similar to very low-gravity objectsfrom <cit.> and <cit.> as well as young (≲10 Myr) companions near and below the deuterium burning limit, as measured from published spectra (Table <ref>). §.§ Physical Properties of ROXs 12 B We implement two methods to assess the effective temperature and surface gravity of ROXs 12 B using atmospheric models.The first approach is based on maximum likelihood fitting of the BT-Settl models (“CIFIST2011bc” version; ) following the general description in <cit.>. In summary, the grid of models are first Gaussian smoothed to the resolving power of the OSIRIS spectrograph (R≈3800) and resampled ontothe same wavelength grid as the data.A reduced chi squared value (χ^2_ν)is calculated for each synthetic spectrum in the grid spanning = 1100–4000 K (Δ= 100 K) and log g = 2.5–5.5 dex [cgs units throughout this work] (Δlog g = 0.5 dex).Similarly, a corresponding radius is derived for each model by making use of the factor that scales the emergent model spectrum to the observed flux-calibrated spectrum, as this quantity is also equal to the ratio ofthe object's radius to its distance, squared.Here we adopt a distance of 137 ± 10 pc, where the mean valuereflects the distance measured to the dark cloudLynds 1688 by <cit.> and the uncertainty is reflects theuncertainty in the membership of this system to the Ophiuchus versus Upper Sco star-forming regions (see Section <ref>).In addition to fitting the entire 1.2–2.4 μm spectrum, we also fit the individual J, H, and K bands using this method.Results are shown in Figure <ref>. The models qualitatively match the data quite well but produce different quantitative resultsdepending on the bandpass, with effective temperatures ranging from 1600 K to 2800 K. The bottom panel shows the entire 1.2–2.4 μm spectrum; the best model does a poor job of reproducing the data by underestimating the flux at shorter wavelengths and over-predicting it in K band. Despite the generally good agreement of the models with the individual bands, the inferred gravities and effective temperatures disagree depending on the spectral region beingconsidered. Two regions of local minima are clearly visible for the effective temperature, one at ≈1500–1700 K and one at ≈2600–2900 K. For the entire spectrum, the best fitting model is visually a poor match to the data despite formally producing the lowest χ^2_ν value. Similarly, the inferred radii span 1.7–5.1 R_Jupdepending on the spectral bandpass used in the fits. Only the H and K bands are consistent with expectations of ≈2 R_Jupfrom hot-start evolutionary models.The implied masses based on surface gravity and radii are also unphysical, ranging from 3.8 for H band to 3300 for the entire spectral fit. These discrepancies make it difficult to interpret the results;systematic errors in the models mean that the data are not standard deviates about the model and therefore the χ^2_ν statistic does not correspond to a maximum likelihood. Instead, these fits are more sensitive to the overall shape of the pseudocontinuum rather than individual absorption features which may be more useful for constraining surface gravity and effective temperature. For this reason we explored another approach based on maximizing the cross correlation between models and data. This method better incorporates information from atomic and molecular lines, which may be more sensitive to temperature and surface gravity, compared with the χ^2 treatment. Cross correlationencodes line positions and line ratios but is mostly insensitive to uncertainties in the calibration of the continuum as well as broad-band spectral variations (e.g., ).The same set of models described above was convolved with a Gaussian kernel to a resolution of R = 10,000. This is high enough to provide a high-enough resolution template for both the OSIRIS and the NIFS spectra, but low enough to avoid interpolation errors. Prior to cross-correlation, a high-pass filterwas applied to both the data and the models to remove broad-band variations. The models were then Doppler-shifted to radial velocities between -900 and +900 km s^-1 (in steps of 30 km s^-1), linearly interpolated to the spectral range of the data, and cross-correlated with our observed spectra. Each spectral channel in the data was weighted by the inverse of its relative error to prevent noisy sections of the spectra from dominating the analysis.The cross-correlation functions (CCFs) obtained with the full set of model spectra are shown in Figure <ref>. Not only does the S/N vary between different bands, but also the peak of the cross-correlation seems to occur at different positions in the spectral sequence, confirming the previous observation that different bands could lead to different best-fitting parameters for the atmosphere.To formally compute confidence intervals on the model parameters, we incorporated the values of the CCFs into a likelihood function L following the approach of <cit.>. Since in this case we are combining four different bands, we adopt the following effective cross-correlation function: C̅^2 = 1 - {∏_i[1- C_i^2] }^1/4 where the product is performed over the cross-correlation function of the four available spectral bands. The effective cross-correlation function and the number of spectral channels N are used to compute a log-likelihood function: log(L) = -N/2log[1-C̅^2] and the corresponding likelihood values L are used to derive confidence intervals (CIs). The bottom-left panel in Figure <ref> shows the two-dimensional map of log(L) (blue shades) and the corresponding 1-, 2-, and 3-σ CIs are labeled. The top-left and bottom-right panels show the projected probability distributions for log(g) and T_eff, respectively. The resulting effective temperature of ROXs 12 B is T_eff = 3100^+400_-500 K, and the bi-modality observed with the previous model comparison is no longer present. However, only an upper limit of log(g) < 4.0 is obtained for the surface gravity. Although this result points to a low-gravity object, the likelihood also rises at high values of log(g). The inferred effective temperature from our cross correlation analysisis warmer than expected for a young L0 object, although the distribution is broad and encompasses the range of anticipated values.For example, using the effective temperature-spectral type relation for young objects from <cit.> yields 2260 ± 60 K, assuming no error on the spectral type.If instead we adopt an uncertainty of 2 subtypes, the median effective temperature of the resulting distribution is 2160 K with a 1-σ highest posterior density range[Also known as the minimum credible interval, this is the region containing the highest values of a posterior distribution and may comprise several intervals for multi-modal distributions. ]of 1810–2770 K.We determine the luminosity of ROXs 12 B using our flux-calibrated OSIRIS spectrum together with bolometric corrections using BT-Settl model synthetic spectra at shorter (λ<1.2 μm) andlonger (λ>2.4 μm) wavelengths. The synthetic spectrum is scaled to the short- and long-wavelength endpoints of our spectrum and then integrated to derive a bolometric flux. Because the results from atmospheric model fits are either inconclusive (for the χ^2 analysis) or have large uncertainties (from the CCF analysis), we utilized the =2300 K model with log g= 4.0 dex based on relations from<cit.> for young L0 objects—although the resulting luminosities were only weakly sensitive to the input model. The bolometric luminosity of ROXs 12 B is log L_bol/L_⊙ = –2.87 ± 0.06 dex, which takes into account uncertainties in individual spectral measurements, the flux calibration scale factor, and the estimated distance to the system. In comparison, using a 3100 K model produces similar bolometric luminosity of–2.82 ± 0.06 dex.Based on the bolometric luminosity of the companion and the systemage (6^+3_-2 Myr; Section <ref>), the inferred mass of ROXs 12 B is 17.5 ± 1.5 from <cit.> hot-start evolutionary models. This mass is consistent with the value of 16 ± 4 found by <cit.>. §.§ 2M1626–2527: A Wide Tertiary Companion 2M1626–2527s appear to constitute a wide tertiary to ROXs 12 ABseparated by 37”, or5100 AU at a distance of about 140 pc. The proper motions of this pair from HSOY ()are consistent with one another: μ_αcosδ = –12.7 ± 2.3 mas yr^-1 and μ_δ = –29.0 ± 2.3mas yr^-1 for ROXs 12, andμ_αcosδ = –11.9 ± 2.3 mas yr^-1 and μ_δ =–30.5 ± 2.3mas yr^-1 for 2M1626–2527. Similarly, we measureconsistent radial velocities for both components from our IGRINS spectra: –6.09 ± 0.16 km s^-1 for ROXs 12 and –6.03 ± 0.16 km s^-1 for 2M1626–2527. This is far less than thevelocity dispersion of ≈ 1 km s^-1 for ρ Ophiuchuscluster members (), offering further evidence that these two stars are not simply members of the same cluster, but arelikely gravitationally bound. The identical radial velocities also suggest that both stars are probably single. Indeed, deep adaptive optics imaging by <cit.> did not reveal any additional members of this system. The kinematic, photometric, and physical properties of this triple system comprising ROXs 12 A, ROXs 12 B, and 2M1626–2527 are listed in Table <ref>. §.§ Disk Properties and Accretion Figure <ref> shows the spectral energy distributions ofROXs 12 A, ROXs 12 B, and 2M1626–2527 based on optical throughinfrared photometry in Table <ref>. Flux density measurements for all three targets are dereddened byA_V=1.8 mag. lcc 0pt 3 Emission Line Equivalent Widths λ_0a LineEW (Å) ID(Å) ROXs 12 A: Keck/HIRES 6562.8Hα –1.41 ± 0.02 8498.0Ca2 –0.40 ± 0.01 8542.1Ca2–0.60 ± 0.01 8662.1Ca2 –0.42 ± 0.02 2M1626–2527: Keck/HIRES 4861.3 Hβ–26.2 ± 0.24921.9 He1–0.53 ± 0.03 5015.7 He1 –0.51 ± 0.03 5875.6 He1–2.54 ± 0.036562.8Hα –53.17 ± 0.13 6678.2He1–1.12 ± 0.028446bO1 –0.47 ± 0.04 8498.0Ca2 –1.1 ± 0.01 8542.1Ca2–1.29 ± 0.02 8662.1Ca2 –0.82 ± 0.01 2M1626–2527: Mayall/RC-Spec6300.3 [O1]–0.9 ± 0.36562.8Hα –46.5 ± 0.5 6678.2 He1 –1.33 ± 0.127002bO1–0.4 ± 0.2 7065.2 He1 –0.9 ± 0.3 8446bO1 –1.09 ± 0.05 8498.0Ca2–2.00 ± 0.058542.1Ca2–1.91 ± 0.04 8662.1 Ca2–1.46 ± 0.042M1626–2527: IRTF/SpeX 8446bO1–0.7 ± 0.3 8498.0Ca2–1.9 ± 0.38542.1Ca2–1.6 ± 0.2 8662.1Ca2 –0.8 ± 0.2 9546.2H1 Pa8 (ϵ)–0.9 ± 0.310049.8 H1 Pa7 (δ)–1.5 ± 0.2 10830.3He1 –1.1 ± 0.2 10938.2H1 Pa6 (γ)–1.7 ± 0.2128218.1 Pa β –1.85 ± 0.13 21661.2 Br γ –1.5 ± 0.2 aNominal air wavelength. bLines are blended in our spectrum.Solar metallicity BT-Settl model atmospheres from <cit.> are overplottedin Figure <ref>. Spectral types are converted to effective temperatures usingempirical calibrations for pre-main sequence stars. ROXs 12 A has a spectral type of M0.0 ± 0.5 (; ), which corresponds toan effective temperature of 3900^+60_-90 K on the <cit.> scale and 3770^+100_-70 K on the <cit.> scale; we adopt an effective temperature of 3900 ± 100 K for ROXs 12 A here. Without a large sample of radius measurements for young stars, it is difficult to assess the size of potential systematic errors in these spectral type-effective temperature scales.However, we note that even systematic errors as largeas 300 K for ultracool dwarfs (e.g., ) do not substantially effect the broad results or conclusions regarding the luminosities, radii, or stellar inclinations throughout this paper.The agreement of the model atmosphere with the dereddened photometry of ROXs 12 A is generally quite good spanning the optical to mid-IR wavelengths, but the SED clearly shows a large excess at 22 μm, presumably from disk emission originating from the host star ROXs 12 A or, more unlikely, a warm disk surrounding the companion ROXs 12 B that is only prominent at 22 μm. If this excess emission originates from ROXs 12 A and peaks at ∼22 μm then the dust temperature is ∼230 K and is located at ≈0.7 AU following the heuristic relations from <cit.>. If it originates from ROXs 12 B then it must be located much closer in at ∼0.05 AU. For ROXs 12 B, dereddened J, H, Kp, and L' photometry from <cit.>are in good agreement with the 2300 K BT-Settl model, which was chosen based on spectral type-effective temperature relations for young brown dwarfs (see Section <ref>). There is no evidence of excess emission from ROXs 12 B that would indicate it hosts a circumsubstellar disk.This contrasts with many other young brown dwarfand planetary-mass companions that harbor accretionsubdisks as evidenced by optical/NIR emission lines and thermal infrared excess(e.g., ; ; ).No spectral type has been published for 2M1626–2527.Our ownlow-resolution optical spectrum shown in Figure <ref> reveals a late-type spectrum with clear signs of active accretion including strong Hα emission (EW=–46 Å) as well as various transitions from He1, [O1],and Ca2, all in emission (Table <ref>). Aside fromHα emission, the optical spectrum of 2M1626–2527 is relatively featureless shortward of ≈7000 Å and appears to be heavily veiled. This likely originates from the boundary layer of an accretion disk (e.g., ; ). The near-infrared spectrum of 2M1626–2527 shown in Figure <ref>also points to a late (M-type) spectrum with strong emission lines from Ca2, He1, and H1(the Paschen series and Br γ; see Table <ref>).The relative depths of the TiO bandheads spanning 7050–7150 Åare highly sensitive to temperature and canbe used as an indicator of spectral type for 2M1626–2527 despite the heavy veiling at shorter wavelengths. Figure <ref> shows a detailed view of this region compared toK4–M4 templates from <cit.> for the M dwarfs and <cit.> for the K dwarfs. The relatively pronounced TiO bands in 2M1626–2527 are stronger than K4–M0 templates and weaker than M2–M4 templates, with M1 being the best match.We therefore assign 2M1626–2527 a spectral type of M1 with an uncertainty of one subclass to reflect the imperfect agreement with this field template. The corresponding effective temperature is 3720^+180_-160 K and 3630 ± 140 K using the<cit.> and<cit.> conversions, respectively. We adopt T_eff = 3700 ± 150 K for 2M1626–2527. A model effective temperature of 3700 K and reddening of A_V=1.8 mag is a good fit to the photometry of 2M1626–2527 in the optical and near-infrared, but beyond about 3 μm there is clearly very strong excess emission from a protoplanetary disk (Figure <ref>).A few selected regions of our HIRES spectra of ROXs 12 A and 2M1626–2527 are shown in Figure <ref>. ROXs 12 A shows relatively weak Hα emission (EW=–1.41 ± 0.02 Å)considering its young age.This is similar to the value of 1.2 Å found by <cit.> and –2.00 ± 0.02 Å from <cit.>. These values are consistent with the lower envelope of Hα emission for pre-main sequence M0 starsspanning the youngest star-forming regions like Taurus () tosomewhat older populations like Upper Scorpius (). 2M1626–2527 shows strong, broadened Hα emission (EW=–53.17 ± 0.13 Å) in our HIRES spectrum,comparable to our measurement with RC-Spec (EW=–46.5 ± 0.5 Å). Both stars show Li1 λ6708 Å absorption; we measure an equivalent width of0.54 ± 0.01 Å for ROXs 12 A— in good agreement with the value of 0.52 ± 0.02 Å from <cit.>— and EW(Li) = 0.32 ± 0.02 Å for 2M1626–2527. The Ca2 infrared triplet is also seen in emission for both sources, with ROXs 12 A exhibiting broad absorption with superimposed central cores in emission.The Ca2 line profiles from 2M1626–2527 show a broad, blueshifted component in emission,offset from the central narrow line emission. This suggests that we are viewing a face-on accretion disk producing a broadened component along our line of sight together with a narrow zero-velocity peak where material hits the star.This geometry is consistent with the mostly pole-on stellar inclination we find for 2M1626–2527 in Section <ref>. §.§ K2 Light Curves The K2 light curves for ROXs 12 A and 2M1626–2527 are shown in Figure <ref>. ROXs 12 A exhibits remarkably strong periodic changes with peak-to-peak semi-amplitudevariations of ≈14%.The most straightforward interpretation of these modulations is that they are caused by large, long-lived regions of inhomogeneous spot coverage coming in and out of view as the star rotates.A Lomb-Scargle (LS) periodogram (; ; )reveals a strong peak period of 9.0998 days, which we adopt as the rotation period of ROXs 12 A.This is near the upper envelope of (but consistent with)rotation period measurements of low-mass pre-main sequence stars during thefirst ≈1–10 Myr of their evolution(e.g., ; ; ). Several flaring events are clearly visible over the course of these observations.The K2 light curve of ROXs 12 A phased to 9.100 d (Figure <ref>) exhibits only slight changes in the shape of the light curve from period to period, indicating minimal evolution of starspots on timescales less than one period.The light curve of 2M1626–2527 shows both low-amplitude (few percent) andhigh-amplitude (≈40%) variations on timescales ranging from hours to days (Figure <ref>). This stochastic behavior is characteristic of young accreting stars and is thought to be caused bytime-dependent mass accretion events that produce transient hot spots on the stellar photosphere (e.g., ; ). <cit.> find this phenomenon is fairly common, occurring among ≈9% ofρ Oph and Upper Sco members with strong infrared excesses. Although there are no obvious signs of periodicity in the light curve,the LS periodogram shows a significant peak power at 3.297 days. To explore whether this represents a real underlying modulation superimposed on stochastic variations, we applied the same periodogram analysis to smaller portions of the light curve which should return the same approximate peak period near 3.3 days if the modulations are authentic and consistent over time. The inset plot in the lower panel of Figure <ref> shows the peak period, second-highest peak, and third-highest peaks starting with 20% of the light curve and progressively including more data in 10% bins.Beginning with 30% of the data, the peak periods range between 3.28 and 3.37 d. This suggests the underlying modulations are real and are probably caused by rotationally modulated starspots. Interestingly, this means that 2M1626–2527 is rotating about three times faster than ROXs 12 A despite having an accreting disk, which has been shown on average to slow the rotation rates of young stars via star-disk interactions (e.g., ; ). This is unusual but not entirely unexpected given the broad spread inperiods for diskless and disk-bearing young stars.We estimate uncertainties for our light curve periods following <cit.> and <cit.>. For a single signal with Gaussian noise, they found that the error in the frequency measurement ν isδν = 3σ/(4√(N)TA), where σ is the standard deviation of the noise about the signal, N is the number of data points, T is the period, and A is the signal semi-amplitude.From this theperiod (P) uncertainty is δP = P^2δν. We fit a sine curve to the phased data to find A from the K2 light curves. The rms noise was approximated by computing the standard deviation of the residuals between the time series photometry and a Gaussian smoothed version of the same data. From this analysis we find formal uncertainties of 0.009 d for ROXS 12 and 0.002 d for 2M1626–2527. The high precision of these errors is primarily a result of the large number of periods sampled during the observing window (9–24 full cycles), the large number of data points in these light curves (≈3400), andthe relatively high semi-amplitudes for both systems (10–14%).Stars exhibit differential rotation between their poles and equator which can make it difficult to infer the equatorial rotation period from light curves alone if the location of their starspots is unknown. For example, the Sun has a longer rotation period of about 34 d at the poles compared to 25 d at its equator. Our period measurements for ROXs 12 A and 2M1626–2527 may therefore be over-estimating the equatorial rotation periods of these objects depending on which latitudinal regions the starspots were located. <cit.> show that thepole-equatorial shear(ΔΩ = 2π(1/P_min – 1/P_max) ) of Kepler stars with comparable effective temperatures to ROXs 12 A and 2M1626–2527 ranges from about 0.01 to 0.10 rad/d, with a typical value of about 0.03 rad/d. This typical shear would imply a range of maximum and minimum periods of8.7–9.5 d for ROXs 12 A and 3.25–3.35 d for 2M1626–2527. To take into account possible effects of differential rotation, we therefore adopt larger rotation period uncertainties of 9.1 ± 0.4 d for ROXs 12 and 3.30 ± 0.05 d for 2M1626–2527 when using these values in the context of equatorial rotation period estimates.§.§ Stellar Luminosities, Masses, and Ages Bolometric luminosities for ROXs 12 A and 2M1626–2527 are derived using the 3900 K and 3700 K BT-Settl models, respectively, shown in Figure <ref>. The models are first scaled to the H-band de-reddened flux and then integratedfrom 0.2–200 μm to find the bolometric flux. A distance of 137 pc is assumed based on VLBA parallax measurements of Ophiuchus members in the Lynds 1688 dark cloud (), and we adopt a ± 10 pc uncertainty to encompass ambiguity in Ophiuchus versus Upper Sco membership. Uncertainties are derived by integrating models 100 K warmer and 100 K cooler than the nominal stellar effective temperatures and calculating the mean difference between these and the stars' bolometric fluxes. Surface gravities of log g = 4.0 dex are chosen based on expectations from evolutionary models (e.g., ), as substantial deviations from this are unphysical. This yields luminosities of log L_bol/L_⊙ = –0.57 ± 0.06 dex for ROXs 12 A and –0.81 ± 0.06 dex for 2M1626–2527.Stellar masses and ages are estimated using <cit.> evolutionary models. For a given effective temperature and luminosity, we identify the corresponding mass and age by finely interpolating the grid of evolutionary models (Figure <ref>).This process is repeated in a Monte Carlo fashion to build a distribution of masses and ages assuming normally-distributed errors inT_eff and log(L_bol/L_⊙). We infer a mass of 0.65^+0.05_-0.09and an age of 6^+4_-2 Myr for ROXs 12 A. For 2M1626–2527 we find a mass of 0.50^+0.10_-0.10and an age of 8^+7_-4 Myr.§ DISCUSSION §.§ Ophiuchus or Upper Scorpius? The question of whether this system belongs to the younger Ophiuchus association or the older, more expansive Upper Sco region hasimplications for the inferred mass ofthe substellar companion ROXs 12 B. ROXs 12 has historically been regarded as a member ofthe ≈0.5–2 Myr Ophiuchus star-forming region since its identification by <cit.>. This region is centered on the dark cloud Lynds 1688 and its >300 members comprisea range of evolutionary stages spanning embedded protostars, accreting T Tauri stars, transition disks, and an older surface population that merges with the≈5–10 Myr Upper Scorpius subgroup (). ROXs 12 and 2M1626–2527 are located about 0.4^∘ from the L1688 cloud core in (or behind) an extended region of modest extinction (A_V = 1–2 mag; ).Unfortunately,because the internal velocity dispersions of cluster members are ≈1.0 km s^-1 (≈1.5 mas yr^-1; ; ) the kinematics of Oph and Upper Sco are nearly indistinguishableusing current proper-motion surveys. <cit.> finds a mean proper motion of μ_αcosδ = –10 ± 1.5 mas yr^-1 and μ_δ = –27 ± 1.5 mas yr^-1 (1 mas yr^-1 systematic error) forOph cloud members. Based on the UVW space velocities of Upper Sco from <cit.>, the expected proper motion of an Upper Sco member at the sky position of ROXs 12 isμ_αcosδ = –11.9 ± 1.5 mas yr^-1 and μ_δ = –24.4 ± 1.5 mas yr^-1. (Note that because of the large angular extent of Upper Sco on the sky, the proper motions of its members vary slightly across this region so we use the association's three-dimensional space velocity at the sky position of ROXs 12to calculate the expected proper motion.) The measured proper motions of ROXs 12 A and 2M1626–2527 (Table <ref>)agree with both of these regions to within about 2 σ. Unsurprisingly, we find Sco-Oph membership probabilities of 99% for both starsfollowing the Bayesian membership analysis described in <cit.> and <cit.>.Regardless of kinematics, the ages inferred for ROXs 12 A and 2M1626–2527 from theirpositions on the HR diagram are most consistent with Upper Sco. <cit.> adopt a median age of 10 ± 3 Myr for Upper Sco with a large intrinsic age spread of ±7 Myr, which is consistent with recent findings by <cit.>.Pecaut et al. also find evidence of an age gradient in this subgroup, with ROXs 12 positioned near the transition point between ≈5–9 Myr ages in the northwest and ≈11–15 Myr ages in the south-east. Our independently-inferred ages for ROXs 12 A (6^+4_-2 Myr) and 2M1626–2527 (8^+7_-4 Myr) from Section <ref> (also see Figure <ref>) appear toagree with typical ages of Upper Sco members in the vicinity of the Ophiuchus cloud. In the near future, this membership analysis and age will be refined with parallactic distances and precise proper motions from Gaia. §.§ Stellar Radii and Line-of-Sight InclinationsThe radii of ROXs 12 A and 2M1626–2527 can be determined with theStefan-Boltzmann law using their effective temperatures and bolometric luminosities. This yields values of 1.14 ± 0.09 R_⊙ for ROXs 12 A and0.96 ± 0.10 R_⊙ for 2M1626–2527, where uncertainties are propagated analytically.A lower limit on the radius of ROXs 12 A can also be inferred from the measuredrotation period (P_rot) and projected rotational velocity(v_p = vsini_*). By equating v and 2πR_*/P_rot,the stellar radius R_* and associated uncertainty σ_R_* are R_* (i_*) = P_rot v_p/2 πsin i_*σ_R_*≈ R_* ( ( σ_P_rot/P_rot)^2 +( σ_v_p/v_p)^2 + ( σ_i_*cos i_*/sin i_*)^2)^1/2. In cases where the inclination is unknown, Equation <ref> becomes a lower limiton R_* if an edge-on orbit is assumed (i_*=90^∘). Adopting P_rot = 9.1 ± 0.4 d and v_p = 8.2 ± 0.7 km s^-1 for ROXs 12 A implies R_* > 1.47 ± 0.14 R_⊙. There is tension between this radius and the radius implied from the Stefan-Boltzmann law at the 2.0 σ level.The most likely origin of this discrepancy is (1) that the inferred effective temperature for ROXs 12 A is too high,(2) our vsini_* measurement is too high,and/or (3) the rotation period samples high-latitude rather than equatorial starspots.The line-of-sight inclination of ROXs 12 A can be determined more preciselyusing joint constraints from the “spectroscopic” (Stefan-Boltzmann) radius and the “rotational” (R_*(i_*)) radius.Equating these two radii and solving for the stellar inclination givesi_* = sin^-1( √(%s/%s)σ_SBπL_bolv_p P_rot T_eff^2),where σ_SB is the Stefan-Boltzmann constant. Uncertainties are incorporated in an Monte Carlo fashion to build a line-of-sight inclination distribution, which peaks at 77^∘ and is truncated at 90^∘ (Figure <ref>). Monte Carlo realizations that result in sin i_* values >1 are excluded. The mode and 68.3% (1-sigma equivalent) highest posterior density interval is 77^+7_-9^∘. The 95.4% (2-sigma equivalent) range spans 60–88^∘.Note that these line-of-sight stellar inclinations aresymmetric about 90^∘ and produce mirrored observational signatures at higher inclinations.Similarly, using P_rot = 3.30 ± 0.05 d and v_p = 4.5 ± 1.0 km s^-1 for 2M1626–2527 implies R_* > 0.29 ± 0.07 R_⊙.This agrees with the spectroscopic radius of 0.96 ± 0.10 R_⊙. The joint constraint on the inclination distribution for 2M1626–2527 is shown in Figure <ref>. The mode and 68.3% highest posterior density interval is 17^+5_-4^∘, andthe 95.4% range spans 9–27^∘.The difference between these two inclination distributions, Δ i_*, is the degree to which these two stars are misaligned. For ROXs 12 A and 2M1626–2527 we find Δ i_* = 60^+7_-11^∘, indicating these stars are strongly misaligned.This is notsurprising for two young stars separated by over 5000 AU, and it adds tomounting evidence that misalignments between the components of wide binaries is a common phenomenon(e.g., ; ; ). This indicates that either the initial fragmenting cloud core that produced this system did not act as a uniformly co-rotating collapsing body, that this wide companion was captured, or thatthe rotational inclinations of these systems to cause them to evolve with time.§.§ Obliquity of ROXs 12 ABROXs 12 B has a projectedseparation of 240 AU, which corresponds to an orbital period of4600 yr and angular orbital motion of 0.08^∘ yr^-1 assuming a circular face-on orbit. Because of the longbaseline since the initial discovery epoch of ROXs 12 B by<cit.> in 2001, enough time elapsed for <cit.> and <cit.> todetect small but significant orbital motion from this companion. Despite the small amount of orbital coverage, <cit.> were able to constrain the orbital elements of ROXs 12 B using the efficient rejection sampling algorithm for Keplerian orbits described in <cit.>.Of particular interest for this study is the posterior distribution of the orbital inclination, i_p, which peaks at about 135^∘ with a broad tail extending from about 90^∘ to 180^∘, as shown in Figure <ref>.The inclination of the stellar rotation axis of ROXs 12 A, i_*, can be compared with i_p to offer clues about the stellar obliquity— the relative orientation of the stellar spin axis andorbital angular momentum vector. The true (de-projected) angle between the stellar spin and planetary orbital axes, ψ,is related to the sky-projected spin-orbit angle, λ, as follows: ψ = cos^-1( cosi_*cosi_p + sini_*sini_pcosλ). Diagrams showing the geometric configuration can be found in, e.g., <cit.> and <cit.>. The three angles λ, i_*, and i_p must be known to determine ψ. The projected obliquity (λ) is regularly measured for large transiting planetsusing the Rossiter-McLaughlin effect (e.g., ; ; ). In those cases, i_p≈90^∘, socosψ≈sini_*cosλ and therefore the projected obliquity (λ)is a measurement of lower bound on the true obliquity (ψ) assuming i_* is unknown. Similarly, if λ is unknown as is the case for ROXs 12 AB, the lower limit on ψ can be determined from the absolute difference between i_p and i_*: ψ ≥ |i_* - i_p| ≡ Δ i (see Appendix). Therefore, a system can have spin-orbit misalignment if Δ i is zero, but a non-zero value of Δ i means the system must be misaligned by at least that amount (). In this sense, knowing both i_p and i_* offers comparable information about ψ as do measurements of the Rossiter-McLaughlin effect.The probability distribution functions for i_*, i_p, and Δ i for ROXs 12 A and B are shown in Figure <ref>. Note that since i_* is symmetric about 90^∘ it has been mirrored from that of Figure <ref> and is therefore bimodal. The distribution of Δ i values peaks at 49^∘ with a 1-σ minimum confidence interval of17–69^∘.From this, we calculate that the probability that Δ i is greater than 10^∘ is 94%. This distribution represents the minimum values of true obliquity angles in this system and indicates that ROXs 12 A and B are likely to have spin-orbit misalignment, although alignment (Δ i=0^∘)cannot be ruled out from the observations.§.§ How Common is Paβ Emission? The presence of disks around widely-separated brown dwarf and planetary-mass companions offers a convenient way to directly study the formation and late stages of growth of these objects. These subdisks produce a variety of observational signatures: ultraviolet excess emission from accretion of hot gas (); hydrogen line emissionlike Hα and Paβ (e.g., ; ); and thermal excess emission in the mid-IR ().Sub-mm emission from warm dust has only been detectedfor the ambiguous companion to FW Tau (; ) despite ongoing searches targeting several other disk-bearing substellar companions (e.g., ; ; ). Although we find no convincing evidence that ROXs 12 B harbors a subdisk, the lack of Paβ emission in this individual object can nevertheless be used to more broadlyassess the global frequency of low-mass companions with accretion rates large enough to result in Paβ line emission. The sample of young companions near and below the deuterium burning limit that also have moderate-resolution J-band spectroscopy hasincreased over the past few yearsto the point where we can begin to quantify the statistical properties of these objects as a population. Note that the strength of the Paβ emission line is expected to vary strongly with effective temperature, which alters the pseudocontinnuum level, as well as mass accretion rate.In addition, both the S/N and resolving power of a spectrum can influence the detection of an emission line. For this analysis we ignore these effects and simply count reported Paβ detections and nondetections as discrete Bernoulli trials.If we isolate a sample of young companions with ages ≲15 Myr, when such subdisks might still be expected to be actively accreting, and companion masses ≲20 , there are 11 low-mass brown dwarfs and planetary-mass objects with publishedmoderate-resolution spectroscopy. Five of these show Paβ in emission:GSC 6214-210 B (; ), CT Cha B (; ), GQ Lup B (), DH Tau B (; ), and FW Tau b (). Six companions do not appear to have Paβ in emission: 1RXS J1609–2105 B (),ROXs 42 Bb (),2M0441+2301 Bb (),SR 12 C (), USco CTIO 108 B (),and ROXs 12 B (this work). Binomial statistics implies a frequency of 46 ± 14% for five detections out of 11 trials.There are several caveats about these accreting companions that are worth noting. Paβ emission was not observed in moderate-resolution spectroscopy of GQ Lup B presented by<cit.> and <cit.>, which suggests that the emission observed by <cit.> was variable.Similarly, <cit.> also find variable emission for DH Tau B.This suggests that other companions without Paβ emissioncould be observed during a quiescent state of low accretion. Published mass estimates for GQ Lup B range from ∼10–40 , sothis companion is consistent with (but may fall above) the cutoff of 20 we use in this analysis. Finally, the mass of FW Tau b is uncertain as its spectrum shows substantial veiling and indicationsof an edge-on disk ().Accretion subdisks are apparently very common among companions near the deuterium-burning limit. The incidence of both accreting and non-accreting circumsubstellar disks in general is likely much higher— and perhaps universal—for these objects.§.§ Implications of Misalignment for the Formation of ROXs 12 B The three-dimensional orbital and rotational architecture of planetary systemsprovides fundamental insight intotheir formation and subsequent dynamical evolution. Massive giant planets (≳1 ) formed in circumstellar disksshould inherit the orientation of the disks'angular momentum vectors, which are expected to initially be aligned with those of the host stars. Similarly, the orbital planes of planets should also align with the equatorial planes of host stars in the absence of internal (other planets) or external (wide binary companions or passing stars) perturbers. There is also now substantial evidence that the tail end of the star formation process canproduce companions in the planetary-mass regime near the opacity limit for fragmentation (e.g., ; ). Just as binary companions that formed from the turbulent fragmentation of molecular cloud cores tend to have misaligned orientations (e.g., ; ; ),the same random spin axes could also be imprinted on planetary-mass companions that form in this fashion. Together these two pathways— formation in a disk versus turbulent cloud core— have direct observational implications forwide substellar companions: in the absence of outside influences,massive isolated objects formed in a disk can be expected to exhibit spin-orbit alignment,whereas misalignment implies formation from turbulent fragmentation. Our result that ROXs 12 B is likely misaligned with the spin axis of its host star suggests it was formed from the cloud fragmentation process as opposed to disk instability,but this interpretation is somewhat complicated by the wide stellar tertiary, as discussed below.It is also possible that ROXs 12 B could have formed from disk instability if it later underwent dynamical interactions with another body, for instance through gravitational scattering or Kozai-Lidov librations with the wide stellar companion 2M1626–2527 (; ; ).The timescale for Kozai-Lidov secular oscillations for a planet with an outer tertiary component isgoverned by the period of the planet, the masses of the host star and tertiary, the semi-major axes of the planet and tertiary, and the eccentricity of the host-tertiary orbit (e.g., ). For the ROXs 12triple system, this characteristic timescale reduces to ≈6×10^7 (1 - e_ter^2)^3/2 yr, where e_ter is the eccentricity of theROXs 12 A and 2M1626–2527 pair. As long as this eccentricity is below about 0.9, oscillation timescales are likely too long to have substantially influenced the orbital inclination of ROXs 12 B given the young age of the system.Similarly, prior to the planet formation process,wide stellar companions can gravitationally torque protoplanetary disks and result in spin-orbit misalignments with planets when they eventually form (e.g., ; ). It is therefore conceivable that at a very early stage,a massive disk around ROXs 12 A could have been torqued by 2M1626–2527 to produce the misalignment we now observe in this system. For a protoplanetary disk around ROXs 12 A, the disk precession period would beroughly 80 Myr following <cit.>— or about 20 Myr for maximal misalignment assuming the outer disk radius ends at the location of ROXs 12 B. A more extended disk or a non-zero eccentricity for 2M1626–2527 canreduce this timescale even further, well within the age of ≈6 Myr of this system,implying that the spin-orbit misalignment we observe for ROXs 12 Bcould plausibly be a result of an initial torque on a disk around ROXs 12 A caused by 2M1626–2527.In addition to ROXs 12 B, only a few other systems with widely separated substellar companions haveinformationavailable about their stellar obliquity angles. There is some evidence that the orbital axis of the brown dwarf companion GQ Lup B may be misaligned with both the stellar rotation axis and the stellar disk inclination (; ; ; ), although it is unclear how significant this potential spin-orbit misalignment is due to discrepancies in the rotation period and radius of the host star. HR 8799 is an excellent example of a system with increasingly robust constraints on the spin axis of the host star, orbital inclination of its four imaged planets, and inclination measurements for its multi-belt debris disk. Interestingly, each of these components appears to be mutually consistent within uncertainties (e.g., ; ; ; ), suggestingaligned angular momentum vectors and planet formation in a disk. As the orbits of directly imaged planets become better constrained in the future, stellar obliquity measurements like the one we have carried out here for ROXs 12 ABwill help identify the dominant formation pathway(s) for these objectsas a population analogous to Rossiter-McLaughlin measurements for transiting planets. § SUMMARY We have carried out a comprehensive analysis of the ROXs 12 triple system comprisingtwo young low-mass stars, ROXs 12 A and 2M1626–2527,and the ≈18 companion ROXs 12 B. Our main results are summarized below:-.1 in * Our moderate-resolution near-infrared spectra of ROXs 12 B show unambiguous signs of low surface gravity associated with youth.We find a spectral type of L0 ± 2, a mass of 17.5 ± 1.5 based on hot-start evolutionary models, and an effective temperature of 3100^+400_-500 K from cross correlation withsynthetic spectra of ultracool objects. * The distant stellar companion 2M1626–2527 shares consistent kinematics with the binary ROXs 12 AB, making this a wide (≈5100 AU) tertiary component. ROXs 12 A has a spectral type of M0 ± 0.5, an effective temperature of 3900 ± 100 K,a mass of0.65^+0.05_-0.09 , and an inferred age of 6^+4_-2 Myr. 2M1626–2527 has a spectral type of M1 ± 1, an effective temperature of 3700 ± 150 K,a mass of0.5 ± 0.1 , and an inferred age of 8^+7_-4 Myr. Both stars show lithium absorption.2M1626–2527 hosts an actively accreting protoplanetary disk, whereasROXs 12 A hosts a passive disk with no signs ofaccretion. Although the pair are located near the young Ophiuchus star-forming region, their older ages instead suggest they are members of Upper Scorpius. * K2 light curves reveal a period of 9.1 ± 0.4 d for ROXs 12 A and 3.3 ± 0.05 d for 2M1626–2527.We combine ourv sin i_* measurements of both stars together with radius constraints to determine their line-of-sight inclinations.The rotation axes of these stars are misaligned by 60^+7_-11^∘, consistent with the pattern of random orientations found for other wide binaries. * The orbit of ROXs 12 B is likely misaligned with the spin axis of its host star by at least 49^+20_-32^∘, suggestingformation via cloud fragmentation, or possibly disk instability if the protoplanetary disk surrounding ROXs 12 A was gravitationally torqued by 2M1626–2527.ROXs 12 B is the lowest-mass imaged companion with evidence of spin-orbit misalignment.Continued orbit monitoring will better constrain the orbital inclination of the companion and will lead to a more precise measurement of the host star's obliquity angle. * ROXs 12 B does not have Paβ emission or thermal (L band) excess and we find no compelling evidence that it harbors a subdisk.However, as a population, the frequency of accreting subdisksis relatively high; 46 ± 14% of companions with masses ≲20 and ages ≲15 Myr show Paβ emission.Since this emission is likely to be variable, and some companions will not be accreting at all, this represents a lower limit on the occurrence rate of circumplanetary disks in general.We thank Trent Dupuy and Konstantin Batygin for helpful discussions about spin-orbit misalignments; Eric Nielsen and Sarah Blunt for the published posterior orbital inclination distribution of ROXs 12 B; and the Keck and Gemini Observatory staff for their exceptional support.Support for this work was provided by NASA through Hubble Fellowship grants HST-HF2-51369.001-A and HST-HF2-51336.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.H.A.K. acknowledges support from the Sloan Foundation.M.C.L. acknowledges support from NSF grant NSF-AST-1518339. A.V. is supported by the NSF Graduate Research Fellowship, grant No. DGE 1144152. L.A.C. acknowledges grant support from CONICYT-FONDECYT number 1171246. This paper includes data taken at The McDonald Observatory of The University of Texas at Austin. It is also based on observations with Mayall/RC-Spec spectrograph were carried out through NOAO Prop. ID 2014A-0019. Based on observations obtained at the Gemini Observatory (NOAO Prop. ID: 2016A-0112; Gemini Program ID: GN-2016A-Q-37; PI: B. Bowler; processed using the Gemini NIFS IRAF package), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. We utilized data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusettsand the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics andSpace Administration and the National Science Foundation.NASA's Astrophysics Data System Bibliographic Services togetherwith the VizieR catalogue access tool and SIMBAD database operated at CDS, Strasbourg, France, were invaluable resources for this work. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This work used the Immersion Grating Infrared Spectrometer (IGRINS) that was developed under a collaboration between theUniversity of Texas at Austin and the Korea Astronomy and Space Science Institute (KASI) with the financial support of theUS National Science Foundation under grant AST- 1229522, to the University of Texas at Austin, and of the Korean GMT Project of KASI. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California,Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Finally, mahalo nui loa to the kama`āina of Hawai`i for their support of Keck and the Mauna Kea observatories. We are grateful to conduct observations from this mountain. Keck:II (NIRC2), Keck:I (OSIRIS, HIRES), Smith (IGRINS), Gemini:Gillett (NIFS), Mayall (RC-Spec), Kepler, IRTF (SpeX) The two relevant angles that have been measured for this system are the line-of-sightinclination of ROXs 12 A's rotation axis, i_*, and the orbital inclination of ROXs 12 B, i_p. These angles are related to the sky-projected obliquity λ and the true (de-projected) spin-orbit obliquity ψ via Equation <ref>. Here we show that the absolute difference between i_p and i_* is a lower limit on the measurement of ψ.For values of λ on the interval [0^∘,180^∘], cosλ ≤ 1.For values of i_* and i_p on the interval [0^∘,180^∘], 1 ≥ sin i_p ≥ 0 and 1 ≥ sin i_* ≥ 0.Therefore their product sin i_*sin i_p is positive andsini_*sini_pcosλ≤sini_*sini_p. Adding the termcosi_*cosi_p to both sides gives cosi_*cosi_p +sini_*sini_pcosλ≤cosi_*cosi_p +sini_*sini_p. Noting that the left side of Equation <ref> is equal to cosψ andmaking use of the identity cos(x - y) =cosxcosy + sinxsiny for the right side, it follows that cosψ≤cos(i_* - i_p), and therefore ψ ≥ i_* - i_p.The same logic holds for ψ ≥ i_p - i_*, so ψ ≥ |i_* - i_p|. The absolute difference between the projected rotational inclination and the orbital inclination places a lower limit on the true spin-orbit angle ψ.A measurement of i_* - i_p that is consistent with zero does not necessarily imply spin-orbit alignment, but a non-zero measurement of i_* - i_p means the system is misaligned by at least that value.[Allard et al.(2011)Allard, Homeier, & Freytag]Allard:2011jx Allard, F., Homeier, D., & Freytag, B. 2011, in ASP Conf Ser 448, 16th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, ed C Johns-Krull, M K Browning, & A A West (San Francisco, CA: ASP), 91[Allard et al.(2012)Allard, Homeier, & Freytag]Allard:2012fp —. 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370, 2765[Allers & Liu(2013)]Allers:2013hk Allers, K. N., & Liu, M. C. 2013, The Astrophysical Journal, 772, 79[Allers et al.(2007)Allers, Jaffe, Luhman, Liu, Wilson, Skrutskie, Nelson, Peterson, Smith, & Cushing]Allers:2007ja Allers, K. N., Jaffe, D. T., Luhman, K. L., et al. 2007, The Astrophysical Journal, 657, 511[Altmann et al.(2017)Altmann, Roeser, Demleitner, Bastian, & Schilbach]Altmann:2017hw Altmann, M., Roeser, S., Demleitner, M., Bastian, U., & Schilbach, E. 2017, A&A, 600, L4[Bailey et al.(2013)Bailey, Hinz, Currie, Su, Esposito, Hill, Hoffmann, Jones, Kim, Leisenring, Meyer, Murray-Clay, Nelson, Pinna, Puglisi, Rieke, Rodigas, Skemer, Skrutskie, Vaitheeswaran, & Wilson]Bailey:2013gl Bailey, V., Hinz, P. M., Currie, T., et al. 2013, The Astrophysical Journal, 767, 31[Bailey et al.(2014)Bailey, Meshkat, Reiter, Morzinski, Males, Su, Hinz, Kenworthy, Stark, Mamajek, Briguglio, Close, Follette, Puglisi, Rodigas, Weinberger, & Xompero]Bailey:2014et Bailey, V., Meshkat, T., Reiter, M., et al. 2014, The Astrophysical Journal, 780, L4[Baraffe et al.(2015)Baraffe, Homeier, Allard, & Chabrier]Baraffe:2015fwa Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42[Barman et al.(2015)Barman, Konopacky, Macintosh, & Marois]Barman:2015dy Barman, T. S., Konopacky, Q. M., Macintosh, B., & Marois, C. 2015, ApJ, 804, 1[Basri & Batalha(1990)]Basri:1990cc Basri, G., & Batalha, C. 1990, Astrophysical Journal, 363, 654[Bate(2009)]Bate:2009br Bate, M. R. 2009, Monthly Notices RAS, 392, 590[Batygin(2012)]Batygin:2012ig Batygin, K. 2012, Nature, 491, 418[Biller et al.(2013)Biller, Liu, Wahhaj, Nielsen, Hayward, Males, Skemer, Close, Chun, Ftaclas, Clarke, Thatte, Shkolnik, Reid, Hartung, Boss, Lin, Alencar, De Gouveia Dal Pino, Gregorio-Hetem, & Toomey]Biller:2013fu Biller, B. A., Liu, M. C., Wahhaj, Z., et al. 2013, ApJ, 777, 160[Blunt et al.(2017)Blunt, Nielsen, De Rosa, Konopacky, Ryan, Wang, Pueyo, Rameau, Marois, Marchis, Macintosh, Graham, Duchene, & Schneider]Blunt:2017eta Blunt, S., Nielsen, E. L., De Rosa, R. J., et al. 2017, The Astronomical Journal, 153, 0[Bochanski et al.(2007)Bochanski, West, Hawley, & Covey]Bochanski:2007it Bochanski, J. J., West, A. A., Hawley, S. L., & Covey, K. R. 2007, The Astronomical Journal, 133, 531[Bonnefoy et al.(2014)Bonnefoy, Chauvin, Lagrange, Rojo, Allard, Pinte, Dumas, & Homeier]Bonnefoy:2014dh Bonnefoy, M., Chauvin, G., Lagrange, A.-M., et al. 2014, A&A, 562, A127[Booth et al.(2016)Booth, Jordan, Casassus, Hales, Dent, Faramaz, Matra, Barkats, Brahm, & Cuadra]Booth:2016iy Booth, M., Jordan, A., Casassus, S., et al. 2016, Monthly Notices RAS Letters, 460, L10[Boss(2001)]Boss:2001vw Boss, A. P. 2001, The Astrophysical Journal Letters, 551, L167[Boss(2006)]Boss:2006ge —. 2006, The Astrophysical Journal, 637, L137[Bouvier & Appenzeller(1992)]Bouvier:1992ww Bouvier, J., & Appenzeller, I. 1992, Astronomy and Astrophysics Supplement Series (ISSN 0365-0138), 92, 481[Bouvier et al.(2014)Bouvier, Matt, Mohanty, Scholz, Stassun, & Zanni]Bouvier:2014ik Bouvier, J., Matt, S. P., Mohanty, S., et al. 2014, Protostars and Planets VI, 433[Bowler(2016)]Bowler:2016jk Bowler, B. P. 2016, Publications of the Astronomical Society of the Pacific, 128, 102001[Bowler et al.(2015)Bowler, Andrews, Kraus, Ireland, Herczeg, Ricci, Carpenter, & Brown]Bowler:2015hx Bowler, B. P., Andrews, S. M., Kraus, A. L., et al. 2015, The Astrophysical Journal Letters, 805, 1[Bowler & Hillenbrand(2015)]Bowler:2015en Bowler, B. P., & Hillenbrand, L. A. 2015, The Astrophysical Journal Letters, 811, L30[Bowler et al.(2009)Bowler, Liu, & Cushing]Bowler:2009hn Bowler, B. P., Liu, M. C., & Cushing, M. C. 2009, The Astrophysical Journal, 706, 1114[Bowler et al.(2014)Bowler, Liu, Kraus, & Mann]Bowler:2014dk Bowler, B. P., Liu, M. C., Kraus, A. L., & Mann, A. W. 2014, ApJ, 784, 65[Bowler et al.(2011)Bowler, Liu, Kraus, Mann, & Ireland]Bowler:2011gw Bowler, B. P., Liu, M. C., Kraus, A. L., Mann, A. W., & Ireland, M. J. 2011, The Astrophysical Journal, 743, 148[Bowler et al.(2017)Bowler, Liu, Mawet, Ngo, Malo, Mace, McLane, Lu, Tristan, Hinkley, Hillenbrand, Shkolnik, Benneke, & Best]Bowler:2017hq Bowler, B. P., Liu, M. C., Mawet, D., et al. 2017, The Astronomical Journal, 153, 1[Brandt et al.(2014)Brandt, McElwain, Turner, Mede, Spiegel, Kuzuhara, Schlieder, Wisniewski, Abe, Biller, Brandner, Carson, Currie, Egner, Feldt, Golota, Goto, Grady, Guyon, Hashimoto, Hayano, Hayashi, Hayashi, Henning, Hodapp, Inutsuka, Ishii, Iye, Janson, Kandori, Knapp, Kudo, Kusakabe, Kwon, Matsuo, Miyama, Morino, Moro-Martin, Nishimura, Pyo, Serabyn, Suto, Suzuki, Takami, Takato, Terada, Thalmann, Tomono, Watanabe, Yamada, Takami, Usuda, & Tamura]Brandt:2014cw Brandt, T. D., McElwain, M. W., Turner, E. L., et al. 2014, ApJ, 794, 159[Brinch et al.(2016)Brinch, Jørgensen, Hogerheijde, Nelson, & Gressel]Brinch:2016ke Brinch, C., Jørgensen, J. K., Hogerheijde, M. R., Nelson, R. P., & Gressel, O. 2016, The Astrophysical Journal Letters, 830, 1[Brogi et al.(2012)Brogi, Snellen, de Kok, Albrecht, Birkby, & de Mooij]Brogi:2012bk Brogi, M., Snellen, I. A. G., de Kok, R. J., et al. 2012, Nature, 486, 502[Bryan et al.(2016)Bryan, Bowler, Knutson, Kraus, Hinkley, Mawet, Nielsen, & Blunt]Bryan:2016eo Bryan, M. L., Bowler, B. P., Knutson, H. A., et al. 2016, ApJ, 827, 1[Burrows et al.(1997)Burrows, Marley, Hubbard, Lunine, Guillot, Saumon, Freedman, Sudarsky, & Sharp]Burrows:1997jq Burrows, A., Marley, M., Hubbard, W. B., et al. 1997, Astrophysical Journal, 491, 856[Caceres et al.(2015)Caceres, Hardy, Schreiber, Canovas, Cieza, Williams, Hales, Pinte, Menard, & Wahhaj]Caceres:2015hg Caceres, C., Hardy, A., Schreiber, M. R., et al. 2015, The Astrophysical Journal Letters, 806, 1[Cambrésy(1999)]Cambresy:1999vv Cambrésy, L. 1999, A&A, 345, 965[Cheetham et al.(2015)Cheetham, Kraus, Ireland, Cieza, Rizzuto, & Tuthill]Cheetham:2015es Cheetham, A. C., Kraus, A. L., Ireland, M. J., et al. 2015, ApJ, 813, 1[Christou et al.(2010)Christou, Neichel, Rigaut, Sheehan, Mcdermid, Trancho, Trujillo, & Walls]Christou:2010hx Christou, J. C., Neichel, B., Rigaut, F., et al. 2010, in Proceedings of the SPIE, Gemini Observatory, United States, 77361R[Chubak et al.(2012)Chubak, Marcy, Fischer, Howard, Isaacson, Johnson, & Wright]Chubak:2012tv Chubak, C., Marcy, G., Fischer, D. A., et al. 2012, arXiv:1207.6212, astro-ph.GA[Cieza & Baliber(2007)]Cieza:2007bi Cieza, L., & Baliber, N. 2007, The Astrophysical Journal, 671, 605[Cody et al.(2017)Cody, Hillenbrand, David, Carpenter, Everett, & Howell]Cody:2017ct Cody, A. M., Hillenbrand, L. A., David, T. J., et al. 2017, ApJ, 836, 1[Cody et al.(2014)Cody, Stauffer, Baglin, Micela, Rebull, Flaccomio, Morales-Calderón, Aigrain, Bouvier, Hillenbrand, Gutermuth, Song, Turner, Alencar, Zwintz, Plavchan, Carpenter, Findeisen, Carey, Terebey, Hartmann, Calvet, Teixeira, Vrba, Wolk, Covey, Poppenhaeger, Günther, Forbrich, Whitney, Affer, Herbst, Hora, Barrado, Holtzman, Marchis, Wood, Guimarães, Lillo-Box, Gillen, McQuillan, Espaillat, Allen, D'alessio, & Favata]Cody:2014el Cody, A. M., Stauffer, J., Baglin, A., et al. 2014, The Astronomical Journal, 147, 82[Cushing et al.(2005)Cushing, Rayner, & Vacca]Cushing:2005ed Cushing, M. C., Rayner, J. T., & Vacca, W. D. 2005, The Astrophysical Journal, 623, 1115[Cushing et al.(2004)Cushing, Vacca, & Rayner]Cushing:2004bq Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004, PASP, 116, 362[Cutri & al(2014)]Cutri:2014wx Cutri, R. M., & al, e. 2014, VizieR On-line Data Catalog, 2328[Cutri et al.(2003)Cutri, Skrutskie, Van Dyk, Beichman, Carpenter, Chester, Cambresy, Evans, Fowler, Gizis, Howard, Huchra, Jarrett, Kopan, Kirkpatrick, Light, Marsh, McCallon, Schneider, Stiening, Sykes, Weinberg, Wheaton, Wheelock, & Zacarias]Cutri:2003tp Cutri, R. M., Skrutskie, M. F., Van Dyk, S., et al. 2003, The 2MASS All-Sky Catalog of Point Sources, University of Massachusetts and Infrared Processing and Analysis Center; IPAC/California Institute of Technology[Deacon et al.(2016)Deacon, Schlieder, & Murphy]Deacon:2016dg Deacon, N. R., Schlieder, J. E., & Murphy, S. J. 2016, Monthly Notices RAS, 457, 3191[Dupuy et al.(2010)Dupuy, Liu, Bowler, Cushing, Helling, Witte, & Hauschildt]Dupuy:2010ch Dupuy, T. J., Liu, M. C., Bowler, B. P., et al. 2010, The Astrophysical Journal, 721, 1725[Evans et al.(2014)Evans, Osborne, Beardmore, Page, Willingale, Mountford, Pagani, Burrows, Kennea, Perri, Tagliaferri, & Gehrels]Evans:2014cp Evans, P. A., Osborne, J. P., Beardmore, A. P., et al. 2014, The Astrophysical Journal Supplement Series, 210, 8[Fabrycky & Winn(2009)]Fabrycky:2009ke Fabrycky, D. C., & Winn, J. N. 2009, The Astrophysical Journal, 696, 1230[Fang et al.(2017)Fang, Herczeg, & Rizzuto]Fang:2017ix Fang, Q., Herczeg, G. J., & Rizzuto, A. 2017, ApJ, 842, 0[Filippazzo et al.(2015)Filippazzo, Rice, Faherty, Cruz, Van Gordon, & Looper]Filippazzo:2015dv Filippazzo, J. C., Rice, E. L., Faherty, J., et al. 2015, ApJ, 810, 1[Fitzpatrick(1985)]Fitzpatrick:1985ih Fitzpatrick, E. L. 1985, Astrophysical Journal, 299, 219[Gaia Collaboration et al.(2016)Gaia Collaboration, Brown, Vallenari, Prusti, de Bruijne, Mignard, Drimmel, Babusiaux, Bailer-Jones, Bastian, Biermann, Evans, Eyer, Jansen, Jordi, Katz, Klioner, Lammers, Lindegren, Luri, O Mullane, Panem, Pourbaix, Randich, Sartoretti, Siddiqui, Soubiran, Valette, van Leeuwen, Walton, Aerts, Arenou, Cropper, H g, Lattanzi, Grebel, Holland, Huc, Passot, Perryman, Bramante, Cacciari, Casta eda, Chaoul, Cheek, De Angeli, Fabricius, Guerra, Hern ndez, Jean-Antoine-Piccolo, Masana, Messineo, Mowlavi, Nienartowicz, Ord ez Blanco, Panuzzo, Portell, Richards, Riello, Seabroke, Tanga, Th venin, Torra, Els, Gracia-Abril, Comoretto, Garcia-Reinaldos, Lock, Mercier, Altmann, Andrae, Astraatmadja, Bellas-Velidis, Benson, Berthier, Blomme, Busso, Carry, Cellino, Clementini, Cowell, Creevey, Cuypers, Davidson, De Ridder, de Torres, Delchambre, Dell Oro, Ducourant, Fr mat, Garc a Torres, Gosset, Halbwachs, Hambly, Harrison, Hauser, Hestroffer, Hodgkin, Huckle, Hutton, Jasniewicz, Jordan, Kontizas, Korn, Lanzafame, Manteiga, Moitinho, Muinonen, Osinde, Pancino, Pauwels, Petit, Recio-Blanco, Robin, Sarro, Siopis, Smith, Smith, Sozzetti, Thuillot, van Reeven, Viala, Abbas, Abreu Aramburu, Accart, Aguado, Allan, Allasia, Altavilla, lvarez, Alves, Anderson, Andrei, Anglada Varela, Antiche, Antoja, Ant n, Arcay, Bach, Baker, Balaguer-N ez, Barache, Barata, Barbier, Barblan, Barrado y Navascu s, Barros, Barstow, Becciani, Bellazzini, Bello Garc a, Belokurov, Bendjoya, Berihuete, Bianchi, Bienaym, Billebaud, Blagorodnova, Blanco-Cuaresma, Boch, Bombrun, Borrachero, Bouquillon, Bourda, Bouy, Bragaglia, Breddels, Brouillet, Br semeister, Bucciarelli, Burgess, Burgon, Burlacu, Busonero, Buzzi, Caffau, Cambras, Campbell, Cancelliere, Cantat-Gaudin, Carlucci, Carrasco, Castellani, Charlot, Charnas, Chiavassa, Clotet, Cocozza, Collins, Costigan, Crifo, Cross, Crosta, Crowley, Dafonte, Damerdji, Dapergolas, David, David, De Cat, de Felice, de Laverny, De Luise, De March, de Martino, de Souza, Debosscher, del Pozo, Delbo, Delgado, Delgado, Di Matteo, Diakite, Distefano, Dolding, Dos Anjos, Drazinos, Duran, Dzigan, Edvardsson, Enke, Evans, Eynard Bontemps, Fabre, Fabrizio, Faigler, Falc o, Farr s Casas, Federici, Fedorets, Fern ndez Hern ndez, Fernique, Fienga, Figueras, Filippi, Findeisen, Fonti, Fouesneau, Fraile, Fraser, Fuchs, Gai, Galleti, Galluccio, Garabato, Garc a Sedano, Garofalo, Garralda, Gavras, Gerssen, Geyer, Gilmore, Girona, Giuffrida, Gomes, Gonz lez Marcos, Gonz lez N ez, Gonz lez Vidal, Granvik, Guerrier, Guillout, Guiraud, G rpide, Guti rrez S nchez, Guy, Haigron, Hatzidimitriou, Haywood, Heiter, Helmi, Hobbs, Hofmann, Holl, Holland, Hunt, Hypki, Icardi, Irwin, Jevardat de Fombelle, Jofr, Jonker, Jorissen, Julbe, Karampelas, Kochoska, Kohley, Kolenberg, Kontizas, Koposov, Kordopatis, Koubsky, Krone-Martins, Kudryashova, Kull, Bachchan, & Lacos...]GaiaCollaboration:2016gd Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2016, A&A, 595, A2[Gauza et al.(2015)Gauza, Bejar, Pérez-Garrido, Rosa Zapatero Osorio, Lodieu, Rebolo, Pallé, & Nowak]Gauza:2015fw Gauza, B., Bejar, V. J. S., Pérez-Garrido, A., et al. 2015, The Astrophysical Journal, 804, 96[Ginski et al.(2014)Ginski, Schmidt, Mugrauer, Neuhäuser, Vogt, Errmann, & Berndt]Ginski:2014ef Ginski, C., Schmidt, T. O. B., Mugrauer, M., et al. 2014, Monthly Notices RAS, 444, 2280[Glebocki & Stawikowski(1997)]Glebocki:1997ul Glebocki, R., & Stawikowski, A. 1997, A&A, 328, 579[Golimowski et al.(2004)Golimowski, Leggett, Marley, Fan, Geballe, Knapp, Vrba, Henden, Luginbuhl, Guetter, Munn, Canzian, Zheng, Tsvetanov, Chiu, Glazebrook, Hoversten, Schneider, & Brinkmann]Golimowski:2004en Golimowski, D. A., Leggett, S. K., Marley, M. S., et al. 2004, The Astronomical Journal, 127, 3516[Gorlova et al.(2003)Gorlova, Meyer, Rieke, & Liebert]Gorlova:2003cs Gorlova, N. I., Meyer, M. R., Rieke, G. H., & Liebert, J. 2003, The Astrophysical Journal, 593, 1074[Gotberg et al.(2016)Gotberg, Davies, Mustill, Johansen, & Church]Gotberg:2016ge Gotberg, Y., Davies, M. B., Mustill, A. J., Johansen, A., & Church, R. P. 2016, A&A, 592, A147[Hale(1994)]Hale:1994gv Hale, A. 1994, The Astronomical Journal, 107, 306[Harris et al.(1996)Harris, Forman, Gioa, Hale, Harnden, Jones, Karakashian, Maccacaro, McSweeney, Primnini, Schwarz, Tananbaum, & Thurman]Harris:1996uk Harris, D. E., Forman, W., Gioa, I. M., et al. 1996, VizieR On-line Data Catalog: IX/13. Originally published in: SAO HEAD CD-ROM Series I (Einstein), 9013[Hartigan et al.(1991)Hartigan, Kenyon, Hartmann, Strom, Edwards, Welty, & Stauffer]Hartigan:1991fy Hartigan, P., Kenyon, S. J., Hartmann, L., et al. 1991, Astrophysical Journal, 382, 617[Henden et al.(2016)Henden, Templeton, Terrell, Smith, Levine, & Welch]Henden:2016uu Henden, A. A., Templeton, M., Terrell, D., et al. 2016, VizieR Online Data Catalog, 2336, 0[Herbst et al.(2002)Herbst, Bailer-Jones, Mundt, Meisenheimer, & Wackermann]Herbst:2002je Herbst, W., Bailer-Jones, C. A. L., Mundt, R., Meisenheimer, K., & Wackermann, R. 2002, A&A, 396, 513[Herczeg & Hillenbrand(2014)]Herczeg:2014is Herczeg, G. J., & Hillenbrand, L. A. 2014, ApJ, 786, 97[Holman et al.(1997)Holman, Touma, & Tremaine]Holman:1997ht Holman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254[Horne & Baliunas(1986)]Horne:1986ds Horne, J. H., & Baliunas, S. L. 1986, Astrophysical Journal, 302, 757[Howell et al.(2014)Howell, Sobeck, Haas, & Still]Howell:2014vua Howell, S. B., Sobeck, C., Haas, M., & Still, M. 2014, PASP, 126, 398[Ireland et al.(2011)Ireland, Kraus, Martinache, Law, & Hillenbrand]Ireland:2011id Ireland, M. J., Kraus, A., Martinache, F., Law, N., & Hillenbrand, L. A. 2011, The Astrophysical Journal, 726, 113[Irwin et al.(2011)Irwin, Berta, Burke, Charbonneau, Nutzman, West, & Falco]Irwin:2011gz Irwin, J., Berta, Z. K., Burke, C. J., et al. 2011, The Astrophysical Journal, 727, 56[Jensen & Akeson(2014)]Jensen:2014hv Jensen, E. L. N., & Akeson, R. 2014, Nature, 511, 567[Johnson et al.(2009)Johnson, Winn, Albrecht, Howard, Marcy, & Gazak]Johnson:2009jr Johnson, J. A., Winn, J. N., Albrecht, S., et al. 2009, The Publications of the Astronomical Society of the Pacific, 121, 1104[Kirkpatrick et al.(2006)Kirkpatrick, Barman, Burgasser, McGovern, McLean, Tinney, & Lowrance]Kirkpatrick:2006hb Kirkpatrick, J. D., Barman, T. S., Burgasser, A. J., et al. 2006, The Astrophysical Journal, 639, 1120[Konopacky et al.(2013)Konopacky, Barman, Macintosh, & Marois]Konopacky:2013jvc Konopacky, Q. M., Barman, T. S., Macintosh, B. A., & Marois, C. 2013, Science, 339, 1398[Konopacky et al.(2016)Konopacky, Marois, Macintosh, Galicher, Barman, Metchev, & Zuckerman]Konopacky:2016gv Konopacky, Q. M., Marois, C., Macintosh, B. A., et al. 2016, The Astronomical Journal, 152, 1[Kovács(1981)]Kovacs:1981bs Kovács, G. 1981, Astrophysics and Space Science, 78, 175[Kozai(1962)]Kozai:1962bo Kozai, Y. 1962, Astronomical Journal, 67, 591[Kraus et al.(2015)Kraus, Andrews, Bowler, Herczeg, Ireland, Liu, Metchev, & Cruz]Kraus:2015fx Kraus, A. L., Andrews, S. M., Bowler, B. P., et al. 2015, The Astrophysical Journal, 798, L23[Kraus et al.(2017)Kraus, Herczeg, Rizzuto, Mann, Slesnick, Carpenter, Hillenbrand, & Mamajek]Kraus:2017bg Kraus, A. L., Herczeg, G. J., Rizzuto, A. C., et al. 2017, ApJ, 838, 0[Kraus & Hillenbrand(2008)]Kraus:2008ix Kraus, A. L., & Hillenbrand, L. A. 2008, The Astrophysical Journal, 686, L111[Kraus et al.(2014)Kraus, Ireland, Cieza, Hinkley, Dupuy, Bowler, & Liu]Kraus:2014tl Kraus, A. L., Ireland, M. J., Cieza, L. A., et al. 2014, ApJ, 781, 20[Kraus et al.(2008)Kraus, Ireland, Martinache, & Lloyd]Kraus:2008bh Kraus, A. L., Ireland, M. J., Martinache, F., & Lloyd, J. P. 2008, The Astrophysical Journal, 679, 762[Kraus et al.(2011)Kraus, Tucker, Thompson, Craine, & Hillenbrand]Kraus:2011ju Kraus, A. L., Tucker, R. A., Thompson, M. I., Craine, E. R., & Hillenbrand, L. A. 2011, The Astrophysical Journal, 728, 48[Lachapelle et al.(2015)Lachapelle, LaFreniere, Gagné, Jayawardhana, Janson, Helling, & Witte]Lachapelle:2015cx Lachapelle, F.-R., LaFreniere, D., Gagné, J., et al. 2015, ApJ, 802, 1[Lafrenière et al.(2010)Lafrenière, Jayawardhana, & van Kerkwijk]Lafreniere:2010cp Lafrenière, D., Jayawardhana, R., & van Kerkwijk, M. H. 2010, The Astrophysical Journal, 719, 497[Lai(2014)]Lai:2014ke Lai, D. 2014, Monthly Notices RAS, 440, 3532[Larkin et al.(2006)Larkin, Barczys, Krabbe, Adkins, Aliado, Amico, Brims, Campbell, Canfield, Gasaway, Honey, Iserlohe, Johnson, Kress, LaFreniere, Lyke, Magnone, Magnone, McElwain, Moon, Quirrenbach, Skulason, Song, Spencer, Weiss, & Wright]Larkin:2006jd Larkin, J., Barczys, M., Krabbe, A., et al. 2006, Proc. SPIE, 6269, 42[Lavigne et al.(2009)Lavigne, Doyon, Lafrenière, Marois, & Barman]Lavigne:2009di Lavigne, J.-F., Doyon, R., Lafrenière, D., Marois, C., & Barman, T. 2009, The Astrophysical Journal, 704, 1098[Lee et al.(2016)Lee, Dunham, Myers, Arce, Bourke, Goodman, rgensen, Kristensen, Offner, Pineda, Tobin, & Vorobyov]Lee:2016fh Lee, K. I., Dunham, M. M., Myers, P. C., et al. 2016, The Astrophysical Journal Letters, 820, 1[Leggett et al.(2006)Leggett, Currie, Varricatt, Hawarden, Adamson, Buckle, Carroll, Davies, Davis, Kerr, Kuhn, Seigar, & Wold]Leggett:2006gg Leggett, S. K., Currie, M. J., Varricatt, W. P., et al. 2006, Monthly Notices RAS, 373, 781[Lidov(1962)]Lidov:1962du Lidov, M. L. 1962, Planetary and Space Science, 9, 719[Lomb(1976)]Lomb:1976wz Lomb, N. R. 1976, Astrophysics and Space Science, 39, 447[Low & Lynden-Bell(1976)]Low:1976wt Low, C., & Lynden-Bell, D. 1976, Royal Astronomical Society, 176, 367[Lucas et al.(2001)Lucas, Roche, Allard, & Hauschildt]Lucas:2001ed Lucas, P. W., Roche, P. F., Allard, F., & Hauschildt, P. H. 2001, Monthly Notices RAS, 326, 695[Mace et al.(2016)Mace, Kim, Jaffe, Park, Lee, Kaplan, Yu, Yuk, Chun, Pak, Kim, Lee, Sneden, Afsar, Pavel, Lee, Oh, Jeong, Park, Kidder, Lee, Nguyen Le, McLane, Gully-Santiago, Oh, Lee, Hwang, & Park]Mace:2016ev Mace, G., Kim, H., Jaffe, D. T., et al. 2016, in SPIE Astronomical Telescopes + Instrumentation, ed. C. J. Evans, L. Simard, & H. Takami (SPIE), 99080C[MacGregor et al.(2017)MacGregor, Wilner, Czekala, Andrews, Dai, Herczeg, Kratter, Kraus, Ricci, & Testi]MacGregor:2017fl MacGregor, M. A., Wilner, D. J., Czekala, I., et al. 2017, ApJ, 835, 1[Mamajek(2008)]Mamajek:2008dx Mamajek, E. E. 2008, Astronomische Nachrichten, 329, 10[Manjavacas et al.(2014)Manjavacas, Bonnefoy, Schlieder, Allard, Rojo, Goldman, Chauvin, Homeier, Lodieu, & Henning]Manjavacas:2014hu Manjavacas, E., Bonnefoy, M., Schlieder, J. E., et al. 2014, A&A, 564, A55[Mann et al.(2013)Mann, Gaidos, & Ansdell]Mann:2013fv Mann, A. W., Gaidos, E., & Ansdell, M. 2013, ApJ, 779, 188[Markwardt(2009)]Markwardt:2009wq Markwardt, C. B. 2009, Astronomical Data Analysis Software and Systems XVIII ASP Conference Series, 411, 251[Martin et al.(2017)Martin, Mace, McLean, Logsdon, Rice, Kirkpatrick, Burgasser, McGovern, & Prato]Martin:2017jf Martin, E. C., Mace, G. N., McLean, I. S., et al. 2017, ApJ, 838, 0[Matthews et al.(2013)Matthews, Kennedy, Sibthorpe, Booth, Wyatt, Broekhoven-Fiene, Macintosh, & Marois]Matthews:2013fd Matthews, B., Kennedy, G., Sibthorpe, B., et al. 2013, ApJ, 780, 97[McCaughrean et al.(2004)McCaughrean, Close, Scholz, Lenzen, Biller, Brandner, Hartung, & Lodieu]Mccaughrean:2004ey McCaughrean, M. J., Close, L. M., Scholz, R.-D., et al. 2004, A&A, 413, 1029[McElwain et al.(2007)McElwain, Metchev, Larkin, Barczys, Iserlohe, Krabbe, Quirrenbach, Weiss, & Wright]McElwain:2007ib McElwain, M. W., Metchev, S. A., Larkin, J. E., et al. 2007, The Astrophysical Journal, 656, 505[McGovern et al.(2004)McGovern, Kirkpatrick, McLean, Burgasser, Prato, & Lowrance]McGovern:2004cc McGovern, M. R., Kirkpatrick, J. D., McLean, I. S., et al. 2004, The Astrophysical Journal, 600, 1020[McGregor et al.(2003)McGregor, Hart, Conroy, Pfitzner, Bloxham, Jones, Downing, Dawson, Young, Jarnyk, & Van Harmelen]McGregor:2003kv McGregor, P. J., Hart, J., Conroy, P. G., et al. 2003, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes. Edited by Iye, 4841, 1581[McLean et al.(2003)McLean, McGovern, Burgasser, Kirkpatrick, Prato, & Kim]McLean:2003hx McLean, I. S., McGovern, M. R., Burgasser, A. J., et al. 2003, The Astrophysical Journal, 596, 561[Mieda et al.(2014)Mieda, Wright, Larkin, Graham, Adkins, Lyke, Campbell, Maire, Do, & Gordon]Mieda:2014dt Mieda, E., Wright, S. A., Larkin, J. E., et al. 2014, PASP, 126, 250[Monet et al.(2003)Monet, Levine, Canzian, Ables, Bird, Dahn, Guetter, Harris, Henden, Leggett, Levison, Luginbuhl, Martini, Monet, Munn, Pier, Rhodes, Riepe, Sell, Stone, Vrba, Walker, Westerhout, Brucato, Reid, Schoening, Hartley, Read, & Tritton]Monet:2003bw Monet, D. G., Levine, S. E., Canzian, B., et al. 2003, The Astronomical Journal, 125, 984[Montmerle et al.(1983)Montmerle, Koch-Miramond, Falgarone, & Grindlay]Montmerle:1983jc Montmerle, T., Koch-Miramond, L., Falgarone, E., & Grindlay, J. E. 1983, Astrophysical Journal, 269, 182[Muiños & Evans(2014)]Muinos:2014ew Muiños, J. L., & Evans, D. W. 2014, Astronomische Nachrichten, 335, 367[Naoz(2016)]Naoz:2016cr Naoz, S. 2016, Annu. Rev. Astro. Astrophys., 54, 441[Naud et al.(2014)Naud, Artigau, Malo, Albert, Doyon, LaFreniere, Gagné, Saumon, Morley, Allard, Homeier, Beichman, Gelino, & Boucher]Naud:2014jx Naud, M.-E., Artigau, E., Malo, L., et al. 2014, ApJ, 787, 5[Offner et al.(2016)Offner, Dunham, Lee, Arce, & Fielding]Offner:2016gl Offner, S. S. R., Dunham, M. M., Lee, K. I., Arce, H. G., & Fielding, D. B. 2016, The Astrophysical Journal Letters, 827, 1[Ohta et al.(2005)Ohta, Taruya, & Suto]Ohta:2005ux Ohta, Y., Taruya, A., & Suto, Y. 2005, The Astrophysical Journal[Ortiz-León et al.(2017)Ortiz-León, Loinard, Kounkel, Dzib, Mioduszewski, Rodriguez, Torres, González-Lópezlira, Pech, Rivera, Hartmann, Boden, Evans, Briceño, Tobin, Galli, & Gudehus]OrtizLeon:2017ce Ortiz-León, G. N., Loinard, L., Kounkel, M. A., et al. 2017, ApJ, 834, 1[Park et al.(2014)Park, Jaffe, Yuk, Chun, Pak, Kim, Pavel, Lee, Oh, Jeong, Sim, Lee, Nguyen Le, Strubhar, Gully-Santiago, Oh, Cha, Moon, Park, Brooks, Ko, Han, Nah, Hill, Lee, Barnes, Yu, Kaplan, Mace, Kim, Lee, Hwang, & Park]Park:2014kn Park, C., Jaffe, D. T., Yuk, I.-S., et al. 2014, in SPIE Astronomical Telescopes + Instrumentation, ed. S. K. Ramsay, I. S. McLean, & H. Takami (SPIE), 91471D[Pecaut & Mamajek(2013)]Pecaut:2013ej Pecaut, M. J., & Mamajek, E. E. 2013, The Astrophysical Journal Supplement Series, 208, 9[Pecaut & Mamajek(2016)]Pecaut:2016fua —. 2016, Monthly Notices RAS, 461, 794[Potter et al.(2002)Potter, Martín, Cushing, Baudoz, Brandner, Guyon, & Neuhäuser]Potter:2002ie Potter, D., Martín, E. L., Cushing, M. C., et al. 2002, The Astrophysical Journal, 567, L133[Queloz et al.(2000)Queloz, Eggenberger, Mayor, Perrier, Beuzit, Naef, Sivan, & Udry]Queloz:2000ui Queloz, D., Eggenberger, A., Mayor, M., et al. 2000, A&A, 359, L13[Rameau et al.(2016)Rameau, Nielsen, De Rosa, Blunt, Patience, Doyon, Graham, LaFreniere, Macintosh, Marchis, Bailey, Chilcote, Duchene, Esposito, Hung, Konopacky, Maire, Marois, Metchev, Perrin, Pueyo, Rajan, Savransky, Wang, Ward-Duong, Wolff, Ammons, Hibon, Ingraham, Kalas, Morzinski, Oppenheimer, Rantakyearo, & Thomas]Rameau:2016dx Rameau, J., Nielsen, E. L., De Rosa, R. J., et al. 2016, The Astrophysical Journal Letters, 822, 1[Ratzka et al.(2005)Ratzka, Köhler, & Leinert]Ratzka:2005jv Ratzka, T., Köhler, R., & Leinert, C. 2005, A&A, 437, 611[Rayner et al.(2003)Rayner, Toomey, Onaka, Denault, Stahlberger, Vacca, Cushing, & Wang]Rayner:2003vo Rayner, J. T., Toomey, D. W., Onaka, P. M., et al. 2003, PASP, 115, 362[Rebull et al.(2004)Rebull, Wolff, & Strom]Rebull:2004ed Rebull, L. M., Wolff, S. C., & Strom, S. E. 2004, The Astronomical Journal, 127, 1029[Reggiani et al.(2016)Reggiani, Meyer, Chauvin, Vigan, quanz, Biller, Bonavita, Desidera, Delorme, Hagelberg, Maire, Boccaletti, Beuzit, Buenzli, Carson, Covino, Feldt, Girard, Gratton, Henning, Kasper, Lagrange, Mesa, Messina, Montagnier, Mordasini, Mouillet, Schlieder, Ségransan, Thalmann, & Zurlo]Reggiani:2016dn Reggiani, M., Meyer, M. R., Chauvin, G., et al. 2016, A&A, 586, A147[Reidemeister et al.(2009)Reidemeister, Krivov, Schmidt, Fiedler, Müller, Löhne, & Neuhäuser]Reidemeister:2009fv Reidemeister, M., Krivov, A. V., Schmidt, T. O. B., et al. 2009, A&A, 503, 247[Reinhold & Gizon(2015)]Reinhold:2015ep Reinhold, T., & Gizon, L. 2015, A&A, 583, A65[Rizzuto et al.(2015)Rizzuto, Ireland, & Kraus]Rizzuto:2015bs Rizzuto, A. C., Ireland, M. J., & Kraus, A. L. 2015, Monthly Notices RAS, 448, 2737[Rizzuto et al.(2011)Rizzuto, Ireland, & Robertson]Rizzuto:2011gs Rizzuto, A. C., Ireland, M. J., & Robertson, J. G. 2011, Monthly Notices RAS, 416, 3108[Rucinski(1999)]Rucinski:1999uu Rucinski, S. 1999, Precise Stellar Radial Velocities, 185, 82[Scargle(1982)]Scargle:1982eu Scargle, J. D. 1982, Astrophysical Journal, 263, 835[Schmidt et al.(2008)Schmidt, Neuhäuser, Seifahrt, Vogt, Bedalov, Helling, Witte, & Hauschildt]Schmidt:2008bf Schmidt, T. O. B., Neuhäuser, R., Seifahrt, A., et al. 2008, A&A, 491, 311[Scholz et al.(2003)Scholz, Mccaughrean, Lodieu, & Kuhlbrodt]Scholz:2003go Scholz, R.-D., Mccaughrean, M. J., Lodieu, N., & Kuhlbrodt, B. 2003, A&A, 398, L29[Schwarz et al.(2016)Schwarz, Ginski, de Kok, Snellen, Brogi, & Birkby]Schwarz:2016fl Schwarz, H., Ginski, C., de Kok, R. J., et al. 2016, A&A, 593, A74[Seifahrt et al.(2007)Seifahrt, Neuhäuser, & Hauschildt]Seifahrt:2007iq Seifahrt, A., Neuhäuser, R., & Hauschildt, P. H. 2007, A&A, 463, 309[Slesnick et al.(2004)Slesnick, Hillenbrand, & Carpenter]Slesnick:2004jy Slesnick, C. L., Hillenbrand, L. A., & Carpenter, J. M. 2004, The Astrophysical Journal, 610, 1045[Stamatellos & Herczeg(2015)]Stamatellos:2015dx Stamatellos, D., & Herczeg, G. J. 2015, Monthly Notices RAS, 449, 3432[Stauffer et al.(2016)Stauffer, Cody, Rebull, Hillenbrand, Turner, Carpenter, Carey, Terebey, Morales-Calderón, Alencar, McGinnis, Sousa, Bouvier, Venuti, Hartmann, Calvet, Micela, Flaccomio, Song, Gutermuth, Barrado, Vrba, Covey, Herbst, Gillen, Guimarães, Bouy, & Favata]Stauffer:2016cu Stauffer, J., Cody, A. M., Rebull, L., et al. 2016, The Astronomical Journal, 151, 1[Todorov et al.(2010)Todorov, Luhman, & Mcleod]Todorov:2010cn Todorov, K., Luhman, K. L., & Mcleod, K. K. 2010, The Astrophysical Journal, 714, L84[Vacca et al.(2003)Vacca, Cushing, & Rayner]Vacca:2003wi Vacca, W. D., Cushing, M. C., & Rayner, J. T. 2003, PASP, 115, 389[Vanderburg & Johnson(2014)]Vanderburg:2014bia Vanderburg, A., & Johnson, J. A. 2014, Publications of the Astronomical Society of Pacific, 126, 948[Voges et al.(2000)Voges, Aschenbach, Boller, Bräuninger, Briel, Burkert, Dennerl, Englhauser, Gruber, Haberl, Hartner, Hasinger, Pfeffermann, Pietsch, Predehl, Schmitt, Trümper, & Zimmermann]Voges:2000wn Voges, W., Aschenbach, B., Boller, T., et al. 2000, IAU Circ., 7432, 3[Vogt et al.(1994)Vogt, Allen, Bigelow, Bresee, Brown, Cantrall, Conrad, Couture, Delaney, Epps, Hilyard, Hilyard, Horn, Jern, Kanto, Keane, Kibrick, Lewis, Osborne, Pardeilhan, Pfister, Ricketts, Robinson, Stover, Tucker, Ward, & Wei]Vogt:1994tb Vogt, S. S., Allen, S. L., Bigelow, B. C., et al. 1994, Proc. SPIE, 2198, 362[Wilking et al.(2008)Wilking, Gagné, & Allen]Wilking:2008wq Wilking, B. A., Gagné, M., & Allen, L. E. 2008, Handbook of Star Forming Regions, I, 351[Wilking et al.(2015)Wilking, Vrba, & Sullivan]Wilking:2015hn Wilking, B. A., Vrba, F. J., & Sullivan, T. 2015, ApJ, 815, 1[Williams et al.(2014)Williams, Mann, Francesco, Andrews, Hughes, Ricci, Bally, Johnstone, & Matthews]Williams:2014fg Williams, J. P., Mann, R. K., Francesco, J. D., et al. 2014, ApJ, 796, 120[Winn et al.(2005)Winn, Noyes, & Holman]Winn:2005tg Winn, J. N., Noyes, R. W., & Holman, M. J. 2005, ApJ, 631, 1215[Wizinowich(2013)]Wizinowich:2013dz Wizinowich, P. 2013, Publications of the Astronomical Society of the Pacific, 125, 798[Wolff et al.(2017)Wolff, Menard, Caceres, Lefevre, Bonnefoy, Canovas, Maret, Pinte, Schreiber, & van der Plas]Wolff:2017ky Wolff, S. G., Menard, F., Caceres, C., et al. 2017, The Astronomical Journal, 154, 0[Wu et al.(2015a)Wu, Close, Males, Barman, Morzinski, Follette, Bailey, Rodigas, Hinz, Puglisi, Xompero, & Briguglio]Wu:2015kh Wu, Y.-L., Close, L. M., Males, J. R., et al. 2015a, ApJ, 801, 4[Wu et al.(2015b)Wu, Close, Males, Barman, Morzinski, Follette, Bailey, Rodigas, Hinz, Puglisi, Xompero, & Briguglio]Wu:2015cz —. 2015b, The Astrophysical Journal Letters, 807, 1[Wu et al.(2017)Wu, Sheehan, Males, Close, Morzinski, Teske, Haug-Baltzell, Merchant, & Lyons]Wu:2017kd Wu, Y.-L., Sheehan, P. D., Males, J. R., et al. 2017, ApJ, 836, 1[Wyatt(2008)]Wyatt:2008ht Wyatt, M. C. 2008, Annu. Rev. Astro. Astrophys., 46, 339[Yelda et al.(2010)Yelda, Lu, Ghez, Clarkson, Anderson, Do, & Matthews]Yelda:2010ig Yelda, S., Lu, J. R., Ghez, A. M., et al. 2010, The Astrophysical Journal, 725, 331[Zhou et al.(2014)Zhou, Herczeg, Kraus, Metchev, & Cruz]Zhou:2014ct Zhou, Y., Herczeg, G. J., Kraus, A. L., Metchev, S., & Cruz, K. L. 2014, The Astrophysical Journal, 783, L17[Zucker(2003)]Zucker:2003it Zucker, S. 2003, Monthly Notice of the Royal Astronomical Society, 342, 1291 | http://arxiv.org/abs/1708.07611v1 | {
"authors": [
"Brendan Bowler",
"Adam Kraus",
"Marta Bryan",
"Heather Knutson",
"Matteo Brogi",
"Aaron Rizzuto",
"Gregory Mace",
"Andrew Vanderburg",
"Michael Liu",
"Lynne Hillenbrand",
"Lucas Cieza"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170825042602",
"title": "The Young Substellar Companion ROXs 12 B: Near-Infrared Spectrum, System Architecture, and Spin-Orbit Misalignment"
} |
Fenchel Dual Gradient Methods for Distributed Convex Optimization over Time-varying Networks Xuyang Wu and Jie LuX. Wu and J. Lu are with the School of Information Science and Technology, ShanghaiTech University, 201210 Shanghai, China. Email: {wuxy, lujie}@shanghaitech.edu.cn.This work has been supported by the National Natural Science Foundation of China under grant 61603254, the Shanghai Pujiang Program under grant 16PJ1406400, and the Natural Science Foundation of Shanghai under grant 16ZR1422500. December 7, 2017 =================================================================================================================================================================================================================================================================================================================================================================================================================================== .5in .5in Résumé français. Des travaux précédents de Helen Wong et de l'auteur ont mis en évidence, quand le paramètre quantiqueq= e^2πiħ est une racine de l'unité, des annulations miraculeuses pour l'application de trace quantique qui relie l'algèbre d'écheveaux du crochet de Kauffman à l'espace de Teichmüller quantique d'une surface. L'article ci-dessous fournit une interprétation plus conceptuelle de ce phénomène, en termes de représentations du groupe quantique U_q(𝔰𝔩_2) et de son algèbre de Hopf dualeSL_2^q.English abstract. In earlier work, Helen Wong and the author discovered certain “miraculous cancellations” for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmüller space, occurring when the quantum parameter q= e^2πiħ is a root of unity. The current paper is devoted to giving a more representation theoretic interpretation of this phenomenon, in terms of the quantum group U_q(𝔰𝔩_2) and its dual Hopf algebra SL_2^q.The equation (X+Y)^n=X^n + Y^nis (unfortunately) very familiar to some of our students, who find it convenient to “simplify” computations. However, it is also well-known that this relation does hold in some cases, for instance in a ring of characteristic n with n prime, or when the variables X and Y satisfy the q-commutativity relation that YX=qXY with q∈ a primitive n–root of unity; see <ref>. The structure of Equation (<ref>) can be described by considering the two-variable polynomial P(X,Y)=X+Y. Then (<ref>) states that the polynomial P(X,Y)^n, obtained by taking the n–th power of P(X,Y), coincides with the polynomial P(X^n, Y^n) obtained by replacing the variables X, Y with their powers X^n, Y^n, respectively. Helen Wong and the author discovered similar identities in their study of the Kauffman bracket skein algebra of a surface <cit.>. These relations involved a 2–dimensional version of the operation of “taking the n–th power”, through the Chebyshev polynomial T_n(t) ∈[t] defined by the property thatA^n = T_n( A)for every 2-by-2 matrix A∈() with determinant 1. A typical consequence of the miraculous cancellations discovered in <cit.> is that, when YX=qXY and q is a primitive n–root of unity, T_n(X + Y + X^-1) = X^n + Y^n + X^-n,which fits the pattern T_n ( P(X,Y) ) = P(X^n, Y^n) for the polynomial P(X,Y)= X + XY + X^-1. The arguments of <cit.> provide many examples of such polynomials, involving several q–commuting variables. In <cit.> these “Chebyshev cancellations” were used to connect, when q is a root of unity, irreducible representations of the Kauffman bracket skein algebra (S) of a surface S to group homomorphisms π_1(S) →(). The skein algebra (S) is a purely topological object whose elements are represented by framed links in the thickened surface S× [0,1]. It draws its origin from Witten's interpretation <cit.> of the Jones polynomial knot invariant within the framework of a topological quantum field theory, and as a consequence it is closely connected to the quantum group . The arguments of <cit.> were often developed by trial and error. The purpose of the current article is to put these constructions into a more conceptual framework, where the connection withand () appears more clearly.Another goal is to emphasize the representation theoretic nature of these phenomena, with the long term objective of generalizing them to quantum knot invariants and skein algebras based on other quantum groups U_q(𝔤), such as the U_q(𝔰𝔩_n)–based HOMFLY polynomial and skein algebra. In addition to the fact that quantum groups are still an acquired taste for many mathematicians, including the author, the connection betweenand () is more intuitive if we replacewith its dual Hopf algebra , in the sense of <cit.>. This will enable us to express our constructions solely in terms of 2-by-2 matrices with coefficients in an arbitrary noncommutative algebra ; in <cit.>, the algebrawas the quantum Teichmüller space of the surface. This point of view is sufficiently close to () that it should be relatively intuitive for those mathematicians who have a long track record in hyperbolic geometry, since PSL_2() is the isometry group of the hyperbolic space ℍ^3. This category includes the author and Jean-Pierre Otal, and it is a pleasure to dedicate this article to him as an acknowledgement of the great influence that he had on the author's work, either through their joint articles <cit.> or through many informal conversations. We now state the main result of this article. Let A_1, A_2, …, A_k be 2-by-2 matrices with coefficients in an algebraover , such that: * each A_i is triangular of the form( [a_ib_i;0 a_i^-1 ]) or ( [a_i0;b_i a_i^-1 ]) with b_ia_i = q a_i b_i for some nonzero number q∈-{0};* a_i and b_i commute with a_j and b_j whenever i ≠ j. Then, if q^2 is a primitive n–root of unity, T_n(A_1A_2… A_k) =A_1^(n)A_2^(n)… A_k^(n)where each A_i^(n) is obtained from A_i by replacing a_i and b_i with their powers a_i^n and b_i^n This statement is easier to understand if we illustrate it by an example. Consider the product of five triangular matricesA=( [a_1b_1;0 a_1^-1 ]) ( [a_2b_2;0 a_2^-1 ]) ( [a_30;b_3 a_3^-1 ]) ( [a_4b_4;0 a_4^-1 ]) ( [a_50;b_5 a_5^-1 ])where b_ia_i = q a_i b_i, and a_i and b_i commute with a_j and b_j whenever i ≠ j.Computing the product and taking the trace straightforwardly givesA= a_1a_2a_3a_4a_5+a_1 a_2a_3b_4b_5 +a_1 b_2b_3a_4a_5 +a_1 b_2b_3b_4b_5 +a_1 b_2a_3^-1a_4^-1b_5 +b_1 a_2^-1b_3a_4a_5 +b_1 a_2^-1b_3b_4b_5 +b_1 a_2^-1a_3^-1a_4^-1b_5 +a_1^-1 a_2^-1b_3b_4a_5^-1 +a_1^-1 a_2^-1a_3^-1a_4^-1a_5^-1. Since A has 10 terms and the Chebyshev polynomial T_n(t) has degree n, one would expect T_n( A) to have about 10^n terms. However, when q^2 is a primitive n–root of unity, many cancellations occur and only 10 terms remain. In factT_n( A)= a_1^n a_2^n a_3^n a_4^n a_5^n +a_1^na_2^n a_3^n b_4^n b_5^n+a_1^nb_2^n b_3^n a_4^n a_5^n+a_1^nb_2^n b_3^n b_4^n b_5^n+a_1^nb_2^n a_3^-na_4^-nb_5^n+b_1^na_2^-nb_3^n a_4^n a_5^n+b_1^na_2^-nb_3^n b_4^n b_5^n+b_1^na_2^-na_3^-na_4^-nb_5^n+a_1^-n a_2^-nb_3^n b_4^n a_5^-n +a_1^-n a_2^-na_3^-na_4^-na_5^-n=A^(n)whereA^(n)=( [a_1^nb_1^n;0 a_1^-n ]) ( [a_2^nb_2^n;0 a_2^-n ]) ( [a_3^n0;b_3^n a_3^-n ]) ( [a_4^nb_4^n;0 a_4^-n ]) ( [a_5^n0;b_5^n a_5^-n ])is obtained from A by replacing each a_i, b_i with a_i^n, b_i^n. When q is transcendental,there are only very fewcancellations and Proposition <ref>shows that T_n( A) is a sum of exactly (5+√(24))^n + (5-√(24))^n >9.89^n monomials.This explicit count is based on a positivity result (Proposition <ref>) which may be of independent interest. Similarly, the example of Equation (<ref>) is provided by applying Theorem <ref> to the matrix A=( [a_1b_1;0 a_1^-1 ]) ( [a_20;b_2 a_2^-1 ]), setting X=a_1a_2 and Y=b_1b_2, and replacing q with √(q). The proof of Theorem <ref> essentially has two parts. The first step is representation theoretic, and connects T_n( A) to the action of the matrix A on the space [X,Y]^q of polynomials in q–commuting variables X and Y and with coefficients in the algebra . This is a relatively easy adaptation to our context of a deep but classical result in the representation theory of the quantum group , the Clebsch-Gordan Decomposition. The second step is a simple computation of traces for this action of A on[X,Y]^q, which is much simpler than the original arguments of <cit.>.Among the hypotheses of Theorem <ref>, some are more natural than others. The q–commutativity relations b_ia_i = q a_i b_i are essentially mandated by the connection of the objects considered to the quantum groupand its dual Hopf algebra . Similarly, the requirement that the matrices A_i be triangular is deeply tied to the structure of the Lie group () and the quantum group(and their Borel subgroups/subalgebras). The commutativity hypothesis that a_i and b_i commute with a_j and b_j whenever i ≠ j is less critical, and was introduced here to define A_1A_2… A_k ∈ (and the product matrix A_1A_2… A_k∈()) in a straightforward way. In fact, it is possible to define such a trace without these commutativity properties, but this requires using the cobraiding of the Hopf algebraas well as additional data that is reminiscent of the original topological context. This was implicitly done in <cit.> for the Kauffman bracket skein algebra of a surface, but a quick comparison between the formulas of <cit.> and <cit.> should make it clear that these arguments canbe expanded to a more representation theoretic framework. The cancellations of Theorem <ref> then extend to this generalized setup, as in <cit.>. § THE EQUATION (X+Y)^N=X^N + Y^NIn spite of the first sentence of this article, most of our students do know the Binomial Formula, which says that(X+Y)^n= X^n + n1X^n-1Y+ + n2X^n-2Y^2 + … + nn-1XY^n-1 +Y^n= ∑_k=0^n nk X^n-kY^kwhere nk is the binomial coefficientnk = n(n-1)(n-2) … (n-k+2)(n-k+1)/k(k-1)(k-2) … 2 1.If we are working in a ring R with characteristic n and if n is prime (which in particular occurs when R is a field), then n=0 in R while k(k-1)(k-2) … 2 1≠ 0 for every k<n. It follows that nk=0 whenever 0<k<n, so that the Frobenius relation holdequation (X+Y)^n = X^n + Y^n holds in this case.Note that the hypothesis that the characteristic n is prime is really necessary. For instance, in the ring /4 of characteristic 4, (X+Y)^4 = X^4 + 6X^2Y^2 + Y^4 ≠ X^4 + Y^4. A less well-known occurrence of Equation (<ref>)involves variables X and Y that q–commute, in the sense that YX = qXY for some q∈. The Quantum Binomial Formula (see for instance <cit.>) then states that(X+Y)^n= X^n + n1qX^n-1Y+ + n2qX^n-2Y^2 + … + nn-1q XY^n-1 +Y^n= ∑_k=0^n nkq X^n-kY^kwhere nkq is the quantum binomial coefficientnkq = nqn-1qn-2q…n-k+2qn-k+1q/kqk-1qk-2q…2q1qdefined by the quantum integersjq = q^j-1/q-1 = 1 +q+q^2 + … +q^j-1. We state the following property as a lemma, as we will frequently need to refer to it.If q is a primitive n–root of unity, in the sense that q^n=1 and q^k≠ 1 whenever 0<k<n, the quantumbinomial coefficient nkq is equal to 0 for every k with 0<k<n.Since q is a primitive n–root of unity, nq=q^n-1/q-1=0 while kq=q^k-1/q-1≠0 whenever 0<k<n. The result follows. If YX=qXY with q∈ a primitive n–root of unity, the Frobenius relation (X+Y)^n = X^n + Y^n holds. As in the characteristic n case, it is necessary that q be a primitive n–root of unity. For instance, whenYX=-XY, q=-1 is a (non-primitive) 4–root of unity and (X+Y)^4 = X^4 + 2X^2Y^2 + Y^4 ≠ X^4 + Y^4. We will now discuss generalizations of Equation (<ref>) arising from properties of the quantum group . As indicated in the introduction, we will express these properties in terms of 2-by-2 matrices with coefficients in an algebra . Incidentally, all algebras considered in this article will be over . Other fields could be used, but the need for primitive n–roots of unitymake this convention more natural. We begin with the classical case of the algebra =, and of the Lie group ().§ THE CLASSICAL ACTION OF () ON [X,Y]The special linear group of order 2 is the group() = {( [ a b; c d ]); a, b,c,d ∈, ad-bc=1 }of 2-by-2 matrices with determinant 1. It has a left action on the plane ^2, and therefore a right action by precomposition on the algebra [X,Y]= {polynomial functions on ^2}= {polynomials ∑_i=0^m ∑_j=0^n a_ij X^i Y^j; a_ij∈}of polynomials in the variables X and Y. More precisely, the action of ( [ a b; c d ])∈() on [X,Y] is such that P(X,Y)( [ a b; c d ]) = P(aX+bY, cX + dY)for every polynomial P(X,Y) ∈[X,Y]. This defines a mapρ() →_( [X,Y] )from () to the algebra of –linear maps [X,Y] →[X,Y]. We collect a few elementary properties in the following lemma. * For every ( [ a b; c d ]) ∈(), ρ( [ a b; c d ]) ∈_( [X,Y] ) is also an algebra endomorphism of [X,Y].* The mapρ is valued in the group _( [X,Y] ) of linear automorphisms of [X,Y], and induces a group antihomomorphism ρ() →_( [X,Y] ), in the sense thatρ( [ a b; c d ][ a' b'; c' d' ]) = ρ[ a' b'; c' d' ]∘ρ[ a b; c d ]for every( [ a b; c d ]),( [ a' b'; c' d' ])∈(). * If [X,Y]_n = {∑_i+j=n a_ij X^iY^j; a_ij∈}≅^n+1 denotes the vector space of homogeneous polynomials of degree n, the representation ρ restricts to a finite-dimensional representation ρ_n () →_( [X,Y] _n) ≅_(^n+1).The order reversal in the second conclusion reflects the fact that () acts on [X,Y] on the right. We could easily turn ρ into a group homomorphism by composing it with any of the standard antiautomorphisms of (), such as( [ a b; c d ]) ↦( [ a b; c d ])^t = ( [ a c; b d ]) or ( [ a b; c d ]) ↦( [ a b; c d ])^-1 = ( [d -b; -ca ]), and many authors dothis. For this reason, we will refer to ρ and its restrictions ρ_n as representations of ().A classical property is that, up to isomorphism, the ρ_n form the collection of all irreducible representations of (). § THE QUANTUM PLANE AND () For a nonzero number q ∈-{0}, thequantum plane is the algebra [X,Y]^q defined by two generators X and Y, and by the relation YX=qXY. Namely, the elements of [X,Y]^q are polynomials P(X,Y)=∑_i=0^m ∑_j=0^n a_ij X^i X^j that are multiplied using the relation YX=qXY. If we want to keep the property thatthe matrices ( [ a b; c d ]) and ( [ a c; b d ]) act as algebra homomorphisms on [X,Y]^q, we need that(cX+dY)(aX + bY)= q (aX + bY)(cX+dY) and(bX+dY)(aX + cY)= q (aX +cY)(bX+dY)to preserve the relation YX=qXY. Identifying the coefficients of X^2, XY and Y^2, this would requireba= qabdb = qbdbc =cbca= qac dc =qcd ad- q^-1 bc= da - qbcwhich is clearly impossible if q≠ 1 and a, b, c, d commute with each other. This leads us to the following definition (see for instance <cit.> for more background).Given an algebra 𝒜 over , let () denote the set of matrices ( [ a b; c d ]) where a, b, c, d∈𝒜 satisfy the relations of (<ref>), as well as ad- q^-1bc =1. More formally, letbe the algebra defined by generators a, b, c, d and by the relations of (<ref>) and (<ref>). Then () can be interpreted as the set of all algebra homomorphisms ^q →.The elements of () are called the 𝒜–points of .Unlike (), the set () is far from being a group. It only comes with a partially defined multiplication. Indeed, if( [ a b; c d ]) and ( [ a' b'; c' d' ]) ∈(), the usual formula( a b c d ) ( a' b' c' d' ) = ( aa'+bc' ab'+bd' ca'+ dc' cb' + dd' )gives an element of () only under additional hypotheses on the entries of these matrices, for instance if a, b, c, d commute with a', b', c', d'. This partially defined multiplication has an identity element( [ 1 0; 0 1 ]) ∈(). However, theoperation of passing to the inverse somewhat misbehaves in the sense that the formal inverse ( [ a b; c d ])^-1 = ( [d-qb; -q^-1ca ]) of ( [ a b; c d ]) ∈() is an element of ^q^-1()=(^) instead of (); here ^ is the opposite algebra of the algebra , consisting of the vector spaceendowed with the new multiplication *_ defined by the property that a*_ b=ba for every a, b∈.In general, the formula (<ref>) gives a globally definedmultiplication () ⊗(ℬ) →(⊗ℬ) for any two algebrasand ℬ. In order to generalize to ()the action of () on [X,Y], we introduce thequantum –plane as the algebra [X,Y]^q= ⊗[X,Y]^q. In practice the elements of [X,Y]^q are polynomials P(X,Y)=∑_i=0^m ∑_j=0^n a_ij X^i X^j with coefficients a_ij∈, and are algebraically manipulated using the relation YX=qXY while the variables X and Y commute with all elements of .The relations defining () are specially designed that an 𝒜–point ofacts as an algebra homomorphism on the quantum –plane [X,Y]^q, by the same formula (<ref>) as in the commutative plane. More precisely, if _ ([X,Y]^q) denotes the space of –linear maps [X,Y]^q →[X,Y]^q, we can define a map ρ() →_ ([X,Y]^q)such that ρ( [ a b; c d ]) (∑_i,ja_ij X^i Y^j ) = ∑_i,ja_ij (aX+bY)^i (cX + dY)^jfor every polynomial ∑_i,ja_ijX^iY^j ∈_q[X,Y]. The map ρ satisfies the following elementary properties.* For every ([ a b; c d ])∈(), the –linear map ρ([ a b; c d ]) ∈_ ([X,Y]^q) is an algebra homomorphism. * For each n, the map ρ([ a b; c d ]) ∈_ ([X,Y]^q) respects the space [X,Y]_n^q = ⊗[X,Y]_n^q of homogeneous polynomials of degree n in X and Y with coefficients in . As a consequence,ρ induces by restrictiona map ρ_n() →_ ([X,Y]^q_n). * If a, b, c, d commute witha', b', c', d' (so that the product ([ a b; c d ]) ([ a' b'; c' d' ]) = ([ aa'+bc' ab' + bd'; ca' + dc' cb'+dd' ]) ∈() makes sense), ρ( [ a b; c d ][ a' b'; c' d' ]) = ρ( a' b' c' d' ) ∘ρ( a b c d ).§ TRACES AND CHEBYSHEV POLYNOMIALS§.§ TracesTraces can misbehave in the noncommutative context. However, we areinterested in endomorphisms of [X,Y]_n^q = ⊗[X,Y]_n^q, which have a natural trace.To emphasize the key property needed, let V be a finite dimensional vector space over , letbe an algebra over , and consider an –linear map f ∈_(⊗ V). The trace of f is defined asf = ∑_i=0^n a_ii∈where e_0, e_1, …, e_n is a basis for the -vector space V, and where the coefficients a_ij∈ are defined by the property that f(e_j) = ∑_i=0^n a_ije_i in ⊗ V for every j=0, 1, …, n. The usual commutative proof immediately extends to this context to give: The trace f∈ is independent of the choice of the basise_0, e_1, …, e_nfor the -vector space V. Note that this property would be false if we only required e_0, e_1, …, e_n to be a basis for the –module ⊗ V ≅^n+1.For practice, let us carry out a few elementary computations for the representations ρ_n () →_([X,Y]_n^q) of Lemma <ref>. For n=1, consider ([ a b; c d ])∈() and its image ρ_1 ([ a b; c d ])∈_([X,Y]_1^q). The polynomials X, Y form a basis for [X,Y]_1^q. Since ρ_1([ a b; c d ])(X) = aX + bY andρ_1([ a b; c d ])(Y) = cX + dY we conclude thatρ_1([ a b; c d ]) is equal to a+d, namely to what we have implicitly called ([ a b; c d ]) in the introduction. For n=2,an elementary computation givesρ_2 ([ a b; c d ]) (X^2)= (aX+bY)^2 = a^2 X^2 + (1+q^2) abXY + b^2 Y^2 ρ_2 ([ a b; c d ]) (XY)= (aX+bY)(cX+dY)-50pt =ac X^2 + (ad+qbc) XY + bd Y^2 ρ_2 ([ a b; c d ]) (Y^2)= (cX+dY)^2=c^2 X^2 + (1+q^2) cdXY + d^2 Y^2so thatρ_2 ([ a b; c d ]) = a^2 + (ad+qbc) + d^2= a^2 + ad + da +d^2 -1= (a+d)^2-1by remembering that da-qbc=1 from the definition of (). For n=3, a longer but similar first step gives thatρ_3 ([ a b; c d ]) = a^3 + ( a^2d+q(1+q^2) abc ) + ( a(1+q^2) bcd + ad^2 ) + d^3= a^3 + a^2d + ada + q^2 ada -a -q^2a + dad + q^2 dad - d -q^2d + ad^2 +d^3,using again the property that da-qbc=1.Since qbc=da-1 and bca=q^2 abc, we have that da^2-a=q^2ada-q^2a. Similarly, because dbc=q^2 bcd, d^2a-a = q^2dad-q^2d. Substituting these values in our first expression for ρ_3 ([ a b; c d ]) givesρ_3 ([ a b; c d ]) = a^3 + a^2d + ada +da^2 + dad +d^2a + ad^2 +d^3-2a-2d= (a+d)^3-2(a+d). In all three cases, we have been able to express the trace ρ_n ([ a b; c d ]) as a polynomial in ([ a b; c d ]) =a+d. We will see in Corollary <ref> that this is a general phenomenon.Part of the purpose in the above calculations was to let the reader experience the fact that this property is not that easy to check by bare-hand computations, which justifies the introduction of the more theoretical constructions of <ref> and <ref> below.§.§ Chebyshev polynomials We will encounter two types of Chebyshev polynomials. The (normalized) Chebyshev polynomials of the first kind are the polynomials T_n(t) ∈ℤ[t] recursively defined by T_n+1(t)= t T_n(t) - T_n-1(t)T_1(t)=tT_0(t) =2. The (normalized) Chebyshev polynomials of the second kind S_n(t) ∈ℤ[t] are remarkably similar, and defined by S_n+1 = t S_n(t) - S_n-1(t)S_1(t)=tS_0(t) =1. In particular, in addition to T_0(t)=2, S_0(t) =1 and T_1(t) = S_1(t)=t, T_2(t)= t^2-2 S_2(t)= t^2-1T_3(t)= t^3 -3t S_3(t)= t^3-2tT_4(t)= t^4-4t^2+2 S_4(t)= t^4-3t^2 +1T_5(t)= t^5-5t^3 +5t S_5(t)= t^5-4t^3 +3tT_6(t)= t^6-6t^4 +9t^2 -2 S_6(t)= t^6-5t^4 + 6t^2 -1 The two types of Chebyshev polynomials are related by the property thatT_n(t) = S_n (t) - S_n-2(t)for every n≥ 2 This is an immediate consequence of the fact that the T_n(t) and S_n(t) satisfy the same linear recurrence relation, and of the initial conditions. The following classical properties connect the Chebyshev polynomials T_n(t) and S_n(t) to the group ().For every A∈(),T_n ( A)=A^n S_n ( A) = ρ_n(A)whereρ_n () →( [X,Y]_n ) is the (n+1)–dimensional representation of <ref>. From the Cayley-Hamilton Theorem (or inspection)A^2 - ( A )A+Id =0for every A=( [ a b; c d ]) ∈(). Multiplying both sides by A^n-1 and taking the trace, we see that A^n satisfies the same recurrence relationA^n+1 = ( A) ( A^n) -A^n-1as T_n( A), as well as the same initial values for n=0 and n=1. It follows that A^n = T_n( A) for every n.For the property that S_n ( A)= ρ_n(A), which is not needed in this article, we can just refer to the special case q=1 of Corollary <ref> below. Note the following closed form expression for the Chebyshev polynomial T_n(t)∈[t].T_n(t) =( t+√(t^2-4)/2)^n + ( t-√(t^2-4)/2)^nLet A ∈() be a matrix such that A=t.If t^2-4 ≠ 0, the characteristic polynomial λ^2 -tλ+1 of A has two distinct roots t±√(t^2-4)/2, which consequently are the eigenvalues of A. Then A^n has eigenvalues (t±√(t^2-4)/2)^n and, by Lemma <ref>,T_n(t) = A^n = ( t+√(t^2-4)/2)^n + ( t-√(t^2-4)/2)^n. By continuity, the property also holds whent^2-4 = 0. §.§ The Hopf algebrasand For most of the article, we are trying to keep the exposition at an elementary level in order to make the algebra more intuitive and to emphasize its connection with geometry. However, we now need deeper algebraic concepts and constructions, which will enable us to apply the well-known Clebsch-Gordan Decomposition for(Theorem <ref>) to obtain a similar statement (Proposition <ref>) for the set () of2-by-2 matrices that we are interested in. We follow here the conventions of <cit.>. We already encountered the –algebra , defined by generators a, b, c, d and by the relations of (<ref>–<ref>). In particular, () is the set ofhomomorphisms fromto the algebra . A better known object is the quantum group , which is a deformation of the enveloping algebra of the Lie algebra 𝔰𝔩_2() of (); see <cit.>. Recall thatis defined by generators E, F, K, K^-1 and by the relationsKK^-1 =K^-1K=1 KE =q^2EK EF-FE =K-K^-1/q-q^-1 KF =q^-2FKThis is a Hopf algebra, whose comultiplication Δ→⊗, counit ϵ→ and antipode map S → are respectively determined by the properties thatΔ(E)= E⊗ K + 1⊗ Eϵ(E)=0 S(E) = -EK^-1 Δ(F)= F⊗ 1 + K^-1⊗ Fϵ(F)=0S(F) = -KF Δ(K)= K ⊗ K ϵ(K)=1S(K) = K^-1. Similarly, is a Hopf algebra with comultiplication Δ→⊗, counit ϵ→ and antipodeS → given byΔ(a)= a ⊗ a + b⊗ cϵ(a) = 1 S(a) = dΔ(b)= a⊗ b + b⊗ dϵ(b) = 0S(b) = -qbΔ(c)= c ⊗ a + d ⊗ cϵ(c) = 0 S(c) = -q^-1c Δ(d)= c ⊗ b + d ⊗ dϵ(d) = 1 S(d) =a. When q=1, the algebra ^1 is just the algebra of regular (= polynomial) functions () → on the algebraic group (). The comultiplication Δ^1 →^1 ⊗^1 then is the algebra homomorphism induced by the group multiplication () ×() →(), the counit ϵ^1 → is induced by the map {*}→() sending * to the identity ([ 1 0; 0 1 ]), and theantipodeS ^1 →^1 is induced by ([ a b; c d ]) ↦([ a b; c d ])^-1. Similarly, as q→ 1, the quantum groupconverges to the enveloping algebra U(𝔰𝔩_2) of the Lie algebra of (), by consideration of H=K-K^-1/q-q^-1.The relationship betweenandcomes in the form of a linear pairing⟨, ⟩ ⊗⟶ U⊗α⟼ ⟨ U , α⟩determined by the properties that⟨ E,a ⟩ =0⟨ E,b ⟩ =1⟨ E,c ⟩ =0⟨ E,d ⟩ =0⟨ F,a ⟩ =0⟨ F,b ⟩ =0⟨ F,c ⟩ =1⟨ F,d ⟩ =0⟨ K,a ⟩ =q⟨ K,b ⟩ =0⟨ K,c ⟩ =0⟨ K,d ⟩ =q^-1and ⟨ U, αβ⟩ = ∑_(U)⟨ U',α⟩⟨ U” ,β⟩ ⟨ UV, α⟩ = ∑_(α)⟨ U, α' ⟩⟨ V, α”⟩for every α, β∈ and U, V∈, using Sweedler's notation thatΔ(U)= ∑_(U) U' ⊗ U”∈⊗ and Δ(α)= ∑_(α)α' ⊗α”∈⊗for the comultiplications ofand . See Lemma <ref> for an interpretation of the formulas of (<ref>), and see <cit.>and <cit.> fordetails. In particular, the duality ⟨ ,⟩ induces a linear map δ→^* fromto the dual of . We will need the following property. The above duality map δ→^* is injective. This is analogous to the property that, because the Lie group () is connected, a regular function on () is completely determined by itsderivatives and higher derivatives at the identity element. In general, the representation theory of a Lie algebra is significantly easier to analyze than that of the corresponding Lie group. The same phenomenon in the quantum world is one of the reasons whyis more popular than . In particular, there is an actionσ⊗[X,Y]^q →[X,Y]^q ofover the quantum plane [X,Y]^q by “quantum derivation”, defined by the property that σ(E ⊗ X^kY^l)= lq X^k+1 Y^l-1σ(F ⊗ X^kY^l) = kq X^k-1 Y^l+1 σ(K ⊗ X^kY^l)= q^k-l X^k Y^lwhere kq denotes the other type of quantum integer kq = q^k - q^-k/q-q^-1 = q^k-1 + q^k-3 + … + q^-k+3 + q ^-k+1= q^-k+1kq^2. This action restricts to the space of homogeneous polynomials of degree n, and gives an (n+1)–dimensional representationσ_n →_( [X,Y]_n^q )for every n. When q is not a root of unity, the σ_n essentially realize all irreducible finite-dimensional representations of , up to isomorphism. To describe all irreducible finite-dimensional representations of , one just need one morefamily of similar representations σ_n', related to σ_n by a simple sign twist. See for instance <cit.>. Similarly, the representation ρ() →_ ([X,Y]^q) of <ref> comes from a coactionτ[X,Y]^q →⊗[X,Y]^q defined by the propertythat τ(P(X,Y) ) = P(a⊗ X + b⊗ Y, c⊗ X + d ⊗ Y)for every polynomial P(X,Y) ∈[X,Y]^q. Indeed, if A ∈() is considered as an algebra homomorphism A →, then ρ(A) ∈_ ([X,Y]^q) is clearly the –linear extension of the –linear map (A ⊗_[X,Y]^q) ∘τ[X,Y]^q →[X,Y]^q. The following statement relates the duality ⟨ , ⟩ of (<ref>) to the 2–dimensional representation σ_1→_( [X,Y]_1^q).For every U∈, the matrix of σ_1(U) ∈_( [X,Y]_1^q) in the basis { X, Y} isσ_1(U) =[ ⟨ U,a⟩ ⟨ U,b⟩; ⟨ U,c⟩ ⟨ U,d⟩ ], where a, b, c, d are the generators of .The property holds for U=E, F or K^±1 by inspection in (<ref>), since σ_1(E) = ( [ 0 1; 0 0 ]), σ_1(F) = ( [ 0 0; 1 0 ]) and σ_1(K) = ( [q0;0 q^-1 ]) in the basis {X,Y}. It then holds for any product of these generators by combining the fact that σ_1 is an algebra homomorphism, the compatibility of the duality ⟨ , ⟩ with the multiplications and comultiplications given by (<ref>), and the definition of the comultiplication Δ→⊗. We now show that the duality ⟨ , ⟩ connects the action ofon [X,Y]^q to the coaction of . To see this, we first rewrite the action σ⊗[X,Y]^q →[X,Y]^q as a linear map Σ[X,Y]^q →^* ⊗[X,Y]^q.The action Σ[X,Y]^q →^* ⊗[X,Y]^q, coaction τ[X,Y]^q →⊗[X,Y]^q and duality map δ→^* are related by the property that the diagram [X,Y]^q @/^20pt/[rrr]^Σ[r]_τ ⊗[X,Y]^q [rr]_δ⊗_[X,Y]^q^* ⊗[X,Y]^qis commutative. Namely, Σ=(δ⊗_[X,Y]^q) ∘τ.The property is equivalent to the commutativity of the diagram60pt⊗[X,Y]^q @/^20pt/[rr]^σ[r]__⊗τ ⊗⊗[X,Y]^q[r]_⟨, ⟩⊗_[X,Y]^q [X,Y]^q To simplify the formulas, write ϕ = _⊗τ and ψ= ⟨, ⟩⊗_[X,Y]^q.We need to prove that ψ∘ϕ (U⊗ P(X,Y) ) = σ (U⊗ P(X,Y) )for every U∈ and P(X,Y)∈[X,Y]^q. We first consider the case where P(X,Y)=X. Then,ψ∘ϕ(U⊗ X)= ψ (U ⊗ a ⊗ X + U ⊗ b ⊗ Y)= ⟨ U , a ⟩X + ⟨ U , b ⟩Y= σ_1(U)(X) = σ(U⊗ X)by Lemma <ref>. So, the property of (<ref>) holds for P(X,Y)=X and every U∈. An almost identical argument shows that (<ref>) holds for P(X,Y)=Y and every U∈. As a consequence,(<ref>) holds for every P(X,Y) ∈[X,Y]_1^q andU∈.We now claim that, if(<ref>) holds for P(X,Y) and Q(X,Y) and every U∈, then it holds for the product P(X,Y)Q(X,Y) and every U∈. A fundamental property of the action σ is that, in the terminology of <cit.>,it makes [X,Y]^q a module algebra over . This means that, in addition to making [X,Y]^q a module over the algebra ,σ satisfies the “quantum product rule” thatσ (U⊗ P(X,Y)Q(X,Y) )= ∑_(U)σ (U'⊗ P(X,Y) ) σ (U”⊗ Q(X,Y) ) ,using Sweedler's notation that Δ(U) = ∑_(U) U' ⊗ U”. See<cit.>.Now, ψ∘ϕ (U⊗ P(X,Y)Q(X,Y) )= ψ(U ⊗ P(a ⊗ X +b ⊗ Y ) Q(a ⊗ X +b ⊗ Y ) )=∑_(U)ψ(U' ⊗ P(a ⊗ X +b ⊗ Y ) ) ψ(U”⊗Q(a ⊗ X +b ⊗ Y ) )= ∑_(U)σ (U'⊗ P(X,Y) ) σ (U”⊗ Q(X,Y) ) = σ (U⊗ P(X,Y)Q(X,Y) ) ,where the second equality comes from (<ref>), the third equality reflects our hypothesis that P(X,Y) and Q(X,Y) satisfy (<ref>) for every U”'∈, and the fourth equality results from (<ref>). This proves our claim that (<ref>) holds for P(X,Y)Q(X,Y) and for every U∈.This inductive step proves that (<ref>) holds in all cases, and concludes the proof of Lemma <ref>. §.§ The Clebsch-Gordan Decomposition forand () A great feature of Hopf algebras is that their comultiplication Δ enables one to take the tensor product of two representations. For , the Quantum Clebsch-Gordan Decomposition expresses the action ofon the tensor product [X,Y]^q_m ⊗[X,Y]^q_n as a direct sum of (irreducible) representations over [X,Y]^q_m+n-2k with 0≤ k ≤inf{m,n}. See <cit.>, and <cit.>. We state here the result for the case we need, when m=1.Recall that the action ofon [X,Y]^q restricts to an algebra homomorphism σ_n →_( [X,Y]_n^q ) for every n.Also, the tensor productσ_1 ⊗σ_n →_([X,Y]^q_1 ⊗[X,Y]^q_n ) = _([X,Y]^q_1)⊗_([X,Y]^q_n )is defined by the property that σ_1 ⊗σ_n(U) = ∑_(U)σ_1(U') ⊗σ_n(U”)for every U∈, using Sweedler's notation that Δ(U) = ∑_(U) U' ⊗ U”. When q is not a k–root of unity with k≤ n, there exists a –linear isomorphism ϕ[X,Y]^q_1 ⊗[X,Y]^q_n →[X,Y]^q_n+1⊕[X,Y]^q_n-1 such that the diagram [X,Y]^q_1 ⊗[X,Y]^q_n [rrr]^σ_1 ⊗σ_n(U)[d]_ϕ^≅ [X,Y]^q_1 ⊗[X,Y]^q_n [d]^ϕ_≅ [X,Y]^q_n+1⊕[X,Y]^q_n-1[rrr]_σ_n+1(U) ⊕σ_n-1(U) [X,Y]^q_n+1⊕[X,Y]^q_n-1 commutes for every U∈.We now consider the coaction τ[X,Y]^q →⊗[X,Y]^q, and more precisely its restriction τ_n [X,Y]^q_n →⊗[X,Y]^q_n. Tensor products of coactions are much simpler to define, and τ_1 ⊗τ_n(P⊗ Q ) =τ_1 (P) ⊗τ_n(Q) ∈⊗[X,Y]^q_1 ⊗[X,Y]^q_nfor every P ∈[X,Y]^q_1 and Q∈[X,Y]^q_n.We now use the duality ⟨ , ⟩ to deduce the following result from Theorem <ref>. Suppose thatq is not a k–root of unity with k≤ n, and consider the –linear isomorphism ϕ[X,Y]^q_1 ⊗[X,Y]^q_n →[X,Y]^q_n+1⊕[X,Y]^q_n-1 of Theorem <ref>. Then,the diagram [X,Y]^q_1 ⊗[X,Y]^q_n [rr]^τ_1 ⊗τ_n[d]_ϕ^≅⊗[X,Y]^q_1 ⊗[X,Y]^q_n [d]^_⊗ϕ_≅ [X,Y]^q_n+1⊕[X,Y]^q_n-1[rr]_τ_n+1⊕τ_n-1⊗[X,Y]^q_n+1⊕[X,Y]^q_n-1 commutes, in the sense that (_⊗ϕ) ∘ (τ_1 ⊗τ_n) = (τ_n+1⊕τ_n-1) ∘ϕTo simplify the notation, set V_k = [X,Y]^q_k. Then the commutative diagram of Lemma <ref> restricts for each k to a commutative diagram V_k @/^20pt/[rrrr]^Σ_k[rr]_τ_k⊗V_k [rr]_δ⊗_V_k^*⊗ V_k where Σ_k is related to the action σ_k →_(V_k)by the property that, for each U∈ and P ∈ V_k, the element σ_k(U)(P) ∈ V_k is obtained by evaluating Σ_k(P) ∈^* ⊗ V_k at U. In the diagram V_1 ⊗ V_n@/^20pt/[rrrr]^Σ_1 ⊗Σ_n[rr]_τ_1 ⊗τ_n[d]_ϕ⊗ V_1 ⊗ V_n[rr]_δ⊗_V_1 ⊗ V_n[d]^_⊗ϕ^* ⊗ V_1 ⊗ V_n[d]^_^*⊗ϕV_n+1⊕ V_n-1@/_20pt/[rrrr]_Σ_n+1⊕Σ_n-1[rr]^τ_n+1⊕τ_n-1⊗ (V_n+1⊕ V_n-1 ) [rr]^δ⊗_V_n+1⊕ V_n-1^* ⊗ (V_n+1⊕ V_n-1 )we want to show that the left-hand square commutes. Here,Σ_1 ⊗Σ_n is definedto be related to the tensor product σ_1 ⊗σ_n →_(V_1 ⊗ V_n) by the property that, for each U∈, P_1 ∈ V_1 and P_n ∈ V_n, the element σ_1 ⊗σ_n(U)(P_1 ⊗ P_n) ∈ V_1 ⊗ V_n is obtained by evaluating Σ_1 ⊗Σ_n(P_1 ⊗ P_n) ∈^* ⊗ V_1 ⊗ V_n at U. The map Σ_n+1⊕Σ_n-1 is similarly associated to the direct sumσ_n+1⊕σ_n-1→_(V_n+1⊕ V_n-1).Because of the way Σ_1 ⊗Σ_n and Σ_n+1⊕Σ_n-1 are respectively associated to σ_1 ⊗σ_n and σ_n+1⊕σ_n-1, Theorem <ref> shows that the outer rectangle of(<ref>) commutes, in the sense that(_^*⊗ϕ) ∘ (Σ_1 ⊗Σ_n) = (Σ_n+1⊕Σ_n-1) ∘ϕ.The lower triangle V_n+1⊕ V_n-1@/_20pt/[rrrr]_Σ_n+1⊕Σ_n-1[rr]^τ_n+1⊕τ_n-1⊗ (V_n+1⊕ V_n-1 ) [rr]^δ⊗_V_n+1⊕ V_n-1^* ⊗ (V_n+1⊕ V_n-1 )of (<ref>) commutes by an immediate application of (<ref>). The commutativity of the upper triangle requires more thought, becausetensor products of actions of algebras are more complicated than direct sums. The diagram V_1 ⊗ V_n@/^20pt/[rrrr]^Σ_1 ⊗Σ_n[rr]_τ_1 ⊗τ_n⊗ V_1 ⊗ V_n[rr]_δ⊗_V_1 ⊗ V_n^* ⊗ V_1 ⊗ V_n commutes.Because of the relationships between Σ_1 ⊗Σ_n and the action σ_1 ⊗σ_n, and between the map δ→^* and the duality ⟨ ,⟩⊗→, it suffices to show that V_1 ⊗ V_n@/^20pt/[rrrr]^σ_1 ⊗σ_n(U)[rr]_τ_1 ⊗τ_n⊗ V_1 ⊗ V_n[rr]_⟨ U,⟩ V_1 ⊗ V_ncommutes for every U∈. Here, for a vector space V, we shorten the notation andwrite ⟨ U,⟩⊗ V → V for the map that we previously denoted by ⟨ U,⟩⊗_V. This property is an immediate consequence of the fact (<ref>–<ref>) that ⟨ ,⟩ establishes a duality between multiplications and comultiplications.Indeed, given two polynomials P_1 ∈ V_1 and P_n ∈ V_n,σ_1 ⊗σ_n(U)(P_1 ⊗ P_n)= ∑_(U)σ_1(U')(P_1) ⊗σ_n(U”) (P_n)=∑_(U)⟨ U', τ_1(P_1)⟩⊗⟨ U”, τ_n(P_n) ⟩= ⟨ U, τ_1(P_1) ⊗τ_n(P_n) ⟩ where the first equality reflects the definition of σ_1 ⊗σ_2, the second equality comes from Lemma <ref> (or (<ref>)), and the third equality follows from (<ref>). The proves the commutativity of (<ref>), and therefore Lemma <ref>.We are now ready to conclude the proof of Proposition <ref>. We proved that, in the diagram (<ref>), the outer rectangle and the two upper and lower triangles commute. By Lemma <ref>, the map δ→^* is injective. It easily follows that the left-hand square V_1 ⊗ V_n[rr]_τ_1 ⊗τ_n[d]^ϕ⊗ V_1 ⊗ V_n[d]^_⊗ϕV_n+1⊕ V_n-1[rr]^τ_n+1⊕τ_n-1⊗ (V_n+1⊕ V_n-1 ) commutes. This is exactly what we needed to prove. After this long digression through the Hopf algebrasand , we now return to our original topic of interest, namely the set () of –points offor some algebra . The representation ρ_n() →_ ([X,Y]^q_n) is related to the coaction τ_n [X,Y]_n^q →⊗[X,Y]_n^qby the property that, if an –point A ∈() is considered as an algebra homomorphism A →, then ρ_n(A) ∈_ ([X,Y]_n^q) isthe –linear extension of the –linear map (A ⊗_[X,Y]_n^q) ∘τ[X,Y]_n^q →[X,Y]_n^q.When q is not a k–root of unity with k≤ n, the representationρ_1 ⊗_ρ_n () →_( [X,Y]_1^q ⊗_[X,Y]_n^q )is isomorphicoverto the direct sum ρ_n+1⊕ρ_n-1 of the representations ρ_n+1() →([X,Y]_n+1^q ) and ρ_n-1() →([X,Y]_n-1^q ). Namely, there exists a –linear isomorphismϕ[X,Y]_1^q ⊗_[X,Y]_n^q→[X,Y]_n+1^q⊕[X,Y]_n-1^q,inducing an –linear isomorphism_⊗_ϕ[X,Y]_2^q ⊗_[X,Y]_n^q →[X,Y]_n+1^q⊕[X,Y]_n-1^q ,such that the diagram[X,Y]_1^q ⊗_[X,Y]_n^q [d]__⊗_ϕ^≅[rrr]^ρ_1(A) ⊗_ρ_n(A) [X,Y]_1^q ⊗_[X,Y]_n^q [d]^_⊗_ϕ_≅ [X,Y]_n+1^q⊕[X,Y]_n-1^q[rrr]_ρ_n+1(A) ⊕ρ_n-1(A) [X,Y]_n+1^q⊕[X,Y]_n-1^qcommutes for every A∈().This is an immediate consequence of Proposition <ref> and of the relationship between the coactions τ_k [X,Y]_k^q →⊗_[X,Y]_k^q and the –linear mapsρ_k(A) ∈_ ([X,Y]_k^q).The fact that the isomorphism [X,Y]_1^q ⊗_[X,Y]_n^q→[X,Y]_n+1^q⊕[X,Y]_n-1^q between ρ_1 ⊗_ρ_n and ρ_n+1⊕ρ_n-1comes from a –linear isomorphism[X,Y]_1^q ⊗_[X,Y]_n^q→[X,Y]_n+1^q⊕[X,Y]_n-1^qwill be crucial for our consideration of traces in the next section.§.§ The trace of ρ_n(A) for A∈()After the hard work of <ref> and <ref>, we now have appropriate tools to compute for A∈() the trace of ρ_n(A) in terms of the trace of A. The few computations that we did at the end of <ref> should convince the reader that this result would be hard to obtain without the heavy machinery of <ref>–<ref>. For every ( [ a b; c d ])∈(),ρ_n ( [ a b; c d ])= S_n (a+d)where S_n is the n–th Chebyshev polynomial of the second kind.The property makes sense for all q, but we first restrict attention to the case where q is not a root of unity in order to apply Proposition <ref>. Consider A= ( [ a b; c d ]) ∈(). By Proposition <ref>, the –linear mapsρ_1(A) ⊗_ρ_n(A) and ρ_n+1(A) ⊕ρ_n-1(A) are isomorphic over . By Lemma <ref>, they consequently have the same trace.Therefore,( ρ_1(A))( ρ_n(A))= (ρ_1(A) ⊗_ρ_n(A))= (ρ_n+1(A) ⊕ρ_n-1(A) )= (ρ_n+1(A)) + ( ρ_n-1(A))and ρ_n(A) therefore satisfies the same recurrence relation (<ref>–<ref>) as the Chebyshev polynomials. For n=0, ρ_0(A)=_ and ρ_0(A)=1. By definition of the Chebyshev polynomial S_n(t) in (<ref>), it follows that ρ_n(A) = S_n( ρ_1(A) ). We already computed ρ_1(A) in <ref>. If A= ( [ a b; c d ]), then ρ_1(A) has matrix ( [ a c; b d ]) in the –basis {X,Y} for [X,Y]_1^q = ⊗[X,Y]_1^q. (Note that this is the transpose matrix.) It follows that ρ_1(A) = a+d, which concludes our computation when q is not a root of unity. The case where q is a root of unity follows from this generic case by continuity. To justify this continuity argument for an arbitrary algebra , first consider the case when =, where we can make sense of continuity with respect to q (for instance by consideringas an algebra over [q, q^-1]). The algebraadmits a tautological –point I ∈(), defined by the identity algebra homomorphism I →. Then, ρ_n(I) = S_n(I ) ∈ for every q by continuity from the case where q is not a root of unity. For a general algebraand an –point A ∈(), a little thought will convince the reader thatρ_n(A)∈ is the image of ρ_n(I)∈ under the algebra homomorphism A →; in particular, A = A(I) by specialization to the case n=1. Then, ρ_n(A)= A ( ρ_n(I) ) = A ( S_n(I ) )= S_n ( A(I ) ) = S_n ( A)for every q, using the fact that A is an algebra homomorphism for the third equality.§ MIRACULOUS CANCELLATIONSWe now prove the main result of this article, namely Theorem <ref> which we rephrase a Theorem <ref> below. Although we just encountered Chebyshev polynomials S_n(t) of the second kind, the property involves the Chebyshev polynomials T_n(t) of the first kind. Note that, if an –point A=( [ a b; c d ]) ∈() ofis upper triangular, namely is such that c=0, then necessarily d=a^-1 by the quantum determinant relation ad-q^-1bc=1 of Relation (<ref>). As a consequence, A can be written as A=( [ab;0 a^-1 ]) with ba=qab. Similarly, any lower triangular element of () is of the form ( [a0;b a^-1 ]) with ba=qab. Let A_1, A_2, …, A_k∈() be 𝒜–points ofsuch that: * each A_i is triangular of the form( [a_ib_i;0 a_i^-1 ]) or ( [a_i0;b_i a_i^-1 ]) for some a_i, b_i∈(with b_ia_i = q a_i b_i);* a_i and b_i commute with a_j and b_j whenever i ≠ j, so that we can make sense ofthe product A_1A_2 … A_n ∈(). Then, if q^2 is a primitive n–root of unity,T_n (A_1A_2… A_k-1A_k) =A_1^(n)A_2^(n)… A_k-1^(n) A_k^(n)where, for each i,A_i^(n) = ( [a_i^nb_i^n;0 a_i^-n ]) or ( [a_i^n0;b_i^n a_i^-n ]) is the 𝒜–point ofSL_2^q^n^2 obtained from A_i =( [a_ib_i;0 a_i^-1 ]) or ( [a_i0;b_i a_i^-1 ]) by replacinga_i and b_i with their powers a_i^n and b_i^n, respectively.Note that b_i^n a_i^n = q^n^2 a_i^n b_i^n since b_ia_i = qa_ib_i. Also, q^n^2 is equal to ±1 since q^2n=1, andis always +1 when n is even. For notational convenience, we will reverse the indexing and prove the equivalent statement thatT_n (A_kA_k-1… A_2A_1) =A_k^(n)A_k-1^(n)… A_2^(n) A_1^(n).For this, we will use Lemma <ref> and Corollary <ref>, so that T_n (A_kA_k-1… A_2A_1)= S_n (A_kA_k-1… A_2A_1) - S_n-2(A_kA_k-1… A_2A_1)=ρ_n ( A_kA_k-1… A_2A_1) - ρ_n-2 (A_kA_k-1… A_2A_1 )for the representations ρ_m() →_([X,Y]^q_m ) of <ref>. We first compute these traces.When A_i is lower triangular,the image of X^n-uY^u ∈[X,Y]^q_n under ρ_n(A_i) is ρ_n(A_i)(X^n-u Y^u ) =ρ_n([a_i0;b_i a_i^-1 ]) (X^n-u Y^u) =(a_iX)^n-u (b_iX + a_i^-1Y)^u= a_i^n-u X^n-u∑_v=0^uuvq^2b_i^u-v X^u-va_i^-vY^v= ∑_v=0^uuvq^2q^-v(u-v) a_i^n-u-v b_i^u-v X^n-vY^v,using the Quantum Binomial Formula (<ref>) of <ref>.In particular, if we express ρ_n(A_i) in the basis {X^n-uY^u; u=0, 1, …, n} for [X,Y]^q_n, the entries of the corresponding matrix areρ_n(A_i)_vu =uvq^2q^-v(u-v) a_i^n-u-v b_i^u-vifv≤ u0 if v>u.Note that this matrix is upper triangular. Similarly, when A_i is upper triangular, ρ_n(A_i)(X^n-u Y^u ) =ρ_n([a_ib_i;0 a_i^-1 ]) (X^n-u Y^u) =(a_iX+ b_iY)^n-u ( a_i^-1Y)^u= ∑_v=u^nn-un-vq^2q^-u(v-u) a_i^n-u-v b_i^v-u X^n-vY^vandρ_n(A_i)_vu = 0 ifv< u n-un-vq^2q^-u(v-u) a_i^n-u-v b_i^v-uif v≥ u.In particular,ρ_n(A_kA_k-1… A_2A_1) = ρ_n(A_1) ∘ρ_n(A_2) ∘…∘ρ_n(A_k-1) ∘ρ_n(A_k) =-10pt ∑_u_1,u_2, …, u_k ∈{0, …, n} -10pt ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3……ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1andρ_n-2(A_kA_k-1… A_2A_1) =-10pt ∑_v_1,v_2, …, v_k ∈{0, …, n-2} -10pt ρ_n-2(A_1)_v_1v_2 ρ_n-2(A_2)_v_2v_3……ρ_n-2(A_k-1)_v_k-1v_k ρ_n-2(A_k)_v_kv_1,where the terms ρ_n(A_i)_vu are given by Equations (<ref>) and (<ref>). We distinguish three types of terms in the sum of Equation (<ref>), according to the corresponding indices u_1, u_2, …, u_k ∈{0, …, n}: (i) no u_i is equal to 0 or n;(ii) some but not all u_i are equal to 0 or n;(iii) all u_i are equal to 0 or n; We begin with the first type.If no u_i is equal to 0 or n, the termU_n(u_1, …, u_k)=ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1of Equation (<ref>) corresponding tou_1, u_2, …, u_k ∈{1, …, n-1} is equal to the termU_n-2(v_1, …, v_k) =ρ_n-2(A_1)_v_1v_2 ρ_n-2(A_2)_v_2v_3…ρ_n(A_k-1)_v_k-1v_k ρ_n-2(A_k)_v_kv_1of Equation (<ref>) corresponding to the indices v_1, v_2, …, v_k ∈{0, …, n-2} with v_i=u_i-1. Set u_k+1=u_1 and v_k+1=v_1 to introduce uniformity in the notation.If A_i is lower triangular, Equation (<ref>) givesρ_n(A_i)_u_iu_i+1 =u_i+1u_iq^2q^-u_i(u_i+1-u_i) a_i^n-u_i+1-u_i b_i^u_i+1-u_iifu_i≤ u_i+10 if u_i>u_i+1while, using the property that v_i=u_i-1,ρ_n-2(A_i)_v_iv_i+1 =u_i+1-1u_i-1q^2q^-(u_i-1)(u_i+1-u_i) a_i^n-u_i+1-u_i b_i^u_i+1-u_iifu_i≤ u_i+10 if u_i>u_i+1.Sinceuvq^2= uq^2/vq^2u-1v-1q^2, it follows thatρ_n(A_i)_u_iu_i+1 =u_i+1q^2/u_iq^2q^u_i-u_i+1ρ_n-2(A_i)_v_iv_i+1when A_i is lower triangular. Similarly, when A_i is upper triangular,ρ_n(A_i)_u_iu_i+1 = 0 ifu_i< u_i+1 n-u_i+1n-u_iq^2q^-u_i+1(u_i-u_i+1) a_i^n-u_i+1-u_i b_i^u_i-u_i+1if u_i≥ u_i+1.andρ_n-2(A_i)_v_iv_i+1 = 0 ifu_i< u_i+1 n-u_i+1-1n-u_i-1q^2q^-(u_i+1-1)(u_i-u_i+1) a_i^n-u_i+1-u_i b_i^u_i-u_i+1if u_i≥ u_i+1,Thereforeρ_n(A_i)_u_iu_i+1 =n-u_i+1q^2/n-u_iq^2q^u_i+1-u_iρ_n-2(A_i)_v_iv_i+1= -q^-2u_i+1u_i+1q^2/-q^-2u_iu_iq^2q^u_i+1-u_iρ_n-2(A_i)_v_iv_i+1= u_i+1q^2/u_iq^2q^u_i-u_i+1ρ_n-2(A_i)_v_iv_i+1using the property thatn-uq^2 = q^2n-2u-1/q^2-1 =-q^-2uq^2u-1/q^2-1 = -q^-2uuq^2since q^2n=1.As a consequence, we get the same formula whether A_i is upper or lower triangular. Taking the product over all i,U_n(u_1, …, u_k)= U_n-2(v_1, …, v_k) ∏_i=1^k u_i+1q^2/u_iq^2q^u_i-u_i+1=U_n-2(v_1, …, v_k) ,where the second equality comes from the fact that u_k+1=u_1. This proves Lemma <ref>.If some but not all indices u_i are equal to 0 or n, the termU_n(u_1, …, u_k)=ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1of Equation (<ref>) corresponding tou_1, u_2, …, u_k ∈{0, …, n} is equal to 0.This is a consequence of Lemma <ref>, which says that, because q^2 is a primitive n–root of unity, the quantum binomial coefficient nuq^2 is equal to 0 for 0<u<n. For convenience, set u_k+1=u_1 as in the proof of Lemma <ref>. By hypothesis, there is then an index i such that 0<u_i<n and u_i+1=0 or n. Consider first the case when 0<u_i<n and u_i+1=0. If A_i is lower triangular, then ρ_n(A_i)_u_iu_i+1=0 by Equation (<ref>), and consequently U_n(u_1, …, u_k)=0.Otherwise, Equation (<ref>) givesρ_n(A_i)_u_iu_i+1 =nn-u_iq^2a_i^n-u_i b_i^u_i =0by Lemma <ref>. This proves that U_n(u_1, …, u_k)=0 in this case. Similarly, if0<u_i<n and u_i+1=n, Equation (<ref>) immediately shows that U_n(u_1, …, u_k)=0 when A_i is upper triangular, and otherwise gives ρ_n(A_i)_u_iu_i+1 =nu_iq^2q^-u_i(n-u_i) a_i^-u_i b_i^n-u_i =0by Lemma <ref>, again proving that U_n(u_1, …, u_k)=0.Lemmas <ref> and <ref> show that, when computingT_n( A_kA_k-1… A_2A_1) = ρ_n(A_kA_k-1… A_2A_1) - ρ_n-2(A_kA_k-1… A_2A_1)using the expressions of Equations(<ref>–<ref>), the only terms left are ∑_u_1,u_2, …, u_k ∈{0, n} -10pt ρ_n(A_1)_u_1u_2 ρ_n(A_2)_u_2u_3…ρ_n(A_k-1)_u_k-1u_k ρ_n(A_k)_u_ku_1whereρ_n(A_i)_u_iu_i+1 = a_i^nifu_i= u_i+1=0b_i^nifu_i=0andu_i+1=n0 if u_i =nandu_i+1=0a_i^-nifu_i= u_i+1=nif A_i is lower triangular, and ρ_n(A_i)_u_iu_i+1 = a_i^nif u_i= u_i+1=00 ifu_i=0andu_i+1=n b_i^nif u_i =nandu_i+1=0a_i^-nif u_i= u_i+1=nif A_i is upper triangular. As a consequence, comparing the general case to the case n=1,T_n( A_kA_k-1… A_2A_1)=-10pt∑_u_1,u_2, …, u_k ∈{0, 1} -10pt ρ_1(A_1^(n))_u_1u_2 ρ_1(A_2^(n))_u_2u_3……ρ_1(A_k-1^(n))_u_k-1u_k ρ_1(A_k^(n))_u_ku_1= ρ_1(A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n))where A_i^(n) = ([ a_i^n b_i^n; b_i^n d_i^n ]) is obtained from A_i = ([ a_i b_i; b_i d_i ])by replacing a_i, b_i, b_i, d_i with a_i^n, b_i^n, b_i^n, d_i^n, respectively. We already observed that, for an –point A = ([ a b; c d ]) ∈(), the matrix of ρ_1(A) in the basis {X,Y} for [X,Y]_1^q is the transpose ([ a c; b d ]), so that ρ_1(A) = a+d =A. It follows that T_n( A_kA_k-1… A_2A_1)=ρ_1(A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n))= A_k^(n) A_k-1^(n)… A_2^(n) A_1^(n).This is exactly the relation (<ref>) that we wanted to prove, which concludes the proof of Theorem <ref>.§ A POSITIVITY PROPERTY Many fewer cancellations occur when q is not a root of unity. This can be precisely quantified by usinga certain positivity property for T_n (A_1A_2… A_k).Let A_1A_2 … A_k be a product of triangular matrices A_i=([a_ib_i;0 a_i^-1 ]) or([a_i0;b_i a_i^-1 ]) ∈() satisfying the hypotheses ofTheorems <ref> or <ref>, except that we are not requiring q to be a root of unity. Then, A_1A_2… A_k can be written as a sum of monomials ∏_i=1^k c_i with c_i∈{a_i, b_i, a_i^-1} and therefore, for every polynomial P(t) ∈[t] with integer coefficients, P (A_1A_2… A_k) can be written as a sum of monomials of the form ± q^ξ∏_i=1^k a_i^α_i b_i^β_i with integer powers ξ, α_i, β_i ∈ (with β_i≥ 0).The following result states that, when P(t) is one of the Chebyshev polynomials S_n(t) or T_n(t),the signs ± can always be taken to be +. Under the hypotheses ofTheorems <ref> or <ref> but without any assumption on the parameter q∈-{0},the evaluationsS_n (A_1A_2… A_k) and T_n (A_1A_2… A_k) ∈ of the Chebyshev polynomials can be written as a sum of positive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with integer powers ξ, α_i, β_i ∈.The case of S_n(t) isrelatively simple. We computedS_n (A_1A_2… A_k) = ρ_n ( A_1A_2… A_k )in the course of the proof of Theorem <ref>. In particular, Equations (<ref>–<ref>) show that, with no assumption on q, ρ_n ( A_1A_2… A_k ) is a sum of positive monomials + q^ξ∏_i=1^k a_i^α_i b_i^β_i. Indeed, it is well-known (and also follows from the Quantum Binomial Formula (<ref>))that the quantum binomial coefficients uvq^2 are polynomials in q^2 with nonnegative integer coefficients. The proof for T_n(t) is more elaborate. We want to show that, when computing T_n (A_1A_2… A_k) = ρ_n ( A_1A_2… A_k ) - ρ_n-2 ( A_1A_2… A_k ),each monomial of ρ_n-2 ( A_1A_2… A_k ) cancels out with a monomial of ρ_n ( A_1A_2… A_k ); this can be seen as a weaker form of Lemma <ref>. For this, we will give a different computation of ρ_n ( A_1A_2… A_k ). This computation goes back to the principles underlying the Quantum Binomial Formula. Let ⟨ X,Y ⟩ be the free algebra generated by the set {X,Y}. Namely, ⟨ X,Y ⟩ consists of all formal polynomials P(X,Y) in noncommutating variables X and Y, and these polynomials are multiplied without simplifications. In particular, the quantum plane [X,Y]^q is the quotient of ⟨ X,Y ⟩ by the ideal generated by YX-qXY, which gives a natural projection π⟨ X,Y ⟩→[X,Y]^q. Similarly, consider ⟨ X,Y ⟩ = ⊗⟨ X,Y ⟩, and the projection _⊗π which we will also denote as π⟨ X,Y ⟩→[X,Y]^q for short.Let ⟨ X,Y ⟩_n be the linear subspace of ⟨ X,Y ⟩ consisting of all homogeneous polynomials of degree n. Namely,⟨ X,Y ⟩ _n consists of all finite sumsP(X,Y) = ∑_u α_u Z_u1 Z_u2… Z_unwherethe coefficients α_u are inand where each variable Z_uv is equal to X or to Y. In particular, ⟨ X,Y ⟩_n is isomorphic to ^2^n as an –module. For comparison, remember that [X,Y]^q_n is isomorphic to ^n+1. The representation ρ_n () →_( [X,Y]^q_n ) lifts to a representation ρ_n () →_( ⟨ X,Y⟩_n ) defined by the property thatρ_n ( [ a b; c d ]) (P(X,Y) ) = P(aX+bY, cX + dY )for every homogeneous polynomial P(X,Y) ∈⟨ X,Y ⟩of degree n in the noncommuting variables X and Y (and with coefficients in ). To give a more combinatorial description of this action, note that every element of ⟨ X,Y⟩_n can be uniquely written as a sum of monomialsα Z_1Z_2 … Z_n where α∈ andeach Z_u ∈{X,Y}. Then, ρ_n([ a b; c d ]) (α Z_1Z_2 … Z_n) ∈⟨ X,Y ⟩_n is the sum of all monomials α' Z_1'Z_2' … Z_n' obtained from α Z_1Z_2 … Z_n by replacing each Z_u with:• either aX or bY, if Z_u=X; •either cX or dY, if Z_u=Y,(and pushing all coefficients ofto the front). As a consequence, ρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n)=ρ_n(A_k) ∘ρ_n(A_k-1) ∘…∘ρ_n(A_1) (α Z_1Z_2 … Z_n)can be described as follows. Let ℳ(α Z_1Z_2 … Z_n) be the set of allsequences M =(M_i)_i=0, 1, …, k of monomials M_i=α_i Z_i1Z_i2… Z_in∈⟨ X,Y ⟩_n such that* M_0 = α Z_1Z_2 … Z_n; * M_i is obtained from M_i-1 by replacing each Z_(i-1)u with• a_i X or b_iY if Z_(i-1)u=X and A_i = ([a_ib_i;0 a_i^-1 ]);• a_i^-1 Y if Z_(i-1)u=Y and A_i = ([a_ib_i;0 a_i^-1 ]);• a_i Xif Z_(i-1)u=X and A_i = ([a_i0;b_i a_i^-1 ]);• b_i Xor a_i^-1 Y if Z_(i-1)u=Y and A_i = ([a_i0;b_i a_i^-1 ]).Then, as we expand ρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n)∈⟨ X,Y⟩_n, we see that itsdecompositioninto monomials is given byρ_n(A_1A_2 … A_k) (α Z_1Z_2 … Z_n) = ∑_M ∈ℳ(α Z_1Z_2 … Z_n) M_kwhere M_k is the last term of the sequence M =(M_i)_i=0, 1, …, k∈ℳ(α Z_1Z_2 … Z_n). As a consequence, if we use the same notation for the monomial X^n-uY^u=X … XY … Y ∈⟨ X,Y⟩_n and for its image X^n-uY^u ∈[X,Y]^q_n under the projection π⟨ X,Y⟩_n →[X,Y]^q_n, ρ_n(A_1A_2 … A_k) ( X^n-uY^u ) = ∑_M ∈ℳ(X^n-uY^u)π(M_k). Finally, let ℳ'(X^n-uY^u) be the set of monomial sequences M ∈ℳ(X^n-uY^u) whose contributionπ(M_k) belongs to X^n-uY^u. For such a monomial sequence M =(M_i)_i=0, 1, …, k, let α(M_k)∈ be the coefficient such that π(M_k) = α(M_k) X^n-uY^u. We can then compute the trace of ρ_n(A_1A_2 … A_k) by usingthe basis { X^n-uY^u; u=0,1,…,n} for [X,Y]^q_n and [X,Y]^q_n = ⊗[X,Y]^q_n, which givesρ_n(A_1A_2 … A_k) = ∑ _u=0^n ∑_M ∈ℳ'(X^n-uY^u)α(M_k). Similarlyρ_n-2(A_1A_2 … A_k) = ∑ _v=0^n-2 ∑_M ∈ℳ'(X^n-v-2Y^v)α(M_k). The expression (<ref>) for ρ_n(A_1A_2 … A_k) was obtained by using the basis { X^n-uY^u; u=0,1,…,n} for [X,Y]^q_n. For comparison with (<ref>), it is more convenient to use the other basis {X^n, Y^n}∪{ X^n-v-2Y^vXY; v=0,1,…,n-2} for [X,Y]^q_n. This gives the expression ρ_n(A_1A_2 … A_k)=∑_M' ∈ℳ'(X^n)α(M_k') +∑_M' ∈ℳ'(Y^n)α(M_k')+∑ _v=0^n-2∑_M' ∈ℳ'(X^n-v-2Y^vXY)β(M_k')where, for M' =(M_i')_i=0, 1, …, k∈ℳ'(X^n-v-2Y^vXY), the coefficient β(M_k') ∈ is defined by the property that π(M_k') = β(M_k') X^n-v-2Y^vXY.Note that, because of the commutativity and q–commutativity properties of the quantities a_i, b_i, X, Y, all terms α(M_k) and β(M_k') in (<ref>–<ref>)are positive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with ξ, α_i, β_i ∈ and β_i≥ 0. .We now compare (<ref>) and (<ref>).Every monomial sequenceM =(M_i)_i=0, 1, …, k∈ℳ'(X^n-v-2Y^v) gives rise to a monomial sequence M'∈ℳ'(X^n-v-2Y^vXY) defined by the property that M_i' = M_iXY for every i. Indeed, rewritingM_i' = M_i (a_iX)(a_i^-1Y)shows that M' =(M_i')_i=0, 1, …, k really satisfies the inductive property defining ℳ'(X^n-v-2Y^vXY). In addition, when M'∈ℳ'(X^n-v-2Y^vXY) is thus associated to M ∈ℳ'(X^n-v-2Y^v), π(M_k') = π(M_kXY) =π(M_k)π(X)π(Y) = α(M_k) X^n-v-2Y^vXY so that β(M_k')= α(M_k). Therefore, when computingT_n (A_1A_2… A_k) = ρ_n ( A_1A_2… A_k ) - ρ_n-2 ( A_1A_2… A_k ),every monomial α(M_k) occurring in (<ref>) cancels out with a monomialβ(M_k') of (<ref>). It follows that T_n (A_1A_2… A_k) is the sum of the remaining coefficients α(M_k) and β(M_k') of (<ref>). We already observed that these monomials are positive, which concludes the proof of Proposition <ref>.Proposition <ref> enables us toprecisely determine the number of monomials in T_n (A_1A_2… A_k) under the hypothesis that there are no extraneous simplifications. This means that q is transcendental and, since we can always assume that the algebrais generated by the entries a_i, b_i of the matrices A_i, thatis the algebra defined by the generators a_i^± 1, b_i and by the relations that b_ia_i = qa_i b_i and that a_i, b_i commute with a_j, b_j whenever i ≠ j. In other words,is the algebra ⊗_i=1^k [a_i^±1, b_i]^q. In this case, every element of =⊗_i=1^k [a_i^±1, b_i]^q has a unique decomposition as a sum of monomials ξ∏_i=1^k a_i^α_i b_i^β_i withξ∈, α_i∈, β_i ∈ and β_i≥ 0. Supposethat q is transcendental, and that =⊗_i=1^k [a_i^±1, b_i]^q. Let triangular matrices A_i=([a_ib_i;0 a_i^-1 ]) or([a_i0;b_i a_i^-1 ]) ∈() be given for i=1, 2, …, k, and consider the positive integer t_0=A_1^(0)A_2^(0)… A_k^(0)where A_i^(0) = ([ 1 1; 0 1 ]) if A_i=([a_ib_i;0 a_i^-1 ]) and A_i^(0)=([ 1 0; 1 1 ]) if A_i=([a_i0;b_i a_i^-1 ]).Then,for every n, T_n (A_1A_2… A_k) is the sum of exactly T_n(t_0) =( t_0+√(t_0^2-4)/2)^n + ( t_0-√(t_0^2-4)/2)^npositive monomials of the form + q^ξ∏_i=1^k a_i^α_i b_i^β_i with ξ, α_i, β_i ∈ and β_i≥0. We already proved in Proposition <ref> that T_n (A_1A_2… A_k) is a sum of monomials of the type indicated. The only issueis to count their number. Because of the positive signs, the number of these monomials can be computed by letting q and the a_i, b_i tend to 1. Under this limiting process, A_1A_2… A_k approaches t_0, and the number of monomials in the expansion for T_n (A_1A_2… A_k) is thereforeequal to T_n(t_0). The formula T_n(t_0) =( t_0+√(t_0^2-4)/2)^n + ( t_0-√(t_0^2-4)/2)^n is provided by Lemma <ref>. amsalpha | http://arxiv.org/abs/1708.07617v2 | {
"authors": [
"Francis Bonahon"
],
"categories": [
"math.QA",
"math.GT",
"20G42"
],
"primary_category": "math.QA",
"published": "20170825055133",
"title": "Miraculous cancellations for quantum $SL_2$"
} |
A Stochastic Analysis of Bike Sharing Systems Shuang Tao School of Operations Research and Information Engineering Cornell University 293 Rhodes Hall, Ithaca, NY [email protected] Jamol Pender School of Operations Research and Information Engineering Cornell University 228 Rhodes Hall, Ithaca, NY [email protected] December 30, 2023 ==============================================================================================================================================================================================================================================================================================================================§ INTRODUCTION In this paper, we discuss “backward simulation,” which traces a time-reversed path from a target region A to the initial configuration (Fig. <ref>). If the outputs of the original simulation (“forward simulation”) are easily restored from those obtained by backward dynamics, we can use backward simulation as a computational tool. In particular, the time required to calculate the probability to reach A from the initial configurations can be significantly reduced when the target region A is small but the initial distribution is broad. An example is a computation of the probability that a typhoon will hit the Tokyo area exactly under a given stochastic model (Sect. <ref>).It is, however, difficult to design backward dynamics with the desired properties. Specifically, consider the forward dynamics of a D-dimensional stochastic process X defined by X_i+1 = g(X_i) + η_i, where η_i is an independent noise that obeys an arbitrary distribution and the function g: ℝ^D→ℝ^D describes noiseless forward dynamics. Then, a naïve way to derive a time-reversed equation isto rearrange Eq. (<ref>) asX_i = g^-1(X_i+1 - η_i). Here, we assume that function g is a one-to-one and onto function and denotes the inverse function of g as g^-1. We can construct a time-reversed path iteratively, using Eq. (<ref>) and the independent realization of η_i, starting from the target region. It defines an apparently natural candidate for backward dynamics.Surprisingly, this naïve method does not work as expected; it does not reproduce the correct probabilities defined by the forward simulation, and the calculation of factors required to correct the bias is often computationally expensive. This becomes clear in Sect. <ref>. Furthermore, the computation of g^-1 in Eq. (<ref>) is time-consuming and reduces the efficiency of the computation.The aim of this research is to draw attention to these facts and propose an algorithm that partially resolves the problem. We named this algorithm the time reverse Monte Carlo method (TRMC). TRMC is based on the ideas of sequential importance sampling (SIS) <cit.> and sequential Monte Carlo (SMC) <cit.>. We discuss TRMC based on SIS in Sects. <ref> and <ref> and its improved version based on SMC in Sect. <ref>.There have been several studies using “path reweighting” in computational chemistry <cit.>. These studies used reweighting for different purposes.TRMC based on SIS is tested for a stochastic typhoon model and the Lorenz 96 model in Sect. <ref>. The improved version with SMC is also tested in Sect. <ref> for simulations with a larger number of steps. In these examples, TRMC provides unbiased estimates of the probabilities without expensive computation. In Sect. <ref>, we discuss its relation to the Bayes formula, as well as the possible improvement and limitations of TRMC. Time-reversed dynamics itself was discussed in several studies <cit.>, mostly from a theoretical viewpoint. On the other hand, related computational problems are found in data science, especially in time-series analysis usingstate-space models <cit.>. Our problem can formally be regarded as a limiting case of the “smoothing” part of these algorithms, where only one observation (“target”) is available at the end of the time series. There are, however, important differences from our problem, which are discussed in Sect. <ref>. Studies related to the statistical inferenceon a discrete state stochastic process, such as genepropagation <cit.> andinformation source detection <cit.> were also reported. These studies, however, did not considerdynamical systems of continuous variables.§ FAILURE OF NAÏVE METHODHere, we provide a detailed discussion of the naïve method and its drawbacks,which form the motivation for our algorithm.Before providing details, we formulate the problem. Let S_T = {0 = t_0 ≤ t_1 ⋯≤ t_N = T} be a partition of the interval [0, T], and let step size Δ t= t_i+1-t_i be a constant; x_i is used to represent the value of stochastic process X at time point t_i. The transition probability density from x_i to x_i+1 defined by Eq. (<ref>) is denoted as p( x_i+1 | x_i). We consider an estimation of the probability that X_N hits a small target region A in the D-dimensional space. The probability is formally written asP(X_N ∈ A) = ∫ dx_0:N1_x_N ∈ A{∏_i=0^N-1p( x_i+1 | x_i) } p(x_0) ,where 1_x ∈ A is the indicator function that takes value 1 when x ∈ A, and 0 otherwise, andp(x_0) is the initial distribution of the forward simulation. Hereafter,dx_k:l indicates dx_kdx_k+1⋯ dx_l for k ≤ l. A naïve method is defined as a repeated simulationwith a uniformly distributed initial condition inthe target region A using Eq. (<ref>). Initially, it appears sufficient to evaluateas . However, there are two problems with this naïve method. First, the exact computation of g^-1 in Eq. (<ref>) is not easy. Computing g^-1 using numerical root-finding techniques such as the Newton-Raphson method is computationally intensive and its severity increases as the dimension increases.Second, this computation does not reproduce the correct probabilityeven with the exact g^-1. To understand this problem, we show the difference between the forward simulation and the naïve method. Let us define Y_i=X_i-η_i-1 = g(X_i-1); i ∈ [1, …, N].Using this definition, we can rewrite Eq. (<ref>) as Y_i + η_i-1 = g^-1(Y_i+1) . Equation (<ref>) can be simplified intoY_i =g^-1(Y_i+1) - η_i-1. The probability calculated using Eq. (<ref>) corresponds tothe equation∫ dy_1:N dx_N1_x_N ∈ Ap̃_f( y_N | x_N) {∏_i=1^N-1p̃( y_i | y_i+1) } p(g^-1(y_1)) ,where p̃_f( y_N | x_N ) is the transition probability density from x_N to y_N defined by Eq. (<ref>) with i=N and p̃( y_i | y_i+1) is the transition probability density from y_i+1 to y_i defined by Eq. (<ref>). An initial condition x_N is uniformly distributed in the target region A.We have to introduce the Jacobian of function g so that Eq. (<ref>) is consistent with Eq. (<ref>). To show this, Eq. (<ref>) is rewritten using equationsp(x_i|x_i-1) dx_i = |det(J_g^-1(y_i+1))| p̃( y_i | y_i+1) dy_i, i∈ [1,⋯, N-1] p(x_0) dx_0 = |det(J_g^-1(y_1))| p( g^-1(y_1) ) dy_1,where |det(J_g^-1(y_i))| is the absolute value of the Jacobian of function g^-1. As a result, the probability P(X_N ∈ A) is calculated as P(X_N ∈ A) = ∫ dy_1:N dx_N 1_x_N ∈ A J(y_1, …, y_N) p̃_f( y_N | x_N) {∏_i=1^N-1p̃( y_i | y_i+1) } p( g^-1(y_1) ) , J(y_1, …, y_N) = {∏_i=0^N-1|det(J_g^-1(y_i+1))| }.We can obtain the correct probability using Eq. (<ref>) instead of Eq. (<ref>). The Jacobian J_g^-1 calculation is, however, computationally expensive.We note that the factor J(y_1, …, y_N) goes toexp( - ∫_0^Tdivf(x_t) dt )in the limit as Δ t → 0 when we assume that g(x) = x + f(x)Δ t. The proof of Eq. (<ref>) is given in Appendix <ref>. This shows that we must include factorJ(y_1, …, y_N) for unbiased estimation even in the limit of infinitesimal Δ t. We can regard the factor written in Eq. (<ref>) as the change inthe infinitesimal volume along each path (Fig. <ref>). § TIME REVERSE MONTE CARLO METHOD To overcome these difficulties, we propose the TRMC method. TMRC essentially involves introducing simplified backward dynamics with a weight. This weight enables the bias of estimators to be corrected. First, we introduce a backward transition probability q( x_i+1→ x_i) from x_i+1 to x_i. We can choose an arbitrary probability density q,while the computation efficiency strongly depends on it. Once we introduce q( x_i+1→ x_i),we can rewrite Eq. (<ref>) as P(X_N ∈ A) = ∫ dx_0:N1_x_N ∈ A/V_A{∏_i=0^N-1 q( x_i+1→ x_i) W_i} V_A p(x_0) , where W_i = p( x_i+1 | x_i) / q( x_i+1→ x_i)is the weight required to correct the bias of estimatorsand V_A is the volume of target region A. Suppose p(x_0) is uniformly distributed on B ⊂ℝ^D; p(x_0)=1/V_B1_x_0 ∈ B, V_B is the volume of B. The efficiency of our algorithm does not depend on the factor V_A when V_B is considerably large. This is the advantage of using our algorithm.The algorithm consists of the following steps.TRMC Algorithm * Draw M samples { x_N^(1), ⋯, x_N^(M)} from the uniform distribution in V_A. * Apply the following steps for j=1, …, M, and for i = N-1, …, 0. * Generate a sample from x^(j)_i+1 to x^(j)_i with transition probability q( x^(j)_i+1→ x^(j)_i). * Calculate weight W_i^(j) using Eq. (<ref>). * Evaluate the unbiased estimates of probability P(X_N ∈ A) as P(X_N ∈ A) ⋍1/M∑_j=1^M W^(j) , where the factor W^(j) = {∏_i=0^N-1 W_i^(j)} V_A p(x_0^(j)) is attached to each simulation path.The inputs of our algorithm are the number of Monte Carlo paths M,the number of time steps N, the initial distribution p(x_0), the target region A, and the transition probability density q. When we actually perform the simulation on our computers, we take the logarithm of these weights to prevent numerical overflow.This algorithm provides unbiased estimates of the desired probabilities. The idea of this scheme is a kind of SIS <cit.>. An advantage of our method is that we do not need to calculate g^-1 or their Jacobian matrices at each i.The remaining problem involves determining the method for choosing the transition probability q( x_i+1→ x_i).The basic idea is to choose the backward dynamics that generates trajectories similar to the forward dynamics defined by Eq. (<ref>). The similarity of the trajectory is measured by W^(j) in Eq. (<ref>).§ IMPLEMENTATION FOR STOCHASTIC DIFFERENCE EQUATION To give concrete examples of the transition probabilityq( x_i+1→ x_i),we assume the forward dynamics to be given in the following form:X_i+1 = X_i + f(X_i)Δ t + ϵ_i√(Δ t).This corresponds to the case wherein g(x) = x + f(x)Δ t in Eq. (<ref>).The noise ϵ_i is assumed to be i.i.d. Gaussian noise with mean zero and the variance-covariance matrix Σ = σσ^T. This class of equations appears in a wide range of problems in many different fields such as physics <cit.>, computational chemistry <cit.>, and mathematical finance <cit.>.In this case, as a simple choice, we can use the following backward dynamics:X_i = X_i+1 - f(X_i+1)Δ t + ϵ_i√(Δ t).This approximation corresponds tosubstituting f(X_i+1) for f(X_i) in Eq. (<ref>).With this choice, weight W_i in Eq. (<ref>) takes the form W_i = p( x_i+1 | x_i) / q( x_i+1→ x_i) = exp[ -1/2(x_i+1 - x_i -f(x_i)Δ t )^T(ΣΔ t)^-1(x_i+1 - x_i -f(x_i)Δ t ) ] /exp[ -1/2(x_i+1 - x_i -f(x_i+1)Δ t )^T(ΣΔ t)^-1(x_i+1 - x_i -f(x_i+1)Δ t ) ] = exp[ -( f(x_i+1) - f(x_i) )^TΣ^-1( (x_i+1 - x_i) - Δ t/2( f(x_i+1) + f(x_i) ) ) ] . As we show in the next section, the resultant algorithm is simple yet effective compared with the forward simulation when the target region A is smaller than the support of the initial distribution p(x_0).We note that the factor ∏_i=0^N-1 W_i goes toexp( - ∫_0^Tdivf(x_t) dt )in the limit as Δ t → 0. The proof of Eq. (<ref>) is given in Appendix <ref>. Note that Eq. (<ref>) coincides with Eq. (<ref>) derived from a different assumption.§ EXPERIMENTAL RESULTS We present the numerical results in this section. Forward simulations (FS) are used to check the consistency and computational efficiency of our result.Using forward and backward dynamics, we simulate sample trajectoriesx={x_1, ⋯, x_N } generated by each model andcompute the probability P(X_N ∈ A) from M independent simulations.We denote a standard error of TRMC to evaluate the computational efficiency by σ_s. We also denote the standard error of FS by σ_s^F. Using these variables, we define a relative value of variance by ρ_1 = ( σ_s^F/σ_s)^2 .The factor ρ_1 indicates the computational efficiency only including the effect caused by the variance of estimators for a fixed sample size. With this definition, more complex algorithms tend to be more efficient while they require more computational time. Then, we also define another measure of the relative computational efficiency ρ_2 asρ_2 = ρ_1τ^F/τ,where τ is the computational time in seconds of the simulation and τ^F is the computational time of FS in seconds.This efficiency is defined in the sense of the actual performance considering both the computational time and the variance of the resulting estimates. §.§ Stochastic typhoon model The first example is a stochastic typhoon model <cit.>, which gives an example of risk estimation by the proposed method.The stochastic typhoon model was designed to reproduce the statistics of typhoons in the northwestern part of the Pacific Ocean. This is a four-dimensional model given byx_i+1 = x_i + v_i, v_i+1 = V(x_i+1)+ w ( v_i - V(x_i) ) + ϵ_i , V(x_i) = a_0 + a_1x_ϕ, i + a_2sin x_λ, i + a_3sin^2 x_λ, i,where we use a global coordinate system defined by the geographic longitude (ϕ) and latitude (λ). We also define the two-dimensional position x = (x_ϕ, x_λ), speed v = (v_ϕ, v_λ) of a typhoon, and function V(x) = (V_ϕ(x), V_λ(x) ).w, a_0, a_1, a_2, and a_3 are constants. The noise ϵ obeys a Gaussian distribution with mean zero and variances σ^2.We fix w=0.93, a_0=(0.792, 0.538), a_1=(0.122, 0.371), a_2=(-0.513, 0.583), a_3=(0.770, -0.387), σ=0.4. The target region A is {(x_ϕ, x_λ); 138.5 ≤ x_ϕ≤ 139.5, 34.5 ≤ x_λ≤ 35.5 }. Since there is no range constraint on the distribution of the final speed v_f at the target,we adopt a uniform distribution with a suitably wide range U_f; here, U_f is defined as the region{ (v_ϕ, v_λ);V_ϕ(x_A)-3 ≤ v_ϕ≤ V_ϕ(x_A)+3, V_λ(x_A) -3 ≤ v_λ≤ V_λ(x_A)+3}, where x_A is the center of target region A.We also assume that the initial condition is uniformly distributed in D = { (x, v);111 ≤ x_ϕ≤ 129,-4 ≤ x_λ≤ 14,v ∈ U_0 }, where U_0 is defined as the region{ (v_ϕ, v_λ);V_ϕ(x_0)-1.5 ≤ v_ϕ≤ V_ϕ(x_0) + 1.5, V_λ(x_0) -1.5 ≤ v_λ≤ V_λ(x_0) + 1.5} andx_0 = (120, 5). This corresponds to the case wherein typhoons that occurred in the Philippines travel to the Tokyo area exactly with a small probability.We set M to 10^8 and N to 16. Examples of Monte Carlo paths for both simulations are given in Figs. <ref> and <ref>.This simulation was carried out on a laptop computer with 2.3 Ghz Intel core i5 and 8 GBytes memory.The computational time of TRMC in this simulation for generating 10^8 Monte Carlo paths is around 4.0 × 10^3 seconds.Table <ref> shows the result of computational experiments for the stochastic typhoon model. It shows that the probabilities of FS and TRMC agree within the error bars. If we ignore the factor defined by Eq. (<ref>), it does not reproduce the unbiased probability as in the case of the stochastic difference equation. Furthermore, it shows that TRMC is 4.2 times in terms of ρ_2 and 7.3 times in ρ1) more efficient than FS.Fig. <ref> shows the convergence of TRMC when the number of Monte Carlo paths M increases.It reveals that our algorithm converges correctly on increasing the number of Monte Carlo paths M. To simulate events with smaller probabilities, we make the target region A smaller as{(x_ϕ, x_λ); 138.75 ≤ x_ϕ≤ 139.25, 34.75 ≤ x_λ≤ 35.25 }. It shows that the smaller the probability, the more efficient our algorithm becomes as compared with the FS.In Fig. <ref>, a few Monte Carlo paths are shown to have moved northward. To prevent this from happening and improve its efficiency, we restrict the velocity distribution of Monte Carlo paths to the tendency to move southward. We change the range U_f of the final speed v_f to{ (v_ϕ, v_λ);V_ϕ(x_A)-3 ≤ v_ϕ≤ V_ϕ(x_A)+3, V_λ(x_A) -2 ≤ v_λ≤ V_λ(x_A)+2}. We call this simulation TRMC (restricted) in Fig. <ref>. Table <ref> shows that the probabilities of TRMC and TRMC (restricted) agree within error bars. Because the number of unnecessary Monte Carlo paths moving northward decreases, TRMC (restricted) is more efficient than TRMC.More severe constraint v_λ≥ 0, however, causes a small bias in the estimated probabilities:see TRMC (v_λ≥ 0) in Table <ref>. Fig. <ref> also shows that our algorithm converges correctly on increasing the number of Monte Carlo paths M.So far, we consider the case that the number of time steps from the initial position to Tokyo (N=16) is precisely known.We can relax this assumption, but we should be careful with a limitation of a discrete time model. A typhoon can pass nearby Tokyo, for example, between N=15 and N=16, which causes an underestimation of the actual risk when we only consider hit at integral time steps.A way to reduce this effect is to develop models with smaller steps, while it is also possible to introduce some initialization or interpolation method into a backward simulation. However, we leave this as a future problem, because this study aims to check whether the concept of backward simulation is mathematically valid. §.§ Lorenz 96 Model As a higher-dimensional example, we evaluate the efficiency of our algorithm for the Lorenz 96 model <cit.>. The Lorenz 96 model is an atmospheric model and was introduced by Edward Lorenz in 1996. It is defined as the set of coupled ordinary differential equations dx_k/dt = f_k(x) + ϵ_k, f_k(x)= -x_k-2x_k-1 + x_k-1x_k+1 - x_k + F, k= 1… K,where x={x_k; k=1… K } is the state of the system and F is a constant.We set K=9 and introduce Gaussian noise ϵ_k with mean zero and variance σ^2. Here, we choose F=8, a value known to cause weak chaotic behavior and often used as a benchmark in data assimilation <cit.>. To simulate Eq. (<ref>),we have to discretize it. While many discretization schemes are available, we focus on the simplest and most common scheme, the Euler scheme. The time-discretized version of Eq. (<ref>) by the Euler scheme isx_k, i+1 = x_k, i + f(x_i)Δ t + ϵ_kΔ t, k=1 … K,where we set Δ t to 0.001 and σ to 0.1/√(Δ t).The target region A is{(x_1, …, x_K)| -5.0 ≤ x_i ≤ 7.0; i=1 … K }.We also assume that the initial state for x_i is uniformly distributed in D={(x_1, …, x_K)| 1.5 ≤ x_i ≤ 8.5; i=1 … K }.We set M to 10^7 and N to 100. We conduct this simulation on the environment described in Sect. <ref>. The computational time of TRMC in this simulation for generating 10^7 Monte Carlo paths is around 3.0 × 10^3 seconds.Table <ref> shows the result of computational experiments for the Lorenz 96 model.It shows that the probabilities of TRMC and FS agree within the error bars. The case where we ignore the factor defined by Eq. (<ref>) does not reproduce the same unbiased probability as the other computational experiments. The result shows that TRMC can perform better for estimating the probabilities in thehigh-dimensional case. TRMC is 5.18 times more efficient than FS in terms of ρ_2, and 8.23 times in ρ1 more efficient in Table <ref>.Fig. <ref> shows the convergence of TRMC when the number of Monte Carlo paths M increases.It reveals that our algorithm converges correctly on increasing the number of Monte Carlo paths M. § IMPROVED SCHEME WITH RESAMPLING Let us consider cases with a larger number of time steps. The proposed algorithm may not always work efficiently in this situation. For example, we consider the case where N is equal to 500 in the Lorenz 96 model (Fig. <ref>); these weights are normalized such that their sum is 1, i.e., ∑ _j=1^MW^(j) = 1. The inset located at the top right of the figure shows the graph with a logarithmic scale on the x-axis. This style is also used in Fig. <ref>. The weight distribution corresponding toN=500in Fig. <ref> has a heavy-tailed distribution. This phenomenon is referred to as degeneracy, and it means that the weights become unbalanced, and a few weights dominate all the others. This consequently causes a decrease in computational efficiency <cit.>. We introduce an improved scheme to solve this problem, which is realized by resampling <cit.>. Hereafter, we denote it as TRMC (RS). This algorithm is effective when both the number of time steps and the amount of noise are large.Note that our algorithm is based on time-reversed dynamicsand uses SMC differently from the previous studies <cit.>on rare event sampling. We assume that the resampling procedure modifies the weight at s time step∏_i=0^s-1 W_iof each Monte Carlo path to an unweighted one by eliminating Monte Carlo paths having small weights and by multiplying Monte Carlo paths having large weights.We denote the jth Monte Carlo path as x^(j)= {x^(j)_0, …, x^(j)_s}. The procedure of resampling is as follows: * Define normalized weights W̃^(j) = ∏_i=0^s-1W_i^(j)/∑_j=1^M∏_i=0^s-1 W_i^(j). * Resample M times with replacement from set { x^(j)}_j=1^M of Monte Carlo paths, where the probability of sampling set of x^(j) is proportional to W̃^(j).After a resampling step,Monte Carlo paths {x^(j)}_j=1^Mand associated weights {W^(j)}_j=1^Mare replaced by the set of replicated Monte Carlo paths with an equal importance weight W^(j)=1/M∑_j=1^M∏_i=0^s-1W_i^(j).Degeneracy is estimated using the effective sample size <cit.>:M_eff = 1/∑_j=1^M (W̃^(j))^2.A small value of M_eff corresponds to high degeneracy.Hence, a resampling procedure is performed when this value is lower than a certain threshold Θ=α M, where α is a relative threshold. That is, a resampling procedure is performed when M_eff/M < α.We can use the Eq. (<ref>) to evaluate probabilities in this case. Figure <ref> shows a graphical scheme of resampling. Using this resampling,we simulate the Lorenz 96 model with σ=0.3/√(Δ t), which is larger than that in Sect. <ref>. We set the threshold α to 0.05, 0.5, and 0.9. The simulations with these threshold values of α are denoted by α=5%,50%, and 90% respectively.Table <ref> shows the result of computational experiments for the Lorenz 96 model. It shows that the probabilities of FS, TRMC, and TRMC (RS, α=50%) agree within the error bars.On the other hand, Fig. <ref> shows that TRMC (RS) is more efficient than TRMC in a wide range of threshold values. We also show the weight distributions of TRMC and TRMC (RS)in Fig. <ref>. The variance of the distribution is much smaller for TRMC (RS) than for TRMC. § DISCUSSIONThe examples provided in the preceding sections show that backward simulations using TRMC provide correct averages and can be more efficient than forward simulations. In these examples, the computational efficiencies of TRMC are 3–16 times higher than those obtained by forward simulation, when the calculated probability of hitting the target is 2 × 10^-3–10^-5.Note that TRMC can be used to calculate the probability for an arbitrarily small target region; this would be impossible by using forward simulation.There are, however, cases in which TRMC is inefficient. First, TRMC is not advantageous if the time-reversed paths rarely encounter a region in which the initial density p(x_0) is high; this can occur when the initial density is not broad. Another case in which TRMC can be inefficient is when the weight in Eq. (<ref>) (or, in the continuous time version, Eq. (<ref>)) is highly time dependent. If paths with smaller weights in the initial stage of backward simulation acquire larger weights in the latter stage, resampling of the path (particle splitting) in SMC may not be effective. In this case, if TRMC with SIS is ineffective, TRMC with SMC also shows poor performance. To discuss the possible improvement of the algorithm, it is useful to introduce optimal backward dynamics. Although it is not easy to obtain these dynamics a priori, the formal definition is derived as follows. First, the marginal probability at step n obtained from the forward simulation is defined asp(x_n) =∫ dx_0:n-1{∏_i=0^n-1p( x_i+1 | x_i) } p(x_0) _,which satisfies the relationp(x_i+1) =∫p(x_i+1|x_i)p(x_i) dx_i _.Using Eqs. (<ref>) and (<ref>), the transition probability q^* of the optimal backward dynamics is defined asq^*(x_i+1→ x_i)= p(x_i+1|x_i)p(x_i)/∫p(x_i+1|x_i)p(x_i) dx_i =p(x_i+1|x_i)p(x_i)/p(x_i+1)_. .Note that Eq. (<ref>) appears similar to the formulas used in Bayesian inference when the probability p(x_i) obtained by forward simulations is regarded as an analog of the prior distribution of x_i. In terms of the selection of Eq. (<ref>) for backward dynamics, the following relation holds:{∏_i=0^n-1p( x_i+1 | x_i) } p(x_0) = p(x_N) ∏_i=N-1^0q^*(x_i+1→x_i),Eq. (<ref>)means that the combined probability of time-reversed paths defined by forward simulation is recovered by the backward dynamics Eq. (<ref>). Specifically, the time-reversed paths initialized by p(x_N) automatically converge to their initial density p(x_0) using the backward dynamics Eq. (<ref>). In this sense, q^*(x_i+1→x_i) in Eq. (<ref>) is considered as the optimal backward dynamics. Implementation of these dynamics, however, requires the probabilities p(x_i), i=1, … N, which are usually not available prior to the simulations. Note that the backward dynamics defined by the Langevin equation in previous studies <cit.> can be derived from Eq. (<ref>) as a continuous-time limit.Equations (<ref>)and(<ref>) were previously discussed <cit.> in the field of time-series data analysis, where approximations of the marginal probabilitiesp(x_i), i=1, … Nare used to define the backward dynamicsq(x_i+1→x_i). In these studies, the observed data were available at many of the time steps i=0, …, N, whereas the target is given only at i=N in our problem. Then, approximations of probabilities p(x_i), i=1, … N are derived using forward simulations constrained with the observed data (“filtering stage”).It is, however, difficult to apply these methods to our problem. If we were to apply a similar method to our problem, we would have to run a number of forward simulations to estimate p(x_i), i=1, … N before executing backward simulations. This would be computationally expensive and seems unrealistic without a highly efficient method for the probability estimation. Methods such as those discussed previously <cit.> may be applied to optimize the backward dynamics in our problem, but this is left for future study.On the other hand, when some observed data are available outside the equations that describe the stochastic process, we may use these data to approximate p(x_i), i=1, … N and hence use them to approximate the optimal backward dynamics.In this case, we avoid the use of a large amount of forward computation to construct the optimized backward dynamics.This seems possible for the stochastic typhoon model, where data from actual observations of real typhoons are available.Note that this idea is different from data assimilation (i.e., inference with simulations combined with observed data), because here we use observed data only to improve the computational efficiency; they do not cause the bias of calculated probabilities. In this paper, we assumed that the number of time steps is fixed. This is not, however, always clear in advance in realistic problems. In the case of a realistic typhoon model, we must consider the case of passing through Tokyo between two discrete time steps. The interpolation method in this case will be developed in a future work. § CONCLUDING REMARKSWe discussed methods for the backward simulation of the stochastic process. These methods trace a time-reversed path from the target region to the initial configuration. A naïve approach to this problem was shown not to function as expected. To resolve the difficulties, the time reverse Monte Carlo method (TRMC) was introduced. The TRMC method is based on SIS and SMC, and is designed to provide the probabilities of events correctly. TRMC with SIS was tested for the stochastic typhoon model and the Lorenz 96 model; it converges more efficiently than forward simulations in some of these examples. For simulations with a larger number of steps, TRMC with SMC was shown to be advantageous. We also discussed the limitation and possible improvement of TRMC and its relation to the Bayes formula. § DEVIATION OF EQ. (<REF>) The aim of this appendix is to prove Eq. (<ref>). Up to the first-order Δ t, the Jacobian det(J_g(x)) is given bydet(J_g(x)) = det( I + ∇ f(x) Δ t ) = 1 + ( ∇ f(x) Δ t ) + O( (Δ t )^2 ) = exp[ div f(x) Δ t ] + O( (Δ t )^2 ),where I is the unit matrix of order D × D. D is the dimension of stochastic process X.Using Eq. (<ref>), we obtain in the limit as Δ t → 0J(y_1, …, y_N) = ∏_i=0^N-1| det(J_g^-1(y_i+1)) | = exp[ ∑_i=1^N - div f(x_i) Δ t ] + O( (Δ t )^2 ) exp[ - ∫_0^Tdivf(x_t) dt ].The above equation is Eq. (<ref>) in the main text.§ DEVIATION OF EQ. (<REF>) The aim of this appendix is to prove (<ref>).Up to the first-order Δ t, the weight at time t_i is given by W_i = exp[ -( f(x_i+1) - f(x_i) )^TΣ^-1( (x_i+1 - x_i) - Δ t/2( f(x_i+1) + f(x_i) ) ) ] = exp[ ( -( f(x_i+1) - f(x_i) )^TΣ^-1( (x_i+1 - x_i) - Δ t/2( f(x_i+1) + f(x_i) ) ) ) ] = exp[ - ( ( ∇ f(x_i) (x_i+1 - x_i) )^TΣ^-1 (x_i+1 - x_i) ) + o(Δ t) ] = exp[ - ( ∇ f(x_i) ^TΣ^-1 (x_i+1 - x_i) (x_i+1 - x_i)^T) + o(Δ t) ]. In the limit as Δ t → 0,Eq. (<ref>) becomes the following stochastic differential equation:dX_t = f(X_t) dt + σ dW_t,where W_t is a standard Brownian motion. Here, we used Ito's rule <cit.>, in which we substitute √(dt) for dW_t and consider up to the order of dt. Using Eq. (<ref>), we obtain the following relation in the limit as Δ t → 0 (x_i+1 - x_i) (x_i+1 - x_i)^T dx_t dx_t^T = (f(x_t) dt + σ dW_t) (f(x_t) dt + σ dW_t)^T = σ dW_t dW_t^T σ^Tdt + O(dt) = Σ dt, where we used the relationships dW_t dW_t^T = dt and σσ^T = Σ.As a result, we obtain using Eq. (<ref>) and (<ref>) ∏_i=0^N-1W_i = exp[ - ∑_i=0^N-1( ∇ f(x_i) ^TΣ^-1 (x_i+1 - x_i) (x_i+1 - x_i)^T) + o(Δ t) ] exp[ - ∫_0^T( ∇ f(x_t) ^T) dt ] = exp[ - ∫_0^Tdivf(x_t) dt ], which is Eq. (<ref>) in the main text.jpsj | http://arxiv.org/abs/1708.08045v4 | {
"authors": [
"Shinichi Takayanagi",
"Yukito Iba"
],
"categories": [
"physics.data-an",
"cond-mat.dis-nn",
"cond-mat.stat-mech",
"nlin.CD",
"stat.ME"
],
"primary_category": "physics.data-an",
"published": "20170827025555",
"title": "Backward Simulation of Stochastic Process using a Time Reverse Monte Carlo method"
} |
C. Sakaridis D. Dai L. Van Gool ETH Zürich, Zurich, SwitzerlandL. Van Gool KU Leuven, Leuven, Belgium Semantic Foggy Scene Understanding with Synthetic DataChristos Sakaridis Dengxin Dai Luc Van Gool Received: date / Accepted: date ========================================================= This work addresses the problem of semantic foggy scene understanding (SFSU). Although extensive research has been performed on image dehazing and on semantic scene understanding with clear-weather images, little attention has been paid to SFSU.Due to the difficulty of collecting and annotating foggy images, we choose to generate synthetic fog on real images that depict clear-weather outdoor scenes, and then leverage these partially synthetic data for SFSU by employing state-of-the-art convolutional neural networks (CNN). In particular, a complete pipeline to add synthetic fog to real, clear-weather images using incomplete depth information is developed. We apply our fog synthesis on the Cityscapes dataset and generate Foggy Cityscapes with 20550 images. SFSU is tackled in two ways: 1) with typical supervised learning, and 2) with a novel type of semi-supervised learning, which combines 1) with an unsupervised supervision transfer from clear-weather images to their synthetic foggy counterparts. In addition, we carefully study the usefulness of image dehazing for SFSU. For evaluation, we present Foggy Driving, a dataset with 101 real-world images depicting foggy driving scenes, which come with ground truth annotations for semantic segmentation and object detection.Extensive experiments show that 1) supervised learning with our synthetic data significantly improves the performance of state-of-the-art CNN for SFSU on Foggy Driving; 2) our semi-supervised learning strategy further improves performance; and 3) image dehazing marginally advances SFSU with our learning strategy.The datasets, models and code are made publicly available.§ INTRODUCTIONCameras and the accompanying vision algorithms are widely used for applications such as surveillance <cit.>, remote sensing <cit.>, and automated cars <cit.>, and their deployment keeps expanding. While these sensors and algorithms are constantly getting better, they are mainly designed to operate on clear-weather images and videos <cit.>. Yet, outdoor applications can hardly escape from “bad” weather. Thus, such computer vision systems should also function under adverse weather conditions. Here we focus on the presence of fog.Fog degrades the visibility of a scene significantly <cit.>. This causes problems not only to human observers, but also to computer vision algorithms. During the past years, a large body of research has been conducted on image defogging (dehazing) to increase scene visibility <cit.>. Meanwhile, marked progress has been made in semantic scene understanding with clear-weather images and videos <cit.>. In contrast, the semantic understanding of foggy scenes has received little attention, despite its importance in outdoor applications. For instance, an automated car still requires a robust detection of road lanes, traffic lights, and other traffic agents in the presence of fog. This work investigates semantic foggy scene understanding (SFSU).High-level semantic scene understanding is usually tackled by learning from many annotations of real images <cit.>. Yet, the difficulty of collecting and annotating images for unusual weather conditions such as fog renders this standard protocol problematic. To overcome this problem, we depart from this traditional paradigm and propose another route, also different from moving to fully synthetic scenes. Instead, we choose to generate synthetic fog into real images that contain clear-weather outdoor scenes, and then leverage these partially synthetic foggy images for SFSU. Given the fact that large-scale annotated data are available for clear-weather images <cit.>, we present an automatic pipeline to add synthetic yet highly realistic fog to such datasets. Our fog simulation uses the standard optical model for daytime fog <cit.> (which has already been used extensively in image dehazing) to overlay existing clear-weather images with synthetic fog in a physically sound way, simulating the underlying mechanism of foggy image formation. We leverage our fog simulation pipeline to create our Foggy Cityscapes dataset, by adding fog to urban scenes from the Cityscapes dataset <cit.>. This has led to 550 carefully refined high-quality synthetic foggy images with fine semantic annotations inherited directly from Cityscapes, plus an additional 20000 synthetic foggy images without fine annotations. The resulting “synthetic-fog” images are used to adapt two semantic segmentation models <cit.> and an object detector <cit.> to foggy scenes. The models are trained in two fashions: 1) by the typical supervised learning scheme, using the 550 high-quality annotated foggy images, and 2) by a novel semi-supervised learning approach, which augments the dataset that is used in 1) with the additional 20000 foggy images and draws the missing supervision for these images from the predictions of the source, clear-weather model on their clear-weather counterparts. For evaluation purposes, we collect and annotate a new dataset, Foggy Driving, with 101 images of driving scenes in the presence of fog. See <ref> for the whole pipeline of our work. In addition, this work studies the utility of three state-of-the-art image dehazing methods for SFSU as well as human understanding of foggy scenes.The main contributions of the paper are: 1) an automatic and scalable pipeline to impose high-quality synthetic fog on real clear-weather images; 2) two new datasets, one synthetic and one real, to facilitate training and evaluation of models used in SFSU; 3) a new semi-supervised learning approach for SFSU; and 4) a detailed study of the benefit of image dehazing for SFSU and human perception of foggy scenes. The rest of the paper is organized as follows. Section <ref> presents the related work. Section <ref> is devoted to our fog simulation pipeline, followed by Section <ref> that introduces our two foggy datasets. Section <ref> describes supervised learning with our synthetic foggy data and studies the usefulness of image dehazing for SFSU in this context. Finally, Section <ref> extends the learning to a semi-supervised paradigm, where supervision is transferred from clear-weather images to their synthetic foggy counterparts, and Section <ref> concludes the paper. § RELATED WORK Our work is relevant to image defogging (dehazing), depth denoising and completion, foggy scene understanding, synthetic visual data, and transfer learning. §.§ Image Defogging/Dehazing Fog fades the color of observed objects and reduces their contrast.Extensive research has been conducted on image defogging (dehazing) to increase the visibility of foggy scenes. This ill-posed problem has been tackled from different perspectives. For instance, in contrast enhancement <cit.> the rationale is that clear-weather images have higher contrast than images degraded by fog. Depth and statistics of natural images are exploited as priors as well <cit.>. Another line of work is based on the dark channel prior <cit.>, with the empirically validated assumption that pixels of clear-weather images are very likely to have low values in some of the three color channels. Certain works focus particularly on enhancing foggy road scenes <cit.>. Methods have also been developed for nighttime <cit.>, given its importance in outdoor applications. Fast dehazing approaches have been developed in <cit.> towards real-time applications. Recent approaches also rely on trainable architectures <cit.>, which have evolved to end-to-end models <cit.>. For a comprehensive overview of defogging/dehazing algorithms, we point the reader to <cit.>. All these approaches can greatly increase visibility. Our work is complementary and focuses on the semantic understanding of foggy scenes. §.§ Depth Denoising and CompletionSynthesizing a foggy image from its real, clear counterpart generally requires an accurate depth map. In previous works, the colorization approach of <cit.> has been used to inpaint depth maps of the indoor NYU Depth dataset <cit.>. Such inpainted depth maps have been used in state-of-the-art dehazing approaches such as <cit.> to generate training data in the form of synthetic indoor foggy images. In contrast, our work considers real outdoor urban scenes from the Cityscapes dataset <cit.>, which contains significantly more complex depth configurations than NYU Depth. Furthermore, the available depth information in Cityscapes is not provided by a depth sensor, but it is rather an estimate of the depth resulting from the application of a semiglobal matching stereo algorithm based on <cit.>. This depth estimate usually contains a large amount of severe artifacts and large holes (cf. <ref>), which render it inappropriate for direct use in fog simulation. There are several recent approaches that handle highly noisy and incomplete depth maps, including stereoscopic inpainting <cit.>, spatio-temporal hole filling <cit.> and layer depth denoising and completion <cit.>. Our method builds on the framework of stereoscopic inpainting <cit.> which performs depth completion at the level of superpixels, and introduces a novel, theoretically grounded objective for the superpixel-matching optimization that is involved. §.§ Foggy Scene UnderstandingSemantic understanding of outdoor scenes is a crucial enabler for applications such as assisted or autonomous driving. Typical examples include road and lane detection <cit.>, traffic light detection <cit.>, car and pedestrian detection <cit.>, and a dense, pixel-level segmentation of road scenes into most of the relevant semantic classes <cit.>. While deep recognition networks have been developed <cit.> and large-scale datasets have been presented <cit.>, that research mainly focused on clear weather. There is also a large body of work on fog detection <cit.>. Classification of scenes into foggy and fog-free has been tackled as well <cit.>. In addition, visibility estimation has been extensively studied for both daytime <cit.> and nighttime <cit.>, in the context of assisted and autonomous driving. The closest of these works to ours is <cit.>, in which synthetic fog is generated and foggy images are segmented to free-space area and vertical objects. Our work differs in that: 1) our semantic understanding task is more complex, with 19 semantic classes that are commonly involved in driving scenarios, 8 of which occur as distinct objects; 2) we tackle the problem with modern deep CNN for semantic segmentation <cit.> and object detection <cit.>, taking full advantage of the most recent advances in this field; and 3) we compile and release a large-scale dataset of synthetic foggy images based on real scenes plus a dataset of real-world foggy scenes, featuring both dense pixel-level semantic annotations and annotations for object detection. §.§ Synthetic Visual DataThe leap of computer vision in recent years can to an important extent be attributed to the availability of large, labeled datasets <cit.>. However, acquiring and annotating such a dataset for each new problem is not (yet) doable. Thus, learning with synthetic data is gaining attention. We give some notable examples. Dosovitskiy <cit.> use the renderings of a floating chair to train dense optical flow regression networks. Gupta <cit.> impose text onto natural images to learn an end-to-end text detection system. Vázquez <cit.> train pedestrian detectors with virtual data. In <cit.> the authors leverage video game engines to render images along with dense semantic annotations that are subsequently used in combination with real data to improve the semantic segmentation performance of modern CNN architectures on real scenes. Going one step further, <cit.> shows that for the task of vehicle detection, training a CNN model only on massive amounts of synthetic images can outperform the same model trained on large-scale real datasets like Cityscapes. By contrast, our work tackles semantic segmentation and object detection for real foggy urban scenes, by adding synthetic fog to real images taken under clear weather. Hence, our approach is based on only partially synthetic data. In the same vein, <cit.> is based on real urban scenes, augmented with virtual cars. A very interesting project is “FOG” <cit.>. Its team developed a prototype of a small-scale fog chamber, able to produce stable visibility levels and homogeneous fog to test the reaction of drivers. §.§ Transfer LearningOur work bears resemblance to works from the broad field of transfer learning. Model adaptation across weather conditions to semantically segment simple road scenes is studied in <cit.>. More recently, a domain adversarial based approach was proposed to adapt semantic segmentation models both at pixel level and feature level from simulated to real environments <cit.>. Our work generates synthetic fog from clear-weather data to close the domain gap. Combining our method and the aforementioned transfer learning methods is a promising direction for future work. The supervision transfer from clear weather to foggy weather in this paper is inspired by the stream of work on model distillation/imitation <cit.>. Our approach is similar in that knowledge is transferred from one domain (model) to another by using paired data samples as a bridge.§ FOG SIMULATION ON REAL OUTDOOR SCENESTo simulate fog on input images that depict real scenes with clear weather, the standard approach is to model the effect of fog as a function that maps the radiance of the clear scene to the radiance observed at the camera sensor. Critically, this space-variant function is usually parameterized by the distance ℓ of the scene from the camera, which equals the length of the path along which light has traveled and is closely related to scene depth. As a result, the pair of the clear image and its depth map forms the basis of our foggy image synthesis. In this section, we first detail the optical model which we use for fog and then present our complete pipeline for fog simulation, with emphasis on our denoising and completion of the input depth. Finally, we present some criteria for selecting suitable images to generate high-quality synthetic fog. §.§ Optical Model of Choice for FogIn the image dehazing literature, various optical models have been used to model the effect of haze on the appearance of a scene. For instance, optical models tailored for nighttime haze removal have been proposed in <cit.>, taking into account the space-variant lighting that characterizes most nighttime scenes. This variety of models is directly applicable to the case of fog as well, since the physical process for image formation in the presence of either haze or fog is essentially similar. For our synthesis of foggy images, we consider the standard optical model of <cit.>, which is used extensively in the literature <cit.> and is formulated asI⃗(x⃗) = R⃗(x⃗)t(x⃗) + L⃗(1 - t(x⃗)),where I⃗(x⃗) is the observed foggy image at pixel x⃗, R⃗(x⃗) is the clear scene radiance and L⃗ is the atmospheric light. This model assumes the atmospheric light to be globally constant, which is generally valid only for daytime images. The transmission t(x⃗) determines the amount of scene radiance that reaches the camera. In case of a homogeneous medium, transmission depends on the distance ℓ(x⃗) of the scene from the camera throught(x⃗) = exp(-βℓ(x⃗)).The parameter β is named attenuation coefficient and it effectively controls the thickness of the fog: larger values of β mean thicker fog. The meteorological optical range (MOR), also known as visibility, is defined as the maximum distance from the camera for which t(x⃗) ≥ 0.05, which implies that if (<ref>) is valid, then MOR=2.996/β. Fog decreases the MOR to less than 1 km by definition <cit.>. Therefore, the attenuation coefficient in homogeneous fog is by definitionβ≥ 2.996×10^-3 m^-1,where the lower bound corresponds to the lightest fog configuration. In our fog simulation, the value that is used for β always obeys (<ref>).Model (<ref>) provides a powerful basis for simulating fog on outdoor scenes with clear weather. Even though its assumption of homogeneous atmosphere is strong, it generates synthetic foggy images that can act as good proxies for real world foggy images where this assumption might not hold exactly, as long as it is provided with an accurate transmission map t. Straightforward extensions of (<ref>) are used in <cit.> to simulate heterogeneous fog on synthetic scenes.To sum up, the necessary inputs for fog simulation using (<ref>) are a color image R⃗ of the original clear scene, atmospheric light L⃗ and a dense transmission map t defined at each pixel of R⃗. Our task is thus twofold: * estimation of t, and * estimation of L⃗ from R⃗. Step <ref> is simple: we use the method proposed in <cit.> with the improvement of <cit.>. In the following, we focus on step <ref> for the case of outdoor scenes with a noisy, incomplete estimate of depth serving as input. §.§ Depth Denoising and Completion for Outdoor ScenesThe inputs that our method requires for generating an accurate transmission map t are: * the original, clear-weather color image R⃗ to add synthetic fog on, which constitutes the left image of a stereo pair,* the right image Q⃗ of the stereo pair,* the intrinsic calibration parameters of the two cameras of the stereo pair as well as the length of the baseline,* a dense, raw disparity estimate D for R⃗ of the same resolution as R⃗, and* a set M comprising the pixels where the value of D is missing.These requirements can be easily fulfilled with a stereo camera and a standard stereo matching algorithm <cit.>. The main steps of our pipeline are the following: * calculation of a raw depth map d in meters, * denoising and completion of d to produce a refined depth map d^' in meters, * calculation of a scene distance map ℓ in meters from d^', * application of (<ref>) to obtain an initial transmission map t̂, and * guided filtering <cit.> of t̂ using R⃗ as guidance to compute the final transmission map t.The central idea is to leverage the accurate structure that is present in the color images of the stereo pair in order to improve the quality of depth, before using the latter as input for computing transmission. We now proceed in explaining each step in detail, except step <ref> which is straightforward. In step <ref>, we use the input disparity D in combination with the values of the focal length and the baseline to obtain d. The missing values for D, indicated by M, are also missing in d.Step <ref> follows a segmentation-based depth filling approach, which builds on the stereoscopic inpainting method presented in <cit.>. More specifically, we use a superpixel segmentation of the clear image R⃗ to guide depth denoising and completion at the level of superpixels, making the assumption that each individual superpixel corresponds roughly to a plane in the 3D scene.First, we apply a photo-consistency check between R⃗ and Q⃗, using the input disparity D to establish pixel correspondences between the two images of the stereo pair, similar to Equation (12) in <cit.>. All pixels in R⃗ for which the color deviation (measured as difference in the RGB color space) from the corresponding pixel in Q⃗ has greater magnitude than ϵ = 12/255 are deemed invalid regarding depth and hence are added to M.We then segment R⃗ into superpixels with SLIC <cit.>, denoting the target number of superpixels as K̂ and the relevant range domain scale parameter as m = 10. For depth denoising and completion on Cityscapes, we use K̂ = 2048. The final number of superpixels that are output by SLIC is denoted by K. These superpixels are classified into reliable and unreliable ones with respect to depth information, based on the number of pixels with missing or invalid depth that they contain. More formally, we use the criterion of Equation (2) in <cit.>, which states that a superpixel T is reliable if and only if(T∖M) ≥max{P, λ(T)},setting P = 20 and λ = 0.6.For each superpixel that fulfills (<ref>), we fit a depth plane by running RANSAC on its pixels that have a valid value for depth. We use an adaptive inlier threshold to account for differences in the range of depth values between distinct superpixels. For a superpixel T, the inlier threshold is set asθ = 0.01_x⃗∈ T∖M{d(x⃗)}.We use adaptive RANSAC and set the maximum number of iterations to 2000 and the bound on the probability of having obtained a pure inlier sample to p = 0.99.The greedy approach of <cit.> is used subsequently to match unreliable superpixels to reliable ones pairwise and assign the fitted depth planes of the latter to the former. Different than <cit.>, we propose a novel objective function for matching pairs of superpixels. For a superpixel pair (s,t), our proposed objective is formulated asE(s,t) = C⃗_s - C⃗_t^2 + αx⃗_s - x⃗_t^2. The first term measures the proximity of the two superpixels in the range domain, where we denote the average CIELAB color of superpixel s with C⃗_s. In other words, we penalize the squared Euclidean distance between the average colors of the superpixels in the CIELAB color space, which has been designed to increase perceptual uniformity <cit.>. On the contrary, the objective of <cit.> uses the cosine similarity of average superpixel colors to form the range domain cost:1-C⃗_s/C⃗_s·C⃗_t/C⃗_t.The disadvantage of (<ref>) is that it assigns zero matching cost to dissimilar colors in certain cases. For instance, in the RGB color space, the pair of colors (δ,δ,δ) and (1-δ,1-δ,1-δ), where δ is a small positive constant, is assigned zero penalty, even though the former color is very dark gray and the latter is very light gray.The second term on the right-hand side of (<ref>) measures the proximity of the two superpixels in the spatial domain as the squared Euclidean distance between their centroids x⃗_s and x⃗_t. By contrast, the spatial proximity term of <cit.> assigns zero cost to pairs of adjacent superpixels and unit cost to non-adjacent pairs. This implies that close yet non-adjacent superpixels are penalized equally to very distant superpixels by <cit.>. As a result, a certain superpixel s can be erroneously matched to a very distant superpixel t which is highly unlikely to share the same depth plane as s, as long as the range domain term for this pair is minimal and all superpixels adjacent to s are dissimilar to it with respect to appearance. Our proposed spatial cost handles these cases successfully: t is assigned a very large spatial cost for being matched to s, and other superpixels that have less similar appearance yet smaller distance to s are preferred.In (<ref>), α > 0 is a parameter that weights the relative importance of the spatial domain term versus the range domain term. Similarly to <cit.>, we set α = m^2/S^2, where S = √(N/K), N denotes the total number of pixels in the image, and m=10 and K are the same as for SLIC. Our matching objective (<ref>) is similar to the distance that is defined in SLIC <cit.> and other superpixel segmentation methods for assigning an individual pixel to a superpixel. In our case though, this distance is rather used to measure similarity between pairs of superpixels.After all superpixels have been assigned a depth plane, we use these planes to complete the missing depth values for pixels belonging to M. In addition, we replace the depth values of pixels which do not belong to M but constitute large-margin outliers with respect to their corresponding plane (deviation larger than θ̂ = 50m) with the values imputed by the plane. This results in a complete, denoised depth map d^', and concludes step <ref>.In step <ref>, we compute the distance ℓ(x⃗) of the scene from the camera at each pixel x⃗ based on d^'(x⃗), using the coordinates of the principal point plus the focal length of the camera.Finally, in step <ref> we post-process the initial transmission map t̂ with guided filtering <cit.>, in order to smooth transmission while respecting the boundaries of the clear image R⃗. We fix the radius of the guided filter window to r = 20 and the regularization parameter to μ = 10^-3, we use the same values as in the haze removal experiments of <cit.>. Results of the presented pipeline for fog simulation on example images from Cityscapes are provided in <ref> for β = 0.01, which corresponds to visibility of ca. 300m. We compare our fog simulation to an alternative implementation, which employs nearest-neighbor interpolation to complete the missing values of the depth map before computing the transmission and does not involve guided filtering as a postprocessing step. §.§ Input Selection for High-Quality Fog SimulationApplying the presented pipeline to simulate fog on large datasets with real outdoor scenes such as Cityscapes with the aim of producing synthetic foggy images of high quality calls for careful refinement of the input.To be more precise, the sky is clear in the majority of scenes in Cityscapes, with intense direct or indirect sunlight, as shown in <ref>fig:sunny:input. These images usually contain sharp shadows and have high contrast compared to images that depict foggy scenes. This causes our fog simulation to generate synthetic images which do not resemble real fog very well, <ref>fig:sunny:output. Therefore, our first refinement criterion is whether the sky is overcast, ensuring that the light in the input real scene is not strongly directional.Secondly, we observe that atmospheric light estimation in step <ref> of our fog simulation sometimes fails to select a pixel with ground truth semantic label sky as the representative of the value of atmospheric light. In rare cases, it even happens that the sky is not visible at all in an image. This results in an erroneous, physically invalid value of atmospheric light being used in (<ref>) to synthesize the foggy image. Consequently, our second refinement criterion is whether the pixel that is selected as atmospheric light is labeled as sky, and affords an automatic implementation.§ FOGGY DATASETSWe present two distinct datasets for semantic understanding of foggy scenes: Foggy Cityscapes and Foggy Driving. The former derives from the Cityscapes dataset <cit.> and constitutes a collection of synthetic foggy images generated with our proposed fog simulation that automatically inherit the semantic annotations of their real, clear counterparts. On the other hand, Foggy Driving is a collection of 101 real-world foggy road scenes with annotations for semantic segmentation and object detection, used as a benchmark for the domain of foggy weather. §.§ Foggy CityscapesWe apply the fog simulation pipeline that is presented in Section <ref> to the complete set of images provided in the Cityscapes dataset. More specifically, we first obtain 20000 synthetic foggy images from the larger, coarsely annotated part of the dataset, and keep all of them, without applying the refinement criteria of Section <ref>. In this way, we trade the high visual quality of the synthetic images for a very large scale and variability of the synthetic dataset. We do not make use of the original coarse annotations of these images for semantic segmentation; rather, we produce labellings with state-of-the-art semantic segmentation models on the original, clear images and use them to transfer knowledge from clear weather to foggy weather, as will be discussed in Section <ref>. We name this set Foggy Cityscapes-coarse.In addition, we use the two criteria of Section <ref> in conjunction to filter the finely annotated part of Cityscapes that originally comprises 2975 training and 500 validation images, and obtain a refined set of 550 images, 498 from the training set and 52 from the validation set, which fulfill both criteria. Running our fog simulation on this refined set provides us with a moderate-scale collection of high-quality synthetic foggy images. This collection automatically inherits the original fine annotations for semantic segmentation, as well as bounding box annotations for object detection which we generate by leveraging the instance-level semantic annotations that are provided in Cityscapes for the 8 classes person, rider, car, truck, bus, train, motorcycle and bicycle. We term this collection Foggy Cityscapes-refined.Since MOR can vary significantly in reality for different instances of fog, we generate five distinct versions of Foggy Cityscapes, each of which is characterized by a constant simulated attenuation coefficient β in (<ref>), hence a constant MOR. In particular, we use β∈{0.005, 0.01, 0.02, 0.03, 0.06}, which correspond approximately to MOR of 600m, 300m, 150m, 100m and 50m respectively. <ref> shows three of the five synthesized foggy versions of a clear scene in Foggy Cityscapes. §.§ Foggy DrivingFoggy Driving consists of 101 color images depicting real-world foggy driving scenes. We captured 51 of these images with a cell phone camera in foggy conditions at various areas of Zurich, and the rest 50 images were carefully collected from the web. We note that all images have been preprocessed so that they have a maximum resolution of 960×1280 pixels.We provide dense, pixel-level semantic annotations for all images of Foggy Driving. In particular, we use the 19 evaluation classes of Cityscapes: road, sidewalk, building, wall, fence, pole, traffic light, traffic sign, vegetation, terrain, sky, person, rider, car, truck, bus, train, motorcycle and bicycle. Pixels that do not belong to any of the above classes or are not labeled are assigned the void label, and they are ignored for semantic segmentation evaluation. At annotation time, we label individual instances of person, rider, car, truck, bus, train, motorcycle and bicycle separately following the Cityscapes annotation protocol, which directly affords bounding box annotations for these 8 classes.In total, 33 images have been finely annotated (cf. the last three rows of <ref>) in the aforementioned procedure, and the rest 68 images have been coarsely annotated (cf. the top three rows of <ref>). We provide per-class statistics for the pixel-level semantic annotations of Foggy Driving in <ref>. Furthermore, statistics for the number of objects in the bounding box annotations are shown in <ref>. Because of the coarse annotation that is created for one part of Foggy Driving, we do not use this part in evaluation of object detection approaches, as difficult objects that are not included in the annotations may be detected by a good method and missed by a comparatively worse method, resulting in incorrect comparisons with respect to precision. On the contrary, the coarsely annotated images are used without such issues in evaluation of semantic segmentation approaches, since predictions at unlabeled pixels are simply ignored and thus do not affect the measured performance.Foggy Driving may have a smaller size than other recent datasets for semantic scene understanding, however, it features challenging foggy scenes with comparatively high complexity. As <ref> shows, the subset of 33 images with fine annotations is roughly on par with Cityscapes regarding the average number of humans and vehicles per image. In total, Foggy Driving contains more than 500 vehicles and almost 300 humans. We also underline the fact that <ref> compares Foggy Driving — a dataset used purely for testing — against the unions of training and validation sets of KITTI <cit.> and Cityscapes, which are much larger than their respective testing sets that would provide a better comparison.As a final note, we identify the subset of the 19 annotated classes that occur frequently in Foggy Driving. These “frequent” classes either have a larger number of total annotated pixels, road, or a larger number of total annotated polygons or instances, pole and person, compared to the rest of the classes. They are: road, sidewalk, building, pole, traffic light, traffic sign, vegetation, sky, person, and car. In the experiments that follow in Section <ref>, we occasionally use this set of frequent semantic classes as an alternative to the complete set of semantic classes for averaging per-class scores, in order to further verify results based only on classes with plenty of examples. § SUPERVISED LEARNING WITH SYNTHETIC FOGWe first show that our synthetic Foggy Cityscapes-refined dataset can be used per se for successfully adapting modern CNN models to the condition of fog with the usual supervised learning paradigm. Our experiments focus primarily on the task of semantic segmentation and additionally include comparisons on the task of object detection, evidencing clearly the usefulness of our synthetic foggy data in understanding the semantics of real foggy scenes such as those in Foggy Driving.More specifically, the general outline of our main experiments can be summarized in two steps: * fine-tuning a model that has been trained on the original Cityscapes dataset for clear weather by using only synthetic images of Foggy Cityscapes-refined, and* evaluating the fine-tuned model on Foggy Driving and showing that its performance is improved compared to the original, clear-weather model. Thus, the reported results pertain to Foggy Driving unless otherwise mentioned.In other words, all models are ultimately evaluated on data from a different domain than that of the data on which they have been fitted, revealing their true generalization potential on previously unseen foggy scenes.We also consider dehazing as an optional preprocessing step before feeding the input images to semantic segmentation models for training and testing, and examine the effect of this dehazing preprocessing on the performance of such a model using state-of-the-art dehazing methods. The effect of dehazing on semantic segmentation performance is additionally correlated with its utility for human understanding of foggy scenes by conducting a user study on Amazon Mechanical Turk. §.§ Semantic SegmentationOur model of choice for conducting experiments on semantic segmentation with the supervised pipeline is the modern dilated convolutions network (DCN) <cit.>. In particular, we make use of the publicly available Dilation10 model, which has been trained on the 2975 images of the training set of Cityscapes. We wish to note that this model was originally trained and tested on 1396×1396 image crops by the authors of <cit.>, but due to GPU memory limitations we train it on 756×756 crops and test it on 700×700 crops. Still, Dilation10 enjoys a fair mean intersection over union (IoU) score of 34.9% on Foggy Driving.In the following experiments of Section <ref>, we fine-tune Dilation10 on the training set of Foggy Cityscapes-refined which consists of 498 images, and reserve the 52 images of the respective validation set for additional evaluation. In particular, we fine-tune all layers of the original model for 3k iterations (ca. 6 epochs) using mini-batches of size 1. Unless otherwise mentioned, the attenuation coefficient β used in Foggy Cityscapes is equal to 0.01.Overall, we consider four different options with respect to dehazing preprocessing: applying no dehazing at all, dehazing with multi-scale convolutional neural networks (MSCNN) <cit.>, dehazing using the dark channel prior (DCP) <cit.>, and non-local image dehazing <cit.>. Unless otherwise specified, no dehazing is applied. Our experimental protocol is consistent with respect to dehazing preprocessing: the same option for dehazing preprocessing is used both at training time and test time. More specifically, at training time we first process the synthetic foggy images of Foggy Cityscapes-refined according to the specified option for dehazing preprocessing and then use the processed images as input for fine-tuning Dilation10. At evaluation time, we process the images in Foggy Driving with the same dehazing preprocessing that was used at training time (if any was), and use the processed images to test the fine-tuned model.Benefit of Fine-tuning on Synthetic Fog. Our first experiment evidences the benefit of fine-tuning on Foggy Cityscapes-refined for improving semantic segmentation performance on Foggy Driving. <ref> presents comparative performance of the original Dilation10 model against its fine-tuned counterparts in terms of mean IoU over all annotated classes in Foggy Driving as well as over frequent classes only. All four options regarding dehazing preprocessing are considered. Note that we also evaluate the original Dilation10 model for all dehazing preprocessing alternatives (only relevant at test time in this case) in the first row of each part of <ref>. Indeed, all fine-tuned models outperform Dilation10 irrespective of the type of dehazing preprocessing that is applied, both for mean IoU over all classes and over frequent classes only. The best-performing fine-tuned model, which we refer to as FT-0.01, involves no dehazing and outperforms Dilation10 significantly, by 3% for mean IoU over all classes and 5% for mean IoU over frequent classes. Note additionally that FT-0.01 has been fine-tuned on only 498 training images of Foggy Cityscapes-refined, compared to the 2975 training images of Cityscapes for Dilation10. Comparison of Fog Simulation Approaches. Next, we compare in <ref> the utility of our proposed fog simulation method for generating useful synthetic training data in terms of semantic segmentation performance on Foggy Driving, against two alternative approaches: the baseline that we considered in <ref> and a truncated version of our method, where we omit the guided filtering step. We consider two different policies for the learning rate when fine-tuning on Foggy Cityscapes-refined: a constant learning rate of 10^-5 and a polynomially decaying learning rate, commonly referred to as “poly” <cit.>, with a base learning rate of 10^-5 and a power parameter of 0.9. Our method for fog simulation consistently outperforms the two baselines and the “poly” learning rate policy allows the model to be fine-tuned more effectively than the constant policy. In all other experiments with DCN, we use the “poly” learning rate policy with the parameters specified above for fine-tuning.Increasing Returns at Larger Distance. As can easily be deduced from (<ref>), fog has a growing effect on the appearance of the scene as distance from the camera increases. Ideally, a model that is dedicated to foggy scenes must deliver a greater benefit for distant parts of the scene. In order to examine this aspect of semantic segmentation of foggy scenes, we use the completed, dense distance maps of Cityscapes images that have been computed as an intermediate output of our fog simulation, given that Foggy Driving does not include depth information. In more detail, we consider the validation set of Foggy Cityscapes-refined, the images of which are unseen both for Dilation10 and our fine-tuned models, and bin the pixels according to their value in the corresponding distance map. Each distance range is considered separately for evaluation by ignoring all pixels that do not belong to it. In <ref>, we compare mean IoU of Dilation10 and FT-0.01 individually for each distance range. FT-0.01 brings a consistent gain in performance across all distance ranges. What is more, this gain is larger in both absolute and relative terms for pixels that are more than 50m away from the camera, implying that our model is able to handle better the most challenging parts of a foggy scene. Note that most pixels in the very last distance range (more than 400m away from the camera) belong to the sky class and their appearance does not change much between the clear and the synthetic foggy images.Generalization in Synthetic Fog across Densities. In order to verify the ability of a model that has been fine-tuned on Foggy Cityscapes-refined for a fixed value β^(t) of the attenuation coefficient, hence fixed fog density, to generalize well to new, unseen fog densities, we evaluate the model on multiple versions of the validation set of Foggy Cityscapes-refined, each rendered using a different value for β which is in general not equal to β^(t). In particular, we use the five different versions of Foggy Cityscapes-refined as described in Section <ref> and obtain five models by fine-tuning Dilation10 on the training set of each version. In congruence with notation in previous experiments, we denote such a fine-tuned model by FT-β^(t), FT-0.02. Afterwards, we evaluate each of these models plus Dilation10 on the validation set of each of the five foggy versions plus the original, clear-weather version where β = 0. The mean IoU performance of the six models is presented in <ref>. Whereas the performance of Dilation10 drops rapidly as β increases, all five fine-tuned “foggy” models are more robust to changes in β across the examined range. Analyzing the performance of each fine-tuned model individually, we observe that performance is high and fairly stable in the range [0, β^(t)] and drops for β > β^(t). This implies that a “foggy” model is able to generalize well to lighter synthetic fog than what was used to fine-tune it. Moreover, all “foggy” models compare favorably to Dilation10 across the largest part of the range of β, with most “foggy” models being beaten by Dilation10 only for clear weather. Note also that the performance gain with “foggy” models under foggy conditions is much larger than the corresponding performance loss for clear weather. Effect of Synthetic Fog Density on Real-world Performance. Our final experiment on semantic segmentation serves two purposes: to examine the effect of varying the fog density of the synthetic training data as well as that of dehazing preprocessing on the performance of the fine-tuned model on real foggy data. To this end, we use three of the versions of Foggy Cityscapes-refined corresponding to the values {0.005, 0.01, 0.02} for β and consider all four options regarding dehazing preprocessing for fine-tuning Dilation10. The performance of the 12 resulting fine-tuned models on Foggy Driving in terms of mean IoU over all annotated classes as well as over frequent classes only is reported in <ref>. We first discuss the effect of varying fog density for each dehazing option individually and defer a general comparison of the various dehazing preprocessing options to the next paragraph.The two conditions that must be met in order for the examined models to achieve better performance are: * a good matching of the distributions of the synthetic training data and the real, testing data, and * a clear appearance of both sets of data, in the sense that the segmentation model should have an easy job in mining discriminative features from the data. Focusing on the case that does not involve dehazing, we observe that the models with β=0.005 and β=0.01 perform significantly better than that with β=0.02, implying that according to point <ref> Foggy Driving is dominated by scenes with light or medium fog. On the other hand, each of the three dehazing methods that are used for preprocessing has its own particularities in enhancing the appearance and contrast of foggy scenes while also introducing artifacts to the output. More specifically, MSCNN is slightly conservative in removing fog, as was found for other learning-based dehazing methods in <cit.>, and operates best under lighter fog, providing a significant improvement in this setting with regard to point <ref>. In conjunction with the light-fog character of Foggy Driving, this explains why fine-tuning on light fog (β=0.005) combined with MSCNN preprocessing delivers one of the two best overall results. By contrast, the more aggressive DCP is known to operate better at high levels of fog, as its estimated transmission is biased towards lower values <cit.>. The performance of models with DCP preprocessing thus peaks at medium rather than low simulated-fog density, which signifies a trade-off between removing fog to the proper extent and minimal introduction of artifacts. Non-local dehazing has also been found to operate best at medium levels of fog <cit.>, which results in a similar performance trend to DCP.Effect of Dehazing Preprocessing on Real-world Performance and Discussion. Comparing the four options regarding dehazing preprocessing via <ref>, we observe thatapplying no dehazing is the best or second best option for both measures and across all three values of β. Only MSCNN marginally beats the no-dehazing option in some cases, while overall these two options are roughly on a par. The absence of a significant performance gain on Foggy Driving when performing dehazing preprocessing can be ascribed to generic as well as method-specific reasons.First, in the real-world setting of Foggy Driving, the homogeneity and uniformity assumptions of the optical model (<ref>) that is used by all examined dehazing methods may not hold exactly. Of course, this model is also used in our fog simulation, however, foggy image synthesis is a forward problem, whereas image defogging/dehazing is an inverse problem, hence inherently more difficult. Thus, the artifacts that are introduced by our fog simulation are likely to be less prominent than those introduced by dehazing. This fact appears to outweigh the potential increase in visibility for dehazed images as far as point <ref> above is concerned. An interesting insight that follows is the use of forward techniques to generate training data for hard target domains based on data from the source domain as an alternative to the application of inverse techniques to transform such target domains into the easier source domain.Second, the optical model (<ref>), on which most of the popular dehazing approaches rely, assumes a linear relation between the irradiance at a pixel and the actual value of the pixel in the processed hazy image. Therefore, these approaches require that an initial gamma correction step be applied before dehazing, otherwise their performance may deteriorate significantly. This in turn implies that the value of gamma must be known for each image, which is not the case for Cityscapes and Foggy Driving. Manually searching for “best” per-image values is also infeasible for these large datasets. In the absence of any further information, we have used a constant value of 1 for gamma as the authors of <cit.> recommend, which is probably suboptimal for most of the images. We thus wish to point out that future work on outdoor datasets, whether considering fog/haze or not, should ideally record the value of gamma for each image, so that dehazing methods can show their full potential on such datasets.Specifically for DCP, performance decreases compared to MSCNN partly due to the light-fog character of Foggy Driving which does not match the optimal operating point of DCP. On the other hand, non-local dehazing uses a different model for estimating atmospheric light than the one that is shared by our fog simulation, MSCNN, and DCP, and thus already faces greater difficulty in dehazing images from Foggy Cityscapes. §.§ Linking the Objective and Subjective Utility of Dehazing Preprocessing in Foggy Scene UnderstandingOur experiments in Section <ref> indicate that using any of the three examined state-of-the-art dehazing methods to preprocess foggy images before feeding them to a CNN for semantic segmentation does not provide a clear benefit over feeding the foggy images directly in the objective terms of mean IoU performance of the trained model. In this section, we complement this objective evaluation with a study of the utility of dehazing preprocessing for human understanding of foggy scenes and show that the comparative results of the objective evaluation generally agree with the comparative results of the human-based evaluation.Both for the objective semantic-segmentation-based and the subjective human-based evaluation, we compare the four aforementioned options with regard to dehazing preprocessing individually on each image of our datasets. <ref> presents examples of the tetrads of images that we consider: the foggy image, which either belongs to the validation set of Foggy Cityscapes-refined with β=0.01 or to Foggy Driving and corresponds to no usage of dehazing, and its dehazed versions using DCP, MSCNN and non-local dehazing. For comparative objective evaluation of the four alternatives on each image, we use the mean IoU scores of the respective fine-tuned DCN models that are considered in the experiment of <ref>, measured on that image. The classes that do not occur in an image are not considered for computing mean IoU on this image. The four alternatives are ranked for each image according to their mean IoU scores on it. Comparative evaluation based on human subjects considers the same tetrads of images but employs a more composite protocol, which is detailed below.User Study via Amazon Mechanical Turk. Humans are subjective and are not good at giving scores to individual images in a linear scale <cit.>. We thus follow the literature <cit.> and choose the paired comparisons technique to let human subjects compare the four options regarding dehazing preprocessing. The participants are shown two images at a time that both pertain to the same scene, side by side, and are simply asked to choose the one which is more suitable for safe driving (easier to interpret). Thus, six comparisons need to be performed per scene, corresponding to all possible pairs.We use Amazon Mechanical Turk (AMT) to perform these comparisons. In order to guarantee high quality, we only employ AMT Masters in our study and verify the answers via a Known Answer Review Policy. Masters are an elite group of subjects, who have consistently demonstrated superior performance on AMT. Each individual task completed by the participants, referred to as Human Intelligence Task (HIT), comprises five image pairs to be compared, out of which three pairs are the true query pairs and the rest two pairs have a known correct answer and are only used for validation. In particular, each known-answer pair consists of two versions of a scene from Foggy Cityscapes-refined with different levels of fog, choosing from three versions of the dataset corresponding to clear weather, β=0.005 and β=0.01. The version with less fog is considered the correct answer. In order to avoid answers based on memorized patterns, the five image pairs in each HIT are randomly shuffled and the left-right order of the images in each pair is randomly swapped. In addition, each HIT is completed by three different subjects to increase reliability. The overall quality of the user survey is shown in <ref>, which demonstrates that the subjects have done a decent job: for 83% of the HITs, both known-answer questions are answered correctly. We only use results from these HITs in our following analysis.Consistency of Subjects' Answers.We first study the consistency of choices among subjects; all subjects are in high agreement if the advantage of one option over the other is obvious and consistent. To measure this, we employ the coefficient of agreement <cit.>:μ = 2σ/m2t2 -1,with σ=∑_i=1^t∑_j=1^ta_ij2,where a_ij is the number of times that option i is chosen over option j, m=3 is the number of subjects, and t=4 is the number of dehazing options. The maximum of μ is 1 for complete agreement and its minimum is -1/3 for complete disagreement. The values of μ for all pairs of options are shown in Table <ref>. The small positive numbers in the table suggest that subjects tend to agree when comparing options pairwise but no single option has dominant advantage over another one.Ranking and Correlation with Objective Evaluation. We finally compute the overall ranking of all four options for each image based on the number of times each option is chosen in all relevant pairwise comparisons. The correlation of these rankings with those induced by mean IoU performance is measured with Kendall's τ coefficient <cit.> with -1 ≤τ≤ 1, where a value of 1 implies perfect agreement, -1 implies perfect disagreement, and 0 implies zero correlation. <ref> provides a complete overview of the comparative results both for our user study and the semantic-segmentation-based evaluation on Foggy Cityscapes-refined and Foggy Driving, including rank correlation results for the two types of evaluation.The results in the top row of <ref> indicate that none of the three examined methods for dehazing preprocessing improves reliably the human understanding of synthetic foggy scenes from Foggy Cityscapes or real foggy scenes from Foggy Driving. In particular, the no-dehazing option beats all other three options in pairwise comparisons on Foggy Cityscapes-refined and loses only to DCP marginally on Foggy Driving, while it is also ranked first on more images than any other option for both datasets.In addition, the rankings obtained with the two types of evaluation are generally in congruence for the real-world case of Foggy Driving. The no-dehazing and DCP options are ranked higher than MSCNN and non-local dehazing both in the user study and in the objective evaluation. The high performance of DCP compared to MSCNN is due to the usage of β=0.01 for Foggy Cityscapes-refined (cf. the discussion in Section <ref>). What is more, the two rankings exhibit a positive correlation on average for Foggy Driving based on the respective distribution of τ in the bottom right chart of <ref>, which supports our conclusion in Section <ref> about the marginal benefit of dehazing preprocessing for foggy scene understanding.§.§ Object DetectionFor our experiment on object detection in foggy scenes, we select the modern Fast R-CNN <cit.> as the architecture of the evaluated models. We prefer Fast R-CNN over more recent approaches such as Faster R-CNN <cit.> because the former involves a simpler training pipeline, making fine-tuning to foggy conditions straightforward. Consequently, we do not learn the front-end of the object detection pipeline which involves generation of object proposals; rather, we use multiscale combinatorial grouping <cit.> for this task.In order to ensure a fair comparison, we first obtain a baseline Fast R-CNN model for the original Cityscapes dataset, similarly to the preceding semantic segmentation experiments. Since no such model is publicly available, we begin with the model released by the author of <cit.> which has been trained on PASCAL VOC 2007 <cit.> and fine-tune it on the union of the training and validation sets of Cityscapes which comprises 3475 images. Fine-tuning through all layers is run with the same configurations as in <cit.>, except that we use the “poly” learning rate policy with a base learning rate of 2×10^-4 and a power parameter of 0.9, with 7k iterations (4 epochs).This baseline model that has been trained on the real Cityscapes with clear weather serves as initialization for fine-tuning on our synthetic images from Foggy Cityscapes-refined. To this end, we use all 550 training and validation images of Foggy Cityscapes-refined and fine-tune with the same settings as before, only that the base learning rate is set to 10^-4 and we run 1650 iterations (6 epochs).We experiment with two values of the attenuation coefficient β for Foggy Cityscapes-refined and present comparative performance on the 33 finely annotated images of Foggy Driving in <ref>. No dehazing is involved in this experiment. We concentrate on the classes car and person for evaluation, since they constitute the intersection of the set of frequent classes in Foggy Driving and the set of annotated classes with distinct instances. Individual average precision (AP) scores for car and person are reported, as well as mean scores over these two classes (“mean frequent”) and over the complete set of 8 classes occurring in instances (“mean all”). For completeness, we note that the original VOC 2007 model of <cit.> exhibits an AP of 2.1% for car and 1.9% for person.Both of our fine-tuned models outperform the baseline model by a significant margin for car. At the same time, they are on a par with the baseline model for person. The overall winner is the model that has been fine-tuned on light fog, which we refer to as FT-0.005: it outperforms the baseline model by 2.4% on average on the two frequent classes and it is also slightly better when taking all 8 classes into account.We provide a visual comparison of FT-0.005 and the baseline model for car detection on example images from Foggy Driving in <ref>. Note the ability of our model to detect distant cars, such as the two cars in the image of the second row which are moving on the left side of the road and are visible from their front part. These two cars are both missed by the baseline model.§ SEMI-SUPERVISED LEARNING WITH SYNTHETIC FOG While standard supervised learning can improve the performance of SFSU using our synthetic fog, the paradigm still needs manual annotations for corresponding clear-weather images. In this section, we extend the learning to a new paradigm which is also able to acquire knowledge from unlabeled pairs of foggy images and clear-weather images. In particular, we train a semantic segmentation model on clear-weather images using the standard supervised learning paradigm, and apply the model to an even larger set of clear but “unlabeled” images (our 20000 unlabeled images of Foggy Cityscapes-coarse) to generate the class responses. Since we have created a foggy version for the unlabeled dataset, these class responses can then be used to supervise the training of models for SFSU. This learning approach is inspired by the stream of work on model distillation <cit.> or imitation <cit.>. <cit.> transfer supervision from sophisticated models to simpler models for efficiency, and <cit.> transfers supervision from the domain of images to other domains such as depth maps. In our case, supervision is transferred from clear weather to foggy weather. The underpinnings of our proposed approach are the following: 1) in clear weather, objects are easier to recognize than in foggy weather, thus models trained on images with clear weather in principle generalize better to new images of the same condition than those trained on foggy images; and 2) since the synthetic foggy images and their clear-weather counterparts depict exactly the same scene, recognition results should also be the same for both images ideally.We formulate our semi-supervised learning (SSL) for semantic segmentation as follows. Let us denote a clear-weather image by 𝐱, the corresponding foggy one by 𝐱^', and the corresponding human annotation by 𝐲. Then, the training data consist of both labeled data 𝒟_l ={(𝐱_i, 𝐱^'_i, 𝐲_i)}_i=1^l and unlabeled data 𝒟_u ={(𝐱_j, 𝐱^'_j)}_j=l+1^l+u, where 𝐲_i^m,n∈{1, ..., K}is the label of pixel (m,n), and K is the total number of classes. l is the number of labeled training images, and u is the number of unlabeled training images. The aim is to learn a mapping functionϕ^': 𝒳^'↦𝒴 from 𝒟_l and 𝒟_u. In our case, 𝒟_l consists of the 498 high-quality foggy images in the training set of Foggy Cityscapes-refined which have human annotations with fine details, and 𝒟_u consists of the additional 20000 foggy images in Foggy Cityscapes-coarse which do not have fine human annotations. Since 𝒟_u does not have class labels, we use the idea of supervision transfer to generate the supervisory labels for all the images therein.To this end, we first learn a mapping function ϕ: 𝒳↦𝒴 with 𝒟_l and then obtain the labels 𝐲̂_j=ϕ(𝐱_j) for 𝐱_j and 𝐱^'_j, ∀ j ∈{l+1, ..., l+u}.𝒟_u is then upgraded to 𝒟̂_u={(𝐱_j, 𝐱^'_j, 𝐲̂_j)}_j=l+1^l+u. The proposedscheme for training semantic segmentation models for foggy images 𝐱^' is to learn a mapping functionϕ^' so thathuman annotations 𝐲 and the transferred labels 𝐲̂ are both taken into account: min_ϕ^'∑_i=1^l L(ϕ^'(𝐱^'_i), 𝐲_i) + λ∑_j=l+1^l+u L(ϕ^'(𝐱^'_j), 𝐲̂_j),where L(.,.) is the Categorical Cross Entropy Loss function for classification, and λ=l/u× w is a parameter for balancing the contribution of the two terms, serving as the relative weight of each unlabeled image compared to each labeled one. We empirically set w=5 in our experiment, but an optimal value can be obtained via cross-validation if needed. In our implementation, we approximate the optimization of (<ref>) by mixing images from 𝒟_l and 𝒟̂_u in a proportion of 1:w and feeding the stream of hybrid data to a CNN for standard supervised training.We select RefineNet <cit.> as the CNN model for semantic segmentation, which is a more recent and better performing method than DCN <cit.> that is used in Section <ref>. The reason for using DCN in Section <ref> is that RefineNet had not been published yet at the time that we were conducting the experiments of Section <ref>. We would like to note that the state-of-the-art PSPNet <cit.>, which has been trained on the Cityscapes dataset similarly to the original version of RefineNet that we use as our baseline, achieved a mean IoU of only 24.0% on Foggy Driving in our initial experiments.We use mean IoU for evaluation, similarly to Section <ref>, and β=0.01 for Foggy Cityscapes. We compare the performance of three trained models: 1) original RefineNet <cit.> trained on Cityscapes, 2) RefineNet fine-tuned on 𝒟_l, and 3) RefineNet fine-tuned on 𝒟_l and 𝒟̂_u. The mean IoU scores of the three models on Foggy Driving are 44.3%, 46.3%, and 49.7% respectively. The 2% improvement of 2) over 1) confirms the conclusion we draw in Section <ref> that fine-tuning with our synthetic fog can indeed improve the performance of semantic foggy scene understanding. The 3.4% improvement of 3) over 2) validates the efficacy of the SSL paradigm.<ref> shows visual results of 1) and 3), along with the foggy images and human annotations. The re-trained model with our SSL paradigm can better segment certain parts of the images which are misclassified by the original RefineNet, the pedestrian in the first example, the tram in the fourth one, and the sidewalk in the last one.Both the quantitative and qualitative results suggest that our approach is able to alleviate the need for collecting large-scale training data for semantic understanding of foggy scenes, by training with theannotations that are already available for clear-weather images and the generated foggy images directly and by transferring supervision from clear-weather images to foggy images of the same scenes.§ CONCLUSIONIn this paper, we have demonstrated the benefit of synthetic data that are based on real images for semantic understanding of foggy scenes. Two foggy datasets have been constructed to this end: the partially synthetic Foggy Cityscapes dataset which derives from Cityscapes, and the real-world Foggy Driving dataset, both with dense pixel-level semantic annotations for 19 classes and bounding box annotations for objects belonging to 8 classes. We have shown that Foggy Cityscapes can be used to boost performance of state-of-the-art CNN models for semantic segmentation and object detection on the challenging real foggy scenes of Foggy Driving, both in a usual supervised setting and in a novel, semi-supervised setting. Last but not least, we have exposed through detailed experiments the fact that image dehazing faces difficulties in working out of the box on real outdoor foggy data and thus is marginally helpful for SFSU. In the future, we would like to combine dehazing and semantic understanding of foggy scenes into a unified, end-to-end learned pipeline, which can also leverage the type of synthetic foggy data we have introduced. The datasets, models and code are available at <http://www.vision.ee.ethz.ch/ csakarid/SFSU_synthetic>. The authors would like to thank Kevis Maninis for useful discussions. This work is funded by Toyota Motor Europe via the research project TRACE-Zürich. spmpsci | http://arxiv.org/abs/1708.07819v3 | {
"authors": [
"Christos Sakaridis",
"Dengxin Dai",
"Luc Van Gool"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170825173602",
"title": "Semantic Foggy Scene Understanding with Synthetic Data"
} |
[email protected] Department of Physics, Brown University,182 Hope St., Providence, Rhode Island 02912, USA Institute for Theory and Computation, Harvard University, 60 Garden Street, Cambridge, Massachusetts, 02138, [email protected] Astronomy Department,Harvard University, 60 Garden Street, Cambridge, Massachusetts, 02138, USA Future generation of gravitational wave detectors will have the sensitivity to detect gravitational wave events at redshifts far beyond any detectable electromagnetic sources. We show that if the observed event rate is greater than one event per year at redshifts z ≥ 40, then the probability distribution of primordial density fluctuations must be significantly non-Gaussian or the events originate from primordial black holes. The nature of the excess events can be determined from the redshift distribution of the merger rate.95.35.+d, 98.80.-k, 04.30.Db, 04.30.?wMaximum redshift ofgravitational wave merger events Abraham Loeb December 30, 2023 =====================================================The discovery of gravitational waves from merging pairs of massive black holes <cit.> has opened a new window to the astrophysics of black holes, their formation, and cosmic evolution. Black holes of stellar masses have been observed with LIGO <cit.> and supermassive black holes in galaxies are expected to be detected by LISA over the next couple of decades <cit.>. The sensitivity of the next generation of ground-based gravitational wave detectors is expected to improve by at least an order of magnitude <cit.>, thus allowing the detection of merging black holes events out to the highest redshifts, potentially exceeding the reach of electromagnetic observations which respond to amplitude squared and not amplitude. The expected rates of black hole mergers has been calculated based on the number and properties of the few events discovered to-date (see, e.g. <cit.>). The rate depends on a multitude of factors: black holes must be formed and they must find a way to get close enough so that gravitational waves can take-over as the dominant energy loss mechanism.The redshift distribution encodes information about the origin of black hole pairs. If black holes originate from massive stellar progenitors then the redshift distribution should relate to the formation, accretion, and cooling of gas in galaxies. If on the other hand the black holes are primordial <cit.>, then the redshirt distribution will extend to earlier cosmic times due to primordial binaries <cit.>. A key difference between these two scenarios is that in the case of a baryonic origin, black holes must form out of cold gas, which accreted into a dark matter gravitational potential well, and then cooled to form black hole progenitors. This path follows the abundance of appropriate potential wells. In this letter we calculate themaximum redshift of expected black hole merger events that have baryonic origin in the standard cosmological model. That is, the black holes are formed in galaxies as opposed to primordial black holes, or black holes that are formed in non-standard cosmological scenarios, e.g., cosmologies with asignificant non-Gaussianity in the primordial density fluctuations of the dark matter.The significance of this calculation is two-fold: first, it defines a maximum redshift over which baryonic structures can form, and second any detection above the derived bound will signify the presence of either non-Gaussianities that controlthe formation of baryonic structures at unexpectedly high redshifts, or that black hole events may be due to primordial black holes.In the following we make two key assumptions in the derivation of a maximum redshift of baryonic black hole gravitational wave events. First, we conservatively assume that black hole pairs merge instantaneously, i.e., there is no time lag between the formation of black holes, the evolution of the binary and the subsequent sequence of events that leads to a merger. Second, we conservatively assume that all gas accreted in dark matter halos end up in stars that end up in black holes. Realistically, both of these assumptions are vastly optimistic. However, their application guarantees that the derived maximum redshift is indeed a very hard limit and thus any observation that violates this bound will be of enormous scientific significance.We begin by calculating the number of observed gravitational wave events per year greater than redshift z as the integral of the rate of black hole mergers per redshift interval N(>z)= ∫_z^∞ d R/dzdz, where d R/dz is the rate of merger events per redshift interval, d R/dz ≡ ∫_(z)^∞dN/dMdV C_NG(M,z) × ⟨ϵ (M,z) ⟩/(1+z)Ṁ_g(M,z)/2 dV/dzdM.Here,dN/dMdV is the comoving density of dark matter halos of mass M at redshift z, C_NG(M,z) is a correction to the mass function in the case where non-Gaussianity is present (with C_NG(M,z) = 1 in the standard ΛCDM cosmology),Ṁ_g(M,z) is the rate of accreted gas in halos of mass M at z,⟨ϵ (M,z) ⟩is the efficiency of converting gas to black holes of mass , dV/dz is the comoving volume per redshift interval and the (1+z) factor in the denominator is to convert the rest frame rate to the observed rate. The integral in Equation (<ref>) is performed from a minimum halo mass (z) to infinity. Throughout the paper we use acosmological model with a power spectrum with a spectral indexn_s = 0.967, a normalization σ_8 = 0.81, a present value of the Hubble parameter H_0 = 70.4 km/s/Mpc and dark matter, baryonic and cosmological constant mass density parameters ofΩ_DM = 0.226, Ω_b = 0.0455, and Ω_Λ = 0.728, respectively <cit.>. Wenext explain how we calculate each of these quantities. Theexpected number of gravitational wave events depends strongly on the halo mass function which declines exponentially at high redshifts for Gaussian fluctuations. Figure <ref>shows the mass function at high redshifts from 15 different numerical simulations <cit.>. The only results that are valid at the high redshifts we consider here are the ones of <cit.>. The halo masses of interest at these high redshifts correspond to extremely rare peaks. The abundance of halos is roughly bounded by two functional forms – the analytic form of Press-Schecter <cit.> (red curve in Figure <ref>)gives the lowest number of halos while the ellipsoidal collapse model of Sheth, Mo & Tormen <cit.> gives the maximum (blue curve in Figure <ref>). All other mass functions, including the more realistic results in <cit.> lie in between these two analytic forms. The presence of a non-Gaussianity in the initial conditions can alter theabundance of dark matter halos, especially in the exponential tail of the mass function (which is the regime of interest here). We therefore modify the mass function to include such features by assuming that the non-Gaussian mass function is the product of the Gaussian mass function multiplied by a correction factor<cit.> that describescosmologies <cit.> (though it is important to emphasize thatis just one possible parametrization of non-Gaussianities), C_NG(M) =[δ_c^2/6 ΔdS_3/d lnσ(M) + Δ]exp( S_3 δ_c^3/ 6 σ^2(M)).Here, Δ≡√( 1 - δ_c S_3 / 3), δ_c = 1.686 √(a), with a = 0.9,and S_3 = 3.15× 10^-4 / σ^0.838(M).The Sheth-Mo-Tormen mass function <cit.>, modified to include the effects ofnon-Gaussianities <cit.> is shown in Figure <ref> as the blue dashed curve. We consider this modified mass function to represent themaximum abundance of dark matter halos (repeating the calculations for cosmologies with g_NL or τ_NL within the currentlimits <cit.> leads to smaller effects than the effects from the current uncertainties in ).It is also important to note that the mass function depends on the normalization of the power spectrum; however, the current percent-level uncertainty of σ_8 is negligible for our purposes. The quantity Ṁ_b(M,z)) represents the rate of gas inflow in halos of mass M at z. It has been predicted in simple theoretical grounds <cit.> and has been measured in hydrodynamical simulations at high redshift <cit.>. We adopt the maximum gas accretion rate<cit.>,Ṁ_g (M,z) ≈ 10^-3 M_⊙yr^-1( M/10^6 M_⊙)^1.127( 1+z /20)^η,where η = 2.5. Assuming a gas accretion rate as given in<cit.>results in a rate that is only slightly smaller(see Figure <ref>). If on the other hand we assume that the redshift dependence is steeper (i.e., η > 2.5) as suggested by high-redshift studies of the growth rate of halos <cit.>, the gas accretion rate can be higher, however the falloff at high redshift becomes much steeper. Both of these assumptions have negligible effects to the scope of this paper. The quantity ⟨ϵ(M,z) ⟩ represents the fraction of infalling gas that turns into black holes. In order to compute an extremely conservative upper bound on the number of black holes, we assume that all starsformed out of infalling gas will be converted to black holes that will merge within the Hubble time at z. We assume that this function is bracketed from above by the ratio of stellar mass to baryon mass in dark matter halos, i.e., ⟨ϵ (M,z) ⟩ = M_stellar / M η, where η = Ω_b / Ω_DM, is the baryon fraction, and we assume the stellar mass as a function of host halo mass and redshift is given byextrapolating (beyond z ≈ 8) the results of <cit.>.In Figure <ref> we show the logarithm ofthe product⟨ϵ(M,z) ⟩Ṁ_g in units of M_⊙yr^-1, from 10^-1 - 10^-15 (in declining factors of ten from left to right).Thin lines correspond to the analytic prediction of <cit.> while thick lines are the numerical results of <cit.>. The contours show that at redshifts z ≤ 30the gas infall rate increases with mass, and that for a fixed mass it decreaseswith increasing redshift atz ≈ 30.At each redshift z, we integrate Equation <ref> from (z) to infinity. The lower mass limit of the integral, (z), is the minimum halo mass in which stars can form. This is set by the requirement of the formation of molecular hydrogen<cit.>. Recently, it has been argued that the tight coupling of baryons to photons prior to recombination gives rise to a velocity component that becomes important once baryons decouple <cit.>. Figure <ref> shows this effect on the minimum mass: the thin solid black curve is the standard case where baryons are assumed to follow dark matter, while the thick solid black curve corresponds to the numerical results of <cit.> where there is a velocity difference between dark matter and baryons.Since the minimum mass of molecular hydrogen cooling is roughly constant with redshift, star formation is severely suppressed at increasing redshifts for two reasons: the minimum mass corresponds to extremely rare peaks in the density field (see dashed grey lines in Figure <ref> that show the rarity of mass scales as a function of redshift) while at the same time the rate of gas infall decreases rapidly. The combination of these two effects introduces a sharp cutoff to the abundance of stars beyond z ≈ 40. Integrating the rate of merger events (Equation <ref>) from redshift z to infinity gives the total number of events per year greater than redshift z (Equation <ref>).Figure <ref> shows the result of this calculation. The blue curve corresponds to the maximal mass function <cit.>, a lower(i.e., ignoring the suppressing effects of a relative speed between dark matter and baryons <cit.>), and the maximal value of gas accretion <cit.>. The dashed blue curve makes the same assumptions as above, but with a modified mass function that includes a correction owing to the presence of non-Gaussianity at the current upper bound of = 43 <cit.>. The red curve is the opposite of the aforementioned case, where the mass function assumed is at its minimum <cit.>, the minimum mass is the largest (including relative velocities between baryons and dark matter <cit.>) and a low gas accretion rate <cit.>. Theshaded area represents everything in between these two extreme cases.We define the maximum redshiftsuch thatthe observed event rate is N(z = ) = 1 yr^-1.We find that the maximum redshift of expected gravitational wave events cannot exceed ≈ 40. All assumptions leading to this result are such so that the maximum redshift is maximized: largest abundance of halos (even including current limits on non-Gaussianity), lowest minimum mass for the formation of stars in halos at high redshifts, the assumed gas infall in halos is the maximum measured in numerical simulations, all stars formed in all halos end up in black hole pairs and all black hole pairs merge instantaneously. This confluence of maximizing all assumptions makes the result that the maximum redshift of expected gravitational wave sources of ≈ 40 a truly hard bound that cannot be violated unless something very drastically different takes place at high redshifts. The aforementioned assumptions can be relaxed and in some cases it is easy to read off the effect on the result (as the vertical axis is a scalable quantity). For example, if all accreted gas ends up in black holes of mass of = 10 (instead of 30) then the solid curves in Figure <ref> simply move up by a factor of 3. If on the other hand only a fraction of0.1% of gas ends in black holes of = 30 then the result of Figure <ref> moves down by a factor of 10^-3.In addition, the assumption of a δ-function mass spectrum of black holes is not realistic. A range of black hole masses is most likely present. The effects of such an assumption have been studied in the context of explaining the current rate of observed black hole mergerevents with LIGO <cit.>. In our case, such a black hole mass function will alter the shape of N(z ), but the effect onis negligible since the factors that give rise to the cutoff remain as discussed earlier (namely the shape of the halo mass function and the decline in gas infall at high redshifts). The prediction of a maximum redshift for black hole merger events can be tested withfuture gravitational wave detectors.In particular, Cosmic Explorer <cit.> will have the ability to detect events at these very high redshifts. Given the current design capabilities, Cosmic Explorer will be able to detect the merger of 30 black hole pairs at 10σ significance out to redshift of z ≈ 36 and at 5σ significance to redshift z ≈ 44 <cit.>. These two limits are shown as vertical dashed lines in Figure <ref>.Any detection of an event rate greater than once a year from a redshift greater than ≈ 40 will have majorimplications for cosmology. It would mean that either structure formation is not proceeding in the way that is currently envisioned, or that black hole mergers aredue to some exotic phenomenon. Two such possibilities exist: a strange non-Gaussianity that is not parametrized in terms of(e.g., decay of cosmic strings <cit.>), or from the merger of primordial black holes <cit.>. The latter idea has received considerable attention recently in light of the spectacular detection of gravitational waves by LIGO; however at present it seems that other astrophysical constraints make such a possibility less likely <cit.>. Nevertheless, if events with redshifts greater than ≈ 40 appear with rates greater than once per year, it may still be possible to disentangle their origin by looking at their redshift distribution as the exact dependence on redshift will be sensitive to the abundance of primordial binaries. We acknowledge useful discussions with Robert Fisher, Zóltan Haiman, Alex Geringer-Sameth, Darren Reed and Kyriakos Vattis. This work was supported by the Black Hole Initiative, which is funded by a grant from the John Templeton Foundation. SMK acknowledges support from DE-SC0017993 and the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics.We acknowledge usage of the HMFCalc online tool <cit.>. | http://arxiv.org/abs/1708.07380v2 | {
"authors": [
"Savvas M. Koushiappas",
"Abraham Loeb"
],
"categories": [
"astro-ph.CO",
"astro-ph.HE",
"gr-qc"
],
"primary_category": "astro-ph.CO",
"published": "20170824125928",
"title": "Maximum redshift of gravitational wave merger events"
} |
Deep Style Match for Complementary RecommendationKui Zhao^†, Xia Hu^, Jiajun Bu^†, Can Wang^††College of Computer Science, Zhejiang UniversityHangzhou, China{zhaokui, bjj, wcan}@zju.edu.cnHangzhou Science & Technology Information Research InstituteHangzhou, [email protected] 30, 2023 ======================================================================================================================================================================================================================================================= Humans develop a common sense of style compatibility between items based on their attributes. We seek to automatically answer questions like “Does this shirt go well with that pair of jeans?” In order to answer these kinds of questions, we attempt to model human sense of style compatibility in this paper. The basic assumption of our approach is that most of the important attributes for a product in an online store are included in its title description. Therefore it is feasible to learn style compatibility from these descriptions. We design a Siamese Convolutional Neural Network architecture and feed it with title pairs of items, which are either compatible or incompatible. Those pairs will be mapped from the original space of symbolic words into some embedded style space. Our approach takes only words as the input with few preprocessing and there is no laborious and expensive feature engineering.§ INTRODUCTIONWe have a common sense of style compatibility between items and can naturally answer questions like “Does this shirt go well with that pair of jeans?” These kind of style compatibility information can be exploited in many commercial applications,such as recommending items to users based on what they have already bought; or generating the wholepurchase outfits (see Figure <ref> for an example of clothes) to users querying certain items, if sufficient compatibilityrelationships between items are provided.To identify these compatibility relationships, existing methods such as frequent itemset mining <cit.>attempt to generate match items automatically by analyzing historical purchasing patterns.However, frequent itemset mining relies on historical purchasing records to find items frequentlypurchased together and new items will inevitably suffer from the “cold start” problem <cit.>.Recently, McAuley et al. <cit.> and Veit et al. <cit.> intend to discover the style match relationships between items using visual information presented in the images of items. However, besides being computationally expensive, image-based matching methodsare frequently plagued by the plentiful contents presented in images.For instance, Figure <ref> shows a part of the image for leggings from Taobao,which is the largest e-commerce platform in China. Besides leggings,the image also contains a coat, a pair of shoes and a very complex background etc.These contents will confuse the learning machines if only leggings are expected.To overcome the limitations of existing methods, we propose in this paper a novel style match approach using the title descriptions of items in online stores.The basic assumption of our work is that online sellers will place mostof the important attributes of a product in its title description,so that the product can be easily found by a keyword-based query.Therefore, the title description is a highly condensed collection ofattribute descriptions for a product.So it is feasible to model compatibility between two items better if we are capable of mapping the title pairs from the original space of symbolic words into some embedded style space.We here design a Siamese Convolutional Neural Network architecturefor matching title sentences. It will map two title sentences from an item pair into low-dimensional vectors respectively in parallel,which are then used to learn the compatibility between these two items in the style space.When designing the amalgamation part of Siamese CNN for computing compatibility,we have considered the ability of our model to be extended tobig data scenarios,which is critical for real-world recommendation applications.We test our approach on two large datasets: a Chinese dataset from Taobao provided byAlibaba Group and an English dataset from Amazon provided by <cit.>.Our approach demonstrates strong performance onboth datasets, which indicates its ability of learning human sense of style compatibility between items. § RELATED WORKFinding complementary items has been studied for a long time.The early works can be traced back to frequent itemset mining <cit.>,which generates match items automatically by analyzing history purchasing patterns.Frequent itemsets such as “beer and diaper” sometimes have nothing to do with compatibility. What's more, they are challenged by the “cold-start” problem, which means new productswith no historical records are invisible to the algorithm <cit.>. Many approaches such as content-based recommendation or social recommendation are proposed to address thisproblem (see <cit.> for a survey).Closely related to our work are <cit.> and <cit.>, in which McAuley et al. and Veit et al. attempt to learn clothing style similarity based on their appearance in images. However, our work differs from <cit.> and <cit.> in the following two aspects: (1) we use title descriptions, which contain rich attribute information instead of item images; (2) the objective of our method is to find matching items from their attribute description instead of learning visual similarities from item images. § PROBLEM FORMULATIONWe here describe the problem in a formal way:given a query item set Q={q_1, ⋯, q_m} and a candidate item set C={c_1, ⋯, c_n},where each query item q_i∈ Q comes together with the compatibilityjudgements {y_i_1, ⋯, y_i_n}.The complementary item c_j∈ C is labeled with y_i_j=1 and y_i_j=0 otherwise.Our goal is to build a model to compute the compatibility probability between q_i and c_j: P(y=1|q_i, c_j) = f(ϕ(q_i,θ_1), ϕ(c_j,θ_1), θ_2),where function ϕ(·) is the sentence model mapping a titlesentence into a low-dimensional representation vectorand function f(·) computes the compatibility probability between two items in the style space.The parameter vectors θ_1 and θ_2 are learned in the training process. § STYLE MATCHThe main building block of our approach is a sentence model based on CNN.This sentence model will map two title sentences from an item pair into low-dimensional vectors respectively in parallel,which are then used to learn the compatibility between two items in the style space.§.§ Sentence modelWe model sentences with function ϕ(·),which is a convolutional architecture as shown in Figure <ref>.In the following, we give a brief explanation of the main components in our Convolutional Neural Network.§.§.§ Sentence matrix.Our sentence model takes a sentence s as the input,where it is treated as a sequence of raw words: [s_1,⋯,s_| s|]and each word s_i is from a vocabulary V. Firstly, each word s_i is represented by a distributional representation vectorw_i ∈ℝ^d, looked up from the word-level embedding matrix W∈ℝ^d× |V|.Then a sentence matrix S∈ℝ^d× | s| can be built for the input sentence s:S= [|||;w_1⋯ w_| s|;||| ],where the i-th column is the distributional representation vector w_ifor the i-th word in s.The values in the embedding matrix W are parameters initialized with an unsupervised neural language model <cit.> and sequentially optimized during training.The embedding dimension d is a hyper-parameter of the model. As shown in Figure <ref>,for the sentence s=[Uniforms, Men's, Short, Sleeve, Polo, Shirt],when the embedding dimension d is 4, the sentence matrix is a matrix in ℝ^4× 6.§.§.§ Convolutional feature maps.Convolution can be seen as a special kind of linear operation and aimed to extract local patterns.We use the one-dimensional convolution to recognize discriminative word sequences from the input sentences.The one-dimensional convolution is an operation betweentwo vectors f∈ℝ^m and s∈ℝ^| s|.The vector f is called as a filter of size m and the vector s is a sequence of size | s|. The specific operation is to take the dot product ofthe vector f with each m-gram sliding along the sequence s and obtaina new sequence c where:c_j= f^ T s_j-m+1:j. In practice, we usually add a bias b to the dot product result:c_j= f^ T s_j-m+1:j+b.There are two types of convolution depending on the allowed range of index j: narrow and wide.The narrow type restricts j in the range [m, | s|] andthe wide type restricts j in the range [1, | s|+m-1].The benefits of wide type over the narrow type in text processing are discussed in detail in <cit.>.Briefly speaking, unlike the narrow convolutionwhere words close to margins are seen fewer times,wide convolution gives equal attention to all words in the sentenceand so is better at handling words at margins.More importantly, a wide convolution always produces a validnon-empty result c even when | s|<m.For these reasons, we use wide convolution in our model. The sentence matrix S is not just a sequence of single values but a sequence of vectors,where the dimension of each vector is d.So when we apply the one-dimensional convolution on the sentence matrix S,we need a filter bank F∈ℝ^d× m consisting of d filters of size m and a bias bank B∈ℝ^d consisting of d baises.Each row of S is convoluted with the corresponding row of Fand then the corresponding row of B is added to the convolution result.After that, we obtain a matrix C∈ℝ^d× (| s|+m-1):conv( S,F,B): ℝ^d× | s|→ℝ^d× (| s|+m-1).The values in filter bank F and bias bank B are parameters optimized during training. The filter size m is a hyper-parameter of the model. In Figure <ref>,after applying a 4× 3 filter bank and a bias bank of size 4 on the 4 × 6 sentence matrix,we obtain an intermediate matrix of size 4 × 8. §.§.§ Activation function.To make the network capable of learning non-linear functions,a non-linear activation α (·) need to be applied in an element-wise wayto the output of the preceding layerand a matrix A∈ℝ^d× (| s|+m-1) is then obtained:α( C): ℝ^d× (| s|+m-1)→ℝ^d× (| s|+m-1). Popular choices of α (·) include: sigmod, tanh and relu (rectified linear defined as max (0, x)).In practice, our experimental results are not very sensitive to the choice of activation, so we choose relu due to its simplicity and computing efficiency.In addition, we can see the bias b in (<ref>) plays the role of settingan appropriate threshold for controlling units to be activated. §.§.§ Pooling.Pooling layer will aggregate the information in the output of preceding layer.This operation aims to make the representation more robust and invariant to small translations in the input.More importantly, pooling helps to handle inputs with varying size, e.g. processing sentences with uncertain length. For a given vectora∈ℝ^| a|, traditional pooling aggregates it into a single value:pooling( a): ℝ^| a|→ℝ.The way of aggregating the information defines two types of pooling operations: average and max.Max pooling is used more widely in practice.Recently, max pooling has been generalized to k-max pooling <cit.>,in which k max values are selected from the vector a and arranged in their original order:k-pooling( a): ℝ^| a|→ℝ^k, where k is a hyper-parameter of the model. When we apply k-max pooling on the matrix A,each row of A is pooled respectively and we obtain a matrix P∈ℝ^d× k: k-pooling( A): ℝ^d× | a|→ℝ^d× k. In Figure <ref>, after applying k-max pooling (with k=5) onthe intermediate matrix of size 4× 8, we obtain a new intermediate matrix of size 4 × 5.§.§.§ Multiple feature maps.After a group of above operations, we obtain the first order representationlearning to recognize the specific m-grams in the input sentence.To obtain higher order representations, we can use a deeper networkby repeating these operations. The higher order representations cancapture patterns of the sentence in much longer range. Meanwhile, we can also extend network to learning multi-aspect representations.Let P^i denote the i-th order representation.We can compute K_i representations P^i_1,⋯, P^i_K_iin parallel at the same i-th order. Each representation P^i_jis computed by two steps.First, we compute convolution on each representation P^i-1_k at the lower order i-1 with the distinct filter bank F^i_j,k and bias bank B^i_j,kand thensum up the results.Second, non-linear activation and k-max pooling are applied to the summation result: P^i_j=k-pooling(α(∑_k=1^K_i-1conv( P^i-1_k,F^i_j,k,B^i_j,k))). In Figure <ref>, there are two representations at the first order and tworepresentations at the second order: P^1_1∈ℝ^4× 5,P^1_2∈ℝ^4× 5and P^2_1∈ℝ^4× 3,P^2_2∈ℝ^4× 3.§.§.§ Full connection.Full connection is a linear operationto combine all representations at the highest order into a single vector.More specifically, for the highest order representationsP^h_1,⋯, P^h_K_h (assume P^h_k∈ℝ^d× l),we first flat them into a vector p∈ℝ^K_h× d× l.Then we transform it with a dense matrix H∈ℝ^(K_h× d× l)× n:x= p^ T H,where x∈ℝ^n is the final representation vector.The values in matrix H are parameters optimized during training.The representation size n is a hyper-parameter of the model.In Figure <ref>, we finally represent the input sentence witha vector of size n=5.§.§ Matching itemsWe compute the compatibility probability between two itemswith function f(·), which is a Siamese Convolutional Neural Network as shown in Figure <ref>.Siamese setup is introduced by Hadsell et al. <cit.> and used widely in learning distance metrics.When designing the amalgamation part of our model,we have considered its scalability for big data scenarios,which is critical for real-world applications. §.§.§ Style space. For two given items q and c, after generatingthe representation vectors x_q∈ℝ^n and x_c∈ℝ^n of their title sentences respectively, we compute the compatibility probability between them as follow:P(y=1|q,c) =σ( x_q^ T M x_c+b)=1/1+e^-( x_q^ T M x_c+b),where M∈ℝ^n× n is a matrix and b is a scalar.Wecall M as the compatibility matrixand the space spanned by M as the style space.After transformation x'_q= x_q^ T M,x'_q represents the item which is most style compatible to q. We seek items whose representations are close to x'_q under linear kernel distance.The values in compatibility matrix M and the bias b are parametersoptimized during the training. On the other hand, x_q^ T M x_c in (<ref>) can beviewed as a noisy-channel model, which has been widely used in the information retrieval andQA system <cit.> <cit.>. § RECOMMENDATIONIn recommendation applications,we are usually given a query item set Q={q_1, q_2,⋯, q_m}and a candidate item set C={c_1, c_2,⋯,c_n},where the query item set is relatively small and the candidate item set is usually very large.For each query item q_i, we intend to query its K most complementary items fromthe candidate set C and rank them from high compatibility to low compatibility.When the item candidate set is very large,it is inefficient and even unacceptable to compute the compatibility for all item pairs (q_i,c_j) and then sort them. Our approach can be easily extended to handle these big data scenarios. Given two items q and c, we first generate their representation vectors x_q and x_c respectively.Then their compatibility probability is computed according to (<ref>).We notice that the function σ (·) in (<ref>) is a monotonic increasing function and b is a learned constant.Thus for a query item q and two candidate items c_1, c_2, we have: P(y=1|q,c_1) ≤ P(y=1|q,c_2)⇔ x_q^ T M x_c_1 ≤ x_q^ T M x_c_2. Based on this property, we transform the original problem of querying the K most complementary itemsof q from the item candidate set Cinto another problem, namely searching K nearest neighbors of x' (x'_q= x_q^ T M)from { x_c_1, ⋯,x_c_n} under the linear kernel distance.It is well known as Maximum Inner Product Search (MIPS).There are many methods solving MIPS efficiently on the large scale data,such as tree techniques <cit.> andhashing techniques <cit.> <cit.> etc.§ TRAININGWe train the model to maximize the likelihood of a observed relationship training set ℛ,where r_ij∈ℛ: r_ij= 1, if itemsi andj are compatible; 0, otherwise. Maximizing the likelihood is equal to minimizing the binary-cross entropy loss function:L=-∑_r_ij∈ℛ[r_ijlog(p)+(1-r_ij)log(1-p)],where p=P(y=1|i,j). The parameters to be optimized in our network are θ_1,θ_2, which have been mentioned above:θ_1={ W,F,B,H} and θ_2={ M, b},namely the word embeddings matrix W, filter bank F, bias bank B,dense matrix H, compatibility matrix M and compatibility bias b.Note that there are multiple filter banks and bias banks to be learned. In the following sections, we present several crucial details for training our deep learning model. §.§ RegularizationTo alleviate the overfitting issue, we use a popular and efficient regularizationtechnique named dropout <cit.>.Dropout is applied to the flatted vector p (presented in (<ref>)) beforetransforming it with the dense matrix H.A portion of units in p are randomly dropped out by setting them to zeroduring the forward phase, which is helpful for preventing the feature co-adaptation.The dropout rate is a hyper-parameters of the model.§.§ Hyper-parameters The hyper-parameters in our deep learning model are set as follows:the embedding dimension is d=100; the size of filters at the first order representation is m=3;the number of max values selected by k-max pooling at the first order representation is k=5;the size of filters at the second order representation is m=2;the number of max values selected by k-max pooling at the second order representation is k=3;the dimension of the vector used to represent the sentence is n=100;the dropout rate is p=0.2.What's more, there are K_1=100 representations computed in parallel at thefirst order representation andK_2=100 representations computed in parallel at the second order representation.§.§ OptimizationTo optimize our network, we use the Stochastic Gradient Descent (SGD) algorithm with shuffled mini-batches.The parameters are updated through the back propagation framework with Adagrad rule <cit.>.The batch size is set to 256 and the network is trained for 20 epochs.The training progress will be early stopped if there is no more updateto the best loss on the validation set for the last 5 epochs. We train our network on a GPU for speeding up.A Python implementation using Keras[http://keras.io]powered by Theano <cit.>can process 428k text pairs per minute on a single NVIDIA K2200 GPU. § EXPERIMENTSWe evaluate our method on two large datasets:a Chinese dataset from Taobao and an English dataset from Amazon. §.§ Datasets§.§.§ Taobao.This dataset is collected from Taobao.com and provideby Alibaba Group[http://tianchi.aliyun.com/datalab/index.htm].It includes a Clothing category and there are about 406k compatibility relationships covering 61k items.The compatibility relationships in this dataset are labelledmanually by clothes collocation experts. §.§.§ Amazon.This dataset is collected from Amazon.com and provided by <cit.>.Though it includes multiple categories,in order to investigate the performance of our approach on both datasets,we mainly focus on the Clothing category.In this category, there are about 12 million compatibility relationships covering 662k items.Unlike the Taobao dataset, the compatibility relationships in Amazon dataset are not labelled manually. They are the co-purchase data from Amazon's recommendations <cit.>.§.§ SetupOur goal is to differentiate compatibility relationships from non-compatibility ones.We consider all positive relationships (compatibility) and generate random non-relationship distractorsof the equal size. That is to say the ratio between positive and negative samples in the dataset is 50:50.Then we separate the whole dataset into training, validation and testing sets according to the ratios 80:10:10.Although we do not expect overfitting to be a serious issue in our experiment with the large training set, we still carefully tune our model on the validation set to avoid overfitting on testing set. We compare our approach against baselines from two aspects:visual one and non-visual ones. §.§.§ Visual baseline.We take the method in <cit.> as the visual comparisonsince it is also in the end-to-end fashion.In particular, we consider the specific setting configured with GoogLeNetand naive sampling for two considerations. First, in all situations of their experiments,GoogLeNet <cit.> outperforms AlexNet <cit.>.Second, naive sampling means sampling randomly from the dataset,which is consistent with the setup in our experiments.We experiment their method on the Taobao dataset andtake the results on the Amazon dataset directly from <cit.>.§.§.§ Non-visual baselines.We take three methods as the non-visual comparison:1) Naive Bayes on Bag of Words (NBBW). We treat the title sentences from an item pair as the bag-of-words andfeed Naive Bayes classifier with it as the feature vector; 2) Random Forest on Bag of Words (RFBW). Random Forest is capable of modeling extremely complex classification surface.We apply Random Forest classifier on the bag-of-words representation of the title sentences from an item pair; 3) Random Forest on Topic Model (RFTM).For the given item pair {q, c}, we first generate the topic representations x_q,x_c of items q,cby LDA model <cit.> respectively, where the topic number is set as 100.Then we concatenate them into a single feature vector x_q,c and process Random Forest classifier on it.The implementation of Naive Bayes and Random Forest is taken from scikit-learn <cit.>.We turned their parameters to obtaine the best loss on the validation set. There is no preprocess on images and texts. All results reported in the following section is on the testing set.§.§ Results§.§.§ Comparison to baselines.Tabel <ref> shows the corresponding areas under the ROC curves of compatibility prediction on the testing set .The results show clearly that our approach outperforms all other baselines. The visual method collapses on the Taobao dataset becauseunlike Amazon, Taobao is a Consumer to Consumer (C2C) platform andhas few strict requirement about quality of item images uploaded by users.A majority of images are like Figure <ref>,where the information is mixed up and confusing to learning machines.In contrast, the title description is a highly condensed collection of more attributes besides appearanceswith few noises. When using title descriptions,a simple method like Naive Bayes on Bag of Words canachieve an acceptable performance anda more sophisticated method like Random Forest on Bag of Wordscan generate competitive results.However, using topic models on title descriptions is not a good idea since most title descriptions are short texts. One important reason why our approach achieves better performanceis that our approach can recognize specific m-grams and more complicated patterns not captured by bag-of-words models.For instance, the complementary styles of the item titled with“white shirt with blue stripes” and “blue shirt with white stripes”are very different, but they have the same bag-of-words representation. The performance upper bound of our approach on Taobao dataset islimited by the segmentation quality of Chinese. This is one of the reasons that all AUC scores onthe Taobao dataset are lower than that on the Amazon dataset. §.§.§ Tuning sentence model.There are several crucial setups in the sentence model:1) the word embedding dimension d;2) the sentence representation dimension n; 3) whether the word embedding matrix W is initialized or not. We show the first ten epochs of training processes of our model on the Taobao dataset with sentence models under different setups inFigure <ref>, where the standard setup means that we set d=100, n=100 andinitialize the word embedding matrix with an unsupervised neural language model <cit.>.We can see that the setup with larger d or n claims better performance.Furthermore, the performance can get better when we continue to increase the value of d or n.In practice, there is a tradeoff between the performance and resources requirementaccording to specific situations.What's more, initializing the word embedding matrix W with an unsupervised neural language modelis indeed benefit to the convergence rate and the final performance.§.§ Discussion §.§.§ Toward general match.While the previous section mainly focuses on clothes matching,we also train classifiers on the other twenty top-level categories from the Amazon datasetand present the results in Table <ref>.As can be seen, we obtain good accuracy in predicting compatibility relationships ina variety of categories.What's more, we have also tried to train a single model to predict compatibility relationships for all categories.There appears to be no “silver bullet” and the result is dissatisfactory: the AUC score of that single model is only 0.694. The comparison across categories is particularly interesting.Our approach performs relatively poor on the categories “CDs & Vinyl”and “Digital Music” since the content of music is too rich to bedescribed very clearly in a short title description.In contrast, the title description is long enough to describean item from the category `Musical Instruments' clearly and thus our approach performs very well on that.In a word, the better titles can describe the attributes of items in a category,the higher performance can be achieved on that category by our approach. § CONCLUSIONSIn this paper, we present a novel approach to model the humansense of style compatibility between items.The basic assumption of our approach is that most of the important attributes fora product in an online store are included in its title description.We design a Siamese Convolutional Neural Network architecture tomap the title descriptions of an item pair from the original space of symbolic words into some embedded style space.The compatibility probability between items can be then computed in the style space.Our approach takes only words as the input with few preprocessing andrequires no laborious and expensive feature engineering.Moreover, it can be easily extended to big data scenarios with KNN searching techniques.The experiments on two large datasets confirm our assumption andshow the possibility of modeling the human sense of style compatibility.There are several interesting problems to be investigated in our future work:(1) we would like to use more sophisticated sentence modelswithout injuring the simplicity of our approach;(2) we are wondering whether it is possible to use the text and image information simultaneously,e.g. hybrid model or mapping the texts and images of items into the same embedded spacefor mutual retrieval and matching.§ ACKNOWLEDGMENTSWe would like to thank Alibaba Group and Julian McAuley for providing the valuable datasets.This work is supported by Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ13F020001),Zhejiang Provincial Soft Science Project (Grant no. 2015C25053),National Science Foundation of China (Grant nos. 61173185, 61173186). aaai | http://arxiv.org/abs/1708.07938v1 | {
"authors": [
"Kui Zhao",
"Xia Hu",
"Jiajun Bu",
"Can Wang"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170826060953",
"title": "Deep Style Match for Complementary Recommendation"
} |
http://arxiv.org/abs/1708.07640v2 | {
"authors": [
"Jun Goryo",
"Yoshiki Imai",
"W. B. Rui",
"Manfred Sigrist",
"Andreas P. Schnyder"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170825075904",
"title": "Surface magnetism in a chiral d-wave superconductor with hexagonal symmetry"
} |
|
|-1.5mm = ℒtreωł[email protected] of Physics, Sun Yat-Sen University, Guangzhou 510275, P. R. ChinaYat Sen School, Sun Yat-Sen University, Guangzhou 510275, P. R. [email protected] Department of Physics, Babes-Bolyai University, Kogalniceanu Street, Cluj-Napoca 400084, RomaniaDepartment of Mathematics, University College London, Gower Street, London WC1E 6BT, United [email protected] of Physics, Sun Yat-Sen University, Guangzhou 510275, P. R. ChinaState Key Laboratory of Optoelectronic Material and Technology, and Guangdong Province Key Laboratory of Display Material and Technology, Guangzhou, P. R. China We investigate the matter creation processes during the reheating period at the end of inflation in the early Universe, by using the irreversible thermodynamic of open systems. The matter content of the Universe is assumed to consist of the inflationary scalar field, which, through its decay, generates relativistic matter, and pressureless dark matter, respectively. At the early stages of reheating the inflationary scalar field transfers its energy to the newly created matter particles, with the field energy decreasing to near zero. The general equations governing the irreversible matter creation during reheating are obtained by combining the thermodynamics description of the matter creation and the gravitational field equations. A dimensionless form of the general system of the reheating equations is also introduced. The role of the different inflationary scalar field potentials is analyzed by using analytical and numerical methods, and the evolution of the matter and scalar field densities, as well as of the cosmological parameters during reheating,are obtained.Typically, the values of the energy densities of relativistic matter and dark matter reach their maximum when the Universe is reheated up to the reheating temperature, which is determined for each case, as a function of the scalar field decay width, the scalar field particle mass, and of the cosmological parameters. An interesting result is that particle production leads to the acceleration of the Universe during the reheating phase, with the deceleration parameter showing a complex dynamics. Once the energy density of the scalar field becomes negligible with respect to the matter densities,the expansion of the Universe decelerates, and inflation has a graceful exit after reheating.04.20.Cv; 04.50.Kd; 04.60.Bc; 04.60.DsIrreversible thermodynamic description of dark matter and radiation creation during inflationary reheating Shi-Dong Liang December 30, 2023 ==========================================================================================================§ INTRODUCTION The inflationary paradigm is one of the corner stones of present day cosmology. Inflation was first proposed in <cit.> to solve the spatial flatness, the horizon and the monopole problems of the early Universe. The main theoretical ingredients in inflation are a scalar field ϕ with self-interaction potential V(ϕ), and its energy density ρ _ϕ=ϕ̇^2/2+V(ϕ), and pressure p_ϕ=ϕ̇^2/2-V(ϕ), respectively <cit.>. In the initial approach the inflation potential arrives at a local minimum at ϕ=0 through supercooling from a phase transition, and then the Universe expands exponentially.However, in this scenario, called "old inflation", there is no graceful exit to the inflationary stage. To solve this problem the "new inflation" model was proposed in <cit.>. In this model the inflationary scalar field is initially at its local maximum ϕ=0. The potential is required to be very flat near the minimum at ϕ≠ 0, so that the field rolls slowly down, without changing much before the Universe expands exponentially. The "flatness" requirement is not easy to obtain, and another problem of this scenario is that the inflation has to begin very late and last a long time, or it may never happen in some cases <cit.>. By extending the new inflation model, the chaotic inflation theory was proposed <cit.>, which solved the problems above. The model does not require a thermal equilibrium state of the early Universe, and various forms of the scalar field potentials are acceptable, so that a large numberpotential forms can be used in this scenario. Later a subtler theory of hybrid inflation with two scalar fields in the model was proposed <cit.>. The potential takes the form of V(σ,ϕ)=(M^2-λσ^2)/(4λ)+m^2ϕ^2/2+g^2ϕ^2σ^2/2, with M, m, λ, σ constants. One scalar field accounts for the exponential growth of the Universe, and the other one is responsible for the graceful exit of inflation.Presently, very precise observations of the Cosmic Microwave Background (CMB) radiation give us the opportunity to test the fundamental predictions of inflation on primordial fluctuations, such as scale independence and Gaussianity <cit.>. One can calculate cosmological parameters from CMB fluctuations, and use them as constraints of the inflationary models. The slow-roll parameters ϵ and η can be calculated directly from the potentials of the inflationary scalar field, so that these parameters can be observed, and the corresponding inflationary models can be tested <cit.>. The slow-roll parameters and the inflationary parameters like scalar spectral index n_s and tensor-to-scalar ratio r of several forms of the inflationary potentials were investigated in <cit.>. Further investigationsof the parameters n_s and r can be found in <cit.>.The prediction of the statistical isotropy, claiming that inflation removes the classical anisotropy of the Universe has been a central prediction of inflationary scenarios, also strongly supportedby the cosmic no-hair conjecture. However, recently several observations of the large scale structure of the Universe have raised the possibility that the principles of homogeneity and isotropy may not be valid at all scales, and that the presence of an intrinsic large scale anisotropy in the Universe cannot be ruled outa priori<cit.>.Even that generally the recent Planck observations tend to confirm the fundamental principles of the inflationary paradigm, the CMB data seem to show the existence of some tension between the models and the observations. For example, the combined Planck andWMAP polarization data show that the index of the power spectrum isn_s = 0.9603± 0.0073<cit.>, at the pivot scale k_0 = 0.05 Mpc^-1, a result which definitely rules out the exact scale-invariance (n_s = 1), as predicted by some inflationary models <cit.>,at more than 5σ. Moreover, this case can also be ruled out based on some fundamental theoretical considerations,since it is connected with a de Sitter phase, and, more importantly,it does not lead to a consistent inflationary model, mainly due to the graceful exit problem. On the other hand the joint constraints on r (tensor-to-scalar ratio) and n_s can put strong restrictions on the inflationary scenarios. For example, inflationary models with power-law potentials of the formϕ ^4 cannot provide an acceptable number of e-folds (of the order of50-60) in the restricted spacer-n_s at a 2σ level <cit.>. In actuality, the theories of inflation have not yet reached a general consensus, because not only the overall inflationary picture, but also the details of the theoretical models are based on physics beyond the Standard Model of particle physics.During inflation, the exponential growth of the Universe lead to a homogeneous, isotropic and empty universe. Elementary particles are believed to be created in the period of reheating at the end of inflation, defrosting the Universe, and transferring energy from the inflationary scalar field to matter. Reheating was suggested along with the new inflationary scenario <cit.>, and further developed in <cit.>. When the inflationary scalar field reaches its minimum after the expansion of the Universe, it oscillates around the minimum of the potential, and decays into Standard Model particles.The newly created particles interact with each other, and reach a thermal equilibrium state of temperature T. However, quantum field theoretical investigations indicate that the reheating period is characterised by complicated nonequilibrium processes, the main characteristic of which are initial, violent particle production via parametric resonance and inflaton decay (‘preheating’) with a highly nonequilibrium distribution of the produced particles, subsequently relaxing to an equilibrium state. Such explosive particle production is due to parametric amplification of quantum fluctuations for the nonbroken symmetry case (appropriate for chaotic inflation) or spinodal instabilities in the broken symmetry phase <cit.>.The reheating temperature T_reh is thought to be the maximum temperature of the radiation dominated Universe after reheating. However, itwas argued that the reheating temperature is not necessarily the maximum temperature of the Universe after reheating <cit.>. In other words, reheating might be just one of the stages of the matter creation, and other substance with larger freeze-out temperature may be created after inflation through other process <cit.>.In the study of reheating,the phenomenological approach introduced in <cit.> has been widely investigated. This approach is based on the introduction of a suitable loss term in the scalar field equation, which also appears as a source term for the energy density of the newly created matter fluid. It is this source term that, if chosen appropriately, is responsible for the reheating process that follows adiabatic supercooling during the de Sitter phase. In this simple modelthe corresponding dynamics is considered within a two component model. The first component is a scalar field, with energy density and pressure ρ _ϕ and p_ϕ, respectively, and with energy momentum tensor ^(ϕ)T_μ^ν=(ρ _ϕ+p_ϕ)u_μu^ν-p_ϕδ _μ^ν. The second component is represented by normal matter, with energy density and pressure ρ _m and p_m, and with energy-momentum tensor ^(m)T_μ^ν=(ρ _m+p_m)u_μu^ν-p_mδ _μ^ν. In the above equations for simplicity we assume that both cosmological fluid components share the same four velocity u_μ, normalized as u_μu^μ=1. Einstein's field equations imply the relation ∇ _μ(^(ϕ)T_μ^ν+^(m)T_μ^ν)=0, which in the case of a flat geometry reduces to ϕ̇ϕ̈+3Hϕ̇^2+V'(ϕ)ϕ̇+ρ̇_m+3H(ρ _m+p_m)=0,where H=ȧ/a is the Hubble function, and a is the scale factor of the Universe. In order to describe the transition process between the scalar field and the matter component it is convenient to introduce phenomenologicallya ‘friction term’ Γ,describing the decay of the scalar field component, and acting as a source term for the matter fluid. Hence, Eq. (<ref>)is assumed to decompose into two separate equations ϕ̈+3Hϕ̇+V'(ϕ)+Γϕ̇=0, ρ̇_m+3H(ρ _m+p_m)-Γϕ̇^2=0. In the framework of the braneworld models, in which our Universe is a 3-brane embedded in a five-dimensional bulk, the reheating era after the inflationary period was analyzed <cit.>,by assuming the possibility of brane-bulk energy exchange. The inflaton field was assumed to decay into normal matter only, while the dark matter is injected into the brane from the bulk.The observational constraint of an approximately constant ratio of the dark and the baryonic matter requires that the dark matter must be non-relativistic (cold). The model predicts a reheating temperature of the order of 3 × 10^6 GeV, and brane tension of the order of 10^25 GeV^4. The composition of the Universe thus obtained is consistent with the observational data. The problem of perturbative reheating and its effects on the evolution of the curvature perturbations in tachyonic inflationary models was investigated in <cit.>. It was found that reheating does not affect the amplitude of the curvature perturbations, and, after the transition, the relative non-adiabatic pressure perturbation decay extremely rapidly during the early stages of the radiation dominated epoch. These behavior ensure that the amplitude of the curvature perturbations remain unaffected during reheating. It was pointed out in <cit.> that among primordial magnetogenesis models, inflation is a prime candidate to explain the current existence of cosmological magnetic fields. Their energy density decreases as radiation during the decelerating eras of the universe, and in particular during reheating. Avoiding magnetic field backreaction is always complementary to CMB, and can give stronger limits on reheating for all high energy models of inflation. The reheating dynamics after the inflation induced by R^2-corrected f(R) models were considered <cit.>. The inflationary and reheating dynamics was analyzed in the Einstein frame. Observational constraints on the model were also discussed.The processes of particle production from the inflaton, their subsequent thermalization and evolution of inflaton/plasma system by taking dissipation of the inflaton in a hot plasma into account were investigated in <cit.>, and it was shownthat the reheating temperature is significantly affected by these effects.In f(R) modified gravity modelsthe reheating dynamics after the inflation is significantly modified,and affects the shape of the gravitational wave background spectrum <cit.>. In <cit.> it was shown that reheating considerations may provide additional constraints to some classes of inflationarymodels. The current Planck satellite measurements of the Cosmic Microwave Backgroundanisotropies constrain the kinematic properties of the reheating era for most of the inflationary models <cit.>. This result is obtained by deriving the marginalized posterior distributions of the reheating parameter for about 200 models. It turns out that the precision of the current CMB data is such that estimating the observational performance of an inflationary model now requires to incorporate information about its reheating history. It was pointed out in <cit.> that the inflaton coupling to the Standard Model particles is generated radiatively, even if absent at tree level. Hence, the dynamics of the Higgs field during inflation can change dramatically. The inflaton decay and reheating period after the end of inflation in the non-minimal derivative coupling gravity model with chaotic potential was investigated in <cit.>. This model provides an enhanced slow-roll inflation, caused by gravitationally enhanced friction. A scale-invariant model of quadratic gravity with a non-minimally coupled scalar field was studied in <cit.>. At the end of inflation, the Hubble parameter and the scalar field converge to a stable fixed point through damped oscillations that can reheat the Universe in various ways. For reviews on the inflationary reheating see <cit.> and <cit.>, respectively.If the importance of the nonequilibrium and dissipative aspects of particle creation during reheating has been already pointed out a long time ago <cit.>, its irreversible and open character has not received a similar attention. Thermodynamical systems in which matter creation occurs belong to the class of open thermodynamical systems, in which the usual adiabatic conservation laws are modified by explicitly including irreversible matter creation <cit.>. The thermodynamics of open systems has been also applied first to cosmology in <cit.>. The explicit inclusion of the matter creation in the matter – energy stress tensor in the Einstein field equations leads to a three stage cosmological model, beginning with an instability of the vacuum. During the first stage the Universe is driven from an initial fluctuation of the vacuum to a de Sitter space, and particle creation occur. The de Sitter space exists during the decay time of its constituents (second stage), and ends, after a phase transition, in the usual FRW phase. The phenomenological approach of <cit.>, was further discussed and generalised in <cit.> through a covariant formulation, allowing specific entropy variations as usually expected for nonequilibrium processes. The cosmological implications of the irreversible matter creation processes in the framework of the thermodynamics of open or quantum systemshave been intensively investigated in <cit.>.From the analysis of some classes of scalar field potentials, including the ϕ ^1 potential, which provide a good fit to the CMB measurements, in <cit.> it was found that the Planck 2015 68% confidence limit upper bound on the spectral index, n_s implies an upper bound on the reheating temperature of T_reh≤ 6× 10^10 GeV. The observations also exclude instantaneous reheating. Hence the low reheating temperatures allowed by some scalar field models open the possibility that dark matter could have been produced during the reheating period, instead of the radiation dominated era in the history of the Universe. Such a possibility could lead to a major change in our understanding of the early evolution stages of the Universe, leading to different predictions for the relic density and momentum distribution of WIMPs, sterile neutrinos, and axions.It is the purpose of the present paper to apply the thermodynamics of open systems, as introduced in <cit.>and <cit.>, to a cosmological fluid mixture, in which particle decay and production occur, consisting of two basic components: scalar field, and matter, respectively. From a cosmological point of view this physical situation is specific to the preheating/reheating period of the inflationary cosmological models. The thermodynamics of irreversible processes as applied to inflationary cosmological models leads to a self-consistent description of the matter creation processes, which determines the whole dynamics and future evolution of the Universe. By combining the basic principles of the thermodynamics of open systems with the cosmological Einstein equations for a flat, homogeneous and isotropic Universe we obtain a set of first order ordinary differential equations describing matter creation due to the decay of the scalar field. As for the matter components we specifically consider a model in which ordinary matter is a mixture of two components, a relativistic one (radiation), and a pressureless fluid, assumed to correspond to dark matter. In order to simplify the analysis of the reheating equations we reformulate them in a dimensionless form, by introducing a set of appropriate dimensionless variables. As a cosmological application of the general formalism we investigate matter creation from a scalar field by adopting different mathematical forms of the self-interaction potential V of the field. More exactly, we will consider power law, exponential and Higgs type potentials. For each case the evolution of the post-inflationary Universe with matter creation is investigated in detail by solving numerically the set of the cosmological reheating evolution equations.The results display the process of inflationary scalar field decaying to matter and dark matter, and the effects of the expansion of the Universe on the decay. We concentrate on the evolution of the dark matter and radiation components, which increase from zero to a maximum value, as well as on the scalar field decay. A simple model, which can be fullysolved analytically, is also presented. As a general result we find that the scalar field potential also plays an important role in the reheating process, and in the decay of the scalar field. Some of the cosmological parameters of each model are also presented, and analyzed in detail.The models are constrained by the observational parameters called the inflationary parameters, including the scalar spectral index n_s, the tensor to scalar ratio r, the number of e-folds N, and the reheating temperature T_reh. These parameters can be calculateddirectly from the potentials used in the models, and they can be used together with the observational data to constrain the free parameters in the potentials.The study of the reheating models, and the development if new formalisms and constraints to the theories are undoubtedly an essential part of the completion to the theory. Analysis of the current experimental data, including the calculation of the inflationary parameters, is able to test the reheating models.The present paper is organized as follows. The basic results in the thermodynamics of open systems are briefly reviewed in Section <ref>. The full set of equations describing matter creation due to the decay of a scalar field are obtained in Section <ref>, where the observational constraints on the inflationary and reheating models are also presented. An exact solution of the system of the reheating equations, corresponding to simple form of the scalar field (coherent wave) is obtained in Section <ref>. Several reheating models, corresponding to different choices of the scalar field self-interaction potential are considered in Section <ref>, by numerically solving the set of reheating evolution equations. The time dynamics of the cosmological scalar field, and of the matter components are obtained, and analyzed in detail. We conclude and discuss our results in Section <ref>.§ IRREVERSIBLE THERMODYNAMICS OF MATTER CREATION In this Section, we briefly review the thermodynamics of matter creation in open systems, which we will use for the study of the post-inflationary reheating processes.In our presentation we start from the fundamental laws of thermodynamics, we proceed to the general covariant formulation of the theory, and then we apply our results to the particular case of homogeneous and isotropic cosmological models. §.§ The First Law of Thermodynamics The main novel element that appears in the thermodynamics of open systems with energy/matter exchange is the creation pressure, related to the variation of the particle number. The expression of the creation pressure can be derived from the fundamental laws of thermodynamics <cit.>. We consider a thermodynamic system consisting of 𝒩 particles,inside a volume 𝒱. For a closed system, 𝒩 is a constant, and the first law of thermodynamics - the energy conservation equation - isdℰ= d𝒬-p d𝒱,where ℰ is the internal energy, 𝒬 is the heat received by the system, and p is the thermodynamic pressure. We can also write the energy conservation asd( ρ/n) = dq-p d( 1/n),where the energy density ρ=ℰ/𝒱, the particle number density n=𝒩/𝒱, and dq= d𝒬/𝒩, respectively. Adiabatic transformations ( d𝒬=0) are described byd(ρ𝒱)+p d𝒱=0. For an open system, the particle number can vary in time, so that 𝒩=𝒩(t). By including the thermodynamic parameters of the newly created particles, the energy conservation equation is modified to <cit.> d(ρ𝒱)= d𝒬-p d𝒱+h/n d(n𝒱),where h=ρ+p is the enthalpy per unit volume. Adiabatic transformations are then expressed asd(ρ𝒱)+p d𝒱-h/n d(n𝒱)=0.The matter creation process is a source of the internal energy of the system. From Eq. (<ref>) we obtain the relationρ̇=(ρ+p)ṅ/n,describing the variation of the matter density due to the creation processes. On the other hand the above equation can also describe physical systems in the absence of matter creation, like, for example, the photon gas,in which, for a particular temperature, the particle number N varies with the volume in a fixed manner, adjusting itself to have a constant density of photons. Hence, for a photon gas the particle number N is a variable, and not a constant quantity, like in an ordinary gas.The physical mechanism by which thermodynamic equilibrium can be established consists in the absorbtion and emission of photons by matter. For a radiation like equation of state, with p=ρ /3, from Eq. (<ref>) we obtain ρ =ρ _0(n/n_0)^4/3, with ρ _0 and n_0 constants,giving the well-known energy density - particle number relation for the ultra-relativistic photon gas.Alternatively, we can reformulate the law of the energy conservation by introducing the matter creation pressure, so that the energy conservation equation for open systems takes a form similar to Eq. (<ref>),with the pressure written as p̃=p+p_c,d(ρ𝒱)=-(p+p_c) d𝒱,where the creation pressure associated describing the creation of the matter isp_c=-ρ+p/n d(n𝒱)/ d𝒱.§.§ The Second Law of ThermodynamicsIn order to formulate the second law of thermodynamics for open systems, we define first the entropy flow d_e𝒮, and the entropy creation d_i𝒮≥ 0, so that the total entropy variation of the system is written as <cit.> d𝒮= d_e𝒮+ d_i𝒮≥ 0.For a closed and adiabatic system, d_i𝒮≡ 0. For an open system with matter creation, in the following we assume thatmatter is created in a thermal equilibrium state d_e𝒮=0. Therefore the entropy increases only due to the particle creation processes.The total differential of the entropy isT d(s𝒱) =d(ρ𝒱)-h/n d(n𝒱)+T d(s𝒱)=d(ρ𝒱)-(h-Ts) d𝒱 = d(ρ𝒱)-μ d(n𝒱),where the entropy density s=𝒮/𝒱≥ 0, and the chemical potential μ=(h-𝒯s)/n≥ 0. Hence the entropy production for an open system isT d_i𝒮=T d𝒮=( h/n)d(n𝒱)-μ d(n𝒱) =T( s/n)d(n𝒱)≥ 0 The matter creation processes can also be formulated in the framework of general relativity. In order to apply the thermodynamics of open systems to cosmology, in the following we consider the formulation of the thermodynamics of open systemsin a general relativistic covariant form <cit.>. Then, by starting from this formulation, weparticularize it to the case of a homogeneous and isotropic Universe, by also deriving the entropy of the system <cit.>. §.§ General relativistic description of the thermodynamic of open systems In the following we denote the energy-momentum tensor of the system as 𝒯^μν, and we introduce the entropy flux vector s^μ, the particle flux vector 𝒩^μ, and the four-velocity u^μ of the relativistic fluid. These variables are defined as𝒯^μν=(ρ+p+p_c)u^μu^ν-(p+p_c)g^μν,s^μ=nσ u^μ, 𝒩^μ=nu^μ,where p_c is the matter creation pressure, σ is the specific entropy per particle, and n is the particle number density.The energy conservation law requires∇_ν𝒯^μν=0,while the second law of thermodynamics imposes the constraint∇_μs^μ≥ 0,on the entropy flux four-vector. The balance equation of the particle flux vector is given by∇_μ𝒩^μ=Ψ, where Ψ>0 represents a particle source, or a particle sink if Ψ<0. For an open system, the Gibbs equation is <cit.> nT dσ= dρ-ρ+p/n dn.In order to derive the energy balance equationwe multiply both sides of Eq. (<ref>) by u^μ, thus obtainingu_μ∇_ν𝒯^μν = u_μ∇_ν(ρ+p+p_c)u^μu^ν +u_μ(ρ+p+p_c)× ∇_ν(u^μu^ν) -u_μ∇^μ(p+p_c)= u^ν∇_ν(ρ+p+p_c) +(ρ+p+p_c)× (u_μu^ν∇_νu^μ+u_μu^μ∇_νu^ν) -(ṗ+ṗ_̇ċ)= ρ̇+(ρ+p+p_c)∇_νu^ν=0,where we have used the relations ρ̇=u^μ∇_μρ= dρ/ ds, u_μu^μ=1, and u_μu^ν∇_νu^μ=0, respectively. Hence we have obtained the energy conservation equation for open systems asρ̇+(ρ+p+p_c)∇_μu^μ=0. In order to obtain the entropy variation we substitute the relation (<ref>) into the Gibbs equation (<ref>), and we use Eqs. (<ref>) and (<ref>) to obtain firstnTu^μ∇_μσ = ρ̇-ρ+p/nu^μ∇_μn T∇_μs^μ-Tσ∇_μ𝒩^μ= -(ρ+p+p_c)∇_μu^μ-ρ+p/n× (∇_μ𝒩^μ-n∇_μu^μ) T∇_μs^μ= -p_c∇_μu^μ-( ρ+p/n-Tσ) ∇_μ𝒩^μ.Thus we obtain the entropy balance equation as∇_μs^μ=-p_cΘ/T-μΨ/T,where μ=(ρ+p)/n-Tσ is the chemical potential, and Θ=∇_μu^μ is the expansion of the fluid. From Eq. (<ref>) we obtain∇_μs^μ=σ u^μ∇_μn+nu^μ∇_μσ =σΨ+nσ̇,where we have constrained the specific entropy per particle to be a constant, σ̇=0. From Eqs. (<ref>) and (<ref>) it follows that-p_cΘ/T-μΨ/T=σΨ,and thus we obtain matter creation pressure asp_c=-ρ+p/nΨ/Θ.One can see from the above expression that if there is no matter creation, Ψ=0, and then the matter creation pressure p_c vanishes, and the entropy becomes a constant. If matter is created with Ψ>0 and Θ>0, then the matter creation pressure will be negative, p_c<0. §.§ Matter creation in homogeneous and isotropic Universes In the case of a homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) geometry, the line element is given byds^2= dt^2-a^2(t)( dx^2+ dy^2+ dz^2),where a(t) is the scale factor. In the comoving frame, the four-velocity is given by u^μ=(1,0,0,0). For any thermodynamic functionℱ its derivative is given by ℱ̇=u^μ∇_μℱ= dℱ/ dt. For thecovariant derivative of the four-velocity,by taking into account that 𝒱=a^3, we find ∇_μu^μ=1/√(-g)∂(√(-g)u^μ)/∂ x^μ=1/a^3∂(a^3u^μ)/∂ x^μ=𝒱̇/𝒱=3H,where H, theexpansion of the fluid (the Hubble function) is given by ∇_μu^μ=𝒱̇/𝒱=ȧ/a. The components of the energy-momentum tensor (<ref>) in the presence of particle creation are𝒯^0_0=ρ , 𝒯^1_1=𝒯^2_2=𝒯^3_3=-(p+p_c). From Eq. (<ref>) we obtain the particle number balance equation asΨ=u^μ∇_μn+n∇_μu^μ=ṅ+n𝒱̇/𝒱.Hence, by substitutingthe above equation into Eq. (<ref>), we obtain the variation of the entropy of the open system due to particle creation processes as𝒮=𝒮_0exp( ∫_t_0^t Ψ(t)/n dt) ,where 𝒮_0=𝒮(t_0) is an arbitrary integration constant. § REHEATING DYNAMICS: IRREVERSIBLE OPEN SYSTEMS THERMODYNAMICS DESCRIPTION Reheating is the period at the end of inflation when matter particles were created through the decay of the inflationary scalar field. The particle creation is essentially determinedbythe energy density ρ _ϕ and pressure p_ϕ of the inflationary scalar field ϕ. For all inflationary models, inflation begins at the energy scale of order 10^14 GeV, or, equivalently,at an age of the Universe of the order of 10^-34 s <cit.>. Hence reheating starts at the end of inflation, at a time of the order of 10^-32 s, and it should end before the hot big bang at 10^-18 s, which means that the energy scale of reheating is from around 10^-7 GeV up to 10^7 GeV. The reheating temperature is an important parameter of the model, which we will discuss in detail.In this Section we derive first the general equations describing the reheating process in the framework of the thermodynamics of open systems. We also review the main observational constraints and parameters of the inflationary cosmological models, prior and during the reheating. §.§ Irreversible thermodynamical description of reheating We consider the Universe in the latest stages of inflation as an open thermodynamic system with matter creation. The three constituents of the Universe are the inflationary scalar field, the ordinary matter, with energy density ρ_m and pressure p_m, and the dark matter, with energy density ρ_DM and pressure p_DM, respectively. Both the ordinary matter and the dark matter are assumed to be perfect fluids. The total energy density ρ of the system isρ=ρ_ϕ+ρ_DM+ρ_m.Similarly, the total pressure p and the total particle number density n of the Universe at the end of inflationare given byp=p_ϕ+p_DM+p_m, n=n_ϕ+n_DM+n_m. In the presence of particle creation processes, the stress-energy tensor 𝒯^μν of the inflationary Universe in obtained as𝒯^μν=(ρ +p+p_c^( total))u^μu^ν-(p+p_c^( total))g^μν,where p_c^( total) is the total creation pressure describing matter creation, and we have assumed that all three fluid components are comoving, thus having the same four velocity u^μ. In the following we will restrict our analysis to the case of the flatFriedmann-Robertson-Walker Universe, with line element given by Eq. (<ref>). The basic equations describing the reheating process in the framework of the thermodynamic of open systems can be formulated as follows. The particle number balance equationIn the FRW geometry the balance equation of the particle flux vector, Eq. (<ref>), can be written for each species of particle with particle numbers n_ϕ, n_ m, n_ DM asΨ _a=ṅ_a+n_a∇_μu^μ=ṅ_a+3Hn_a, a=ϕ, m,DM. We assume thatthe particle source function Ψis proportional to the energy density of the inflationary scalar field ρ_ϕ,with the proportionality factor Γ representing a decay (creation) width. Hence the balance equations of the particle number densities are given byṅ_ϕ+3Hn_ϕ=-Γ_ϕ/m_ϕρ_ϕ,ṅ_DM+3Hn_DM=Γ_DM/m_ DMρ_ϕ,ṅ_m+3Hn_m=Γ_m/m_ mρ_ϕ,where Γ_ϕ, Γ_DM and Γ_m are generally different, nonzero functions, and m_ϕ, m_ DM and m_ m denotes the masses of the scalar field, dark matter, and ordinary matter particles, respectively. For the whole system, the particle balance equation isṅ+3Hn=(Γ_m+Γ_DM-Γ_ϕ)ρ_ϕ.The energy balance equationsFrom Eq. (<ref>) we obtain the equations of the energy density balance of the cosmological fluids asρ̇_ϕ+3H(ρ_ϕ+p_ϕ)=-Γ_ϕ(ρ_ϕ+p_ϕ)ρ_ϕ/m_ϕn_ϕ,ρ̇_DM+3H(ρ_DM+p_DM)=Γ_DM(ρ_DM+p_DM)ρ_ϕ/m_ DMn_DM,ρ̇_m+3H(ρ_m+p_m)=Γ_m(ρ_m+p_m)ρ_ϕ/m_ mn_m,and ρ̇+3Hρ = ρ_ϕ[ Γ_m(ρ_m+p_m)/m_ mn_m+Γ_DM(ρ_DM+p_DM)/m_ DMn_DM- Γ_ϕ(ρ_ϕ+p_ϕ)/m_ϕn_ϕ],respectively. The matter creation pressures From the expression of the creation pressure associated to the newly created particles Eq. (<ref>) we findp_c^(a)=-ρ^(a)+p^(a)/n^(a)Ψ^(a)/Θ^(a)=-(ρ^(a)+p^(a))Ψ^(a)/3Hn^(a),where subscript (a) denotes the different cosmological fluids. During the reheating phase, for each component, the decay and creation pressures arep_c^(ϕ)=Γ_ϕ(ρ_ϕ+p_ϕ)ρ_ϕ/3Hm_ϕn_ϕ,p_c^(DM)=-Γ_DM(ρ_DM+p_DM)ρ_ϕ/3Hm_ DMn_DM,p_c^(m)=-Γ_m(ρ_m+p_m)ρ_ϕ/3Hm_ mn_m.The total creation pressure isp_c^( total) = ρ_ϕ/3H[ Γ_ϕ/m_ϕn_ϕ(ρ_ϕ+p_ϕ)-Γ_DM/m_ DMn_DM× (ρ_DM+p_DM)- Γ_m/m_ mn_m(ρ_m+p_m) ].The gravitational field equations The Einstein gravitational field equations in a flat FRW universe with inflationary scalar field, ordinary matter and dark matter fluids in the presence of matter creation can be written as follows3H^2=1/m_Pl^2(ρ_ϕ+ρ_DM+ρ_m),2Ḣ+3H^2=-1/m_Pl^2(p_ϕ+p_DM+p_m+p_c^( total)).In the present paper we use natural units with c =ħ=k_B=1, in which 8π G = 1/m_Pl^2 = 1.687 × 10^-43 MeV^-2, where m_Pl is the ”reduced” Planck mass.The entropy generation The entropy generated by the newly created matter is𝒮_m(t)/𝒮_m(t_0)=exp( ∫^t_t_0Γ_mρ_ϕ/m_ mn_mdt) .§.§ The models of reheating Presently, the accepted paradigm for reheating is based on particle production and parametric amplification, as extensively discussedin <cit.> and <cit.>. According to this model,the main process of particle production is parametric amplification in combination with a slowing expansion. This mechanism entails a description in terms of Mathieu equations, either for the scalar or fermion fields. These equations feature unstable bands with the concomitant parametric instabilities. This mechanism cannot be described in terms of a simple Γ functions, and different bands feature different Floquet exponents (in Minkowski or near Minkowski space time), and correspond to different wave-vector bands.In this picture, matter is created in a highly non-thermal state (determined primarily by the position and widths of the unstable parametric bands), and thermalizes (at least in principle, although never quite proven directly) via scattering. This last step requires a clear knowledge of relaxation times, which entail understanding the microscopic processes and cross sections.In the following we will consider models of reheating involving energy transfer between the inflationary scalar field, and the newly created ordinary matter, and dark matter, respectively. We assume that particles are created through an irreversible thermodynamic process. In this Subsection, we present explicitlythe general equations governing the matter and dark matter creation during inflationary reheating. In our reheating models,the energy density and pressure for the inflationary scalar field have the formρ_ϕ=1/2ϕ̇^2+V(ϕ), p_ϕ=1/2ϕ̇^2-V(ϕ),where V(ϕ) is the self-interaction potential of the scalar field. Substituting Eqs. (<ref>) into Eq. (<ref>), we obtain the generalized Klein-Gordon equation for the scalar field in the presence of matter creation. The equation of state of the dark matter is defined as w_DM=p_DM/ρ_DM≡ 0, since we assume that the pressure of the newly created dark matter particles is p_DM=0. The energy density of dark matter is proportional to its particle number density, ρ_DM∼ n_DM, so that ρ_DM=m_DMn_DM, where m_DM is the mass of the dark matter particles. For pressureless matter from Eq. (<ref>) we obtain the general relationnρ̇=ρṅ. The newly created matter is considered to be in form of radiation (an ultra-relativistic fluid), with the equation of state w_m=p_m/ρ_m=1/3. The relation between the energy density and the particle number density of radiation is ρ_m∼ n_m^4/3.In summary, the set of equations governing the matter creation process during reheating is written below as3H^2= 1/m_Pl^2(ρ_ϕ+ρ_DM+ρ_m),2Ḣ+3H^2=-1/m_Pl^2(p_ϕ+p_m+p_c^(total)), ϕ̈+V'(ϕ)+3Hϕ̇+Γ_ϕρ_ϕ/m_ϕn_ϕϕ̇=0, ρ̇_DM+3Hρ_DM=Γ_DMρ_ϕ,ρ̇_m+4Hρ_m=Γ_mρ_ϕ, ṅ_ϕ+3Hn_ϕ+Γ_ϕ/m_ϕρ_ϕ=0,where Γ_ϕ, Γ_DM, Γ_mare considered as constants.Eq. (<ref>) is the generalized Klein-Gordon equation, describing the evolution of the scalar field. The last term on the left hand side of this equation represents an effective friction term,describing the effects of irreversible matter creation (decay) on the scalar field dynamics. If ρ _ϕ/n_ϕ= constant, we recover Eq. (<ref>), introduced on a purely phenomenological basis in <cit.>.§.§.§ Dimensionless form of the reheating equations In order to simplify the mathematical formalism we introduce a set of dimensionless variables τ, h, r_ϕ, r_DM, r_m, N_ϕ, N_m, γ _DM, γ _m, and v(ϕ), respectively, defined as t=1/M_ϕτ,H=M_ϕ h,ρ_a=M_ϕ ^2m_Pl^2r_a,p_a=M_ϕ ^2m_Pl^2P_a,a=ϕ,DM, m,c, v(ϕ)=1/M_ϕ ^2V(ϕ), n_a=M_ϕ ^2N_a,a=ϕ,DM, m, Γ _b/m_b=M_ϕγ _b, b=ϕ, DM, m,where M_ϕ is a constant with the physical dimension of a mass.In the new variables the equations (<ref>)-(<ref>) take the dimensionless form3h^2=r_ϕ+r_DM+r_m,2dh/dτ+3h^2=-P_ϕ-P_DM-P_c^(total), d^2ϕ/dτ ^2+v'(ϕ)+3hdϕ/dτ+γ _ϕr_ϕ/N_ϕdϕ/dτ=0, dr_DM/dτ+3hr_DM=γ _DMr_ϕ,dr_m/dτ+4hr_m=γ _mr_ϕ, dN_ϕ/dτ+3hN_ϕ+γ _ϕr_ϕ=0. Eqs. (<ref>) and (<ref>) can be solved exactly, and we obtain the following exact representation of the scalar field particle number and matter densities,N_ϕ(τ )=N_ϕ(0)a(0)/a^3(τ )-∫γ _ϕρ _ϕa^3(τ)dτ/a^3(τ ),r_i=∫γ _aρ _ϕa^3(τ)dτ/a^n(τ ),{i= DM, n=3}, {i= m, n=4,}.When ρ _ϕ→ 0, the particles creation processes end, and the Universe enters in the standard expansionary phase, with its chemical composition consisting of two matter components, as well as a possible background of scalar field particles. After reheating, the energy density of the dark matter and radiation varies asr_ DM=r_ DM0/a^3, r_ m=r_ m0/a^4,where r_ DM0 and r_ m0 are the dark matter and radiation energy densities at the end of reheating. The evolution of the scale factor is described by the equationh(a)=1/ada/dτ=1/√(3)√(r_ DM0/a^3+r_ m0/a^4),giving τ (a)-τ_0=2 (a r_DM0-2 r_m0) √( r_DM0a+r_m0)/√(3)r_DM0^2.The deceleration parameter for the post-reheating phase is given byq(a)=3r_DM0a+4 r_m0/2(a r_DM0a+r_m0)-1.In the limit r_DM0a>>r_m0, the deceleration parameter tends to the value q=1/2, indicating a decelerating expansion of the Universe. §.§ Observational tests of inflationary models In order to facilitate the comparison of the theoretical models of the reheating based on the thermodynamics of open systemswith the observational results, we also present and calculate various cosmological parameters, either directly from the inflationary models, or from the potentials of the inflationary scalar field. Slow-roll parametersThe slow-roll parameters were introduced in 1992, and are based on the slow-roll approximation <cit.>. Their calculations are presented in <cit.>. During the period before reheating, the inflationary scalar field has not yet decayed, and the field equation and the equation of the scalar field can be written as3H^2=1/m_Pl^2ρ_ϕ=1/m_Pl^2(1/2ϕ̇^2+V(ϕ)), ϕ̈+V'(ϕ)+3Hϕ̇=0.The slow-roll approximation requires that the terms ϕ̇^2/2 and ϕ̈ are sufficiently small, so that the above equations can be simplified to3H^2≃1/m_Pl^2V(ϕ), 3Hϕ̇≃ -1/m_Pl^2V'(ϕ).The slow-roll parameters are defined as ϵ (ϕ)=(1/2)(V'/V)^2, and η(ϕ)=V”/V- (1/2)(V'/V)^2, respectively <cit.>. The slow-roll approximation only holds when the following conditions are satisfied, ϵ(ϕ)≪ 1 and η(ϕ)|≪ 1, respectively. Inflationary ParametersDuring inflation, the Universe expands exponentially. The scale of inflation is represented by the scale factor a. The ratio of the scale factor at the end of inflation and at its beginning is always large, so that the logarithm of this quantity is of physical interest. The number of e-folds is defined as N(t)=ln[a(t_end)/a(t)], where t_end is the time when inflation ends. The number of e-folds can also be obtained from the comoving Hubble length <cit.>. Under the slow-roll approximation, it can be calculated asN=∫^t_end_tH dt≃∫^ϕ_CMB_ϕ_endV/V' dϕ=∫^ϕ_CMB_ϕ_end1/√(ϵ(ϕ')) dϕ',where ϕ_end is defined as ϵ(ϕ_end)=1, when the slow-roll approximation is violated, and inflation ends. Typically N≃ 50. Sometimes the integral is multiplied by a factor of 1/√(2).The scalar spectral index n_s is defined by the slow-roll parameters n_s=1-4ϵ(ϕ_CMB)+2η(ϕ_CMB). The index n_s is usually a function of the parameters of the model. The value of n_s can be obtained from the observations <cit.>, and is used to constrain the inflationary models. The tensor-to-scalar ratio r is similarly defined as r=13.7×ϵ(ϕ_CMB). One can see from their definition that the two parameters n_s and r are not independent. Cosmological parametersThe deceleration parameter is defined asq= d/ dt( 1/H) -1=-aä/ȧ^2.It indicates thenature of the acceleration of the expansion of the Universe. q<0 means that the expansion of the Universe is accelerating, while a positive q indicates deceleration.The density parameters of the inflationary scalar field, matter and dark matter are defined as Ω_ϕ=ρ_ϕ/ρ, Ω_DM=ρ_DM/ρ, Ω_m=ρ_m/ρ. The evolution of density parameters of each matter component in the early Universe gives us a hint on the matter creation process during inflationary reheating.§.§.§ Model parameters In studying the reheating models, we consider different scalar field potentials that influence the dynamics of the early Universe. During inflation the energy density of the scalar field is potential energydominated. Scalar field potentials We picked six potentials of cosmological interest we are going to investigate, and presented themin Table <ref>.Some of these potentials are simple, and satisfy the slow-roll conditions, like the large-field, small-field, and linear potentials. However, since the slow-roll condition is a sufficientbut not necessary condition for inflation <cit.>, some other forms of the potentials are also considered. We have also selected the well-studied exponential potential, which is easy to analyze and constrain observationally. Finally, as the only scalar field in the Standard Model of particle physics, the Higgs type potentials are also worth to investigate. The observational constraints of the six potentials are not exactly the same. Each potential has its own slow-roll and inflationary parameters, while they share similar constraints from the cosmological parameters and the reheating temperature. The values of the constants are discussed in the next paragraphs. Slow-roll parametersThe slow-roll parameters are obtained for each potential. Their values are shown in Table <ref>. Inflationary Parameters Similarly, the inflationary parameters, scalar spectral index n_s and tensor-to-scalar ratio r are presented in Table <ref>. Mass scale M_ϕIn order to compare the different potentials we assume that in all of them the mass scale M_ϕ is the same. We constrainM_ϕ in the potentials by estimating the values of V at the beginning and ending time of reheating. Reheating begins at the end of inflation at around 10^-32 s and ends before 10^-16 s <cit.>. Thus the value of the mass scale M_ϕ is 10^18 s^-1≤ M_ϕ≤ 10^32 s^-1, 6.58× 10^-7 GeV≤ M_ϕ≤ 6.58× 10^7 GeV.In the set of reheating equations, M_ϕ is defined asM_ϕ=τ/t=τ/t(s)× 6.58× 10^-25 GeV=0.658τ keV.When the dimensionless time variable is in the range 0.3≤τ≤ 3, the value of M_ϕ varies as0.1974 keV≤ M_ϕ≤ 1.974 keV.In our irreversible thermodynamic study of reheating models we adopt the value M_ϕ≈ 1 keV. Reheating TemperatureThe reheating temperature is usually referred to as the maximum temperature during reheating period. However, it was argued that the reheating temperature is not necessarily the maximum temperature of the Universe <cit.>. In the radiation dominated Universe the relation between energy density and the temperature of the matter (radiation) is ρ_m∼ T^4. With ρ_m=M_ϕ ^2m_Pl^2r_m we have T∼ M_ϕ^1/2m_Pl^1/2√(r_m). When the dimensionless variable is approximately 0.1≤ r_m≤ 0.5, we obtainT∼ 0.5 m_Pl^1/2M_ϕ^1/2.Hence, for the maximum temperature of the reheating period we findT_reh∼ 3.29× 10^7 GeV.Decay widthΓ_ϕ of the inflationary scalar fieldThe relation between decay width of inflationary scalar field and M_ϕ is given by <cit.> Γ_ϕ=α _ϕM_ϕ√(1-( T/M_ϕ) ^2)≃α _ϕM_ϕ=α _ϕ× 1 keV,where T is the temperature of the decay product, and α _ϕ is a constant. As we have previously seen, the maximum temperature during reheating is of the order of 10^7 GeV. The mass scale of the inflationary scalar field is of the order of the Planck Mass, which leads to T≪ M_ϕ, so the decay width of inflationary scalar field is similarly proportional toM_ϕ. Hence for the decay width in the reheating model we adopt the value Γ_ϕ≈α _ϕ× 1 keV.Mass of the dark matter particle The mass of dark matter particles is investigated model independently in <cit.>. Two kinds of dark matter particles, and their decoupling temperature from the dark matter galaxies, were obtained. One is possibly the sterile neutrino, gravitino, the light neutralino or majoron, with mass about 1∼ 2 keV and decoupling temperature above 100 GeV. The other one are the Weakly Interacting Massive particles (WIMP), with mass ∼ 100 GeV and decoupling temperature ∼ 5 GeV. From the analysis of reheating temperature above, we adopt the value of the mass of the dark matter particles as m_DM∼ 1 keV. Other parametersFor the reduce Planck mass we adopt the value m_Pl=2.435× 10^18 GeV. If not otherwise specified, the numerical value of the spectral index is taken as n_s=0.968. In order to numerically solve the system of the reheating equations we use the initial conditions a(0)=0.33, r_DM(0)=r_m(0)=n_m(0)=0, N_ϕ (0)=N_(ϕ)0, ϕ (0)=ϕ _0, and .(dϕ /dt)|_τ =0=u_0, respectively. § COHERENT SCALAR FIELD MODEL As a first example of the analysis of the reheating process in the framework of the thermodynamics of open systems we assume that the inflationary scalar field can be regarded as a homogeneous oscillating coherent wave, with the energy density of the wave ϕ(t) given by <cit.> ρ_ϕ=m_ϕn_ϕ,where m_ϕ is the mass of the scalar field particle (inflaton). Then from relations (<ref>) and (<ref>) we obtainρ_ϕ=ϕ̇^2, p_ϕ=0,givingV(ϕ)=1/2ϕ̇^2. In the following we assume that the energy density of the newly created matter and dark matter particles are small enough, and they do not affect the evolution of the energy density of the inflationary scalar field. That is, we assume the approximationsρ_DM,ρ_m≪ρ_ϕ, n_DM,n_m≪ n_ϕ, p_DM,p_m≪ p_ϕ. Then Eqs. (<ref>)-(<ref>) take the simplified form2Ḣ+3H^2=Γ_ϕm_ϕH, ṅ_DM+3Hn_DM=3Γ_DMH^2,ṅ_m+3Hn_m=3Γ_mH^2.Eqs. (<ref>)-(<ref>) can be transformed into dimensionless form by introducing the dimensionless variables τ, h, N_DM and N_m, defined as t=2/Γ_ϕm_ϕτ, H=Γ_ϕm_ϕ/3h, n_DM=2Γ_ϕm_ϕΓ_DM/3N_DM, n_m=2Γ_ϕm_ϕΓ_m/3N_m.Hence we obtain the dimensionless equationsdh/ dτ+h^2+h=0,dN_DM/ dτ+2hN_DM=h^2, dN_m/ dτ+2hN_m=h^2.The solution for h is obtained ash(τ)=1/e^τ-1,while the time variations of the particle numbers areN_a=( e^τ_0-1/e^τ-1)^2 N_a0 e^2(τ-τ_0)+e^2(τ-τ_0)-1/2(e^τ-1)^2, a= DM,m,where for dark matter N_a0=N_DM0=N_DM(τ_0), and N_a0=N_m0=N_m(τ_0) for the ordinary matter.The scale factor in this model is given by,a(τ)=a_0 ( e^τ-1/e^τ)^2/3,where a_0 is an integration constant, while the deceleration parameter q is found to beq=3/2e^τ-1.Since q>0 for all τ≥ 0, it follows that the expansion rate of the universe continues to decrease. Inflation ends at the beginning of the reheating period.Lets' consider now the beginning of the inflationary reheating, when τ is small. Then the solutions (<ref>)-(<ref>) can beapproximated ash≃1/τ, a≃ a_0τ^2/3, ρ_ϕ≃1/τ^2,N_a≃τ-τ_0+N_a0τ_0^2/τ^2, a= DM,m.When τ reaches the value τ_max=2τ_0-2N_0τ_0^2, the particle number reaches its maximum value,n_(max)a=4Γ^2_a/12Γ _at_0-9n_a0t_0^2,a= DM,m,where n_a0=2Γ_ϕm_ϕΓ_a/3. For dark matter, Γ _a=Γ_DM, and for matter Γ _a=Γ_m.The entropy production of matter can also be calculated <cit.>, and for small times it shows a linear increase, so that𝒮_m(t)/S_m(t_0)=τ-τ_0/N_0τ_0^2+1. § EFFECT OF THE SCALAR FIELD POTENTIAL ON THE REHEATING DYNAMICS In the present Section we will investigate the reheating and generation of the matter content of the Universe, as described by the thermodynamics of open systems. In order to perform this study we will investigate the role played by the different scalar field potentials on the reheating dynamics. §.§ Large-field polynomial and linear potentials We will begin our analysis by assuming that the scalar field potential is a large-field polynomial potential, of the formV(ϕ)=M_ϕ^2( ϕ/μ) ^p,where p≥ 1, M_ϕ is a parameter of the dimension of mass, while μ is a dimensionless constant. This kind of large-field polynomial potential is used in chaotic inflation models. A special case of the large-field polynomial potentials are the linear potentials, corresponding to p=1. These types of potentials are usually suggested by particle physics models, with dynamical symmetry breaking or nonperturbative effects <cit.>.The inverse power law potentialsappear in a globally supersymmetric SU(N_c) gauge theory with N_c colors and the condensation of N_f flavors. Positive power-law potentials can drive an inflationary expansion, with the scalar field becoming progressively less important as the cosmological evolution proceeds.Hence the use of power law scalar field potentials can convincingly justify the neglect of the scalar field terms during the post-inflationary and late times cosmological evolution in the Friedmann equations.The dimensionless energy density andpressure of the inflationary scalar field take then the formr_ϕ=1/2(dϕ/dτ)^2+( ϕ/μ) ^p, P_ϕ=1/2(dϕ/dτ)^2-( ϕ/μ) ^p. The dimensionless equations Eqs. (<ref>)-(<ref>) describing the evolution of the Universe during the reheating phase can then be written asdϕ/dτ=u, du/dτ = -p/μ( ϕ/μ) ^p-1- √(3)√(u^2/2+( ϕ/μ) ^p+r_DM+r_m)u- α _ϕ/N_ϕ[ u^2/2 +( ϕ/μ) ^p] u,dN_ϕ/dτ = -√(3)√(u^2/2+( ϕ/ μ) ^p+r_DM+r_m)N_ϕ- α _ϕ[ u^2/2 +( ϕ/μ) ^p] ,dr_DM/dτ = -4/√(3)√(u^2/2+( ϕ/ μ) ^p+r_DM+r_m)r_DM+ γ _DM[ u^2/2 +( ϕ/μ) ^p] ,dr_m/dτ = -4/√(3)√(u^2/2+(ϕ/μ) ^p+r_DM+r_m)r_m+ γ _m[ u^2/2 +( ϕ/μ) ^p] .The system of Eqs. (<ref>)-(<ref>) must be integrated with the initial conditions ϕ (0)=ϕ _0, u(0)=u_0, N_ϕ( 0) =N_ϕ 0, r_DM(0)=0, and r_m(0)=0, respectively.The variations with respect to the dimensionless time τ of the energy density of the dark matter, of the radiation, of the scalar field particle number, and of the Hubble function, are presented in Figs. <ref> and <ref>, respectively. In order to numerically integrate Eqs. (<ref>)-(<ref>) we have fixed the parameters of the potential as p=1.8 and μ =3.5, respectively. For the initial conditions of the scalar field and of its variation we have adopted the values ϕ (0)=1.1, and u(0)=-1.0. The initial number of the scalar field particles was fixed to N_ϕ( 0)=3, while for the coefficients γ _DM and γ _m we have used the values γ _DM=(4/5)α _ϕ and γ _m=(1/5)α _ϕ. As one can see from Figs. <ref>, the evolution of the matter component (dark plus radiation) of the Universe can be divided into two distinct phases. In the first phase, the matter energy densities increase from zero to a maximum value r_a^(max), a= DM,m, which is reached at a time interval τ _max, due to the energy transfer from the scalar field. For time intervals τ >τ _max, the expansion of the Universe becomes the dominant force describing the matter dynamics, leading to a decrease in the matter energy density. The energy density of the scalar field,shown in the left panel of Fig. <ref>, as well as the associated particle number decreases rapidly during the reheating process, and tends to zero in the large time limit. The energy density of the scalar field is not affected significantly by the variation of the numerical values of the parameter α _ϕ. The deceleration parameter of the Universe, depicted in the right panel of Fig. <ref>, shows a complex evolution, depending on the nature of the particle creation processes. For the adopted initial values the particle creation phase in the inflationary history of the Universe begins with values of the deceleration parameter of the order of q≈ -0.20. During the particle creation processes the Universe continues to accelerate, a process specifically associate to the presence of the negative creation pressure, that induces a supplementary acceleration. This acceleration periods ends once the number of the newly created particles declines significantly, and at values ofq of the order of q=-0.25, the Universe starts decelerating, with the deceleration parameter tending to zero, corresponding to a marginally expanding Universe. When ≈ 0, the energy density of the scalar field also becomes extremely small, and the Universe enters in the standard, matter dominated cosmological phase. with dark matter and radiation as the major components, plus a small remnant of the inflationary scalar field, which may play the role of the dark energy, and become again dominant in the latest stages of expansion.We define the reheating temperature T_reh as the temperature corresponding to the maximum value of the energy density of the relativistic matter r_ m^(max, whose energy density is defined as ρ _ m=(π^2/30))g_rehT^4, where g_reh≈ 100 is the number of relativistic degrees of freedom. The maximum matter energy density r_ m^(maxis obtained from the condition dr_ m/dτ =0,which gives the equation4h(τ_max,T_reh)π ^2/30g_rehT_reh^4/M_ϕ^2m_Pl^2=1/5α _ϕr_ϕ(τ _max),or, equivalently,T_reh= 1.56× 10^9×[ 24 /π ^2g_rehα _ϕr_ϕ( τ _max ) /h( τ _max ) ] ^1/4× M_ϕ^1/2GeV.Eqs. (<ref>) and (<ref>) determine the reheating temperature as a function of M_ϕ, the decay width of the scalar field, the value of the scalar field at maximum reheating, and its potential, respectively. If the reheating temperature is known from observations, from Eq. (<ref>) one can constrain the parameters of the inflationary model, and of the scalar field potential. By adopting the bound T_reh≤ 6× 10^10<cit.>, we obtain the restriction on the parameter M_ϕ of the potential asM_ϕ≤ 1.6× 10^3 ×[ 24 /π ^2g_rehα _ϕr_ϕ( τ _max ) /h( τ _max ) ] ^-1/4GeV.The functions r_ϕ and h are very little influenced by the modifications of the parameters of the potential or of α _ϕ (but they depend essentially on the initial conditions). Hence for this class of potentials we approximate r_ϕ( τ _max )≈ 0.20, and h( τ _max)≈ 0.70. Hence we can approximate the coefficient in the scalar field parameter as [ 24 /π ^2g_rehα _ϕr_ϕ( τ _max ) /h( τ _max ) ] ^-1/4≈ 3.46×α _ϕ^-1/4.Therefore the second important parameter restricting the reheating temperature, or the potential parameters, is the decay width of the scalar field.For α _ϕ=10^-3, α _ϕ=10^3/4=5.62, while for α _ϕ=10^3, α _ϕ=10^-3/4=0.17. Hence the uncertainty in the knowledge of α _ϕ can lead to variations over two orders of magnitude in the potential parameters. §.§ Small-field power law and small-field quadratic potentialsThe small-field polynomial potentials are of the formV(ϕ)=M_ϕ^2[ 1-( ϕ/μ) ^p],where ϕ≪μ and p≥ 2. M_ϕ is a parameter of the dimension of mass, while μ is a dimensionless constant. These potentials appear as a result of a phase transition from spontaneous symmetry breaking. They are usually used in the "new inflation" models. A particular case of this class of potentials are small-field quadratic potentials, corresponding to p=2. Natural inflation models usually use a cosine potential, and the small-field quadratic potentials are its expansion near ϕ =0 for p=2. For this class of potentials the dimensionless energy density and pressure of the scalar field takes the formr_ϕ = 1/2(dϕ/dτ)^2+[ 1-( ϕ/μ) ^p],P_ϕ = 1/2(dϕ/dτ)^2-[ 1-( ϕ/μ) ^p].The dimensionless equations describing particle production from inflationary scalar fields with small field polynomial potentials aredϕ/dτ=u, du/dτ = p/μ( ϕ/μ) ^p-1- √(3)√(u^2/2+[ 1-( ϕ/μ) ^p ] +r_DM+r_m)u- α _ϕ/N_ϕ[ u^2/ 2+1-( ϕ/μ) ^p] u, dN_ϕ/dτ = -√(3)√(u^2/2+[ 1-( ϕ/μ) ^p] +r_DM+r_m)N_ϕ- α _ϕ[ u^2/2+1-( ϕ/μ) ^p] , dr_DM/dτ = -√(3)√(u^2/2+[ 1-(ϕ/μ) ^p] +r_DM+r_m)r_M+ α _DM[u^2/2+1-( ϕ/μ) ^p] , dr_m/dτ = -4/√(3)√(u^2/2+[ 1-( ϕ/μ) ^p] +r_DM+r_m)r_m+ α _m[ u^2/2+1-( ϕ/μ) ^p] .The system of Eqs. (<ref>)-(<ref>) must be integrated with the initial conditions ϕ (0)=ϕ _0, u(0)=u_0, N_ϕ( 0) =N_ϕ 0, r_DM(0)=0, and r_m(0)=0, respectively.The variation of the dark matter energy density, of the radiation energy density, of the scalar field particle number, and of the Hubble function, respectively, are depicted in Figs. <ref> and <ref>. In order tonumerically integrate the system of Eqs. (<ref>)-(<ref>) we have fixed the value of p at p=3, and the value of μ as μ =10^3. We have slightly varied the numerical values of the parameter α _ϕ, which resulted in significant variations in the time evolution of the cosmological quantities. The initial conditions used for the numerical integration of the evolution equations are u(0)=-4, and ϕ (0)=1, N_ϕ=5, while the initial values of the energy density of the dark matter and of radiation were taken as zero. As one an see from the left panel of Fig. <ref>, the energy density of the dark matter is initially a monotonically increasing function of time, reaching its maximum value r_ DM at a time τ _max. For time intervals τ >τ _max, the energy density of the dark matter decreases due to the expansion of the Universe. The time variation of the radiation, presented in the right panel of Fig. <ref>, shows a similar behavior as the dark matter component. The energy density of the scalar field decreases, shown in the left panel of Fig. <ref>, tends to zero in a finite time interval. The deceleration parameter of the Universe, presented in the right panel of Fig. <ref>, shows an interesting behavior. For the chosen initial values the Universe is at τ =0 in a marginally accelerating state, with q≈ 0. However, during the particle creation phase, the Universe starts to accelerate, and reaches a de Sitter phase with q=-1, at a time interval around τ =τ _fin≈ 1, when the energy density of the scalar field becomes negligibly small. Since ρ _ϕ≈ 0 for τ >τ _fin, the particle creation processes basically stop, and the Universe enters in the matter dominated, decelerating phase. For the chosen range of initial conditions the numerical values of ρ _ϕ(τ _max) and h_τ_max are roughly the same as in the previous case, thus leading to the same restrictions on the potential parameter M_ϕ. §.§ The exponential potential The exponential potentials are of the formV(ϕ)=M_ϕ^2exp( ±√(2/p)ϕ)where M_ϕ is a parameter of the dimension mass, and p is a constant. Exponential potentials are sometimes used in extra dimensions models and superstring theories.For example, an exponential potential is generated from compactification of the higher-dimensional supergravity or superstring theories in four-dimensional Kaluza-Klein type effective theories <cit.>. The moduli fields associated with the geometry of the extra-dimensions can have in string or Kaluza-Klein theories effective exponential type potentials, which aredue to the curvature of the internal spaces, or, alternatively, to the interaction on the internal spaces of the moduli fields with form fields. Non-perturbative effects such as gaugino condensation could also lead to the presence of exponential potentials <cit.>.The dimensionless energy density and pressure of the inflationary scalar field arer_ϕ=1/2(dϕ/dτ)^2+e^±√(2/p)ϕ, P_ϕ=1/2(dϕ/dτ)^2+e^±√(2/p)ϕ. The dimensionless equations Eqs. (<ref>)-(<ref>) describing the evolution of the Universe in the presence of irreversible matter creationcan then be formulated asdϕ/dτ=u, du/dτ = ∓√(2/p)e^±√(2/p)ϕ- √(3)√(u^2/2+e^±√(2/p)ϕ+r_DM+r_m)u- α _ϕ/N_ϕ[ u^2/2+e^±√(2 /p)ϕ] u, dN_ϕ/dτ = -√(3)√(u^2/2+e^±√(2 /p)ϕ+r_DM+r_m)N_ϕ- α _ϕ[ u^2/2 +e^±√(2/p)ϕ] , dr_DM/dτ = -√(3)√(u^2/2+e^±√(2/p )ϕ+r_DM+r_m)r_M+ α _DM[ u^2/2+e^±√( 2/p)ϕ] , dr_m/dτ = -4/√(3)√(u^2/2+e^±√( 2/p)ϕ+r_DM+r_m)r_m+ α _m[ u^2/2+e^±√(2/p)ϕ] . The system of Eqs. (<ref>)-(<ref>) must be integrated with the initial conditions ϕ (0)=ϕ _0, u(0)=u_0, N_ϕ( 0) =N_ϕ 0, r_DM(0)=0, and r_m(0)=0, respectively. In order to numerically integrate the cosmological evolution equations we have fixed the values of p as p=5, and we have adopted the negative sign for the scalar field exponential potential. The initial conditions used for the numerical study are ϕ (0)=15, u(0)=-5.5, N_ϕ( 0) =20, together with the null initial conditions for the matter energy densities. As one can see from the left panel of Fig. <ref>, the amount of dark matter increases very rapidly during the first phase of the reheating. For the chosen values of the parameters and initial conditions, there is a sharp peak in the dark matter distribution, followed by a decreasing phase. A similar dynamic is observed for the time evolution of the radiation, shown in the right panel of Fig. <ref>. The energy density of the scalar field, presented in the left panel of Fig. <ref> decreases rapidly, and reaches the value ρ _ϕ≈ 0 at τ≈ 1. The evolution of the scalar field energy density is basically independent of the variation of the numerical values of the decay width of the scalar field α _ϕ. The numerical values of the deceleration parameter (right panel in Fig. <ref>) do depend significantly on α _ϕ. At the beginning of the particle creation epoch the Universe is in a marginally inflating phase with q≈ 0. Particle creation induces a re-acceleration of the Universe, with the deceleration parameter reaching values of the order of q≈ -0.15 at the moment when the matter energy density reaches its peak value. However, the acceleration of the Universe continues, and at the moment the energy density of the scalar field becomes (approximately) zero, the deceleration parameter has reached values of the order of q≈ -0.40. Once the energy density of the scalar field is zero, the Universe starts its matter dominated expansion. For the chosen initial conditions ρ _ϕ(τ _max) and h_τ_max have significantly greater values than in the cases of the power law potentials. However, since the model parameters depend on their ratio, no significant variations in the reheating temperature or model parameters are expected. §.§ The Higgs Potential The Higgs potentials are of the formV(ϕ)=M_ϕ^2[ 1-( ϕ/μ) ^2+( ϕ/ν) ^4]where M_ϕ is a parameter of the dimension mass,and μ and ν are constants. The Higgs field is the only scalar field in the Standard Model of particle physics. The Higgs boson is a very special particle physics and in the Standard Model <cit.>. It provides a physical mechanism for the inclusion of the weakly interacting massive vector bosons in the Standard Model, and for giving masses to the fundamental particle of nature like quarks and leptons. The Higgs field may have also played an important role in cosmology.It could have been the basic field making the Universe flat, homogeneous and isotropic, and it could have generated the small scale fluctuations that eventually led to cosmic structure formation. The Higgs field could have also allowed the occurrence of the radiation-dominated phase of the hot Big Bang <cit.>.It is an intriguing question whether the initial scalar field driving the inflation is the same field as the Higgs field <cit.>. The dimensionless energy density and pressure of the inflationary Higgs scalar field are given by ρ_ϕ = 1/2(dϕ/dτ)^2+[ 1-( ϕ/μ) ^2+( ϕ/ν) ^4],P_ϕ = 1/2(dϕ/dτ)^2-[ 1-( ϕ/μ) ^2+( ϕ/ν) ^4]. The equations describing irreversible matter creation processes from a decaying scalar field in the presence of the Higgs type potential aredϕ/dτ=u, du/dτ = ϕ( 2/μ ^2-4ϕ ^2/ν ^4 ) - √(3)√(u^2/2+1-( ϕ/μ) ^2+( ϕ/ν) ^4+r_DM+r_m)u- α _ϕ/N_ϕ[ u^2/2+1-( ϕ/μ ) ^2+( ϕ/ν) ^4] u,dN_ϕ/dτ=-√(3)× √(u^2/2+1-( ϕ/μ) ^2+( ϕ/ν) ^4+r_DM+r_m) N_ϕ- α _ϕ[ u^2/2+1-( ϕ/μ ) ^2+( ϕ/ν) ^4] ,dr_DM/dτ=-√(3)× √(u^2/2+1-( ϕ/ μ) ^2+( ϕ/ν) ^4+r_DM+r_m) r_M+ α _DM[ u^2/2+1-( ϕ/μ) ^2+( ϕ/ν) ^4] ,dr_m/dτ=-4/√(3)× √(u^2/2+1-( ϕ/μ) ^2+( ϕ/ν) ^4+r_DM+r_m)r_m+ α _m[ u^2/2+1-(ϕ/μ) ^2+( ϕ/ν) ^4] . In order to obtain the numerical evolution of the Universe in the presence of irreversible matter creation triggered by the decay of a cosmological scalar field with Higgs type potential we fix the parameters of the potential as μ =2 and ν =3, respectively. For the initial conditions of the scalar field and of the scalar field particle number we have adopted the numerical values phi (0)=10.5 and N_ϕ(0)=100, respectively, while for u(0) we have taken the value u(0)=-0.05. The time variations of the dark matter and radiation energy densities, of the scalar field potential, and of the Hubble function are presented in Figs. <ref> and <ref>, respectively. For the adopted set of the parameters, the dark matter energy density, depicted in the right panel of Fig. <ref>, increases from zero to a maximum value, and then begins to decrease, due to the combine effects of the expansion of the Universe, and of the decay of the scalar field. The radiation energy density, presented in the right panel of Fig. <ref>, shows a similar behavior like dark matter. The evolution of both matter components is strongly dependent on the numerical values of the model parameters, as well as on the initial conditions. The energy density of the Higgs type scalar field, shown in the left panel of Fig. <ref>,becomes negligibly small at τ≈ 1. Its evolution also strongly depends on the model parameters. The deceleration parameter q, presented in the right panel of Fig. <ref>, indicates that in the case of scalar field with Higgs potential the irreversible matter creation process starts when the Universe is in a de Sitter stage, with q=-1. During the first phase of the particle creation the Universe decelerates, but not significantly, and it continues its decelerating trend up to the moment when the energy density of the scalar field vanishes, at around τ≈ 1, and standard matter dominated evolution takes over the expansionary dynamics of the Universe. However, at this moment in time the deceleration parameter still has a large negative value of the order of q≈ -0.7. For the chosen set of initial conditions the numerical values of ρ _ϕ(τ _max) and h_τ_max are similar to the case of the exponential potential, and hence no significant differences in the predicted reheating temperature or in model parameter constraints can take place.§ DISCUSSIONS AND FINAL REMARKS In the present paper we have investigated the implications of the irreversible thermodynamic description of particle creation during the reheating period of the early Universe. By interpreting the post-inflationary Universe as an open system with energy-matter transfer between the fluid components, a complete thermodynamic description of the basic physical processes taking place at the end of inflation can be obtained.The basic feature of our model is the decay of the inflationary scalar field and the transfer of its energy to the newly created particles. As for the matter content of the post-inflationary Universe we have assumed that it consists of pressureless dark matter, and a relativistic, radiation type component. By using the fundamental principles of the thermodynamics of open systems and the Einstein field equations, a systematic and consistent approach describing the time evolution of the matter components after inflation can be obtained. We have investigated the effects on particle creation of various forms of the scalar field potential, which generallylead to similar qualitative results for the reheating scenario.What we have found from our numerical analysis is that the choice of the functional form of the scalar field potential has an essential influence on the cosmological dynamics. This influence can be seen in the time interval for which the maximum energy densities of the dark matter and radiation are obtained, in the decay rate of the energy density of the scalar field, and its conversion to matter, as well as in the evolution of the deceleration parameter. Moreover, the cosmological dynamics also depends on the numerical values of the potential parameters.However, the scalar field potentials have a unique quantitative imprint on the matter creation during reheating. By adopting a set of initial conditions for the physical variables,the general equations describing reheating can be solved numerically.During reheating, the particle number density of the scalar field drops to near zero, while the number of the dark matter and matter particles increases from zero to a maximum value, thus heating up the Universe to a maximum reheating temperature T_reh. The reheating temperature depends on the physical parameters of the inflation (scalar field energy density, field decay width, potential parameters), and on the cosmological quantities, like the value of the Hubble function at the moment when the matter density reached its maximum value. By using the general constraint on the reheating temperature as obtained in <cit.>, we have obtained some general constraints on the mass parameter in the scalar field potential. However, in our investigation these results are only of qualitative nature, since a significant increase in the accuracy of the observational data is still needed to obtain an accurate reconstruction of the physical conditions in the early Universe.Large-field potential models lead to similar results as the kinetic dominated ones, since the observation of the scalar spectral index n_s constrains the value of p to be relatively small, withobservations preferring the potentials with p=2<cit.>.The exponential potential models speed up the matter creation process, specifically the preheating process to τ<0.2, and keeps creating new particles even after the matter energy density has reached its maximum value. The process is similar to the one induced by the Higgs potentials model. In our present approach we have neglected the backreaction of the newly created particles on the dynamics of the Universe.We have also carefully investigated the behavior of the deceleration parameter during the reheating period. A general result is that for all considered potentials the Universe still accelerates during the reheating phase. This is not an unexpected result from a physical point of view, since in the formalism of the thermodynamics of open systems particle creation is associated with a negative creation pressure, that accelerates the system. Irreversible particle creation has been proposed as a possible explanation of the late time acceleration of the Universe <cit.>. A similar process takes place in the irreversible thermodynamic description of the reheating, with the Universe acquiring a supplementary acceleration. Moreover, this acceleration extends beyond the moment of the reheating (when the maximum temperature is reached), and indicates that the Universe does not enter the standard, matter and radiation dominated phase, in a decelerating state. Hence the use of the thermodynamic of open systems with irreversible matter creation leads to an accelerating expansion of the Universe during the particle production stage, corresponding to the post-inflationary reheating. This effect is strongly scalar field potential dependent, and while for some models the Universe acceleration is marginal, with q≈ 0 (coherent scalar field and large field polynomial potential with V(ϕ)∝ϕ ^p), for other types of potentials (small field power law, exponential, Higgs), the deceleration parameter at the end of the reheating period could have negative values in the range q∈ (-1,-0.4). These extremelyaccelerating models may be unrealistic, since they would indeed pose some serious problems to the nucleosynthesis and structure formation, and contradict some well-established cosmological bounds on the chemical composition of the Universe, and the time frame the first chemical elements were produced.On the other hand, in the present model the cosmological behavior and particle creation rates also strongly depend on the scalar field potential parameters, whose full range has not been systematically and completely investigated in the present study. Therefore the use of the thermodynamic of open processes for the description of the post-inflationary reheating may also provide some strong constraints on the admissible types of scalar field potentials. In this context we would like to note that model independent constraints on the cosmological parameters can be obtained within the framework of theso-called cosmography of the Universe. Cosmography is essentially a Taylor expansion as applied to cosmology. Model independent constraints on the evolution of the deceleration parameter have been found by using the cosmographic approach in <cit.>.When the temperature of the decay products T approaches the mass scale M_ϕ in the scalar field potentials, T→ M_ϕ, then, as one can see from Eq. (<ref>), the decay width Γ _ϕ of the scalar field tends to zero, Γ _ϕ→ 0. Moreover, the energy density of the scalar field becomes negligibly small, meaning that after the end of the reheating phase, corresponding to Γ _ϕ≈ 0 and ρ _ϕ≈ 0, the number of the dark matter particles, and of the photons, is conserved. Hence in the post reheating phase the cosmological evolution of the Universe is governed by the equations3H^2=1/m_Pl^2(∑ _iρ _i1/2ϕ̇^2+V(ϕ)), ρ̇_i+3H(ρ _i+p_i)=0, i=1,2,.., ϕ̈+3Hϕ̇+V'(ϕ)=0,where ρ _i and p_i,i=1,2,.. denotes the thermodynamic densities and pressures of the matter components (radiation, dark matter, baryonic matter etc.) of the Universe.Eqs. (<ref>)-(<ref>) represents a cosmological model in which the Universe is composed from matter and dark energy, described by a scalar field with a self-interaction potential V(ϕ). In this way we have recovered the quintessence dark energy models,in which dark energy is described by a canonical scalar field ϕ minimally coupled to gravity <cit.>. As compared to other scalar-field models, like phantom and k-essence dark energy scenarios, the quintessence model is the simplest scalar-field approach not having major theoretical shortcomings such as the presence of ghosts and of the Laplacian instabilities. Moreover, a slowly varying scalar field together with a potential V(ϕ) can lead to the late-time acceleration of the Universe. The quintessence scalar field has the important property of being very weakly coupled to matter (baryonic and dark),but it contributes a negative pressure to the global equation of state.The magnitude of the energy scale of the quintessence potential must be of the order of ρ _DE≈10^-47GeV^4 at the present time,a value that is much smaller than that of the inflaton potential triggering the inflationary expansion. In order to achieve a late time cosmic acceleration, the mass m_ϕof the quintessence field, given by m_ϕ^2=d^2V (ϕ)/dϕ ^2 must be extremely small, of the order of|m_ϕ| ∼H_0 ≈10^−33 eV <cit.>, where H_0 is the present day value of the Hubble function. The evolution of the Universe as described by the Eqs. (<ref>)-(<ref>) starts from the initial conditions fixed at the end of reheating, including the initial matter densities, scalar field and Hubble function values. In the early stages of the evolution of the Universe (post-reheating phase) the quintessence scalar field had a very small contribution to the energy balance of the Universe, but during the later cosmological evolution it decreases slower with the scale factor as compared to the matter and radiation densities,and consequently it is dominant at the present time, triggering an exponential increase of the scale factor.Therefore even the Universe enters in the conservative phase in an accelerating state, in the first stages of the conservative evolution, in which the effect of the scalar field can be neglected, the cosmological evolution will strongly decelerate. By assuming that the energy density of the scalar field is negligibly small, and the Universe is matter dominated, then the scale factor is given by a(t)=t^2/3, with the corresponding deceleration parameter having the value q=1/2.An interesting and important question is the physical nature of the dark matter particles that could have beencreated during reheating. Presently, the true physical nature of the dark matter particle is not known. One possibility is that dark matter may consist of ultra-light particles, with masses of the order of m≈ 10^-24 eV (see <cit.> and references therein). Such an ultra-light dark matterparticle may represent, from a physical point of view, apseudo Nambu-Goldstone boson. Other important ultra-light dark matter particle candidates are the axions, with masses of the order ofm≤ 10^-22 eV <cit.>. Very low mass particles can be easily created even in very weak gravitational fields, and hence their production in large numbers during reheating could have taken place. Another hypothesis about the nature of dark matter is representedby the so-called Scalar Field Dark Matter (SFDM) models <cit.>. In these models dark matter is described as a real scalar field, minimally coupled to gravity, with the mass of the scalar field particle having a very small value, of the order of m <10^-21 eV. An important property of scalar field dark matter models is that at zero temperature all particles condense to the same quantum ground state, thus forming a Bose-Einstein Condensate (BEC). Hence, from a physical point of view it turns out that SFDM models are equivalent to the BEC dark matter models <cit.>. On the other hand, in the open irreversible thermodynamicmodel of reheating introduced in the present paper, dark matter particle creation can take place also in the form of a scalar field, one we assume that the scalar field does not fully decays into some ordinary matter.Hence, scalar field dark matter could be a particular result of the inflationaryscalar field dynamics during the reheating phase in the evolution of the Universe.Particle creation can also be related to what is called the arrow of time: a physical process that provides a direction to time, and distinguishes the future from the past. There are two different arrows of time: the thermodynamical arrow of time, the direction, in which entropy increases,and the cosmological arrow of time, the direction in which the Universe is expanding. Particle creation introduces asymmetry in the evolution of the Universe, and enables us to assign a thermodynamical arrow of time, which agrees, in the presentmodel, with the cosmological one. This coincidence is determined, in a natural way, by the physical nature of inflation.The present day observations cannot accurately constrain the main predictions of the present model. However, even without a full knowledge of the true cosmological scenario one can make some predictions about reheating. In the models we are considering the energy density of the inflationary scalar field does not drop completely to zero during reheating. This may suggest that the dark energy that drives the accelerating expansion of the Universe today may have also originated from the original inflationary scalar field. Newly created dark matter particles may cluster with matter through gravitational interactions, and form galaxies, with baryonicmatter clustering in its center. The initial conditions of the reheating as well as the physics of the decay of the scalar fields are not fully known, and they may need physics beyond the Standard Model of particle physics to account for the dark matter creation process. And certainly we need more accurate observational data as well, to constrain the models and find out what truly happens in the early Universe.§ ACKNOWLEDGMENTS We would like to thank to the three anonymous reviewers for comments and suggestions that helped us to significantly improve our manuscript. T. H. would like to thank to the Institute of Advanced Studies of the Hong Kong University of Science and Technology for the kind hospitality offered during the preparation of this work.S.-D. L. would like to thank to the Natural Science Funding of Guangdong Province for support (2016A030313313).99guth1981inflationary A. Guth, Physical ReviewD 23, 347 (1981).B3 A. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic Publishers, London, UK, 1992B31 E. Kolb and M. Turner, The Early Universe, Westview Press, New York, USA, 1994B1 A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, Cambridge, UK, 2000B2 V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Cambridge, UK, 2005linde1982new A. Linde, Physics LettersB 108, 389 (1982).linde1982coleman A. Linde, Physics LettersB 114, 431 (1982).linde1982scalar A. Linde, Physics LettersB 116, 329 (1982).albrecht1982cosmology A. Albrecht and P. Steinhardt, Phys. Rev. Lett.48, 1220 (1982).linde2000inflationary A. Linde, Physica ScriptaT85, 168 (2000).linde1983chaotic A. Linde, Physics Letters B 129, 177 (1983).linde1994hybrid A. Linde, Physical ReviewD 49, 748 (1994).Pl1P. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. I. Overview of products and scientific results",arXiv:1502.01582 (2015).Pl2 P. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. XIII. Cosmological parameters", arXiv:1502.01589 [astro-ph.CO] (2015).Pl3 P. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. XVII. Constraints on primordial non-Gaussianity",arXiv:1502.01592 [astro-ph.CO] (2015).Pl4 P. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. XVIII. Background geometry & topology",arXiv:1502.01593 [astro-ph.CO] (2015).Pl5 P. A. R. Ade et al. [Planck Collaboration], “Planck 2015 results. XX. Constraints on inflation”, arXiv:1502.02114 [astro-ph.CO] (2015).liddle1992cobe A. Liddle and D. Lyth, Physics LettersB 291, 391 (1992).dodelson1997cosmic S. Dodelson, W. Kinney and E. Kolb, Physical ReviewD 56, 3207 (1997).kinney1998constraining W. Kinney, Physical ReviewD 58, 123506 (1998).kinney2000new W. Kinney, A. Melchiorri and A. Riotto, Physical ReviewD 63, 023505 (2001).kinney2008latest W. Kinney, E. Kolb, A. Melchiorri and A. Riotto, Physical ReviewD 78, 087302 (2008).Haw S. W. Hawking,Physics LettersB, 115,295 (1982).Muk V. F. Mukhanov, Journal of Experimental and Theoretical Physics Letters 41, 493 (1985). AnC. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, Adv. Astron.2010, 847541 (2010).Alb A. Albrecht, P. J. Steinhardt, M. S. Turner, and F. Wilczek,Phys. Rev. Lett.48, 1437 (1982).Kof L. Kofman, A. Linde, and A. Starobinsky, Phys. Rev. Lett.73, 3195 (1994).kofman1997towards L. Kofman, A. Linde and A. Starobinsky, Phys. Rev.D 56, 3258 (1997).Pre1 J. Repond and J. Rubio, Journal of Cosmology and Astroparticle Physics07, 043 (2016).Pre2 K. D. Lozanov and M. A. Amin, Journal of Cosmology and Astroparticle Physics06, 032 (2016).Pre3 S. Antusch, D. Nolde, and S. Orani, Journal of Cosmology and Astroparticle Physics06,009 (2015).Pre4 I. Rudenok, Y. Shtanov, and S. Vilchinskii, Phys. Lett.B 733, (2014).scherrer1985decaying R. Scherrer and M. Turner, Phys. Rev.D 31, 681 (1985).giudice2001largest G. Giudice, E. Kolb and A. Riotto, Phys. Rev.D 64, 023508 (2001).Re11 T. Harko, W. F. Choi, K. C. Wong, and K. S. Cheng, Journal of Cosmology and Astroparticle Physics06, 002 (2008).Re10 R. K. Jain, P. Chingangbam, and L. Sriramkumar, Nucl. Phys.B 852, 366 (2011).Re9 V. Demozzi and C. Ringeval, Journal of Cosmology and Astroparticle Physics05,009 (2012).Re8 H. Motohashi and A. Nishizawa, Phys. Rev.D 86, 083514 (2012).Re7 K. Mukaida and K. Nakayama, Journal of Cosmology and Astroparticle Physics03, 002 (2013).Re6 A. Nishizawa and H. Motohashi, Phys. Rev.D 89, 063541 (2014).Re5 L. Dai, M. Kamionkowski, and J. Wang, Phys. Rev. Lett.113, 041302 (2014).Re4 J. Martin, C. Ringeval, and V. Vennin, Phys. Rev. Lett.114, 081303 (2015).Re3 C. Gross, O. Lebedev, and M. Zatta, Phys. Lett.B 753, 178 (2016).Re2 Y. S. Myung and T. Moon, Journal of Cosmology and Astroparticle Physics07,014 (2016).Re1 M. Rinaldi and L. Vanzo, Phys. Rev.D 94, 024009 (2016).allahverdi2010reheating R. Allahverdi, R. Brandenberger, F. Cyr-Racine and A. Mazumdar, Annu. Rev. Nucl. Part. Sci.60, 27 (2010).RehRev M. A. Amin, M. P. Hertzberg, D. I. Kaiser, and J. Karouby, Int. J. Mod. Phys.D 24,1530003 (2015).Zim W. Zimdahl, J. Triginer, and D. Pavon, Phys. Rev.D 54, 6101 (1996).prigogine1988thermodynamics I. Prigogine, J. Geheniau, E. Gunzig and P. Nardone, Proceedings Of The National Academy Of Sciences85, 7428 (1988).Calv M. Calvão, J. Lima and I. Waga, Physics LettersA 162, 223 (1992).Calv1J. A. S. Lima and A. S. M. Germano, Physics LettersA 170, 373 (1992).op1 T. Harko and M. K. Mak, Astrophys. Space Science253, 161 (1997). op2 T. Harko and M. K. Mak, Class. Quantum Grav.16, 2741 (1999).op3 T. Harko and M. K. Mak, Gen. Relativ. Grav.31, 849 (1999).op4 M. K. Mak and T. Harko, Class. Quantum Grav.16, 4085 (1999).op5 M. K. Mak and T. Harko, Aust. J. Phys.52, 659 (1999).op5a V. V. Papoyan, V. N. Pervushin, and D. V.Proskurin, Astrophysics 46, 92 (2003).op5b R. Brandenberger and A. Mazumdar, JCAP0408,015 (2004).op5c Y. Qiang, T.-J. Zhang, and Z.-L. Yi, Astrophys. Space Sci.311, 407 (2007). op6 S. K. Modak and D. Singleton, Phys. Rev.D 86, 123515 (2012).op7 S. K. Modak and D. Singleton, Int. J. Mod. Phys.D 21, 1242020 (2012).op7a J. Chen, F. Wu, H. Yu, and Z. Li, The European Physical JournalC 72, 1861 (2012). op8 T. Harko and F. S. N. Lobo, Phys. Rev.D 87, 044018 (2013). op9 R. O. Ramos, M. V. dos Santos, and I. Waga, Phys. Rev.D 89, 083524 (2014). op10, S. Chakraborty, Phys. Lett.B 732, 81 (2014). op10a J. Quintin, Y.-F. Cai, and R. H. Brandenberger, Phys. Rev.D 90, 063507 (2014). op11 S. Chakraborty and S. Saha, Phys. Rev.D 90, 123505 (2014).op12 J. C. Fabris, J. A. de Freitas Pacheco, and O. F. Piattella, JCAP06, 038 (2014).op13 J. A. S. Lima and I. Baranov, Phys. Rev.D 90, 043515 (2014).op14 T. Harko, Phys. Rev.D 90, 044067 (2014).op15 S. Pan and S. Chakraborty, Adv. High Energy Phys.2015 654025 (2015).op16 T. Harko, F. S. N. Lobo, J. P. Mimoso, and D. Pavón, Eur. Phys. J.C 75, 386 (2015).op17 R. C. Nunes and S. Pan, Monthly Notices of the Royal Astronomical Society459, 673 (2016).op18 V. Singh and C. P.Singh, International Journal of Theoretical Physics55, 1257 (2016).op19 S. Shandera, N. Agarwal, and A. Kamal, arXiv:1708.00493 [hep-th] (2017).Reh1 T. Rehagen and G. B. Gelmini, Journal of Cosmology and Astroparticle Physics06,039, (2015).kolb2003reheating E. W. Kolb, A. Notari and A. Riotto,Phys. Rev.D 68, 123505 (2003).de2010model H. De Vega and N. Sanchez, Monthly Notices Of The Royal Astronomical Society404, 885 (2010).P1P. J. E. Peebles and B. Ratra, Ap. J. Lett.325, L17 (1988).P2B. Ratra and P. J. E. Peebles, Phys. Rev.D 37, 3406 (1988).57C. G. Callan, E. J. Martinec, M. J. Perry and D. Friedan, Nucl. Phys.B 262, 593 (1985).58 B. de Carlos, J. A. Casas and C. Munoz, Nucl. Phys.B 399, 623 (1993).H1 M. Shaposhnikov, Phil. Trans. R. Soc.A 373, 20140038 (2015).H2 F. L. Bezrukov and M. Shaposhnikov, Phys. Lett.B 659, 703 (2008).a1 M. P. Freaza, R. S. de Souza, and I. Waga,Phys. Rev.D 66, 103502 (2002).a2 J. A. S. Lima, J. F. Jesus, and F. A. Oliveira, Journal of Cosmology and Astroparticle Physics11,027 (2010).a3 R. O. Ramos, M. V. dos Santos, and I. Waga, Phys. Rev.D 89, 083524 (2014).a4 S. Chakraborty and S. Saha, Phys. Rev.D 90, 123505 (2014).a5 J. A. S. Lima, S. Basilakos, and J Solà, Eur. Phys. J.C 76, 228 (2016).a6 C. Pigozzo, S. Carneiro, J. S. Alcaniz, H. A. Borges, and J. C. Fabris, Journal of Cosmology and Astroparticle Physics05, 022 (2016).q1 C. Cattoen and M. Visser, Class. Quant. Grav.24, 5985, (2007).q2 A. Aviles, C. Gruber, O. Luongo, and H. Quevedo, Phys. Rev.D 84, 103520, (2012).q3 A. de la Cruz-Dombriz, P. K. S. Dunsby, O. Luongo, and L. Reverberi, Journal of Cosmology and Astroparticle Physics12, 042, (2016).q4 O. Luongo, G.B. Pisani, and A. Troisi, International Journal of Modern Physicsbf D 26, 1750015 (2017).q5 A. Al Mamon and S. Das, Eur. Phys. J.C 77, 495 (2017).Tsu S. Tsujikawa, Class. Quant. Grav.30,214003 (2013).Lee J.-W. Lee, Phys. Lett.B 681, 118 (2009). Park C.-G. Park, J.-C. Hwang, and H. Noh, Phys. Rev.D 86, 083535 (2012).Mat L. A. Martinez-Medina, V. H. Robles, and T. Matos, Phys. Rev.D 91, 023519 (2015).Bose1 C. G. Boehmer and T. Harko, JCAP 06, 025 (2007).Bose2 V. H. Robles and T. Matos, Mon. Not. Roy. Astron. Soc. 422, 282 (2012).Bose3 T. Rindler-Daller and P. R. Shapiro, Mon. Not. Roy. Astron. Soc. 422, 135 (2012).Bose4 H. Velten and E. Wamba, Phys. Lett.B709, 1 (2012).Bose5T. Harko, Phys. Rev. D 89, 084040 (2014).Bose6 M.-H. Li and Z.-B. Li, Phys. Rev. D 89, 103512 (2014). | http://arxiv.org/abs/1708.08004v1 | {
"authors": [
"Juntong Su",
"Tiberiu Harko",
"Shi-Dong Liang"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170826181426",
"title": "Irreversible thermodynamic description of dark matter and radiation creation during inflationary reheating"
} |
<ref>We deal with NS_ω_1 and 𝒟_ω_1 under Martin's maximum and the proper forcing axiom. Topological π-junctions from crossed Andreev reflection in the Quantum Hall regime F. Finocchiaro^1,2, F. Guinea^2,3 and P. San-Jose^1 December 30, 2023 ==================================================================================§ INTRODUCTION In this paper we try to study the influence of forcing axioms on the non-stationarty ideal over ℵ_1 (denoted by NS_ω_1) and its dual, the club filter 𝒟_ω_1. In the first section we introduce a general reflection theorem from which we derive a saturation property of NS_ω_1. In the second section we deal with Galvin's property with respect to 𝒟_ω_1.Saturation. An ideal 𝒥 over κ is (μ,λ,θ)-saturated iff for every collection {A_α:α<μ}⊆𝒥^+ there is a sub-collection {A_α_ε:ε∈λ} such that ⋂{A_α_ε:ε∈ y}∈𝒥^+ for every y∈[λ]^θ. An ideal ℐ over a successor cardinal κ^+ is Laver iff ℐ is uniform (i.e., contains all the bounded subsets of κ^+), κ^+-complete and (κ^++,κ^++,κ)-saturated. If we require only (κ^++,κ^+,κ)-saturation then ℐ is called weakly Laver. If we have just (κ^++,κ^+,<κ)-saturation then ℐ is almost weakly Laver.Laver ideals can be forced over ℵ_1 by collapsing sufficiently large cardinals. In the original work of Laver, <cit.>, a huge cardinal was used. Other concepts of saturation can be forced with the same basic idea, i.e. to collapse a large cardinal and preserve some of its qualities. An interesting question is what happens under classical forcing axioms. One expects ℵ_1 to behave like a large cardinal under these axioms.The simplest notion of saturation for an ideal ℐ over κ^+ says that if {A_α:α<κ^++}⊆ℐ^+ then there are α<β<κ^++ so that A_α∩ A_β∈ℐ^+. Such an ideal is called Kunen, and the following theorem from <cit.> embodies the above expectation:Under MM, the ideal NS_ω_1 is Kunen. thmkunenSo strong forcing axioms yield saturation properties at ℵ_1. However, the situation is more involved. Laverness, for example, is excluded from ℵ_1 under any classical forcing axiom. The theorem below is typical, its first assertion is due to Larson, <cit.>, and its second part appears in <cit.>.Let 𝒥 be a uniform ideal. (ℵ) If MA + 2^ω=ω_2 and 𝒥 is an ideal over ℵ_1 then 𝒥 is not Laver.(ℶ) If BA + 2^ω_1=ω_3 and 𝒥 is an ideal over ℵ_2 then 𝒥 is not Laver.thmmabaA natural question, raised by Larson, is about the intermediate concept of weak Laverness at ℵ_1 under some forcing axiom. Larson asked, in particular, about NS_ω_1. In the first section we obtain the following result:If MM holds then NS_ω_1 is almost weakly Laver. thmawlaverWe indicate that this result follows in a different way from <cit.>, Lemma 3.10. In the second section we address the cardinal characteristic 𝔤𝔭, related to 𝒟_ω_1. We also deal with Galvin's property under the proper forcing axiom. Regarding this property we prove the consistency of PFA with the strong negation of Galvin's property. We believe that this strong negation holds in any model of PFA.We shall use the standard notation. If ℐ is an ideal over κ^+ then ℐ will be uniform and κ^+-complete unless otherwise stated. The set 𝒫(κ^+)∖ℐ is denoted by ℐ^+ (the ℐ-positive sets). We employ the Jerusalem forcing notation, i.e. p≤ q reads p is weaker than q.If g:κ→ S and β∈κ then g”β = {g(α):α<β}. The set 𝒫_κλ is the collection of subsets of λ whose cardinality is less than κ. Let M,N be models of (enough) ZFC. We say that N is a κ-end-extension of M iff for every ζ∈ N∩κ either ζ∈ M or ζ≥sup(M∩κ). This concept will be used below with respect to κ=ℵ_2.Let A be a set and let F:[A]^<ω→𝒫_κ(A) be a function. An element x of 𝒫_κ(A) will be called a closure point of F iff F(e)⊆ x for every e∈[x]^<ω. The set of closure points of F will be denoted by cℓ_F. It is straightforward that this set is a club in 𝒫_κ(A). Moreover, every club C of 𝒫_κ(A) can be coded by a function F in the sense that there exists a function F:[A]^<ω→𝒫_κ(A) such that cℓ_F⊆ C.A central concept from <cit.> is ℵ_1-reflection. Suppose that S⊆[ℋ(θ)]^ω. We shall say that S reflects to a set of size ℵ_1 iff there exists x⊆ℋ(θ), |x|=ℵ_1, ω_1⊆ x such that S∩[x]^ω is stationary in [x]^ω. The following is labeled in <cit.> as Theorem 13:MM and ℵ_1-reflection. Assume MM and let θ be a regular uncountable cardinal. Every stationary set S⊆ [ℋ(θ)]^ω reflects to a set of size ℵ_1. thmreflection§ ALMOST WEAK LAVERNESS Our first statement is a strengthening of typical properties which can be proved under MM. It says, roughly, that we have a good control on M∩ω_1 for countable elementary submodels of ℋ(θ). We shall need, however, a simple lemma ahead of the main claim.Let ℚ be an ℵ_2-cc forcing notion, and let G⊆ℚ be generic. Let {q_ε:ε∈ω_2} be any collection of conditions in ℚ. There exists a condition p∈ℚ which forces ℵ_2-many members from this collection into the generic set G. Proof. Fix a collection {q_ε:ε∈ω_2}⊆ℚ. We shall prove that some q_ε forces that |{ζ:q_ζ∈G}|=ℵ_2. Toward contradiction assume that there is no such condition. Hence for every ε∈ω_2 there are p_ε∈ℚ and an ordinal α_ε such that q_ε≤ p_ε and p_ε⊩α_ε is the first ordinal such that q_ζ∉G for every ζ∈[α_ε,ω_2). Let δ = ⋃_ε∈ω_2α_ε.Case 1: δ=ω_2.In this case there is an increasing sequence ⟨ε_ζ: ζ∈ω_2⟩ of ordinals in ω_2 such that ⟨α_ε_ζ:ζ∈ω_2⟩ is strictly increasing. We claim that the corresponding set {p_α_ε_ζ : ζ∈ω_2} is an antichain in ℚ, contradicting the ℵ_2-chain condition. For proving this fact assume that η<ζ<ω_2. So p_α_ε_η⊩ q_ζ∉G for every ζ∈[α_ε_η,ω_2) while p_α_ε_ζ⊩∃τ∈ [α_ε_η,ω_2), q_τ∈G. It means that p_α_ε_η⊥ p_α_ε_ζ, so we are done.Case 2: δ<ω_2.In this case there is an ordinal α<ω_2 such that |{ε∈ω_2: α_ε=α}|=ℵ_2. Choose any ζ∈ω_2 such that α<ζ and α_ζ=α. On the one hand p_ζ⊩ q_ζ∈G since q_ζ≤ p_ζ. On the other hand, p_ζ forces that q_τ∉G whenever τ∈[α_ζ,ω_2)=[α,ω_2). In particular, p_ζ⊩ q_ζ∉G, a contradiction.lemchainThe main claim, to be proved below, says that one can find nice extensions of countable elementary submodels. We shall prove it in two steps, first relativized to some club subset and then in a general form. Ahead of the claims (and proofs) we define the properties that we shall need.Nice extensions. Assume that M is countable and M≺ℋ(θ) for some regular cardinal θ. Assume further that {A_β:β∈ω_2}⊆ NS^+_ω_1. An elementary submodel N≺ℋ(θ) will be called a nice extension of M iff: (ℵ) M⊆ N.(ℶ) N is countable.(ℷ) N is an ω_2-end-extension of M.(ℸ) ∃β∈ N∖ M such that M∩ω_1 = N∩ω_1∈ A_β.The third requirement, namely N is an ω_2-end-extension of M, can be verified while concentrating on ordinals of ω_1. This is the content of the following:Suppose that θ=(θ) and M,N are countable structures which satisfy M≺ N≺ℋ(θ). If M∩ω_1=N∩ω_1 then N is an ω_2-end-extension of M. Proof. Assume that ξ∈ N∩ω_2 and ζ∈ M∩ω_2 for some ζ>ξ. We need showing that ξ∈ M. If ξ∈ω_1 then ξ∈ M as we are assuming that M∩ω_1 = N∩ω_1. So we may assume that ω_1≤ξ, and hence ζ>ω_1.Now ζ∈ω_2∩ M and ζ>ω_1 so there is a bijection h:ζ→ω_1, h∈ M. Since ξ<ζ we see that h(ξ)∈ω_1. Since ξ∈ N and h∈ M⊆ N we see that h(ξ)∈ N. Therefore, h(ξ)∈ N∩ω_1 = M∩ω_1, hence h(ξ)∈ M as well. By elementarity h^-1∈ M and hence ξ=h^-1(h(ξ))∈ M, so we are done.lemextensionWe are ready now for the first assertion:Assume MM. Suppose that {A_γ: γ∈ω_2}⊆ NS^+_ω_1, A∈ NS^+_ω_1 and every stationary subset of A intersects ω_2-many A_γ-s positively. Then for every sufficiently large regular cardinal θ there is a club C⊆𝒫_ω_1(ℋ(θ)) which consists of countable elementary submodels of ℋ(θ) such that for every M∈ C if M∩ω_1∈ A then there exists a nice extension N of M. Proof. Toward contradiction assume that such a club C does not exist. Fix a stationary subset S of 𝒫_ω_1(ℋ(θ)) such that every M∈ S is a countable elementary submodel of ℋ(θ) satisfying M∩ω_1∈ A and there is no nice extension of M.By the reflection principle of Theorem <ref> there is some I⊆ℋ(θ), |I|=ℵ_1, ω_1⊆ I such that S∩𝒫_ω_1(I) is stationary in 𝒫_ω_1(I). Fix a bijection g:ω_1→ I. Define T = {β∈ω_1: g”β∈ S}. We claim that T is a stationary subset of ω_1, and moreover T∩ A is stationary. For suppose not, and choose a club D⊆ω_1 such that D∩(T∩ A)=∅. Without loss of generality if α∈ D then ω_1∩ g”α=α, as we can cut D with the club of ordinals α for which ω_1∩ g”α=α.Let E = {M∈𝒫_ω_1(I): M∩ω_1∈ D, g”(M∩ω_1)⊆ M ∧ g^-1[M]⊆ M}. Notice that E is a club in 𝒫_ω_1(I). Likewise, M=g”(M∩ω_1) whenever M∈ E. For this, if x∈ M then x=g(g^-1(x)). But g^-1(x)∈ M∩ω_1 since M is closed under g^-1, so g(g^-1(x))∈ g”(M∩ω_1). It means that M⊆ g”(M∩ω_1), and g”(M∩ω_1)⊆ M follows from the fact that M is closed under g.Pick any element M∈ E∩ S. On the one hand M∩ω_1∈ D since M∈ E. On the other hand M∩ω_1∈ T∩ A since M∈ S and M=g”(M∩ω_1). In other words, the ordinal M∩ω_1 belongs to D∩(T∩ A), a contradiction.Define η=sup(I∩ω_2). Let D' = {β∈ω_1: ω_1∩ g”β = β}, and let B' = (T∩ A)∩ D'. The set B'⊆ A is stationary, and by the properties of A there is an ordinal γ∈ω_2 such that η<γ and B = B'∩ A_γ is stationary in ω_1. Choose N≺ℋ(θ), |N|=ℵ_0 such that g,γ∈ N and β=N∩ω_1∈ B. Set M_0=g”β. Notice that M_0∈ S since β∈ B⊆ T. We claim that N is a nice extension of M_0, which is an absurd as M_0∈ S.First observe that M_0⊆ N since g∈ N and β⊆ N. Second, N∩ω_1=β=M_0∩ω_1 since β∈ D', so Lemma <ref> ensures that N is an ω_2-end-extension of M_0. Finally, γ∈ N by the choice of N but γ∉ M_0. Indeed, M_0=g”β⊆ I so if γ∈ M_0 then γ=g(α) for some α<β. It means that γ∈ I (as g:ω_1→ I) and γ∈ω_2, which is impossible since γ>η=sup(I∩ω_2). Summing up, N is a nice extension of the element M_0∈ S, a contradiction.clmclubWe proceed to the second step, which is basically the same statement but without applying to the club C⊆𝒫_ω_1(ℋ(θ)).Nice extensions. Assume MM and Suppose that {A_γ: γ∈ω_2}⊆ NS^+_ω_1, A∈ NS^+_ω_1, and for every S∈ NS^+_ω_1, S⊆ A there are ω_2-many ordinals β for which S∩ A_β∈ NS^+_ω_1. Then for every large enough regular cardinal χ and every countable M≺ℋ(χ) so that M∩ω_1∈ A one can find a nice extension N≺ℋ(χ). Proof. We shall assume that ℋ(χ) is augmented by a fixed well-ordering and any required functions or predicates, e.g. definable Skolem functions and the sequence {A_γ: γ∈ω_2} as well as the set A as predicates. Let θ=(θ) be large enough to satisfy Claim <ref>. We may assume that every function from the finite subsets of ω_2 into ω_2 belongs to ℋ(θ), upon noticing that this requires θ≥ω_3. Choose χ=(χ)>θ so that ℋ(θ)∈ℋ(χ). The latter is needed in order to guarantee that the club C from Claim <ref> belongs to ℋ(χ).Assume that M≺ℋ(χ), |M|=ℵ_0 and M∩ω_1∈ A. It follows that M∩ℋ(θ)∈𝒫_ω_1(ℋ(θ)), and we may assume that M∩ℋ(θ)∈ C. Indeed, we can assume that the club C is coded by a function F which appears as a predicate in the augmented structure of ℋ(χ). Now F is a function from ℋ(θ)^<ω into ℋ(θ) and M is an elementary submodel of ℋ(χ) with this predicate, so together M∩ℋ(θ) is closed under F and hence belongs to C.By virtue of Claim <ref> there is a nice extension N_0 of M∩ℋ(θ), and in particular N_0≺ℋ(θ). We wish to extend N_0 to an elementary submodel of ℋ(χ) which will be a nice extension of M.Define N= Sk^ℋ(χ)(M∪(N_0∩ω_2)). Notice that N≺ℋ(χ), and we shall prove that N is as desired. Being the Skolem hull of a countable set, N is virtually countable. Clearly, M⊆ N. Likewise, there exists an ordinal β∈ N_0∖(M∩ℋ(θ)) such that (M∩ℋ(θ))∩ω_1 = N_0∩ω_1∈ A_β. Since θ is large enough, (M∩ℋ(θ))∩ω_1 = M∩ω_1, and β∈ N_0∩ω_2⊆ N. So if we prove that N_0∩ω_2 = N∩ω_2 we will conclude that N_0∩ω_1 = N∩ω_1∈ A_β. Moreover, N_0∩ω_2 = N∩ω_2 accomplishes all the requirements listed in the definition of a nice extension.Let us prove, therefore, that N_0∩ω_2 = N∩ω_2. Clearly, N_0∩ω_2 ⊆ N∩ω_2, so assume that ξ∈ N∩ω_2 with the goal of showing that ξ∈ N_0 as well. By the definition of N there exists a Skolem function f, a finite set {a_1,…,a_k}⊆ M and a finite set of ordinals {γ_1,…,γ_ℓ}⊆ N_0∩ω_2 such that:ξ = f(a_1,…,a_k,γ_1,…,γ_ℓ).We may assume that Rang(f)⊆ω_2 (if not, replace f by g which agrees with f whenever f(e)∈ω_2 and assumes zero otherwise). Define h:[ω_2]^ℓ→ω_2 by h(t) = f(a_1,…,a_k,t).We claim that h∈ M. Indeed, M≺ℋ(χ) and ℋ(χ) contains a complete set of Skolem functions, including the specific function f. The function h is definable from f and {a_1…,a_k}⊆ M hence belongs to M. Likewise, assuming that θ is sufficiently large we may assume that all the functions from [ω_2]^ℓ into ω_2 are in ℋ(θ), in particular h∈ℋ(θ) and hence h∈ M∩ℋ(θ)⊆ N_0. Since {γ_1,…,γ_ℓ}⊆ N_0 we conclude that ξ = f(a_1,…,a_k,γ_1,…,γ_ℓ) = h(γ_1,…,γ_ℓ)∈ N_0, and the proof is accomplished.mclmThe ability to produce nice extensions of countable elementary submodels resembles similar theorems which hold at measurable cardinals. It means that ℵ_2 behaves like a measurable cardinal under MM. This is coherent with other statements which show that small accessible cardinals possess large cardinal properties under strong forcing axioms. rmeasurableAs indicated in the introduction, the fact that NS_ω_1 is almost weakly Laver under MM follows from Lemma 3.10 of <cit.>. It can also be derived from the main claim above:The ideal NS_ω_1 is (ℵ_2,ℵ_1,<ℵ_0)-saturated under MM. Proof. Suppose that {A_α:α∈ω_2}⊆ NS^+_ω_1. Let A⊆ω_1 be as guaranteed by Lemma <ref>. By induction on γ∈ω_1 we choose (for sufficiently large χ) a model N_γ≺ℋ(χ) as follows:γ=0: Let N_0 be a countable elementary submodel of ℋ(χ) so that N_0∩ω_1∈ A. We also assume that {A_α:α∈ω_2}∈ N_0.γ+1: Denote N_γ by M, and choose N as described in Claim <ref> with respect to M. Let α_γ∈ N∖ M be the ordinal given by the definition of nice extensions. Set N_γ+1 = N.γ is limit: Let N_γ = ⋃_δ<γN_δ.Denote the union ⋃_γ∈ω_1N_γ by N. Notice that {A_α_γ:γ∈ω_1}⊆ N. We claim that this collection exemplifies the almost weak Laverness of NS_ω_1. For this, let u be a finite subset of ω_1 and S = ⋂_γ∈ uA_α_γ. Notice that S∈ N as u is finite.Assume toward contradiction that S∈ NS_ω_1, and let C be a club subset of ω_1 so that C∈ N and N C∩ S=∅. Let τ be the characteristic ordinal N_0∩ω_1. By the nature of the N_γ-s we see that N_γ∩ω_1=τ for every γ∈ω_1 and hence N∩ω_1=τ.On the one hand, τ∈ C since ℋ(χ) believes that C is unbounded below τ and C is closed. On the other hand, τ∈ A_α_γ for each γ∈ u since N_γ∩ω_1 = N∩ω_1 = τ for every γ∈ω_1 and by the definition of nice extensions. In particular τ∈ S, a contradiction.thmmtWe make the comment that the finite number of A_α_β-s is required only in order to make sure that S = ⋂_β∈ uA_α_β∈ N. This seems to be an essential obstacle while considering the possibility of strengthening the above theorem. Indeed, no closure assumption on N can be made (apart from the closure to finite sequences which follows from elementarity).The question of weak Laverness under MM remains open, but almost weak Laverness is quite settled. It is interesting to compare MM with PFA at this point:Forcing axioms and the non-stationary ideal. (ℵ) NS_ω_1 is almost weakly Laver under MM.(ℶ) The almost weak Laverness of NS_ω_1 is independent over the PFA.Proof. The first statement is the above theorem, and PFA is consistent with NS_ω_1 being almost weakly Laver since MM implies PFA. The consistency of PFA with the failure of almost weak Laverness of NS_ω_1 and even with the failure of the simplest saturation property is due to Velickovic, <cit.> (and as noted there, this result was obtained independently by Shelah).corawlFor the main result we used MM, but actually almost weak Laverness follows from the ℵ_1-reflection principle and the fact that NS_ω_1 is Kunen. This might be meaningful when other axioms are deemed with respect to various saturation properties. § THE CLUB FILTER A celebrated theorem of Galvin which appeared in <cit.> says that under the continuum hypothesis any collection of ℵ_2 many clubs of ℵ_1 contains a sub-collection of size ℵ_1 whose intersection is a club subset of ℵ_1. Remark that the collection of end-segments of ℵ_1 is a family of ℵ_1 many clubs of ℵ_1, each sub-family of which of size ℵ_1 has an empty intersection. This means that one can gather a small number of clubs of ℵ_1 with empty intersection of each ℵ_1 of them, but (under the continuum hypothesis) one cannot collect many clubs in this way. Here, many clubs means ℵ_2 clubs of ℵ_1.Galvin's result can be generalized in several directions, and one of them is based on replacing the continuum hypothesis by Devlin-Shelah's weak diamond. It has been proved in <cit.> that if 2^ω<2^ω_1 (this is equivalent to the weak diamond at ℵ_1) then any collection of λ clubs of ℵ_1 contains a sub-collection of size ℵ_1 whose intersection is a club subset of ℵ_1 where λ=(2^ω)^+. This is quite a surprising result, since 2^ω_1 can be much larger than λ and yet Galvin's property holds here with respect to any collection of λ sets. Actually, this gives rise to defining the first cardinal which satisfies this property as a cardinal characteristic of the continuum:The cardinal characteristic 𝔤𝔭. We define 𝔤𝔭 as the minimal κ such that every family {C_α:α∈κ^+} of club subsets of ℵ_1 contains a subfamily {C_α_β:β∈ω_1} whose intersection is closed and unbounded in ℵ_1. The definition comes from <cit.> and it has been shown there that 𝔤𝔭 is well-defined and assumes a value between ℵ_1 and 2^ℵ_0. The behavior of 𝔤𝔭 when related to other cardinal characteristics is very peculiar. It is consistent that 𝔤𝔭<𝔪, but it is also consistent that 𝔤𝔭>𝔦. Dealing with 𝔤𝔭 by itself, it has been asked in <cit.> what are the possible cofinalities of 𝔤𝔭 and, in particular, is it consistent that (𝔤𝔭)=ω. In this paper we try to deal with this question. Like many other cardinal characteristics, the definition of 𝔤𝔭 generalizes to higher cardinals. This is central in our context, since our knowledge on (𝔤𝔭_κ) for κ=(κ)≥ℵ_3 is a bit better.The cardinal characteristic 𝔤𝔭_κ. Let κ=(κ). We define 𝔤𝔭_κ as the minimal λ such that every family {C_α:α∈λ^+} of club subsets of κ^+ contains a subfamily {C_α_β:β∈κ^+} whose intersection is closed and unbounded in κ^+. A straightforward modification of the statements in <cit.> shows that 𝔤𝔭_κ is well-defined and that κ^+≤𝔤𝔭_κ≤ 2^κ. For basic background regarding pcf theory we suggest <cit.>, and for further information we refer to <cit.>.We commence with showing that under some pcf assumptions one can prove that (𝔤𝔭)>ω. The assumption is a consequence of the existence of a good scale, which in turn can be proved to exist from instances of weak square. However, this assumption cannot be established in ZFC, and actually it fails if one assumes certain instances of Chang's conjecture. We shall prove our first theorem based on a pcf assumption, and later we shall connect this assumption to known combinatorial concepts like weak squares and Chang's conjecture.Let μ be a singular cardinal. (A) Suppose that: (a) ω=(μ)<μ<2^ω and (μ_n:n∈ω) is an increasing sequence of regular cardinals such that μ=⋃_n∈ωμ_n.(b) J⊇ J^ bd_ω and tcf(∏_n∈ωμ_n,J)=μ^+, as witenessed by the scale (f_α:α∈μ^+).(c) For every A⊆μ^+ such that |A|=ℵ_1 it is true that |{f_α(n):α∈ A,n∈ω}|=ℵ_1.Then 𝔤𝔭≠μ.(B) Suppose that: (a) ω_1=(μ)<μ≤ 2^ω and (μ_γ:γ∈ω_1) is an increasing sequence of regular cardinals such that μ=⋃_γ∈ω_1μ_γ.(b) J⊇ J^ bd_ω_1 and tcf(∏_γ∈ω_1μ_γ,J)=μ^+, as witenessed by the scale (g_α:α∈μ^+).(c) For every A⊆μ^+ such that |A|=ℵ_1 there is δ∈ω_1 such that |{g_α(γ):α∈ A,γ∈δ}|=ℵ_1.Then 𝔤𝔭≠μ.Proof. For the first part, assume toward contradiction that 𝔤𝔭=μ. Fix a collection 𝒟={D_α:α∈μ} such that ⋂{D_α_β:β∈ B} is bounded in ω_1 for every B∈[μ]^ℵ_1. For every α∈μ^+ let C_α=⋂{D_f_α(n):n∈ω}. Each C_α is a club subset of ℵ_1, being the intersection of but ℵ_0-many clubs of ℵ_1. Set 𝒞={C_α:α∈μ^+}.Since 𝔤𝔭=μ, one can choose A⊆μ^+,|A|=ℵ_1 such that C=⋂{C_α:α∈ A} is a club subset of ℵ_1. By our assumptions, |{f_α(n):α∈ A, n∈ω}|=ℵ_1 and hence {D_f_α(n):α∈ A,n∈ω}⊆ [𝒟]^ℵ_1. By the nature of the collection 𝒟, the set D=⋂{D_f_α(n):α∈ A, n∈ω} is bounded in ω_1. However, D=C, a contradiction.For the second part, assume toward contradiction that 𝔤𝔭=μ and fix a family of clubs 𝒟={D_α:α∈μ} which exemplifies this fact. For every α∈μ^+ we define C_α = Δ{D_g_α(γ):γ∈ω_1}, so every C_α is a club subset of ℵ_1. Let 𝒞={C_α:α∈μ^+}. Choose a set A⊆μ^+,|A|=ℵ_1 such that ⋂{C_α:α∈ A} is a club of ℵ_1. Fix an ordinal δ∈ω_1 such that |{g_α(γ):α∈ A, γ<δ}|=ℵ_1. Define C=⋂{C_α:α∈ A} - (δ+1), so C is still a club of ℵ_1. Let D=⋂{D_g_α(γ):α∈ A,γ<δ}, so D is bounded in ω_1. But if ζ∈ C then for every α∈ A and every γ<δ we see that ζ∈ D_g_α(γ), since ζ>δ>γ. This means that C⊆ D, a contradiction.thmcofgpFor placing the above theorem in a familiar combinatorial land we must analyze the conditions under which the assumption on A can be made. Let (f_α:α<δ) be a scale with respect to J=J^ bd_ω (this is our typical case). We shall say that γ is a good point if there is an unbounded u⊆γ and n∈ω such that ⟨ f_α(m):α∈ u⟩ is strictly increasing for every m≥ n. A scale (f_α:α<δ) is good iff every γ∈δ of uncountable cofinality is a good point.Assume that ω=(μ)<μ<2^ω, and there exists a good scale (f_α:α∈μ^+) with respect to some product of regular cardinals and J=J^ bd_ω. Then 𝔤𝔭≠μ. Proof. For every β∈μ^+ such that (β)>ω let u_β be an unbounded subset of β as guaranteed by the assumption that (f_α:α∈μ^+) is a good scale. Assume that A⊆μ^+,|A|=ℵ_1 and β=sup(A). Without loss of generality, (β)=ω_1. By shrinking A and u_β if needed we may assume that the elements of A and u_β are cofinally interleaved.For every α∈ A choose δ^α_0,δ^α_1∈ u_β such that δ^α_0<α and δ^α_1 is the first element of u_β such that α<δ^α_1. Choose n=n(α)∈ω such that f_δ^α_0(m)<f_α(m) <f_δ^α_1(m) for every m∈[n,ω). By shrinking A once more we may assume that for every α∈ A we have n(α)=n_0 for some fixed n_0∈ω.Since (f_α:α∈μ^+) is a good scale, and (β)>ω, there exists n_1∈ω such that ⟨ f_δ(m):δ∈ u_β⟩ is strictly increasing for every m∈[n_1,ω). Set n=max{n_0,n_1}. It follows that |{f_α(ℓ):α∈ A, ℓ∈[n,ω)}|=ℵ_1, as required.clmvgsThe above claim enables us to derive conclusions about the cofinality of 𝔤𝔭 from a weak form of the square principle. It is known that □^*_μ implies the existence of a good scale (f_α:α∈μ^+). Actually, it gives a bit more. It is shown in <cit.> that a better scale can be obtained from the weak square.Assume □^*_μ for every μ>(μ)=ω below 2^ω. Then (𝔤𝔭)>ω. corweaksThe above conclusions are stated with respect to the possibility that (𝔤𝔭)=ω, but the real point here seems to be different. If μ>(μ)=ω and pp(μ) is large then mild assumptions rule out the possibility of 𝔤𝔭=ν for many cardinals ν∈[μ, pp(μ)). The following is a typical statement:Assume that: (a) ω=(μ)<μ<2^ω.(b) (μ_n:n∈ω) is a sequence of regular cardinals, μ=⋃_n∈ωμ_n and J⊇ J^ bd_ω.(c) λ= tcf(∏_n∈ωμ_n,J) and (f_α:α∈λ) is a scale which witnesses this fact.(d) μ≤ν<λ.(e) For every A∈[λ]^ℵ_1 it is true that |{f_α(n):α∈ A, n∈ω}|=ℵ_1.Then 𝔤𝔭≠ν. Proof. Follow the proof of Theorem <ref>, upon replacing μ by ν and μ^+ by λ.thmppIt follows that the consistency of (𝔤𝔭)=ω requires strong assumptions. We turn to 𝔤𝔭_κ when κ=(κ)>ℵ_0. If κ>ℵ_0 then theoretically it may occur that (𝔤𝔭_κ)<κ. Weak assumptions about good scales imply (𝔤𝔭_κ)>κ by a straightforward generalization of the previous statements. The purpose of the following theorem is to show, in ZFC, that (𝔤𝔭_κ) cannot drop down dramatically without any special assumption.Martin's maximum and the cofinality of 𝔤𝔭_κ. Assume that κ=θ^+3. Then (𝔤𝔭_κ)>θ, and moreover if κ<μ and (μ)≤θ then 𝔤𝔭_κ∉[μ, pp(μ)). Similarly, Martin's maximum implies that (𝔤𝔭_θ^+)>θ for every infinite cardinal θ. Proof. Fix any μ∈(κ,2^κ) such that (μ)≤θ. For notational simplicity we assume that (μ)=θ, the proof of the case in which (μ)<θ is identical. Let λ∈[μ, pp(μ)) be any cardinal, and assume toward contradiction that 𝔤𝔭_κ=λ. Notice that λ^+≤ pp(μ) and hence there is a set 𝔞 = {λ_i:i∈θ}⊆ Reg∩μ such that sup(𝔞)=μ and tcf(∏𝔞,J)=λ^+ for some J⊇ J^ bd_κ.Choose an increasing cofinal sequence f̅ = (f_α:α∈λ^+) which exemplifies this fact and satisfies S^ gd_θ(f̅) =_𝒟_λ S^λ_θ. Such a sequence of functions exists by results of Shelah, see e.g. <cit.>, footnote 5. By shrinking f̅ if needed we may assume that if A⊆λ^+, |A|=κ^+ then |{f_α(i):α∈ A, i∈θ}|=κ^+. Now the argument of the proof of Theorem <ref> leads to a contradiction. The additional part under Martin's maximum follows from <cit.>.thmgpkappaChang's conjecture (μ^+,μ)↠(ℵ_1,ℵ_0) implies the existence of sets A⊆μ^+, |A|=ℵ_1 such that |{f_α(n): α∈ A,n∈ω}|=ℵ_0 for some scale (f_α:α∈μ^+) in the relevant context. But this fact by itself doesn't imply immediately (𝔤𝔭)=ω. The point is that even under strong instances of Chang's conjecture we might be able to find a set A of size ℵ_1 for which |{f_α(n): α∈ A,n∈ω}|=ℵ_1. If we can make sure that such A gives rise to a collection of ℵ_1-many clubs with unbounded intersection in ω_1 then (𝔤𝔭)>ω. One way to guarantee the above property is captured in the following definition:Strong Galvin's property. A collection {C_α:α∈λ} of clubs of κ=(κ)>ℵ_0 exemplifies the strong Galvin's property iff there exists a sequence ⟨ H_ε:ε<κ⟩ so that H_ε∈[λ]^λ for every ε<κ, and for every h∈∏_ε<κH_ε the set ⋂{C_h(ε):ε<κ} is a club subset of κ. The proof of Galvin from <cit.> as well as the proof under the weak diamond in <cit.> give the strong property for λ=(2^ω)^+ and every family of λ clubs in ω_1. Let us prove that this strong property implies (𝔤𝔭)>ω.Assume that 𝔤𝔭_κ=μ. If every family {C_α:α∈μ^+} of clubs of κ^+ has the strong Galvin's property then (μ)>κ. Proof. Assume toward contradiction that θ=(μ)≤κ. Choose a collection 𝒟={D_α:α<μ} which exemplifies 𝔤𝔭_κ=μ. Fix a scale (f_α:α∈μ^+) in (∏_η∈θμ_η,J^ bd_θ). By chopping an initial segment and thinning out the sequence we can assume that κ<μ_0 and ζ<η⇒μ_ζ<μ_η. Denote the ideal J^ bd_θ by J.For every α∈μ^+ let C_α = ⋂{D_f_α(η):η∈θ}, and let 𝒞 = {C_α:α∈μ^+}. By our assumptions there is a sequence of sets ⟨ H_ε:ε<κ^+⟩, each H_ε belongs to [μ^+]^μ^+ and for every h∈∏_ε<κ^+H_ε the set ⋂{C_h(ε):ε<κ^+} is a club of κ^+.We define a set A⊆μ^+,|A|=κ^+ as follows. By induction on ε<κ^+ we choose an ordinal α_ε∈ H_ε such that sup{f_α_ζ: ζ<ε}<_J f_α_ε. Since κ<μ_η for every η, the function sup{f_α_ζ: ζ<ε} is an element of the product. The existence of f_α_ε follows from the fact that |H_ε|=μ^+ and hence {f_α:α∈ H_ε} is cofinal in the product (∏_η∈θμ_η,J).Let A={α_ε:ε∈κ^+}. By the choice of each α_ε we see that |{f_α(η): α∈ A, η∈θ}|=κ^+. As before, let C=⋂{C_α:α∈ A}, D=⋂{D_f_α(η): α∈ A, η∈θ}, so C=D. However, D is bounded in κ^+ while C is a club of κ^+, a contradiction.clmsgpThe above claim invites an additional parameter within the definition of 𝔤𝔭. For κ=(κ)>ℵ_0 and μ≥κ we shall say that a family {C_α:α∈λ} of clubs of κ has μ-Galvin's property iff there is a sequence ⟨ H_ε:ε<κ⟩, H_ε∈[λ]^μ for every ε<κ, such that for every h∈∏_ε<κH_ε the set ⋂{C_h(ε): ε<κ} is a club subset of κ.𝔤𝔭_κ with a parameter. Let κ be a regular cardinal, μ∈[κ^+,(2^κ)^+]. The characteristic 𝔤𝔭_κ with parameter μ is the minimal cardinal λ such that every collection {C_α:α∈λ^+} of clubs of κ^+ has μ-Galvin's property. If μ=(2^κ)^+ then we call it the strong 𝔤𝔭_κ and denote it by 𝔰𝔤𝔭_κ. In this light, Claim <ref> says that the cofinality of 𝔰𝔤𝔭_κ is always larger than κ. Therefore, if we can force (𝔤𝔭_κ)=κ then we can distinguish 𝔰𝔤𝔭_κ from 𝔤𝔭_κ. This can be phrased in a general manner:Is it consistent that 𝔤𝔭 is strictly smaller than 𝔰𝔤𝔭? More generally, for κ=(κ) and κ^+≤μ_0<μ_1≤(2^κ)^+, is it consistent that 𝔤𝔭_κ with parameter μ_0 is strictly smaller than 𝔤𝔭_κ with parameter μ_1? In closing this paper we address another problem from <cit.>, about the connection between the proper forcing axiom and Galvin's property. Here we refer to the original property proved by Galvin under the continuum hypothesis, namely that every family of ℵ_2 many clubs of ℵ_1 admits a sub-family of size ℵ_1 whose intersection is a club of ℵ_1. We shall prove that consistently this property fails under PFA. For this end, we shall use the usual way to force PFA from a supercompact cardinal. It is plausible that the negation of Galvin's property follows from the PFA no matter how it is forced.PFA and Galvin's property. It is consistent that PFA holds and there is a family {B_α:α∈ω_2} of clubs of ω_1 for which any subfamily of size ℵ_1 has finite intersection. Proof. We denote the ground model ahead of forcing the proper forcing axiom by V. Let κ be supercompact in V, and force PFA in the usual way described first by Baumgartner. In the generic extension, κ becomes ℵ_2 and the PFA holds. Let 𝔹 be Baumgartner's forcing from <cit.> to add a club of ℵ_1 with finite conditions as defined in V. Let f be the Laver function employed in the iteration which forces PFA. Denote the iteration which forces PFA by ⟨ℙ_α,ℚ_β:α≤κ,β<κ⟩, and fix a generic subset G⊆ℙ_κ.Let A={ζ∈κ: ⊩_ℙ_ζ f(ζ)=𝔹̌}. If ζ∈ A then ℚ_ζ adds a club of ℵ_1, denoted by 𝔹_ζ. We claim that the sequence ⟨𝔹_ζ:ζ∈ A⟩ exemplifies the negation of Galvin's property in the generic extension by ℙ_κ. Remark that in the generic extension this is a set of size ℵ_2, so this will accomplish the proof.Assume, therefore, that X is a name of an element of [A]^ℵ_1. Let τ = ⋂_ζ∈X𝔹_ζ. Assume toward contradiction that τ is forced to be uncountable. Fix an element p of ℙ_κ, with the goal to extend p to a condition which forces the opposite statement. Since p is arbitrary, we will be done. For a sufficiently large regular χ we fix M≺ℋ(χ) such that p,X,ℙ_κ,τ∈ M and M is countable. By properness, let q be an (M,ℙ_κ)-generic condition such that p≤ q. Define:σ = {(r,γ)∈ M: r⊩γ̌∈τ}.The focal point is that q⊩σ=τ∩ M. We prove the inclusion of τ∩ M in σ, the opposite inclusion being proved exactly in the same manner. Assume, therefore, that γ∈ M. For the current direction we assume that there is a condition s≥ q such that s⊩γ̌∈τ. Since q is (M,ℙ_κ)-generic and the set of conditions which decide the statement γ̌∈τ is dense, one can find t∈ M such that t∥ s ∧ t⊩γ̌∈τ. By the definition of σ we see that t⊩γ̌∈σ and hence some extension of s forces this statement as well, giving the desired inclusion. The opposite inclusion is proved with respect to the opposite statement, namely s⊩γ̌∉τ. Remark that q⊩σ=τ∩ M along with the assumption toward contradiction imply that σ is forced by q to be an infinite set.Let η=sup(X). We may assume that this fact is forced by q. By cutting off the upper part of X if necessary, we may assume that ^V(η)>ω. Let δ=sup(M∩η). Recall that M is countable, and hence δ<η. Choose a condition q^+≥ q such that q^+⊩γ̌∈X∧γ∈[δ,η). We shall argue that q^+⊩(σ⊆𝔹_γ∩ M), thus arriving at a contradiction.Our strategy is to prove the above statement in a larger universe, and then using absoluteness in order to show that this holds already in the generic extension by ℙ_κ. Let 𝕋 be the termspace forcing of ℙ_[η,κ) with respect to ℙ_[δ,η) as computed in the universe obtained by the generic extension with ℙ_δ. Let ℝ = ℙ_δ∗(ℙ_[δ,η)×𝕋). Observe that ℝ has a natural projection onto ℙ_κ, namely π(p,q,r)=p^⌢ q^⌢ r. In particular, there are suitable conditions (a,b,c) such that π(a,b,c)=q^+. Choose a generic set H for the forcing notion ℙ_δ∗𝕋, such that (a,c)∈ H. Notice that the name σ is interpreted by H since there are no coordinates of σ in the interval of ℙ_[δ,η) by the definition of σ.We claim that in V[H] it is true that ∅_ℙ_[δ,η)⊩(σ⊆B_γ∩ M). Remark that maybe ℵ_1 is collapsed in V[H] (though not in V[G]), but our claim is still valid. Indeed, if p is a finite condition in the forcing which adds 𝔹_γ then we can extend p by adding a pair which will be forced to be in σ but not in 𝔹_γ∩ M. For this end, let α=sup(σ). Since p is finite, there is a maximal pair (ν,ξ) in p. Choose an ordinal ρ∈σ∩(ξ,α) such that ρ+1 will not be in σ (such an ordinal always exists) and add the pair (ν+1,ρ+1) to p. This shows that (σ⊆𝔹_γ∩ M) in the generic extension by H. Since this statement is Δ_0 we conclude that V[G](σ⊆𝔹_γ∩ M) as well, so we are done.thmpfaWe make the comment that an additional argument shows that Galvin's property fails in a strong sense in the above model, namely the intersection of any family of ℵ_1 Baumgartner clubs is finite. The important feature of the usual iteration to force PFA that we used is the fact that the set A of names for Baumgartner clubs along the iteration is unbounded in κ. amsplain | http://arxiv.org/abs/1708.08049v2 | {
"authors": [
"Shimon Garti",
"Yair Hayut",
"Haim Horowitz",
"Menachem Magidor"
],
"categories": [
"math.LO",
"03E50"
],
"primary_category": "math.LO",
"published": "20170827042622",
"title": "Martin's maximum and the non-stationary ideal"
} |
1]Wilson Ye Chen 2]Richard H. Gerlach [1]School of Mathematical and Physical Sciences, University of Technology Sydney [1]Australian Research Council Centre for Excellence in Mathematical and Statistical Frontiers [2]Discipline of Business Analytics, The University of SydneySemiparametric GARCH via Bayesian model averaging [ August 24, 2017 ================================================= As the dynamic structure of the financial markets is subject to dramatic changes, a model capable of providing consistently accurate volatility estimates must not make strong assumptions on how prices change over time. Most volatility models impose a particular parametric functional form that relates an observed price change to a volatility forecast (news impact function). We propose a new class of functional coefficient semiparametric volatility models where the news impact function is allowed to be any smooth function, and study its ability to estimate volatilities compared to the well known parametric proposals, in both a simulation study and an empirical study with real financial data. We estimate the news impact function using a Bayesian model averaging approach, implemented via a carefully developed Markov chain Monte Carlo (MCMC) sampling algorithm. Using simulations we show that our flexible semiparametric model is able to learn the shape of the news impact function from the observed data. When applied to real financial time series, our new model suggests that the news impact functions are significantly different in shapes for different asset types, but are similar for the assets of the same type.Keywords: volatility, news impact function, Markov chain Monte Carlo, functional coefficient, regression spline, heavy tail§ INTRODUCTION Extensions of the GARCH model have been actively developed in the literature for the past three decades since the seminal studies by <cit.> and <cit.>. As the normality assumption of the innovation distribution of the GARCH model is often inadequate for describing the excess kurtosis of the daily returns, one stream of studies focuses on GARCH models with flexible parametric innovation distributions. Examples include the Student's t distributions <cit.>, generalised error distribution (GED) <cit.>, skewed-t distribution <cit.>, normal inverse Gaussian (NIG) distribution <cit.>, exponential generalized beta distribution of the second kind (EGB2) <cit.>, asymmetric Laplace distribution <cit.>, and the two-sided Weibull distribution <cit.>. Another line of research focuses on developing flexible conditional volatility dynamics. Examples include the EGARCH <cit.>, GJR-GARCH <cit.>, threshold-GARCH <cit.>, NAGARCH <cit.>, smooth transition GARCH <cit.>, GARCH model with slowly-varying volatility component <cit.>, Beta-t-GARCH <cit.>, and the more general family of score driven models <cit.>.If either the innovation distribution or the conditional volatility dynamics is allowed to be of unknown form and estimated nonparametrically, the resulting model is referred to as a semiparametric GARCH model. To avoid efficiency loss or inconsistency due to misspecification, several studies propose to estimate the innovation density with a nonparametric estimator, such as the kernel density estimator. For example, see <cit.>, <cit.>, <cit.>, and <cit.> for details. Let us introduce a few notations before proceeding with the semiparametric GARCH models of the latter kind, which is the focus of this paper.Consider a discrete-time real-valued stochastic process denoted by {y_tt ∈ℤ} (The abbreviated notation {y_t} is used hereafter.), such that y_t = σ_tϵ_t, where {ϵ_t} is an i.i.d. sequence with (ϵ_t) = 0 and (ϵ_t) = 1. By construction, {y_t} is a martingale difference sequence. It follows that σ^2_t is the conditional variance of y_t; i.e., (y_tℱ_t-1) = σ^2_t, where ℱ_t-1 = σ({y_ss ≤ t-1}) denotes the smallest σ-algebra containing the past observations of the process, and represents the available information at t-1. Assume that σ^2_t is the value of a function of possibly infinitely many past observations,σ^2_t = h(y_t-1,y_t-2, … ad inf),where h(·) is referred to as the volatility function.Most extensions of the GARCH model, including those with nonparametrically estimated densities, make particular parametric assumptions about the form of the volatility function h(·), which plays a key role in the modelling of the conditional volatility process. With the hope of not having the need to choose amongst the various alternative parametric proposals, several studies focused on the nonparametric generalisations of the volatility function. Studies such as <cit.> and <cit.> assumed that h(·) is a smooth but unknown J-dimensional function for some J < ∞,σ^2_t = h(y_t-s_1 y_t-s_J),with the lags {s_1 s_J} selected a priori, and estimated h(·) nonparametrically using a J-dimensional smoother. For example, notice that (<ref>) can be written in the formy^2_t = h(y_t-s_1 y_t-s_J) + u_t,where u_t = σ^2_t(ϵ^2_t - 1) and {u_t} is a martingale difference sequence, which suggests that one can estimate h(·) using a kernel regression method by regressing y^2_t on the lagged observations y_t-s_1 y_t-s_J. In practice, the model in (<ref>) can only be estimated for a small J as it suffers from the so called “curse of dimensionality"; the optimal rate of convergence of a consistent nonparametric estimator of h(·) decreases rapidly as the dimensionality J increases; see for example <cit.> and <cit.>. As the conditional variance processes estimated from many empirical datasets are highly persistent, a large value of J is typically needed for the model in (<ref>) to adequately capture the empirically observed high persistence. In order to improve the best achievable rate of convergence, studies such as <cit.>, <cit.>, <cit.>, and <cit.> focused on additive models of the formσ^2_t = ∑_i=1^Jψ_i(θ)m(y_t-i),for some J < ∞, where the coefficient functions {ψ_1(θ) ψ_J(θ)} are assumed known up to a parameter vector θ, and the smooth but unknown univariate function m(·) is to be estimated nonparametrically. Studies such as <cit.>, <cit.>, and <cit.> considered a more general case where J = ∞ in (<ref>). Their models include the popular GARCH(1,1) model as a special case, while (<ref>) is a semiparametric ARCH(J) model. Notice that in (<ref>) if we let J = ∞, θ = β, and ψ_i(β) = β^i-1, with β∈ (0,1), then the volatility function in (<ref>) can be written in the form σ^2_t = βσ^2_t-1 + m(y_t-1). The function m(·) in (<ref>) is often referred to as the “news impact function" in the literature, as it updates the conditional volatility based on the new information contained in the observation y_t-1. Another alternative strategy for modelling the volatility function h(·) was investigated in <cit.> and <cit.> where h(·) was assumed to be a smooth but unknown bivariate functionσ^2_t = h(y_t-1, σ^2_t-1),and was estimated nonparametrically using a bivariate smoothing method.In this paper, we propose a novel semiparametric GARCH model, where the sequence of unobserved conditional variances {σ^2_t} follows a functional coefficient autoregressive (FAR) process. We model the coefficient function by a quadratic regression spline, where both the number and the positions of the knots are selected using a Bayesian model averaging (BMA) approach. A crucial step in BMA is to compute the expectations of random variables of interest with respect to the joint posterior distribution of the model parameters and the knot placements. For this purpose, we develop an adaptive Markov chain Monte Carlo (MCMC) algorithm to simulate from such joint posterior distribution. Similar to the additive model in (<ref>), the proposed approach only requires the estimation of an unknown univariate function, however, unlike (<ref>), our model nests the GARCH(1,1) model. Compared with a kernel regression method with a global bandwidth, a regression spline with strategically placed knots has the advantage of being locally adaptive <cit.>, and thus can accommodate varying degrees of smoothness across regions of the domain.The rest of the paper is structured as follows. Section <ref> presents our semiparametric GARCH model, shows how various parametric models can be written as special cases of the proposed model, and defines the spline functional coefficient. Section <ref> describes the model in a Bayesian context, and explains how the scalar parameters and the functional coefficient are estimated using a MCMC sampling algorithm. Section <ref> presents a simulation study comparing the performance of the proposed model to the popular parametric alternatives. Section <ref> demonstrates the proposed model by applying it to ten real financial time series, and presents a model comparison study based on the averaged DIC – a slight generalisation of the usual single model DIC.§ THE PROPOSED MODEL§.§ Functional coefficient semiparametric GARCH Continuing with the notations introduced in Section <ref>, let {r_tt ∈ℤ} denote a real-valued discrete-time stochastic process, and ℱ_t = σ({r_ss ≤ t}) denote the natural filtration of {r_t}. Our proposed functional coefficient semiparametric GARCH (denoted by SP-GARCH hereafter) model is defined as follows.r_t = μ + y_t, y_t = σ_tϵ_t,σ^2_t = ω + g(ϵ_t-1)σ^2_t-1,where {ϵ_t} is an i.i.d. sequence of innovations with (ϵ_t) = 0 and (ϵ_t) = 1. It follows that μ and σ^2_t are, respectively, the conditional mean and conditional variance of r_t; i.e., (r_tℱ_t-1) = μ and (r_tℱ_t-1) = σ^2_t. In (<ref>), we make the assumption that the conditional mean of r_t is constant over time. The constant mean assumption is often reasonable in empirical applications such as when {r_t} is a model for a sequence of daily returns of a financial asset <cit.>. The distribution of the innovation ϵ_t can either be chosen from a wide range of parametric families with finite variance such as those mentioned in Section <ref> or modelled nonparametrically. In this paper, we take a parametric approach and assume that ϵ_t∼ F_sdt,ν, for ν > 2, where F_sdt,ν denotes a standardised Student-t distribution with ν degrees of freedom. (A random variable z follows F_sdt,ν if it is given by z = x[(ν - 2) / ν]^1/2, where x follows a Student-t distribution with ν degrees of freedom. It follows that (z) = 0 and (z) = 1.) The typical assumption that ϵ_t follows a heavy-tailed distribution is motivated by the stylised fact that the distribution of daily returns, standardised using an estimated volatility model, is heavy-tailed and approximately symmetric for many financial assets <cit.>.The proposed model in (<ref>) admits the interpretation that the sequence of unobserved conditional variances {σ^2_t} follows an autoregressive process with a functional coefficient g(·). It can be seen that the function g(·) plays a similar role to that of the news impact function m(·) in (<ref>), as it determines how σ^2_t-1 is updated given the value of the innovation ϵ_t-1, which can be considered as news. Notice that the argument of g(·) is the lagged innovation ϵ_t-1, as opposed to the lagged centred observation y_t-1. We consider this as an advantage of the new model over the previous semiparametric proposals. For example, from a modeller's perspective, since the distribution of ϵ_t is either fully known or at least known up to its third moment (by construction), and is usually not time dependent, one can utilise such prior knowledge to improve the statistical properties of an estimator by imposing a certain structure on g(·). Furthermore, because the scale of ϵ_t is the same across different datasets, the estimates of g(·) can be directly compared in empirical applications.Notice that g(ϵ_t) is an i.i.d. random variable. Let us suppose that g(ϵ_t) is replaced by a constant γ and the process {σ^2_t} has an initial value of σ^2_0 > 0, so that {σ^2_t} becomes a fully deterministic process. For γ∈ (0,1), we can see that γ controls the rate at which σ^2_t approaches its lower bound ω / (1 - γ) as t →∞. Since g(ϵ_t) is a random variable, the quantity [g(ϵ_t)] controls the average rate of decay of {σ^2_t}, and is referred to as the persistence of the conditional variance process. The process {r_t} defined in (<ref>) is covariance stationary if[g(ϵ_t)] ∈ [0, 1).If {r_t} is covariance stationary, its unconditional variance is given by(r_t) = (σ_t^2) = ω/1 - [g(ϵ_t)]. The GARCH(1,1) model and many of its popular parametric extensions can be written as special cases of the new SP-GARCH model in (<ref>) with particular forms of the coefficient function g(·). For example, Table <ref> lists the forms of the coefficient function when the first order GARCH, GJR-GARCH, NAGARCH, and Beta-t-GARCH models are written as special cases of (<ref>), where β, α, α_1, α_2, c, and ν are parameters, taking their values typically in a subset of ℝ, and I_(-∞,0)(ϵ) takes the value 1 if ϵ < 0 and 0 otherwise. Figure <ref> shows the corresponding plots of these functions in the interval [-4, 4] and how their shapes change for the various values of their parameters.Notice that all the models except for the GARCH allow a negative innovation to contribute differently to the conditional variance from a positive innovation of the same magnitude. For the GJR-GARCH and the Beta-t-GARCH models, the asymmetry of g(·) is created by scaling an even function differently on each side of zero. The scaling factor is α_1 + α_2 for a negative argument, and is α_1 for a positive argument. Moreover, the NAGARCH model is the only one that allows the minimum of g(·) to occur away from 0. For the Beta-t-GARCH model, the shape of g(·) is directly linked to the innovation distribution, which is assumed to be F_sdt,ν.The Beta-t-GARCH model can be considered as a specific member of the family of score driven models known as generalised autoregressive score (GAS) models in <cit.> and dynamic conditional score (DCS) models in <cit.>. In a score driven volatility model, g(·) is a transformed gradient function (i.e., score) of the conditional distribution of r_t with respect to σ_t^2. The gradient function of the Student-t distribution dampens the effect of extreme observations on conditional variance when the degree of freedom parameter is small. Notice that u →ϵ^2 as ν→∞. Hence, the Beta-t-GARCH model is reduced to GJR-GARCH when the innovation distribution is Gaussian.It is straightforward to show that the stationarity condition and the unconditional variance of each of the parametric models is consistent with those of the new model in (<ref>) and (<ref>). §.§ Spline coefficient function In this paper, we generalise the various parametric proposals by assuming that the coefficient function g(·) in (<ref>) is an unknown but smooth univariate function. We model g(·) nonparametrically by a polynomial regression spline. A pth degree polynomial spline, for p ∈ℕ_0 = {0, ℕ}, is a function constructed piece-wise where each piece is a pth degree polynomial function. The set of points in the domain at which the pieces are joint together are referred to as knots. It is worth mentioning at this point that we consider a so called regression spline approach where there are significantly fewer number of knots than the number of observations, and the knot positions are selected as part of the modelling task. For example, a simple strategy is to place the knots either equally spaced or at specific sample quantiles. This is different from the smoothing spline approach (See <cit.>, for example.), where the knots are the observations themselves and a roughness penalty is added explicitly to reduce the effective degrees of freedom.Specifically, the coefficient function in our model is given byg(ϵ) = ∑_i=0^p b_iϵ^i + ∑_i=1^Kβ_i (ϵ - k_i)^p_+,where p ∈ℕ_0 is the degree of the spline, K ∈ℕ_0 is the number of knots, {k_1 k_K} is the sequence of knots satisfying k_1 < ⋯ < k_K, and (ϵ - k_i)^p_+ is known as a pth degree truncated power function defined by(ϵ - k_i)^p_+ = (ϵ - k_i)^pI_[k_i,∞)(ϵ),with I_[k_i,∞)(ϵ) taking the value 1 if ϵ≥ k_i and 0 otherwise. In (<ref>), the pth degree spline is represented as a linear combination of the functionsϵ^0ϵ^p,(ϵ - k_1)^p_+ (ϵ - k_K)^p_+,where b_0 b_p, β_1β_K are the coefficients in ℝ. It can be shown that the set of functions in (<ref>) is a basis for the space of pth degree splines with knots k_1 k_K (See <cit.> and <cit.>, for example.). Furthermore, any linear combination of the functions in (<ref>) has p - 1 continuous derivatives. The particular basis whose members are given by (<ref>) is known as a truncated power basis. In this paper, we restrict our attention to quadratic splines where p = 2 andg(ϵ) = b_0 + b_1ϵ + b_2ϵ^2 + ∑_i=1^Kβ_i (ϵ - k_i)^2_+.It can be shown that, when p = 2, several well known parametric models can be expressed exactly in terms of the spline coefficients. For example, following from the definitions in Table <ref>, the first order GARCH corresponds to the case where b_1 = 0 and K = 0; GJR-GARCH corresponds to the case where b_0 = β, b_1 = 0, b_2 = α_1 + α_2, β_1 = -α_2, K = 1, and k_1 = 0; NAGARCH corresponds to the case where b_0 = β + α c^2, b_1 = -2 α c, b_2 = α, and K = 0.Assume that the distribution of ϵ_t is both standardised and symmetric. Using the definition in (<ref>) and the distributional assumption of ϵ_t, we can derive the persistence of the SP-GARCH conditional variance process as follows.[g(ϵ_t)] = ∫_-∞^∞g(ϵ_t)f(ϵ_t)dϵ_t = b_0 + b_2 + ∑_i=1^Kβ_i c_i,where f(·) is the density function of the innovation distribution, and c_i is given byc_i = ∫_k_i^∞ (ϵ_t - k_i)^2 f(ϵ_t) d ϵ_t.The integral for c_i in (<ref>) can be obtained in closed form if f(·) is the density function of the standard normal distribution.c_i = - 1/√(2 π)exp(-k_i^2/2) k + 1/2 (1 + k_i^2) ( k_i/√(2)),where (x) = 1 - (x) denotes the complementary error function. If f(·) is instead the density function of F_sdt,ν, as per our assumption in Section <ref>, the expression for c_i is rather complicated, and the evaluation of such expression is subject to numerical difficulties. In such case, as a practical alternative, the value of c_i can be approximated with high accuracy using a numerical integration algorithm, such as the univariate adaptive quadrature. Figure <ref> shows plots of c_1 c_9 computed using the adaptive quadrature for F_sdt,ν with ν∈{2.2,3,8}, and using the closed-form expression in (<ref>) for the standard normal distribution, where the corresponding knots k_1 k_9 are placed at the quantile levels 0.10.9 of F_sdt,8. It can be seen that, even at ν = 8, the values of {c_i} are already very close to those computed analytically for the standard normal distribution, which suggests that, for the t distribution, equation (<ref>) can be used to approximate the integral for c_i when ν is moderate or large (for example, ν≥ 8). § BAYESIAN INFERENCE§.§ Bayesian model averaging Our goal in Section <ref> is to estimate the degrees of freedom ν, the mean μ, the constant parameter ω of the volatility function, and the coefficient function g(·) of the SP-GARCH model defined in (<ref>), where g(·) is modelled as a quadratic regression spline with a sequence of knots {k_1 k_K} defined in (<ref>). The choice of the knot sequence is important in estimating functions by regression splines, as it imposes a certain structure on the estimates. Ideally, we would want a small number of well positioned knots, so that we have a parsimonious model that can also account for the important details in the data. We employ a Bayesian variable selection approach where model averaging occurs naturally, and in doing so, we account for two types of uncertainty: the uncertainty in the knot sequence {k_1 k_K} and the uncertainty in the parameter values conditional on a particular knot sequence. For a detailed treatment on Bayesian model selection and model averaging, see <cit.>, <cit.>, and the references therein.Suppose that there are M ∈ℕ candidate models, indexed by τ∈{1M}. Let θ_τ∈Θ_τ denote the parameter vector associated with the τth model, where the set Θ_τ defines the parameter space, and let 𝐫 = (r_1 r_T) denote a vector of observations. Further suppose that we are interested in estimating a quantity ϕ, such that conditional on 𝐫, ϕ = η(θ_τ, τ), for some function η(·). For example, ϕ may be the one-day-ahead forecast of the conditional volatility, or the value of g(ϵ_t) at some specific value of ϵ_t. Under the assumption of model uncertainty, the posterior mean of ϕ can be taken as a Bayesian point estimate, which is obtained by averaging over all the candidate models;ϕ̂_Bayes = (ϕ𝐫)= ∑_τ=1^M[∫_Θ_τη(θ_τ, τ)p(θ_τ, τ𝐫)dθ_τ] = ∑_τ=1^M[∫_Θ_τη(θ_τ, τ)p(θ_ττ, 𝐫)dθ_τ]p(τ𝐫) = ∑_τ=1^M(ϕτ, 𝐫)p(τ𝐫).While for some problems one can obtain the closed-form expressions for both the conditional expectation (ϕτ, 𝐫) and the posterior model probabilities p(τ𝐫) (For example, see <cit.>.), for many realistic models they are analytically intractable, and approximation methods must be used. Even in the case of <cit.>, the summation in (<ref>) is infeasible to compute due to a large number of candidate models. We develop an adaptive MCMC algorithm to simulate from the joint posterior distribution p(θ_τ, τ𝐫) defined on the space⋃_τ∈{1M}Θ_τ×{τ}.If {(θ_τ^[i], τ^[i])i ∈{1N}} denotes a sample from p(θ_τ, τ𝐫) generated by a MCMC sampling algorithm, an approximation of ϕ̂_Bayes can then be computed asϕ̂_MCMC = 1/N∑_i=1^Nη(θ_τ^[i], τ^[i]).It can be seen from (<ref>) that model averaging is achieved without additional efforts via a single run of the MCMC sampler.An advantage of the basis representation in (<ref>) is that candidate models can be represented as submodels (or restricted models) nested within a supermodel. If we consider a model with a large number of knots (i.e., a large K) as a supermodel, then a candidate model with a specific knot sequence can be considered as a submodel with a set of knot coefficients {β_i} restricted to zero for some i ∈{1K}. In a Bayesian approach, the knot selection can be achieved by allowing a positive prior probability that each knot coefficient β_i is exactly zero. §.§ Joint posterior In this section, we define the the joint posterior distribution of the model parameters and the knot sequences. We adopt the paradigm where there is a large number of potential knots, and a subset of which is assigned to a candidate model. Let θ = (ν, μ, ω,b_0,b_1,b_2, β_1β_K) denote the parameter vector of the full model, and 𝐤 = (k_1 k_K) denote the vector of the potential knots. Each candidate model is indexed by a binary vector 𝐦 = (m_1 m_K) such thatm_i =0if β_i = 0,1if β_i 0.The joint posterior distribution of the pair (θ, 𝐦) given data is then proportional to the likelihood multiplied by the joint prior;p(θ, 𝐦𝐫) ∝ p(𝐫θ, 𝐦) p(θ, 𝐦). Following from the model definition in (<ref>) and (<ref>), the likelihood is given byp(𝐫θ, 𝐦) = p(𝐫θ) = I_Θ(θ) ∏_t=1^Tp(r_tℱ_t-1, θ), where the conditional distribution of each observation r_t is a Student-t distribution with ν degrees of freedom, whose mean is μ and whose variance is σ^2_t, and I_Θ(θ) is equal to one if θ∈Θ and zero otherwise. It should be made clear in (<ref>) that 𝐫 and 𝐦 are independent conditional on θ; the vector θ encodes all the information needed to generate 𝐫, as a result of the nested model set-up. We assume that ℱ_0 = σ(∅), and set the initial conditional variance σ^2_1 to the sample variance of the realisation of 𝐫. We use the indicator function I_Θ(·) to restrict the parameter space, by setting the likelihood to zero when θ∉Θ. The restricted parameter space Θ is defined asΘ = {θ 2 < ν≤ 200,ω > 0,0 < b_0 + b_2 + ∑_i=1^Kβ_ic_i < 1,σ^2_t > 0, for t ∈{1T+1}},so that when θ∈Θ, the process {r_t} is covariance stationary with a well defined unconditional variance given by (<ref>), and the process {σ^2_t} is positive for t ∈{1T+1}. Notice that the positivity restriction on the conditional variance process in (<ref>) is data-dependent, as σ^2_t is a function of both θ and 𝐫. This type of implicit restrictions was used in <cit.> to replace the sufficient conditions, which are explicitly available in terms of the model parameters, but more restrictive. The constants {c_i} are given by (<ref>). Since the density function f(·) in (<ref>) is of F_sdt,ν, the integral is evaluated numerically using adaptive quadrature. From an implementation perspective, these constants can be computed ex-ante on a fine grid for a range of degrees of freedom ν, and stored in a lookup table. During the evaluation of the indicator function I_{θ∈Θ}(θ), a table-lookup operation is computationally less expensive than running a quadrature algorithm.The joint prior can be constructed asp(θ, 𝐦) = p(θ𝐦) p(𝐦).When 𝐦 is known, we define the conditional prior for θ asp(θ𝐦) ∝ν^-2∏_i=1^K[ m_if_Gauss(β_i;0, σ^2_β) + (1 - m_i)δ(β_i)],where f_Gauss(β_i;0, σ^2_β) is a Gaussian density, with mean zero and variance σ^2_β, the delta function δ(β_i) represents a point probability mass at β_i = 0. Conditional on 𝐦, the parameters are assumed to be independent a priori. The conditional prior in (<ref>) implies that β_i is known to be exactly zero a priori when m_i = 0. As the prior is centred at zero for each β_i, the value of σ^2_β controls the level of shrinkage of the nonzero knot coefficients. We treat σ^2_β as a tuning parameter whose value is set by the user. From (<ref>), we can see that the prior for ν behaves similarly to a half-Cauchy prior with a scale parameter equal to one. It can be shown that this prior is flat on ν^-1. Similar priors for ν were employed in studies such as <cit.> and <cit.> in the context of the GARCH models. For all the parameters other than ν and β_1β_K, the conditional prior in (<ref>) is flat. Finally, a flat prior is used for the model probabilities,p(𝐦) = 0.5^K∝ 1,so that each model is given an equal prior probability. More generally, one can use the independent Bernoulli priors, p(𝐦) = ∏_i=1^K[π_i^m_i(1 - π_i)^(1 - m_i)], with different values of π_1π_K to express a prior preference for certain knots over the others. The main feature of the the joint prior specified in (<ref>) to (<ref>) is that each knot coefficient β_i is given a positive probability a priori to be exactly zero. This type of prior for β_i is known in the literature as the spike-and-slab prior <cit.>. §.§ Adaptive MCMC algorithm As the closed-form expressions are unavailable for the exact Bayesian estimates of the type in (<ref>) based on the joint-posterior distribution of our model, we develop an adaptive MCMC sampling algorithm to simulate from p(θ, 𝐦𝐫) defined in (<ref>) to (<ref>), and compute the approximate estimates using (<ref>). The aim is to construct an ergodic Markov chain whose stationary distribution is p(θ, 𝐦𝐫), and then generate a sample path {(θ^[i], 𝐦^[i])i ∈{1N}} of the chain. In general, our sampling algorithm involves the following steps: * Let 𝐦^[1] be an arbitrary binary vector of length K, and i = 1.* Generate a proposal (θ^*, 𝐦^*) from the joint distribution q(θ^*, 𝐦^*𝐦^[i]) by * generating 𝐦^* from q(𝐦^*𝐦^[i]),* generating θ^* from q(θ^*𝐦^*, 𝐦^[i]). * Let (θ^[i+1], 𝐦^[i+1]) = (θ^*, 𝐦^*) with probability u, or (θ^[i+1], 𝐦^[i+1]) = (θ^[i], 𝐦^[i]) with probability 1 - u.* Increment i by one, and repeat from step 2 until i = N. In step 2, the joint proposal distribution is constructed in a similar way to that of the prior distribution;q(θ^*, 𝐦^*𝐦^[i]) = q(θ^*𝐦^*, 𝐦^[i])q(𝐦^*𝐦^[i]),where θ^* = (ν^*, μ^*, ω^*,b_0^*,b_1^*,b_2^*, β_1^*β_K^*) and 𝐦^* = (m_1^* m_K^*). We can generate a realisation (θ^*, 𝐦^*) from (<ref>) by first generating a knot configuration 𝐦^*, followed by generating a parameter vector θ^* conditional on 𝐦^*, as in the steps 2(a) and 2(b). In step 2(a), conditional on 𝐦^[i], we generate a knot configuration 𝐦^* from the following Bernoulli mixture proposal density.q(𝐦^*𝐦^[i]) = ∏_j=1^K[m^[i]_jf_Bern(m^*_j;1-π) + (1-m^[i]_j)f_Bern(m^*_j; π)],where f_Bern(m^*_j; π) = π^m^*_j(1-π)^(1-m^*_j). By employing the proposal density in (<ref>), the resulting chain {𝐦^[i]} is a random walk in the K-dimensional binary vector space, where the parameter π controls the step-size. When π < 0.5, the realisations of m^[i]_j and m^[i+1]_j are more likely to be the same, and when π > 0.5, the opposite is true.Let θ_𝐦^* denote a (6 + ∑_i=1^Km_i^*)-dimensional vector containing only those elements of θ^* that are known a priori to be nonzero (I.e., θ_𝐦^* is the parameter vector of the submodel implied by 𝐦^*), and θ_𝐦^* denote a (∑_i=1^K(1 - m_i^*))-dimensional vector containing only those knot coefficients that correspond to the zero elements of 𝐦^*. In step 2(b), when 𝐦^* is known, we write the proposal density for θ^* asq(θ^*𝐦^*, 𝐦^[i]) = q(θ^*𝐦^*) = q(θ_𝐦^*𝐦^*)q(θ_𝐦^*𝐦^*),where q(θ_𝐦^*𝐦^*) = δ(θ_𝐦^*) is a multivariate delta function representing a point mass at θ_𝐦^* = (00). Notice that θ^* and 𝐦^[i] are conditionally independent given 𝐦^*. We generate θ_𝐦^* from a mixture of multivariate Gaussian distributions with a different scale for each component,q(θ_𝐦^*𝐦^*) = ∑_j=1^N_mix w_j f_mvn(θ_𝐦^*; θ̂_𝐦, ς_jΣ̂_𝐦),where ∑_j=1^N_mix w_j = 1 and f_mvn(θ_𝐦^*; θ̂_𝐦, ς_jΣ̂_𝐦) is a multivariate Gaussian density with mean vector θ̂_𝐦 and covariance matrix ς_jΣ̂_𝐦. The number of components N_mix, the vector of weights 𝐰 = (w_1 w_N_mix), and the vector of scales ς = (ς_1ς_N_mix) are tuning parameters that can be adjusted to help the sampling algorithm efficiently explore the posterior distribution.The mean vector θ̂_𝐦 and the base covariance matrix Σ̂_𝐦 must be set carefully to ensure that the proposal density for θ_𝐦^* in (<ref>) is a good approximation of the posterior density of the submodelp(θ_𝐦𝐦^*, 𝐫) ∝ p(𝐫θ_𝐦, 𝐦^*)p(θ_𝐦𝐦^*).We first generate a sample path from a pilot MCMC run, targeting the submodel posterior in (<ref>), which is easily obtained by imposing zero restrictions on the conditional posterior of the full model p(θ𝐦^*, 𝐫). For this pilot run, we employ the random walk Metropolis (RMW) kernel and a multivariate Gaussian proposal distribution, whose covariance matrix is adaptively estimated, up to a scaling factor, using the realised states of the chain, while the scaling factor is adaptively tuned to target an acceptance rate of 0.234, as in <cit.>. After discarding an initial burn-in sample, we set θ̂_𝐦 and Σ̂_𝐦 to be the sample mean and sample covariance matrix of the pilot chain. It is still acceptable if the stationary distribution of this pilot chain is not exactly (<ref>) due to adaptation <cit.>, since the sole purpose of the pilot chain is to learn good parameter values of θ̂_𝐦 and Σ̂_𝐦. To reduce computational cost, we save the values of θ̂_𝐦 and Σ̂_𝐦 for each newly visited knot configuration so that the rerunning of the RWM sampler is not required when the same configuration is revisited.Note that conditional on 𝐦^*, the proposal distribution is fixed, and the sampling algorithm effectively becomes an independence sampler <cit.> targeting (<ref>). Similar to the method of importance sampling, the independence sampler is known to be efficient if the proposal density is a good approximation of the target while being heavier-tailed <cit.>. We set N_mix = 3, 𝐰 = (0.85,0.1,0.05), and ς = (1,10,100), with the hope that Gaussian mixture proposal density is heavier-tailed than the target posterior in (<ref>), while we approximate the location, scale, and orientation of target by choosing θ̂_𝐦 and Σ̂_𝐦 using the pilot chain.In step 3, as (<ref>) is a density with respect to the natural measure on the product space ℝ^6+K×{0, 1}^K, the standard Metropolis-Hastings acceptance probability is a valid option for u <cit.>, and is given byu= min[p(θ^*, 𝐦^*𝐫) q(θ^[i], 𝐦^[i]θ^*, 𝐦^*)/p(θ^[i], 𝐦^[i]𝐫) q(θ^*, 𝐦^*θ^[i], 𝐦^[i]),1 ] = min[p(θ^*, 𝐦^*𝐫) q(θ^[i]𝐦^[i], θ^*, 𝐦^*) q(𝐦^[i]θ^*, 𝐦^*)/p(θ^[i], 𝐦^[i]𝐫) q(θ^*𝐦^*, θ^[i], 𝐦^[i]) q(𝐦^*θ^[i], 𝐦^[i]),1 ].The set of conditional independencies implied by the construction of the proposal distribution in step 2 is represented graphically in Figure <ref>, using a directed acyclic graph (DAG). Using the DAG, the expression for u can be simplified tou = min[ p(θ^*, 𝐦^*𝐫) q(θ^[i]𝐦^[i]) q(𝐦^[i]𝐦^*)/p(θ^[i], 𝐦^[i]𝐫) q(θ^*𝐦^*) q(𝐦^*𝐦^[i]),1 ].Using the fact that the proposal density for the knot configuration 𝐦^*, given by (<ref>), is symmetric, the acceptance probability is further reduced tou = min[ p(θ^*, 𝐦^*𝐫) q(θ^[i]𝐦^[i])/p(θ^[i], 𝐦^[i]𝐫) q(θ^*𝐦^*),1 ].§ SIMULATION STUDY A simulation study is conducted to investigate the comparative performance of the proposed SP-GARCH model and the popular parametric alternatives. The parametric models considered are the first order GARCH, GJR-GARCH (GJR), and Beta-t-GARCH (Beta-t). The aim is to investigate whether the SP-GARCH model is able to offer more accurate estimates of the conditional volatilities {σ_tt ∈{1T}} such that σ_t = [(r_tℱ_t-1)]^1/2 for certain family of processes followed by {r_t}. The focus is on comparing structural features of the estimated coefficient functions and measures of loss in both the in-sample and out-of-sample estimates of conditional volatilities. The model definitions of the comparing parametric models are given by (<ref>) and Table <ref>. The innovation distribution for each model is F_sdt,ν. The parameters are constrained to ensure the stationarity of {r_t} and the positivity of {σ_t^2}, by truncating the likelihood, similar to (<ref>). For each parametric model, the flat prior is used for most of the parameters with an exception of ν, where the prior is proportional to ν^-2.Independent sequences of T observations are generated from four data generating processes (DGPs), labelled DGP 1, DGP 2, DGP 3, and DGP 4. The DGPs can be written in the functional coefficient form of (<ref>), with DGP i, for i ∈{14}, given byr_t = σ_tϵ_i,t,σ^2_t = 0.1 + g_i(ϵ_i,t-1) σ^2_t-1,where {ϵ_i,t} is an i.i.d. sequence from F_sdt,8 for i ∈{1,2}, and from F_sdt,5 for i ∈{3,4}. The coefficient functions g_1(·)g_4(·) are specified as follows.g_1(ϵ)= 1.1 - 0.48(ϵ + 0.77)^2_+ + 0.58(ϵ + 0.473)^2_+; g_2(ϵ)= 0.85 + 0.1ϵ^2; g_3(ϵ)= 0.82 + 0.15[6ϵ^2/(3 + ϵ^2)]; g_4(ϵ)= 0.8 + [0.1 + 0.15 I_(-∞,0)(ϵ)]ϵ^2.The function g_1(·) is a quadratic spline with two knots at -0.77 and -0.473. The motivation for g_1(·) is to construct a coefficient function with structural features that cannot be fully captured by most parametric GARCH models, yet are not completely unrealistic from an empirical perspective. For example, g_1(·) implies that a relatively small negative return impacts the next day conditional volatility differently from a small positive return of the same magnitude, while the impact of a negative return stops to grow with its magnitude once the return is beyond a certain threshold. The functions g_2(·), g_3(·), and g_4(·) correspond to the coefficient functions of the first order GARCH, Beta-t, and GJR models, respectively. It can be checked that all DGPs satisfy the covariance stationarity condition in (<ref>), with [g_1(ϵ_1,t)] = 0.977, [g_2(ϵ_2,t)] = 0.95, [g_3(ϵ_3,t)] = 0.97, and [g_4(ϵ_4,t)] = 0.975.We generate N_sim = 500 independent sequences of length T = 4001 from each of the two DGPs. The posterior mean estimates of the conditional volatilities (as well as other quantities of interest presented in this section) are computed using the first 4000 observations of each generated sequence; the last observation of each sequence is left out for quantifying the out-of-sample performance.To compute the posterior mean estimates for the SP-GARCH model via (<ref>), the adaptive MCMC sampling algorithm detailed in Section <ref> is employed. The sampling algorithm is run for 5.5 × 10^5 iterations, where the first 0.5 × 10^5 burn-in iterations are ignored when computing the posterior mean estimates. The potential knots k_1 k_9 are chosen as quantiles of the standardised Student-t distribution with 8 degrees-of-freedom, at levels 0.10.9, respectively; the step-size parameter π in (<ref>) is set to 0.1; the shrinkage parameter σ^2_β in (<ref>) is set to 2500.To compute the posterior mean estimates for the parametric models, we use an independence sampler with a multivariate Gaussian mixture proposal configured in the same way as (<ref>). The mean vector and the base covariance matrix of the Gaussian mixture are set to the sample mean and covariance matrix of a pilot chain generated by the same adaptive RWM sampler as the one used in step 2(b) of the sampling algorithm in Section <ref>. The independence sampler is shown to work well for parametric GARCH models in <cit.> and <cit.>.The Monte Carlo (MC) mean of the 500 posterior mean estimates of the coefficient functions is plotted for each model in Figure <ref>, along with the 95% MC confidence band. Both the MC mean and the MC confidence band are computed pointwise using an equally spaced grid of points over [-4,4] with a spacing of 0.01. For DGP 1, only the SP-GARCH is able to capture the essential features of the true coefficient function. It is interesting to see that both the GJR and Beta-t models are trying to capture the asymmetry of g_1(·) near zero, where most of the innovations are found, while missing the important features for large innovations. For DGP 2, which corresponds to the the GARCH model, all the models except for the Beta-t are able to approximate the simple quadratic function. The Beta-t model is performing poorly due to the fact that its coefficient function deviates from the quadratic from if the innovation distribution is heavier tailed than the Gaussian. Conversely, when the true model is Beta-t, as for DGP 3, the quadratic models (GARCH and GJR) provide poor approximations as g_3(·) increases faster than quadratic for small innovations, but grows much slower than quadratic for large innovations. The SP-GARCH is able to offer a good approximation to g_3(·) except for the tails of the innovation distribution where the data is sparse. Lastly, when the true model is GJR, SP-GARCH is able to learn the shape of g_4(·) reasonably well. Expectedly, the approximation becomes increasingly biased as ϵ moves deeper into the tails of the innovation distribution. To compare the quality of the posterior mean estimates of the conditional volatilities between the models for each of the two DGPs, both the in-sample and out-of-sample versions of the L_p loss are considered. For the ith simulated sequence, the in-sample L_p loss is defined, for p ≥ 1, asL_p,i^(In) = [1/T-1∑_t=1^T-1|σ̂_i,t - σ_i,t|^p]^1/p,where σ̂_i,t is the posterior mean estimate of the volatility for tth observation of the ith simulated sequence, and σ_i,t is the corresponding true volatility; for the holdout observations across different sequences, the out-of-sample L_p loss is defined, for p ≥ 1, asL_p^(Out) = [1/N_sim∑_i=1^N_sim|σ̂_i,T - σ_i,T|^p]^1/p.Notice that the out-of-sample L_p loss is calculated cross-sectionally rather than time-series-wise. The cases p ∈{1, 2} are considered in this study: When p = 2, the loss measure L_p^(·) corresponds to the usual root-mean-square-error (RMSE); when p = 1, L_p^(·) corresponds to the usual mean-absolute-error (MAE).The MC mean of the 500 realisations of the in-sample L_p loss is reported for each p ∈{1, 2}, each model, and each DGP in Table <ref>, where L̅_p^(In) denotes (1/500)∑_i=1^500L_p,i^(In). Table <ref> is the out-of-sample analogue of Table <ref>. We make some observations: (i) Comparing Table <ref> with Figure <ref>, it is evident that conditional volatility estimates are sensitive to mis-specification of the coefficient function. (ii) Compared to the parametric models, SP-GARCH is robust to such mis-specification as it can adapt to a wide range of coefficient functions. (iii) For certain class of coefficient functions, the incurred efficiency loss (by not using the correct parametric model) is very small. For example, when the true DGP is GARCH, SP-GARCH is as efficient as the GJR model, given the fact that GJR has only one more degree-of-freedom than that of GARCH, and that SP-GARCH is a much more flexible model. (iv) The relative accuracy of SP-GARCH is retained out-of-sample. This suggests that the accuracy gain is not caused by overfitting.§ EMPIRICAL STUDY As an empirical study, the SP-GARCH model is applied to the daily returns of five stock market indices and five individual stocks. The aim is to gain insights into the conditional volatility process of each series by analysing the estimated coefficient function. The five stock market indices are: S&P 500 (US), FTSE 100 (UK), DAX (Germany), Nikkei 225 (Japan), and Hang Seng (Hong Kong); the five individual stocks are: Apple, ARM, Intel, Nvidia, and SanDisk. A sample of daily log-returns is obtained for each index or stock. Most of the samples start in January 1996, with exceptions of ARM and Nvidia, for which the samples start in April 1998 and January 1999, respectively. All of the samples end in November 2015. Nvidia has the smallest sample size of 4236 observations, while FTSE has the largest sample size of 5176 observations.For each sample, the posterior mean estimate of the coefficient function and the 95% credible band are computed using the MCMC sampling algorithm detailed in Section <ref>. The sequence of potential knots, the configuration of the sampling algorithm, and the value of the hyperparameter are the same as those used for the simulation study in Section <ref>.In Figure <ref>, the posterior mean estimate of the coefficient function is plotted for each of the ten samples, together with the pointwise 95% credible band. It is striking to see that the samples can be easily separated into two groups based on the structural features of the coefficient functions; there appear to be a clear distinction between the coefficient functions for the stock indices and those for the individual stocks. For the stock indices, the functions appear to be quadratic, with asymmetries created by shifting the minimums away from zero; for the individual stocks, the functions appear to be piecewise-linear-like, with asymmetries created by having slopes with varying degrees of steepness for negative and positive innovations. There also appear to be a “threshold effect" for large negative innovations for some individual stocks such as Intel and SanDisk. The posterior summaries of ν and μ are reported in Table <ref>. For each parameter-sample combination, we report the posterior mean (Mean) and the 95% credible interval (Lower, Upper). One can observe that the innovation distributions for the individual stocks are heavier tailed than those for the stock indices, as indicated by the estimates of ν. The estimates of μ indicate that the long-run (unconditional) means of the daily return processes are all positive but small in magnitude, with Apple generating the largest expected daily return of around 0.1%. One may be interested in estimating the long-run (unconditional) volatilities of the return processes as quantitative measures of the long-run daily risk; σ = (r_t)^1/2. This can be done using (<ref>) and (<ref>) for the SP-GARCH model. In Figure <ref>, a scatter plot is generated by plotting each sample in the product space of unconditional volatility and unconditional mean; {σ}×{μ}. Such plot is useful for comparing the long-run daily risk-return profiles of the samples, which could correspond to assets in an investment portfolio. It can be seen from the plot that both DAX and Apple show favourable risk-return characteristics compared to the rest of the samples. As an illustration of the mixing properties of the constructed Markov chain when the sampling algorithm is applied to real data, the realised values of the chain in dimensions b_2 and β_3, after discarding the initial burn-in iterations, are plotted against the iteration counts minus 5.5 × 10^5 in panels (a) and (b) of Figure <ref>, when the posterior distribution is conditional on the Apple data. It appears that the chain is settled down to rapid mixing in both dimensions. In panel (b), it is reassuring to see that the algorithm appears to be able to effectively switch between multiple modes including a point mass at exactly zero. The scatter plot in panel (c) reveals the complex landscape of the joint marginal posterior distribution of b_2 and β_3. It is useful to also examine the mixing properties of the Markov chain of the binary vectors, i.e., the Markov chain in model space. One way to visualise the sample path (or trace) of such chain is to first map each realisation of the K-dimensional binary vector 𝐦^[i] to a scalar τ^[i], for i ∈{1N}, such that the mapping is one-to-one, and then plot τ^[i] against i, for i ∈{1N}. One such mapping is easily found by treating each binary vector as a K-bit binary number. The mapped scalar is then the decimal equivalent of the binary number. To illustrate the idea, this mapping is used to generate the trace plots in Figure <ref> for S&P 500 and Apple. For both samples, the chain appears to be rapidly mixing in model space.A simple measure of model complexity for the SP-GARCH model is the number of knots chosen for the regression spline, given byK_𝐦 = ∑_i=1^Km_i.Using the realised binary vectors of the Markov chain, one can easily compute the marginal posterior probability of K_𝐦. Such posterior probabilities are reported in Table <ref> for each sample for K_𝐦≤ 3. For stock indices, the most favoured model is the one without any knots; for individual stocks, more complex models are preferred, as most of the posterior mass is on K_𝐦 = 2. It is possible that the structural features of the estimated coefficient functions shown in Figure <ref> are not unique, due to multimodality of the joint posterior distributions, and that our MCMC sampler is not effectively exploring all the modes. This possibility can potentially render our conclusions concerning the shapes of the coefficient functions invalid. For example, there could be a parametric GARCH model that fits the data equally well as the SP-GARCH, but giving a structurally different estimate of the coefficient function. One way to guard against this possibility is to compare SP-GARCH against the parametric models using a measure of model adequacy. We need to ensure that SP-GARCH is indeed a more adequate description of the data when the inference about the coefficient function is different under SP-GARCH.We perform a model comparison study based on the deviance information criterion (DIC) of <cit.> which can be considered as a Bayesian generalisation of the Akaike information criterion (AIC) <cit.>. Successful applications of the DIC include <cit.> and <cit.> for comparing stochastic volatility models, and <cit.> for comparing Markov-switching GARCH models. When computing the DIC, one obtains both a measure of model fit and model complexity. The model with the smallest DIC achieves the best fit-complexity trade-off.Usually, a flexible semiparametric model like the SP-GARCH would not perform well under the DIC due to its high complexity. However, the proposed BMA approach will cause the posterior mass to concentrate over simpler models when more complex models do not significantly improve the likelihood (i.e., model fit). It would thus be interesting to investigate how SP-GARCH performs against the parametric models when both fit and complexity are taken into consideration.A few studies have investigated the definition and the computation of the DIC for latent variable models <cit.>, however the DIC has not been examined in the context of BMA. For example, it is not clear how one may compute the DIC using the output of the reversible jump MCMC sampler of <cit.>, where the models are sampled together with their parameters. A natural generalisation of the DIC in the context of BMA may be the averaged DIC defined as follows (using the more general notation in Section <ref>).DIC_ave = _τ(DIC_τ𝐫) = ∑_τ=1^MDIC_τ p(τ𝐫),where τ∈{1M} is the model index, p(τ𝐫) is the posterior probability of the model τ. The DIC for model τ, denoted by DIC_τ, is the usual single model DIC given byDIC_τ = D̅_τ + p_D^τ.The first term is the conditional posterior mean of the devianceD̅_τ = _θ_τ[D(θ_τ, τ) τ, 𝐫],which can be considered as a Bayesian measure of model fit. The deviance is defined asD(θ_τ, τ) = -2 log p(𝐫θ_τ, τ),where p(𝐫θ_τ, τ) is the likelihood. The second term in (<ref>) is the effective number of parameters – a measure of model complexity, and is defined asp_D^τ = D̅_τ - D(θ̅_τ, τ),where θ̅_τ is the conditional posterior mean of θ_τ; θ̅_τ = _θ_τ(θ_ττ, 𝐫). It can be seen from the definition in (<ref>) that, for any consistent model selection procedure, the averaged DIC in (<ref>) is asymptotically equivalent to the single model DIC of <cit.>.It is straightforward to compute an estimate of DIC_ave whenever the posterior sample {(θ_τ^[i], τ^[i])i ∈{1N}} is available. Let the index set ℐ_τ⊆{1N} be defined as follows.ℐ_τ = {i_1 i_N_τ τ^[i_1] = ⋯ = τ^[i_N_τ] = τ}.The estimate of DIC_τ can be computed using the posterior draws {(θ_τ^[i], τ^[i])i ∈ℐ_τ}, where D̅_τ≈ N_τ^-1∑_i ∈ℐ_τ D(θ_τ^[i], τ^[i]) and θ̅_τ≈ N_τ^-1∑_i ∈ℐ_τθ_τ^[i]. The posterior model probability p(τ𝐫) is approximated by N_τ / N. For any τ such that ℐ_τ = ∅, it can be assumed that p(τ𝐫) ≈ 0; thus such τ is excluded during computation. Alternatively, one can compute DIC_ave by separately computing the averaged posterior mean of devianceD̅_ave = _τ(D̅_τ𝐫) = _τ{_θ_τ[D(θ_τ, τ) τ, 𝐫] 𝐫} = _θ_τ, τ[D(θ_τ, τ) 𝐫],and the averaged effective number of parametersp_D^ave = _τ(p_D^τ𝐫) = D̅_ave - ∑_τ=1^M D(θ̅_τ, τ) p(τ𝐫).The averaged DIC is then given by DIC_ave = D̅_ave + p_D^ave. Similarly, both terms can be approximated using a sample of posterior draws. For example, D̅_ave≈ N^-1∑_i=1^N D(θ_τ^[i], τ^[i]).In Table <ref>, for each parametric model, we report the estimates of the single model DIC, the posterior mean of deviance, and the effective number of parameters; for the SP-GARCH model, we report the estimates of the averaged DIC, the averaged posterior mean of deviance, and the averaged effective number of parameters. For the table and the rest of the section, the notations DIC, D̅, and p_D are overloaded to mean either the single model or the averaged version of the measures, depending on the model in question.For all the stock indices, the DIC ranks SP-GARCH as the best model, followed by (in the order of increasing DIC) Beta-t, GJR, and GARCH models. Both measures D̅ and DIC agree on the ranking of the models, indicating that the differences in model fit dominate the differences in complexity when ranked by the DIC. For all the models that allow g(·) to be asymmetric, the DIC estimates are exceedingly lower than those for the GARCH model. This supports the well known finding that it is important to model asymmetry for equity returns.The fact that the SP-GARCH model stands out as the best amongst the asymmetric models calls for further investigation. Recall that, in Figure <ref>, the coefficient function estimates (a) – (e) are minimised at points between 0 and 2, and that, in Table <ref>, most of the marginal posterior mass is on K_𝐦 = 0 for the indices. In other words, the most adequate model for the stock indices is obtained when g(·) is a simple shifted quadratic function. This leads to the conclusion that, when modelling for stock indices, asymmetry is perhaps best modelled by allowing the minimum of g(·) to shift away from zero, as opposed to by allowing each side of zero to scale differently. To contrast the coefficient functions estimated under these two approaches, Figure <ref>(a) shows the posterior mean estimates of g(·) of both the GJR and SP-GARCH models for the S&P 500 returns.For the individual stocks, both the DIC and D̅ estimates for the Beta-t and SP-GARCH models are substantially lower than those for the GARCH and GJR models. Unlike the index case, the DIC and D̅ estimates for the GJR model are very similar to those for the GARCH model. This suggests either that asymmetry is of less importance for the chosen stocks, or that the type of asymmetry present in these returns can not be adequately captured by the GJR model.For Apple, ARM, and Intel, the estimates of D̅ for Beta-t are almost identical to those for SP-GARCH, while Beta-t achieves slightly lower DIC estimates. For Nvidia and SanDisk, SP-GARCH is ranked ahead of Beta-t by both DIC and D̅.By comparing the plots of the estimated coefficient functions of the SP-GARCH model to those of the Beta-t model, we find that the estimates given by the two models are very similar in their shapes. An example of such comparison is shown in Figure <ref>(b) for the Intel returns. Notice that the two functions are almost identical for ϵ_t-1∈ [-2,2], where most of the observations are found. Moreover, for large negative innovations, the coefficient function of the SP-GARCH shows a similar “damping" behaviour to that of the Beta-t model. This finding provides evidence, at least for the chosen stocks, supporting the empirical validity of the Beta-t model.The BMA approach allows SP-GARCH to adapt its complexity according to the complexity of the data. This is evident in the estimated effective number of parameters p_D. For the indices, the p_D estimates for SP-GARCH are all close to 6, which are similar to those for the GJR and Beta-t models; for the individual stocks, the p_D estimates for SP-GARCH are increased slightly to approximately 7, indicating that more complexity is needed to accommodate the more complex dynamics of the data.§ CONCLUSION In its original form, the volatility function governing the dynamics of the GARCH model has a very restricted form. However, financial markets may be subjected to dramatic changes, calling for flexible models that can adapt to the changes. The main contribution of this paper is to propose the SP-GARCH model, where the sequence of conditional variances {σ^2_t} follows a functional coefficient autoregressive (FAR) model. Flexibility is achieved by allowing the coefficient function to be an arbitrary smooth function of the the lagged innovation. This smooth function is then approximated using regression spline basis functions based on a certain knot configuration. A strong feature of the proposed model is that the GARCH model and many of its popular extensions are special cases of the SP-GARCH model, differentiated by only the shape of the coefficient function g(·); inference on g(·) allows implicit testing of the simpler specifications. To perform inference, while accounting for the uncertainty in knot configuration, an approach based on Bayesian model averaging (BMA) is employed. To implement the BMA based approach, a carefully designed MCMC algorithm is developed to sample from the joint posterior distribution over the knot configurations and the parameter space. As illustrated by the trace plots, the sampler is able to move rapidly through the space on which the joint posterior distribution is defined. A simulation study shows that, when flexibility is required, the SP-GARCH model is, indeed, able to learn the shape of the coefficient function from the data. On the other hand, when the true data generating process is simple (namely, the original GARCH model), the SP-GARCH model, as a conditional volatility estimator, is as efficient as the simpler parametric models. Applications to ten real financial time series reveal that the coefficient functions for the stock indices are different in structural features from those for the individual stocks. A model comparison study based on the DIC confirms that the SP-GARCH model is, indeed, the most adequate description of the data. Furthermore, the SP-GARCH model is able to adapt its complexity according to the complexity of the data; when applied to real return series, the SP-GARCH model is often as parsimonious as the popular parametric models, which makes it very interpretable.The SP-GARCH model can be naturally extended to incorporate realised measures of volatility constructed using high-frequency intra-daily data, such as realised variance, realised kernel, and realised range (See, for example, <cit.>, <cit.>, <cit.>, and <cit.>). For example, the volatility function of the SP-Realised-GARCH model can be written as σ^2_t = ω + g(x_t-1) σ^2_t-1, where x_t-1 is a lagged realised measure.§ ACKNOWLEDGEMENT We thank Gareth W. Peters and Chris J. Oates for their comments on the manuscript. WYC was supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). chicago | http://arxiv.org/abs/1708.07587v1 | {
"authors": [
"Wilson Ye Chen",
"Richard H. Gerlach"
],
"categories": [
"stat.ME",
"q-fin.RM",
"stat.AP"
],
"primary_category": "stat.ME",
"published": "20170825005620",
"title": "Semiparametric GARCH via Bayesian model averaging"
} |
P-PC#1#2^#1#2 c/(ROI t yr)usc]V.E. Guiseppecor1 lbnl]N. Abgrall uw]S.I. Alvis pnnl]I.J. Arnquist usc,ornl]F.T. Avignone III ITEP]A.S. Barabash usd]C.J. Barton ornl]F.E. Bertrand mpi]T. Bode lbnl]A.W. Bradley JINR]V. Brudanin duke,tunl]M. Busch uw]M. Buuck unc,tunl]T.S. Caldwell lbnl]Y-D. Chan sdsmt]C.D. Christofferson lanl]P.-H. Chu uw,clara]C. Cuesta uw]J.A. Detwiler sdsmt]C. Dunagan ut,ornl]Yu. Efremenko ou]H. Ejiri lanl]S.R. Elliott unc,tunl]T. Gilliss princeton]G.K. Giovanetti ncsu,tunl,ornl]M.P. Green uw]J. Gruszko uw]I.S. Guinnunc,tunl]C.R. Haufe lbnl]L. Hehn unc,tunl]R. Henning pnnl]E.W. Hoppe unc,tunl]M.A. Howe blhill]K.J. Keeter ttu]M.F. Kidd ITEP]S.I. Konovalov pnnl]R.T. Kouzes ut]A.M. Lopez queens]R.D. Martin lanl]R. Massarczyk unc,tunl]S.J. Meijer mpi,tum]S. Mertens lbnl]J. Myslik unc,tunl]C. O'Shaughnessy unc,tunl]G. Othman lbnl]A.W.P. Poon ornl]D.C. Radford unc,tunl]J. Rager unc,tunl]A.L. Reine lanl]K. Rielage uw]R.G.H. Robertson uw]N.W. Rouf unc,tunl]B. Shanks JINR]M. Shirchenko sdsmt]A.M. Suriano usc]D. Tedeschi unc,tunl]J.E. Trimble ornl]R.L. Varner JINR]S. Vasilyev lbnl]K. Vetter unc,tunl]K. Vorren lanl]B.R. White unc,tunl,ornl]J.F. Wilkerson usc]C. Wiseman usd]W. Xu JINR]E. Yakushev ornl]C.-H. Yu ITEP]V. Yumatov JINR]I. Zhitnikov lanl]B.X. Zhu[usc]Department of Physics and Astronomy, University of South Carolina, Columbia, SC, USA [lbnl]Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA [uw]Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA, USA [pnnl]Pacific Northwest National Laboratory, Richland, WA, USA [ornl]Oak Ridge National Laboratory, Oak Ridge, TN, USA [ITEP]National Research Center “Kurchatov Institute” Institute for Theoretical and Experimental Physics, Moscow, Russia [usd]Department of Physics, University of South Dakota, Vermillion, SD, USA[mpi]Max-Planck-Institut für Physik, München, Germany [JINR]Joint Institute for Nuclear Research, Dubna, Russia [duke]Department of Physics, Duke University, Durham, NC, USA [tunl]Triangle Universities Nuclear Laboratory, Durham, NC, USA [unc]Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, USA [sdsmt]South Dakota School of Mines and Technology, Rapid City, SD, USA [lanl]Los Alamos National Laboratory, Los Alamos, NM, USA [ut]Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USA [ou]Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka, Japan [princeton]Department of Physics, Princeton University, Princeton, NJ, USA [ncsu]Department of Physics, North Carolina State University, Raleigh, NC, USA [blhill]Department of Physics, Black Hills State University, Spearfish, SD, USA [ttu]Tennessee Tech University, Cookeville, TN, USA [queens]Department of Physics, Engineering Physics and Astronomy, Queen's University, Kingston, ON, Canada[tum]Physik Department, Technische Universität, München, Germany [clara]Present Address: Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, 28040, Madrid, Spain[cor1]Corresponding author: [email protected] Status and Initial Results of the Majorana Demonstrator Experiment [ December 30, 2023 ====================================================================== Neutrinoless double-beta decay searches play a major role in determining the nature of neutrinos, the existence of a lepton violating process, and the effective Majorana neutrino mass. The Majorana Collaboration assembled an array of high purity Ge detectors to search for neutrinoless double-beta decay in ^76Ge. The Majorana Demonstrator is comprised of 44.1 kg (29.7 kg enriched in ^76Ge) of Ge detectorsdivided between two modules contained in a low-background shield at the Sanford Underground Research Facility in Lead, South Dakota, USA. The initial goals of the Demonstrator are to establish the required background and scalability of a Ge-based next-generation ton-scale experiment. Following a commissioning run that started in 2015, the first detector module started low-background data production in early 2016.The second detector module was added in August 2016 to begin operation of the entire array. We discuss results of the initial physics runs, as well as the status and physics reach of the fullMajorana Demonstrator experiment.§ EXPERIMENTAL OVERVIEW The discovery of Majorana neutrinos would have profound theoretical implications in extending the standard model of particle physics while yielding insights into the origin of mass itself. Experimental evidence of neutrinoless double-beta () decay would definitively reveal the Majorana nature of neutrinos and establish a lepton violating process.TheCollaboration is pursuing R&D aimed at a ton-scale, ^76Ge -decay experiment through the operation of aarray <cit.>on the 4850-foot level of the Sanford Underground Research Facility (SURF) <cit.>in Lead, SD, USA. The goals are to 1) demonstrate backgrounds are low enough to justify moving towards a ton-scaleexperiment, 2) establish the feasibility of constructing and fielding modular arrays of Ge detectors, and 3) search for additional physics beyond the standard model. The background goal in thepeak region of interest (ROI) is 3after analysis cuts.Thecontains a total of 44.1 kg of Ge detectors with 29.7 kg of the detector mass enriched to 88% in the isotope ^76Ge.The p-type, point contact detectors <cit.> in use offer excellent energy resolution, background rejection methods, and low energy thresholds. The detectors are split between two separate modules (1 and 2) that are surrounded by a low-background passive shield with active muon veto. §.§ Material Purity and Background RejectionA key to thedesign is the strict material purity requirements for parts used in the construction of the array. Prior to construction, a comprehensive assay program sampled candidate materials in order to identify acceptable detector components. A full Monte Carlo simulation of thegeometry converts the assay radio-purity levels or limits into a total background projection of ≤3.5 . A listing of all simulated background contributions can be found in Ref. <cit.>. In addition to its excellent thermal and mechanical properties, the purity of underground electroformed Cu makes it the ideal choice for the components in proximity to the detector:the detector unit and string assemblies, the cryostat, and the innermost passive shield. Our underground electrofoming baths produced 2474 kg of ultra pure Cu (≤ 0.1 μBq/kg Th and ≤ 0.1 μBq/kg U <cit.>).All Cu and plastics parts were fabricated in a dedicated underground machine shop at SURF, cleaned in an adjacent wet-lab cleanroom, and tracked through a custom database <cit.> during assembly of the array.A validation of cleaning procedures ensured clean parts remained clean through the handling and processing of parts.The remaining nearby components went through a careful selection and design process to ensure low mass and acceptable purity. Custom front-end boards were designed and fabricated to house the first stage of detector readout electronics. Fine coaxial cables were selected for the signal and high voltage wiring using custom miniature connectors. With the innermost passive shield layer made from underground electroformed Cu, the additional layers of the passive shield contain commercially sourced Cu and Pb that also underwent an aggressive surface cleaning procedure to remove contaminants from fabrication and handling. The background rejection strategies of thestart with the Ge detectors. The process of enrichment, zone refining, and crystal pulling provides thenecessary purification of the Ge metal from unwanted radioactive isotopes. The cosmogenic activation of the purified enriched Ge was controlled through underground storage and shielding during all stages of Ge detector fabrication <cit.>. The excellent energy resolution (2.4 keV FWHM at 2039 keV) of of a p-type, point-contact Ge detector allows discrimination of thespectrum of events from the narrowregion. Further, the slow drift velocity and localized weighting potential allows separation in time of multi-site events occurring in a detector due to gamma-ray interactions.The short range of the emitted electrons allows double-beta decay to be characterized as a single-site event. This multi-site event cut is achieved through Pulse Shape Discrimination (PSD) of the digitized waveformby comparing the maximum Amplitude (A) of the current pulse to the reconstructed Energy (E) through the calculation of an `AvsE' parameter <cit.>. Alpha-emitting contamination on the surface of the Ge detector or nearby materials contribute a background that can be identified by a signature detector response. An alpha striking the passivated surface of the Ge detector results in prompt collection of a fraction of the alpha energy while the remainder is collected more slowly. Due to their separate time components, only the fast component is reconstructed as the event energy. A Delayed Charge Recovery (DCR) parameter <cit.> relates the greater slope of the waveform tail to the slow charge collection component and the presence of a surface alpha event.Additional background suppression is achieved through a passive shield containing Cu and Pb within a sealed aluminum enclosure that defines a nitrogen gas purge volume to displace radon gas. Two layers of plastic scintillating panels tag penetrating muons to enable an active veto of prompt muon-induced backgrounds. Finally, a layer of high density polyethylene aids in moderating neutrons.§.§ Operation and Data SetsThehas been operating with enriched Ge detectors since May 2015. Changes in configuration or operational status are marked by distinct Data Sets (DS). Module 1 (M1) occupied the shield prior to the installation of the inner electroformed Cu shield for DS-0. DS-1 (operated in a data prescaling blindness mode between Dec. 31, 2015 - May 24, 2016) achieved lower backgrounds due to the presence of the innermost shield. The background levels of DS-0 and DS-1 are reported in Ref. <cit.>. Analysis is ongoing for M1 within DS-2 (May 24 - July 14, 2016) where multi-sampling is enabled to take a longer waveform to improve the DCR cut. With the addition of Module 2 (M2)to the shield, the results within the DS-3 and DS-4 full-array operation (as separate data streams) from Aug. 25 - Sep. 27, 2016 are presented here. Since that time, the array has been collecting physics data within DS-5 (as a single data stream) and DS-6 (a return to blindness mode). Analysis is ongoing over all data sets with a combined background andlimit to be released soon. § INITIAL RESULTS The first analysis of data from DS-0 and DS-1 M1 data are reported in Ref. <cit.>. Since that release, the Collaboration has been improving the PSD algorithms and added the second module. The latest analysis of data is from 1.39 kg·y of enriched exposure in DS-3 & DS-4, which contain both modules operating in the low background shield configuration. The energy spectrum, after data cleaning cuts, is displayed in Fig. <ref>(a) along with the sequential effect of performing the AvsE cut to remove multiple site events, such as background gamma rays, and the DCR cut to remove surface alpha backgrounds. The resulting energy spectrum (in Fig. <ref>(b) at a finer energy binning) leaves one main feature: thespectrum. Though theregion of interest is <3-keV wide centered at 2039 keV, a background index determination is made from a larger 400 keV window on the same center. The projected background rate is 5.1^+8.9_-3.2where the ROI is pinned at 2.9 keVand 2.6 keV, respectively, for M1 and M2 based on the average detector energy resolutions. The corresponding background index is 1.8×10^-3 c/(keV kg y).The excellent energy resolution and low energy thresholds of p-type, point contact detectors open up sensitivity to additional low energy physics searches. The controlled surface exposure of the enriched Ge material limited the production of internal cosmogenic activation, which would be the dominant source of backgrounds at low energy in a shielded low-background experiment.A low-background, low-energy spectrum allows searches for pseudoscalar dark matter, vector dark matter, solar axions, and other exotic physics beyond the standard model. Our analysis of DS-0 achieved a background rate of around 0.04 c/(kev kg d) near 20 keV with limits on low energy physics reported in Ref. <cit.>. Data cleaning cuts and improved data analysis continue in subsequent data sets where the inner shield offers lower backgrounds. The improved background rate can be seen in Fig. <ref>(c) with a background rate <0.02 c/(kev kg d) near 20 keV. The efficiency of the detectors below 5 keV is under study in order to open up the spectrum and cut non-physics events down to the detector threshold. § OUTLOOK Thecollaboration is operating itsarray as R&D for a ton-scale -decay experiment. Initial analysis shows that we have achieved the best energy resolution (2.4 keV FWHM) of anyexperiment. The PSDprovides a reduction of background events leaving only the feature of aspectrum. The PSD algorithms and stability checks are being improved as analysis of the ongoing data sets are finalized. The backgrounds from the initial data sets are amongst the lowest backgrounds in aregion of interest and are approaching the values set by the GERDA experiment <cit.>. As the analysis of the later data sets are finalized, a greater statistics background determination can be made with, at the time of writing, over 10 kg·y of exposure. Given the expected background rate, we project a half life sensitivity (90%) of 10^26 y from 3-5 years of data. At low energies, the low energy threshold and excellent energy resolution allow sensitive tests for new physics beyond the standard model. Thesuccess is due to its detector technology, the selection and careful handing of ultra-low-activity materials, and its low mass design. The combined strengths of the GERDA anddesigns are the basis for the newly formed LEGEND collaboration <cit.> and its goals for a ton-scale ^76Geexperiment. Based on the success of the two designs, LEGEND aims to reach a background on the order of 0.1and energy resolution necessary for discovery level sensitivities in the inverted neutrino mass ordering region. § ACKNOWLEDGMENTSThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, the Particle Astrophysics and Nuclear Physics Programs of the National Science Foundation, and the Sanford Underground Research Facility.aipnum-cp | http://arxiv.org/abs/1708.07562v1 | {
"authors": [
"V. E. Guiseppe",
"N. Abgrall",
"S. I. Alvis",
"I. J. Arnquist",
"F. T. Avignone III",
"A. S. Barabash",
"C. J. Barton",
"F. E. Bertrand",
"T. Bode",
"A. W. Bradley",
"V. Brudanin",
"M. Busch",
"M. Buuck",
"T. S. Caldwell",
"Y-D. Chan",
"C. D. Christofferson",
"P. -H. Chu",
"C. Cuesta",
"J. A. Detwiler",
"C. Dunagan",
"Yu. Efremenko",
"H. Ejiri",
"S. R. Elliott",
"T. Gilliss",
"G. K. Giovanetti",
"M. P. Green",
"J. Gruszko",
"I. S. Guinn",
"C. R. Haufe",
"L. Hehn",
"R. Henning",
"E. W. Hoppe",
"M. A. Howe",
"K. J. Keeter",
"M. F. Kidd",
"S. I. Konovalov",
"R. T. Kouzes",
"A. M. Lopez",
"R. D. Martin",
"R. Massarczyk",
"S. J. Meijer",
"S. Mertens",
"J. Myslik",
"C. O'Shaughnessy",
"G. Othman",
"A. W. P. Poon",
"D. C. Radford",
"J. Rager",
"A. L. Reine",
"K. Rielage",
"R. G. H. Robertson",
"N. W. Rouf",
"B. Shanks",
"M. Shirchenko",
"A. M. Suriano",
"D. Tedeschi",
"J. E. Trimble",
"R. L. Varner",
"S. Vasilyev",
"K. Vetter",
"K. Vorren",
"B. R. White",
"J. F. Wilkerson",
"C. Wiseman",
"W. Xu",
"E. Yakushev",
"C. -H. Yu",
"V. Yumatov",
"I. Zhitnikov",
"B. X. Zhu"
],
"categories": [
"physics.ins-det",
"nucl-ex"
],
"primary_category": "physics.ins-det",
"published": "20170824213808",
"title": "The Status and Initial Results of the MAJORANA DEMONSTRATOR Experiment"
} |
Gaziosmanpasa University, Physics Dept., Tokat, TurkeyUniversity of California at Irvine, Physics and Astronomy Dept., Irvine, CA, USACherenkov detectors are in use today in small experiments as well as modern ones as those at the LHC. This short note is about the construction of a small Cherenkov detector with limited resources, which could be used to observe the cosmic rays. The Cherenkov light obtained in this particular detector is read with a photo multiplier tube and its signal is observed on an oscilloscope. The detector construction can be achieved by relatively simple means and can be used as an educational tool.A simple Cherenkov detector for educational purposes G. Unel version of December 30, 2023 ====================================================§ INTRODUCTION Cherenkov radiation is electromagnetic radiation when a charged particle passes through in a dielectric medium at a speed greater than the speed of light in that medium. When a charged particle travels, it disturbs the electric field in the medium and medium becomes electrically polarized. If a charged particles has enough speed, a coherent shockwave is emitted. The similar analogy can be though as a sonic boom of a supersonic aircraft. Cherenkov emitting angle can be easily expressed as cos(θ_c)=1/nβ , where n is the refractive index of the material and β is v/c. If β is greater than 1/n, Cherenkov radiation will be emitted as shown in Fig.<ref> left side. Using relativistic kinematics, the threshold energy of Cherenkov radiation can be given by: E_threshold(n,m)=mc^2n/√(n^2-1) <cit.>. At this energy, the Cherenkov light is emitted along the path of the particle. The same figure right side contains the blueish Cherenkov light observed at the Reed Research Reactor originating from fast electrons. In case of Δ n=n-1≪1 approximation, the threshold energy can be rewritten as E_threshold(n,m)=mc^2/√(2Δ n) <cit.>. Therefore, such a detector could be used to trigger on various particles at various energies for small experiments or for educational purposes, depending on the radiation medium.§ DETECTOR DESIGN AND CONSTRUCTION The simplest Cherenkov detector would consist of a barrel with enough volume to produce Cherenkov light and reflective inner surfaces to channel that light into photon counting detectors. In high energy physics such a detector would be a Photo Multiplier Tube (PMT) <cit.> requiring about 1.5-2 kV for operation. The speed of light in the material to be installed into the barrel has to be smaller the vacuum value (c) to permit Cherenkov radiation. For example water (v_γ=0.75c) would be the a cheap and easily accessible material allowing detection of Cherenkov light for cosmic muons with energy larger than 160 MeV. However using CO2 (Air) would allow detection of cosmic muons of energy threshold of 3.52 (4.4) GeV.To build the simplest possible detector in the cheapest possible way, possible barrel alternatives have been considered such as a beer keg and a large water dispenser. The simplest solution was found to be a 20L volume plastic trash bin since it permits about 50 cm water depth, large enough to produce Cherenkov light. The inner side of the bin was covered with kitchen grade aluminum foils to improve its reflectivity. A view of the Al covered bin can be seen in Fig. <ref>. The available PMTs were R7525HA-2 from Hamamatsu <cit.> with 25mm photocathode diameter and requiring about 1500V for proper operation. Since these PMTs can not operate in water, a simple setup from foam and cardboard was prepared to keep the single PMT above the water level. The signal output from the PMT base can be readout directly with an oscilloscope without needing a pre-amplifier circuit. The light tightness of the finished product was provided by multiple layers of black felt blankets.§ DETECTOR TESTS The detector was first tested with air inside the bin to have an understanding of the background values. With a threshold as low as -7.2 mV, the background rate is measured to be less than 10Hz. This value is consistent with a background rate originating from the PMT's own dark current and from cosmic rays passing through the PMT window. A screenshot from the oscilloscope in accumulating display mode can be seen in Fig. <ref>. After determination of the rate and pulse hight of the background events, the bin was filled with tap water providing a water height of about 50cm as seen in Fig.<ref>left side. The signal threshold was set to -30mV eliminating most of the backgrounds. The solid line pulse in the same figure right side acquired after about 10 mins is consistent with a cosmic ray passing through the photocathode, both rate-wise and pulse height-wise which is about -60 mV.The expected Cherenkov signal on the other hand has to be rarer and should have a much higher pulse hight. Such an event was observed after about 15 mins as shown in Fig.<ref>. Here the pulse hight is about -150 mV as expected from Cherenkov light from a high energy cosmic ray. It is difficult to further elaborate on the cosmic ray's properties without calibrating the detector in a test beam. However, it is thought that by improving the quality of the inner reflecting surfaces, i.e. by collecting more light, the sensitivity of the setup can be increased. Of course, adding at least one scintillator to the system and measuring the timing between the scintillator and Cherenkov would largely improve the setup.§ CONCLUSIONS AND OUTLOOK Although this particular detector was produced with the minimum of investment, aiming a proof of the principle, a more developed detector using not tap water but Gadolinium doped pure water has been submitted to the Turkish Research and Technical Council, TUBITAK. The aim is to measure the anti-neutrino flux from the soon-to-be-build nuclear reactors in Turkey <cit.>. This current setup would be extremely useful for educational and demonstration purposes, if PMT, high voltage and oscilloscope requirements can be eased. For the first two, one could consider the utilization of Silicon photo multipliers. The last one can be matched by a small portable computer running an oscilloscope and display application. There are such low cost USB devices <cit.>and computers capable of running these <cit.>. With such a setup it would be possible to display the cosmic rays and measure their energies in real-time by adapting various materials with ϵ>1. Furthermore, if batteries can be used as both low voltage and high voltage power sources, the detector would become cable-independent, a major step for using it in particle physics education.The authors would like to thank B.Bilki for useful discussions, helping detector system setup and a careful reading of the manuscript, and Y.Onel for providing the PMT. 1 thesis Eric Antonio Quintero, The Cosmic Ray Muon Energy Spectrum via Cherenkov Radiation, MIT Bachelor of Science Thesis, 2010pmt0 Photomultiplier Tubes Basics and Applications, Hamamatsu, 2007pmt https://wiki.jlab.org/cuawiki/images/7/72/R7525-TPMH1244E02.pdfpub Sertac Ozturk et al., Monitoring Akkuyu Nuclear Reactor Using Antineutrino Flux Measurement, Turk J Phys,10.3906/fiz-1604-20, 2017 , arXiv:1602.04646 [physics.ins-det].bitscope BitScope: Test, measurement & data acquisition, http://www.bitscope.comraspberry Raspberry Pi: credit-card sized computer, http://www.raspberrypi.org. | http://arxiv.org/abs/1708.08772v1 | {
"authors": [
"Sertac Ozturk",
"Gokhan N. Unel"
],
"categories": [
"physics.ed-ph",
"physics.ins-det"
],
"primary_category": "physics.ed-ph",
"published": "20170827122425",
"title": "A simple Cherenkov detector for educational purposes"
} |
compatibility=false | http://arxiv.org/abs/1708.07802v2 | {
"authors": [
"E. A. Garcés",
"M. Hernández Villanueva",
"G. López Castro",
"P. Roig"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170825163338",
"title": "Effective-field theory analysis of the $τ^- \\to η^{(\\prime)} π^- ν_τ$ decays"
} |
Normandie Univ, ENSICAEN, UNICAEN, CNRS, CRISMAT, 14000 Caen, France [Corresponding author: ][email protected] Normandie Univ, ENSICAEN, UNICAEN, CNRS, CRISMAT, 14000 Caen, France 71.10.Fd, 72.15.Nj, 71.30.+h The Kotliar and Ruckenstein slave-boson representation of the Hubbard model allows to obtain an approximation of the charge dynamical response function resulting from the Gaussian fluctuations around the paramagnetic saddle-point in analytical form. Numerical evaluation in the thermodynamical limit yields charge excitation spectra consisting of a continuum, a gapless collective mode with anisotropic zero-sound velocity, and a correlation induced high-frequency mode at ω≈ U. In this work we show that this analytical expression obeys the particle-hole symmetry of the model on any bipartite lattice with one atom in the unit cell. Other formal aspects of the approach are also addressed.Particle-hole symmetry of charge excitation spectra in the paramagnetic phase of the Hubbard model Raymond Frésard December 30, 2023 ====================================================================================================§ INTRODUCTION Most peculiar properties of transition metal oxides that attract a lot of attention are believed to result from strong electronic correlations. A great variety of physical phenomena has been evidenced <cit.>, with prominent examples being thestriking metal-to-insulator transitions in vanadium sesquioxide <cit.>, high-T_c superconductivity in the cuprates <cit.>, non-Fermi liquid behavior in the vanadates <cit.>, stripes in nickelates <cit.> andcuprates <cit.>, or the colossal magnetoresistance observed in the manganites <cit.>. In addition, a whole series of promising materials for thermoelectric applications has been discovered <cit.>.The infancy of the microscopical modeling of strongly correlated systems dates to the early sixties with the introduction of the so-called one-band Hubbard Model <cit.>, H = -t ∑_⟨ i,j⟩ ,σf_iσ^† f^†_jσ+ U ∑_i n_i↑^† n_i↓^† ,that describes interacting fermions hopping on a lattice between nearest neighbor sites with amplitude -t. The screened Coulomb interaction is assumed local, and its strength on each site i is given by U. It was later on extended by Oleś to multiband systems to better embrace the diversity of transition metal oxides <cit.>. A fundamental consequence of strong correlations was already recognized by Hubbard, who showed that they split the non-interacting tight-binding bandand give rise to additional features in the excitation spectra including the upper Hubbard band (UHB) <cit.>. In fact, more recent investigations of the one-band Hubbard Model within dynamical mean-field theory revealed that its one-particle excitation spectra generically consist of lower and upper Hubbard bands, together with a quasi-particle peak <cit.>. These genuine interaction-driven features are reflected in two-particle excitation spectra. For instance, as shown in Fig. <ref> the charge excitation spectra consist of a continuum, a zero-sound collective mode, and a high-frequency collective mode originating from the upper Hubbard band <cit.>. The latter escapes a description within perturbation theory. The Kotliar and Ruckenstein slave-boson representation of the Hubbard model <cit.> is a convenient tool to obtain these spectra at one-loop order in the paramagnetic phase <cit.>. Such a symmetry-breaking free calculation may be performed in the thermodynamical limit. It thus allows to resolve the full momentum dependence of the spectra. Since incommensurate magnetic instabilities are strongly suppressed with increasing temperature, neglecting magnetic instabilities does not severely constrain the parameter range where the calculation may be meaningfully performed. For instance, on the square lattice, they essentially disappear for T≈ t/6 <cit.>. The Gutzwiller approximation, which entails the interaction driven Brinkman-Rice metal-to-insulator transition <cit.>, is reproduced on the saddle-point level by the Kotliar and Ruckenstein slave-boson representations <cit.>. The zero-temperature Brinkman-Rice metal-to-insulator transition is secondorder <cit.>.It turns first order at finite temperature as the latter may destabilize a poorlycoherent Fermi Liquid <cit.>. It is located in a coexistence region betweena metallic and an insulating phase. Furthermore this first order transition extendsto finite doping, where it is replaced by a transition from a good metal (with largequasi-particle residue) to a poor metal (with small quasi-particle residue). Itextends up to a critical endpoint that depends on temperature <cit.>.In contrast, a zero-temperature first order transition has been found in the two-bandmodel in the vicinity of half-filling <cit.>. The role of the lattice geometry on the metal-to-insulator transitionwas also investigated <cit.>. In particular, a very good agreementon the location of the metal-to-insulator transition has been found withQuantum Monte Carlo simulations on the honeycomb lattice <cit.>.Further comparisons of groundstate energiesto existing numerical solutions have also been carried out for the squarelattice. Regarding groundstate energies, for instance for U=4t, it couldbe shown that the slave-boson result is larger than its counterpart by lessthan 3% <cit.>. For larger values of U, the slave-boson groundstateenergy exceeds the exact diagonalization data by less than4% (7%) for U=8t (20t) and doping larger than15%. The discrepancy increases when the doping is lowered <cit.>.The saddle-point approximation is exact in the large degeneracy limit,and the Gaussian fluctuations provide 1/N corrections <cit.>.Moreover it obeys a variational principle in the limit of large spatialdimensions where the Gutzwiller approximation becomes exact for theGutzwiller wave function <cit.>.Let us finally emphasize that a quantitative agreement was establishedbetween the charge structure factors calculated from Gaussian fluctuationswithin the slave-boson approach and quantum Monte Carlosimulations <cit.>. Numerous valuable results have been obtained with Kotliar andRuckenstein <cit.> and related slave-boson representations <cit.>. Special attention has been paid to anti-ferromagnetic <cit.>, spiral <cit.>, and striped <cit.> phases. In addition,the competition between the latter two has been addressed as well <cit.>. It has also been obtained that the spiral order continuously transforms to the ferromagnetic order in the large U regime (U ≳ 60t) <cit.> so that its experimental realization is unlikely.Furthermore, in the two-band model on the square lattice, ferromagnetism was predictedas a possible groundstate in the doped Mott insulating regime only <cit.>.However, the ferromagnetic instability line could be brought down to the intermediate coupling regime when taking into account a ferromagnetic exchangecoupling <cit.>. A sufficiently large next-nearest-neighbor hoppingamplitude <cit.>, as well as going to the fcc lattice <cit.>, results in asimilar effect. In addition, this formalism extended to the Hubbard model with inter-siteCoulomb interaction has been applied to address the strongly inhomogeneouspolaronic states observed in correlated heterostructures <cit.>.Most recently the possibility to enhance the capacitance by strongcorrelation effects in the metallic plates of a capacitor has been investigatedwithin this approach <cit.>. Yet it needs to be verified that the approximate analytical expression used to calculate the charge excitation spectra complies with thesymmetries of the model. Clearly, translational invariance and spin-rotational invariance are satisfied in a paramagnetic phase. However the less obvious particle-hole symmetry remains to be established. Furthermore, there is a certain degree of arbitrariness inherent to this representation as to how to perform the one-loop calculation: While the internal gauge symmetry group of the representation allows to simplify the problem, as the phase of three of the four slave-boson fields may be gauged away by promoting the Lagrange multipliers to time-dependent fields, there is no prescription to determine which one of them must remain a complex field. In this paper, we not only show that the charge excitation spectra computed in <cit.> are indeed particle-hole symmetric, but also that they do not depend on whether the selected complex slave-boson field describes doubly occupied sites or empty sites.§ MODEL AND METHOD In the spin-rotation invariant (SRI) Kotliar and Ruckenstein slave-bosonrepresentation <cit.> the Hubbard Hamiltonian is expressed withpseudo-fermion operators f_iσ and auxiliary boson operatorse_i, p_iμ, d_i (for atomic states with respectively zero, single anddouble occupancy) asH = -t ∑_⟨ i,j⟩∑_σ, σ', σ” z_iσ”σ^† f_iσ^† f_jσ' z_jσ' σ” + U ∑_i d_i^† d_i.In this form the on-site Coulomb interaction has the advantage to be bilinear in bosonic operators. The canonical operators p_iμbuild a 2×2 matrix in spin space in order to preserve spin rotationsymmetry <cit.>. It is expanded into the identity matrixτ^0 and the Pauli matrices asp_i = 1/2∑_μ=0^3 p_iμτ^μ.The operator z_i takes into account the occupancy change that occurs during a hopping process. In the spin space it is also a matrix defined asz_i = e_i^†L_i M_i R_ip_i + p̃_i^†R_i M_iL_i d_iwith M_i = [ 1 + e_i^† e_i + ∑_μ=0^3 p_iμ^† p_iμ+d_i^† d_i ]^1/2, L_i = [ (1 -d_i^† d_i) τ^0- 2 p_i^†p_i ]^-1/2, R_i = [ (1 - e_i^† e_i) τ^0- 2 p̃_i^†p̃_i ]^-1/2wherep̃_i = 1/2 ( p_i0τ^0-p_i ·τ ). The subspace of physical states in the augmented Fock space generated by theauxiliary fermion and boson operators is the intersection of the kernels ofoperators A_i= e_i^† e_i + ∑_μ=0^3 p_iμ^† p_iμ +d_i^† d_i - 1, B_i0= ∑_μ=0^3 p_iμ^† p_iμ + 2 d_i^† d_i - ∑_σ f_i σ^† f_i σ,B_i =p_i0^† p_i +p_i^ † p_i0 -p_i^ †× p_i- ∑_σ, σ'τ_σσ' f_iσ'^† f_iσ,i.e. in this subspace A_i = 0 that is the constraint of oneatomic state per site, and B_iμ = 0 which equates thenumber of fermions to the number of p and d bosons. Functional integration is used to calculate the partition function <cit.>with the effective LagrangianL =L^ B +L^ F. Here the purelybosonic part isL^ B = ∑_i[e_i^†∂_τ e_i+∑_μ=0^3 p_iμ^†∂_τ p_iμ+ d_i^† (∂_τ + U) d_i +α_iA_i + ∑_μ=0^3 β_iμ B_iμ^ B]with B_iμ^ B being the bosonic part of the operatorB_iμ. After the integration of the fermion fields the mixed fermion-bosonpart may be written asL^ F = -tr{ln[ (∂_τ - μ +β_i0)δ_σσ'δ_ij + β_i ·τ_σσ'δ_ij + t_ij∑_σ_1 z_jσσ_1^† z_iσ_1σ'] },with μ the chemical potential. The constraints that define the physical states are enforcedwith Lagrange multipliers α_i and β_iμ. The problem may be simplifiedthanks to the internal gauge symmetry group of the representation. Indeedby promoting the Lagrange multipliers to time-dependent fields <cit.>, thephases of e and p_μ can be gauged away. This leaves us with radialslave-boson fields <cit.>. Their saddle-point values may be viewed asan approximation to their exact expectation values that aregenerically non-vanishing <cit.>. In fact, disproving earlierclaims <cit.>, the slave-boson fieldcorresponding to double occupancy d_i = d'_i +i d”_i howeverhas to remain complex as emphasized by several authors <cit.>.Furthermore the dynamics of the, now, real e_i and p_iμ fields drops out ofL^ B due to the periodic boundary conditions on boson fields.Within the approximation of Gaussian fluctuations, the action isexpanded to second order in field fluctuations ψ(k) = (δ e(k),δ d'(k),δ d”(k),δ p_0(k), δβ_0(k), δα(k), δ p_1(k),δβ_1(k),δ p_2(k),δβ_2(k), δ p_3(k), δβ_3(k) )around the paramagnetic saddle-point solution ψ_ MF = (e, d, 0, p_0,β_0,α, 0,0,0,0,0,0)as∫ dτ L(τ) =S_ MF + ∑_k,μ,νψ_μ(-k) S_μν(k) ψ_ν(k)(the matrix S is given in Appendix A of <cit.>). We have introduced thenotation k=( k,ν_n), with the Matsubara frequencies ν_n=2π nT, and ∑_k = T ∑_ν_n L^-1∑_ k with L the number of lattice sites. The correlation functions of boson fields are thenGaussian integrals which can be obtained from the inverse of the fluctuationmatrix S as⟨ψ_μ(-k) ψ_ν(k) ⟩ = 1/2 S^-1_μν(k).Using the density fluctuationδ N = δ(d^† d - e^† e),the charge susceptibility isχ_c(k) =⟨ N(-k)N(k) ⟩= 2 e^2 S_1,1^-1(k) - 4 e d S_1,2^-1(k) + 2 d^2S_2,2^-1(k) .Dynamical response functions are eventually evaluated within analyticalcontinuation iν_n →ω +i 0^+. § SYMMETRY OF THE SADDLE-POINT SOLUTIONThe Hubbard model possesses the particle-hole symmetry on a bipartitelattice. We show below that the symmetry is preserved at the saddle-pointlevel in the specific case of the square lattice.At first we present the general results of the paramagneticsolution, which do not presume any property of the lattice. At thesaddle-point level, the boson values can be expressed withthe hole doping from half-filling δ and the variable x =e +d ase = x^2 + δ/2x,d = x^ 2 - δ/2 x ,p_0^2 = 1 - x^4 + δ^ 2/2 x^2.The bare quasiparticle dispersion t_ k is renormalized as E_ k = z_0^2 t_ k - μ_ effwith the factorz_0^2= 2 p_0^2 (e+d)^2/1-δ^2,and μ_ eff the effective chemical potential.The paramagnetic solution for fixed values of doping δ andcoupling U is found by determining the chemical potentialμ_ eff via the filling condition2/L∑_ k n_F(E_ k) = 1 - δ and the solution x of the saddle-point equation(1-x^2) x^4/x^4 - δ^2=U/U_0.Here the coupling scaleU_0 = - 8 ε_0/(1-δ^2)has been introduced in terms of the semi-renormalized kinetic energyε_0 =2/L∑_ kt_ k n_F(E_ k) and the Fermi function n_F(ϵ) = 1/(exp(ϵ/T) + 1).Solving the saddle-point equation (Eq. <ref>) at half-filling yieldsz_0^2=1-(U/U_0)^2, in which case the effective mass of the quasiparticles diverges when U reaches U_0. This is the Brinkman-Rice mechanism of the metal-to-insulator transition, as it also arises in the Gutzwillerapproximation <cit.>.In order to establish the particle-hole symmetry of the saddle-pointapproximation, we show that the solution for the opposite doping is obtained by swapping the values of the empty and double occupancy, eand d, and reversing the sign of the effective chemical potentialμ_ eff, while keeping unchanged p_0. Hence the saddle-point value x and the renormalization factor z_0 are even functions ofδ. The boson expressions (<ref>) comply with thetransformation, so it remains to check that the latter (i) leaves invariantthe saddle-point equation and (ii) yields the filling condition for theopposite doping.This can be achieved using the parity of the quasiparticledensity of state N(-ϵ) = N(ϵ).Alternatively, the dispersion on the square lattice t_ k = -2t(cos k_x +cos k_y)yields the property t_ k+ Q = - t_ k, with Q = (π,π), that will be used in the next section.For (i) one can remarkthat both sides of Eq. (<ref>) are even in δ sinceε_0(-δ) =2/L∑_ kt_ k n_F(z_0^2 t_ k + μ_ eff) =2/L∑_ k't_ k'+ Q n_F(z_0^2 t_ k'+ Q+ μ_ eff) =2/L∑_ k' t_ k'[n_F(z_0^2 t_ k' - μ_ eff) - 1] = ε_0(δ)(the last equality results from the vanishing of ∑_ k t_ k over the Brillouin zone) so the saddle-point equation is invariant.As for (ii), the density of the transformed solution2/L∑_ k n_F(z_0^2 t_ k + μ_ eff) =2/L∑_ k' n_F(z_0^2 t_ k'+ Q + μ_ eff)=2/L∑_ k'[ 1 - n_F(z_0^2 t_ k' - μ_ eff) ]= 2 - (1 - δ) = 1 + δindeed corresponds to the opposite doping. § SYMMETRY OF THE QUASIPARTICLE RESPONSE FUNCTIONSOn the square lattice the quasiparticle response function is transformed under the reversal of the doping sign asχ_m^-(k) = (-1)^m χ_m^+(k).Here we have introduced the notation for the generalized quasiparticleresponse functions at doping ±δ χ_m^±(k) = 2/L∑_ q(t_ q +k + t_ q )^m n_F(E_ q +k^±) - n_F(E_ q^±)/ω - (E_ q +k^± - E_ q^±)with E_ q^± = z_0^2 t_ q∓μ_ eff.The relation between the expressions at opposite dopings can be derived bysumming instead on p = - q -k +Q. This yieldsχ_m^-(k) =2/L∑_ p(t_ -p +Q + t_- p- k+Q )^m×n_F(E_- p+Q^-) - n_F(E_- p- k+Q^-)/ω - (E_- p+Q^- - E_- p- k+Q^-)=2/L∑_ p(- t_ p - t_ p+ k )^m n_F(- E_ p^+) - n_F(-E_ p+ k^+)/ω - (- E_ p^+ + E_ p+ k^+)wherein we have used the equalities t_- p +Q = -t_ pand E_- p +Q^- = -z_0^2 t_ p + μ_ eff = - E_ p^+. Finally the sum can be written asχ_m^-(k) = (-1)^m 2/L∑_ p(t_ p+ k + t_ p )^m n_F(E_ p+ k^+) - n_F(E_ p^+)/ω - (E_ p+ k^+ - E_ p^+)since n_F(-ϵ) = 1 - n_F(ϵ). En passant the above relation shows the particle-hole symmetry of the RPA chargeresponse functionχ_ RPA(k) = χ_0^(0)(k)/1 + U/2χ_0^(0)(k)where χ_0^(0)(k) is the charge response function of a Fermi gas,i.e. χ_0^(0)(k) = χ_0(k)|_z_0 = 1.§ SYMMETRY OF THE SLAVE-BOSON CHARGE RESPONSEThe charge dynamical response function has been given in Ref. <cit.>. In order to demonstrate its particle-hole symmetry, we cast it into an expression which is explicitly invariant under the transformation {δ↦ -δ, e ↦ d, d ↦ e, χ_m(k) ↦ (-1)^m χ_m(k) } undergone when reversing the doping sign. After a lengthy but straightforward expansion, it can be written as χ_c(k) = A(k) + B(k)ω^2/C(k) + D(k)ω^2whereA(k) =2p_0^2 ε_0 /1 - δ^2[ ( p_0^2(d^2 S_11 + e^2 S_22) + 4 e^2 d^2 S_44 + 2edp_0^2 S_12 - 4edp_0(d S_14 + e S_24)) χ_0(k) + 2 ( e Δ_1 - d Δ_2 )^2 ],B(k) = ed p_0^2 χ_0(k),C(k) =2p_0^2 ε_0 /1 - δ^2[ ( p_0^2 S_11 S_22 - (p_0 S_12 - d S_14 - e S_24)^2 + 4 ed S_14 S_24 - 2 p_0( d S_11 S_24 + e S_22 S_14) + (d^2 S_11 + e^2 S_22 - 2ed S_12) S_44) χ_0(k)/2 + S_11Δ_1^2 + S_22Δ_2^2 + S_44Δ_4^2+ 2 S_12Δ_1Δ_2 -2 S_14Δ_1Δ_4 - 2 S_24Δ_2Δ_4],D(k) = ed/(e+d)^2[ ( p_0^2 (S_11 + S_22) + (e-d)^2 S_44 - 2 p_0^2 S_12 + 2 (e-d) p_0 (S_24 - S_14) ) χ_0(k)/2 + ( Δ_1 + Δ_2 )^2 ].Here the elements of the fluctuation matrix areS_11= -ε_0 z_0/e∂ z/∂ e + s_11(k), S_22= -ε_0 z_0/d∂ z/∂ d' + s_22(k), S_44= -ε_0 z_0/p_0∂ z/∂ p_0 + s_44(k), S_μν= s_μν(k) for μ,ν=1,2,4 with μ≠ν wheres_μν(k) = ε_0 z_0 ∂^2 z/∂ψ_μ∂ψ_ν + ∂ z/∂ψ_μ∂ z/∂ψ_ν[ ε_ k - z_0^2/2χ_2(k) ]withε_ k = 2/L∑_ q t_ q+ k n_F(E_ q), andΔ_1 =d p_0 + ( p_0 ∂ z/∂ d' - d ∂ z/∂ p_0) z_0/2χ_1(k), Δ_2 =e p_0 - ( p_0 ∂ z/∂ e - e ∂ z/∂ p_0)z_0/2χ_1(k),Δ_4 = 2 e d + ( e∂ z/∂ d' - d ∂ z/∂ e)z_0/2χ_1(k).The expressions of the derivatives of z may be gathered from Ref. <cit.>.The first-order derivatives are∂ z/∂ e= √(2)η p_0 ( 1 + 2 x e/1-δ), ∂ z/∂ d'= √(2)η p_0 ( 1 + 2 x d/1+δ),∂ z/∂ p_0= √(2)η x ( 1 + 2 p_0^2/1-δ^2),with η =1/√(1-δ^2). The second-order derivatives are∂^2 z/∂ e^2= 2√(2)η p_0/1-δ( x + 2e + 6 x e^2/1-δ),∂^2 z/∂ d'^2= 2√(2)η p_0/1+δ( x + 2d + 6 x d^2/1+δ),∂^2 z/∂ p_0^2= 2√(2)η^3 x p_0 (3 + (6η^2 - 2) p_0^2 ),∂^2 z/∂ e ∂ d'= 2√(2)η p_0 ( e/1-δ + d/1+δ + 2 η^2 x e d ),∂^2 z/∂ e ∂ p_0= √(2)η( 1 + 2η^2 p_0^2 (1 + x e)+ 2 x e/1-δ + 6 x e p_0^2/(1-δ)^2),∂^2 z/∂ d' ∂ p_0= √(2)η( 1 + 2η^2 p_0^2 (1 + x d)+ 2 x d/1+δ + 6 x d p_0^2/(1+δ)^2).It should be emphasized that due to the symmetry of the saddle-point solution, i.e. e(-δ) = d(δ) and p_0(-δ) = p_0(δ), the values of the partialderivatives of z by e are interchanged with those by d' when reversing the sign of the doping, e.g. (∂^2 z/∂ e∂ p_0) (-δ) = (∂^2 z/∂d'∂ p_0) (δ). The values of the fluctuation matrixelements S_μν with indices 1 and 2 are thus interchanged, e.g.S_11(-δ) = S_22(δ) or S_14(-δ)= S_24(δ).Furthermore one can check that Δ_4(-δ) = Δ_4(δ) andΔ_1(-δ) = Δ_2(δ). As a result, the coefficients (<ref>),and then the response function, are actually invariant. As an example we plotted the numerical evaluation of Eq. (<ref>) in Fig. <ref>, where the characteristic features of the inelastic charge response function are clearly visible. While neither the continuum nor the zero-sound collective mode above the upper edge of the continuum could be brought into a simple analytical form, an approximate expression of the UHB pole for all momenta applicable in the strong coupling regime could be derived <cit.>. It readsω_ UHB( k) ≈ U √(1 - U_0/2U( 1-3|δ| + (1-|δ|) ε_ k/ε_0)) .In Fig. <ref> we compare the evaluation of Eq. (<ref>) with the approximation Eq. (<ref>) for U=20t and 40t. The smallest energy of the UHB mode peak is obtained for k = (0,0), and the highest one for k = (π,π). The dispersion of this mode is approximately given by -2 (1-|δ|) ε_ k, and is accordingly largest at half-filling and vanishingly small in the limit of empty/full system. Furthermore, the difference between the approximation Eq. (<ref>) and Eq. (<ref>) tends to vanish already for U=40t.Let us finally mention an alternative derivation of Eq. (<ref>). We recall that there is some arbitrariness to its derivation as we chose here to gauge away the phase of the e-boson (on top of the one of the p_μ-bosons). Alternatively one may chose to gauge away the phase of the d-boson while keeping the e-boson as a complex field. This obviously leads to a S-matrix that differs from the one derived in <cit.> and used here. Yet, tedious work shows that χ_c(k) obtained this way is nevertheless given by Eq. (<ref>), thereby overcoming the above mentioned arbitrariness and putting Eq. (<ref>) on a firmer ground. § CONCLUSION Summarizing, the particle-hole symmetry of the charge response function obtained for the Hubbard model using various approximations has been considered. It has first been established that this symmetry is obeyed in the random phase approximation for a bipartite lattice such as the square lattice that we explicitly addressed. We then considered the expression of χ_c resulting from the SRI Kotliar and Ruckenstein slave boson representation calculated to one-loop order again on the square lattice. In this case we succeeded to cast its rather involved expression into a form that is manifestly particle-hole symmetric. The latter also applies to the Kotliar and Ruckenstein representation. Our arguments can easily be extended to other bipartite lattices with one atom in the unit cell, such as the simple cubic one.§ ACKNOWLEDGMENTS The authors acknowledge the financialsupport of the French Agence Nationale de la Recherche (ANR), through the program Investissements d'Avenir (ANR-10-LABX-09-01), LabEx EMC3, the Région Basse-Normandie, the Région Normandie, and the Ministère de la Recherche. § BIBLIOGRAPHY99Ima98 M. Imada, A. Fujimori, and Y. Tokura,Rev. Mod. Phys. 70, 1039 (1998)10.1103/RevModPhys.70.1039. McW73 D. B. McWhan, A. Menth, J. P. Remeika, W. F. Brinkman andT. M. Rice, Phys. Rev. B 7, 1920 (1973). 10.1103/PhysRevB.7.1920.Hel01 K. Held, G. Keller, V. Eyert, D. Vollhardt, and V. I. Anisimiov, Phys. Rev. Lett. 86, 5345 (2001). 10.1103/PhysRevLett.86.5345.Lim03 P. Limelette, A. Georges, D. Jérome, P. Wzietek, P. Metcalf,and J. M. Honig, Science 302, 89 (2003). 10.1126/science.1088386. Gry07 C. Grygiel, Ch. Simon, B. Mercey, W. Prellier, R. Frésard, and P. Limelette, Appl. Phys. Lett. 91, 262103 (2007). 10.1063/1.2824465. Bed86 J. G. Bednorz and K. A. Müller, Z. Phys. B: Condens. Matter 64, 189 (1986). 10.1007/BF01303701. Sch93 A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, Nature (London) 363, 56 (1993). 10.1038/363056a0. Ina95 F. Inaba, T. Arima, T. Ishikawa, T. Katsufuji, and Y. Tokura Phys. Rev. B 52, R2221(R) (1995).10.1103/PhysRevB.52.2221. Sac95n V. Sachan, D. J. Buttrey, J. M. Tranquada, J. E. Lorenzo,and G. Shirane, Phys. Rev. B 51, R12742 (1995). 10.1103/PhysRevB.51.12742. Tra17 R. Zhong, B. L. Winn, G. Gu, D. Reznik, and J. M. Tranquada,Phys. Rev. Lett. 118, 177601 (2017).10.1103/PhysRevLett.118.177601. Tra95c J. M. Tranquada,B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature (London) 375, 561 (1995).10.1103/10.1038/375561a0. Tra17b Z. Guguchia, B. Roessli, R. Khasanov, A. Amato, E. Pomjakushina, K. Conder, Y. J. Uemura, J. M. Tranquada, H. Keller, and A. Shengelaya, Phys. Rev. Lett. 119, 087002 (2017).10.1103/PhysRevLett.119.087002. Hel93 R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, andK. Samwer, Phys. Rev. Lett. 71, 2331 (1993). 10.1103/PhysRevLett.71.2331. Tom95 Y. Tomioka, A. Asamitsu, Y. Moritomo, H. Kuwahara, and Y. Tokura, Phys. Rev. Lett. 74, 5108 (1995).10.1103/PhysRevLett.74.5108. Rav95 B. Raveau, A. Maignan, and V. Caignaert, J. Solid State Chem. 117, 424 (1995). 10.1006/jssc.1995.1297. Mai95 A. Maignan, Ch. Simon, V. Caignaert, and B. Raveau, Solid State Commun. 96, 623 (1995).10.1016/0038-1098(95)00538-2. Ter97 I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 56, R12685 (1997).10.1103/PhysRevB.56.12685. Mas00 A. C. Masset, C. Michel, A. Maignan, M. Hervieu, O. Toulemonde, F. Studer, andB. Raveau, Phys. Rev. B62, 166(2000). 10.1103/PhysRevB.62.166. Mat01 I. Matsubara, R. Funahashi, T. Takeuchi, S. Sodeoka, T. Shimizu, and K. Ueno, Appl. Phys. Lett. 78, 3627 (2001). 10.1063/1.1376155.I. Matsubara, R. Funahashi, T. Takeuchi, S. Sodeoka, T. Shimizu, and K. Ueno, Appl. Phys. Lett. 78, 3627 (2001). 10.1063/1.1376155.Mic07 M. Miclau, J. Hejtmanek, R. Retoux, K. Knizek, Z. Jirak, R. Frésard, A. Maignan, S. Hébert, M. Hervieu, and C. Martin,Chem. Matter. 19, 4243 (2007). 10.1021/cm0710958.Ohta07 H. Ohta et al, Nat. Mater. 6, 129 (2007). 10.1038/nmat1821.Mai09b A. Maignan, V. Eyert, C. Martin, S. Kremer, R. Frésard, andD. Pelloquin, Phys. Rev. B 80, 115103. (2009). 10.1103/PhysRevB.80.115103.Wang13 N. Wang, H. J. Chen, H. C. He, W. Norimatsu, M. Kusunoki, and K. Koumoto, Sci. Rep. 3, 3449 (2013). 10.1038/srep03449. Hub63 J. Hubbard,Proc. Roy. Soc. London A 276, 238 (1963). 10.1098/rspa.1963.0204.Hub64 J. Hubbard,Proc. Roy. Soc. London A 281, 401 (1964). 10.1098/rspa.1964.0190.Kan63 J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).10.1143/PTP.30.275.Gut63 M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963). 10.1103/PhysRevLett.10.159.Ole83 A. M. Oleś, Phys. Rev. B 28, 327 (1983). 10.1103/PhysRevB.28.327.Geo96 A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys.68, 13 (1996).10.1103/RevModPhys.68.13.Bul01 R. Bulla, T. A. Costi, and D. Vollhardt. Phys. Rev. B 64, 045103 (2001). 10.1103/PhysRevB.64.045103.Dao17V. H. Dao and R. Frésard, Phys. Rev. B 95, 165127 (2017). 10.1103/PhysRevB.95.165127.Kot86 G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986). 10.1103/PhysRevLett.57.1362. Bri70 W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970). 10.1103/PhysRevB.2.4302.FWR. Frésard and P. Wölfle, Int. J. of Mod. Phys. B 6, 685 (1992)10.1142/S0217979292000414;ibid. 6, 3087 (1992). Doll3 R. Frésard and K. Doll, Proceedings of the NATO ARW The Hubbard Model: Its Physics and Mathematical Physics, eds. D. Baeriswyl, D. K. Campbell, J. M. P. Carmelo, F. Guinea, and E. Louis, San Sebastian (1993) (Plenum Press, 1995), p. 385.FK R. Frésard and G. Kotliar, Phys. Rev. B 56, 12 909 (1997). 10.1103/PhysRevB.56.12909. Mez17 A. Mezio and R. H. McKenzie,Phys. Rev. B 96, 035121 (2017).10.1103/PhysRevB.96.035121.Fre02 R. Frésard and M. Lamboley,J. Low Temp. Phys. 126, 1091 (2002).10.1023/A:1013815313109. Kot00 G. Kotliar, E. Lange, and M. J. Rozenberg, Phys. Rev. Lett. 84, 5180 (2000).10.1103/PhysRevLett.84.5180.Fre91 R. Frésard, M. Dzierzawa, and P. Wölfle, Europhys. Lett. 15, 325 (1991).10.1209/0295-5075/15/3/016.Fre92 R. Frésard and P. Wölfle, J. Phys.: Condens. Matter 4, 3625 (1992). 10.1088/0953-8984/4/13/022.Met89 W. Metzner, and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).10.1103/PhysRevLett.62.324.Met88 W. Metzner, and D. Vollhardt, Phys. Rev. B 37, 7382 (1988).10.1103/PhysRevB.37.7382.Met89b W. Metzner, Z. Phys. B 77, 253 (1989).10.1007/BF01313669.Zim97 W. Zimmermann, R. Frésard, and P. Wölfle, Phys. Rev. B 56, 10097 (1997). 10.1103/PhysRevB.56.10097.Li89T. C. Li, P. Wölfle, and P. J. Hirschfeld, Phys. Rev. B 40, 6817 (1989).10.1103/PhysRevB.40.6817.Lil90 L. Lilly, A. Muramatsu, and W. Hanke, Phys. Rev. Lett. 65, 1379 (1990).10.1103/PhysRevLett.65.1379.Igo13 P. A. Igoshev, M. A. Timirgazin, A. K. Arzhnikov, and V. Y. Irkhin,JETP Lett. 98, 150 (2013).10.1134/S0021364013160054.Doll2 B. Möller, K. Doll, and R. Frésard, J. Phys.: Condensed Matter 5, 4847 (1993).10.1088/0953-8984/5/27/029.SeiSi G. Seibold, E. Sigmund, and V. Hizhnyakov, Phys. Rev. B 57, 6937 (1998). 10.1103/PhysRevB.57.6937.Fle01 M. Fleck, A. I. Lichtenstein, and A. M. Oleś, Phys. Rev. B 64, 134528 (2001).10.1103/PhysRevB.64.134528.Sei02 J. Lorenzana and G. Seibold, Phys. Rev. Lett. 89, 136401 (2002).10.1103/PhysRevLett.89.136401.Sei03 J. Lorenzana and G. Seibold, Phys. Rev. Lett. 90, 066404 (2003).10.1103/PhysRevLett.90.066404.Sei05 J. Lorenzana and G. Seibold, Phys. Rev. Lett. 94, 107006 (2005).10.1103/PhysRevLett.94.107006.Rac06 M. Raczkowski, R. Frésard, and A. M. Oleś, Phys. Rev. B 73, 174525 (2006).10.1103/PhysRevB.73.174525.Rac07M. Raczkowski, M. Capello, D. Poilblanc, R. Frésard, and A. M. Oleś, Phys. Rev. B 76, 140505(R) (2007).10.1103/PhysRevB.76.140505.RaEPL M. Raczkowski, R. Frésard, and A. M. Oleś, Europhys. Lett. 76, 128 (2006).10.1209/epl/i2006-10227-1. lhoutellier15 G. Lhoutellier, R. Frésard, and A. M. Oleś,Phys. Rev. B 91, 224410 (2015). 10.1103/PhysRevB.91.224410.FW98 R. Frésard and W. Zimmermann, Phys. Rev. B 58, 15288 (1998). 10.1103/PhysRevB.58.15288.Igo15 P.A. Igoshev, M.A. Timirgazin, V.F. Gilmutdinov, A.K. Arzhnikov, V.Yu. Irkhin,J. Phys: Condens. Matter 27, 446002 (2015). 10.1088/0953-8984/27/44/446002.Pav06 N. Pavlenko and T. Kopp,Phys. Rev. Lett. 97, 187001 (2006). 10.1103/PhysRevLett.97.187001.Ste17 K. Steffen, R. Frésard, and T. Kopp, Phys. Rev. B 95, 035143 (2017).10.1103/PhysRevB.95.035143.fresard12 R. Frésard, J. Kroha, and P. Wölfle,in Theoretical Methods for Strongly Correlated Systems,edited by A. Avella and F. Mancini,Springer Series in Solid-State Sciences Vol. 171(Springer-Verlag, Berlin, 2012), pp. 65-101. 10.1007/978-3-642-21831-6.li91 T. Li, Y. S. Sun, and P. Wölfle,Z. Phys. B 82, 369 (1991).10.1007/BF01357181.Fre01 R. Frésard and T. Kopp,Nucl. Phys. B 594, 769 (2001). 10.1016/S0550-3213(00)00657-X.Kop07 R. Frésard, H. Ouerdane, and T. Kopp,Nucl. Phys. B 785, 286 (2007). 10.1016/j.nuclphysb.2016.04.005.Ras88 J. W. Rasul and T. Li, J. Phys. C: Solid State Phys. 21, 5119 (1988). 10.1088/0022-3719/21/29/009. Lav90 M. Lavagna, Phys. Rev. B 41, 142 (1990).10.1103/PhysRevB.41.142.Li94 T. Li and P. Bénard, Phys. Rev. B 50, 17837 (1994).10.1103/PhysRevB.50.17837.Jol91 Th. Jolicœur and J. C. Le Guillou,Phys. Rev. B 44, 2403 (1991). 10.1103/PhysRevB.44.2403.Kot92 Y. Bang, C. Castellani, M. Grilli, G. Kotliar, R. Raimondi, and Z. Wang, Int. J. of Mod. Phys. B 6, 531 (1992); Proceedings of the Adriatico Research Conference and Miniworkshop Strongly Correlated Electrons Systems III, eds. Yu Lu, G. Baskaran, A. E. Ruckenstein, E. Tossati (World Scientific Publishing Co., Singapore, 1992).10.1142/S0217979292000311.Vol84 D. Vollhardt,Rev. Mod. Phys. 56, 99 (1984).10.1103/RevModPhys.56.99.Vol87 D. Vollhardt, P. Wölfle, and P. W. Anderson, Phys. Rev. B 35, 6703 (1987).10.1103/PhysRevB.35.6703. | http://arxiv.org/abs/1708.07760v2 | {
"authors": [
"Vu Hung Dao",
"Raymond Frésard"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170824154657",
"title": "Particle-hole symmetry of charge excitation spectra in the paramagnetic phase of the Hubbard model"
} |
http://arxiv.org/abs/1708.07990v1 | {
"authors": [
"M. Ruiz-Garcia",
"L. L. Bonilla",
"A. Prados"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170826161229",
"title": "Bifurcation analysis and phase diagram of a spin-string model with buckled states"
} |
|
codeindent[1][0.25cm]#1mechanismMechanism privacytestPrivacy Test remarkRemark theoremTheorem theoremrecTheorem corollaryCorollary lemmaLemma definitionDefinition observationObservationcopyrightspace3 Plausible Deniability for Privacy-Preserving Data Synthesis (Extended Version)Vincent [email protected] Reza ShokriCornell [email protected] Carl A. [email protected] 30, 2023 =================================================================================================================================================Releasing full data records is one of the most challenging problems in data privacy. On the one hand, many of the popular techniques such as data de-identification are problematic because of their dependence on the background knowledge of adversaries.On the other hand, rigorous methods such as the exponential mechanism for differential privacy are often computationally impractical to use for releasing high dimensional data or cannot preserve high utility of original data due to their extensive data perturbation.This paper presents a criterion called plausible deniability that provides a formal privacy guarantee, notably for releasing sensitive datasets: an output record can be released only if a certain amount of input records are indistinguishable, up to a privacy parameter. This notion does not depend on the background knowledge of an adversary. Also, it can efficiently be checked by privacy tests. We present mechanisms to generate synthetic datasets with similar statistical properties to the input data and the same format. We study this technique both theoretically and experimentally. A key theoretical result shows that, with proper randomization, the plausible deniability mechanism generates differentially private synthetic data. We demonstrate the efficiency of this generative technique on a large dataset; it is shown to preserve the utility of original data with respect to various statistical analysis and machine learning measures. § INTRODUCTION There is tremendous interest in releasing datasets for research and development.Privacy policies of data holders, however, prevent them from sharing their sensitive datasets.This is due, to a large extent, to multiple failed attempts of releasing datasets using imperfect privacy-preserving mechanisms such as de-identification. A range of inference attacks on, for example, AOL search log dataset <cit.>, Netflix movie rating dataset <cit.>, Genomic data <cit.>, location data <cit.>, and social networks data <cit.>, shows that simple modification of sensitive data by removing identifiers or by generalizing/suppressing data features results in major information leakage and cannot guarantee meaningful privacy for data owners.These simple de-identification solutions, however, preserve data utility as they impose minimal perturbation to real data.Rigorous privacy definitions, such as differential privacy <cit.>, can theoretically guarantee privacy and bound information leakage about sensitive data.However, known mechanisms, such as the Laplacian mechanism <cit.> or the exponential mechanism <cit.>, that achieve differential privacy through randomization, have practical limitations.The majority of scenarios, where they have been applied, are limited to interactive count queries on statistical databases <cit.>.In a non-interactive setting for releasing generic datasets, these mechanisms are either computationally infeasible on high-dimensional data, or practically ineffective because of their large utility costs <cit.>.At best, these methods are used to release some privacy-preserving statistics (e.g., histograms <cit.>) about a dataset, but not full data records.It is not obvious how to protect the privacy of full records as opposed to that of aggregate statistics (by adding random noise).Despite all these obstacles, releasing full data records is firmly pursued by large-scale data holders such as the U.S. Census Bureau <cit.>. The purpose of this endeavor is to allow researchers to develop analytic techniques by processing full synthetic data records rather than a limited set of statistics.Synthetic data could also be used for educational purpose, application development for data analysis, sharing sensitive data among different departments in a company, developing and testing pattern recognition and machine learning models, and algorithm design for sensitive data.There exists some inference-based techniques to assess the privacy risks of releasing synthetic data <cit.>.However, the major open problem is how to generate synthetic full data records with provable privacy, that experimentally can achieve acceptable utility in various statistical analytics and machine learning settings.In this paper, we fill this major gap in data privacy by proposing a generic theoretical framework for generating synthetic data in a privacy-preserving manner.The fundamental difference between our approach and that of existing mechanisms for differential privacy (e.g., exponential mechanism) is that we disentangle the data generative model from privacy definitions.Instead of forcing a generative model to be privacy-preserving by design, which might significantly degrade its utility, we can use a utility-preserving generative model and release only a subset of its output that satisfies our privacy requirements.Thus, for designing a generative model, we rely on the state-of-the-art techniques from data science independently from the privacy requirements.This enables us to generate high utility synthetic data.We formalize the notion of plausible deniability for data privacy <cit.>, and generalize it to any type of data.Consider a probabilistic generative model that transforms a real data record, as its seed, into a synthetic data record.We can sample many synthetic data records from each seed using such a generative model.According to our definition, a synthetic record provides plausible deniability if there exists a set of real data records that could have generated the same synthetic data with (more or less) the same probability by which it was generated from its own seed.We design a privacy mechanism that provably guarantees plausible deniability.This mechanism results in input indistinguishability: by observing the output set (i.e., synthetics), an adversary cannot tell for sure whether a particular data record was in the input set (i.e., real data).The degree of this indistinguishability is a parameter in our mechanism.Plausible deniability is a property of the overall process, and similar to differential privacy, it is independent of any adversary's background knowledge.In fact, we prove that our proposed plausible deniable data synthesis process can also satisfy differential privacy, if we randomize the indistinguishability parameter in the privacy mechanism.This is a significant theoretical result towards achieving strong privacy using privacy-agnostic utility-preserving generative models.Thus, we achieve differential privacy without artificially downgrading the utility of the synthesized data through output perturbation.The process of generating a single synthetic data record and testing its plausible deniability can be done independently from that of other data records.Thus, millions of data records can be generated and processed in parallel.This makes our framework extremely efficient and allows implementing it at a large scale.In this paper, we develop our theoretical framework as an open-source tool, and run it on a large dataset: the American Community Survey <cit.> from the U.S. Census Bureau which contains over 3.1 million records. In fact, we can generate over one million privacy-preserving synthetic records in less than one hour on a multi-core machine running 12 processes in parallel. We analyze the utility of synthetic data in two major scenarios: extracting statistics for data analysis, and performing prediction using machine learning.We show that our privacy test does not impose high utility cost.We also demonstrate that a significant fraction of candidate synthetic records proposed by a generative model can pass the privacy test even for strict privacy parameters.We show that a strong adversary cannot distinguish a synthetic record from a real one with better than 63.0% accuracy (baseline: 79.8%). Furthermore, when it comes to classification tasks, the accuracy of the model learned on a synthetic dataset is only slightly lower than that of model trained on real data. For example, for Random Forest the accuracy is 75.3% compared to 80.4% when trained on real data (baseline: 63.8%); whereas for AdaBoostM1 the accuracy is 78.1% compared to 79.3% when trained on real data (baseline: 69.2%). Similar results are obtained when we compare logistic regression (LR) and support vector machine (SVM) classifiers trained on our synthetic datasets with the same classifiers trained (on real data) in a differential private way (using state-of-the-art techniques).Concretely, the accuracy of classifiers trained on our synthetic data is 77.5% (LR) and 77.1% (SVM); compared to 76.3% (LR) and 78.2% (SVM) for objective-perturbation ε-DP classifiers.Contributions.We introduce a formal framework for plausible deniability as a privacy definition.We also design a mechanism to achieve it for the case of generating synthetic data.We prove that using a randomized test in our plausible deniability mechanism achieves differential privacy (which is a stronger guarantee).We also show how to construct generative models with differential privacy guarantees. The composition of our generative model and plausible deniability mechanism also satisfies differential privacy.We show the high accuracy of our model and utility of our generated synthetic data.We develop a generic tool and show its high efficiency for generating millions of full data records. § PLAUSIBLE DENIABILITY In this section, we formalize plausible deniability as a new privacy notion for releasing privacy-preserving synthetic data.We also present a mechanism to achieve it.Finally, we prove that our mechanism can also satisfy differential privacy (which is a stronger guarantee) by slightly randomizing our plausible deniability mechanism. Informally, plausible deniability states that an adversary (with any background knowledge) cannot deduce that a particular record in the input (real) dataset was significantly more responsible for an observed output (synthetic record) than was a collection of other input records.A mechanism ensures plausible deniability if, for a privacy parameter k>0, there are at least k input records that could have generated the observed output with similar probability. Unlike the majority of existing approaches (e.g., to achieve differential privacy), designing a mechanism to satisfy plausible deniability for generative models does not require adding artificial noise to the generated data.Instead, we separate the process of releasing privacy-preserving data into running two independent modules: (1) generative models, and (2) privacy test.The first consists in constructing a utility-preserving generative data model.This is ultimately a data science task which requires insight into the type of data for which one wants to generate synthetics.By contrast, the privacy test aims to safeguard the privacy of those individuals whose data records are in the input dataset.Every generated synthetic is subjected to this privacy test; if it passes the test it can be safely released, otherwise it is discarded.This is where the plausible deniability criterion comes into the frame: the privacy test is designed to ensure that any released output can be plausibly denied.In this section, we assume a generic generative model that, given a data record in the input dataset as seed, produces a synthetic data record.In Section <ref>, we present a generic generative model based on statistical models, and show how it can be constructed in a differentially-private manner, so that it does not significantly leak about its own training data.Plausibly deniable mechanisms protect the privacy of the seeds, and are not concerned about how the generative models are constructed. Let ℳ be a probabilistic generative model that given any data record d can generate synthetic records y with probability y = ℳ(d). Let k ≥ 1 be an integer and γ≥ 1 be a real number. Both k and γ are privacy parameters.For any dataset D with |D| ≥ k, and any record y generated by a probabilistic generative model ℳ such that y = ℳ(d_1) for d_1 ∈ D, we state that y is releasable with (k,γ)-plausible deniability, if there exist at least k-1 distinct records d_2, ..., d_k ∈ D ∖{ d_1 } such that γ^-1≤y = ℳ(d_i)/y = ℳ(d_j)≤γ , for any i,j ∈{1, 2, …, k}. The larger privacy parameter k is, the larger the indistinguishability set for the input data record.Also, the closer to 1 privacy parameter γ is, the stronger the indistinguishability of the input record among other plausible records. Given a generative model ℳ, and a dataset D, we need a mechanism ℱ to guarantee that the privacy criterion is satisfied for any released data. Specifically ℱ produces data records by using ℳ on dataset D. The following mechanism enforces (k,γ)-plausible deniability by construction. [ℱ with Plausible Deniability]Given a generative model ℳ, dataset D, and parameters k, γ, output a synthetic record y or nothing. * Randomly sample a seed record d ∈ D. * Generate a candidate synthetic record y = ℳ(d). * Invoke the privacy test on (ℳ,D,d,y,k,γ). * If the tuple passes the test, then release y.Otherwise, there is no output. The core of Mechanism <ref> (ℱ) is a privacy test that simply rejects a candidate synthetic data record if it does not satisfy a given privacy criterion.We can think of Definition <ref> as a privacy criterion that can be efficiently checked and enforced. So, instead of trying to measure how sensitive the model ℳ is with respect to input data records, we test if there are enough indistinguishable records in the input dataset that could have (plausibly) generated a candidate synthetic data record. [Deterministic test 𝒯] Given a generative model ℳ, dataset D, data records d and y, and privacy parameters k and γ, output pass to allow releasing y, otherwise output fail. * Let i ≥ 0 be the (only) integer that fits the inequalitiesγ^-i-1 < y = ℳ(d)≤γ^-i . * Let k' be the number of records d_a ∈ D such thatγ^-i-1 < y = ℳ(d_a)≤γ^-i . * If k' ≥ k then return pass, otherwise return fail. Step 2 counts the number of plausible seeds, i.e., records in D which could have plausibly produced y. Note that for a given y, there may exist some records d_a ∈ D such that y = ℳ(d_a) = 0. Such records cannot be plausible seeds of y since no integer i ≥ 0 fits the inequalities.Remark that Privacy Test <ref> (𝒯) enforces a stringent condition that the probability of generating a candidate synthetic y given the seed d and the probability of generating the same record given another plausible seed d_a both fall into a geometric range [γ^-i-1,γ^-i], for some integer i ≥ 0, assuming γ > 1. Notice that, under this test, the set of k-1 different d_as plus d satisfies the plausible deniability condition (<ref>).Informally, the threshold k prevents releasing the implausible synthetics records y.As k increases the number of plausible records which could have produced y also increases.Thus, an adversary with only partial knowledge of the input dataset cannot readily determine whether a particular input record d was the seed of any released record y. This is because there are at least k-1 other records d_i ≠ d in the input dataset which could plausibly have been the seed.However, whether y passes the privacy test itself reveals something about the number of plausible seeds, which could potentially reveal whether a particular d is included in the input data. This can be prevented by using a privacy test which randomizes the threshold k (as Section <ref> shows) in which case the mechanism achieves (ε,δ)-differential privacy. §.§ Relationship with Differential Privacy We show a connection between Plausible Deniability and Differential Privacy, given the following definition. Mechanism F satisfies (ε,δ)-differential privacy if for any neighboring datasets D, D', and any output S ⊆Range(F): F(D') ∈ S≤ e^εF(D) ∈ S + δ .Typically, one chooses δ smaller than an inverse polynomial in the size of the dataset, e.g., δ≤ |D|^-c, for some c > 1.In this section, we prove that if the privacy test is randomized in a certain way, then Mechanism <ref> (ℱ) is in fact (ε,δ)-differentially private for some δ > 0 and ε > 0.Privacy Test <ref> simply counts the number of plausible seeds for an output and only releases a candidate synthetic if that number is at least k.We design Privacy Test <ref> which is identical except that it randomizes the threshold k.[Randomized test 𝒯_ϵ_0] Given a generative model ℳ, dataset D, data records d and y, privacy parameters k and γ, and randomness parameter ϵ_0, output pass to allow releasing y, otherwise output fail. * Randomize k by adding fresh noise: k̃ = k + Lap(1/ϵ_0). * Let i ≥ 0 be the (only) integer that fits the inequalitiesγ^-i-1 < y = ℳ(d)≤γ^-i . * Let k' be the number of records d_a ∈ D such thatγ^-i-1 < y = ℳ(d_a)≤γ^-i . * If k' ≥k̃ then return pass, otherwise return fail. Here z ∼Lap(b) is a sample from the Laplace distribution 1/2bexp(-|z|/b) with mean 0 and shape parameter b > 0. Let ℱ denote Mechanism <ref> with the (randomized) Privacy Test <ref> and parameters k ≥ 1, γ > 1, and ε_0 > 0. For any neighboring datasets D and D' such that |D|, |D'| ≥ k, any set of outcomes Y ⊆𝒰, and any integer 1 ≤ t < k, we have: ℱ(D') ∈ Y≤ e^εℱ(D) ∈ Y + δ , for δ = e^-ε_0 (k-t) and ε = ε_0 + ln(1+γ/t). The privacy level offered by Theorem <ref> is meaningful provided k is such that δ is sufficiently small. For example, if we want δ≤1/n^c for some c > 1, then we can set k ≥ t + c/ε_0lnn. Here t provides a trade-off between δ and ε.The proof of Theorem <ref> can be found in Appendix <ref>. Roughly speaking, the theorem says that, except with some small probability δ, adding a record to a dataset cannot change the probability that any synthetic record y is produced by more than a small multiplicative factor. The intuition behind this is the following.Fix an arbitrary synthetic record y produced by the mechanism on some dataset. Remark that given y, records are partitioned into disjoint sets according to their probabilities of generating y (with respect to ℳ). That is, partition i for i=0,1,2…, contains those records d such that γ^-(i+1) < y = ℳ(d)≤γ^-i. (We ignore records which have probability 0 of generating y.)Now suppose we add an arbitrary record d' to the dataset. The probability of producing y changes in two ways: (1) the probability that y is generated increases because d' may be chosen as seed, and (2) the probability that y passes the privacy test increases because d' is an additional plausible seed. Remark that this change only impacts whichever partition d' falls into because the probability of passing the privacy test depends only on the number of plausible seeds in the partition of the seed.Thus, we focus on the partition in which d' falls. If that partition contains a small number of records compared to k then introducing d' could increase the probability of generating y significantly, but the probability of passing the privacy test is very small. (The likelihood of passing the privacy test decreases exponentially the fewer plausible seeds are available compared to k.) In contrast, if the partition contains a number of records comparable to k or larger, then the probability of generating y increases only slightly (because there are already a large number of plausible seeds with similar probability of generating y as d'). And, the probability of passing the privacy test increases by a multiplicative factor of at most e^ε_0 due to adding Laplacian noise. In both cases, the increase to the probability of producing y due to adding d' is small and can be bounded. § GENERATIVE MODEL In this section, we present our generative model, and the process of using it to generate synthetic data.The core of our synthesizer is a probabilistic model that captures the joint distribution of attributes.We learn this model from training data samples drawn from our real dataset .Thus, the model itself needs to be privacy-preserving with respect to its training set.We show how to achieve this with differential privacy guarantees.Let S, T, and P be three non-overlapping subsets of dataset .We use these datasets in the process of synthesis, structure learning, and parameter learning, respectively.§.§ Model Let {x_1, x_2, ..., x_m } be the set of random variables associated with the attributes of the data records in .Let 𝒢 be a directed acyclic graph (DAG), where the nodes are the random variables, and the edges represent the probabilistic dependency between them.A directed edge from x_j to x_i indicates the probabilistic dependence of attribute i to attribute j.Let (i) be the set of parents of random variable i according to the dependency graph 𝒢.The following model, which we use in Section <ref> to generate synthetic data, represents the joint probability of data attributes. x_1, ..., x_m= ∏_i=1^m x_i {x_j }_∀j ∈(i)This model is based on a structure between random variables, captured by 𝒢, and a set of parameters that construct the conditional probabilities.In Section <ref> and Section <ref>, we present our differentially-private algorithms to learn the structure and parameters of the model from , respectively. §.§ Synthesis Using a generative model, we probabilistically transform a real data record (called the seed) into a synthetic data record, by updating its attributes.Let {x_1, x_2, ..., x_m } be the values for the set of data attributes for a randomly selected record in the seed dataset S.Let ω be the number of attributes for which we generate new values.Thus, we keep (i.e., copy over) the values of m - ω attributes from the seed to the synthetic data.Let σ be a permutation over {1, 2, ..., m} to determine the re-sampling order of attributes. We set the re-sampling order σ to be the dependency order between random variables.More precisely, ∀j ∈(i): σ(j) < σ(i).We fix the values of the first m - ω attributes according to σ (i.e., the synthetic record and the seed overlap on their {σ(1), ..., σ(m-ω) } attributes).We then generate a new value for each of the remaining ω attributes, using the conditional probabilities (<ref>).As we update the record while we re-sample, each new value can depend on attributes with updated values as well as the ones with original (seed) values. We re-sample attribute σ(i), for i > m-ω, as x'_σ(i)∼x_σ(i) {x_σ(j) = x_σ(j)}_∀j ∈(i), j ≤ m-ω, {x_σ(j) = x'_σ(j)}_∀j ∈(i), j > m-ω In Section <ref>, we show how to protect the privacy of the seed data record using our plausible deniability mechanisms.Baseline: Marginal Synthesis. As a baseline generative model, we consider a synthesizer that (independently from any seed record) samples a value for an attribute from its marginal distribution.Thus, for all attribute i, we generate x_i ∼x_i.This is based on an assumption of independence between attributes' random variables, i.e., it assumes x_1, ..., x_m = ∏_i=1^m x_i.§.§ Privacy-Preserving Structure Learning Our generative model depends on the dependency structure between random variables that represent data attributes.The dependency graph 𝒢 embodies this structure.In this section, we present an algorithm that learns 𝒢 from real data, in a privacy-preserving manner such that 𝒢 does not significantly depend on individual data records.The algorithm is based on maximizing a scoring function that reflects how correlated the attributes are according to the data.There are multiple approaches to this problem in the literature <cit.>.We use a method based on a well-studied machine learning problem: feature selection.For each attribute, the goal is to find the best set of features (among all attributes) to predict it, and add them as the attribute's parents, under the condition that the dependency graph remains acyclic. The machine learning literature proposes several ways to rank features in terms of how well they can predict a particular attribute.One possibility is to calculate the information gain of each feature with the target attribute.The major downside with this approach is that it ignores the redundancy in information between the features.We propose to use a different approach, namely Correlation-based Feature Selection (CFS) <cit.> which consists in determining the best subset of predictive features according to some correlation measure.This is an optimization problem to select a subset of features that have high correlation with the target attribute and at the same time have low correlation among themselves.The task is to find the best subset of features which maximizes a merit score that captures our objective.We follow <cit.> to compute the merit score for a parent set (i) for attribute i as score((i)) = ∑_j∈(i)corr(x_i, x_j)/√(|(i)| +∑_j,k∈(i)corr(x_j, x_k)) , where |(i)| is the size of the parent set, and corr() is the correlation between two random variables associated with two attributes.The numerator rewards correlation between parent attributes and the target attribute, and the denominator penalizes the inner-correlation among parent attributes.The suggested correlation metric in <cit.>, which we use, is the symmetrical uncertainty coefficient: corr(x_i, x_j) = 2 - 2 H(x_i, x_j)/H(x_i) + H(x_j) , where H() is the entropy function.The optimization objective in constructing 𝒢 is to maximize the total score((i)) for all attributes i.Unfortunately, the number of possible solutions to search for is exponential in the number of attributes, making it impractical to find the optimal solution.The greedy algorithm, suggested in <cit.>, is to start with an empty parent set for a target attribute and always add the attribute (feature) that maximizes the score.There are two constraints in our optimization problem.First, the resulting dependency graph obtained from the set of best predictive features (i.e., parent attributes) for all attributes should be acyclic.This would allow us to decompose and compute the joint distribution over attributes as represented in (<ref>).Second, we enforce a maximum allowable complexity cost for the set of parents for each attribute.The cost is proportional to the number of possible joint value assignments (configurations) for the parent attributes.So, for each attribute i, the complexity cost constraint is cost((i)) = ∏_j∈(i) |x_j| ≤ where |x_j| is the total number of possible values that the attribute j takes.This constraint prevents selecting too many parent attribute combinations for predicting an attribute.The larger the joint cardinality of attribute i's parents is, the fewer data points to estimate the conditional probability x_i {x_j }_∀j ∈(i) can be found.This would cause overfitting the conditional probabilities on the data, that results in low confidence parameter estimation in Section <ref>.The constraint prevents this.To compute the score and cost functions, we discretize the parent attributes.Letbe a discretizing function that partitions an attribute's values into buckets.If the attribute is continuous, it becomes discrete, and if it is already discrete,might reduce the number of its bins.Thus, we update conditional probabilities as follows. x_i {x_j }_∀_j ∈(i)≈x_i {x_j}_∀_j ∈(i) where the discretization, of course, varies for each attribute.We update (<ref>) and (<ref>) according to (<ref>).This approximation itself decreases the cost complexity of a parent set, and further prevents overfitting on the data. §.§.§ Differential-Privacy Protection In this section, we show how to safeguard the privacy of individuals whose records are in , and could influence the model structure (which might leak about their data).All the computations required for structured learning are reduced to computing the correlation metric (<ref>) from .Thus, we can achieve differential privacy <cit.> for the structure learning by simply adding appropriate noise to the metric.As, the correlation metric is based on the entropy of a single or a pair of random variables, we only need to compute the entropy functions in a differentially-private way.We also need to make sure that the correlation metric remains in the [0, 1] range, after using noisy entropy values.Let H̃(z) be the noisy version of the entropy of a random variable z, where in our case, z could be a single or pair of random variables associated with the attributes and their discretized version (as presented in (<ref>)).To be able to compute differentially-private correlation metric in all cases, we need to compute noisy entropy H̃(x_i), H̃(x_i), H̃(x_i, x_j), and H̃(x_i, x_j), for all attributes i and j.For each of these cases, we generate a fresh noise drawn from the Laplacian distribution and compute the differentially-private entropy as H̃(z) = H(z) + Lap(Δ_H/ε_H) where Δ_H is the sensitivity of the entropy function, and ε_H is the differential privacy parameter.It can be shown that if z is a random variable with a probability distribution, estimated from n_T = | T |data records, then the upper bound for the entropy sensitivity is Δ_H≤1/n_T [2 + 1/ln(2) + 2 log_2n_T] = O(log_2n_T/n_T) The proof of (<ref>) can be found in Appendix <ref>.Remark that Δ_H is a function of n_T (the number of records in T) which per se needs to be protected.As a defense, we compute Δ_H in a differentially-private manner, by once randomizing the number of records ñ_T = n_T + Lap(1/ε_n_T)By using the randomized entropy values, according to (<ref>), the model structure, which will be denoted by 𝒢̃, is differentially private.In Section <ref>, we use the composition theorems to analyze the total privacy of our algorithm for obtaining a differentially-private structure.§.§ Privacy-Preserving Parameter Learning Having a dependency structure 𝒢̃, we need to compute the conditional probabilities for predicting each of the attributes given its parent set (see (<ref>)).This is a well-known problem in statistics.In this section, we show how to learn the parameters that represent such conditional probabilities, from P, in a differentially private manner.The problem to be solved is to first learn a prior distribution over the parameters of the conditional probabilities.To do so, we learn the hyper-parameters (the parameters of the prior distribution over the model's parameters) from data.Only then, we can compute the parameters that form the conditional probabilities from the prior distribution.Let us take the example of computing the parameters for predicting discrete/categorical attributes.In this case, we assume a multinomial distribution over the attribute's values (that fall into different bins).The conjugate prior for multinomials comes from a Dirichlet family.The Dirichlet distribution assigns probabilities to all possible multinomial distributions, according to the statistics obtained from a set of data records.Let |x_i| be the number of distinct values that attribute i can take.The probability of some multinomial distribution parameters p⃗^c_i = p^c_i,1, p^c_i,2, ..., p^c_i,|x_i| to predict attribute i, under configuration c for (i), is p⃗^c_i 𝒢̃, P = Dir(α⃗^c_i + n⃗^c_i) where α⃗^c_i is the vector of default hyper-parameters for the Dirichlet distribution, and n⃗^c_i is the vector for the number of data records in P with (i) configuration c with different values for attribute i (i.e., element n ^c_i,l is the number of records for which x_i = l and (i) configuration is c).The Dirichlet distribution is computed as Dir(α⃗^c_i + n⃗^c_i) = ∏_i=1^m ∏_c=1^#cΓ(α^c_i + n^c_i) ∏_l=1^|x_i|(p^c_i)^α^c_i,l + n^c_i,l-1/Γ(α^c_i,l + n^c_i,l) where α^c_i = ∑_l α^c_i,l, and n^c_i = ∑_l n^c_i,l, and the number of configurations #c is ∏_j∈(i) |x_j|, which according to constraint (<ref>) can at most be .Learning the parameters of the model, in the case of a Dirichlet prior for multinomial distribution, is simply computing n⃗^c_i from the data records in P.Given the probability distribution (<ref>) over the multinomial parameters, we can compute the most likely set of parameters asp^c_i,l = α^c_i,l + n^c_i,l/α^c_i + n^c_i or, we can sample a set of multinomial parameters according to (<ref>).This is what we do in our generative model, in order to increase the variety of data samples that we can generate. Note that for computing the marginal distributions, that are needed for the baseline, we perform the same computations by setting the parent sets to be empty.If an attribute is continuous, we can learn the parameters of a Normal distribution or learn a regression model from our data to construct its conditional probability.We omit the details here (as in the dataset we evaluate in Section <ref> all attributes are discrete).§.§.§ Differential-Privacy Protection The parameters of the conditional probabilities depend on the data records in P, thus they can leak sensitive information about individuals who contributed to the real dataset.In this section, we show how to learn parameters of the attribute conditional probabilities (i.e., p⃗^c_i values) with differential privacy guarantees.Note that in (<ref>), the only computations that are dependent on P are the n⃗^c_i counts (for all c and i).To find the variance of the noise to be added to these counts, to achieve differential privacy, we need to compute their sensitivity with respect to one individual's data record.Suppose we are computing the parameters associated with predicting a given attribute i given its parent set (i).Note that adding a record to P increases exactly a single component n^c_i,l, for which it matches value l for attribute i and configuration c for its parent set.So, only one single element among all #c × |x_i| elements of n⃗_i = n⃗^1_i, n⃗^2_i, ..., n⃗^#c_i changes.This implies that the L1 sensitivity of n⃗_i is 1.Consequently, a random noise drawn from Lap(1/ε_p) can be added to each component of n⃗_i independently.More precisely, for any attribute i value l, and configuration c, we randomize counts as ñ^c_i,l = max(0, n^c_i,l + Lap(1/ε_p)) and use them to compute (<ref>) with differential privacy.§.§ Differential Privacy Analysis In this section, we compute the differential privacy level that we can guarantee for the whole datasetfor learning the structure and the parameters of the model.We compute the total (ϵ, δ) privacy by composing the differentially-private mechanisms in Section <ref> and Section <ref>.Remark that we often protect the output f(x) of some function f by adding noise from Lap(Δ_f/ε), where Δ_f is the L1 sensitivity of f. This mechanism is known as the Laplace mechanism and it satisfies ε-differential privacy (Theorem 3.6 of <cit.>).Thus the m (m+1) entropy values, H̃(z), needed for structure learning (Section <ref>) are obtained in an a way that satisfies ε_H-differential privacy. This is also the case for the number of records n_T, i.e., it satisfies ε_n_T-differential privacy. Similarly, the counts n^c_i,l parameters learned for each configuration (Section <ref>) satisfy ε_p-differential privacy.For structure learning, we make use of both sequential composition (Theorem <ref>) and advanced composition (Theorem <ref>).Specifically, we use advanced composition for the m (m+1) entropy values and sequential composition with the number of records. That is, the overall privacy achieved (of structure learning) is (ε_L,δ_L)-differential privacy for a fixed δ_L ≪1/n_T and ε_L = ε_n_T + ε_H √(2 m(m+1) ln(δ_L^-1))+ m(m+1) ε_H (e^ε_H - 1).For the parameter learning (as explain in Section <ref>), for a given attribute i, the L1 sensitivity of all configurations of the parent set of i, i.e., (i), is 1. The overall privacy achieved (of parameter learning) is (ε_P,δ_P)-differential privacy using advanced composition over the m attributes. Here, δ_P ≪1/n_p (where n_p is the number of records in P) and ε_P = ε_p √(2 m ln(δ_P^-1)) + m ε_p (e^ε_p - 1).Given that T and P are non-overlapping, the privacy obtained for the generative model is differentially private with parameters (max{ε_L, ε_P}, max{δ_L,δ_P}).Due to random subsampling of T and P from , the privacy parameters can be further improved by using the amplification effect of sampling (Theorem <ref>) to obtain (ε,δ)-differential privacy. § DATA For validation, we use the 2013 American Community Survey (ACS) <cit.> from the U.S. Census Bureau. The dataset contains upwards of 3 million of individual records. Each record includes a variety of demographics attributes such as age, sex, race, as well as attributes related to the individual's income such as yearly income in USD.The ACS dataset has been used for various purposes ranging from examining the relationship between education and earnings <cit.> to looking at current language use patterns of children of immigrants <cit.>. Furthermore, the prominent UCI Adult dataset, which provides a well-established benchmark for machine learning tasks, was extracted from the 1994 Census database. The 2013 ACS dataset contains similar attributes so we process it in a manner similar to how the Adult dataset was extracted. In particular, we extract the same attributes whenever possible.As pre-processing, we discard records with missing or invalid values for the considered attributes (Table <ref>). Table <ref> shows some statistics of the data cleaning and extracted dataset. This is a highly dimensional dataset despite having only 11 attributes, there are more than half a trillion possible records and out of the roughly 1.5 million records obtained after cleaning, approximately 2/3 are unique.We bucketize (Section <ref>) values of the age attribute in bins (i.e., buckets) of 10, i.e., 17 to 26, 27 to 36, etc. (Following the rules used to extract the Adult dataset, we only consider individuals older than 16.) We also bucketize the values of: hours worked per week (HPW), in bins of 15 hours; education, to aggregate education level below a high-school diploma in a single bin, and high-school diploma but not college into (another) single-bin. Bucketization is performed based on the data format and the semantics of attributes (and thus is privacy-preserving). It is done only for structured learning (Section <ref>); both the input and output data format remain the same. § SYNTHETICS GENERATOR TOOL The synthetic generator <cit.> is implemented as C++ tool which takes as input: a dataset represented as a CSV file, a few metadata text files describing the dataset, and a config file. As output, the tool produces a synthetic dataset of the requested size and some metadata.The generation process is defined by the config file, i.e., parameters defined within in control various aspects of the generation process. The parameters are the privacy parameters k, γ, ε_0, and also parameters of the generative model such as ω. In addition, the tool takes two optional parameters to control the privacy test:and , which allow the test to terminate early. Specifically, the implementation initially sets k' = 0, and iterates over the records of D in a random order, incrementing k' for each plausible seed record d_a encountered. The process terminates whenever k' ≥ or ifrecords have been examined (whichever occurs first). Note that this affects performance (but not privacy); lower values lead to faster generation time in cases where plausible seeds are abundant, at the cost of fewer synthetics passing the test (potentially lowering utility).The synthesis process, given a chosen seed, is independent of other seeds (Section <ref>); so the generation process itself is embarrassingly parallel. One hurdle with running multiple concurrent instances is implementing differentially-private parameters learning (Section <ref>). In general, the number of configuration (in the sense of Section <ref>) of the model is too large (i.e., exponential in the number of attributes) to learn the model as a pre-processing step. So we design the tool to learn the model for each configuration as it encounters it. To ensure that the privacy guarantee holds we set the RNG seed number to be a deterministic function (i.e., a hash) of the configuration. § EVALUATION We feed the 2013 ACS dataset (Section <ref>) as input to our tool and generate millions of synthetic records. We start with a description of the experimental setup. The evaluation itself is divided into four logical parts: (Section <ref>) statistical measures (how good are the synthetics according to well-established statistical metrics); (Section <ref>) machine learning measures (how good are the synthetics for machine learning tasks, specifically classification); (Section <ref>) distinguishing game (how successful is an adversary at distinguishing between a real record and a synthetic one); and (Section <ref>) performance measures (how computationally complex it is to generate synthetics). §.§ Setup To achieve differential privacy we sampled the input dataset into disjoint sets of records. Each of T and P contains roughly 280,000 records, whereas S contains roughly 735,000 records (Section <ref>). For differential privacy of the generative model, we set ε=1 (though we give some results for ε=0.1) and always set δ to be at most 2^-30≈ 10^-9.We typically compare the quality of our generated synthetics with real records (coming from the input dataset) and privacy-preserving marginals (Section <ref>) which we refer to as reals and marginals, respectively. The synthetics we generate are referred by their generation parameters (e.g., ω = 10). Unless otherwise stated, we set k=50, ε_0=1, γ = 4, and ω is set to vary between 5 and 11. We maintain a testing set of roughly 100,000 records. Evaluation of classifiers (in this section) uses at least 100,000 records for training and a (disjoint) testing set of size at least 30% of the size of the aforementioned training set.§.§ Statistical Measures We evaluate the quality of the synthetics in terms of their statistical utility, i.e., the extent to which they preserve the statistical properties of the original (input) dataset. We can do this at the level of the generative model (Section <ref>) itself. Concretely, we directly quantify the error of the privacy-preserving generative model before any synthetic record is generated. We do this for each attribute by repeatedly selecting a record from the input dataset (uniformly at random) and using the generative model to find the most likely attribute value (of that attribute) given the other attributes. The generative model error is then measured as the proportion of times that the most likely attribute value is not the correct (i.e., original) one. We repeat this procedure millions of times to quantify the average error of the model for each attribute. Because the generative model is made differentially private by adding noise (Section <ref>) we additionally repeat the whole procedure 20 times (learning a different private model each time) and take the average.The results are shown in Figures <ref> and <ref>. Figure <ref> shows the relative decrease in model error (i.e., improvement of model accuracy) over the (privacy-preserving) marginals; it shows this improvement for the un-noised, (ε=1)-differential private, and (ε=0.1)-differential private generative models. There is a clear accuracy improvement over marginals, in addition to a low decrease in improvement between the un-noised model and the ε=1 and ε=0.1 noisy versions. Figure <ref> shows the accuracy of the un-noised generative model against the (un-noised) marginals, random guessing (baseline), and the best classifier we could find (trained on as many records as the generative model), the random forest (RF). While RF's accuracy is sometimes higher than that of the generative model, the accuracy of the latter is in many cases significantly higher than that of marginals and random guessing.We conclude that while the proposed generative model does not perform as well as RF (though making RF differentially private would certainly lower its performance) it does perform significantly better than marginals (or random guessing). In addition to the error of the generative model, we can more directly evaluate the extent to which the generated synthetics preserve the statistical properties of the original (input) dataset. To do this, we compare the probability distributions of the synthetics with the reals and marginals. Specifically, for reals, marginals and synthetics datasets, we compute the distribution of each attribute and of each pair of attributes. We compare each of these distributions to those computed on (other) reals and quantify their distance. We use a well-established statistical distance metric called “the” statistical distance (a.k.a. total variation distance <cit.>).The results are shown in Figures <ref> and <ref>, where Figure <ref> shows box-and-whisker plots for the distance of the distributions of each attribute separately, and Figure <ref> shows box-and-whisker plots for the distance of the distributions of all pairs of attributes. While marginals do well for single attribute and sometimes outperform our synthetics (though the statistical distance for all datasets is small), synthetics clearly outperform marginals for pairs of attributes. We conclude that the generated synthetics preserve significantly more statistical information than marginals. §.§ Machine Learning Measures In addition to preserving statistical properties of the original (input) dataset, the synthetics should also be suitable to various machine learning tasks. In particular, given a learning task, we can evaluate the extent to which synthetics are suitable replacements for a real dataset. For the ACS dataset, a natural and well-establish classification task is to predict a person's income class (i.e., ≥ 50K or < 50K) using the other attributes as features (Section <ref>). We train various classifiers on the synthetic datasets and on the real (input) dataset. We then compare: the classification accuracy obtained, and the agreement rate of the learned classifiers. Specifically, for two classifiers trained on different datasets (but with the same classification task), we define the agreement rate to be the percentage of records for which the two classifiers make the same prediction (regardless of whether the prediction is correct). Given that we look at the agreement rate of classifiers trained on reals and synthetics, the agreement rate reveals the extent to which the classifier trained on synthetic data has learned the same model as the classifier trained on real data.Table <ref> shows the obtained results for three (best) classifiers: Classification Tree (Tree), Random Forest (RF), and AdaBoostM1 (Ada). The accuracy and agreement rate are calculated as the average over 5 independent runs, that is, for each run, we use different (randomly sampled) training and testing datasets. Overall, we see that both the accuracy and the agreement rates of the synthetics are significantly closer to that of the reals than the marginals are.In addition to comparing the best classifiers trained on real data versus those trained on synthetic data, we can also compare privacy-preserving classifiers trained on real data versus non-private classifiers trained on (privacy-preserving) synthetic data.In particular, Chaudhuri et al. <cit.> propose two techniques based on empirical risk minimization to train logistic regression (LR) and support vector machines (SVM) binary classifiers: output perturbation (noise is added to the learned model), and objective perturbation (noise is added to the objective function of the minimization problem). To train such classifiers, we first pre-process our datasets following the instructions in <cit.>: we transform each categorical attribute into an equivalent set of binary attributes, and normalize features so that each feature takes values in [0,1] and subsequently further normalize each training example such that its norm is at most 1. The target attribute for classification is again the person's income class. The method proposed in <cit.> has two parameters: the privacy budget ε which we set to 1 (the same as for our generative model), and λ which is a regularization parameter. We use the code of <cit.>, which we obtain courtesy of the authors, to train the LR and SVM classifiers. Because the classification models vary greatly depending on λ, we vary its value in the set {10^-3,10^-4,10^-5, 10^-6} and (optimistically) pick whichever value maximizes the accuracy of the non-private classification model.We report the accuracy obtained in each case in Table <ref>, where we contrast non-private, output perturbation DP, and objective perturbation DP classifiers trained on real data with non-private classifiers trained on our synthetic datasets (for various values of ω). Remark that for the case ω=11, for example, this is a fair comparison as the obtained LR and SVM classifiers are ε=1-DP and thus provides the exact same privacy guarantee as the output perturbation and objective perturbation LR and SVM classifiers. Non-private LR and SVM classifiers trained on our (privacy-preserving) synthetic datasets are competitive with differentially private LR and SVM classifiers trained on real data. We emphasize that the results should be interpreted in favor of our proposed framework. Indeed, the classifiers trained on our privacy-preserving synthetics outperforms ε-DP LR classifier and only achieves about 1% lower accuracy than the objective-perturbation ε-DP SVM.This is significant because the technique to train the ε-DP LR and SVM is specifically optimized for that task. In contrast, our synthetics are not specifically generated to optimize any particular classification task; instead the general objective is to preserve the statistical properties of real data. §.§ Distinguishing Game A different way to evaluate the quality of synthetic datasets is to quantify the extent to which the synthetics can “pass off” as real records. In other words, we can imagine a game in which the participant is given a random record either from a real dataset or a synthetic dataset (but doesn't know which) and is asked to distinguish between the two possibilities. In this case, the utility is measured by how likely a sophisticated participant (e.g., a well-established learning algorithm) is to make a mistake, i.e., confuse a synthetic record with a real record or vice-versa.For our purpose the role of the participant is played by the two best classifiers (those that best distinguish synthetics from reals): Random Forest (RF) and Classification Tree (Tree). Specifically, we provide 50,000 records from both a real dataset and a synthetic dataset (i.e., 100,000 total) as training examples to the (binary) classifier. We then evaluate the accuracy on a 50% mix of real and synthetic records which were not part the training set. Table <ref> shows the results: both classifiers obtain reasonably high (79.8% and 73.2%) accuracy in distinguishing marginals from real records. However, both classifiers obtain much lower accuracy (i.e., 63%) when trying to distinguish synthetics from reals. §.§ Performance Measures In addition to how much utility they preserve, synthetics also need to be easy to generate. The generation is a parallel process, so we measure the time taken for both the learning of the privacy-preserving generative model (model learning) and the synthetics generation (synthesis) itself. Figure <ref> shows the time taken to produce various number of synthetics records (totaling over 1 million). The parametersand(Section <ref>) were set to 100 and 50000 respectively. The machine used for the experiment runs Scientific Linux and is equipped with an Intel Xeon E5-2670 processor (2.60GHz) with 32 processing units and 128GB of RAM. We ran 96 instances (16 in parallel at a time) and picked a random maximum runtime for each instance between 3 and 15 minutes.The generator outputs all synthetics produced regardless of whether they pass the privacy test. Naturally, only those which pass the test should to be released. Thus, the extent to which we can synthesize large (privacy-preserving) datasets depends on how easy it is to find synthetics that pass the privacy-test (Section <ref>). To evaluate this, we set γ = 2 and = 100,000, and vary k and ω. We measure the proportion of synthetics which pass the privacy test. The results are shown in Figure <ref>: even for stringent privacy parameters (e.g., k=100) a significant proportion (i.e., over 50% for ω∈_R [5-11]) of synthetics pass the test.§ RELATED WORK Data synthesis is the process of creating artificial data that can be used to serve some key purposes of real data. For example, it can be used to test software <cit.> when there is an inadequate supply of real tests available from operations. It can be used to evaluate the performance of database systems <cit.> when obtaining real data at sufficient scale is difficult or expensive.It can also be used to protect privacy. In 1993, Rubin <cit.> proposed the idea of creating synthetic data based on multiple imputation, that is, on repeated use of a function that proposes values for missing fields in a record. The generative model we presented in Section <ref> uses a similar technique. This and related work have given rise to a substantial body of research on the release of synthetic data <cit.>. Such techniques have achieved significant deployment; for example, they have been adopted by the U.S. Census Bureau <cit.>.An alternative to data synthesis sometimes called syntactic privacy protection transforms the sensitive data using a combination of aggregation, suppression, and generalization, to achieve criteria such as k-anonymity <cit.> or l-diversity <cit.>. Although these techniques support privacy protected data publishing without synthesis, the degree of privacy protection they provide depends on the background knowledge of adversaries. The key difference between (k,γ)-plausible deniability and k-anonymity is that the latter is a syntactic condition on the output of a sanitization process, whereas plausible deniability is a condition on a synthetic generator mechanism with respect to its input data.Plausible deniability as a privacy notion for synthetic data was proposed by Bindschaedler and Shokri in <cit.> which describes a technique to synthesize location trajectories in a privacy-preserving way. The use of plausible deniability in <cit.> is specific to location privacy as it is defined in terms of a semantic distance between two location trajectories. In contrast, this work generalizes the notion of plausible deniability for general data synthesis by establishing it as a privacy criterion in terms of the underlying synthesis probabilities. Consequently, this criterion is applicable to any system and any (kind of) data. The generative framework described in this paper enables us to formally connect plausible deniability to differential privacy (Theorem <ref>).Differential privacy provides guarantees even against adversaries with (almost) unlimited background knowledge. However, popular differentially private mechanisms such as the Laplacian mechanism target the release of aggregate statistics. By contrast, we focus on synthesizing data with the same format as the (sensitive) input data. Preserving the data format is valuable for many reasons, such as enabling the use of applications and code that are targeted at raw or sanitized data. There is a line of work on mechanisms that are differentially private and provide data as an output. Some of these techniques have theoretical properties that may make them impractical in important cases <cit.>.A prominent example is the Exponential Mechanism <cit.>.Informally, the mechanism induces a distribution on the space of output records by assigning a weight to each such record and then producing output records by sampling from that distribution. The mechanism is of great importance for algorithm design due to its generality. However, as several researchers have pointed out <cit.>, a direct application is too costly to be practical for high-dimensional datasets due to the complexity of sampling, which grows exponentially in the dimension of the data records. Concretely, a straightforward implementation of the exponential mechanism to generate synthetic records from the ACS dataset (Section <ref>) would sample from a universe of records of size roughly 2^39 (Table <ref>). This would require pre-computing that many probabilities. If we assume we can store each value in four bytes this would require about 2TB of memory. In contrast, the complexity of synthesizing a record with our framework depends only on the number of records in the dataset and on the complexity of our generative model and thus the process is very efficient in practice (Section <ref>).There is a growing collection of mechanisms and case studies for differentially private release of data <cit.>, although some of these are based on a broad view of data release, such as the release of histograms or contingency tables. Our use of plausible deniability to achieve differentially private data adds to this body of work. The typical approach to protect privacy in this context is to add noise directly to the generative model. For example, this is the approach taken by <cit.>. In particular, <cit.> constructs a generative model based on Bayesian networks similar to the generic generative model of Section <ref>.Our work takes a novel approach: instead of trying to achieve differential privacy directly, we design the generative framework to achieve plausible deniability. A major step towards achieving plausible deniability and a key novelty is the idea of testing privacy. That is, instead of designing the mechanism so it achieves that notion by design, we use a privacy test which rejects “bad” samples. As a side effect, the generative model is decoupled from the framework. Privacy is guaranteed in a way that is oblivious to the specifics of the generative model used for synthesis. Furthermore, the guarantee offered by our proposed (k,γ)-plausibly deniable mechanism is surprisingly close to that of differential privacy, as evidenced by the fact that merely randomizing the threshold yields a differentially private mechanism.The idea of running data synthesis and then testing privacy has been used before. For example, Reiter et al. in <cit.> and <cit.> use inference to evaluate privacy risk for a synthetic data release. However, there is no proof of privacy, and it is not efficient to run a set of inference attacks to estimate the risk before releasing a dataset.§ DISCUSSION Regardless of whether one intends to release a set of aggregate statistics or a synthetic dataset, there is no privacy protection technique that can preserve utility for all meaningful utility functions. However, one key feature of our generative framework is that, unlike other approaches based on differential privacy, any generative model ℳ can be used while keeping the same privacy guarantees.As a result, designing ℳ is a data science problem which need not involve considerations of the privacy of the seeds.Parameter ω (Section <ref>) controls the closeness of synthetics to their seeds. (Lower ω means more dependence on the seed but it is harder to pass the privacy test.) A pragmatic approach is to generate synthetics for various values of ω and then randomly sample a subset of those synthetics which pass the privacy test (this is evaluated in Section <ref>). Note that no matter what the value of ω is, the privacy of the seeds is ensured because of the privacy test.In the special case where the generative model ℳ is independent of the seed, the privacy guarantee applies to any output from Mechanism <ref> because the privacy test always passes. However, for a seed-dependent generative model, the privacy of the seeds is safeguarded by rejecting synthetics which do not pass the privacy test. So, when generating several synthetics using the same input dataset, the privacy obtained depends on the number of synthetics released.In fact, when Privacy Test <ref> is used, the (ε,δ)-differential privacy guarantee applies to a single (released) synthetic record y. That said, the composition theorems for differential privacy can be used to extend the guarantee to arbitrarily large synthetic datasets, provided the privacy budget is appropriately increased. We leave as future work the design of improved composition strategies.§ CONCLUSIONS We have presented the first practical data synthesis tool with strong privacy guarantees. We have formalized plausible deniability for generating generic data records, and have proven that our mechanisms can achieve differential privacy without significantly sacrificing data utility.Acknowledgments. This work was supported in part by NSF CNS grants 13-30491 and 14-08944. The views expressed are those of the authors only. abbrv § COMPOSING DIFFERENTIAL PRIVACY Let ℱ_i be an (ε_i,δ_i)-differentially private mechanism, for i=1,…,m. Then, for any dataset D, the mechanism which releases outputs: (ℱ_1(D), ℱ_2(D), …, ℱ_m(D) ) is an (ε,δ)-differentially private mechanism for ε = ∑_i=1^mε_i, and δ = ∑_i=1^mδ_i. Let ℱ_i be an (ε,δ)-differentially private mechanism, for i=1,…,m. Then the mechanism represented by the sequence of k queries over ℱ_1(·), ℱ_2(·), …, ℱ_m(·) with potentially different inputs is (ε',δ')-differentially private for: ε' = ε√(2 k ln1/δ”) + k ε ( e^ε - 1 )andδ' = kδ + δ” . Let mechanism ℱ be an (ε,δ)-differentially private mechanism. The mechanism which first sub-samples each record of its input dataset with probability δ < p < 1 and then runs ℱ on the sub-sampled dataset is (ε',δ')-differentially private for: ε' = ln(1 + p (e^ε - 1)) andδ' = pδ . § PROOF: SENSITIVITY OF ENTROPYLet z be a discrete random variable with a probability distribution estimated from n data records. The sensitivity of H(z) is: Δ_H≤1/n [2 + 1/ln(2) + 2 log_2n] . Lemma <ref> is used in Section <ref> (<ref>). Let z and z' be the random variables associated with two histograms (of dimension m) computed from two neighboring datasets D and D', respectively. Both datasets have n records but differ in exactly one record. Let z_c = (c_1, c_2, …, c_m) and z'_c = (c'_1, c'_2, …, c'_m) represent the histograms over the m values of the attribute for z and z', respectively. The entropy is computed over the probability distribution represented by a histogram. Remark that the histograms of the considered neighboring datasets D and D' can only differ in two positions. If they do not differ in any position, then the Lemma trivially holds (Δ_H = 0). That is, without loss of generality, there exists j_1 and j_2 (with j_1 ≠ j_2) such that c'_j_1 = c_j_1 + 1 and c'_j_2 = c_j_2 - 1. Also, n-1 ≥ c_j_1≥ 0 which means that n ≥ c'_j_1≥ 1, and n ≥ c_j_2≥ 1 which means that n-1 ≥ c'_j_2≥ 0. Furthermore for i ≠ j_1, j_2, we have c'_i = c_i, and also:∑_i=1^m c_i = ∑_i=1^m c'_i = n. Now: H(z)= -∑_i=1^mc_i/nlog_2c_i/n= -1/n[ ∑_i=1^m c_i log_2c_i - n log_2n] = log_2n - 1/n( c_j_1log_2c_j_1 + c_j_2log_2c_j_2 + ∑_i ≠ j_1,j_2 c_i log_2c_i). Similarly, H(z')= log_2n - 1/n∑_i ≠ j_1,j_2 c_i log_2c_i- 1/n[ (c_j_1+1) log_2(c_j_1+1) + (c_j_2-1) log_2(c_j_2-1)]. We have that Δ_H = max_c_j_1,c_j_2|H(z) - H(z') |, but for brevity we omit the max and analyze this quantity with respect to the values of c_j_1 and c_j_2 to show that the lemma holds in each case. Observe that: Δ_H = |H(z) - H(z') |= 1/n| c_j_1log_2c_j_1 - (c_j_1+1) log_2(c_j_1+1). + . c_j_2log_2c_j_2 - (c_j_2-1) log_2(c_j_2-1)| . * Case 1: c_j_1 = 0. We haveΔ_H = 1/n | c_j_2log_2c_j_2 - (c_j_2-1) log_2(c_j_2-1)|. Clearly, if c_j_2 = 1 then Δ_H = 0 and the Lemma trivially holds. So assume c_j_2 > 1. We have: Δ_H = 1/n | c_j_2log_2c_j_2 - (c_j_2-1) log_2(c_j_2-1) | = 1/n| c_j_2log_2(c_j_2/c_j_2 - 1) + log_2(c_j_2 - 1)|≤1/n| c_j_2log_2(c_j_2/c_j_2 - 1)| + 1/nlog_2(c_j_2 - 1)≤1/nlog_2(n-1) + 1/n| (a+1) log_2(1+1/a)| , where a = c_j_2-1 ≥ 1. It is easy to see that (a+1) log_2(1+1/a) ≤ 2. We conclude that: Δ_H≤1/n (2 + log_2n). * Case 2: c_j_2 = 1. We haveΔ_H = 1/n | c_j_1log_2c_j_1 - (c_j_1+1) log_2(c_j_1+1)|. Again if c_j_1 = 0, then the Lemma trivially holds. So assume c_j_1 > 0. We have: Δ_H = 1/n | c_j_1log_2c_j_1 - (c_j_1+1) log_2(c_j_1+1)| = 1/n|c_j_1log_2(c_j_1/c_j_1+1) - log_2(c_j_1+1)|≤log_2n/n +1/n|c_j_1log_2(c_j_1/c_j_1+1)|= log_2n/n + 1/n c_j_1log_2(c_j_1+1/c_j_1)= log_2n/n + 1/n c_j_1log_2(1 + 1/c_j_1) . Using L'Hopital's rule we have c_j_1log_2(1 + 1/c_j_1)≤1/ln2. We conclude that: Δ_H≤1/n (1/ln2 + log_2n). * Case 3: c_j_1≥ 1, c_j_2≥ 2. We have: Δ_H = 1/n | c_j_1log_2c_j_1 - (c_j_1+1) log_2(c_j_1+1)+ c_j_2log_2c_j_2 - (c_j_2-1) log_2(c_j_2-1) | ≤1/n | c_j_1log_2c_j_1 - (c_j_1+1) log_2(c_j_1+1)| + 1/n | c_j_2log_2c_j_2 - (c_j_2-1) log_2(c_j_2-1) |, where it is seen that the two terms have been bounded for cases 1 and 2. Thus, putting it all together, we conclude that Δ_H≤1/n( 2 + 1/ln2 + 2 log_2n). § PROOF: CONNECTION WITH DIFFERENTIAL PRIVACY In this section, we prove Theorem <ref> of Section <ref>.[Differential Privacy of ℱ] Let ℱ denote Mechanism <ref> with the (randomized) Privacy Test <ref> and parameters k ≥ 1, γ > 1, and ε_0 > 0. For any neighboring datasets D and D' such that |D|, |D'| ≥ k, any set of outcomes Y ⊆𝒰, and any integer 1 ≤ t < k, we have: ℱ(D') ∈ Y≤ e^εℱ(D) ∈ Y + δ , for δ = e^-ε_0 (k-t) and ε = ε_0 + ln(1+γ/t). We start with some notation. Let 𝒰 denote the universe of data records. All data records, i.e., those from the datasets D and D', including synthetic records produced by ℳ (and ℱ) are elements of 𝒰. Let D and D' denote two neighboring datasets, i.e., either D = D' ∪{ d } for some d ∈𝒰, or D' = D ∪{ d' } for some d' ∈𝒰. We assume that both D and D' have at least k records, and we have parameters k ≥ 1, γ > 1, ε_0 > 0. For convenience we write p_d(y) = y = ℳ(d), and refer to ℳ only implicitly.Given a dataset D^⋆, we want to reason about the probability that synthetic record y is released: ℱ(D^⋆) = y. Observe that given synthetic record y ∈𝒰, the records of D^⋆ can be partitioned (into disjoint sets) by the privacy criterion. Concretely, let I_d(y) be the partition number of a record d ∈ D^⋆ with respect to y. The partition number I_d(y) is the unique non-negative integer such that γ^-(I_d(y)+1) < p_d(y) ≤γ^-I_d(y). In other words, I_d(y) = ⌊ - log_γ p_d(y) ⌋. If p_d(y) = 0 then the partition number is undefined. Similarly, we define the partition (or partition set) for i ≥ 0as C_i(D^⋆,y) = { d : d ∈ D^⋆, I_d(y) = i }. That is, partition i is the set of records with partition number i. A key step is to express the probability ℱ(D^⋆) = y in terms of: (1) the probability of generating y from a specific partition (i.e., the seed is in that partition) and (2) the probability of passing the test. For (2) remark that the probability of passing the privacy test depends only on the partition of the seed (see Privacy Test <ref>). For any dataset D^⋆, if the seed is in partition i, the probability of passing the privacy test is given by: pt(D^⋆, i, y) = L ≥ k - |C_i(D^⋆,y)|, where L ∼Lap(1/ε_0). For any dataset D^⋆, the probability of producing y from partition i is: q(D^⋆, i, y) = pt(D^⋆, i, y) ∑_s ∈ C_i(D^⋆,y) p_s(y).As the following demonstrates, ℱ(D^⋆) = y is readily expressed in terms of (1) and (2). For any dataset D^⋆ and any synthetic record y ∈𝒰 we have: ℱ(D^⋆) = y = 1/|D^⋆|∑_i ≥ 0 q(D^⋆, i, y) . In other words, the probability of releasing y (from D^⋆) can be expressed as the sum, over all partitions, of the probability of generating y from a given partition and then releasing it. Fixing a y and following the description of Mechanism <ref>, we have: ℱ(D^⋆) = y = ∑_s ∈ D^⋆sis seed, ℱ(D^⋆) = y= 1/|D^⋆|∑_s ∈ D^⋆ p_s(y)(D^⋆,s,y)passes test , given the fact that the seed is sampled uniformly at random. Further, observe that terms of the sum for which p_s(y) = 0 can be omitted and that we can partition the set { d : d ∈ D^⋆, p_d(y) > 0 } with respect to the partition number of its elements. That is: { d : d ∈ D^⋆, p_d(y) > 0 } = ∪_i ≥ 0 C_i(D^⋆, y), where for each d ∈ D^⋆ such that p_d(y) > 0, there exists a unique non-negative integer j such that d ∈ C_j(D^⋆, y). Thus: ℱ(D^⋆) = y= 1/|D^⋆|∑_s ∈ D^⋆ p_s(y)(D^⋆,s,y)passes test= 1/|D^⋆|∑_i ≥ 0∑_s ∈ C_i(D^⋆, y) p_s(y)(D^⋆,s,y)passes test= 1/|D^⋆|∑_i ≥ 0 L ≥ k - |C_i(D^⋆,y)|∑_s ∈ C_i(D^⋆, y) p_s(y) = 1/|D^⋆|∑_i ≥ 0 pt(D^⋆, i, y) ∑_s ∈ C_i(D^⋆, y) p_s(y) = 1/|D^⋆|∑_i ≥ 0 q(D^⋆, i, y), given that the privacy test depends only on the partition of the seed (and not on the seed itself). (See the description of Privacy Test <ref> with L being drawn from Lap(1/ε_0).) Remark that q(D^⋆, i, y) = 0 if and only if C_i(D^⋆, y) = ∅. Also, observe that if a record is added to or subtracted from D^⋆ then only one partition changes. As a result, we can analyze case-by-case the change in the probability of releasing y from partition i, when adding or removing a record to partition i. The following shows that adding a record to some partition only increases the probability of passing the privacy test by at most e^ε_0. (This is a consequence of adding Laplacian noise to the threshold.) Given any y ∈𝒰, any two neighboring datasets D and D' such that D' = D ∪{ d' }. For any partition i we have: pt(D, i, y) ≤pt(D', i, y) ≤ e^ε_0pt(D, i, y). To prove Lemma <ref>, we use the following observation (which comes from the CDF of the Laplace distribution). For any x ∈ℝ, if L is a Laplace random variable with shape parameter b and mean 0, then we have: L ≥ x≤L ≥ x - 1≤ e^1/bL ≥ x . There are two cases: i = I_d'(y) or i ≠ I_d'(y). If i = I_d'(y) then d' falls into partition i and so C_i(D',y) = C_i(D,y) ∪{ d' }. We have: pt(D', i, y)= L ≥ k - |C_i(D',y)|≤ e^ε_0L ≥ k - |C_i(D',y)| + 1= e^ε_0L ≥ k - |C_i(D,y)| = e^ε_0pt(D, i, y), Also, we have that: pt(D', i, y) > pt(D, i, y). Otherwise, if i ≠ I_d'(y) then C_i(D',y) = C_i(D,y), and so: pt(D', i, y) = L ≥ k - |C_i(D,y)| = pt(D, i, y). Putting it together yields the result. To quantify the change in q(D^⋆, i, y) due to adding a record to partition i we need to separate two cases: (1) the partition is initially empty (or more generally has initially less than t records) and (2) the partition is not empty (or more generally has at least t records). For any y ∈𝒰 and any dataset D. Let D' = D ∪{ d' } for some d' ∈𝒰. Let j be the partition number of d' (i.e., I_d(y) = j). The following holds. (a) For all i ≠ j, we have q(D',i,y) = q(D,i,y). (b) If |C_j(D, y)| < t: q(D, j, y) < q(D', j, y),and q(D', j, y) ≤ e^-ε_0 (k - t)∑_s ∈ C_j(D',y) p_s(y) ≤ te^-ε_0 (k - t) . If |C_j(D, y)| ≥ t: q(D', j, y)/q(D, j, y)≤ e^ε_0[ 1 + γ/t] . For any y ∈𝒰 and any dataset D. Let D' = D ∪{ d' } for some d' ∈𝒰. We have q(D, i, y) ≤ q(D', i, y), for all i ≥ 0. Fix y and let j be the partition that d' falls into. For part (a), remark that for i≠ j, we have C_i(D,y) = C_i(D',y). So q(D, i, y) = q(D', i, y). For part (b), we have that: q(D, j, y)= pt(D, j, y) ∑_s ∈ C_j(D,y) p_s(y) < pt(D, j, y) [ ∑_s ∈ C_j(D,y) p_s(y) + p_d'(y) ]= pt(D, j, y) ∑_s ∈ C_j(D',y) p_s(y) ≤pt(D', j, y) ∑_s ∈ C_j(D',y) p_s(y) = q(D', j, y), given that p_d'(y) > 0 and pt(D', j, y) ≥pt(D, j, y) (Lemma <ref>). Now, if |C_j(D, y)| < t, then: q(D', j, y)= pt(D', j, y) ∑_s ∈ C_j(D',y) p_s(y) ≤ e^-ε_0 (k - t)∑_s ∈ C_j(D',y) p_s(y) ≤ te^-ε_0 (k - t) , given that |C_j(D', y)| ≤ t and p_d(y) ≤ 1 for any d. Here, the first inequality follows from the fact that pt(D', j, y) = L ≥ k - |C_j(D',y)|≤L ≥ k - t = 1/2 e^-ε_0 (k - t). If |C_j(D, y)| ≥ t, we have: q(D', j, y)= pt(D', j, y)∑_s ∈ C_j(D',y) p_s(y) = pt(D', j, y)[ ∑_s ∈ C_j(D,y) p_s(y) + p_d'(y) ] ≤ e^ε_0 pt(D, j, y)[ ∑_s ∈ C_j(D,y) p_s(y) + p_d'(y) ] ≤ e^ε_0 [1 + γ/t]pt(D, j, y) ∑_s ∈ C_j(D,y) p_s(y) = e^ε_0 [1 + γ/t]q(D, j, y), given Lemma <ref> and the fact that p_d'(y) ≤γ p_s(y) for any s ∈ C_j(D,y) and so p_d'(y) ≤γ/t∑_s ∈ C_j(D,y)p_s(y). The following Lemma is the core result underlying Theorem <ref>. Let ℱ denote Mechanism <ref> with the (randomized) Privacy Test <ref> and parameters k ≥ 1, γ > 1, and ε_0 > 0. Take any dataset D with |D| ≥ k and let D' = D ∪{d'} for any d' ∈𝒰. Then for any integer 1 ≤ t < k and synthetic record y ∈𝒰, we have: ℱ(D) = y≤ e^εℱ(D') = y , and ℱ(D') = y≤ e^εℱ(D) = y + δ , where δ = δ(D', d', y) ≤ e^-ε_0 (k-t) and ε = ε_0 + ln(1+γ/t). Here, δ(D',d',y) = e^-ε_0 (k - t) |D'|^-1∑_s ∈ C_j(D',y) p_s(y), with j = I_d'(y). Fix an arbitrary synthetic record y ∈𝒰 and an arbitrary dataset D with |D| ≥ k. Let D' = D ∪{ d' } for some arbitrary d' ∈𝒰. Applying Lemma <ref> to D we have: ℱ(D) = y = 1/|D|∑_i ≥ 0 q(D, i, y). Also, from Corollary <ref> we have q(D, i, y) ≤ q(D', i, y) for all i. Thus: ℱ(D) = y = 1/|D|∑_i ≥ 0 q(D, i, y) ≤1/|D|∑_i ≥ 0 q(D', i, y) = |D'|/|D|ℱ(D') = y≤( 1 + 1/k)ℱ(D') = y . Observe that since (by assumption) γ > 1 and 1 ≤ t ≤ k, we have: 1/k≤1/t, and so 1+1/k < 1 + γ/t≤ e^ε_0 (1 + γ/t) = e^ε. This shows the first part. For the second part, apply Lemma <ref> to D', and let j be the partition number of d', i.e., j = I_d'(y). We have: ℱ(D') = y = 1/|D'|∑_i ≥ 0 q(D', i, y) = 1/|D'|[ ∑_i ≥ 0 : i ≠ j q(D', i, y) + q(D', j, y)] = 1/|D'|∑_i ≥ 0 : i ≠ j q(D, i, y) + q(D', j, y)/|D'| . The last equality follows from Lemma <ref> part (a). Applying Lemma <ref> part (b), we obtain two cases. * Case 1: |C_j(D, y)| < t. We have: ℱ(D') = y = 1/|D'|∑_i ≥ 0 : i ≠ j q(D, i, y) + q(D', j, y)/|D'|≤1/|D'|∑_i ≥ 0 : i ≠ j q(D, i, y) + δ(D',j,y)≤1/|D'|∑_i ≥ 0 q(D, i, y) + δ(D',j,y) = |D|/|D'|ℱ(D) = y + δ(D',j,y) ≤ℱ(D) = y + δ , where δ(D',j,y) = 1/|D'| e^-ε_0 (k - t)∑_s ∈ C_j(D',y) p_s(y). * Case 2: |C_j(D, y)| ≥ t. We have: ℱ(D') = y= 1/|D'|[ ∑_i ≥ 0 : i ≠ j q(D, i, y) + q(D', j, y) ] ≤1/|D'|[ ∑_i ≥ 0 : i ≠ j q(D, i, y) +e^ε_0[1 + γ/t] q(D, j, y) ] ≤ e^ε_0[1 + γ/t] 1/|D'|∑_i ≥ 0 q(D, i, y) = e^ε_0[1 + γ/t] |D|/|D'|ℱ(D) = y≤ e^εℱ(D) = y , for ε = ε_0 + ln(1 + γ/t) given that |D|/|D'| < 1. Letting δ(D',d',y) = δ(D',I_d'(y),y) finishes the proof.With this, we are in a position to prove Theorem <ref>. Fix dataset D with |D| ≥ k and any record d' ∈𝒰. Let D' = D ∪{d'}. The range of ℱ is 𝒰 and so any outcome Y is a non-empty subset of 𝒰. Fix an arbitrary Y ⊆𝒰 with Y ≠∅. We will show that F(D_1) ∈ Y≤ e^εF(D_2) ∈ Y + δ, whether D_1 = D and D_2 = D', or D_1 = D' and D_2 = D. Consider first the case D_1 = D and D_2 = D'. Applying Lemma <ref>, we obtain: ℱ(D) ∈ Y = ∑_y ∈ Yℱ(D) = y≤∑_y ∈ Y e^εℱ(D') = y= e^εℱ(D') ∈ Y . Now, consider the case D_1 = D' and D_2 = D. Define c(d',y) = |C_I_d'(y)(D, y)|. Given d', we can partition Y between those y ∈ Y such that the partition in which d' falls has at least t and those such that the partition has less than t. That is, Y = Y_t-∪ Y_t+, with Y_t- = { y : y ∈ Y, c(d',y) < t } and Y_t+ = { y : y ∈ Y, c(d',y) ≥ t }. We have: ℱ(D') ∈ Y = ∑_y ∈ Yℱ(D') = y≤∑_y ∈ Y_t+e^εℱ(D) = y+ ∑_y ∈ Y_t-[ e^εℱ(D) = y + δ(D', I_d'(y), y) ] = e^ε∑_y ∈ Yℱ(D) = y +∑_y ∈ Y_t-δ(D', d', y) = e^εℱ(D) ∈ Y + ∑_y ∈ Y_t-δ(D', d', y), where the inequality applies cases of Lemmas <ref> and <ref> separately to each y depending on whether y ∈ Y_t- (case 1 in the proof of Lemma <ref>) and y ∈ Y_t+ (case 2 in the proof of Lemma <ref>). It remains to show that ∑_y ∈ Y_t-δ(D', d', y) ≤ e^-ε_0 (k - t). For this define C(D', d', y) = C_I_d'(y)(D', y). We have: ∑_y ∈ Y_t-δ(D', d', y)= ∑_y ∈ Y_t-e^-ε_0 (k - t)/|D'|∑_s ∈ C(D', d', y) p_s(y) = e^-ε_0 (k - t)/|D'|∑_s ∈ D'∑_y ∈ Y_t-1_I_d'(y) = I_s(y) p_s(y) ≤ e^-ε_0 (k - t) , given that ∑_y ∈ Y_t-1_I_d'(y) = I_s(y)p_s(y) ≤∑_y ∈𝒰 p_s(y) ≤ 1 for all s ∈ D'. | http://arxiv.org/abs/1708.07975v1 | {
"authors": [
"Vincent Bindschaedler",
"Reza Shokri",
"Carl A. Gunter"
],
"categories": [
"cs.CR",
"cs.DB",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.CR",
"published": "20170826141328",
"title": "Plausible Deniability for Privacy-Preserving Data Synthesis"
} |
[email protected] of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USAMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA G. S. Gautam and P. Canepa contributed equally to the work G. S. Gautam and P. Canepa contributed equally to the workMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, [email protected], [email protected] of Materials Sciences and Engineering, University of California Berkeley, CA 94720, USAMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAMagnesium oxide and sulfide spinels have recently attracted interest as cathode and electrolyte materials for energy-dense Mg batteries, but their observed electrochemical performance depends strongly on synthesis conditions.Using first principles calculations and percolation theory, we explore the extent to which spinel inversion influences Mg^2+ ionic mobility in MgMn_2O_4 as a prototypical cathode, and MgIn_2S_4 as a potential solid electrolyte.We find that spinel inversion and the resulting changes of the local cation ordering give rise to both increased and decreased Mg^2+ migration barriers, along specific migration pathways, in the oxide as well as the sulfide. To quantify the impact of spinel inversion on macroscopic Mg^2+ transport, we determine the percolation thresholds in both MgMn_2O_4 and MgIn_2S_4. Furthermore, we analyze the impact of inversion on the electrochemical properties of the MgMn_2O_4 cathode via changes in the phase behavior, average Mg insertion voltages and extractable capacities, at varying degrees of inversion.Our results confirm that inversion is a major performance limiting factor of Mg spinels and that synthesis techniques or compositions that stabilize the well-ordered spinel structure are crucial for the success of Mg spinels in multivalent batteries.Influence of inversion on Mg mobility and electrochemistry in spinelsGerbrand Ceder December 30, 2023 ======================================================================= § INTRODUCTION Multivalent (MV) batteries, such as those based on Mg^2+,<cit.> can potentially achieve high volumetric energy density via facile non-dendritic stripping/deposition on an energy-dense metal anode.<cit.> However, the development of viable MV technology is hindered by poor Mg diffusivity in oxide cathodes as well as poor Coulombic efficiencies in liquid electrolytes.<cit.> One pathway to improve Mg migration in solids is to utilize host structures where Mg occupies an unfavorable coordination environment.<cit.> Spinels with composition AM_2X_4 (A = Mg, M = metal cations, X = O or S) are appealing structures in this regard because of their tetrahedrally-coordinated Mg sites, rather than the preferred octahedral coordination of Mg. Theoretical calculations indeed predict reasonable Mg^2+ migration barriers (∼ 550 - 750 meV) in both oxide and sulfide spinels.<cit.>Note that oxide spinels have long been used as cathodes and anodes in commercial Li-ion batteries.<cit.> Spinel-Mn_2O_4 is a particularly promising, energy-dense, MV cathode, as it is one of the few oxides<cit.> to have shown electrochemically reversible Mg^2+ intercalation.<cit.>However, the cyclable Mg content, i.e., the observed capacity, seems to depend strongly on the synthesis conditions.<cit.> Several studies on the MgMn_2O_4 structure<cit.> have indicated that the spinel is prone to inversion, i.e., Mg/Mn antisite disorder (see Section <ref>), where the degree of inversion can range from 20%<cit.> to 60%.<cit.> It has further been argued that the propensity of Mn^3+ to disproportionate into Mn^2+ and Mn^4+ promotes spinel inversion and phase transformations.<cit.>Since inversion directly affects the local cation arrangement, it may significantly impact the Mg^2+ ionic mobility.<cit.>For the rational design of improved Mg battery cathodes it is, therefore, crucial to understand how inversion in oxide spinels affects Mg^2+ migration. Inversion is not a phenomenon unique to oxides, and other chalcogenide spinels such as sulfides, which are also important cathode materials in MV technology,<cit.> are also known to exhibit inversion.<cit.> A recent combined theoretical and experimental study has identified ternary sulfide and selenide spinels as promising Mg-ion conductors with potential applications as solid electrolytes in MV batteries.<cit.>Solid electrolytes combine the advantage of improved safety with a high Mg transference number.Three promising compounds were reported, namely, MgSc_2Se_4, MgSc_2S_4, and MgIn_2S_4.<cit.> MgIn_2S_4 spinel had previously been reported,<cit.> and the available literature as well as our own synthesis attempts (Figure S1 in Supporting Information, SI[^# Electronic Supporting Information available free of charge online at <http://dx.doi.org/10.1021/acs.chemmater.7b02820>]^#) indicate that the compound is prone to inversion, where the degree of inversion can be as high as ∼ 85% (Table S2 in SI). In the present work, motivated by the importance of the spinel structure for MV battery technology, we explore the influence of spinel inversion on Mg mobility in ternary oxides and sulfides, using MgMn_2O_4 and MgIn_2S_4 as the prototype for each class of spinels.We consider all possible local cation environments that arise due to inversion and compute the activation barriers for Mg migration in each scenario using first-principles calculations. The high requirement for the ionic conductivity in solid electrolytes typically demands migration barriers to be < 500 meV, as observed in solid Li-conductors,<cit.> while cathodes can operate under lower ionic mobilities (barriers ∼ 750 meV, see Section <ref>)<cit.> as the required length is less than for a conductor. Hence, we limit accessible Mg^2+ migration paths to those with a barrier less than 500 meV and 750 meV for operation as a solid electrolyte and cathode, respectively. We will use MgMn_2O_4 as the prototype cathode for which we restrict barriers to 750 meV and MgIn_2S_4 as an example of an electrolyte (barriers < 500 meV). Our results indicate that inversion, in both solid electrolytes and cathodes, can simultaneously cause a decrease in activation barriers across certain migration trajectories while increasing the barriers across others, leading to a complex interplay of opening and closing of specific Mg migration pathways.To quantify the impact of these variations in the microscopic activation barriers on macroscopic Mg diffusion, we estimate the critical Mg concentrations (percolation thresholds) required to facilitate Mg^2+ diffusion through the structure at different degrees of inversion.Note that Mg extraction from the cathode material creates Mg-vacancies that can affect the percolation properties. For example, vacancies can cause migration pathways that are inactive in the fully discharged composition to become accessible. Hence, for a cathode, we examine the variation of the percolation threshold with vacancy content in the spinel lattice. In electrolytes, the Mg concentration does not significantly vary and we do not consider the effect of Mg-vacancies in MgIn_2S_4. Our estimates indicate that stoichiometric MgMn_2O_4 and MgIn_2S_4 spinels remain percolating up to ∼ 55–59% and 44% inversion, respectively.Finally, we discuss the impact of spinel inversion on Mg-electrochemistry in the Mn_2O_4 cathode by evaluating the 0 K phase diagram, average voltages and the accessible Mg capacity at various degrees of inversion. While previous studies have analyzed the impact of inversion on structural, thermal, electronic, and magnetic properties,<cit.> the effect on Mg mobility in spinels has not yet been investigated. Understanding the influence of inversion on ion mobility will provide guidelines to tune the synthesis and electrochemical conditions of both cathodes and solid electrolytes, not only in MV systems but also in existing Li-ion architectures.<cit.>Finally, our results emphasize the importance of the topology of cation sites in setting the migration behavior within a general anion framework.<cit.> § STRUCTURE A spinel configuration is a specific ordering of cation sites (A and M in AM_2X_4) in a face-centered cubic (FCC) packing of anion sites (X), as shown in Figure <ref>. In a “normal" spinel, half of the octahedral (oct) sites, i.e., 16d, are occupied by M atoms (Mn/In, blue octahedra in Figure <ref>), while 1/8 of the tetrahedral (tet) sites (8a) are occupied by A (Mg, orange tetrahedra) cations.Polyhedra in the spinel structure share faces, edges and corners, as summarized in Table <ref>. For example, the 8a sites that are occupied by A are face-sharing with vacant (Vac) 16c oct sites (dashed red square in Figure <ref>a), edge-sharing with vacant 48f tet (dashed red triangle) and corner-sharing with vacant tet (48f, 8b) and M-containing 16d oct sites.<cit.>Face-sharing polyhedra have the lowest cation–cation distance, leading to the highest level of electrostatic repulsion, followed by edge-sharing and subsequently corner-sharing polyhedra.<cit.> Indeed, the 16c, 48f and 8b sites are vacant in spinel lattices (8b not shown in Figure <ref>) since they face-share with occupied 8a or 16d sites.Inversion in a spinel structure refers to the collection of anti-site defects in the 8a (A) and 16d (M) sub-lattices, as shown in Figure <ref>b. The degree of inversion, i, isdefined as the fraction of 8a sites occupied by M cations, with a value of 0 (or 0%) and 1 (100%) indicating a normal and a fully inverted spinel, respectively.Thus, cations A and Mare exchanged in inverted spinels (green arrows in Figure <ref>b), leading to a stoichiometry of A_1-iM_i[A_i/2M_1-(i/2)]_2X_4, compared toAM_2X_4in normal spinels.§.§ Possible Mg-hops Figure <ref> and Table <ref> summarize the possible local cation arrangements in a spinel structure that can originate from inversion. The orange, blue, and green polyhedra in Figure <ref> correspond to Mg, M, and mixed (Mg/M) occupation, respectively, with the arrows in each panel indicating the Mg migration trajectory. The dashed rectangles and triangles signify vacancies. Grey polyhedra correspond to 8a sites that are either cation occupied or vacant. While Figure <ref>a indicates the migration trajectory in a normal spinel, panels b, c, d, and e depict the possible Mg-hops that can occur in an inverted spinel. The sub-panels in Figure <ref>b correspond to slices along perpendicular directions, i.e., the 8a sites in the left sub-panel of Figure <ref>b are perpendicular to the plane of the paper in the right sub-panel.In a normal spinel, the rate for Mg diffusion is determined by the hop between adjacent 8a tet sites face-sharing with a 16c octahedron, as shown in Figure <ref>a. Hence, the migration topology is tet-oct-tet, and referred to as “Hop 1" in our work. The intermediate 16c site in Hop 1 shares edges with six 16d oct sites (“ring” sites) that are occupied by M cations (2 out of 6 ring sites are shown in Figure <ref>a). It was recentlyproposed<cit.> that the migration barrier in normal spinels, both oxides and sulfides, is predominantly set by the size of the shared triangular face (not shown in Figure <ref>a) between the 8a tet and 16c oct sites.Along the tet-oct-tet migration pathway in inverted spinels (referred to as “Hop 2”) the 16d ring sites can be occupied by both M and Mg cations, as indicated by the six green polyhedra in the right sub-panel of Figure <ref>b. To evaluate Mg^2+ migration along Hop 2, we considered multiple configurations from 1 ring site occupied by Mg to all 6 ring sites being occupied by Mg. Since each ring site occupancy (e.g., 2/6 or 3/6 Mg) corresponds to a large number of possible cation decorations on the ring sites, we used the decoration that had the lowest electrostatic energy, as obtained by minimizing the Ewald energy of the unit cell<cit.> using classical charges in the spinel framework. The specific cation arrangements used to evaluate the Mg migration barriers along Hop 2 are displayed in Figure S13.As inversion leads to Mg^2+ occupancy of 16d sites, Mg-hopping across 16d sites must also be considered. A 16d-16d hop can occur through two possible tetrahedral intermediate sites, the 8b and 48f. The 8b sites typically share all their triangular faces with occupied 16d sites and are therefore not open to Mg^2+ migration due to high electrostatic repulsion, as shown by previous studies.<cit.> However, the 48f sites share 2 triangular faces with vacant 16c sites, enabling them to act as viable intermediate sites for Mg^2+ hopping. As such, we only consider the 16d-16d hop via the 48f asintermediate site, leading to a 16d-48f-16d topology (Figures <ref>c, d, and e). The 48f shares one of its edges with an 8a tet site (Table <ref>), where the “edge-8a” can be occupied by Mg^2+ (“Hop 3", Figure <ref>c), M^3+/4+ (“Hop 4", Figure <ref>d) or a vacancy (“Hop 5", Figure <ref>c). Additionally, across Hops 3, 4, and 5, we consider two scenarios where the 8a sites that share a corner with the 48f (“corner-8a”, grey polyhedra in Figure <ref>) are either occupied by cations or left vacant.§.§ Percolation theory While activation barriers for the various cation arrangements in Figure <ref> determine the active Mg^2+ migration hops (or channels) on the atomic scale, the macroscopic diffusion of Mg^2+, which is essential for (dis)charge of cathodes or ionic conduction in solid electrolytes, depends on the existence of a percolating network of active migration channels. As the 8a-16c-8a channels form a percolating network throughout the spinel structure, stoichiometric normal spinels with Mg in 8a enable macroscopic diffusion of Mg^2+ as long as the 8a-16c-8a hop is open, i.e., the migration barrier for Hop 1 is below a threshold value. However, inversion leads to mixing of cation occupancies in both the 8a and 16d sites, potentially causing some 8a-16c-8a channels to close (due to higher Mg^2+ migration barriers along Hop 2) while opening new channels typically closed in a normal spinel (e.g., Hops 3, 4, or 5). Hence, in addition to identifying facile microscopic hops, it is important to consider whether a percolating network of low-barrier migration channels exists. Analogous studies have been done on Li^+ percolation in rocksalt lattices.<cit.>In percolation theory, the site percolation problem<cit.> identifies the critical concentration, x = x_crit, at which an infinite network of contiguous connected sites exists in an infinite lattice of randomly occupied sites. In terms of ionic diffusion, x_crit sets the “percolation threshold", above which percolating channels exist in a given structure and macroscopic ion diffusion is feasible. While percolation thresholds are accessible analytically for 2D lattices,<cit.> Monte-Carlo (MC) simulations need to be used to estimate x_crit in 3D structures.<cit.>The existence of a percolating diffusion network in a structure at a certain x (> x_crit) does not imply that all ions in the structure can be (reversibly) extracted. Mg sites that are not part of a percolating network will form isolated clusters throughout the structure so that the amount of extractable ions is lower than the total concentration, i.e., x_ext < x. The quantity x_ext can be assumed to correspond to the capacity of a cathode material. Numerically, x_ext is also estimated from MC simulations.<cit.>In summary, the two central quantities obtained from percolation MC simulations are the Mg concentration beyond which macroscopic diffusion is feasible (x_crit) and the fraction of extractable Mg ions in a percolating structure (x_ext). In order tostudy Mg diffusion in spinels, we modified the nearest neighbor model (normally considered in site percolation estimations) to include occupancies up to the 3^rd nearest neighbor (i.e., corner-sharing sites in Table <ref>). Two Mg sites in a given spinel arrangement are considered connected only if the migration channel linking them is open (i.e., the migration barrier is below an upper-limit). Thus, a percolating network of Mg sites is formed solely via open migration channels. Whether a channel is considered open will depend on the migration barrier for Mg hopping through it.§ METHODS The computational approaches to predict properties relevant to cathode materials have recently been reviewed by Urban et al.<cit.> Also, the ability of Density Functional Theory (DFT)<cit.> methods to predict materials with novel properties has been amply demonstrated.<cit.> As a result, all calculations in this work are done with DFT as implemented in the the Vienna Ab Initio Simulation Package,<cit.> and employing the Projector Augmented Wave theory. <cit.> An energy cut-off of 520 eV is used for describing the wave functions, which are sampled on a well-converged k-point (4×4×4) mesh.The electronic exchange-correlation is described by the semi-local Perdew-Burke-Ernzerhof (PBE)<cit.>functional of the Generalized Gradient Approximation (GGA). Calculations on Mg_ xMn_2O_4 are always initialized with an ideal cubic structure while allowing for potential tetragonal distortions during the geometry relaxation as the spinel can be either cubic (x_ Mg∼ 0) or tetragonal (x_ Mg∼ 1) based on the concentration of Jahn-Teller active Mn^3+ ions. The computed c/a ratio for the tetragonal-MgMn_2O_4 structure is in excellent agreement with experimental reports<cit.> (see Section S12). For voltage and 0 K phase diagram calculations of Mg_ xMn_2O_4, the PBESol exchange-correlation functional<cit.> is used to improve the description of the energetics,<cit.> while a Hubbard U correction of 3.9 eV is added to remove spurious self-interaction of the Mn d-electrons.<cit.>The activation barrier calculations are performed with the Nudged Elastic Band (NEB) method.<cit.> The barriers are calculated in a conventional spinel cell (32 anions), which ensures a minimum distance of ∼ 8 Å between the elastic bands and reduces fictitious interactions with periodic images. We verified that migration barriers do not change appreciably (< 3% deviation) when equivalent calculations are performed in larger supercells (see Figure S2).Seven images are introduced between the initial and final end points to capture the saddle point and the migration trajectory. All NEB results are based on the PBE functional, without Hubbard U.<cit.> The migration barriers in spinel-MgIn_2S_4 are calculated with compensating electrons added as a background charge to ensure charge-neutrality of the structure at non-stoichiometric Mg concentrations.As migration barriers are calculated in the conventional spinel cell, the degree of inversion (i) that can be modeled is constrained by the migration trajectory under consideration, in both the oxide and the sulfide. For example, along Hop 2 (Figure <ref>b), a 3/6 Mg ring site occupancy leads to 3 Mn/In atoms in the 8a sites, and consequently results in i ∼ 3/8 = 0.375. Similarly, the barrier calculations along the 16d-48f-16d topology (Hops 3, 4, and 5), which require a minimum of 2 Mg atoms in the 16d sites (or 2 Mn/In sites in the 8a), correspond to i ∼ 0.25. Monte-Carlo simulations are used to estimate the Mg percolation thresholds (x_crit) and the fraction of extractable Mg ions (x_ext). A 6×6×6 supercell of the primitive spinel structure is used, which corresponds to 1728 anion atoms (Figure S6 plots convergence behavior with supercell size). In MC simulations, a network of Mg sites is considered percolating when it spans the periodic boundaries of the simulation cell in one or more directions.<cit.> Inversion in the spinel is introduced during MC sweeps by labelling a number of random 8a and 16d sites, corresponding to the degree of inversion, as part of the “Mg sub-lattice”. For example, the Mg sub-lattice in a normal spinel consists of all 8a sites. However, in an inverted spinel (with the degree of inversion i) the Mg sub-lattice will be composed of (1-i)% of all 8a sites and (i/2)% of all 16d sites.To evaluate the M composition at which percolation occurs, a MC sweep is performed with the following steps:<cit.> (i) the supercell is initialized with M atoms in both M and Mg “sub-lattices”, corresponding to a M_3X_4 (X = O, S) stoichiometry, (ii) M atoms on the Mg sub-lattice are randomly changed to Mg, (iii) after all Mg sub-lattice sites are changed (i.e., a stoichiometry of MgM_2X_4 is attained), M atoms on the M sub-lattice are randomly flipped to Mg. During an MC sweep, once a Mg atom replacement results in the formation of a percolating network, the current Mg concentration (x_ Mg) is taken as an estimate of the percolation threshold (x_crit), while for x > x_crit, the fraction of sites within the percolating network, x_ext, is stored. The values of x_crit and x_ext are averaged over 2000 MC sweeps to guarantee well-converged estimates. The effect of vacancies on Mg percolation in the Mn-spinel is captured by initializing the Mg sub-lattice with varying vacancy concentrations, at a given degree of inversion, corresponding to a Vac_ zMn_ 3-zO_4 stoichiometry (z ≤ 1). Whenever vacancies are initialized in a supercell, only the Mn atoms are changed to Mg during a MC sweep. § RESULTS§.§ MgMn_2O_4Figure <ref> plots the ranges of Mg^2+ migration barriers in Mg_ xMn_2O_4 (y-axis) for all hops of Figure <ref> and Table <ref>, while the raw data is included in Figure S3 of the SI. The migration barriers are calculated with respect to the absolute energies of the end points, nominally identical for a given Mg^2+ hop. However, there are a few cases where the end point energies are different, since the local symmetry of the cation decoration is broken differently across the end points (e.g., 3/6 hop in Figure S3b). In such cases, the barrier is reported with respect to the end point with the lowest energy. The dotted black line in Figure <ref> is the upper-limit of the Mg migration barrier, as required for reasonable battery performance,<cit.> and is used to determine the percolation thresholds (see Section <ref>). For a Mg_ xMn_2O_4 cathode particle of size ∼ 100 nm being (dis)charged at a C/3 rate at 60^∘C, the migration barrier upper-limit is ∼ 750 meV (the upper-limit decreases to ∼ 660 meV at 298 K).<cit.> Since full-cell Mg batteries so far have displayed superior performance at ∼ 60^∘C than at 25^∘C,<cit.> the value of ∼ 750 meV has been used as the cut-off to differentiate “open” and “closed” Mg^2+ migration channels. In terms of notations, the fractions used in Hop 2 (e.g., 1/6, 2/6, etc., yellow rectangle in Figure <ref>) correspond to the fraction of 16d ring sites (Figure <ref>b) that are occupied by Mg^2+. The terms “8a empty" and “8a full" along Hops 3, 4, and 5 in Figure <ref> indicate that the corner-8a sites (Figures <ref>c, d, and e) are vacant and occupied by cations, respectively.x_ Mg in Figure <ref> is the Mg concentration in the cell used for the barrier estimation, corresponding to the “dilute Mg" (x_ Mg∼ 0, solid red lines) and “dilute vacancy" (x_ Mg∼ 1, dashed blue lines) limits.Mg migration barriers along Hop 1 (tet-oct-tet, normal spinel) at the dilute Mg and dilute vacancy limits are ∼ 717 meV and ∼ 475 meV, respectively (red rectangle in Figure <ref>), in good agreement with previous studies.<cit.> Note that the dilute Mg (vacancy) limit for Hop 1 corresponds to the regime when no 8a sites, other than those required to model the hop, are occupied by Mg (vacancies). Since the migration barriers at both Mg concentration limits are below ∼ 750 meV, Hop 1 is always open for Mg migration. Barriers along Hop 2 (yellow rectangle in Figure <ref>) decrease initially with Mg occupation of the 16d ring sites (∼ 393 meV at 2/6 vs. 536 meV at 1/6) before increasing beyond 750 meV at 5/6 and 6/6 Mg. The non-monotonic variation of the migration barriers along Hop 2 is due to the gradual destabilization of the 16c site. The increasing instability of the 16c also changes the migration energy profile (Figure S3b) from “valley”-like<cit.> at 1/6 Mg to “plateau”-like at 5/6 Mg. Figure S14 shows the Mg migration barriers along Hop 2 when the ring sites are occupied by vacancies instead of Mg^2+. In the case of the oct-tet-oct Hops 3 and 4 (green and cyan rectangles in Figure <ref>), which respectively have tet Mg and Mn edge-sharing with the intermediate 48f site, the barriers vary drastically based on Mg content and occupancy of the corner-8a sites. For example, at (i)x_ Mg∼ 0 and vacant corner-8a, the barrier along Hop 3 (∼ 592 meV) is well below the upper-bound of 750 meV, while the barrier is comparable along Hop 4 (∼ 743 meV). At (ii) x_ Mg∼ 0 and cation-occupied corner-8a, the barriers along Hops 3 and 4 increase significantly (∼ 1388 meV and ∼ 1418 meV) and surpass the upper-limit set for open channels. Eventually, at (iii) x_ Mg∼ 1 (cation-occupied corner-8a), the barriers decrease to ∼ 845 meV and ∼ 784 meV along Hops 3 and 4, respectively. Note that the barriers along Hops 3 and 4 in Figure <ref> are calculated at a degree of inversion, i ∼ 0.25. At a higher degree of inversion (i ∼ 1) and x_ Mg∼ 1 (cation-occupied corner-8a), the barrier is ∼ 1039 meV along Hop 4 (Figure S5). Hence, from the data of Figure <ref>, Hop 3 is considered closed for Mg migration whenever the corner-8a sites are cation-occupied, while Hop 4 is always considered a closed channel. Mg migration barriers decrease significantly if the edge-8a is vacant (i.e., along Hop 5). For example, the migration barriers along Hop 5 (purple rectangle in Figure <ref>) are well below that of Hops 3 and 4 across the scenarios of (i) low Mg, vacant corner-8a (319 meV for Hop 5 vs. 592 and 743 meV for Hops 3 and 4, respectively), (ii) low Mg, cation-occupied corner-8a (703 meV vs. 1388 and 1418 meV), and (iii) high Mg, cation-occupied corner-8a (570 meV vs. 845 and 784 meV). Hence, Hop 5 is always open for Mg migration, since the barriers are below the upper limit of 750 meV.In summary, the tet-oct-tet pathway (Hops 1 and 2) remains open for Mg migration in MgMn_2O_4 until a high degree of Mg occupation on the 16d ring sites (i.e., ≥ 5/6 Mg) is present, which corresponds to high degrees of inversion (i > 0.625). The oct-tet-oct pathway is open only when the edge-8a is vacant (Hop 5) or when the corner-8a are vacant with Mg in the edge-8a (Hop 3).§.§ MgIn_2S_4 Figure <ref> plots the Mg^2+ migration barriers in MgIn_2S_4 for the hops of Figure <ref> (the raw data are shown in Figure S4). Since we consider MgIn_2S_4 as an ionic conductor, off-stoichiometric Mg concentrations are not of interest. Hence, all hops in Figure <ref> are evaluated at the dilute vacancy limit (x_ Mg∼ 1, dashed blue lines in Figure <ref>).The fractions used (1/6, 2/6, etc.) in Figure <ref> are the number of 16d ring sites occupied by Mg^2+ in Hop 2. Along Hops 3 – 5, we use cation-occupied corner-8a sites (i.e., “8a full” in Figures S4c, d, and e). The upper-limit of the Mg migration barrier for classifying open and closed migration channels (as indicated by the dotted black line in Figure <ref>) is set to ∼ 500 meV, based on migration barriers of ∼ 400 – 500 meV observed in fast Li-ion conductors, such as Garnets and Si-based thio-LISICONs.<cit.> In the case of Hop 1, the barrier is ∼ 447 meV, well below the upper limit of ∼ 500 meV. Mg migration barriers along Hop 2 (yellow rectangle in Figure <ref>) follow trends similar to that of MgMn_2O_4 (Figure <ref>). For example, at low Mg occupation of the ring sites (1/6 or 2/6 Mg), the barrier is below the limits for percolating diffusion, before increasing beyond 500 meV at higher Mg content in the ring sites (> 3/6 Mg). Also, the shape of the migration energy curve changes from a “valley" at 1/6 Mg (solid black line in Figure S4b) to a “plateau" beyond 2/6 Mg (solid red line in Figure S4b), indicating that the 16c site becomes progressively unstable with increasing Mg occupation of the ring 16d.Along the 16d-48f-16d pathways (Hops 3, 4 and 5), the migration barriers are always higher than 500 meV, irrespective of the occupancy of the edge-8a. Indeed, the magnitude of the barriers are ∼ 683 meV, ∼ 531 meV, and ∼ 504 meV for Mg-occupied, In-occupied and vacant edge-8a, respectively, indicating that the oct-tet-oct pathway will not be open for Mg^2+ migration. §.§ Percolation thresholds Based on the data of Figures <ref> and <ref>, and the upper limits of Mg migration barriers set for MgMn_2O_4 (750 meV) and MgIn_2S_4 (500 meV), we compiled a list of conditions that enable the opening of the possible hops in Table <ref>. For example, Hop 1 (8a-8a) is open for all values of x_ Mg and i for both Mg_ xMn_2O_4 and MgIn_2S_4. Both the oxide and the sulfide spinel exhibit high barriers (> 1 eV) for a 16d-8a hop (Figure S8), which would limit Mg transfer between an octahedral 16d site and an adjacent tetrahedral 8a site. Thus, in our percolation simulations, the 8a-8a (Hops 1 and 2) and the 16d-16d (Hops 3, 4, and 5) channels remain decoupled, and a percolating network consists solely of either 8a-8a or 16d-16d channels.Figures <ref>a and b plot the percolation threshold (x_crit, black lines), at various degrees of inversion (i) in Mn_ 3-xO_4 and In_ 3-xS_4. The dashed yellow lines indicate the stoichiometric spinel, i.e., M:X = 2:4. The blue (red) shaded region corresponds to Mg concentration ranges which do (do not) exhibit percolation. The x-axis in Figure <ref> begins at a M_3X_4 (i.e., 50% M-excess or 100% Mg-deficient) configuration and spans concentrations up to Mg_1.5M_1.5X_4 (i.e., 25% M-deficient, 50% Mg-excess). Generally, percolation thresholds in the M-excess domain (i.e., x_crit < 1) are desirable as this implies that the stoichiometric spinel will possess percolating networks and will facilitate macroscopic Mg transport.In the case of cathodes (Mn_2O_4), Mg deintercalation from the framework creates vacancies, which can facilitate the formation of Mg percolating networks by opening certain migration channels (e.g., Hop 5 in MgMn_2O_4, Figure <ref>). Therefore, we explored the variation of the percolation threshold with vacancy concentration (“z” in Figure <ref>a) in the Mn-spinel. For the sake of simplicity, x in Figure <ref> refers to the sum of Mg and vacancy concentrations. For example, x = 1 and z = 0.5 (green circle on the dashed black line) in Figure <ref>a indicates a composition of Mg_0.5Vac_0.5Mn_2O_4, while x = 0.6 and z = 0 (green square) corresponds to the M-excess spinel-Mg_0.6Vac_0Mn_2.4O_4. The percolation threshold in the absence of vacancies (z = 0) is indicated by the solid black line in Figure <ref>a. When vacancies are introduced, the threshold decreases, as indicated by the x_crit at z = 0.4 (dotted black line) or 0.5 (dashed) consistently exhibiting lower values than x_crit at z = 0 in Figure <ref>a. For example, at x = 0.8 and i = 0.5 (indicated by the green star in Figure <ref>a), the spinel does not form a percolating network when there are no vacancies (z = 0, Mg_0.8Mn_2.2O_4), since x_crit∼ 0.88 > 0.8. However, the structure can percolate Mg when vacancies are introduced (z = 0.5, Mg_0.3Vac_0.5Mn_2.2O_4), as x_crit reduces to ∼ 0.52 < 0.8. In a case such as this, the initial cathode structure may not be percolating, but introducing vacancies in the initial part of the charge can create a percolating zone on the cathode particle surface through which further Mg-removal can occur. However, upon discharge the percolating structure could easily become non-percolating if polarization increases the surface Mg concentration too rapidly.At any degree of inversion, the magnitude of x_crit varies non-monotonically and reduces only up to a vacancy content, z = 0.4 or 0.5 (see Figure S7a). Indeed, at x = 0.8 and i = 0.5 (green star), an increase in z beyond 0.5 (such as z = 0.6, Mg_0.2Vac_0.6Mn_2.2O_4), causes the x_crit to increase to ∼ 0.6, but the spinel continues to percolate. Thus, the shaded grey region in Figure <ref>a, which is bound by the z = 0.4, 0.5 and 0 lines represents the extent of variation of x_crit with vacancy content in the cathode. Notably, the lowest value of x_crit is obtained at z = 0.4 for 0 ≤ i ≤ 0.35 and 0.595 ≤ i < 0.77, and at z = 0.5 for 0.35 ≤ i ≤ 0.595, respectively.The stoichiometric {Mg/Vac}Mn_2O_4 spinel at i = 0 (dashed yellow line in Figure <ref>a), permits macroscopic Mg diffusion, since the percolation threshold (x_crit∼ 0.44 for z = 0 – 0.4) is in the Mn-excess domain (i.e., x_crit < 1). When vacancies are absent in the stoichiometric spinel (z = 0), which corresponds to the discharged MgMn_2O_4 composition, the structure percolates Mg up to i ∼ 0.55. Upon charging, the presence of vacancies (z = 0.5) enables Mg percolation within Mg_0.5Vac_0.5Mn_2O_4 up to i ∼ 0.59.At higher degrees of inversion (0.59 < i < 0.77), the oxide spinel requires Mn-deficient concentrations (i.e., x > 1) to facilitate Mg percolation, as illustrated by x_crit∼ 1.05 - 1.13 (z = 0.5 – 0) at i = 0.6. At i > 0.77, the oxide does not form a percolating Mg network at any level of Mn-deficiency (for z ≤ 1) in the lattice. In stoichiometric ionic conductors, such as MgIn_2S_4, the vacancy concentration is low and therefore vacancies are not expected to play a major role in macroscopic Mg transport. Specifically in MgIn_2S_4, vacancies do not open additional migration channels, as indicated by the closed Hop 5 in Figure <ref>.Indeed, the percolation threshold in the In-spinel does not change up to a vacancy content, z = 0.2 in the structure (see Figure S7b). At z = 0, the x_crit in In_ 3-xS_4 (solid black line in Figure <ref>b) increases continuously with increase in inversion, with x_crit∼ 0.435, and 0.74 at i = 0, and 0.4, respectively. Thus, at low i, stoichiometric MgIn_2S_4 should exhibit significant ionic conductivity. However, at higher degrees of inversion (i > 0.44), the sulfide spinel does not form percolating networks at any Mg-concentration, owing to the absence of open 16d-16d channels in combination with the 8a-8a channels being closed beyond 2/6 Mg ring site occupancy (Table <ref>). In general, mobility requirements in an ionic conductor are more stringent than in a cathode, consistent with the stricter cut-off of 500 meV we applied to the migration barriers in MgIn_2S_4.<cit.> Indeed, a sulfide spinel Mg-cathode (such as Mg_ xTi_2S_4<cit.>) exhibiting similar activation barriers with inversion as MgIn_2S_4 will not suffer from any percolation bottlenecks, since the barriers across all cation arrangements are well below the milder 750 meV cut-off set for cathodes (Figure <ref>). §.§ Impact of inversion on cathode electrochemistry Under ideal conditions, the structure of an ionic conductor (such as MgIn_2S_4) should not undergo significant changes during operation. Thus, the extent of inversion should, in principle, be measured using characterization experiments post-synthesis (the calculated formation energies of various inverted configurations in spinel-MgIn_2S_4 are plotted in Figure S11). However in a cathode material such as Mg_ xMn_2O_4, which can generate mobile Mn^2+ ions (Figure S9) through disproportionation of Mn^3+, the degree of inversion (i) can change during electrochemical cycling.<cit.> Consequently, structural changes in a cathode during cycling should manifest themselves as changes in the voltage profile and observed capacity, which can be benchmarked with theoretical predictions.<cit.> To evaluate the effect of inversion on thevoltage profile of Mg_ xMn_2O_4, we calculated the phase diagram and energy of the intercalation system at 0 K as a function of Mg content under various degrees of inversion.<cit.>To evaluate the ground state hull of the Mg_ xMn_2O_4 system, we enumerated over 400 Mg-vacancy configurations, at different Mg concentrations (x_ Mg = 0, 0.25, 0.5, 0.75 and 1) and different degrees of inversion (i= 0, 0.25, 0.5, 0.75 and 1).Figure <ref>a displays structures with formation energies (y-axis) below 200 meV/Mn_2O_4 at different Mg concentrations (x-axis), and the formation energies of all the Mg-vacancy configurations considered are plotted in Figure S10 of the SI. Notably, formation energies in Figure <ref>a have been referenced to the non-inverted (i = 0), empty Mn_2O_4 and magnesiated (MgMn_2O_4) spinel configurations. For each configuration, the degree of inversion is indicated by the corresponding symbol used, ranging fromi = 0 (black circles) to i = 1 (red stars).Overall, the Mg_ xMn_2O_4 system is phase separating at 0 K across non-inverted (i = 0) MgMn_2O_4 and Mn_2O_4 domains, since the ground state hull of the system (dashed black line in Figure <ref>a) only exhibits two configurations (i.e., MgMn_2O_4 and Mn_2O_4). Some solubility at low Mg content may be possible given the low positive mixing energy at x_ Mg = 0.25 for the non-inverted spinel (E_formation∼ 14 meV/Mn_2O_4). At higher Mg content, the formation energies are very high for the non-inverted spinel (Figure S10), making a solid solution behavior very unlikely. Inversion becomes likely to occur at intermediate Mg compositions, as the low positive formation energies are on the scale of the configurational entropy. For example, E_formation∼ 11 meV/Mn_2O_4 at i = 0.25 and x_ Mg = 0.5 (green square at x = 0.5 in Figure <ref>a). Hence, inversion at intermediate states of magnesiation is likely. While Mg by definition has to be mobile in Mn_2O_4 to operate as a cathode, Mn mobility, which is required for spinel inversion to occur, depends strongly on its valence state.<cit.> Typically, Mn^3+ can be mobile through a temporary disproportionation mechanism, generating mobile Mn^2+ (Figure S9).<cit.> Figure <ref>b plots the average voltages as a function of x_ Mg at different i by taking the lowest E_formation configuration at each i and x_ Mg.<cit.> The average voltage for Mg insertion in the non-inverted (i = 0) configuration is ∼ 2.84 V (dashed black line in Figure <ref>b), in agreement with previous theoretical estimates.<cit.> Inversion does increase the average insertion voltage (averaged over x_ Mg = 0 to 1) marginally compared to the normal spinel, with specific values of ∼ 2.92, 2.99, 2.97 and 2.99 V at i = 0.25, 0.5, 0.75 and 1, respectively. Notably, the phase behavior of the Mg_ xMn_2O_4 system under inversion will be different compared to the normal spinel due to the formation of metastable inverted states at intermediate Mg compositions. The extractable Mg content (x_ext, see Section <ref>), obtained as a function of inversion from our Monte-Carlo simulations, indicates the extractable capacity of a cathode particle, and is shown in Figure <ref>c for stoichiometric MgMn_2O_4. The y-axis indicates the % of the cathode's theoretical capacity (∼ 270 mAh/g for MgMn_2O_4), that can be cycled reversibly. At low degrees of inversion, the extractable capacity in the stoichiometric spinel decreases roughly linearly with the degree of inversion, reaching ∼ 41% (∼ 110 mAh/g) at i = 0.4. The extractable Mg content decreases more rapidly from i = 0.4 to i = 0.5, before stabilizing around ∼ 15% (∼ 40 mAh/g) between i = 0.5 and 0.6. Eventually, none of the Mg becomes extractable beyond i = 0.61, reflecting the trends in the percolation thresholds (x_crit∼ 0.59 at stoichiometric MgMn_2O_4, Figure <ref>a) at high degrees of inversion. Note that, the overall amount of cyclable Mg from a cathode particle is influenced both by the extractable Mg (shown in Figure <ref>c) and by the phase behavior as a function of x_ Mg. For example, if the Mg removal occurs via a two-phase reaction (as is the case for the non-inverted spinel), then the presence of anon-percolating layer on the surface may prevent extraction of Mg from the bulk, even if percolation conditions are still favorable in the bulk material. § DISCUSSION In this work, we have used DFT-based NEB calculations to assess the changes in the activation barrier for Mg^2+ migration arising from inversion in both oxide (MgMn_2O_4) and sulfide (MgIn_2S_4) structures. From our results (Figures <ref> and <ref>), we can conclude that inversion has a significant impact on both oxides and sulfides, by opening and closing specific migration trajectories. In order to extrapolate the impact of the various Mg^2+ migration barriers on macroscopic Mg diffusion, we estimated the percolation thresholds under different degrees of spinel inversion. Furthermore, we analyzed the impact of spinel inversion on cathode properties of Mg_ xMn_2O_4 by evaluating the average voltages and practical capacities at different degrees of inversion.§.§ Factors influencing barriers in MgMn_2O_4 Trends from activation barriers of Figure <ref> suggest that Mg migration along the 8a-16c-8a pathways (Hops 1 and 2) can improve significantly with Mg occupation of the 16d ring sites (up to 4/6 Mg), at low degrees of inversion. Additionally, the 16d-48f-16d channels open for Mg migration whenever the edge-8a is vacant. However, high degrees of inversion detrimentally affect Mg^2+ motion, due to the closing of both 16d-16d (corner- and edge-8a become occupied by the metal cation) and 8a-8a channels (high migration barriers at high Mg in the ring sites). Although we have specifically considered the case of spinel-Mg_ xMn_2O_4, similar trends can be expected for other oxide spinels, given the similarity in Mg migration barriers along Hop 1 with different 3d-metals.<cit.>Previous studies have used electrostatic considerations to partially explain trends in Li^+ activation barriers in a Mn_2O_4 spinel.<cit.> Indeed, the reduction in Mg migration barriers along Hops 1, 3, 4, and 5 (Figure <ref>) with increasing Mg concentration can be attributed to lower electrostatic repulsions at the corresponding intermediate sites caused by the reduction of Mn^4+ to Mn^3+. For example, the barrier reduces from 717 to 475 meV along Hop 1 and 1388 to 845 meV along Hop 3, as x_ Mg increases from ∼ 0 to ∼ 1. However, Mg^2+ activation barriers generally depend on steric and bonding constraints in addition to electrostatics, which are often difficult to deconvolute over a range of NEB calculations. For example, the Mg^2+ activation barriers across Hop 2 (yellow bar in Figure <ref>) at low Mg occupation in the ring sites (1/6, 2/6) are lower than Hop 1 (red bar, Figure <ref>), which may be attributed to reduced electrostatic repulsion on the intermediate 16c (due to Mg^2+ replacing higher valent Mn in the ring sites). However, barriers along Hop 2 increase beyond Hop 1 and eventually beyond the limit of ∼ 750 meV at higher Mg in the ring sites (5/6, 6/6), despite lower electrostatic repulsion. Thus, the high Mg content in the ring sites decreases the stability of the intermediate 16c. One possible reason for the instability of the 16c site could arise from charge-deficient oxygen atoms being shared with adjacent, Mg^2+-occupied (instead of Mn^3+/4+) 16d sites. Indeed, the instability of the 16c (e.g., in the case of 6/6 Mg in Hop 2) is quantified by longer (DFT-based) ∼ 2.3 Å Mg–O bonds, compared to ∼ 2.08 Å in 16d with Mg (along the same hop) and ∼ 2.13 Å in rocksalt MgO.<cit.> For the 16d-48f-16d hops in Figure <ref> (Hops 3–5), electrostatic effects are more dominant than for the tet-oct-tet hops (Hops 1, 2), primarily due to the intermediate 48f edge-sharing with an 8a. Indeed, the cation centers in edge-sharing tetrahedra are closer (∼ 2.15 Å experimentally between 48f and 8a in an ideal LiMn_2O_4-spinel<cit.>) than in edge-sharing octahedra (∼ 2.88 Å between 16c and 16d). Consequently, the Mg barriers are consistently lower with a vacant edge-8a (Hop 5, Figure <ref>) compared to Mg/Mn-filled edge-8a (Hops 3, 4 in Figure <ref>). Also, Mg^2+ activation barriers (at x_ Mg∼ 0) increase significantly when the corner-8a sites are cation-occupied rather than vacant (Figure <ref>). A closer look at the cation-cation distances across corner-sharing 48f and 8a (∼ 2.88 Å in ideal LiMn_2O_4) reveals that the corner-sharing tetrahedra within a spinel framework may experience electrostatic repulsion as high as edge-shared octahedra (i.e., 16c and 16d). Thus, the combination of cation-cation repulsion arising from both edge- and corner-8a sites results in the high barriers along Hops 3 and 4.§.§ Barriers in sulfides vs. oxidesActivation barriers calculated in MgIn_2S_4 (Figure <ref>) exhibit similar trends to MgMn_2O_4 (Figure <ref>), resulting from analogous trends in electrostatics, steric and bonding environments. However, the absolute changes in barriers in the sulfide are remarkably lower than the oxide. For example, the absolute difference between the lowest and the highest Mg migration barriers of MgMn_2O_4 (at x_ Mg∼ 1) across Hops 1 through 5 is ∼ 662 meV (1055 – 393 meV), while this is a much lower ∼ 236 meV (683 – 447 meV) for MgIn_2S_4. Similarly, the barriers along the 16d-48f-16d trajectory are far less sensitive to the edge-8a occupancy in the sulfide (504–673 meV) than in the oxide(570–845 meV at x_ Mg∼ 1).Surprisingly, the migration barrier with an edge-8a occupied by Mg^2+ is higher (∼ 683 meV) than when the edge-8a is occupied by In^3+ (∼ 531 meV), suggesting that the In–S bonding environment screens the higher In^3+ charge better than the Mg–S bonds screen Mg^2+.Lower activation barriers for Mg in sulfides have been reported before,<cit.> which have been assigned to robust electrostatic screening, high polarizability, higher degree of covalency and large volume per anion ofS^2- compared to O^2-.<cit.> For example, a Mg_ xTi_2S_4<cit.> cathode will not suffer from any percolation bottlenecks, if the barriers across all cation arrangements are similar to the calculated values in MgIn_2S_4 (i.e., < 750 meV, Figure <ref>). But a more stringent upper-bound of ∼ 500 meV on the barrier in a solid-state conductor<cit.> indicates that inversion can significantly affect a sulfide ionic conductor by closing all 16d-16d channels and several 8a-8a channels with high Mg in the 16d ring (Figure <ref>). Since ionic mobility is expected to improve with larger anions and higher covalency (such as Se^2- compared to S^2- and O^2-), inversion is expected to affect Mg-mobility to a lesser extent in Mg-containing Se-spinels, such as MgSc_2Se_4, compared to oxides and sulfides.§.§ Percolation under inversionEstimations of percolation thresholds (x_crit) in the Mg_ xMn_ 3-xO_4 system (Figure <ref>a) indicate that spinel inversion should not detrimentally affect macroscopic Mg^2+ diffusion across the structure up to a fairly high degree of inversion, i ∼ 0.55 - 0.59. However, Mg-excess concentrations are required to ensure percolating networks form at i = 0.6 - 0.7, while the spinel completely ceases to percolate Mg beyond i = 0.77 (Figure <ref>a). Given the preponderance of conversion reactions under Mg-excess concentrations in the oxide spinel, specifically the decomposition of Mg_ xMn_ 3-xO_4 (x>1) into MgO and MnO,<cit.> it is of paramount importance that the chemically synthesizable, stoichiometric {Mg/Vac}Mn_2O_4 remains percolating. Efforts should be made to reduce or precisely control the amount of inversion (i.e., i < 0.6), by carefully tuning synthesis temperature and cooling rate<cit.> during MgMn_2O_4 synthesis. Higher Mg conductivity, as is required for a solid state electrolyte, demands a lower cut-off for the migration barrier along a pathway. In the case of MgIn_2S_4, where we used a 500 meV cut-off, MC simulations indicate that the stoichiometric spinel should remain percolating up to i ∼ 0.44. However, high degrees of inversion (i ∼ 0.85) can be observed during MgIn_2S_4 synthesis (Figure S1). As a result, strategies to limit inversion (i.e., i < 0.44) in sulfide spinel ionic conductors, such as chemical doping and careful calibration of synthesis conditions, need to be sought.§.§ Voltages and capacitiesInversion can also significantly impact electrochemical properties, such as phase behavior, average voltages and extractable capacities in an oxide-spinel cathode (Figure <ref>). For example, the average voltage for Mg intercalation, across x_ Mg = 0 - 1 in the Mn_2O_4-spinel, is higher in an inverted spinel compared to a normal spinel (Figure <ref>b). Mg intercalation experiments in spinel-Mn_2O_4 have reported a marginally higher average voltage (∼ 2.9 V)<cit.> than predicted for the normal spinel (∼ 2.84 V), with extraction voltages as high as ∼ 3.5 V during the charging cycle, which might be an indication of the spinel inverting during electrochemistry. Also, the calculated 0 K phase diagram of the Mg-Mn_2O_4 system (Figure <ref>a) suggests that the tendency to invert is the highest at an intermediate Mg concentration, as indicated by low E_formation (< 50 meV/Mn_2O_4) configurations with i = 0.25 at x_ Mg = 0.5. Hence, the degree of inversion in the Mn-spinel can indeed change dynamically during electrochemical Mg cycling, especially due to the presence of mobile Mn^2+ ions (Figure S9). As reported by Ling et al.<cit.>, the mobility of Mn^2+ within the spinel can also depend on the local arrangement of Mg^2+ ions. Thus, from the data in Figure <ref>a, we expect the degree of inversion to vary largely between 0 and 0.25 during Mg-cycling. Also, previous Mg-cycling experiments in spinel-Mn_2O_4 have reported solvent co-intercalation based phase transformations,<cit.> which can be aided by the presence of mobile Mn^2+ ions. Additionally, the first Mg-site that will be (de-)intercalated in the spinel will depend on the degree of inversion on the surface of the cathode particle. For example, if the degree of inversion is ∼ 0 on the surface, then Mg^2+ ions present in the 8a sites will be de-intercalated first from magnesiated-MgMn_2O_4. Similarly, in a partially inverted surface of a discharged cathode, the Mg^2+ ions in 16d sites that are connected via Hop 3 channels will be extracted as well as those in 8a sites connected via Hop 1 and open Hop 2 channels. In the case of a partially inverted surface in a charged-Mn_2O_4 cathode, the Mg^2+ ions are more likely to first insert into 16d channels connected via Hop 5, since Hop 5 exhibits lower Mg migration barriers compared to Hop 1 (Figure <ref>) at x_ Mg∼ 0.Since the percolation threshold in the oxide cathode can change with the vacancy concentration during Mg (de)intercalation (Figure <ref>a), a dynamic change in the degree of inversion during Mg-cycling can cause polarization within the cathode particle. For example, if i changes from 0.55 to 0.59 while charging the MgMn_2O_4 cathode, during the following discharge the spinel is percolating only up to Mg_0.5Vac_0.5Mn_2O_4 (z = 0.5 in Figure <ref>a) at i = 0.59. For further discharge into the structure, i.e., from Mg_0.5Vac_0.5Mn_2O_4 toMgMn_2O_4, a reduction in i to 0.55 is necessary, which can lead to hysteresis in the voltages during the charge and discharge cycles. Importantly, the extractable Mg content in stoichiometric MgMn_2O_4 decreases continuously with inversion, reaching values of ∼ 63% (171 mAh/g) and ∼ 17% (46 mAh/g) at i = 0.25 and 0.5 (Figure <ref>c), respectively. Thus, strategies to minimize changes in i, during Mg^2+ cycling, such as cation-doping of Mn to prevent Mn^2+ generation, should be employed to ensure reversible Mg (de)intercalation.§ CONCLUSION Spinels are promising materials in the development of multivalent battery electrodes and solid electrolytes but are prone to antisite disorder in the form of spinel inversion.With the example of two prototypical oxide and sulfide spinels, MgMn_2O_4 (cathode) and MgIn_2S_4 (solid electrolyte), we demonstrated that inversion can significantly impact both Mg-ion mobility and electrochemical properties.Using first-principles calculations, we analyzed the migration barrier for Mg^2+ hopping in different local cation arrangements, and found that inversion can both open and close select migration pathways on the atomic scale.To quantify the influence of local barrier changes on the macroscopic transport of Mg^2+ ions, we determined the minimal M-deficiency x in Mg_ xM_ 3-xX_4 required for percolation. Using a cut-off of 750 meV and 500 meV for cathodes and solid electrolytes, respectively, we found that the stoichiometric MgMn_2O_4 and MgIn_2S_4 compositions are Mg percolating up to ∼ 55–59% and 44% inversion.Since the degree of inversion in the spinels considered in this work may vary between 20% and 85% depending on the method of preparation,<cit.> a careful calibration of the synthesis conditions is essential to ensure sufficient Mg transport and to reduce the resultant impedance.In addition, spinel inversion can affect the electrochemical properties of cathode materials by changing the phase behavior, average voltage, and extractable capacities.Specifically, we find that the degree of inversion can change dynamically during electrochemical Mg cycling, as indicated by the 0 K phase diagram of the Mg_ xMn_2O_4 system and the activation barriers for Mn^2+ hopping. Notably, even low degrees of inversion (i < 0.4) can detrimentally reduce the extractable capacity in stoichiometric MgMn_2O_4, with an estimated 15% decrease in capacity with every 10% increase in inversion. Thus, spinel inversion can hinder the electrochemical performance of both cathodes and solid electrolytes in MV systems and synthesis efforts must always be made to stabilize the normal spinel structure.Given that the Mg^2+ migration barriers over a range of oxide<cit.> and sulfide spinels<cit.> show similar trends, we expect similar behavior upon inversion in other spinel materials.Finally, the framework developed in this work, particularly the data reported on percolation thresholds and extractable Mg, is readily transferable to other spinels that have potential applications in Li-ion, Na-ion, Ca/Zn-multivalent and other battery fields. The current work is fully supported by the Joint Center for Energy Storage Research (JCESR), an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science and Basic Energy Sciences. This study was supported by Subcontract 3F-31144. The authors thank the National Energy Research Scientific Computing Center (NERSC) for providing computing resources. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The authors declare no competing financial interests. GSG is thankful to Daniel C. Hannah at Lawrence Berkeley National Laboratory for a thorough reading of the manuscript. | http://arxiv.org/abs/1708.07458v1 | {
"authors": [
"Gopalakrishnan Sai Gautam",
"Pieremanuele Canepa",
"Alexander Urban",
"Shou-Hang Bo",
"Gerbrand Ceder"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170824153250",
"title": "Influence of inversion on Mg mobility and electrochemistry in spinels"
} |
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China. Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China.Centre for Modern Physics, Chongqing University, Chongqing 400044, China Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia [][email protected] State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, AustraliaCenter for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China 71.10.Fd, 75.40.Cx,02.30.Ik The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here by solving the thermodynamic Bethe ansatz equations we study the universal thermodynamics, quantum criticality andmagnetism of the 1D attractive Hubbard model.We showthat the compressibility and the susceptibilityof the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of twofree quantum fluids.The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluidsat low temperatures, a non-Fermi liquid at high temperatures and the quantum fluid to non-Fermi liquid crossover in between.The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D.Furthermore, we showthat the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the twofreefluids of the FLLO-like state.The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent z=2 and correlation critical exponent ν=1/2 in the quantum critical region whenever a phase transition occurs.Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.Universal thermodynamics of the one-dimensional attractive Hubbard model Xi-Wen Guan December 30, 2023 ========================================================================§ INTRODUCTION How to capture the essential features of many-body physics through a simple model is always of great importance in condensed matter physics.In this regard,the Hubbard model <cit.> has long provided an active area of research since it was put forward as an instance of a Mott insulator and later considered as a potential high-T_c superconductor.The Hubbard model has thus become a prototypical strongly correlated system which provides rich many-body phenomena,such as a Mott transition,superconductivity, spin-charge separation and a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. However, the Hubbard model, as a simplification of interacting fermions on realistic lattices, can be analytically resolved in neither two-dimensions (2D) nor three-dimensions (3D).The one-dimensional (1D) case within a single band is integrable, firstly solved by Lieb and Wu in terms ofthe Yang-Baxter equation <cit.> and the nested Bethe ansatz <cit.> (see Ref. Ess05 for an extensive review).More specifically, since Lieb and Wu's seminal work, the 1D repulsive Hubbard model has been investigated in various aspects, including, but not restricted to, thermodynamic properties in the ground state <cit.>, low-lying excitations <cit.>, finite temperature thermodynamics <cit.> and correlation functions <cit.>. The thermodynamics of the 1D Hubbard modelis accessible through two alternative approaches – the thermodynamic Bethe ansatz (TBA) equations <cit.> and the quantum transfer matrix method <cit.>.The former is established on the so-called `stringhypothesis' and Yang-Yang grand canonical ensemble approach <cit.>, whereasthe latterstems from the lattice path integral formulations for the partition function <cit.>. In principle, the low-lying excitations can be constructed with the help of the TBA equations in the zero temperature limit and by the logarithmof the Lieb-Wu equations <cit.>. Despite these systematic approaches and other methodsemployed for the study of the ground state properties <cit.> and low-lying excitations <cit.>, a complete understanding of the universal thermodynamics and quantum criticality of the 1D Hubbard model has not yet been achieved. The key reason for preventing the solution of this problem is thedifficulty of findinga suitable generating function for the equation of state at low temperatures. On the other hand, the correlation functions are also extremely difficult to calculatedirectly using the Bethe wave function.Fora 1D conformally invariantsystem, the critical exponents determining the power law decay of correlation functions are connected with finite-size corrections to the ground state energy <cit.>.The 1D repulsive Hubbard model is conformally invariant only in the vicinity of Fermi points. The conformal field theory (CFT) approach provides one method to obtain the asymptotics of correlation functions <cit.>.The low-lying excitations provide a practicable opportunity foran investigation of long distance asymptotics of correlation functions <cit.>, where the finite-size corrections areaccessible through the Bethe ansatz method <cit.>.However, difficulties involved in the actual calculations of correlation functions usuallyprevent full access to the many-body correlations <cit.>.The mechanism of Cooper pairing in the 1D attractive Hubbard model has attracted attention <cit.> due to the discovery of high-temperature superconductors. In particular, the FFLO-like pair correlation and spin correlations are consequentlyinvestigated by various methods,such as density-matrix renormalization group <cit.>, quantum Monte Carlo <cit.> and CFT <cit.>. The nature of the FFLO-like pair correlation was predicted in expansion dynamics of the attractive Hubbard modeltrapped in 1D <cit.>. Very recently,trapping cold atoms onoptical lattices becomes a promisingmethod to simulate the many-body physics ofthe Hubbard model <cit.>.In particular,ultracoldatoms offer an ideal platform for testing results predicted from1D exactly solvablemodels <cit.>. It is understood that the macroscopic behaviour of 1D materials, such asthe spin compound Cu(C_4H_4N_2)(NO_3)_2 <cit.> and the heavy fermion material YbNi_4P_2 <cit.>, demonstrates a type of 3D Fermi liquid behaviour <cit.>.The motivation of the present work is to provide understanding of free fluidnature and quantum criticality in the context of the 1D attractive Hubbard model.Firstly, the 1D attractive Hubbard model plays an important role in understanding many-body phenomena such as superconductivity, BEC-BCS crossover and FFLO-likecorrelation <cit.>, with several publications touching upon it <cit.>. Secondly, one expects to find universal behaviour for this model, including thermodynamics, quantum criticality and Luttinger liquid properties.Thirdly, regardingthe complicated FFLO state, it is highly desirable to obtain simple rules to describe the nature of quantum liquidsin the attractive Hubbard model.Last, but not least, the interplay of this work with experiments with ultracold atoms <cit.> may broaden our knowledge of many-body physics through 1D exactly solvable models. This paper is organized as follows. In section II, we presenta derivation of the TBA equations for the 1D attractive Hubbard model and determine the ground state phase diagram.In section III, we derive the equation of state in the strong coupling regime.In section IV, usingthe equation of state, we obtain various analytical results for the thermodynamics and magnetism which are relevant to experimental study.We also investigate quantum criticality and obtain the universal scaling forms of thermodynamic quantities.In section V, we demonstratethe free fluid nature of the FFLO phase throughthe simple additivity rules ofthe thermodynamic quantities.We find thatthe compressibility Wilson ratio isvery powerful in identifying the Fermi liquid/Tomonaga-Luttinger liquid phases in the low temperature phase diagram.The last section VI is reserved fora summaryand conclusion.We conclude this section by noting that this article provides a fuller and more detailed account of our key results presented elsewhere <cit.>.§ THERMODYNAMICS: THE YANG-YANG APPROACH§.§ Thermodynamic Bethe Ansatz equationsThe 1D Hubbard model is described by the HamiltonianH= -∑_j=1^L ∑_a=↑,↓( c_j,a^† c_j+1,a + c_j+1,a^† c_j,a) + u ∑_j=1^L ( 1-2n_j,↑) ( 1-2n_j,↓),where c_j,a^† and c_j,a are the creation and annihilation operators of fermionswith spin a (a=↑ or a=↓) at site j in a 1Dperiodic lattice of length L, n_j,a=c_j,a^† c_j,a is the corresponding particle number operator, and u represents anon-site interaction between particles (u>0 for repulsion and u<0 for attraction).By means of theBethe ansatz, the eigenenergies of the Hamiltonian are given by E=-2∑_j=1^N cos k_j+u(L-2N), where thequasimomenta{k_j}satisfy the Lieb-Wu equations <cit.>exp(ik_jL) =∏_α=1^M sin k_j-Λ_α+iu/sin k_j-Λ_α - iu,∏_j=1^N sin k_j-Λ_β + iu/sin k_j-Λ_β - iu =-∏_α=1^M Λ_α-Λ_β+2i u/Λ_α-Λ_β-2iu,where {Λ_β} denote spin rapidities, j=1,2,…,N, β=1,…,M, with N and M the total particle number and spin down particle number, respectively.Similar to the analysis<cit.> used forthe repulsive case u>0, one finds that the roots to the Bethe ansatz equations (<ref>) and (<ref>) forthe attractive Hubbard modelcan bedivided into three categories: single real k, k-Λ string and Λ-Λ string, which constitute thestring hypothesis.They are given by <cit.>* single real k's. * the α-th k-Λ string of length-m, for which there are2m k's,k_α^1 =arcsin(Λ_α^'^m+ im |u|),k_α^2 =arcsin(Λ_α^'^m+ i (m-2)|u|),k_α^3 =π-k_α^2, ⋮k_α^2m-2 =arcsin(Λ_α^'^m- i (m-2)|u|),k_α^2m =arcsin(Λ_α^'^m-im |u|),accompanied by m spin-rapiditiesΛ_α^'^m,j = Λ_α^'^m+ i(m+1-2j)|u|, inΛ space, wherej=1,2,3,…, m and Λ_α^'^m is the real center of the k-Λ string, see Fig. <ref>.* the β-th Λ-Λ string of length-m,Λ_β^m,j=Λ_β^m+ i(m+1-2j)|u|,where j=1,2,3, …, m, and Λ_α^m is the real center of the Λ string.The Λ strings represent the spin wave bound states in the spin sector.In the above equations wedenoted M_m, M_m^', and ℳ_e as the number of Λ strings of length m, of k-Λ strings of length-m, and of single real k's, respectively.It is easy to see that M=∑_m=1^∞ m(M_m+M_m^') and N=ℳ_e+∑_m=1^∞ 2m M_m^'. Substituting the string hypothesis into the Lieb-Wu equations and taking logarithms leads to the discrete nested BA equations k_jL =2πI_j+∑_m=1^∞∑_α=1^M_m^'θ( sin k_j-Λ_α^'^m/m |u|) +∑_m=1^∞∑_α=1^M_mθ( sin k_j-Λ_α^m/m |u|), ∑_j=1^N-2M^'θ( Λ_α^n-sin k_j/n |u|) =2πJ_α^n+∑_m=1^∞∑_β=1^M_mΘ_n m( Λ_α^n-Λ_β^m/n |u|),2L [ arcsin(Λ_α^'^n+in |u|) ] =2π J_α^'^n+∑_j=1^N-2M^'θ( Λ_α^'^n-sin k_j/n |u|) +∑_m=1^∞∑_β=1^M_m^'Θ_nm( Λ_α^'^n-Λ_β^'^m/|u|), where M^'=∑_m=1^∞ m M_m^' is the total number of Λ's involved in the k-Λ strings, θ(x)=2 arctan(x), and Θ_nm(x)=θ(x/|n-m|) + 2θ( x/|n-m|+2)+⋯+2θ( x/n+m-2) + θ( x/n+m)if n≠ m2θ( x/2)+2θ( x/4)+⋯+2θ( x/2n-2)+θ(x/2n)if n=m.The quantum numbers I_j, J_α^n and J_α^'^n are either integers or half-odd integers, stemming from the multivaluedness of the log functions. They are determined by therelationsI_j= integers if ∑_m=1^∞ (M_m^'+M_m) is even half-odd integers if ∑_m=1^∞ (M_m^'+M_m) is odd,J_α^n= integers if N-M_n is odd half-odd integers if N-M_n is even, J_α^'^n= integers if L-N+M_n^' is odd half-odd integers if L-N+M_n^' is even.With the help of the string hypothesis, the eigenenergies areE = -2∑_j=1^N-2M^'cos k_j-2uN+uL-4∑_n=1^∞∑_α=1^M_n^'[ √(1-(Λ_α^'^n+i n |u|)^2)]. We now introduce counting functions for the quantum numbers, y(k_j)=2πI_j/L, z_n(Λ_α^n)=2πJ_α^n/L and z_n^'(Λ_α^'^n)=2π J_α^'^n/L.Considering the thermodynamic limit, N, M, L→∞ with N/L,M/L finite, we further define the distributionsy(k)/k =2π[ρ^p(k)+ρ^h(k) ], z_n(Λ)/Λ =2π[σ_n^p(Λ)+σ_n^h(Λ)], z_n^'(Λ)/Λ =2π[σ_n^'^p(Λ)+σ_n^'^h(Λ)],where ρ^p, σ_n^p, σ_n^'^p (ρ^h, σ_n^h, σ_n^'^h) are root densities of particles (holes) in quasimomenta of excess fermions, Λ-string parameter spaceand k-Λ string space, respectively.Then one can derive the densities of excess fermions, Λ-spin strings and k-Λ strings, withρ^p(k)+ρ^h(k)=1/2π -cos k∑_n=1^∞∫_-∞^∞Λa_n(sin k-Λ) [ σ_n^p(Λ)+σ_n^'^p(Λ) ],σ_n^h(Λ)= ∫_-π^πk a_n(sin k-Λ) ρ^p(k) -∑_m=1^∞ A_nm∗σ_m^p(Λ), σ_n^'^h(Λ) =1/π[1/√(1-(Λ+in|u|)^2)]-∑_m=1^∞ A_nm∗σ_m^'^p (Λ) -∫_-π^πk a_n(sin k-Λ)ρ^p(k),where the functiona_n(x)=1/2π2n|u|/(n|u|)^2+x^2. As usual, ∗ stands for the convolution (f∗ g)(Λ)=∫_-∞^∞f(Λ-Λ')g(Λ')dΛ', namely,A_nm∗ f (x)=δ_n, mf(x)+∫_-∞^∞y/2π/xΘ_nm( x-y/|u|)f(y).Here we denoted the derivative of the functionΘ_nm as1/2π/xΘ_nm(x) =a_|n-m|(x)+2a_|n-m|+2(x) +…+a_n+m(x) if n≠ m 2a_2(x)+2a_4(x)+… +2a_2n-2(x)+a_2n(x)if n=m. The rootdistribution functions (<ref>)-(<ref>) determine spin and charge excitations, spin dynamics and full energy spectra. Inthe grand canonical ensemble, the Gibbs free energy per sitecan be expressed in terms of these root densities in different sectorsf = e-μn_c-2B m-Ts = ∫_-π^πk (-2cos k-μ-2u-B)ρ^p(k)-∑_n=1^∞∫_-∞^∞Λσ_n^'^p(Λ) [ 4√(1-(Λ_α^'^n+ in |u|)^2).. + n(2μ+4u) ]+∑_n=1^∞∫_-∞^∞Λ2nBσ_n^p(Λ)-Ts+u,where μ is the chemical potential, B the magnetic field and T the temperature.In the above equations n_c is the particle density, m=N-2M/2L the magnetization and s the entropy per site. Following the Yang-Yang grand canonical description <cit.>, the entropy per site is explicitly given by s = ∫_-π^πk{( ρ^p(k)+ρ^h(k))ln( ρ^p(k)+ρ^h(k)) -ρ^p(k)lnρ^p(k)-ρ^h(k)lnρ^h(k) }+∑_n=1^∞∫_-∞^∞Λ {(σ_n^'^p(Λ)+σ_n^'^h(Λ)) ln(σ_n^'^p(Λ)+σ_n^'^h(Λ)) -σ_n^'^p(Λ)lnσ_n^'^p(Λ) -σ_n^'^h(Λ)lnσ_n^'^h(Λ)}+∑_n=1^∞∫_-∞^∞Λ {( σ_n^p(Λ)+σ_n^h(Λ)) ln( σ_n^p(Λ)+σ_n^h(Λ)) -σ_n^p(Λ)lnσ_n^p(Λ) -σ_n^h(Λ) lnσ_n^h(Λ) }.In the following, we only consider the physics with B≥ 0 and μ≤ 0.In thethermodynamic equilibrium, the true equilibrium state can be determined by the minimization of the free energy with respect tothe densities.Carrying out avariation of (<ref>) under the restrictionof(<ref>)-(<ref>), we obtainthe TBA equations for the attractive Hubbard model in the formε^u(k) = -2cos k-μ-2u-B +∑_n=1^∞∫_-∞^∞Λa_n(sin k-Λ)ε_n^' -(Λ) -∑_n=1^∞∫_-∞^∞Λa_n(sin k-Λ)ε_n^ -(Λ),ε_n(Λ) = ∫_-π^πkcos k a_n(sin k-Λ)ε^u -(k) + 2nB+∑_m=1^∞ T_nm∗ε_m^ -(Λ), ε_n^'(Λ) = -4√(1-(Λ+ i n|u|)^2)-n(2μ+4u) +∫_-π^πkcos k a_n(sin k-Λ)ε^u -(k) +∑_m=1^∞ T_nm∗ε_m^' -(Λ), where we have denotedε^u -(x)=Tln(1+e^-ε^u(x)/T), ε_n^' -(x)=Tln(1+e^-ε_n^'(x)/T), ε_n^ -(x)= Tln(1+e^-ε_n(x)/T). In the above equations,wedefined the dressed energiesε^u(k) = Tlnζ(k)=Tlnρ^h(k)/ρ^p(k), ε_n(Λ) = Tlnη_n(Λ)=Tlnσ_n^h(Λ)/σ_n^p(Λ), ε_n^'(Λ) = Tlnη_n^'(Λ)=Tlnσ_n^'^h(Λ)/σ_n^'^p(Λ). Theconvolution T_nm∗ f(x)=A_nm∗ f(x)-δ_n,m f(x) is defined by convention.The TBA equations (<ref>)-(<ref>) indicate that the dressed energies ε^u(k), ε_n(Λ), ε_n^'(Λ) describe the excitation energies which are subject to interactions among the bound states of electrons, spin wave fluctuations, magnetic field and chemical potential. They contain fullthermal and magnetic fluctuations in both spin and charge degrees of freedom.Therefore from these equations we can determine the thermal and magnetic properties of the model in full temperature regimes.After some algebra, the Gibbs free energy per site is consequently given by f=u-∫_-π^πk/2πε^u -(k)-∑_n=1^∞∫_-∞^∞Λ/π[1/√(1-(Λ+ i n|u|)^2)]ε_n^' -(Λ). This result builds up analytical access to the full thermodynamics of the model.§.§ Zero Temperature Phase DiagramInthe zero temperature limit, most dressed energies are nonnegative and thus make no significant contributions to the free energy (<ref>).We observe thatin the ground state, there exist only unpaired fermions and bound pairs of fermions.The spin Λ-Λ strings Eq.(<ref>)are suppressed due to the fact that in the FFLO-like phase IV, the spin wave bound states ferromagnetically couple to the Fermi sea ofthe unpaired fermions.The driving term in the TBA equation (<ref>) is positive due to this ferromagnetic ordering. At T→ 0, theΛ-Λ stringsare gapped.The driving term in the TBA equation (<ref>) can be positive when n≥ 2 due to the negative chemical potential.Taking thelimit T→ 0, the corresponding TBA equations (<ref>) and (<ref>) thus reduce to coupled linear integral equations, called the dressed energy equations,ε^u (k)= -2cos k-μ-2u-B -∫_-A^A Λa_1(sin k-Λ) ε_1^' (Λ), ε_1^' (Λ)= -2μ-2∫_-π^πkcos^2 ka_1(sin k-Λ)-∫_-Q^Q kcos k a_1(sin k-Λ) ε^u (k) - ∫_-A^A Λ^'a_2(Λ-Λ^')ε_1^'(Λ^'),where the integration boundaries Q and A represent the Fermi points of these two kinds of states (pairs and single fermions).In Eq. (<ref>) we used the expression4√(1-(Λ-in |u|)^2)-4n|u|=∫_-π^πk/πcos^2 k 2 n |u|/(nu)^2+(sin k- Λ)^2.The integration boundariesaredetermined byε^u (± Q)=0 and ε^'_1(± A)=0.Within the intervals [-Q,Q] and [-A,A], the dressed energiesare negative, i.e., ε^u(k)≤ 0 and ε^'_1(Λ)≤ 0.This means that particlestates occupy all vacancies in the two Fermi seas. With the help of (<ref>)-(<ref>), the root densities for quasimomentumk and spin rapidity Λ in the k-Λ string of length-1 atzero temperature are expressed asρ(k)= 1/2π-cos k ∫_-A^A Λa_1(sin k-Λ) σ_1^'(Λ), σ_1^'(Λ)= 1/π1/√(1-(Λ+ i |u|)^2)-∫_-Q^Q k a_1(sin k-Λ)ρ(k)-∫_-A^A Λ^'a_2(Λ-Λ^')σ_1^'(Λ^'). In the grand canonical ensemble, we explicitly write down the above root densities, whichsatisfy the two conditions ∫_-Q^Q kρ(k) + 2∫_-A^A Λ σ_1^'(Λ)=N/L and ∫_-A^A Λ σ_1^' (Λ)=M/L=N_↓/L.Thus the total particle density is given by n_c = N/L =∫_-Q^Q kρ(k) + 2∫_-A^A Λ σ_1^'(Λ)and the magnetization per site by m=(N-2M)/(2L)=1/2∫_-Q^Q kρ(k).By varying the integration boundaries Q and A, the system possesses different fillings and quantum phases. A phase transition occurs when the dressed energies exactly satisfy ε^u(0)=0, ε^u(π)=0 or ε_1^'(0)=0.Consequently we can determinefive phases, () vacuum, () fully polarized state, () half-filling state, () partially polarized state, i.e., FFLO-like state, () fully paired state.The phase boundary between () and () is determined by ε_1^'(0)=0together with the conditionQ=0.Then the TBA equation (<ref>) leads to the critical field value μ_c=2|u|-2√(1+u^2).The phase boundarybetween () and () and between () and () are determined by the conditions A=0, ε^u(0)=0 and by A=0, ε^u(π)=0, respectively.With regard to the boundaries for the FFLO-like phase, the situation is much moresubtle.The phase boundary between () and () is determined by ε_1^'(0)=0 and ε^u(Q)=0, while the phase boundary between () and () is determined by ε^u(0)=0 and ε_1^'(A)=0. The phase boundaries in the ground state phase diagram Fig. <ref> are summarized as follows * (-)μ_c1=2|u|-2√(1+u^2). * (-)μ_c2=-B-2u-2. * (-)μ_c3 = 2-B-2u. * (-)μ_c4= 2|u|-2√(1+u^2)-∫_-Q^Q kcos k a_1(sin k)[cos Q-cos k],B_c4= 2√(1+u^2)-2cos Q -∫_-Q^Q kcos k a_1(sin k)[cos Q-cos k],with Q∈ [0,π]. * (IV-V) This phase transition occurs if the critical magnetic field is sufficient to break the bound state of fermions, whose boundary in principle is fixed byε_1^' (Λ)= -2μ-2∫_-π^πkcos^2 ka_1(sin k-Λ) -∫_-A^A Λ^'a_2(Λ-Λ^')ε_1^'(Λ^'),ε_1^'(A)= 0,μ=-2-2u-B-∫_-A^A Λa_1(Λ) ε_1^' (Λ).WhenA ≪ 1,the density for pairs of fermions is low, the phase boundary could be obtainedby iteration,i.e., by applying Taylor expansion to (<ref>) with respect to Λ, it can be approximately resolved by iteration.Thesolution of (<ref>) givesA in terms of μ and B, then we derive the phase boundary by substituting the above results for A into the Eqs. (<ref>) and(<ref>). By iteration, we finally obtainμ_c5≈ 2|u|-B-2 +4√(2)/π |u|α_1[μ_c5+2(√(1+u^2)-|u|)]^3/2.Here at low energy physics only length-1 k-Λ stringsare involved.From theTBA equations (<ref>)-(<ref>),we may introduce theparameters α_nand β_n to indicate the interacting effect of the length-n k-Λ bound states on a lattice in the low density regiem.They are given byα_n= ∫_-π^π k2n |u| cos^2 k (n^2u^2-3 sin^2 k)/π(n^2u^2+sin^2 k)^3, β_n = ∫_-π^π k a_n(sin k).In general, α_n represents the lattice effect in the length-n k-Λ strings.Meanwhile, if A ≫ 1, the phase boundary is givenby (<ref>) and (<ref>) in Appendix (<ref>), where we have used the Wiener-Hopf method to solve the TBA integral equations.From the dressed energy equations (<ref>) and (<ref>) the complete phase diagram at zero temperature is shown in Fig. <ref>.This phase diagram was also obtained by the Shiba transformation, whichbuildsup a mapping between repulsive and attractive regions in the ground state of the Hubbard model <cit.>.However, once we are concerned with the low temperature thermodynamics, correlation functions and quantum criticality, the Shiba transformation does not work in actual calculations, see the analysis of the ground state properties of the attractive Hubbard model <cit.>.This is mainly because the differentspin-spin strings, k-Λstrings and excess fermionshave differentcut-offprocesses (the cut-off strings, see Appendix B) at low temperature physics.For example, the spin fluctuation term (the third term)in the unpaired dressed energy can be safelyignored in the strongly attractive Hubbard model at low temperatures.However, the counterpart of such a spinfluctuation term inthe repulsive Hubbard model essentially determines theantiferromagnetic ordering.Even in the repulsive regime, such spin string dynamics, quantum criticality and scaling functions still lack an analytical calculation.The ground stateproperties of the attractive Hubbard model were initially studied by Woynarovich <cit.>.In this paper, using the TBA equations (<ref>)-(<ref>), we obtain exact results for the FFLO pairing correlation, universal thermodynamics and quantum criticality of the 1D attractive Hubbard model.Our study provides a preciseunderstanding of the universallow energy physics of interacting fermions with pairing and depairing ona 1D lattice.§ EQUATION OF STATE The TBA equations describe the fullthermodynamics of the model.At low temperatures quantum liquid behavior and critical scaling in the thermodynamics should be obtained from the TBA equations (<ref>)-(<ref>).However, the analysis of such coupled nonlinear integral equations provides a formidable challenge.In particular, it is challenging to solve infinitely many coupled nonlinear integral TBA equations, i.e., the desired analytical or numerical solution is not achievable by solving the whole set of TBA equations.This obstacle prevents us to understand the microscopic Cooper pairing mechanism and many-body phenomena for this model.On the other hand,in the FFLO-like phase IV, the spin wave bound states ferromagnetically couple to the Fermi sea ofthe unpaired fermions.In this phase, except two gapless excitations in the sectors of bound pairs and excess fermions, there exist a spin waveferromagnetic fluctuation, which is no longer a linear dispersion.Bosonization or Tomonaga-Luttinger liquid (TLL) theory <cit.> are not available once such aferromagnetic ordering is involved in the low temperature physics.A similar situation was studied in the 1D two-component Bose gas<cit.>.Moreover, the TLLis notapplicable to the quantum critical region near a phase transition. Here we proceed with an analyticalinvestigation of the low energy physics of the 1D attractive Hubbard model beyond the scope of the TLL approaches. In order to obtain the universal thermodynamics and quantum criticality of the 1D attractive Hubbard model, we first solve the TBA equations(<ref>)-(<ref>) analytically in the strong coupling regime.We willderive the equation of state which is crucial for the investigation of the quantum criticality of the model.These results can be helpful to understand current experimental developments in ultra-cold atoms <cit.>. In the following discussion wemainly concentrate on the low density regime.In general, it is very difficult to find universalcharacteristics ofquantum liquids in quantum many-body systems, for example, forthe Gaudin-Yang Fermi gas <cit.>.Under the assumption that the density ofpairs and the bound states of multiple fermions are low and the interaction is strong, the TBA equations (<ref>)-(<ref>)can be rewritten asε^u(k)=-2cos k + 2a̅cos^2 k - μ -2u -B +∑_n=1^∞ p_n^b + a̅ - T e^-2B/T e^-K̅ I_0(K̅)+o(1/|u|^4), ε_n^'(Λ)= -2nμ+η_n-d_1/π n |u| - d_2/π (n|u|)^3+Λ^2 [ d_1/π(n|u|)^3-φ_n ] +o(1/|u|^4),where K̅=∫_-π^πk/2π cos kln(1+e^-ε^u(k)/T) and I_0(x) is the zeroth order modified Bessel function, which stemsfrom the spin-wave contributions.In the above equations, we denotedd_1 = 2π-∫_-π^πkcos kε^u -(k), d_2 = -π/2-∫_-π^πkcos ksin^2 kε^ u-(k), η_n = ∑_m=1^∞∫_-∞^∞ΛT_nm(Λ)ε_m^' -(Λ), a̅ = 1/2∑_n=1^∞∫_-∞^∞Λ[b_n(Λ)-4 b_n(Λ)Λ^2/(n u)^2+Λ^2] ε_n^'-(Λ), φ_n = ∑_m=1^∞∫_-∞^∞ΛQ_nm(Λ)ε_m^'-(Λ), withb_n(Λ)=a_n(Λ)/(n u)^2+Λ^2 andQ_nm(x)=b_|n-m|(x)+2b_|n-m|+2(x)+…+2b_n+m-2(x)+b_n+m(x) if n ≠ m2b_2(x)+2b_4(x)+…+2b_2n-2(x)+b_2n(x) if n=m.The results (<ref>) and (<ref>) are valid for the low density limit and strong interaction regime. Substituting <ref> into (<ref>), the pressure per unit length is given byp=p^u+∑_n=1^∞ p_n^b+|u| with p^u=T∫_-π^πk/2πln(1+ e^-ε^u(k)/T),p_n^b=T∫_-∞^∞Λ/π[1/√(1-(Λ+ in|u|)^2)]ln(1+e^-ε_n^'(Λ)/T).Using the results (<ref>) and (<ref>) and taking integration by parts within the above expressions for the effective pressures (<ref>), we then obtain the set of coupled equationsp^u = T ln( 1+e^(μ+2u+B-∑_n=1^∞ p_n^b + a̅ -2 )/T)-2a̅/π∫_-1^1 xx^2/√(1-x^2)/1+e^2x/T/z +2a̅/1+e^4/T/z^2 +2/π∫_-1^1 xarccos(-x)/1+e^2x/T/z +o(1/u^4), p_n^b = T[1-1/4(nu)^2]ln(1+e^2nμ/T)+2 D_n/π[1-1/4(nu)^2] ∫_0^∞xarctan√(x)/1+e^D_n x/T/ζ_n +o(1/u^4),which serve as the equations of state. In the above equations,D_n = d_1/π n|u|-(n u)^2φ_n,z = e^( μ+2u+B-∑_n=1^∞ p_n^b +a̅)/T, ζ_n = e^( 2nμ-η_n+d_1/π n |u|+d_2/π(n|u|)^3) /T. We alsodefinedthe auxiliary functionsd_1 =2π-4 ∫_-1^1 x√(1-x^2)/1+e^2x/T/z - 4a̅∫_-1^1 xx^3/√(1-x^2)/1+e^2x/T/z+o(1/u^4),d_2 = -π/2 - 4/3∫_-1^1 x(1-x^2)^3/2/1+e^2x/T/z+o(1/u^4), a̅ =∑_n=1^∞D_n/π (n u)^2∫_0^∞x√(x)/(1+x)^2/1+e^D_n x/T/ζ_n+o(1/u^4), η_n = ∑_m=1^∞𝔗_nm^ξ (m)+o(1/u^4),φ_n = ∑_m=1^∞𝔗_nm^ϕ (m)+o(1/u^6).In these equations, we define 𝔗_nm^x (m)= x_|n-m|^m+2x_|n-m|+2^m + ⋯ + 2x_n+m-2^m + x_n+m^m, with x_0^m=0 (x=η,ϕ) andauxiliary functionsξ_p^m= Tln(1+e^2mμ/T) + 2 D_m/π∫_0^∞xarctan( m/p√(x))/1+e^D_m x/T/ζ_m, ϕ_p^m=T/2(p u)^2(1+e^2mμ/T) + m/pD_m/π u^2∫_0^∞x√(x)/(p^2+m^2 x)/1+e^D_m x/T/ζ_m+ D_m/π (p u)^2∫_0^∞xarctan( m/p√(x))/1+e^D_m x/T/ζ_m. These functions are indicative of thesophisticated many-body effects induced by k-Λ strings of different lengths.A more detailed derivation of the above result is presented in Appendix <ref>. In order to conceive the universal behavior of the system, we need to further simplify the equations of state (<ref>) and(<ref>). To this end, we utilize the conditions |μ/T| ≫ 1 and strong interaction |u|≫ 1, which suppress the large lengthk-Λ strings inthis physical regime.We observethatno larger length-nk-Λ bound states than n=1 exist in the FFLOphase IV at low temperatures.Then the pressure per unit lengthsimplifies to p= p^u + p^b + |u|, wherep^u and p^b are given byp^u= Tln(1+e^(μ+2u+B-p^b-2)/T) +2/π∫_-1^1 xarccos(-x)/1+e^2x/T/z_1+o(1/u^2), p^b=2 D_1/π∫_0^∞xarctan√(x)/1+e^D_1 x/T/ζ+o(1/u^2),where z_1=e^( μ+2u+B-p^b )/T, ζ=e^( 2μ-η+d_1/π |u|)/T and the aboveauxiliary functions with n=1 readD_1 = d_1/π |u|-u^2φ,d_1 =2π-4 ∫_-1^1 x√(1-x^2)/1+e^2x/T/z_1+o(1/u^2), η =2 D_1/π∫_0^∞xarctan( 1/2√(x))/1+e^D_1 x/T/ζ+o(1/u^2),φ = D_1/2π u^2∫_0^∞x√(x)/(4+x)/1+e^D_1 x/T/ζ + D_1/4π u^2∫_0^∞xarctan( 1/2√(x))/1+e^D_1 x/T/ζ+o(1/u^4). Here we only consider the corrections up to order 1/|u| in the strong coupling regime |u|≫ 1.The equations of state (<ref>) and (<ref>)give a very good approximation of the low energy physics.In Fig. <ref>, we demonstrate theaccuracy of theseequationscompared to the numerical results obtained from the TBA equations (<ref>)-(<ref>).The peaks in the susceptibility and the discontinuities of the first derivativeofthe density reveal important behavior of the model near quantum phase transitions. The pressures (<ref>) and (<ref>) could be further approximately resolved by appropriate iteration.For the low density regime n_c≪ 1,weexpand the numeratorsin the pressure p^b and the auxiliary functions η and φwith respect to a small value of x in these integrals. Then we can represent η and φ in terms of p^b.After iteration wethus obtainp^b≈ -T^3/2f_3/2/√(d_1/|u|-π p^b/8), where we have defined f_s=_s [-exp( 1/T( 2μ-p^b/2+d_1/π |u|) )] in terms of the polylog function _s(x).Using this expression for p^b and after some lengthy algebra,we finally obtain the closed form expressionsp^b= -1/√(π D_0)T^3/2f̃_3/2+o(1/u^2,T^2 ),p^u=Tln( 1+e^(μ+2u+B- p^b-2)/T)+2/π∫_-1^1 xarccos(-x)/1+e^2x/T/(z_0e^-p^b/T)+o(1/u^2,T^2 )for the two pressures, with the auxiliary functionsd_0= 2π-4 ∫_-1^1 x√(1-x^2)/1+e^2x/T/z_0+o(1/u^2,T^2 ),D_0= d_0/|u|π+1/8√(|u|/d_0)T^3/2g_3/2+o(1/u^2,T^2 ). In results (<ref>) and (<ref>), f̃_s=g_s-1/2√(|u|/d_0)T^1/2 g_sg_s-1, z_0=e^(μ+2u+B)/T and g_s=Li_s[ -e^(2μ+d_0/|u|π)/T].The pressures (<ref>) and (<ref>) give deep insight into quantum scaling in the critical regimes. § QUANTUM CRITICALITYQuantum phase transitions occur in the attractive Hubbard model at zero temperature as the external magnetic field and chemical potentialare varied across any phase boundary in Fig. <ref>. In general, near a quantum critical point, the modelis expected to show universal scaling behaviour in the thermodynamic quantities due to the collective nature of many-body effects <cit.>.We see that the 1D attractive Hubbard model is an ideal model to explore such a universalscale-invariant description on a 1D lattice, whichcan be determined by the power-law scaling of the various thermodynamic properties.The behavior of the thermodynamic quantitiesis governed by scaling functions with critical exponents in the V-shaped region fanning out tofinite temperatures from the quantum critical point.In order to calculate the thermodynamic quantities which contain enough thermal and quantum fluctuations to describe quantum criticality,we here use the form of the equation of state with the results given in (<ref>), (<ref>), and (<ref>)-(<ref>) for the pressure terms.We observe thatfirst-order derivatives of these pressureswith respect to μ or B form a set of linear equations.Solution to this set of linear equations directly leadsto the particle density n_c=(∂ p/∂μ)_B and magnetization m=1/2(∂ p/∂ B)_μ.Similarly, one canderive the second-order derivatives of the pressures, the compressibility κ=(∂ n/∂μ)_B andthe susceptibility χ=(∂ m/∂ B)_μ. The corresponding scaling laws can be obtained in the different physical regimes. At very low temperatures, spin fluctuation in the FFLO-like phase is suppressed, as are the bound states of higher k-Λ strings for |μ/T| ≫ 1.In this regime, the thermodynamics of the modelis governed by a two-component TLL or say two-component Fermi liquid consisting of excess fermions and of hard-core bosoniccharge bound states. The leading low-temperature correction to the free energy is given byf ≈ f_0 - π T^2/6( 1/v_1 + 1/v_2),where f_0 is the ground state free energy and v_1 (v_2) is the sound velocity of excess fermions (bound pairs). This result is valid for arbitrary interaction strength.Whenthe particle density is very low, i.e., n_1,2≪ 1,we explicitly obtain the two velocitiesv_1≈ 2π n_1 [1+4 n_2/|u|+12(n_2/|u|)^2], v_2≈πn_2 √(2α_1)/β_1[1+1/β_1(2 n_1/|u|+n_2/|u|) . . + 3/β_1^2(2 n_1/|u|+n_2/|u|)^2 ],where the latticeparametersα_1 = ∫_-π^πk2|u|cos^2 k (u^2-3sin^2 k)/π(sin^2 k+ u^2)^3, β_1 = ∫_-π^πka_1(sin k), arefunctions of |u| representing the lattice effect <cit.>.In the above equations, n_1,2 stands for the densities of excess fermions and the bound pairs, respectively.We plot the two lattice parameters against interaction strength in Fig. <ref>.We shall see that thecritical exponents and thermodynamics of the model are subject to these two parameters.The susceptibility is independent of temperature so that the dimensionless Wilson ratio reaches a constant (we will study this nature of the Fermi liquid in the next section).The TLL validates only in the region below the crossover temperatures, where the entropy or specific heat retains a linear temperature-dependence, see the dashed lines in Fig. <ref>.The entropy in the temperature-magnetic field plane displays the visible areas of the critical regions (QC) near different critical points.In what follows, we will derive the scaling functions for the critical regions.Using the equation of state with the pressures (<ref>), (<ref>), and (<ref>)-(<ref>), we can further derive the scaling forms of the thermodynamic quantities in thecritical regimes. Analytic results for the scaling functions help to understand themicroscopic origin of quantum criticality ofthe 1D attractive Hubbard model. For convenience, we first simplify the auxiliary functionsδ= 1/π∫_-1^1 x1/1+e^2x/T/z̅_0x/√(1-x^2), γ= 1/π∫_-1^1 x1/1+e^2x/T/z̅_01/√(1-x^2), γ'= 1/π T∫_-1^1 xe^2x/T/z̅_0/(1+e^2x/T/z̅_0)^21/√(1-x^2), δ'= 1/π T∫_-1^1 xe^2x/T/z̅_0/(1+e^2x/T/z̅_0)^2x/√(1-x^2),where z̅_0=exp( μ+2u+B+ T^3/2 g_3/2 / √(π D_0)) with D_0 given in (<ref>).By virtue of results (<ref>), (<ref>), and (<ref>)-(<ref>), the closed form expressions for thermodynamic quantities can be derived.For strong attraction, we have the relationsn_c=γ+(1-γ)∂ p^b/∂μ, m=1/2[ γ+(1-γ)∂ p^b/∂ B], κ=(1- ∂ p^b/∂μ)^2 γ' + (1-γ) ∂^2 p^b/∂μ^2, χ=1/2[ (1- ∂ p^b/∂ B)^2 γ' + (1-γ) ∂^2 p^b/∂ B^2],for thermodynamic quantities. Here we calculated the derivatives of the pressures∂ p^b/∂μ =-τ^1/2(2 f_1/2-τ f_3/2)/Δ_t , ∂ p^b/∂ B =-2 δτ^1/2(f_1/2 - τ f_3/2)/|u|Δ_t , ∂^2 p^b/∂μ^2 =- f_-1/2(16 π - √(π)τ^3/2 f_3/2)/4D_0 τ^1/2Δ_t^3, ∂^2 p^b/∂ B^2 =-2δ' τ^1/2(f_1/2-τ f_3/2)/|u|Δ_t -πδ^2 f_-1/2/u^2 D_1 τ^1/2(4 f_1/2-τ f_3/2) Δ _t^4 ×( 16 √(π)f_1/2 -8 τ^1/2 f_1/2^2 -5τ ^3/2 f_1/2 f_3/2 -4 √(π)τf_3/2)with τ=T/D_0 and Δ_t=√(π)- 1/2τ^1/2 f_1/2+1/8τ^3/2 f_3/2. These results constitute very accurate results for the thermodynamics.The asymptotic results for the thermodynamic properties (<ref>)-(<ref>) have been demonstrated in Fig. <ref>.The universality class of quantum criticality is determined by the critical exponents.As we have seen in Fig. <ref>, the 1D attractive Hubbard model has a rich phase diagram.At least one branch of the density of states shows sudden change when the driving parameters vary across the phase boundary in the phase diagram.The singular behavior of thermodynamic properties is uniquely determined by the critical exponents,which are independent of the microscopic details of the system.Indeed, quantum criticality of quantum many-body systemsdependssolelyon thedimensionality and the symmetry of the Hamiltonian.Here we expand the above equations of state for the thermodynamic quantities in the limit |μ-μ_c| ≪ T.Wederive the scaling forms of the thermodynamics at quantum criticality and thus read off the critical exponents. We find that the suddenly changed density of state usually results in a quantum phase transition,so that the thermodynamical properties can be cast intothe forms of universal quantum scaling functions in the critical region.For example,for the phase transition from the fully-paired phase V to the FFLO-like state IV,thermodynamic quantities ofexcess fermionsdisplay the singular parts in the scaling functions,whereas the thermodynamic properties of the bound pairs presenttheregular parts.In contrast to the attractive SU(2) Fermi gas, the half-fillingphase in the attractive Hubbard model contributes a constant regular part to the thermodynamic quantities due to its unique band-filling. Our results for the scaling functions of particle density, magnetization, compressibility and susceptibility are summarized as follows: ∙ phase transition(I-V),n_c= -√(2|u|/π)T^1/2 Li_1/2( -exp( 2μ-2μ_c1/T) ),m≈ 0 , κ=-2 √(2|u|/π)T^-1/2 Li_-1/2( -exp( 2μ-2μ_c1/T) ), χ≈ 0.∙ phase transition (I-II),n_c=-1/2√(π) T^1/2Li_1/2(-exp( μ-μ_c2/T)), m= -1/4√(π) T^1/2Li_1/2(-exp( μ-μ_c2/T)),κ= -1/2√(π)T^-1/2Li_-1/2( -exp( μ-μ_c2/T) ),χ=-1/4√(π)T^-1/2Li_-1/2( -exp( μ-μ_c2/T) ).∙ phase transition (II-III),n_c= 1+1/2√(π)T^1/2 Li_1/2( - exp( -μ-μ_c3/T) ),m= 1/2+1/4√(π)T^1/2 Li_1/2( - exp( -μ-μ_c3/T) ), κ=-1/2√(π)T^-1/2Li_-1/2( - exp( -μ-μ_c3/T) ), χ=-1/4√(π)T^-1/2Li_-1/2( - exp( -μ-μ_c3/T) ).∙ phase transition (II-IV),n_c= n_b4+λ_1 T^1/2Li_1/2( -exp( 2(μ-μ_c4)/T) ),m= m_b4+λ_2 T^1/2Li_1/2( -exp( 2(μ-μ_c4)/T) ), κ= κ_b4+λ_3 T^-1/2Li_-1/2(-exp( 2(μ-μ_c4)/T) ), χ= χ_b4+λ_4 T^-1/2Li_-1/2( -exp(2(μ-μ_c4)/T) ).∙ phase transition (V-IV),n_c= n_b5+λ_5 T^1/2Li_1/2( -exp( μ-μ_c5/T) ),m=-1/4√(π)T^1/2 Li_1/2( -exp( μ-μ_c5/T) ), κ= κ_b5+ λ_6 T^-1/2 Li_-1/2( -exp( μ-μ_c5/T)), χ= -1/4√(π)T^-1/2Li_-1/2( -exp( μ-μ_c5/T) ). In the above scaling forms some constants are given in Appendix <ref>.Thesescaling forms can be cast into the form of well known universal scaling laws.For example, the universal scaling laws forthe density and compressibility read <cit.>n(μ,B,T)= n_0(μ,B,T)+T^d/z+1-(1/ν z)𝒢( μ-μ_c/T^1/ν z),κ(μ,B,T)=κ_0(μ,B,T)+T^d/z+1-(2/ν z)ℱ( μ-μ_c/T^1/ν z), where n_0 and κ_0 are the regular parts,i.e., the background values before the phase transition.Meanwhile 𝒢(x)=Li_1/2(x), ℱ(x)=Li_-1/2(x) give the scaling functions in the singular parts.From the above scaling forms, weread off the dynamical exponent z=2 and correlation critical exponent ν=1/2.This scaling theory is valid for allphase transitions across the phase boundaries in the phase diagram <ref>.Such universal scaling laws are demonstrated in Fig. <ref> for various phase transitions. The above scaling forms are observed to give the same critical exponents which characterize the universality class of free-fermion criticality.An intuitive explanation for this result is that the phase transitions occurred in the 1D Hubbard modelhave a common feature:at least one branch of Fermi sea vanishing, namely ε^u,b(0) =0.This naturally leads to a change in dispersion, i.e., a linear dispersion vanishes while a quadratic dispersion is created when the phase transition occurs.This change in dispersion underlies a universality class of quantum criticality, see also the recent studies of the 1D interacting Bose gas <cit.> and the 1D Heisenberg spin chain <cit.>.Moreover, the phase V in the phase diagram Fig. <ref> shows a gapped phase (fully paired phase), where the susceptibility reveals a particular exponential decay at low temperatures.Using the equation of state with the pressures (<ref>), (<ref>), and (<ref>)-(<ref>), we furthershowthat the susceptibility decays exponentially with the energy gap induced by the ferromagnetic ordering, namely χ≈T^-1/2/4√(π)e^-Δ/T, where the energy gap is given by Δ=ε^u(0)=-2-μ-2u-B+p^b withp^b=4(2π-q^3/3)/3|u|π( 1+2|u|πμ/2π-q^3/3)^3/2.This result can also be obtained by applying Sommerfeld expansion in<ref>.Weapproximately obtain the susceptibility χ≈1/2γ' ≈ -T^-1/2/2√(π)Li_-1/2(-e^-Δ/T) from <ref>.In the next section, we further demonstrate the macroscopic nature of the susceptibility in the FFLO phase.§ FREE FLUIDS AND ADDITIVITY RULES Fermi liquid theory is believed to break down in 1D strongly correlated systems due to the absence of well defined quasi-particles <cit.>.Consequently the TLL theory is generally believed to describe the collective low-lying excitations in 1D many-body systems.Despite such a big difference in the microscopic origins of the two low-energy theories, both the Fermi liquid and the TLL share a common feature – a smalldistortion of the Fermi surface or Fermi points results in the universal low-energy physics of many-body systems.From the results of the last section, we observed that at very low temperatures the low-energy physics of the FFLO-likestate is governed by the universality class of a two-component TLL.However, in view ofthe macroscopic properties of the 1D attractive Hubbard model, we argue that such a universality class of two-component TLL reveals an important nature offree fluids.In order to show this elegant nature, we will introduce twoeffective chemical potentials for the excess fermions and bound pairs on a 1D lattice.Then we will showthat the thermodynamic properties in the FFLO-like phase behave liketwo independent free fluids.In particular, we find simpleadditivity rules for the compressibility and susceptibility which represent a universal characteristic of quantum liquids at the renormalization fixed point. Prior to a discussion of the free fluids, we first make an approximation for the zero temperature TBA equations <ref> in the low density regime,ε^1(k)=k^2-μ_1 -a_1 ⋆ε^2(k),ε^2(Λ)=α_1 Λ^2-α_1 μ_2 - a_1 ⋆ε^1(Λ) - a_2 ⋆ε^2(Λ), where a_m⋆ε^n(x)=∫_-y_c^y_cy a_m(x-y)ε^n(y) with y_c being the Fermi point of ε^n(y), i.e., ε^n(y_c)=0.In the above equations, we introduced the effective chemical potentials for excess fermions and bound pairs asμ_1= μ+B+2u+2, μ_2= 2/α_1(μ+2√(u^2+1)-2|u| ). The effective chemical potential of the bound pairsreveals a deep physical insight into the crossover from Bose-Einstein condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) superconductor.Later we shall see these effective chemical potentials reveal an important free quantum liquid nature.We will showthat for the balanced case the effective chemical potential μ_2 varies from the kinetic energy of bound pairs to the free Fermi energy when the interaction changes from negative infinity to zero.This reveals a 1D analogue of the BEC-BCS crossover.This form ofthe TBA equations is useful to access the ground state properties, such as sound velocities, stiffness and effective chemical potentials.By virtue of Eqs. (<ref>) and(<ref>) we rewrite the free energy per site (<ref>) asf=u+∫_-k_c^k_ck/2πε^1 (k) + ∫_-Λ_c^Λ_cΛ/2πβ_1 ε^2(Λ),where ε^u=ε^1 and ε_1^'=ε^2. We now proceed tocalculatethe TLL parameters of the model and compare them with those of the 1Dattractive SU(2) Fermi gas <cit.>.The basic idea is to express the effective chemical potentials in terms of the Fermi points by employing iteration of <ref>.Usingthe factthat the twodressed energies vanish at their corresponding Fermi points, weexpress those Fermi points in terms ofthe densities of excess fermions and bound pairs.We shallsee that this process leads to a separation of two free fluids in the ground state energy per site. In order to simplify the lengthy iterations, we firstly rescale the TBA equations (<ref>) and (<ref>) by defining ε̃^n=ε^n/u^2, μ̃_n=μ_n/u^2, ỹ_c=y_c/|u|, and ã_n(x)=n/π1/n^2+x^2. Then we introduce a vector presentation of the rescaled TBA equations.In view of the properties of even functions, we utilize the base {k^2n} and {Λ^2n} (n=0,1,2,…) to expand these scalar equations, thus wehave,respectivelyε⃗^ 1=V⃗^1-𝐀^1(Λ̃_c) ε⃗^ 2, ε⃗^ 2=V⃗^2-𝐀^1(k̃_c) ε⃗^ 1-𝐀^2(Λ̃_c) ε⃗^ 2.The vectorsV⃗^1=[-μ̃_1,1,0,…]^t and V⃗^2=[-α_1μ̃_2,α_1,0,…]^t are the driving terms and thesuperscript t represents transpose operation.The matrix 𝐀^n(ỹ_c) ε⃗ corresponds to the integral ∫_-ỹ_c^ỹ_cyã_n(x-y) ε̃(y). Furthermore, as we only retain the first few leading terms, ε⃗^ n and 𝐀^n(ỹ_c) can be expanded as sums of a few leading orders withrespect to y_c, i.e., ε⃗^ n=ε⃗_(0)^ n+ε⃗_(1)^ n+ε⃗_(2)^ n+… and 𝐀^n(ỹ_c)=𝐀_(1)^n(ỹ_c)+𝐀_(3)^n(ỹ_c)+𝐀_(5)^n(ỹ_c)+….More details of the latter expansion are presentedin <ref>.Substitutingthese expansions into the TBA equations of vectorial form <ref>, and sortingterms order by order,leads to the set ofequationsε⃗_(0)^ 1=V⃗^1, ε⃗_(0)^ 2= V⃗^2,ε⃗_(1)^ 1= -𝐀^1_(1)(Λ̃_c) ε⃗_(0)^ 2,ε⃗_(1)^ 2= -𝐀^1_(1)(k̃_c) ε⃗_(0)^ 1 - 𝐀^2_(1)(Λ̃_c) ε⃗_(0)^ 2,ε⃗_(2)^ 1= -𝐀^1_(1)(Λ̃_c) ε⃗_(1)^ 2,ε⃗_(2)^ 2= -𝐀^1_(1)(k̃_c) ε⃗_(1)^ 1 - 𝐀^2_(1)(Λ̃_c) ε⃗_(1)^ 2,ε⃗_(3)^ 1= -𝐀^1_(1)(Λ̃_c) ε⃗_(2)^ 2 - 𝐀^1_(3)(Λ̃_c) ε⃗_(0)^ 2, ε⃗_(3)^ 2= -𝐀^1_(1)(k̃_c) ε⃗_(2)^ 1 - 𝐀^2_(1)(Λ̃_c) ε⃗_(2)^ 2 - 𝐀^1_(3)(k̃_c) ε⃗_(0)^ 1- 𝐀^2_(3)(Λ̃_c) ε⃗_(0)^ 2.It is easy to solve the above vectorial forms ε⃗_(r)^ 1 and ε⃗_(r)^ 2 with r=1,2,3. We then substitute these results into the scalar expression of the rescaled TBA equations.Together withε̃^n(ỹ_c)=0 and the expansion μ̃_n=μ̃_n^(2)+μ̃_n^(3)+μ̃_n^(4)+⋯, we then obtain a set of recurrence equations for μ̃_n^(2), μ̃_n^(3) and μ̃_n^(4).Here we observe that the expansions for chemical potentials begin from n=2 due to the fact that ε̃^1(k̃_c)=-μ̃_1+k̃_c^2+o( k̃_c^3)=0, i.e.,μ̃_1=k̃_c^2+o( k̃_c^3). Similarly for μ̃_2.We solve these equations and then express the solution as the vectorial equation[ [ μ̃_1; α_1 μ̃_2;]] =(𝐈+2/3𝐓) [ [ k̃_c^2; α_1 Λ̃_c^2;]],where the matrix 𝐓 is given by𝐓 =1/π[ [ 0 2Λ̃_c; 2k̃_cΛ̃_c; ]].Finally, the free energy per site <ref>is expressed in terms of Fermi points k_c and Λ_c, with resultf=-2/3π(k_c^3+α_1β_1 Λ_c^3)+u. We now proceedto obtain the particle densitiesin terms ofthe Fermi points.To this end,we turn to the total particle density n_c and magnetization m̅ per site based on <ref>,n_c= -∂ f/∂μ= -∫_-k_c^k_ck/2π∂ε^1/∂μ - β_1 ∫_-Λ_c^Λ_cΛ/2π∂ε^2/∂μ,m̅= -∂ f/∂ B= -∫_-k_c^k_ck/2π∂ε^1/∂ B - β_1 ∫_-Λ_c^Λ_cΛ/2π∂ε^2/∂ B. In order to getclosed forms for these two properties, wefirst take partial derivatives of <ref> with respect to μ and B, respectively. Then we rewrite these integral equations in terms of the vectorial forms similar to<ref>.Finally, by lengthyiteration and after some manipulations, we obtain[ [ ñ_1; ñ_2; ]] =1/π( 𝐈-𝐓+𝐓^2)^t [ [ k̃_c; β_1 Λ̃_c;]],where ñ_r=n_r/|u| (r=1,2) with n_1=m̅ and n_2=(n_c-n_1)/2 being respectively the densities for the excess fermions and bound pairs.Here the redefined m̅=2m is introduced according to the original TBA equations. An inverse of <ref> gives the cut-off momenta in terms of the densities n_1,2k_c ≈ π n_1 ∑_n=0^3 ( 2n_2/|u|)^n,Λ_c≈ π n_2/β_1∑_n=0^3 [ 2n_1+n_2/β_1|u|]^n.For the next step, substituting Eq. (<ref>) into (<ref>), leads to separating the ground state energy per site into the energies of excess fermions and bound pairs, with resulte=e_1+e_2+e_b.Here e_b is the binding energy and the subscripts 1 and2 denote theexcess fermions andbound pairs, respectively.The terms are given explicitly bye_1= π^2/3n_1^3 [1+2(2n_2/|u|)+3(2n_2/|u|)^2], e_2= π^2/3 α_1 n_2^3/β_1^2[1+2(2n_1+n_2/β_1|u|)+3(2n_1+n_2/β_1|u|)^2], e_b= -(2u+2)n_1-4(u+√(u^2+1))n_2.As usual,we define a dimensionless interaction strengthγ_s=2|u|/n_s (s=1,2) <cit.>. Using the relationK_s=π/√(3e(γ_s)-2γ_s e(γ_s)/γ_s+1/2γ_s^2^2 e(γ_s)/γ_s^2), the Luttinger parameters for the excess fermions and bound pairs can be directly worked out to beK_1=1,K_2=2√(2) β_1/√(α_1)[1-2/β_1 γ_2+1/(β_1 γ_2)^2].We note that the Luttinger parameter K_2 in the fully paired phase depends explicitly on the lattice parameters α_1 and β_1.This behavior is different from the constant value K_2=4 for the bound pairs phase of the strongly attractive SU(2) Fermi gas <cit.>.In the limitsu→ 0 and n_s/|u|small,the lattice parameters α_1→ 2, β_1→ 2.Thuswe haveK_2=4 which is the same as for the SU(2) Fermi gas.The two limits u → 0 and n_s/|u|≪ 1represent the lattice-gas mappingbetween 1D attractive Hubbard model and SU(2) Fermi gas <cit.>. Beside this framework of the TLL theory, we also find that forthelow density case, the chemical potentials for the unpaired fermions and pairs are given explicitly byμ_1 = π n_1^2A_1^2+4π^2α_1/3β_1^3|u|n_2^3A_2^3,μ_2 = π^2n_2^2/β_1^2A_2^2+4π^2/3α_1|u|n_1^3A_1^3+2π^2/3β_1^3|u|n_2^3A_2^3 ,whereA_1=1+2n_2/|u|+(2n_2/|u|)^2 and A_2=1+2n_1+n_2/β_1 |u|+(2n_1+n_2/β_1 |u|)^2, whichindicateinteracting effects among pairs and unpaired fermions.We observe that the chemical potential μ_2 tends tothe kinetic energy of bound pairsin the BEC limit |u| →∞.Whereas in the weak coupling limit, |u| → 0, μ_2 tends tothe Fermi energy of the free fermions on a 1D lattice.The effective chemical potentials (<ref>) and (<ref>) revealthat the thermodynamic quantities could be separable, i.e., the total is equal to a sum of the effective thermodynamic quantities oftwo individual constituents. Here we further derive the additivity rules for the compressibility and susceptibility.For the compressibility, using the standard thermodynamic relation κ=(∂ n_c/∂μ)_B, the derivatives of the density and effective chemical potentials for fixed magnetic field could be further expressed as n_c=n_1+2 n_2 and μ_1=α_1/2μ_2=μ, respectively.Inserting these relations into the definition of compressibility, κ=∂ n_c/∂μ|_B=n_1+2n_2/μ, we thus obtainκ=κ_1+2/α_1κ_2.Here the effective compressibilities of excess fermions and bound pairs are defined as κ_1=(∂ n_1/∂μ_1)_B and κ_2=2(∂ n_2/∂μ_2)_B.Details are given in see in Appendix<ref>.The additivity rule (<ref>) for the compressibility can be confirmed numerically, as shownin Fig. <ref>(a).For the susceptibility in the canonical ensemble, defined as χ̅=(∂m̅/∂ B)_n_c, it is straightforward to see n_c=n_1+2n_2=0 and B=μ_1-α_1/2μ_2, and thus the additivity rule1/χ̅=1/χ̅_1+α_1/21/χ̅_2. Here χ̅_1=(∂ n_1/∂μ_1)_n_c and χ̅_2=2(∂ n_2/∂μ_2)_n_c are the effective susceptibilities for excess fermions and bound pairs, respectively.These explicit expressions for the effective thermodynamic quantities can be found in Appendix <ref>.The additivity rule (<ref>) for the susceptibility canalso be confirmednumerically, as shown in Fig. <ref>(b).Similar to the observation concerning TLL parameters, the additivity rules for the 1D attractiveHubbard model also reduce to those for the SU(2) Fermi gas through the lattice-gas mapping.In Appendix<ref> we calculate the individual compressibility and susceptibility explicitly.The simple additivity nature of the thermodynamics at low temperatures characterizes the universal low energy physics of the FFLO-like state of the 1D attractive Hubbard model.In this sense, the additivity rules reflect a universal nature of the multicomponentTLL in 1D.The simple additivity rule thus reveals the significant two free fluid nature of the FFLO phase, as predicted in expansion dynamics of the FFLO state in 1D <cit.>.Themacroscopic magnetic properties in the FFLO-like phase show the properties of the ordinary higher-dimensional Fermi liquid, see Fig. <ref>. This figure shows that inthe free fluidsregion the magnetization is nearly temperature independent. In the non-Fermi liquid region thermal fluctuations gradually overwhelm quantum fluctuations.Thus the magnetization has a uniform temperature dependence for different magnetic fields, indicating paramagnetism.The non-Fermi liquid crossover region reveals a scaling invariance, which was studied in Section IV.Such Fermi liquid-like features have been found in the spin compound Cu(C_4H_4N_2)(NO_3)_2 <cit.> and the heavy fermion material YbNi_4P_2 <cit.>.The study of Fermi and non-Fermi liquids in 1D has received significant recent interest <cit.>.Using the explicit expressions for the compressibility (<ref>) and susceptibility (<ref>), we may calculate the Wilson ratio, which is a dimensionless ratio defined as the susceptibility or compressibility over the specific heat divided by the temperature.The Wilson ratio is the ratio describing quantum fluctuations and energy thermal fluctuations.Both the Fermi liquid and TLL give a constant Wilson ratio <cit.>, i.e., two types of fluctuations are on equal-footing in temperature scaling.However, near a critical point, the dimensionless Wilson ratios exhibits a sudden enhancement indicating a sudden change in the density of state.Therefore the Wilson ratios serves as a powerful tool for distinguishing the phases of a quantum liquidand for determining the finite temperature phase diagram as well. The compressibility Wilson ratio R_W is determined byR_W^κ = π^2 k_B^2/3 κ/C_v/T= π(κ_1+2/α_1κ_2)/( 1/v_1+1/v_2),where we have used <ref> to calculate the specific heat and set the Boltzmann constant to k_B=1.This Wilson ratio vanishes in both phases I (vacuum) and III (half-filling phase).In the limit n_c/|u| → 0 the compressibility Wilson ratio for phases II and IV are respectively, R_W^κ=1 and R_W^κ=2√(2)β_1/√(α_1).These results turn out to be the same as for thestrongly attractive SU(2) Fermi gas <cit.> whenthe limit u → 0 is applied. On the other hand, the susceptibility Wilson ratio is defined by R_W^χ=4/3(π k_B/μ_Bg_L)^2 χ/C_v/T with Bohr magneton μ_B and Lande factor g_L.Fig. <ref> shows a contour plot of each type of Wilson ratio which demonstrates the macroscopic feature of the Fermi liquid nature.This figure also presents the low-temperature phase diagram in the B-μ plane. § CONCLUSIONIn summary we have presented a framework to determine the nature of quantum criticality and quantum liquids in the 1D attractive Hubbard model.We have obtained the universal thermodynamics of themodel by solvingthe TBA equations.In particular, we have analytically derived the equation of state at low temperatures, from which we have obtained effective chemical potentials of excess fermions and bound pairs, along with the density, compressibility, susceptibility and specific heat in terms of the chemical potential μ, magnetic field B,temperature T and interaction strength constant.At quantum criticalitythescaling forms of these thermal and magnetic properties have been obtained.The dynamical exponent z=2 and correlation critical exponent ν=1/2, indicating the universality class of criticality of free fermion theory. Our results provide strong evidence for the existence of two free fluids of bound pairs and of unpaired fermions, which were noticed in the expansion dynamics of the FFLO state in 1D <cit.>.Regarding the nature of the two fluids in the attractive Hubbard model,we haveshownthat in the low-density regime the interaction effect resulting from the paired and unpaired fermions can be absorbed intoeffective chemical potentials of two non-interacting ideal gases.Consequently, theadditivity rules in the compressibility and susceptibility of the 1D attractive Hubbard model hold as long asthe dimensionless Wilson ratio remains a constant.This behavior significantly reflects the free fluidsnature inthermodynamic properties ofthe model.In this phase, the FFLO pair correlation functionG_p(x,t) = ⟨Ψ_↑^†(x,t) Ψ_↓^†(x,t) Ψ_↑(0,0) Ψ_↓(0,0) ⟩≈A_p,1cos( π(n_↑-n_↓)x )/|x+iv_1 t|^2θ_1 |x+iv_2 t|^2θ_2 + A_p,2cos( π(n_↑-3n_↓)x )/|x+iv_1 t|^2θ_3 |x+iv_2 t|^2θ_4, shows a typical spatial oscillation whichis a characteristicof the FFLO state.In the above equation, the exponents θ_1 ≈ 1/2, θ_2 ≈ 1/2+n_2/|u|β_1,θ_3 ≈1/2-4n_2/|u|β_1 and θ_4 ≈5/2-4 n_1/|u|-3n_2/|u|β_1depend essentially on the lattice parameter β_1.Here n_2,1=N_2,1/L arethe dimensionlessdensitiesof pairs andunpaired fermions, with the sound velocities v_1,2 given in (<ref>).The study of the FFLO pair correlation is presented elsewhere <cit.>.To conclude, we note that our work provides benchmark physics of the 1D attractive Hubbard model of relevance to experiments with ultracoldfermionic atoms onlattices.Acknowledgments.The authors SC and YCY contributed equally to the calculations in this paper. The authorsthank R. Huletfor helpful discussion. This work is supported by Key NNSFC grant number 11534014, MOST grant number 2017YFA0304500, NNSFC grant numbers 11374331, 11174375 and ARC Discovery Projects DP130102839, DP170104934. Afigure Atable Aequation Alabel § APPENDICES§.§ Wiener-Hopf methodThe phase boundary between phases IV and V is determined by the conditions ε^u(0)=0 and ε^b(0)< 0, which imply that Q=0 and A is finite.Thus at zero temperature the TBA equations are simplified toε^u(k)= -2cos k-μ-2u-B -∫_-A^A Λa_1(sin k-Λ)ε_1^'(Λ),ε_1^'(Λ)= -2μ-2∫_-π^πkcos^2 ka_1(sin k -Λ) -∫_-A^A Λ^'a_2(Λ-Λ^') ε_1^'(Λ^').Particularly, if A=∞, it follows that the chemical potentialμ=0. The intersection of the phase boundary withthe B-axiscould be calculated exactly by theFourier transformationB_c1=2|u|-2+2∫_0^∞ω J_1(ω)exp(-|u|ω)/wcosh(u ω). Now weconsider the more general case A ≫ 1, forwhichthe phase boundary can be resolved using the Wiener-Hopf method.By applying Fourier transformation on <ref> and after some algebraic manipulations, we have ε_1^'(Λ) = -μ-∫_-∞^∞ω J_1(ω)/ωcosh(u ω)exp(i ω Λ) +∫_0^∞Λ^' ε_1^'(Λ^'+A)[R(Λ-Λ^'-A)+R(Λ+Λ^'+A)], where we have introduced the functionR(x)=∫_-∞^∞w/2π exp(i w x)/1+exp(2|u w|). Substitutingy(Λ)=ε_1^'(Λ+A) and expandingy(Λ)=∑_n=0^∞ y_n(Λ) in terms of powers of Λ in <ref>, the result can beseparatedinto a series of Wiener-Hopf integral equations in terms of the functionsy_n(Λ), namelyy_n(Λ)=g_n(Λ)+∫_0^∞Λ^'R(Λ-Λ^') y_n(Λ^').Here we denote the driving termsg_0(Λ) = -μ-∫_-∞^∞ω J_1(ω) e^iω(Λ+A)/ωcosh(u ω), g_n(Λ) = ∫_0^∞Λ^'R(Λ+Λ^'+2A) y_n-1(Λ^'). To solve these integral equations for y_n(Λ), we begin by definingỹ_n^±(ω)= ∫_-∞^∞Λ θ_H(±Λ)y_n(Λ) e^iωΛ,where ỹ_n^+(ω) (ỹ_n(ω)) is an analytic function inthe upper (lower) half-plane.It isobvious thatthe Fourier transformation of y_n(x) satisfiesthe relation ỹ_n(ω)=ỹ_n^+(ω)+ỹ_n^-(ω).From <ref> it follows thatỹ_n^+(ω)1/1+exp(-2|u| |ω|) + ỹ_n^-(ω)=g̃_n(ω),by applying Fourier transformation. We further decompose the denominator1+exp(-2|u| |ω|) into a product of two pieces,1+exp(-2|u||ω|)=G^+(ω)G^-(ω),where G^+(ω) (G^-(ω)) is an analytic function in the upper (lower) half-plane.Then substituting this last equation into <ref> results in theformỹ_n^+(ω)/G^+(ω)+G^-(ω)ỹ_n^-(ω)=G^-(ω)g̃_n(ω). Furthermore, we decomposeG^-(ω)g̃_n(ω) into a sum of two pieces,G^-(ω)g̃_n(ω)=Q_n^+(ω)+Q_n^-(ω),where similarly Q_n^+(ω) (Q_n^-(ω)) is an analytic function in the upper (lower) half-plane.Then substitution of this last equation into <ref> givesỹ_n^+(ω)=G^+(ω)Q_n^+(ω), ỹ_n^-(ω)=Q_n^-(ω)/G^-(ω).In this way we canwork out the Fourier transformation of y_0(Λ) andy_n(Λ) itself.To this end, recalling (<ref>), we firstly decompose 1+exp(-2|u||ω|) asG^+(ω)=G^-(-ω) = √(2π)/Γ(1/2-i|u|ω/π)( -i|u|ω/π)^-i |u|ω/πexp(i |u|ω/π),where we should note that lim_ω→∞ G^±(ω)=1, along with the special values G^±(0)=√(2) and G^±(±iπ/2|u|)=√(π/e) of these functions.The decomposition for G^-(ω)g̃_n(ω) in general is subtle, however the leading case G^-(ω)g̃_0(ω) is accessible. We start analysis from the Fourier transformation of g_0(Λ),g̃_0(ω)=-μ2πδ_D(ω)-2π J_1(ω)exp(-i ω A)/ωcosh(uω),where on the rhs the δ_D function could be decomposed as2πδ_D(ω)=i( 1/ω+i ϵ-1/ω-i ϵ) (ϵ→ +0 ).The second term on the rhs is a meromorphic function of ω with poles located atω_n=iπ/2|u|(2n+1) (n ∈𝐙)originating from the term 1/cosh(uω), implying the decomposition1/cosh(uω)=χ^+(ω)+χ^-(ω), χ^+(ω)= i/|u|∑_n=0^∞ (-1)^n 1/ω+ω_n, χ^-(ω)= 1/cosh(uω)-i/|u|∑_n=0^∞ (-1)^n 1/ω+ω_n,where χ^+(ω) and χ^-(ω) are analytic functions in the upper and lower half-planes, respectively.With the help of <ref>, as for any analytic and bounded function f^-(ω) in the lower half-plane, the decomposition of f^-(ω)/cosh(u ω) isf^-(ω)/cosh(u ω)=F^+(ω)+F^-(ω), F^+(ω)=i/|u|∑_n=0^∞ (-1)^n f^-(-ω_n)/ω+ω_n, F^-(ω)=f^-(ω)/cosh(u ω)-F^+(ω). By virtue of <ref>, we make the following decomposition for G^-(ω)g̃_0(ω),Q_0^+(ω)= -i μG^-(0)/ω+i ϵ-q(ω)Q_0^-(ω)= i μG^-(0)/ω+i ϵ - 2π J_1(ω)exp(-i ω A)G^-(ω)/ωcosh(u ω) + q(ω)where q(ω)=4i∑_n=1^∞ (-1)^n G^-(-i h_n)I_1(h_n)exp(-h_n A)/(2n+1)(ω+i h_n), I_1(z) is the first order modified Bessel function, h_n=π/2|u|(2n+1), with the series converging only if A>1.If A ≫ 1, using <ref>, we havey_0^+(ω)=G^+(ω)[ -i μ G^-(0)/ω+i ϵ- q(ω) ].Obviously, we know y(0)=ε_1^'(A)=0, which implies0=y(0)=lim_ω→∞ -i ω ỹ^+(ω).Hereafter we replace y(ω) with y_0(ω), which is a reasonable approximation if A ≫ 1. Therefore <ref> give rise toμ=-4∑_n=0^∞G^-(-i h_n)I_1(h_n)exp(-h_n A)/(2n+1)G^-(0). Sincewe have obtained a parametric expression for the critical chemical potential, we turnto the expression for the magnetic field. Due to the fact thatthe phase boundary is determined by ε^u (0)=0, we thus use<ref> to determine the magnetic field.For simplicity, we rewrite <ref> asε^u(k)= -2cos k-μ-2u-B + ∫_A^∞Λ [ a_1(sin k - Λ) + a_1(sin k + Λ) ] ε^'_1(Λ) -∫_-∞^∞Λa_1(sin k - Λ) ε_1^'(Λ),ε_1^'(Λ)=ε_1^' (0)(Λ) - ∫_-A^A Λ^'a_2(Λ-Λ^') ε_1^'(Λ^'),where we have denotedε_1^'(0)(Λ)=-2μ-2∫_-π^πkcos^2 ka_1(sin k - Λ). Substituting<ref> into the last term on the rhs of <ref> givesε^u(k) = -2cos k-μ-2u-B +∫_0^∞Λ [s(Λ+A-sin k)+ s(Λ+A+sin k) ] y(Λ) -∫_-∞^∞Λs(Λ-sin k) ε_1^' (0)(Λ),where we have introduced the function s(x)=1/4|u|cosh(π x/2|u|) and made use of the two identities1/4|u|cosh(π x/2|u|) = ∑_n=0^∞ (-1)^n a_2n+1(x),∫_-∞^∞y a_n(x-y) a_m(y-z) =a_m+n(x-z). Substituting the expansion s(x)=1/2|u|∑_n=0^∞ (-1)^n exp(-h_n x), where | π x/u| < 1 and <ref> into <ref>, and after some algebraic manipulations, we arrive at the resultε^u(k)= -2cos k-2u-B +∑_n=0^∞(-1)^n/|u| ỹ^+(i h_n) cosh(h_n sin k) exp(-h_n A) + 2∫_0^∞ω J_1(ω) cos(ωsin k) exp(-|u|ω) /ωcosh(uω), Using <ref> and ε^u (0)=0 we derive theexpressionB= -2+2|u|+∑_n=0^∞(-1)^n/|u| ỹ^+(i h_n) exp(-h_n A) +2∫_0^∞ω J_1(ω) exp(-|u|ω) /ωcosh(uω)for determining the critical magnetic field.Here we denotedh_n= π/2|u|(2n+1).The equation (<ref>) sets up a relation between the magnetic field and the chemical potential.In summary, the phase boundary between phase IV and V is determinedby <ref>forA ≫ 1.Bequation §.§Derivation of the Equation of State The derivation of the equation of state is rather involved.Here we sketch the calculations for the terms p^u and p^b_n.Prior to substituting the dressed energies into the definitions of p^u and p^b_n integrating by parts, we first need to find a suitable form of the TBA equations for this procedure, i.e., (<ref>) and (<ref>).For simplicity in later discussion, we approximate the definition of p^b_n asp^b_n=∫_-∞^∞Λ/π1/√(Λ+in |u|)ε_n^' -(Λ)=T∫_-∞^∞Λ/2π∫_-π^πk a_n(Λ-sin k) =∫_-∞^∞Λ ε_n^' -(Λ)Δ_n(Λ) + o( 1/|u|^4), where Δ_n(Λ)=a_n(Λ)-1/2 b_n(Λ)+ 2Λ^2 b_n(Λ).In the above equations, we used the abbreviationsε_n^' -(x)= Tln(1+e^-ε_n^'(x)/T), ε_n^ -(x)= Tln(1+e^-ε_n(x)/T).To obtain the result (<ref>), regarding the first series of integral terms on the rhs of (<ref>), we expand them in the strong coupling regime as∑_n=1^∞∫_-∞^∞Λa_n(sin k-Λ)ε_n^' -(Λ) = ∑_n=1^∞∫_-∞^∞Λ Δ_n(Λ) ε_n^' -(Λ)×a_n(Λ){ 1+ 2Λsin k - sin^2 k/(nu)^2+Λ^2 + [2Λsin k - sin^2 k/(nu)^2+Λ^2]^2 }/Δ_n(Λ) + o( 1/|u|^4) = ∑_n=1^∞ p_n^b + a̅ + 2a̅cos^2 k + o( 1/|u|^4),where we have inserted (<ref>) and a̅ has been defined in <ref>.While for the second series of integral terms, it is easy to see that under the assumption B/T ≫ 1, these spin-wave contributions are no more than -Te^-2B/Te^-K̅ I_0(K̅), which is accessible through simple iteration of (<ref>). In fact, the spin degree of freedom is frozen here, and thus this term could be neglected in later discussion.We can rewrite (<ref>) asε^u(k)=ε^u_0(k)-A^u,where ε^u_0(k)=-2cos k + 2 a̅cos^2 k and A^u=μ+2u+B-∑_n=1^∞ p^b_n + a̅.Integrating by parts in p^u, we obtainp^u= T ln( 1+e^(μ+2u+B-∑_n=1^∞ p_n^b + a̅ -2)/T)+1/π∫_2a̅-2^2a̅+2ε^u_0 k(ε_0^u)/1+e^ε_0^u/T/z,where z=e^A^u/T and k(ε_0^u)=arccos( 1-√(1+2a̅ε_0^u)/2a̅) represents the inverse function of ε_0^u (k).By taking account of a̅∼∑_n=1^∞p^b_n/(nu)^2 for thestrong coupling regime, the integral in the above equation can be further simplified,1/π∫_2a̅-2^2a̅+2ε^u_0 k(ε_0^u)/1+e^ε_0^u/T/z =2/π∫_a̅-1^a̅+1x k(2x)/1+e^2x/T/z = 2/π∫_-1^1x arccos(-x)/1+e^2x/T/z-2a̅/π∫_-1^1xx^2/√(1-x^2)/1+e^2x/T/z+2a̅/1+e^2ε^u(π)/T+o(1/|u|^4),where we have changed the integration variable ε^u_0=2x and then applied Taylor expansion with respect to a̅. See ε^u(π) in <ref>. The result (<ref>) is therefor achieved.We then turn to the transformation of ε_n^'(Λ). Similar to the treatment for ε^u(k), we employ Taylor expansion to expand (<ref>) inthe strong coupling region, with resultε^'_n(Λ) = -2nμ- a_n(Λ) [ 2π-∫_-π^πk cos k ε^u -(k) ] - b_n(Λ) [ -π/2-∫_-π^πk cos ksin^2 kε^u -(k) ]+ ∑_m=1^∞ T_nm∗ε_m^' -(Λ) + o( 1/|u|^4), where the integral terms in the brackets are denoted as d_1 and d_2, respectively.Here d_1 and d_2 can be calculated via integration by parts, similar to that done for p^u above, see the explicit expressions in <ref>.With respect to the convolution term, due to the condition of low density, the cut-off of the dressed energy ε^'_n(Λ) is small, thus in general we can make the approximations∫_-∞^∞Λ^'a_p(Λ-Λ^')ε_q^' -(Λ^')= ∫_-∞^∞Λ^'a_p(Λ^') ε_q^' -(Λ^')-Λ^2 ∫_-∞^∞Λ^'b_p(Λ^') ε_q^' -(Λ^') +o ( 1/|u|^4),which results in <ref> in the main text. We next rewrite (<ref>) asε^'_n(Λ)=D_n (Λ/n|u|)^2 - A^b_n,where A^b_n=2nμ-η_n+d_1/π n |u|+d_2/2π(n|u|)^3 and D_n was defined in <ref>.<ref> is arrived at by substituting the above equation into the definition of p^b_n and integrating by parts.Lastly, a̅, ξ_p^m=T∫_-∞^∞Λ^'a_p(Λ^') ε_m^' -(Λ^') and ϕ_p^m=T∫_-∞^∞Λ^'b_p(Λ^') ε_m^' -(Λ^') are calculated in a similar way.Cequation §.§ Some constants in the scaling functionsThe constants usedin thescaling forms for the phase transition (II-IV)are given explicitly byλ_1 = -2√(|u|)(1-q_c4/π)/√(2π-q_c4^3/3),λ_2 = (1-q_c4/π)q_c4/π/√(|u|)√(2π-q_c4^3/3),n_b4 = γ,m_b4=1/2γ, κ_b4 = γ' (1+ 4/√(π)τ^1/2f̃_1/2 + 6/πτf̃_1/2^2 + 5/π^3/2τ^3/2f̃_1/2^3 - 2/√(π)τ^3/2f̃_3/2),λ_3 = -4√(|u|)(1-q_c4/π)/√(2π-q_c4^3/3),χ_b4 = 1/2γ^ ' + 2δ γ^ '-(1-γ)δ'/|u|√(π)( τ^1/2f̃_1/2 + 1/2√(π)τf̃_1/2^2 + 1/4πτ^3/2f̃_1/2^3 ..- τ^3/2f̃_3/2),λ_4 = -2(q_c4/π)^2(1-q_c4/π)/√(|u|^3)√(2π-q_c4^3/3).Here the parameter q_c4=√(B+2-2√(1+u^2)) +1/3π|u|( B+2-2√(1+u^2)).The constants usedin thescaling forms for the phase transition (V-IV)are given explicitly byn_b5 = -2/√(π)τ^1/2f̃_1/2 -1/πτf̃_1/2^2 - 1/2π^3/2τ^3/2f̃_1/2^3 + 1/√(π)τ^3/2f̃_3/2,λ_5 =-1/2√(π)( 1- 4/π√(1+2π|u|μ̃_c5/2π-q_c5^3/3)), λ_6 = -1/2√(π)( 1 - 8/π√(1+2π|u|μ̃_c5/2π-q_c5^3/3)) κ_b5 =-(1-γ) f̃_-1/2/D_0 τ^1/2( 4/√(π) + 6/πτ^1/2f̃_1/2 + 6/π^3/2τf̃_1/2^2. .- 7/4πτ^3/2f̃_3/2+ 5/π^2τ^3/2f̃_1/2^3 ),whereμ̃_c5≈ 2|u|-B-2+8√(2)/3π|u|α_1(2√(1+u^2)-B-2)^3/2 and q_c5=√(μ̃_c5+2u+B+2). Dequation§.§Explicit forms of the additivity rules The vectorial forms of <ref> are accessible by expanding the rescaled TBA equations in terms of {k^2n} and {Λ^2n} (n=0,1,2…).We here give the explicit expression for the matrix 𝐀^n(ỹ_c) (n=1,2) with the elements{𝐀^n (ỹ_c) }_jl=2/π∑_0≤ j ≤ i < ∞(-1)^i C_2i^2j ỹ_c^2i-2j+2l+1/n^2i+1(2i-2j+2l+1),with j,l=0,1,2,…. Thus the first two orders of 𝐀_(q)^n (y_c) are written as 𝐀^1_(1)(ỹ_c)= 2/π[ [10⋯; -10⋯;⋮ ;]]ỹ_c,𝐀^2_(1)(ỹ_c)=1/π[ [10⋯; -1/40⋯;⋮ ;]]ỹ_c,𝐀^1_(3)(ỹ_c)= 2/π[ [ -1/31/3⋯;21/3⋯;⋮ ;]]ỹ_c^3,𝐀^2_(3)(ỹ_c)= 1/π[ [ -1/12 1/3 ⋯; 1/8 -1/12 ⋯; ⋮; ]]ỹ_c^3.Next, the partial derivatives of <ref> read ∂ε⃗^ 1/∂μ̃ = ∂V⃗^1/∂μ̃ -𝐀^1(Λ̃_c) ∂ε⃗^ 2/∂μ̃, ∂ε⃗^ 2/∂μ̃ = V⃗^2 -𝐀^1(k̃_c) ∂ε⃗^ 1/∂μ̃ -𝐀^2(Λ̃_c) ∂ε⃗^ 2/∂μ̃. With the help of the explicit forms of𝐀^n (ỹ_c), i.e., (<ref>) and (<ref>), we can obtain <ref>, which relates the densities and the cutoffs.Together with <ref>, we then obtain the relation between the densities and the effective chemical potentials (<ref>) and (<ref>). On this basis, we now proceed toderive the explicit expressions for the effective compressibility and susceptibility in terms of densities of bound pairs and excess fermions.Apparently, the densities of bound pairs and excess fermions rely on the chemical potential and the magnetic field, and vice versa, which in fact indicates under fixed magnetic field one could obtain the following results through the total derivatives,κ_1=( ∂ n_1/∂μ_1)_B=n_1/μ, κ_2=2( ∂ n_2/∂μ_2)_B=α_1n_2/μ,where we keep d B=∂ B/∂ n_1d n_1+∂ B/∂ n_2d n_2=0. Thus we haven_1/μ=1/J( ∂ B/∂ n_2)_n_1, n_2/μ=-1/J( ∂ B/∂ n_1)_n_2.Here the Jacobian determinantJ= ( ∂μ/∂ n_1)_n_2( ∂ B/∂ n_2)_n_1 - ( ∂ B/∂ n_1)_n_2( ∂μ/∂ n_2)_n_1= -α_1/2[ ( ∂μ_1/∂ n_1)_n_2( ∂μ_2/∂ n_2)_n_1 - ( ∂μ_2/∂ n_1)_n_2( ∂μ_1/∂ n_2)_n_1],where we have used <ref>.Similarly, the magnetic field is dependent on the effective chemical potentials while the latter is dependent on densities of bound pairs and excess fermions.Therefore by application of chain rule we have(∂ B/∂ n_1)_n_2= (∂μ_1/∂ n_1)_n_2 -α_1/2(∂μ_2/∂ n_1)_n_2,(∂ B/∂ n_2)_n_1= (∂μ_1/∂ n_2)_n_1 -α_1/2(∂μ_2/∂ n_2)_n_1. It is obvious that once the explicit expression of μ_r in terms of n_s (r,s=1,2) is known, our goal of the effective compressibilities is easy to achieve. We use <ref> to deriveμ_1=π^2 n_1^2[1+2(2n_2/|u|)+3(2n_2/|u|)^2] +4π^2α_1/3β_1^3|u|n_2^3[1+32n_1+n_2/β_1|u|], μ_2=π^2 n_2^2/β_1^2[1+22n_1+n_2/β_1|u|+3(2n_1+n_2/β_1|u|)^2] +4π^2/3α_1 |u|n_1^3[1+3(2n_2/|u|)] +2π^2/3β_1^3 |u|n_2^3[1+32n_1+n_2/β_1|u|],and thusκ_1= π^2/J[ -α_1 n_2/β_1^2-4 α_1 n_1 n_2/|u|β_1^3+4 n_1^2/|u|-4 n_1^3/u^2+24 n_1^2 n_2/u^2. .-12 α_1 n_1^2 n_2/u^2 β_1^4+6 α_1 n_2^3/u^2 β_1^4],κ_2= -2α_1π^2/J[ n_1-n_1^2/|u|+4 n_1 n_2/|u|-6 n_1^2 n_2/u^2-α_1 n_2^2/u β_1^3. .+12 n_1 n_2^2/u^2-6 α_1 n_1 n_2^2/u^2 β_1^4],J= -2π^4α_1/β_1^2n_1 n_2 [ 1+4 n_1/|u|β_1+12 n_1^2/u^2 β_1^2+4 n_2/|u|+4 n_2/|u| β_1. .+24 n_1 n_2/u^2 β_1^2+8 n_1 n_2/u^2 β_1+12 n_2^2/u^2+10 n_2^2/u^2 β_1^2+16 n_2^2/u^2 β_1]. The situation for effective susceptibilities is rather simple. With fixed total particle density, one confirms that n_1+2n_2=0, and thus the total derivative of the effective chemical potentials with respect to n_r (r=1,2) isμ_1 = [ ( ∂μ_1/∂ n_1)_n_2 -1/2( ∂μ_1/∂ n_2)_n_1]n_1,μ_2 = -2[(∂μ_2/∂ n_1)_n_2 -1/2(∂μ_2/∂ n_2)_n_1] n_2. After some algebraic manipulations we then obtainχ̅_1= 1 / (∂μ_1/∂ n_1 -1/2∂μ_1/∂ n_2),χ̅_2= -1/ (∂μ_2/∂ n_1 -1/2∂μ_2/∂ n_2),which together with <ref> result in χ̅_1= 1/(2π^2)/n_1-n_1^2/|u|+4 n_1 n_2/|u|-6 n_1^2 n_2/u^2-α_1 n_2^2/|u| β_1^3+12 n_1 n_2^2/u^2-6 α_1 n_1 n_2^2/u^2 β_1^4, χ̅_2= 1/π^2/n_2/β_1^2-4 n_1^2/|u| α_1+4 n_1^3/u^2 α_1 +4 n_1 n_2/|u| β_1^3-24 n_1^2 n_2/u^2 α_1+12 n_1^2 n_2/u^2 β_1^4-6 n_2^3/u^2 β_1^4. 99Hubbard1963J. Hubbard, J. Proc. R. Soc. A 276, 237 (1963); ibid 281 401 (1964).Yang:1967C. N. Yang,Phys. Rev. Lett. 19, 1312 (1967).Baxter:1972R. J. Baxter,Ann. Phys. (N.Y.) 70, 193 (1972). Lieb1968 E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968). Ess05F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümperand V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2005).MT1969s M. Takahashi, Prog. Theo. Phys. 42, 1098 (1969); ibid 43, 860 (1970); ibid 43, 1619 (1970).MT1971M. Takahashi, Prog. Theo. Phys. 45, 756 (1971).Shiba1972 H. Shiba, Phys. Rev. B 6, 930 (1972). Krivnov1975 V. Ya. Krivnov and A. A. Ovchinnikov, Zh. Eksp. Teor. Fiz. 67, 1568 (1974).Usuki1989 T. Usuki, N. Kawakami and A. Okiji, Phys. Lett. A 135, 476 (1989).Woyna1991 F. Woynarovich and K. Penc, Z. Phys. B 85, 269 (1991).Coll1974 C. F. Coll, Phys. Rev. B 9, 2150 (1974).Ovchi1970 A. A. Ovchinnikov, Sov. Phys. JETP 30, 1160 (1970).Woyna1982s F. Woynarovich, J. Phys. C 15, 85 (1982); ibid 15, 97 (1982); ibid 16, 5293 (1983).Klumper1990 A. Klümper, A. Schadschneider and J. Zittartz, Z. Phys. B 78, 99 (1990).Degu2000 T.Deguchi, F. H. L. Essler, F.Göhmann, A. Klümper, V. E. Korepin and K. Kusakabe, Phys. Rep. 331, 197 (2000).Woyna1983 F. Woynarovich, J. Phys. C 16, 6593 (1983). Ess1994a F. H. L. Essler and V. E. Korepin, Phys. Rev. Lett. 72, 908 (1994).Ess1994b F. H. L. Essler and V. E. Korepin, Nucl. Phys. B 426, 505 (1994).MT1972 M. Takahashi, Prog. Theo. Phys. 47, 69 (1972). MT1974 M. Takahashi, Prog. Theo. Phys. 52, 103 (1974). Lee1988 K.-J.-B. Lee and P. Schlottmann, Phys. Rev. B 38, 11566 (1988).Okiji1989 N. Kawakami, T. Usuki and A. Okiji, Phys. Lett. A 137, 287 (1989); T. Usuki, N. Kawakami and A. Okiji, J. Phys. Soc. Japan 59, 1357 (1990). Sacra1995 P. D. Sacramento, J. Phys. C 7, 143 (1995). Frahm1990 H. Frahm and V. E. Korepin, Phys. Rev. B 42, 10553 (1990); ibid 43, 5653 (1991). Woyna1987F. Woynarovich and H. P. Eckle, J. Phys. A 20, L443 (1987); F. Woynarovich, J. Phys. A 22 4243 (1989). Ogata1990 M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 (1990); M. Ogata, T. Sugiyama and H. Shiba, Phys. Rev. B 43, 8401 (1991).Parola1990 A. Parola and S. Sorella, Phys. Rev. Lett. 64, 1831 (1990); Phys. Rev. Lett. 45, 13156 (1992); Phys. Rev. B 57, 6444 (1997).Penc1995 K. Penc, F. Mila and H. Shiba, Phys. Rev. Lett. 75, 894 (1995); K. Penc, K. Hallberg, F. Mila and H. Shiba, Phys. Rev. Lett. 77, 1390 (1996); Phys. Rev. B 55, 15475 (1997).Gohmann1998 F. Göhmann, A. R. Its and V. E. Korepin, Phys. Lett. A 249, 117 (1998); F. Göhmann and V. E. Korepin, Phys. Lett. A 260, 516 (1999). Ess2000 E. Jeckelmann, F. Gebhard and F. H. L. Essler, Phys. Rev. Lett. 85, 3910 (2000); F. H. L. Essler and A. M. Tsvelik, Phys. Rev. B 65, 115117 (2002); D. Controzzi and F. H. L. Essler, Phys. Rev. B 66, 165112 (2002); B. Doyon and S. Lukyanov, Nucl. Phys. B 644, 451 (2002); F. H. L. Essler and A. M. Tsvelik, Phys. Rev. Lett. 90, 126401 (2003). Bogoliubov1988 N. M. Bogoliubov and V. E. Korepin, Mod. Phys. Lett. B 1, 349 (1988); Teor. i Mat. Fiz. 82, No. 3, 331 (1990); Int. J. Mod. Phys. B 3, 427 (1994). DMRG A. E. Feiguin, and F. Heidrich-Meisner, Phys. Rev. B 76, 220508(R) (2007); A. Lüscher, R. M. Noack and A. M. Läuchli, Phys. Rev. A 78, 013637 (2008); M. Rizzi, M. Polini, M. A. Cazalilla, M. P. Tosi and R. Fazio, Phys. Rev. B 77, 245105 (2008); M. Tezuka and M. Ueda, Phys. Rev. Lett. 100, 110403 (2008); M. Tezuka and M. Ueda, New J. Phys. 12, 055029 (2010). QMC G. G. Batrouni, M. H. Huntley, V.G. Rousseau and R. T. Scalettar, Phys. Rev. Lett. 100, 116405 (2008); S. K. Baur, J. Shumway and E. J. Mueller, Phys. Rev. A 81, 033628 (2010); M. J. Wolak, V. G. Rousseau, C. Miniatura, B. Gémaud, R. T. Scalettar and G. G. Batroun, Phys. Rev. A 82, 013614 (2010).SC S. Cheng, Y.-Z. Jiang, Y.-C. Yu, M. T. Batchelor and X.-W. Guan, Nucl. Phys. B 929, 353 (2018).Klumper1996 A. Klümper and R.Z. Bariev, Nucl. Phys. B 458, 623 (1996); G. JüKttner, A. Klümper and J. Suzuki, Nucl. Phys. B 522, 471 (1998).Yang1969 C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969).Koma1987 T. Koma, Prog. Theor. Phys. 78, 1213 (1987); ibid 81, 783 (1989). Cardy1984 J. L. Cardy, J. Phys. A 17, L385 (1984); Nucl. Phys. B 270 186, (1986). Blote1986 H. W. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986). Aff1986 I. Affleck, Phys. Rev. Lett. 56, 746 (1986). Belavin1984 A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984). ZhaoE:2008E. Zhao and W. Vincent Liu, Phys. Rev. A 78, 063605 (2008).Kajala:2011J. Kajala, F. Massel and P. Törmä, Phys. Rev. A 84, 041601 (R) (2011). Singha2011 A. Singha, M. Gibertini, B. Karmakar, S. Yuan, M. Polini, G. Vignale, M. I. Katsnelson, A. Pinczuk, L. N. Pfeiffer, K. W. West and V. Pellegrini, Science332, 1176 (2011).Hart2015 R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse and R. G. Hulet, Nature 519, 211 (2015).Greif2016 D. Greif, M. F. Parsons, A. Mazurenko, C. S. Chiu, S. Blatt, F. Huber, G. Ji and M. Greiner, Science 351, 953 (2016).Parsons2016 M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif and M. Greiner, Science 353, 1253 (2016).Lawrence2016 L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol and M. W. Zwierlein, Science 353, 1260 (2016).Boll2016 M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch and C. Gross, Science 353, 1257 (2016).Zhang:2015 R. Zhang, Y. Cheng, H. Zhai and P. Zhang, Phys. Rev. Lett. 115, 135301 (2015).Mazurenko:2017 A. Mazurenko,C.S. Chiu, G. Ji, M. F. Parsons, M.Kanász-Nagy, R.Schmidt, F.Grusdt, E.Demler, D.Greif andM.Greiner, Nature 545, 462 (2017).Kono:2015 Y. Kono, T. Sakakibara, C. P. Aoyama, C. Hotta, M. M. Turnbull, C. P. Landee and Y. Takano,Phys. Rev. Lett. 114, 037202 (2015).Krellner:2016C. Krellner, S. Lausberg, A. Steppke, M. Brando, L. Pe- drero, H. Pfau, S. Tencé, H. Rosner, F. Steglich and C Geibel, New J. Phys. 13, 103014 (2011).Carmelo:1992J. M. P. Carmelo, P. Horsch and A. A. Ovchinnikov, Phys. Rev. B 45, 7899 (1992).Shaginyan:2016V. R. Shaginyan, V. A. Stephanovich, K. G. Popov, E. V. Kirichenko and S. A. Artamonov, Ann. Phys. (Berlin) 528, 483 (2016).short S. Cheng, Y.-Z. Jiang, M. T. Batchelor and X.-W. Guan, “FFLO correlation and free fluids in the one-dimensional attractive Hubbard model", arXiv:1708.07776.Giamarchi:2004T. Giamarchi, Quantum Physics in onedimension (Oxford University Press, Oxford, 2004).Matveev:2008K. A. Matveev and A. Furusaki, Phys. Rev. Lett. 101, 170403 (2008).YCY2016 Y.-C. Yu, Y.-Y. Chen, H.-Q. Lin, R. A. Roemer and X.-W. Guan, Phys. Rev. B 94, 195129 (2016).Fisher1989 M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. S. Fisher, Phys. Rev. B 40, 546 (1989).note1 The sound velocity here is accessible by using the formula v_r=√(L/m_pn_r r∂^2 E_r/∂ L^2), where the particle mass has been rescaled as m_p=1/2 and E_r=e_r L. See e_r in <ref>.Sachdev S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999).Ho2010 Q. Zhou and T.-L. Ho, Phys. Rev. Lett. 105, 245702 (2010).Hazzard2011 K. R. A. Hazzard and E. J. Mueller, Phys. Rev. A 84, 013604 (2011). Yang:2017B. Yang, Y.-Y. Chen, Y.-G. Zheng, H. Sun, H.-N. Dai, X.-W. Guan, Z.-S. Yuan and J.-W. Pan,Phys. Rev. Lett. 119, 165701 (2017)He:2017 F. He, Y.-Z. Jiang, Y.-C. Yu, H.-Q. Lin and X.-W. Guan,arXiv:1702.05903.Rozhkov:2014A. V. Rozhkov, Phys. Rev. Lett. 112, 106403 (2014).Lebed:2015 A. G. Lebed, Phys. Rev. Lett. 115, 157001 (2015). | http://arxiv.org/abs/1708.07784v3 | {
"authors": [
"Song Cheng",
"Yi-Song Yu",
"Murray T Batchelor",
"Xi-Wen Guan"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170825154317",
"title": "Universal thermodynamics of the one-dimensional attractive Hubbard model"
} |
Steward Observatory, University of Arizona, Tucson, AZ 85721, USA Accepted for Publication in ApJLDynamical mass estimates down to the planet-mass regime can help to understand planet formation. We present Atacama Large Millimeter/submillimeter Array (ALMA) 1.3 mm observations of FW Tau C, a proposed ∼10 M_ Jup planet-mass companion at ∼330 au from the host binary FW Tau AB. We spatially and spectrally resolve the accretion disk of FW Tau C in ^12CO (2–1). By modeling the Keplerian rotation of gas, we derive a dynamical mass of ∼0.1 M_. Therefore, FW Tau C is unlikely a planet, but rather a low-mass star with a highly inclined disk. This also suggests that FW Tau is a triple system consisting of three ∼0.1 M_ stars.§ 1. INTRODUCTIONIn the past ∼10 years, many planet-mass companions at wide separations (few to tens of M_ Jup; tens to hundreds of astronomical units from host stars) have been discovered in direct imaging surveys. Their masses are usually determined by comparing observables like luminosity and effective temperature to predictions from evolutionary and atmospheric models (e.g., ), which vary widely depending on the formation pathway (e.g., core accretion versus gravitational instability). As a result, it would be valuable if masses could be dynamically measured. As some of these substellar companions are young (≲10 Myr) and have features associated with active accretion (e.g., ), their dynamical masses can be measured if their accretion disks can be spatially and spectrally resolved.Among all of the known wide-separation companions, the proposed planet-mass object FW Tau C is the prime target to search for a Keplerian-rotating disk. FW Tau C is a tertiary companion located at ∼23 projected separation (∼330 au) to FW Tau AB, a close binary (∼008) of nearly equal-mass stars (∼0.1 M_) in the 2 Myr Taurus–Auriga star-forming region. FW Tau C was discovered by <cit.> in their survey of binary stars, and its common proper motion was recently confirmed <cit.>. Studies have shown that FW Tau C has a rather flat near-infrared continuum owing to accretion-induced veiling, as well as many emission lines indicative of outflow and accretion activities <cit.>. The accretion disk was previously detected by the Atacama Large Millimeter/submillimeter Array (ALMA) in 1.3 mm dust continuum <cit.> and ^12CO (2–1) <cit.>, and a 1–2 M_⊕ of dust was inferred. Despite these observational efforts, FW Tau C's nature remains enigmatic because strong veiling inhibits accurate mass estimates. <cit.> derived a planetary mass of 10±4 M_ Jup from its dereddened K' flux, while <cit.> suggested that FW Tau C could be a 0.03–0.15 M_ brown dwarf or low-mass star embedded in an edge-on disk in order to explain its flat K-band spectrum and faint optical and near-infrared brightness.Accurate mass measurement is therefore key to distinguishing the planet-mass scenario from the stellar-mass scenario. Here, we present the new ALMA 1.3 mm data from Cycle 3. With a ∼02 beam and ∼0.3 velocity resolution, we spatially and spectrally resolve the gas disk and derive a dynamical mass of FW Tau C by modeling the Keplerian rotation. Parameters of the FW Tau system are summarized in Table <ref>. @lccl@Properties of FW Tau Parameter FW Tau AB FW Tau C References Distance (pc) 2c∼140 1 Age (Myr) 2c∼2 2 Separation () 2c∼2.3 3, 4 PA () 2c∼296 3, 4 SpT M6 ± 1 ⋯ 3 A_V (mag) ∼0.4 ⋯ 3 log(L/L_) ∼-1.1a ⋯ 5 Mass (M_) ∼0.12a ∼0.1 5, 6 ^a For each component. (1) <cit.>, (2) <cit.>, (3) <cit.>, (4) <cit.>, (5) <cit.>, (6) This work. § 2. METHODOLOGY §.§ 2.1. Observations and Data ReductionFW Tau was observed with ALMA Band 6 during Cycle 3, on 2016 September 14. During the observations thirty-six 12 m antennas were available, with baselines ranging from 15 to 3247 m. The Band 6 receiver was configured to have three basebands set up for dust continuum observations, centered at 233.0, 246.0, and 248.0 GHz and each with 2 GHz of bandwidth. The last baseband was configured with 3840 0.122 MHz channels centered at 230.538 GHz (0.32 velocity resolution; Hanning smoothed) in order to spatially and spectrally resolve ^12CO (2–1) emission from the disk. J0510+1800 was used as the bandpass and flux calibrator, and J0433+2905 was used as the the gain calibrator. The on-source time was ∼13 minutes. The data were reduced using the ALMA pipeline in thepackage. We employed one iteration of phase-only self-calibration to the 2 GHz continuum basebands. The baseband containing CO emission was much narrower, so it had much lower signal-to-noise ratio in continuum emission and we were unable to obtain good self-calibration solutions. We then CLEANed the calibrated data using the multi-frequency synthesis mode and natural weighting to enhance sensitivity in the images. The continuum map (left panel of Figure <ref>) has a beam size of 029×015 with a position angle (PA) of 1608, and an rms of 35 μJy beam^-1. The CO channel maps (top panel of Figure <ref>) have a beam size of 030×016 with a PA of 1603, and a mean rms of 4.5 mJy beam^-1 in signal-free channels. The integrated moment-zero and moment-one maps of the CO emission are shown in Figure <ref>. @lc@FW Tau C Disk PropertiesParameter Value M_* (M_⊙) 0.098 ± 0.015 M_disk,dust (M_) 1.15 ± 0.01M_disk,gas (M_) 0.58±0.13R_disk (au) 141.9 ± 6.6γ1.63 ± 0.14T_0 (K)62.1 ± 34.6q0.14 ± 0.10ξ ()0.65 ± 0.08v_ sys ()6.10 ± 0.02 i () 62.8 ± 2.4PA () 40.6 ± 1.5§.§ 2.2. Disk Modeling: Continuum The continuum emission from FW Tau C is, at best, only marginally resolved by our observations, so we use a simple geometrical model to fit the data. We assume that the continuum emission traces a uniform brightness disk, with a 1.3 mm brightness F_ν, a radius R_ disk, and some inclination i and position angle PA. We also allow the centroid of the emission to vary in our fit. We fit the model directly to the continuum visibilities using the Markov Chain Monte Carlo (MCMC) package<cit.>.In order to measure the dust mass of the system, we assume that the continuum emission traces optically thin dust so that we can estimate the disk mass fromM_ disk = F_ν D^2/κ_νB_ν(T)<cit.>. We use standard assumptions of T = 20 K and κ_ν = 2.3 cm^2 g^-1 and a distance to FW Tau C of 140 pc.§.§ 2.3. Disk Modeling: Keplerian Rotation Unlike the 1.3 mm continuum emission, FW Tau C's gas disk is well resolved by our ^12CO (2–1) observations (see Figure <ref>), including a clear detection of spatially resolved Keplerian rotation. To model the data and determine disk and stellar parameters, we follow the modeling procedure described in <cit.> to fit our channel maps with synthetic channel maps produced from radiative transfer models. Such models can be used to measure disk parameters such as radius and inclination, as well as the stellar (or planetary) mass <cit.>. Although we follow the procedure outlined by <cit.>, we have developed our own codes to run and fit these models to our data set.We assume that the ^12CO (2–1) emission comes from a flared accretion disk, with a density profile described byρ(R,z) = Σ(R)/√(2π)h(R) exp[-1/2(z/h(R))^2],where R and z are defined in cylindrical coordinates, and Σ(R) and h(R) are the surface density and disk scale height, respectively. We assume that the disk has a power-law surface density profile: Σ(R) = Σ_0(R/R_0)^-γ,and we calculate the CO column density from this surface density profile asN_ CO(R) = X_ CO Σ(R)/μm_ H.Here, X_ CO = 1 × 10^-4 is the CO mass abundance fraction, and μ = 2.37 is the mean molecular weight. The disk is truncated at an inner radius of 0.1 au and an outer radius of R_ disk, beyond which the density drops to zero.We also assume that the disk is in local thermodynamic equilibrium and is vertically isothermal, with a radial temperature profile of T(R) = T_0(R/1 au)^-q.Under these assumptions, the scale height in the disk is set by the balance of thermal pressure and gravity such thath(R) = ( k_b R^3 T(R)/G M_*μm_ H)^1/2,where k_b is the Boltzmann constant, and G is the gravitational constant.Finally, rotation in the disk is assumed to be Keplerian, with an azimuthal velocity ofv_k = √(G M_*/r).We assume that the velocities in the radial and vertical directions are zero. We also include microturbulent line broadening, which we assume is uniform throughout the disk, with a value of ξ in units of . Finally, we allow the star to have a systemic velocity, v_ sys, which Doppler shifts the velocity center away from zero.In all, the density, temperature, and velocity structure of the system are described by the following parameters: M_ disk, R_ disk, γ, T_0, q, ξ, M_*, and v_ sys. We also allow the viewing geometry of the system, the inclination, and position angle to vary. In our model, the position angle of the disk is defined as the angle east of north of the projection of the disk angular momentum vector on the sky and ranges from 0^∘ to 360^∘. As our detection of CO (2–1) is not high sensitivity, we allow the inclination of the fit to range from 0^∘ to 90^∘. With higher sensitivity observations, however, it may be possible to distinguish between i<90^∘ and i>90^∘ <cit.>. Initially, we center the CO data based on the results of our geometrical continuum modeling, but allow for a small deviation from that centering in our model. We fix the distance to the disk at 140 pc (see Table 1). The uncertainty in the distance, however, translates linearly into an uncertainty on the measured stellar mass <cit.>, so we add the uncertainty on the distance estimate in quadrature with the uncertainty derived from our fit. Here, we assume a conservative distance uncertainty of 20 pc following <cit.>.We use the 3D radiative transfer modeling package<cit.> to calculate the level populations in each cell and produce synthetic ^12CO (2–1) channel maps for a given set of model parameters. Those synthetic channel maps are Fourier transformed and fit directly to the visibilities using the MCMC fitting package<cit.>, with uniform priors for all parameters.Although these models are computationally expensive, they can be run on powerful supercomputers so that the computations are spread out over a large number of central processing units (CPUs). For this particular instance, we run the fit over 128 CPUs on the University of Arizona El Gato supercomputer, and the modeling took a few days to converge. The models were determined to be converged when thewalkers had reached a steady state with measured best-fit values changing minimally over a large number of steps.§ 3. RESULTS Figure <ref> shows the 1.3 mm dust continuum, the ^12CO (2–1) integrated intensity map (moment 0), and the intensity-weighted velocity map (moment 1) of the FW Tau C disk. Similar to <cit.> and <cit.>, no signal is found from the close binary FW Tau AB, and the 3σ upper limit for a unresolved source suggests a dust mass ≲0.07 M_⊕. It is known that close binaries can shorten disk lifetimes (e.g., ), so the disk around FW Tau AB may be depleted already. For the companion FW Tau C, its dust disk is compact and likely unresolved; in contrast, the gas disk is more extended and clearly shows a Keplerian rotation.Figure <ref> is the position–velocity diagram constructed along the major axis of the gas disk. On top of the PV diagram we also plot the Keplerian rotation curves for 10 M_ Jup and 0.1 M_ objects. It is clear that the velocity profile of the FW Tau C disk is incompatible with the 10 M_ Jup rotation curve, but more consistent with that of a 0.1 M_ star. Our disk modeling (see Figure <ref> and Table <ref>) also indicates that FW Tau C's mass is ∼0.1 M_, about 10 times higher than the 10 M_ Jup suggested by <cit.>. Hence, FW Tau C is not a planetary-mass object, but is rather a low-mass star with an inclined disk, as suggested by <cit.>. The high mass of FW Tau C is also in agreement with some features in its spectrum that closely resemble T Tauri stars <cit.>. Our observations therefore suggest that FW Tau is a young triple system in which each of the stars has a mass of ∼0.1 M_.Searches for accretion disks in millimeter around directly imaged planet-mass companions have not yielded positive results (e.g., ), although the disk around the free-floating planet OTS 44 has recently been imaged <cit.>.Current flux upper limits imply that these wide companion disks might have <0.1 M_⊕ of dust, which in turn could imply a very short disk lifetime. Future deep imaging down to a sublunar mass regime is perhaps needed to detect these disks, or place stringent constraints on their masses, and further elucidate the mass growth history of the planetary-mass companions at wide orbits. Finally, we summarize our disk modeling results as below and also in Table <ref>. Modeling the FW Tau C's dust disk as a circular Gaussian, we find that it has a 1.3 mm flux of ∼2.06 mJy, which equates to a dust mass of ∼1.15 M_⊕ and is consistent with previous measurements in <cit.> and <cit.>. For the gas disk, our radiative transfer modeling, as demonstrated in Figure <ref>, finds that the gas disk has a radius of ∼140 au, inclination of ∼63, PA of ∼41, systemic velocity of ∼6.1 , and gas mass of ∼0.58 M_⊕. Although recent studies have shown a low gas-to-dust ratio in protoplanetary disks (e.g., ), we caution that with a single optically thick CO line, we are very likely to underestimate the true gas mass by a factor of few or even orders of magnitude (e.g., ). As <cit.> pointed out, to accurately derive the gas mass, one should observe low-transition lines to ameliorate the optical-depth effects and should also use multiple CO isotopologues to characterize the chemical depletion and abundance variation across the disk.§ 4. SUMMARYFW Tau C is a wide-orbit companion at ∼330 au from the close binary FW Tau AB that has been suggested to have a planet-like mass of 10 M_ Jup. Here, we have used ALMA to detect FW Tau C's accretion disk in a 1.3 mm dust continuum and ^12CO (2–1) emission. We find that the gas motion is both spatially and spectrally resolved and clearly follows Keplerian rotation, enabling a dynamical mass estimate of the central object. We show that FW Tau C's mass is in fact ∼0.1 M_, so it is more likely a low-mass star embedded in an inclined accretion disk. We thank the referee for very helpful comments. We thank Laird Close, Johanna Teske, Yu-Cian Hong, and Jing-Hua Lin for discussions. Y.-L.W. is supported by the NASA Origins of Solar Systems award and the TRIF fellowship. This Letter makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.00773.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.[Andrews et al.(2013)]A13Andrews, S. M., Rosenfeld, K. A., Kraus, A. L., & Wilner, D. J. 2013, , 771, 129 [Ansdell et al.(2016)]A16Ansdell, M., Williams, J. P., van der Marel, N., et al. 2016, , 828, 46 [Baraffe et al.(2015)]Baraffe15Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, , 577, A42 [Bayo et al.(2017)]B17Bayo, A., Joergens, V., Liu, Y., et al. 2017, , 841, L11 [Beckwith et al.(1990)]B90Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, , 99, 924 [Bowler et al.(2015)]B15Bowler, B. P., Andrews, S. M., Kraus, A. L., et al. 2015, , 805, L17 [Bowler et al.(2014)]B14Bowler, B. P., Liu, M. C., Kraus, A. L., & Mann, A. W. 2014, , 784, 65 [Bowler et al.(2011)]Bowler11Bowler, B. P., Liu, M. C., Kraus, A. L., Mann, A. W., & Ireland, M. J. 2011, , 743, 148 [Caceres et al.(2015)]C15Caceres, C., Hardy, A., Schreiber, M. R., et al. 2015, , 806, L22 [Chabrier et al.(2000)]C00Chabrier, G., Baraffe, I., Allard, F., & Hauschildt, P. 2000, , 542, 464 [Cieza et al.(2009)]Cieza09Cieza, L. A., Padgett, D. L., Allen, L. E., et al. 2009, , 696, L84 [Czekala et al.(2015)]Cz15Czekala, I., Andrews, S. M., Jensen, E. L. N., et al. 2015, , 806, 154 [Czekala et al.(2016)]Cz16Czekala, I., Andrews, S. M., Torres, G., et al. 2016, , 818, 156 [Dullemond(2012)]Dullemond12Dullemond, C. P. 2012, RADMC-3D: A Multi-purpose Radiative Transfer Tool, Astrophysics Source Code Library, ascl:1202.015 [Foreman-Mackey et al.(2013)]FM13Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, , 125, 306 [Isella et al.(2014)]I14Isella, A., Chandler, C. J., Carpenter, J. M., Pérez, L. M., & Ricci, L. 2014, , 788, 129 [Kenyon et al.(1994)]K94Kenyon, S. J., Dobrzycka, D., & Hartmann, L. W. 1994, , 108, 1872 [Kraus et al.(2015)]K15Kraus, A. L., Andrews, S. M., Bowler, B. P., et al. 2015, , 798, L23 [Kraus & Hillenbrand(2009)]K09Kraus, A. L., & Hillenbrand, L. A. 2009, , 704, 531 [Kraus et al.(2014)]K14Kraus, A. L., Ireland, M. J., Cieza, L. A., et al. 2014, , 781, 20 [Lynden-Bell & Pringle(1974)]LBP74Lynden-Bell, D., & Pringle, J. E. 1974, , 168, 603 [MacGregor et al.(2017)]M17MacGregor, M. A., Wilner, D. J., Czekala, I., et al. 2017, , 835, 17 [Marley et al.(2007)]M07Marley, M. S., Fortney, J. J., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2007, , 655, 541 [Ricci et al.(2017)]R17Ricci, L., Cazzoletti, P., Czekala, I., et al. 2017, , 154, 24 [Rosenfeld et al. (2013)]R13Rosenfeld, K. A., Andrews, S. M., Hughes, A. M., et al. 2013, , 774, 16 [Spiegel & Burrows(2012)]SB12Spiegel, D. S., & Burrows, A. 2012, , 745, 174 [White & Ghez(2001)]WG01White, R. J., & Ghez, A. M. 2001, , 556, 265 [Williams & Best(2014)]WB14Williams, J. P., & Best, W. M. J. 2014, , 788, 59 [Wolff et al.(2017)]Wolff17Wolff, S. G., Ménard, F., Caceres, C., et al. 2017, , 154, 26 [Wu et al.(2017)]W17Wu, Y.-L., Sheehan, P. D., Males, J. R., et al. 2017, , 836, 223 [Yu et al.(2017)]Y17Yu, M., Evans, N. J., Dodson-Robinson, S. E., Willacy, K., & Turner, N. J. 2017, , 841, 39 [Zhou et al.(2014)]Z14Zhou, Y., Herczeg, G. J., Kraus, A. L., Metchev, S., & Cruz, K. L. 2014, , 783, L17 | http://arxiv.org/abs/1708.08122v1 | {
"authors": [
"Ya-Lin Wu",
"Patrick D. Sheehan"
],
"categories": [
"astro-ph.EP",
"astro-ph.SR"
],
"primary_category": "astro-ph.EP",
"published": "20170827184640",
"title": "An ALMA Dynamical Mass Estimate of the Proposed Planetary-mass Companion FW Tau C"
} |
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USANational Radio Astronomy Observatory, Socorro, NM 87801, USA We present a daytime thermal image of Europa taken with the Atacama Large Millimeter Array. The imaged region includes the area northwest of Pwyll Crater, which is associated with a nighttime thermal excess seen by the Galileo Photopolarimeter Radiometer and with two potential plume detections. We develop a global thermal model of Europa and simulate both the daytime and nighttime thermal emission to determine if the nighttime thermal anomaly is caused by excess endogenic heat flow, as might be expected from a plume source region. We find that the nighttime and daytime brightness temperatures near Pwyll Crater cannot be matched by including excess heat flow at that location. Rather, we can successfully model both measurements by increasing the local thermal inertia of the surface. § INTRODUCTIONEuropa may be one of the most habitable worlds in the Solar System. Beneath a relatively thin ice shell, it hosts a liquid water ocean in contact with a rocky core <cit.>. Europa's young surface age, surface geology, and salty surface composition point to a history of geologic activity that may have facilitated contact between the ocean and the surface environments <cit.>. If such activity continued today, then direct study of the oceanic composition may be possible, but the question of modern geologic activity remains. Recent observations using the Hubble Space Telescope (HST) have hinted at the possibility of active water vapor plumes at Europa <cit.>. However, the detections have been sporadic and tenuous, making confirming the existence of plumes difficult. It is possible that sites of recent or ongoing geologic activity would cause persistent spatially localized thermal anomalies, similar to the so-called "tiger stripes" of Enceladus <cit.>. Therefore, high-resolution thermal data may present another, perhaps more robust, way of identifying active regions in the case of Europa. <cit.> and <cit.> detected potential off-limb absorption near the crater Pwyll in HST images of Europa as it transited Jupiter. This location is coincident with a nighttime thermal excess in brightness temperatures measured by the Galileo Photopolarimeter-Radiometer (PPR) <cit.>. If the thermal excess were caused by increased subsurface heat flow, this association could corroborate the interpretation that the off-limb features are due to subsurface geologic activity and the venting of plume material. However, endogenic heating is not the only potential cause of thermal anomalies; they can also be due to localized variations in surface properties, such as albedo or thermal inertia. In the case of nighttime thermal anomalies, like the one in question, thermal inertia becomes particularly important. Indeed, <cit.> cite thermal inertia as a potential explanation for the nighttime brightness temperatures near Pwyll. Anomalies caused by variations in thermal inertia and in endogenic heat flow should have diurnal temperature curves that behave differently over the course of a Europa day, making distinguishing between these explanations possible with temperature measurements at more than one time of day. The published Galileo PPR maps include only a single nighttime observation of the region surrounding Pwyll Crater <cit.>. We present a complementary thermal observation obtained using the Atacama Large Millimeter Array (ALMA), which captures the region of interest during the daytime. Using a thermal model, we fit both the ALMA and Galileo PPR observations and evaluate whether the anomaly is best explained by variation in the thermal inertia or whether it is truly indicative of an endogenic hot spot. § ALMA OBSERVATIONS AND DATA REDUCTIONThe observations described herein were undertaken with the 12-m array of the Atacama Large Millimeter Array (ALMA).This synthesis array is a collection of radio antennas, each 12 m in diameter, spread out on the Altiplano in the high northern Chilean Andes.Each of the pairs of antennas acts as a two-element interferometer, and the combination of all of these individual interferometers allows for the reconstruction of the full sky brightness distribution, in both dimensions.ALMA is tunable in 7 discrete frequency bands, from ∼ 90 to ∼ 950 GHz.The observation described in this paper was taken in Band 6, near 230 GHz, in the “continuum” (or “TDM”) mode, with the standard frequency tuning.For band 6, this yields four spectral windows in the frequency ranges: 224–226 GHz; 226–228 GHz; 240–242 GHz; and 242–244 GHz.In the final data analysis, we averaged over the entire frequency range in both bands, and we use 233 GHz as the effective frequency in our modeling.We observed Europa with ALMA on November 27 of 2015 from 10:00 to 10:40 UTC. At the center time of the observation, the sub-Earth longitude was 319.5 and the sub-Earth latitude was -1.54, capturing Pwyll Crater in the afternoon at ∼ 60 past local noon. ALMA had 50 available antennas in its C36-7 configuration, with a maximum useable antenna spacing of ∼ 5 km.Fully calibrated visibility data were provided by ALMA. We performed several iterations of self-calibration <cit.> on the visibility data to create a deconvolved Europa image with the 0.05" effective resolution of the interferometer. Figure <ref> shows this image with a pixel sampling of 10 times the full spatial resolution. With this resolution, and given Europa's projected diameter of 0.77" on the sky, we obtained ∼15 resolution elements across the disk.§ THERMAL MODELINGWe develop a global thermal diffusion model for Europa, similar to those developed in the past for several solar system bodies <cit.>. The model begins with calculations of solar insolation across the surface, where the solar flux absorbed at each point on the disk is given by F_abs=(1-A)μF_solar/r^2.Here, μ is the cosine of the solar incidence angle with respect to the local surface normal, F_solar is the solar constant at 1 AU, r is the solar distance in AU, and A is the hemispherical albedo at that point.In the absence of anomalous endogenic heating and in the limit of very low thermal inertia, the surface temperatures are the result of instantaneous radiative equilibrium with the absorbed flux. However, real bodies will have a finite thermal inertia, I=√(ρ c_pK),where ρ is the density, c_p is the specific heat capacity, and K is the thermal conductivity, resulting in a diurnal thermal wave with depth. Temperature as a function of time, t, and depth, z, is then given by the one-dimensional heat equation ρ c_p ∂T/∂t = ∂/∂z(K∂T/∂z). The model achieves a numerical solution to this equation by computing finite differences across depth elements. We assume a global heat flux of 20 mW/m^2 <cit.> as a lower boundary condition and an outgoing surface flux of ϵσ T^4 as an upper boundary condition, where ϵ and σ are the bolometric emissivity and Stefan-Boltzmann constant, respectively. We simulate a total of 5 diurnal skin depths, where the skin depth is given by d=√(2KP/ρ c_p) and P is the rotational period of Europa. We define the thickness of the top layer to be d/30, with the thickness of each subsequent layer increasing by a factor of 1.2.We run the model using a time step no greater than 1/500 of a Europa day and allow it to equilibrate until the maximum difference in surface temperature across 5 Europa days at any point on the disk is less than 1% (usually 10 - 15 Europa days). Further equilibration results in differences ≪1 K everywhere on the disk. The product is a temperature map of the surface of Europa, which can be output at any point in time throughout the Europa day.We use this simple thermal model combined with an approximated high-resolution albedo map of the surface to simulate the ALMA and Galileo observations and establish a baseline against which to assess thermal anomalies. Since no published high-resolution albedo map of the surface exists, we construct a high-resolution map by using discrete Voyager normal albedos as tie-points to the USGS Voyager/Galileo greyscale basemap of Europa (at 2 pixel/degree resolution) <cit.>. We take the normal albedo points of <cit.> in the Voyager green, blue, violet, and ultraviolet filters, weight them by the width of each filter and the magnitude of the solar flux at the relevant wavelengths, and then add them to get approximate wavelength-integrated normal albedos. We then find the greyscale value of each corresponding point in the USGS basemap and use the resulting linear correlation to get an approximate wavelength-integrated normal albedo for each point in the USGS map. The greyscale values and approximate normal albedos correlate linearly with an R^2 of .92 and a standard deviation of .03, which we take as the statistical error in our albedos. As the phase integral of Europa is 1.01 <cit.>, we take these normal albedos as approximate hemispherical albedos and use them in model calculations of solar flux absorbed across the surface of Europa. In modeling the observations, we take the thermal inertia and emissivity as free parameters and treat the entire disk as homogenous in these properties. We assume a snow-like constant regolith density of 500 kg/m^3 <cit.> and a c_p of 900 J/(K · kg), which is appropriate for water ice near 100 K <cit.>. After equilibration, we halt the simulation at the time specified by the sub-solar longitude of the observation. We then convert surface temperatures into flux units via Planck's Law and project the model output based on the viewing geometry of the observation, such that the central point on the disk corresponds to the sub-observer coordinates. When modeling the ALMA observation, we apply a Gaussian filter with full widths at half maximum (FWHMs) corresponding to those of the elliptical ALMA beam to smooth the output to match ALMA resolution. When modeling the Galileo PPR observation, we apply a Gaussian filter consistent with a 140-km linear resolution, within the 80–200-km resolution of the PPR observations <cit.>. Finally, we convert the smoothed images into brightness temperature, again using Planck's Law, and compare them to the actual observations. The nighttime PPR observation of Pwyll was taken in the open filter position (sensitive from 0.35 to ∼ 100 ), but these brightness temperatures generally agreed to < 1 K with those taken in the 27.5filter <cit.>. Thus, in modeling the PPR observation, we output brightness temperatures for a wavelength of 27.5<cit.>. In modeling the ALMA observation, we calculate brightness temperatures at a wavelength of 1.3 mm (233 GHz). We treat the emissivities of the surface at the ALMA and PPR wavelengths as equal. This assumption is reasonable in the case of water ice under laboratory conditions, as the optical constants are similar at both wavelengths <cit.>. However, the relevant emissivities for Europa-like conditions and compositions are not known.It should be noted that, for emissivities less than 1, the resulting brightness temperatures at these two wavelengths will be significantly different. While the ALMA wavelength is nearly in the Rayleigh-Jeans limit, the Galileo PPR was sensitive to Europa's ∼ 30blackbody peak. In both cases, Planck's law can be used to find the brightness temperature in terms of the physical temperature T_b = hν/k(ln[1+e^hν/kT-1/ϵ])^-1, where ν is the frequency, h is Planck's constant, k is Boltzmann's constant, and ϵ is the emissivity. This equation gives different results for the two wavelength regimes. For instance, an emissivity of 0.8 and a physical temperature of 125 K, produce a brightness temperature of 119 K at 27.5and a brightness temperature of 101 K at 1.3 mm. It should also be noted that the two observations were taken at very different solar distances. During the PPR observation, Jupiter was near perihelion (at 4.96 AU). However, it was near aphelion (at 5.4 AU) in 2015 when the ALMA image was taken. We account for this effect in our model.Some caveats do apply to our very simple model, however. First, we do not include the effects of surface roughness. Rough topography has the tendency to enhance surface temperatures, with the largest effects appearing at the limbs. However, Europa is thought to be relatively smooth compared to other solar system bodies <cit.>. We tested a roughness model with rms slopes up to 20, using a similar implementation of surface roughness to <cit.>, and found that the effects did not significantly affect our results. Second, our model assumes that the thermal emission imaged in the ALMA and Galileo observations originates from the topmost model layer. For the Galileo PPR observations, which were sensitive to the ∼ 30blackbody peak of Europa, this is a valid assumption. However, ALMA senses slightly deeper into the surface at a wavelength of 1.3 mm. Thus, model ALMA brightness temperatures for a given ϵ and I are slightly warmer than they would be if this effect were included. In testing a variation of our simple model, which included sensing beneath the surface with an e-folding of 1 cm, we found that much of this brightness temperature variation was captured by a slight change in the model emissivity, ϵ.Finally, our model assumes that all of the absorbed solar flux is captured in the topmost layer. This is the standard assumption in many thermal models <cit.>, and is valid for solar system bodies with low bolometric albedos. However, it is possible that sunlight is able to penetrate to significant depths beneath a high albedo surface, such as that of much of Europa <cit.>. As this effect can create a heat reservoir at depth, it can be difficult to distinguish from a change in thermal inertia <cit.>. We found this to be true in testing a version of our model that also included sunlight propagation with an e-folding of 2 cm, and this effect did not improve our fits to the data. For this analysis, we are primarily interested in relative local variation in the thermal parameters near Pwyll Crater, rather than in accurately determining the true global values. Thus, we choose to present the simplest model, with the knowledge that some of the caveats discussed here may manifest as changes in our model parameters. § FITS TO ALMA AND GALILEO PPR OBSERVATIONS In order to obtain simultaneous best-fit parameters to both the ALMA and Galileo PPR data, we run our model for each observation over a wide grid of thermal inertias and emissivities and minimize the sum of the squares of the residuals for the region covered by both datasets. By fitting both observations at once, we find that an emissivity of 0.8 and a thermal inertia of 95 J/(m^2 · K · s^1/2) provide the best result for the overlapping region. This thermal inertia is slightly higher than the value of 70 J/(m^s · K · s^1/2) reported by <cit.> for the equatorial latitudes, but is within the range of 30–140 J/(m^s · K · s^1/2) calculated by <cit.>. Our ALMA observation, best-fit model ALMA image, and the corresponding residuals are shown in Figure <ref>, where the approximate location of the Galileo thermal anomaly and potential plume source region is circled. While the homogenous thermal model is able to reproduce the large-scale structure of the ALMA image well, there are significant localized discrepancies, which are not necessarily surprising given the inhomogeneous nature of Europa's surface. This observation includes much of the dark trailing hemisphere of Europa, which is compositionally diverse <cit.>. Therefore, we do not expect the surface to be well-represented by a single thermal inertia or emissivity. However, it is interesting that the area associated with the Galileo nighttime thermal excess is actually colder in the ALMA data than the model predicts. For the Galileo PPR observation, shown in Figure <ref> alongside the best-fit model image and the resulting residuals, the same location is indeed anomalously warm, as noted by <cit.> and <cit.>. In fact, the entire Pwyll Crater region, not just the potential plume source location slightly northwest of the crater, shows up as anomalously hot at night and cold during the day. This pattern is suggestive of a variation in the local thermal inertia. If the thermal anomaly were instead due to an endogenic hot spot with an excess subsurface heat flux, one would expect the area to have elevated brightness temperatures throughout the diurnal cycle. To investigate whether the ALMA and PPR brightness temperatures are best explained by an endogenic hot spot or a thermal inertia anomaly, we model the location of the anomaly under both scenarios over the course of a diurnal cycle and attempt to fit both data points. We simulate an area 156 km in radius (corresponding to our ALMA resolution in this region) centered on 276 W and 16.8 S, which is coincident with the Galileo thermal anomaly in the potential plume source region <cit.>. We model the case of an endogenic thermal anomaly by raising the geothermal heat flux beneath the lowest layer of our simulation (5 diurnal skin depths ≈ 0.75 m at our best-fit parameters). We define the best fits by minimizing the sum of the squares of the differences between the models and the two data points. The results of these fits are shown in Figure <ref>, where the ALMA data point is taken to be 95.6 K, the ALMA brightness temperature at 276 W and 16.8 S, and the PPR data point is 95.1 K, the brightness temperature given by averaging the measured flux over an area 156 km in radius centered on the same location.Overall, we find that the ALMA and Galileo measurements are best explained by invoking a thermal inertia anomaly, and that the anomalous region cannot be solely attributed to endogenic heating. We successfully match both measurements by increasing the thermal inertia by 47% from 95 to 140 J/(m^2 · K · s^1/2) and increasing the albedo of the region by 5% from 0.56 to 0.59, which is within our albedo uncertainties. However, we are unable to successfully fit both brightness temperatures with a subsurface hot spot. Reproducing the Galileo nighttime brightness temperature requires raising the subsurface heat flux from 0.02 W/m^2 to 0.66 W/m^2, which produces a daytime brightness temperature much higher than we observed with ALMA (Figure <ref>).Similarly, a combination of subsurface heating and a thermal inertia anomaly cannot explain the two measurements. As endogenic heating increases both day and night temperatures, fitting one of the two data points in this manner always overestimates the brightness temperature of the other. We can only invoke endogenic heating in matching the data if we also significantly raise the local albedo. A local heat flux of 1.6 W/m^2 can account for both the ALMA and Galileo PPR brightness temperatures, but only when combined with a local albedo increase of 23% from 0.56 to 0.69, a 4σ deviation from our albedo model. Discrepancies of this magnitude would only result from systematic biases, rather than occur in isolation at one location. Systematic albedo biases would affect the entire albedo map and be largely absorbed by changes in the best-fit parameters. Thus, we argue that the simplest and most likely explanation for the Galileo nighttime thermal anomaly near Pwyll Crater is a moderate increase in the local thermal inertia.Spatially localized thermal inertia variations can result from a number of causes, including compositional differences and changes in the average grain size of the surface material. An elevated thermal inertia near Pwyll Crater and the anomaly in question, as originally noted by <cit.>, may result from higher average regolith particle sizes in the ejecta blanket. This possibility seems particularly plausible as the anomalous temperatures are not just constrained to the relatively small potential plume source area <cit.>, but are observed across the entirety of the Pwyll region (Figures <ref> and <ref>). <cit.> also suggest the possibility that impact-exposed water ice may allow for deeper sunlight penetration, which, as discussed in Section <ref>, can mimic the effects of increased thermal inertia.One final potential explanation warrants mentioning. We cannot rule out the possibility that the region associated with the thermal anomaly was anomalously warm due to endogenic heat at the time of the Galileo observation in 1998, but has since cooled. For instance, if a hot spot were not actively heated, but rather were caused by a singular upwelling of liquid water or warm ice at or near the surface, then detectable heat signatures need not necessarily last the 17 years between the Galileo and ALMA observations <cit.>. However, our ALMA observation was taken in 2015, prior to the 2016 potential plume detection of <cit.>. Thus, if the hot spot had dissipated by the time of our observation, then the same anomaly cannot be linked to that of <cit.>.§ CONCLUSIONSUsing ALMA, we obtained a daytime thermal measurement of the Galileo PPR nighttime thermal excess <cit.> near Pwyll Crater, which is associated with two potential plume detections <cit.>. If the thermal excess were due to an endogenic hot spot, then it could support the idea that the region northwest of Pwyll exhibits modern geologic activity. Using a global one-dimensional thermal diffusion model, we fit both the ALMA and PPR observations. However, while the location in question does appear hot relative to our model at night, it appears colder in our ALMA daytime image than the model predicts. We suggest that this pattern is indicative of a locally elevated thermal inertia. To investigate whether we can simultaneously explain both temperature measurements with endogenic heating or need to invoke a thermal inertia anomaly, we model the potential plume source location over the entire course of a Europa day under both scenarios and attempt to fit the two measured brightness temperatures. While we can explain the Galileo nighttime brightness temperature with an endogenic heat source, this situation results in a daytime brightness temperature that is too hot. However, we successfully fit both observations by raising the local thermal inertia by 47% and adjusting the albedo by an amount within our uncertainties. We therefore conclude that the nighttime Galileo thermal anomaly is most likely explained by a variation in the local surface thermal inertia, which may result from its proximity to the crater Pwyll.This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.01302.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research was supported by Grant 1313461 from the National Science Foundation. The authors thank John R. Spencer for kindly providing the Galileo PPR data used within this paper.natexlab#1#1[Abramov & Spencer(2008)]Abramov2008 Abramov, O., & Spencer, J. R. 2008, Icarus, 195, 378[Anderson et al.(1998)Anderson, Schubert, Jacobson, Lau, Moore, & Sjogren]Anderson1998 Anderson, J., Schubert, G., Jacobson, R., et al. 1998, Science, 281, 2019[Barr & Showman(2009)]BarrShowman2009 Barr, A. C., & Showman, A. P. 2009, Europa, ed. R. T. Pappalardo, W. B. McKinnon, & K. Khurana (Tucson, AZ: The University of Arizona Press), 405–430[Bierhaus et al.(2009)Bierhaus, Zahnle, & Chapman]Bierhaus2009 Bierhaus, E. B., Zahnle, K., & Chapman, C. R. 2009, Europa, ed. R. T. Pappalardo, W. B. McKinnon, & K. Khurana (Tucson, AZ: The University of Arizona Press), 161–180[Brown & Matson(1987)]Brown1987 Brown, R. H., & Matson, D. L. 1987, Icarus, 72, 84[Carlson et al.(2009)Carlson, Calvine, Dalton, Hansen, Hudson, Johnson, McCord, & Moore]Carlson2009 Carlson, R. W., Calvine, W. M., Dalton, J. B., et al. 2009, Europa, ed. R. T. Pappalardo, W. B. McKinnon, & K. Khurana (Tucson, AZ: The University of Arizona Press), 283 –328[Domingue & Verbiscer(1997)]Domingue1997 Domingue, D., & Verbiscer, A. 1997, Icarus, 128, 49[Feistel & Wagner(2006)]FeistelWagner2006 Feistel, R., & Wagner, W. 2006, Journal of Physical and Chemical Reference Data, 35, 1021[Fischer et al.(2015)Fischer, Brown, & Hand]Fischer2015 Fischer, P. D., Brown, M. E., & Hand, K. P. 2015, The Astronomical Journal, 150, 164[Grundy et al.(2007)Grundy, Buratti, Cheng, Emery, Lunsford, McKinnon, Moore, Newman, Olkin, Reuter, et al.]Grundy2007 Grundy, W. M., Buratti, B. J., Cheng, A. F., et al. 2007, Science, 318, 234[Hayne & Aharonson(2015)]Hayne2015 Hayne, P. O., & Aharonson, O. 2015, Journal of Geophysical Research: Planets, 120, 1567[Kattenhorn & Hurford(2009)]Kattenhorn2009 Kattenhorn, S. A., & Hurford, T. 2009, Europa, ed. R. T. Pappalardo, W. B. McKinnon, & K. Khurana (Tucson, AZ: The University of Arizona Press), 199–236[Kivelson et al.(2000)Kivelson, Khurana, Russel, Volwerk, Walker, & Zimmer]Kivelson2000 Kivelson, M. G., Khurana, K. K., Russel, C. T., et al. 2000, Science, 289, 1340[McCord et al.(1999)McCord, Hansen, Matson, Johnson, Crowley, Fanale, Carlson, Smythe, Martin, Hibbitts, Granahan, & Ocampo]McCord1999 McCord, T. B., Hansen, G. B., Matson, D. L., et al. 1999, Journal of Geophysical Research: Planets, 104, 11827[McEwen(1986)]McEwen1986 McEwen, A. S. 1986, Journal of Geophysical Research: Solid Earth, 91, 8077[Mitri & Showman(2005)]MitriShowman2005 Mitri, G., & Showman, A. P. 2005, Icarus, 177, 447[Moore et al.(2009)Moore, Black, Buratti, Phillips, Spencer, & Sullivan]Moore2009 Moore, J. M., Black, G., Buratti, B. J., et al. 2009, Europa, ed. R. T. Pappalardo, W. B. McKinnon, & K. Khurana (Tucson, AZ: The University of Arizona Press), 329–349[Rathbun et al.(2010)Rathbun, Rodriguez, & Spencer]Rathbun2010 Rathbun, J. A., Rodriguez, N. J., & Spencer, J. R. 2010, Icarus, 210, 763[Roth et al.(2014a)Roth, Retherford, Saur, Strobel, Feldman, McGrath, & Nimmo]Roth2014b Roth, L., Retherford, K. D., Saur, J., et al. 2014a, PNAS, 111, E5123[Roth et al.(2014b)Roth, Saur, Retherford, Strobel, Feldman, McGrath, & Nimmo]Roth2014 Roth, L., Saur, J., Retherford, K. D., et al. 2014b, Science, 343, 171[Sparks et al.(2016)Sparks, Hand, McGrath, Bergeron, Cracraft, & Deustua]Sparks2016 Sparks, W. B., Hand, K. P., McGrath, M. A., et al. 2016, The Astrophysical Journal, 829, 121[Sparks et al.(2017)Sparks, Schmidt, McGrath, Hand, Spencer, Cracraft, & Deustua]Sparks2017 Sparks, W. B., Schmidt, B. E., McGrath, M. A., et al. 2017, The Astrophysical Journal, 839, L18[Spencer(1987)]Spencer1987 Spencer, J. R. 1987, PhD thesis, University of Arizona, Tuscon, AZ[Spencer(1990)]Spencer1990 —. 1990, Icarus, 83, 27[Spencer et al.(1989)Spencer, Lebofsky, & Sykes]Spencer1989 Spencer, J. R., Lebofsky, L. A., & Sykes, M. V. 1989, Icarus, 78, 337[Spencer et al.(1999)Spencer, Tamppari, Martin, & Travis]Spencer1999 Spencer, J. R., Tamppari, L. K., Martin, T. Z., & Travis, L. D. 1999, Science, 284, 1514[Spencer et al.(2006)Spencer, Pearl, Segura, Flasar, Mamoutkine, Romani, Buratti, Hendrix, Spilker, & Lopes]Spencer2006 Spencer, J. R., Pearl, J. C., Segura, M., et al. 2006, Science, 311, 1401[Taylor et al.(1999)Taylor, Carilli, & Perley]Taylor1999 Taylor, G. B., Carilli, C. L., & Perley, R. A. 1999, in Synthesis Imaging in Radio Astronomy II, Vol. 180[Urquhart & Jakosky(1996)]Urquhart1996 Urquhart, M. L., & Jakosky, B. M. 1996, Journal of Geophysical Research: Planets, 101, 21169[USGS(2002)]USGSmap USGS. 2002, Controlled photomosaic map of Europa, Je 15M CMN: U.S. Geological Survey Investigations Series I-2757, ,[Warren & Brandt(2008)]Warren2008 Warren, S. G., & Brandt, R. E. 2008, Journal of Geophysical Research: Atmospheres, 113 | http://arxiv.org/abs/1708.07922v1 | {
"authors": [
"Samantha K. Trumbo",
"Michael E. Brown",
"Bryan J. Butler"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170826025147",
"title": "ALMA Thermal Observations of a Proposed Plume Source Region on Europa"
} |
Hamiltonian Maker-Breaker games on small graphs Miloš Stojaković [Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia. Partly supported by Ministry of Education and Science, Republic of Serbia, and Provincial Secretariat for Science, Province of Vojvodina. {milos.stojakovic, nikola.trkulja}@dmi.uns.ac.rs] Nikola Trkulja [1] ================================================================================================================================================================================================================================================================================================================================================We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph K_n, where Maker's goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for n = 8 and n = 9. Then we use an inductive argument to show that, independent of who starts, Maker can win the game if and only if n ≥ 8. This, in particular, resolves in the affirmative the long-standing conjecture of Papaioannou from <cit.>.We also study two standard positional games related to Hamiltonicity game. For Hamiltonian Path game, we show that Maker can claim a Hamiltonian path if and only if n ≥ 5, independent of who starts. Next, we look at Fixed Hamiltonian Path game, where the goal of Maker is to claim a Hamiltonian path between two predetermined vertices. We prove that if Maker starts the game, he wins if and only if n ≥ 7, and if Breaker starts, Maker wins if and only if n ≥ 8. Using this result, we are able to improve the previously best upper bound on the smallest number of edges a graph on n vertices can have, knowing that Maker can win the Maker-Breaker Hamiltonicity game played on its edges. To resolve the outcomes of the mentioned games on small (finite) boards, we devise algorithms for efficiently searching game trees and then obtain our results with the help of a computer. § INTRODUCTION A positional game is a hypergraph (X,ℱ), where X is a finite set representing the board of the game, and ℱ⊆ 2^X is a family of sets that we call winning sets. In Maker-Breaker positional games, the players are called Maker and Breaker. In the course of the game, Maker and Breaker alternately claim unclaimed elements of the board X, one element at a time, until all of the elements are claimed. Maker wins the game if he claims all elements of a winning set, while Breaker wins if he claims an element in every winning set. Each game can be observed in two variants, depending on which player is to play first. One of the main questions in the theory of positional games is the existance of a winning strategy for one of the players, when both are playing optimally. The player that has a strategy to win the game is referred to as the winner of the game.The positional games we intend to study are played on graphs, and in particular, on the edge set of a complete graph K_n. Our prime interest lies with Hamiltonicity game _n, where the winning sets are the edge sets of all Hamiltonian cycles in K_n. The game was first introduced by Chvátal and Erdős in <cit.>, and since then it has been one of the most studied positional games on graphs, see <cit.>. It was shown in <cit.> that Maker has a winning strategy for all sufficiently large n. Papaioannou <cit.> later proved that Maker wins the game for all n ≥ 600, and at the same time conjectured that the smallest n for which Maker can win is 8. Hefetz and Stich <cit.> further improved the upper bound by showing that Maker wins for all n ≥ 29. We note that these statements hold under the assumption that Maker is the first to play.In the present paper, we determine the outcome of Hamiltonicity game for every value of n, and for each of the players starting the game. In particular, this resolves the mentioned long-standing conjecture of Papaioannou in the affirmative. As the trivial cases n ≤ 3 can be handled directly, from now on we assume n ≥ 4. In the Maker-Breaker _n game on E(K_n), Maker, as first or second player, wins if and only if n ≥ 8. Next, we look at two games where Maker's goal is to claim a Hamiltonian path. In Hamiltonian Path game ℋ𝒫_n, first introduced in <cit.>, the winning sets are the edge sets of all Hamiltonian paths in K_n. We are able to show the following. In the Maker-Breaker Hamiltonian Path game ℋ𝒫_n on E(K_n), Maker, as first or second player, wins if and only if n ≥ 5.This theorem is a strengthening of Papaioannou's result from <cit.> where he proved that Maker, as first player, can win ℋ𝒫_n if and only if n ≥ 5. In Fixed Hamiltonian Path game ℱℋ𝒫_n, the goal of Maker is to claim a Hamiltonian path between two fixed (predetermined) vertices, u,v∈ V(K_n). Even though in the literature this game does not draw as much interest as _n and ℋ𝒫_n, it has appeared as an auxiliary game when studying some other games on graphs, see e.g. <cit.>. In the Maker-Breaker Fixed Hamiltonian Path game ℱℋ𝒫_n on E(K_n), Maker, as first player, wins if and only if n ≥ 7, and as second player he wins if and only if n ≥ 8.Note that in the game ℱℋ𝒫_n the edge between the fixed vertices u and v actually does not participate in any winning set, so right away we obtain the same result for the game played on E(K_n) ∖{(u,v)}.Here we also want to mention two other standard positional games, Connectivity and Perfect Matching, where Maker's goal is to claim, respectively, a spanning tree and a perfect matching. For both of thesegames it is straightforward to determine the outcome for every size of the board. Maker wins the Connectivity game as first player for every n, and as second player if and only if n ≥ 4. In Perfect Matching game, where n is even, Maker can win as first or second when n is at least 6. On top of that, he wins also for n=2 when he plays first.Let us note that even though answering the question of who wins a game on a small board (with n fixed) is a finite problem, it still may have a greater scientific impact for several reasons. First of all, the approaches we use to resolve standard positional games when n is large often do not apply for small n, and in that case in order to determine the outcome we need to develop new methods. As we saw, resolving the “small cases” is straightforward for some games, but not for all.Also, standard positional games, like Hamiltonicity, Hamiltonian Path, Connectivity, etc. are often used as auxiliary games when studying other positional games on graphs. Sometimes these auxiliary games have boards of fixed size, and knowing their outcome is essential for completing the analysis. An example of that can be found in <cit.>, where one of the problems tackled was to estimate the smallest number of edges m̂(n) a graph on n vertices can have, knowing that Maker as first player can win the Maker-Breaker Hamiltonicity game played on its edges. It was proved in <cit.> that 2.5n ≤m̂(n) ≤ 21n, for all n ≥ 1600. We show how to apply our results about Hamiltonicity game in Theorem <ref> and Fixed Hamiltonian Path game in Theorem <ref> to improve this upper bound, eventually obtaining the following. For n ≥ 336, we have m̂(n) ≤ 4n. Positional games are combinatorial games (sequential two player games with perfect information and no randomness involved), and it is well known (see, e.g., <cit.>) that we can find out which of the players has a strategy to win by simply traversing the whole game tree of the game. But this fact alone is of limited practical use, knowing that already for games on relatively small boards the game trees are way too big (they are exponentially large in the size of the game board) to be completely traversed by a computer. In particular, if the board of the game is E(K_n), its size is n2, so there are n2! different game plays.Often there is no need to search through the whole game tree, as some moves are “analogue” to the others, we may arrive to the same game position more than once, and on top of that some game positions are “similar” to the others. In Section <ref> we devise a sophisticated set of algorithms that formalizes and exploits these “similarities” as part of the optimization of the brute force search algorithm. This enables us to write a computer program that efficiently calculates the outcome of all three mentioned games for enough initial values of n to inductively determine the outcome of the game for every n.When implementing the algorithms we aimed at doing it in a generic way to make our code easily adaptable for other positional games on graphs. Even though some algorithms are tailored to fit the particularities of the games we analyzed, most of them are generally applicable fordetermining the outcome of positional games on graphs with the help of a computer.Having that in mind, our software can be seen as a general framework for computer based attacks on positional games. §.§ Preliminaries and organization Our graph-theoretic notation is standard and follows the one from <cit.>. In particular, we use the following. For a graph G, let V(G) denote its set of vertices and E(G) its set of edges, where v(G) = |V(G)| and e(G) = |E(G)|. For a set A ⊆ V(G), let G[A] denote the subgraph of G induced on the vertex set A and let E_G(A) denote the set of edges of G with both endpoints in A. Results presented in this paper rely on the following statement, see e.g. <cit.>. If Maker (or Breaker) has a winning strategy in some positional game as second player, then he also has a winning strategy if he starts the game. If a Maker-Breaker game played on edges of a graph is in progress, Maker's (respectively, Breaker's) vertex degree of a vertex v, denoted by d_M(v) (respectively, d_B(v)), is the number of edges incident to v that are claimed by Maker (respectively, Breaker). Maker's edge degree of an edge e = (u,v), denoted by ed_M(e), is defined to be ed_M(e) := d_M(u) + d_M(v), and Breaker's edge degree is ed_B(e) := d_B(u) + d_B(v).The rest of the paper is organized as follows. In Section <ref> we prove Theorem <ref>, in Section <ref> we prove Theorem <ref> and in Section <ref> we prove Theorem <ref>. Section <ref> contains the description of the algorithms and optimization methods used in our computer programs, as well as the results we obtain with their help. In Section <ref> we show how our results can be applied in solving other problems in theory of positional games, by proving Theorem <ref>. Finally, in Section <ref> we give some open problems and ideas for future work.§ HAMILTONICITY GAMEThe following lemma lies in the foundation of our proof of Theorem <ref>. Independent of who starts the game _n, Breaker wins for n ∈{ 4, 5, 6, 7 }, and Maker wins for n = 8 and n = 9. We prove this result with the help of a computer. Description of the approach used is postponed to Section <ref>.The previous result gives us a firm induction base for proving that Maker can win _n for all n ≥ 8. We note that Papaioannou already proved the induction step of size 2 for Maker as first player, which can be paired up with our Lemma <ref> to resolve the case when Maker is first. Inspired by his work, in the following lemma we prove the same induction step for Maker playing as second player, which eventually completes the proof of Theorem <ref>. If Maker, as second player, wins _n, he can also win _n+2 as second player.Suppose that Maker, as second player, has a strategy to win _n. We will present Maker's winning strategy for _n+2. At the beginning of the game, Breaker will claim some edge (u,v) and Maker's response will be to claim an edge m=(v,w) for some w. Let V_1:= V(K_n+2) ∖{v,w}. We can now partition the set of all unclaimed edges into two sets, the first set being E_1 = E(V_1) and the second one E_2 = E(K_n+2) ∖ (E_1 ∪{ m, (u,v) }). These edge sets are shown in the Figure <ref>. From this point on, Maker will try to respond to every move of Breaker by claiming the edge in the same edge set as Breaker e.g. whenever Breaker claims an edge in E_i Maker will also try to claim an edge from E_i, i= 1, 2. If this is not possible, he will claim an arbitrary edge in the other set. It is generally true that such “extra” edges can be “forgotten”, i.e. if Maker can accomplish something in the other set without the extra edge, he will still manage to do it with the extra edge, see e.g. <cit.>. Maker will play in E_1 using his winning strategy for _n, meaning that by the time all edges in E_1 are claimed he will fully occupy some n-cycle H_n in E_1. When playing in E_2 Maker's goal is to make sure that by the time all the edges are claimed (in both E_1 and E_2) he claimed a Hamiltonian cycle. This Hamiltonian cycle will be an extension of H_n to a cycle on two more vertices, now also containing the edge m. More precisely, in order to extend the cycle H_n=(x_1, x_2, …, x_n, x_1), Maker wants to claim edges (x_i, y_1), (x_i+1, y_2) ∈ E_2, such that the edge (x_i,x_i+1) ∈ H_n and(y_1,y_2) = m. That way, Maker would claim a Hamiltonian cycle H_n+2=(x_1, …, x_i, y_1, y_2, x_i+1, …, x_n, x_1), as shown in Figure <ref>. During the game, if for a vertex x ∈ V_1 both edges (x,v) and (x,w) are claimed by Maker (Breaker), we say the vertex x is isolated by Maker (Breaker). If one of the edges (x,v) and (x,w) is claimed by Maker (Breaker) and the other one is unclaimed, we say the the vertex x is half isolated by Maker (Breaker). Finally, if both edges (x,v) and (x,w) are unclaimed, we say that the vertex x is free. When all the edges are claimed, we observe the following two situations. Situation 1: Breaker isolated at most one vertex, and Maker claimed an edge in E_2 incident to each of vertices v and w.Situation 2: Breaker isolated exactly two vertices, Maker has one isolated vertex a, and Maker claimed an edge from E_2 ∖{(a,v), (a,w)} incident to each of vertices v and w.First, let us prove that in each of the two above mentioned situations Maker created a Hamiltonian cycle and thus won the game. We can apply simple labeling procedure, assigning a label set to each vertex x ∈ H_n. This set contains 0 if and only if edge (x,v) is claimed by Maker and 1 if and only if edge (x,w) is claimed by Maker.In Situation 1, since Maker claimed an edge in E_2 incident to each of vertices v and w, at least one vertex is labeled with 0, and at least one vertex is labeled with 1. Also, there is at most one vertex with empty label set, so in every Hamiltonian cycle H_n there must exist two adjacent vertices x_i and x_i+1 containing labels 0 and 1, respectively. Maker will then use these vertices x_i and x_i+1 to extend the cycle H_n.In Situation 2 there is a vertex a with label set {0,1}. If this vertex is adjacent to some vertex x ∈ H_n whose label set is not empty, Maker can extend the cycle H_n using a and x. Otherwise, if vertex a is adjacent to vertices whose label set is empty (two vertices isolated by Breaker), then Maker will be able to extend H_n for the same reasons as in Situation 1, since he claimed an edge from E_2 ∖{(a,v), (a,w)} incident to each of vertices v and w. Next, we are going to give a strategy for Maker that will enable him to always end up in one of these two situations. Maker's strategy on E_2. In each move, Maker will apply the first applicable rule in the following list: * If there is only one free vertex x ∈ V_1 left, and Maker has not claimed any edges incident to a vertexy ∈{v, w}, he claims the edge (x,y). * If there exists a vertex half isolated by Breaker, Maker picks an arbitrarily vertex x half isolated by Breaker and claims the other edge incident to x. * If there exists a free vertex, Maker picks an arbitrarily free vertex and claims one of its free edges. Of the two edges, he prefers the edge incident with the vertex in {v, w} incident with less edges claimed by Maker (if such vertex exists). * Maker claims a free edge incident to a vertex half isolated by Maker. Of all such edges, he prefers edges incident with the vertex in {v, w} incident with less edges claimed by Maker (if such vertex exists), breaking ties arbitrarily. As any free edge is incident either with a vertex half isolated by Breaker, or a free vertex, or a vertex half isolated by Maker, this is a valid strategy.Initially, Breaker has claimed one edge, the edge (u,v). For his first move in E_2 he essentially has only three different choices, shown in Figure <ref> (along with the move of Maker that will follow in each of the cases). Now, let us analyze each of the three cases from Figure <ref> individually. Case (a): In this case, Breaker first has an option to isolate a vertex x_i_1. If he indeed does so, Maker will apply Rule 3 and claim an edge incident to a vertex v, otherwise Maker will apply Rule 2 and claim an edge incident to v. When Breaker isolates a vertex, Maker will apply Rule 2 in the remainder of the game to prevent Breaker from isolating more vertices. The proposed strategy thus inevitably puts Maker in Situation 1. Case (b): Again, Breaker can start by isolating a vertex x_i_1 which inevitably puts Maker in Situation 1 as this is analogous to Case (a). Otherwise, Maker will wait (by applying Rule 2) for Breaker to isolate some vertex, until there is only one free vertex left. If Breaker isolates a vertex during this period of the game Maker will apply Rule 3 claiming an edge incident to v and he will spend the remainder of the game applying Rule 2 in order to prevent Breaker from isolating more vertices. It can happen that Breaker does not isolate any vertex by the time there is only one free vertex left, in which case Maker will end the game by applying the Rule 1 as shown in Figure <ref>. Therefore, Maker will inevitably find himself in Situation 1. Case (c): In this case we will need to further analyze each of the three possible Breaker's moves. Breaker can either claim an edge incident to the only vertex half isolated by Maker, or for some free vertex x ∈ V_1 he can claim (x,w) or (x,v). These three possibilities are shown in Figure <ref> (along with the move of Maker that will follow in each of the cases). In cases (i) and (ii), which are essentially the same, Maker claimed an edge incident to each of vertices v and w while Breaker already isolated a vertex u. Maker will continue the game with only goal to prevent Breaker from isolating more vertices, by applying Rule 2 whenever it is possible, and hence the game will eventually end up in Situation 1. In the third case we will first analyse the game until there is only one free vertex left. Maker is again focused on preventing the Breaker from isolating a vertex by applying Rule 2 whenever possible. If during this period of the game Maker claims some edge incident to v, Rule 1 will not be applied, and the game will finish in Situation 1. If it happens that Rule 1 needs to be applied, Maker will claim an edge (v,x_i_n-1), where x_i_n-1 is the last free vertex. Depending if Breaker now isolates a vertex x_i_n-2 or not, the game can end up in different situations. If Breaker does not isolate x_i_n-2 the game ends in Situation 1. Otherwise, after vertex x_i_n-2 is isolated by Breaker, Maker isolates x_i_1 and the game ends in Situation 2. Hence, Maker can always create one of the two winning situations. Applying mathematical induction, with Lemma <ref> as the base case and Lemma <ref> as the induction step we get that Maker, as second player, wins _n for all n ≥ 8. Combining this result with the result from Theorem <ref> we complete the proof of Theorem <ref>. § HAMILTONIAN PATH GAMEIn order to complete the proof of Theorem <ref> we used our computer program to obtain following result. Independent of who starts the game ℋ𝒫_n, Breaker wins for n = 4, and Maker wins for n ∈{5,6,7}. Description of the proof and the approach used is postponed to Section <ref>.With the help of the previous lemma and Theorem <ref>, the proof of Theorem <ref> is straightforward. It is easy to see that Maker's win in Hamiltonicity game implies Maker's win in Hamiltonian path game since each Hamiltonian cycle contains a Hamiltonian path. This means that Theorem <ref> also tells us that Maker can win ℋ𝒫_n for all n ≥ 8 as second player. When we combine this with the result from Lemma <ref> proof of Theorem <ref> is complete. § FIXED HAMILTONIAN PATH GAME The following lemma will cover several finite cases in the proof of Theorem <ref>. When Maker starts the game ℱℋ𝒫_n, Breaker wins for n∈{4, 5, 6}, and Maker wins for n ∈{7,8,9}.When Breaker is the one who starts, he wins for n∈{4, 5, 6, 7}, while Maker wins for n ∈{8,9}. We prove this result with the help of a computer. Description of the approach used is postponed to the following section. It remains to determine the outcome of the game ℱℋ𝒫_n for larger values of n. We will not perform an induction step, but rather rely on our knowledge of the outcome of _n. Combining the following result with Theorem <ref>, along with Lemma <ref> that covers the few initial cases, we readily prove Theorem <ref>. If Maker, as second player, wins _n, he can also win ℱℋ𝒫_n+2 as second player. Suppose that Maker, as second player, has a strategy to win _n. We will present Maker's winning strategy for ℱℋ𝒫_n+2, with two fixed vertices u and v. As we already noted, the edge e = (u,v) is not interesting to either of the players since it is not contained in any of the winning sets. Let V_1:= V(K_n+2) ∖{u,v}. We can partition the set of all unclaimed edges into two sets, the first set being E_1 = E(V_1) and the second one E_2 = E(K_n+2) ∖ (E_1 ∪{ e }).Now the goal of Maker is exactly the same as in the proof of Lemma <ref> – he wants to build an n-cycle on E_1, and he wants to play on E_2 to connect vertices u and v to two neighboring vertices of that n-cycle. As the structure of E_1 and E_2 here is exactly the same, with exception of E_2 having one more unclaimed edge (which is only beneficial for Maker), he can follow the same strategy as in the proof of Lemma <ref> and win the game. § ALGORITHMIC ANALYSIS In this section we present the techniques and approaches used in our computer program to solve games _n, ℋ𝒫_n and ℱℋ𝒫_n when game boards are relatively small, as well as some of the major challenges we encountered during the process. Our program is available at <http://people.dmi.uns.ac.rs/ nikola.trkulja/ham.html>, where one can find all the information on how to run and use it, along with the source code. §.§ Traversing the game tree As we already noted, in order to determine the winner of a particular game, it is enough to completely traverse the game tree, which essentially comes down to simulation of all possible plays. This means that in each round we need to explore all valid moves a player can play. To achieve this we used the following algorithm which represents an adaptation of the standard and well-known game theoretic algorithm – Minimax <cit.>. Main procedure of our algorithm is procedure Play, in charge of recursively exploring the whole game subtree starting from the given tree node (containing the current game board G, and the identity of the player P whose turn it is). The value returned by procedure Play corresponds to the player winning the game on board G, when player P moves first. As the name suggests, procedures DoMove(e,G,P) and RevertMove(e,G,P) are used to mark and unmark, respectively, the edge e as claimed by player P on board G.The winner detection is an essential part of the algorithm, done by procedure PlayerWins. Every time the winner is not immediately detected by procedure PlayerWins, we continue recursively exploring the game tree. The algorithm stops processing the remaining possible moves for current player P as soon there is a move which guaranties the win. On the other hand, if it turns out that there is no such move, we know that the other player wins the game. §.§ Isomorphism pruning Basic version of the algorithm we just described can be improved so that the same game subtrees are not explored multiple times. This is where transposition table <cit.> jumps into picture. Straightforward application of transposition table would be to store the winner for each game tree node, and then use these values to skip processing identical nodes. But the exponential size of the game tree renders such solution impossible because it would require huge amounts of computer memory. Much more efficient way of handling this situation is to recognize the isomorphic nodes of the game tree. For this task we implemented Brendan McKay's canonical labeling algorithm <cit.>. Canonical labeling of a graph is defined in the following way. Canonical labeling is a function C mapping graphs to graphs, such that every graph G_1 is isomorphic to C(G_1), and for every graph G_2 isomorphic with G_1 we have C(G_2)=C(G_1). In other words, canonical labeling gives a unique representative for each class of isomorphic graphs, and with McKay's algorithm we can define and compute this graph in an efficient manner. That enables us to prune the game tree significantly, as we can store the winner only for canonical labelings. §.§.§ Edge colored graphs When a Maker-Breaker game is played on the edge set of a graph, the situation on the game board can be seen as a 3-coloring of the edges, as each edge is either unclaimed (black), claimed by Maker (red), or claimed by Breaker (blue). One of the issues in application of McKay's canonical labeling algorithm emerges right here because the original version only handles non-colored and vertex colored graphs out of the box. In <cit.> the author of the algorithm described how edge colored graph can be transformed into equivalent vertex colored graph which can then be fed as an input to canonical labeling algorithm (an interested reader can find more details in <cit.>). Unfortunately this transformation produces a graph with more vertices than the original graph (even when we have just two colors, the number of vertices doubles). Because of this, we did the implementation from scratch, having <cit.> in mind, where authors adapted the canonical labelling algorithm to work with edge colored graphs by distinguishing the vertices according to the colors of incident edges. §.§.§ Non-transitive winning sets Another issue with detection of isomorphic game tree nodes emerges when dealing with Fixed Hamiltonian path game since this game has an additional property compared to the other two games we worked on – there are two predefined vertices that require special attention. Therefore, we have to incorporate an additional condition, only taking into account the isomorphisms which map those two vertices to themselves.We further slice down each isomorphism class into subclasses defined in the following way. We say that graph G_1 with predefined vertices u_1, v_1 ∈ V(G_1) is in the same isomorphism subclass as graph G_2 with predefined vertices u_2, v_2 ∈ V(G_2) if and only if C(G_1) = C(G_2), and isomorphism functions f_1 and f_2 transforming G_1 and G_2 into C(G_1), respectively, are such that { f_1(u_1), f_1(v_1) } = { f_2(u_2), f_2(v_2) }. Applying this definition to our problem, we chose sets { u_1, v_1 } and { u_2, v_2 } to be sets of endpoints of winning sets in ℱℋ𝒫_n, meaning that Maker wins the game on G_1 if and only if he wins on G_2. §.§ Move ordering Our algorithm is not processing the whole game tree but rather skipping some of the subtrees whenever it is sure about the outcome of that particular subtree. For example, when Maker is the one to claim an edge, we will test all the possible moves that he can play, one after another, until we find the one that guarantees his win (provided that such a move exists). Of course, it would be better that we probe a winning move as early as possible. Having this in mind we tested a number of heuristics for defining the order of the iteration over free edges in line <ref> of Algorithm <ref>. We relied on sorting the free edges according to a predefined criterion. Particularly, free edges are sorted in non-decreasing order according to Maker's edge degree ed_M for Maker's moves, and in non-increasing order according to Breaker's edge degree ed_B for Breaker's moves. The idea behind this is that, informally speaking, Maker may want to prefer playing where his “weakest link” is, while Breaker may prefer choosing to enforce his “strongest disconnecting point” first. It turned out this optimization technique, implemented as described, is one of the most important ones, and it had a tremendous impact on the performance of our algorithm making the number of traversed game tree nodes significantly smaller. §.§ Winner detection Procedure PlayerWins(G,P) is essential for checking if the board of the game G is such that the player P won the game[If P is Maker, he won if he claimed a whole winning set, and if P is Breaker, he won if he claimed an edge in each of the winning sets.]. In order to efficiently implement this task we always did pre-computation of the set ℱ of winning sets that are free of Breaker's edges. E.g. in game _n we pre-generate the set ℱ of all Hamiltonian cycles in K_n, and then we constantly update ℱ so that in each moment it contains only the winning sets without any edges claimed by Breaker. §.§ Optimization of memory usage Several optimization techniques applied in our software are on the technical i.e. implementation level, and majority of them are computer memory related. This is natural since we rely on transposition table in order to keep track of calculated values for visited game tree nodes, or more precisely their isomorphism classes, and even for relatively small graphs the number of these classes is extremely large. Utilizing sets in many components of our program we are able to achieve optimal ratio of time and memory efficiency. Sets are used to implement graphs, canonical labelings, winning sets and most importantly transposition tables. Of course, we needed efficient implementations of sets and for that job we used bitsets and high-performance 3rd party set implementations (in the form of Trove library <cit.>). After all this effort, our transposition tables were still not able to cope with the required amount of entries because of the limited computer memory. To solve this problem, we applied various heuristics to clean the tables i.e. to remove the “less important” entries and free up the memory for the “more important” ones. One efficient technique to do this was to focus mainly on entries coming from lower depths of the game tree, say depths less than or equal to a fixed integer d, so whenever we detect that the table is full all entries coming from depths larger than d are removed from the table. §.§ Implementation, testing and results Implementation was done using Java programming language, relying only on standard Java libraries with just one additional component, the already mentioned 3rd party high performance collections library – Trove <cit.>. There are two main parts of our computer program, Minimax algorithm and Brendan McKay's canonical labeling algorithm. We chose to implement canonical labeling algorithm on our own in order to be more flexible and tailor the algorithm fully to our needs. We note that there are implementations of related algorithms available online (e.g. <cit.>). While implementing the algorithm we followed the idea of doing it in a generic way to make our code easily adaptable for solving other positional games on graphs as well. Examples of this can be seen when browsing through our code since for all three games we exploited this approach, using the same code base with just minor modifications for each particular game. For testing purposes we used various revisions of Java version 6 and 7, and on top of that we ran the program on a number of different versions of three different operating systems, Windows, Mac OSX and Linux, all in the effort to minimize the possibility of error coming from any of these layers. We further conducted thorough testing of both of the main parts of our implementation using unit tests, in order to eliminate the possibility of errors coming from smaller components of our program. In the end, we put a lot of additional effort to verify that our implementation of Brendan McKay's algorithm is correct by comparing output of our implementation with the one provided by the author <cit.> for many different types of graphs. The machine used for testing was Fujitsu Celsius M730 with Intel Xeon E5-1620 processor running at 3.7 GHz, 32GB of RAM and with Ubuntu operating system. We were able to prove following statements using this machine. 2-7 1c| _4 _5 _6 _7 _8 _9 Running timeMaker as first player 0s 0s 0.1s 1s 2s 30s Winner Breaker Breaker Breaker Breaker Maker Maker 2-7 1c| _4 _5 _6 _7 _8 _9 Running timeMaker as second player 0s 0s 0.1s 1s 3s 68s Winner Breaker Breaker Breaker Breaker Maker Maker 2-5 1c| ℋ𝒫_4 ℋ𝒫_5 ℋ𝒫_6 ℋ𝒫_7 Running timeMaker as first player 0s 0s 0s 0.3s Winner Breaker Maker Maker Maker 2-5 1c| ℋ𝒫_4 ℋ𝒫_5 ℋ𝒫_6 ℋ𝒫_7 Running timeMaker as second player 0s 0s 0.1s 0.3s Winner Breaker Maker Maker Maker 2-7 1c| ℱℋ𝒫_4 ℱℋ𝒫_5 ℱℋ𝒫_6 ℱℋ𝒫_7 ℱℋ𝒫_8 ℱℋ𝒫_9 Running timeMaker as first player 0s 0s 0.5s 3s 110s 3990s Winner Breaker Breaker Breaker Maker Maker Maker 2-7 1c| ℱℋ𝒫_4 ℱℋ𝒫_5 ℱℋ𝒫_6 ℱℋ𝒫_7 ℱℋ𝒫_8 ℱℋ𝒫_9 Running timeMaker as second player 0s 0s 0.1s 2s 230s 16286s Winner Breaker Breaker Breaker Breaker Maker Maker § AN APPLICATIONGenerally speaking, the most direct way to show that a player wins a positional game is to exhibit an explicit strategy for the player, proving that it is a winning one. There are numerous cases (see <cit.> for examples) that such a strategy relies on strategies for one or more auxiliary positional games, where the auxiliary games are often confined to some part of the board and/or some stage of the game play.Our results are applicable when such auxiliary games have Hamiltonian cycles or Hamiltonian paths as winning sets, and they are played on boards of fixed size. An example of that can be found in <cit.>, when estimating the smallest number of edges m̂(n) a graph on n vertices can have, knowing that Maker as the first player can win the Maker-Breaker Hamiltonicity game played on its edges. Using our results from Theorem <ref> and Theorem <ref>, we are able to improve the upper bound on m̂(n). In <cit.> it was showed that m̂(n) ≤ 21n for n ≥ 1600, by explicit construction of a graph G'_n with n vertices and 21n edges, along with a proof that Maker can win the game playing on E(G'_n). The graph G'_n was constructed by dividing n vertices into sets V_0, V_1, … , V_m-1, V', with m ≥ 40, such that d ≤ |V_i| ≤ d+1, where d = 38, and V' = { u_0, …, u_m-1} containsexactly m vertices. As for the edges of G_n, for every 0 ≤ i <m, every two vertices in V_i are connected with an edge (i.e. the graph induced on V_i is a clique), and on top of that, for every 0 ≤ i < m, there is an edge between u_i and every vertex of both V_i and V_i+1 (indices are observed modulo m). All of this totals to no more than (d/2 + 2)n = 21n edges.To prove that Maker can win Hamiltonicity game on E(G'_n), the authors of <cit.> relied on the fact that Maker wins _n on E(K_n) for all n ≥ 38, as n=38 was the smallest known such n at the time. This was later improved by Hefetz and Stich <cit.> to n=29, and now our Theorem <ref> gives the optimal n=8. Having this in mind we can repeat the exact same proof with an adjusted construction of G'_n, reducing the size of all V_i by setting d=8, and getting an immediate improvement of the upper bound to m̂(n) ≤ 6n. But, it turns out that with the help of Fixed Hamiltonian Path game we are able to alter the construction and get an even better upper bound. We will first describe our construction of the graph G_n whose edge set will be the board of the game, and then we will prove that on this board Maker indeed can win Hamiltonicity game.Let d = 7, and let m and 0 ≤ r < d be integers such that n = md + r. The vertices of G_n are partitioned into m sets V_0, V_1,… , V_m-1, such that |V_i|=d+1, 0 ≤ i < r, and |V_i|=d, r ≤ i < m. Next, we arbitrarily pick vertices a_i ∈ V_i, 0 ≤ i < m, and define W_i:= { a_i-1}∪ V_i, where indices are observed modulo m. The set of edges of G_n is defined withE(G_n) = ⋃_i=0^m-1{ (v_1, v_2)| v_1, v_2∈ W_i,v_1 ≠ v_2, { v_1, v_2}≠{ a_i-1, a_i}}.In other words, the graph G_n consists of a “cycle” of cliques, where each two consecutive cliques overlap on a vertex, and each clique is missing exactly one edge (between the two vertices it shares with the neighboring cliques). The number of edges in G_n isE(G_n)= r92 + (m-r)82 - m =m(82 -1 ) + 8r ≤n/7(4· 7 - 1 ) + 48.If n ≥ 336, this is upper bounded by 4n.To win Hamiltonicity game on E(G_n), Maker simply plays m Fixed Hamiltonian Path games in parallel, one on each of graphs G_n[W_i], 0 ≤ i < m. More precisely, in each of those games Maker, as the second player, plays the game ℱℋ𝒫_|W_i|, with fixed vertices a_i-1 and a_i. As we previously observed, the edge (a_i-1, a_i) does not participate in any winning sets, so Theorem <ref> guarantees Maker's win in all those games. Hence, for every 0 ≤ i < m, Maker can claim a Hamiltonian path on G_n[W_i] with endpoints a_i-1 and a_i. These paths together form a cycle that covers all vertices of G_n, so Maker wins the Hamiltonicity game on E(G_n). Even though our proof works for n ≥ 336, we can similarly obtain a non-trivial upper bound for any n ≥ 14. Indeed, we can construct a graph consisting of a “cycle” of (two or more) cliques in the exact same way as in the previous proof, just making sure that each clique has at least 8 vertices. Then we can borrow Maker's strategy from the previous proof, as well as the argument that he can claim a Hamiltonian cycle playing on that graph.§ FUTURE WORK We have resolved three unbiased games played on the edges of a complete graph, Hamiltonicity game, Hamiltonian Path game and Fixed Hamiltonian Path game, for every n and for each of the players starting the game. One related game that remains out of reach is the so-called Hamiltonian-Connectivity game, where Maker's goal is to claim all edges of some spanning Hamiltonian-connected graph (a graph in which every two vertices are connected by a Hamiltonian path). Our approach was not efficent enough to handle Hamiltonian-Connectivity game since the nature of the winning sets is too complex making our pruning and winner detection procedures much less effective. This game may be utilized in a similar way as Fixed Hamiltonian Path game, but with more freedom – what were two fixed (predetermined) vertices in Fixed Hamiltonian Path game, become any two vertices in Hamiltonian-Connectivity game. For that reason, we think that knowing the outcome of the game for every n would be valuable.One consequence of finding the winner of a game using a computer is that typically we do not have a compact graph-theoretic description of a winning strategy for the winner. This is true for all games on small boards that we looked at – we know who wins, but this knowledge is not accompanied by a coherent strategy (other than a huge list of winning moves, one for each game position). Because our computer program is implemented in a generic way it could be fairly easily adapted for attacking other positional games when played on small (enough) boards. We are hoping that other positional games can be solved using the tools developed in this paper. abbrv-pages | http://arxiv.org/abs/1708.07579v3 | {
"authors": [
"Miloš Stojaković",
"Nikola Trkulja"
],
"categories": [
"math.CO",
"cs.AI",
"cs.DM"
],
"primary_category": "math.CO",
"published": "20170825000920",
"title": "Hamiltonian Maker-Breaker games on small graphs"
} |
@addto@macromaketitle 3D Object Reconstruction from a Single Depth View with Adversarial Learning Bo YangUniversity of Oxford [email protected] WenUniversity of Warwick [email protected] WangHeriot-Watt University [email protected] Ronald ClarkImperial College London [email protected] Andrew MarkhamUniversity of Oxford [email protected] Niki TrigoniUniversity of Oxford [email protected] 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================== In this paper, we propose a novel 3D-RecGAN approach, which reconstructs the complete 3D structure of a given object from a single arbitrary depth view using generative adversarial networks. Unlike the existing work which typically requires multiple views of the same object or class labels to recover the full 3D geometry, the proposed 3D-RecGAN only takes the voxel grid representation of a depth view of the object as input, and is able to generate the complete 3D occupancy grid by filling in the occluded/missing regions. The key idea is to combine the generative capabilities of autoencoders and the conditional Generative Adversarial Networks (GAN) framework, to infer accurate and fine-grained 3D structures of objects in high-dimensional voxel space. Extensive experiments on large synthetic datasets show that the proposed 3D-RecGAN significantly outperforms the state of the art in single view 3D object reconstruction, and is able to reconstruct unseen types of objects. Our code and data are available at:<https://github.com/Yang7879/3D-RecGAN>. § INTRODUCTIONThe ability to reconstruct the complete and accurate 3D geometry of an object is essential for abroad spectrum of scenarios, from AR/VR applications <cit.> and semantic understanding, to robot grasping <cit.> and obstacle avoidance. One class of popular approaches is to use the off-the-shelf low-cost depth sensing devices such as Kinect and RealSense cameras to recover the 3D model of an object from captured depth images. Most of those approaches typically sample multiple depth images from different views of the object to create the complete 3D structure <cit.> <cit.> <cit.>. However, in practice it is not always feasible to scan all surfaces of the object, which leads to incomplete models with occluded regions and large holes. In addition, acquiring and processing multiple depth views require significant computational power, which is not ideal in many applications that require real-time response.In this paper, we aim to tackle the problem of inferring the complete 3D model of an object using a single depth view. This is a very challenging task, since the partial observation of the object (i.e. a depth image from one viewing angle) can be theoretically associated with infinite number of possible 3D models. Traditional reconstruction approaches typically use interpolation techniques such as plane fitting <cit.> or Poisson surface estimation <cit.> <cit.> to estimate the underlying 3D structure. However, they can only recover very limited occluded/missing regions, e.g. small holes or gaps due to quantization artifacts, sensor noise and insufficient geometry information. Interestingly, humans are surprisingly talent at such ambiguity by implicitly leveraging prior knowledge. For example, given a view of achair with two rear legs occluded by front legs, humans are easily able to guess the most likely shape behind the visible parts. Recent advances in deep neural nets and data driven approaches are suitable to deal with such a task. In this paper, we aim to acquire the complete 3D geometry of an object given a single depth view. By utilizing the high performance of 3D convolutional neural nets and large open datasets of 3D models, our approach learns a smooth function to map a 2.5D view to a complete 3D shape. Particularly, we train an end-to-end model which estimates full volumetric occupancy from only one 2.5D depth view of an object, thus predicting occluded structures from a partial scan.While state-of-the-art deep learning approaches <cit.> <cit.> <cit.> <cit.> <cit.> for 3D shape reconstruction achieve encouraging and compelling results, they are limited to a very small resolution, typically less than 40^3 voxel grids. As a result, the learnt 3D shape tends to be coarse and inaccurate. However, to increase the model resolution without sacrificing recovery accuracy is challenging, as even a slightly higher resolution would exponentially increase the search space of potential 2.5D to 3D mapping functions, resulting in difficulties in convergence of neural nets.Recently, deep generative models achieve impressive success in modeling complex high-dimensional data distribution, among which Generative Adversarial Networks (GANs) <cit.> and Variational Autoencoders (VAEs) <cit.> emerge as two powerful frameworks for generative learning, including image and text generation <cit.> <cit.>, and latent space learning <cit.> <cit.>. In the past two years, a number of works <cit.> <cit.> <cit.> <cit.> apply such generative models to learn latent space to represent 3D object shapes, and then to solve simple discriminative tasks such as new image generation, object classification, recognition and shape retrieval. However, 3D shape reconstruction, as a more difficult generative task, has yet to be fully explored. In this paper, we propose 3D-RecGAN, a novel model that combines both an autoencoder and GAN to generate a full 3D structure conditioned on a single 2.5D view. Particularly, our model first encodes the 2.5D view to a low-dimensional latent space vector which implicitly represents general 3Dgeometric structures, then decodes it back to recover the most likely complete 3D structure. The rough 3D structure is then feed into a conditional discriminator which is adversarially trained to distinguish whether the coarse 3D shape is plausible or not.The autoencoder is able to approximate the corresponding shape, while the adversarial training tends to add fine details to the estimated shape. To ensure the final generated 3D shape corresponds to the input single partial 2.5D view, adversarial training of our model is based on conditional GAN <cit.> instead of random guessing. Our contributions are as follows:(1) We formulate a novel generative model to reconstruct the full 3D structure using a single arbitrary depth view. By drawing on both autoencoder and GAN, our approach is end-to-end trainable with high level of generality. Particularly, our model consumes a simple occupancy grid map without requiring object class labels or any annotations, while predicting a compelling shape with a high resolution of 64^3 voxel grid.(2) We exploit conditional GAN during training to refine 3D shape estimates from autoencoder. Key contribution here is the use of a latent distribution rather than a binary variable from the discriminator to train both discriminator and autoencoder. Using a latent distribution of high-dimensional real or fake 3D reconstructed shapes from discriminator significantly stabilizes the training of GAN, while using the standard binary variable 0/1 for training leads to the GAN crash easily.(3) We conduct extensive experiments for single category and multi-category reconstruction, outperforming the state of the art. Besides, our approach is also able to generalize previously unseen object categories.We evaluate our approach on synthetic datasets from virtually scanned 3D CAD models. Ideally, this task should be evaluated on real world 2.5D depth views, but it is very challenging to obtain the ground truth of 3D shape with regard to a specific 2.5D view for both training and evaluation. To the best of our knowledge, there are no good open datasets which have the ground truth for occluded/missing parts and holes for each 2.5D view in real world. Extensive experiments demonstrate that our 3D-RecGAN outperforms the state of the art by a large margin. Our reconstruction results are not only quantitatively more accurate, but also qualitatively with more details. An example of chair completion is shown in Figure <ref>.§ RELATED WORKWe review different pipelines for 3D reconstruction or shape completion. Both conventional geometry based and the state-of-the-art deep learning based approaches are covered.(1) 3D Model/Shape Fitting. <cit.> uses plane fitting to complete small missing regions, while <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> applies shape symmetry to fill in holes. Although these methods show good results, relying on predefined geometric regularities fundamentally limits the structure space to hand-crafted shapes. Besides, these approaches are likely to fail when missing or occluded regions are relatively big. Another similar fitting pipeline is to leverage database priors. Given a partial shape input, <cit.> <cit.> <cit.> <cit.> <cit.> try to retrieve an identical or most likely CAD model and align it with the partial scan. However, these approaches explicitly assume the database contains identical or very similar shapes, thus being unable to generalize novel objects or categories. (2) Multi-view Reconstruction. Traditionally, 3D dense recovery requires a collection of images <cit.>. Geometric shape is recovered by dense feature extraction and matching <cit.>, or by directly minimizing reprojection errors <cit.>. Basically, these methods are used for traditional SfM and visual SLAM, which is unable to build 3D structures for featureless regions such as white walls. Recently, <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> leverage deep neural nets to learn a 3D shape from multiple images. Although most of them do not directly require 3D ground-truth labels for supervision during training, they rely on additional signals such as contextual or camera information to supervise the view consistency. Obviously, extra efforts are required to acquire such additional signals. Additionally, resolution of the recovered occupancy shape is usually up to a small scale of 32^3.(3) Single-view Reconstruction. Predicting a complete 3D object model from a single view is a long-standing and very challenging task. When reconstructing a specific object category, model templates can be used. For example, morphable 3D models are exploited for face recovery <cit.> <cit.>. This concept was extended to reconstruct simple objects in <cit.>. For general and complex object completion, recent machine learning approaches achieve promising results. Firman et al. <cit.> trained a random decision forest to predict unknown voxels. 3D ShapeNets <cit.> is amongst the early work using deep networks to predict multiple 3D solutions from a single partial view. Fan et al. <cit.> also adopted a similar strategy to generate multiple plausible 3D point clouds from a single image. However, that strategy is significantly less efficient than directly training an end-to-end predictor <cit.>. VConv-DAE <cit.> can be used for shape completion, but it is originally designed for shape denoising rather than partial range scans. Wu et al. proposed 3D-INN <cit.> to estimate a 3D skeleton from single image, which is far from recovering an accurate and complete 3D structure. Dai et al. developed 3D-EPN <cit.> to complete an object's shape using deep nets to both predict a 32^3 occupancy grid and then synthesize a higher resolution model based on a shape database. While it achieves promising results, it is not an end-to-end system and it relies on a prior model database. Perspective Transformer Nets <cit.> and the recent WS-GAN <cit.> are introduced to learn 3D object structures up to a 32^3 resolution occupancy grid. Although they do not need explicit 3D labels for supervision, it requires a large number of 2D silhouettes or masks and specific camera parameters. In addition, the training procedure of <cit.> is two-stage, rather than end-to-end. Song et al. <cit.> proposed SSCNet for both 3D scene completion and semantic label prediction. Although it outputs a high resolution occupancy map, it requires strong voxel-level annotations for supervision. It also needs special map encoding techniques such as elimination of both view dependency and strong gradients on TSDF. <cit.> <cit.> use tree structures, while <cit.> applies Hibert Maps for 3D map representation to recover the 3D shape, thus being able to produce a relatively higher resolution of 3D shape. However, their deep networks only consist of a 3D encoder and decoder, without taking advantage of adversarial learning. Varley et al. <cit.> provides an architecture for 3D shape completion from a single depth view, producing an up to 40^3 occupancy grid. Although reconstruction results are encouraging, the network is not scalable to higher resolution 3D shape because of the heavy fully connected layers.§ 3D-RECGAN§.§ Overview Our method aims to predict a complete 3D shape of an object, which takes only an arbitrary single 2.5D depth view as input. The output 3D shape is automatically aligned with the corresponding 2.5D partial scan. To achieve this task, each object model is represented in a 3D voxel grid. We only use the simple occupancy information for map encoding, where 1 represents an occupied cell and 0 remains an empty cell. Specifically, both the input, denoted as I, and output 3D shape, denoted as Y, are 64^3 occupancy grids in our networks. The input shape is directly calculated from a single depth image. To generate ground true training and evaluation pairs, we virtually scan 3D objects from ModelNet40 <cit.>. Figure <ref> is the t-SNE visualization of partial 2.5D views and the corresponding full 3D shapes for multiple general chair and bed models. Each green dot represents the t-SNE embedding of a 2.5D view, whilst a red dot is the embedding of corresponding 3D shapes. It can be seen that multiple categories inherently have similar 2.5D to 3D mapping relationships. Essentially, our neural network is to learn a smooth function, denoted as f, which maps green dots to red dots in high dimensional space as shown in Equation <ref>. The function f is parametrized by convolutional layers in general. Y = f(I) ( I,Y ∈ Z_2^64^3, where Z_2={0,1}) After generating training pairs, we feed them into our networks. The first part of our network loosely follows the idea of an autoencoder with U-net architecture <cit.>. The autoencoder serves as a generator which is followed by a conditional discriminator <cit.> for adversarial learning.Instead of reconstructing the original input and learning an efficient encoding, the autoencoder in our network aims to learn a correlation between partial and complete 3D structures. With the supervision of complete 3D labels, the autoencoder is able to learn a function f and generate a reasonable 3D shape given a brand new partial 2.5D view. In the testing phase, however, the results tend to be graining and without fine details. To address this issue, in the training phase, the reconstructed 3D shape from the autoencoder is further fed into a conditional discriminator to verify its plausibility. In particular, a partial 2.5D input view is paired with its corresponding complete 3D shape, which is called the “real reconstruction", while the partial 2.5D view is paired with its corresponding output 3D shape from autoencoder, which is called “fake reconstruction". The discriminator aims to discriminate all “fake reconstruction" against “real reconstruction". In the original GAN framework <cit.>, the task of discriminator is to simply classify real and fake input, but its Jensen-Shannon divergence-based loss function is difficult to converge. The recent WGAN <cit.> leverages Wasserstein distance with weight clipping as a loss function to stabilize the training procedure, whilst the extended work WGAN-GP <cit.> further improves the training process using a gradient penalty with respect to its input. In our 3D-RecGAN, we apply WGAN-GP as the loss function of our conditional discriminator, which guarantees fast and stable convergence. The overall network architecture for training is shown in Figure <ref>, while the testing phase only needs the well trained autoencoder as shown in Figure <ref>.Overall, the main challenge of 3D reconstruction from an arbitrary single view is to generate new information including filling the missing and occluded regions from unseen views, while keeping the estimated 3D shape corresponding to the specific input 2.5D view. In the training phase, our 3D-RecGAN firstly leverages the autoencoder to generate a reasonable “fake reconstruction", then applies adversarial learning to refine the “fake reconstruction" to make it as similar to “real reconstruction" through jointly updating parameters of autoencoder. In the testing phase, given a novel 2.5D view as input, the jointly trained autoencoder is able to recover a full 3D model with satisfactory accuracy, while the discriminator is no longer used.§.§ Architecture Figure <ref> shows the detailed architecture of our proposed 3D-RecGAN. It consists of two main networks: the generator as in the top block and the discriminator as in the bottom block. The generator is based on autoencoder with skip-connections between encoder and decoder. Unlike the vanilla GAN generator which generates data from arbitrary latent distributions, our 3D-RecGAN generator synthesizes data from latent distribution of 2.5D views. Particularly, the encoder has five 3D convolutional layers, each of which has a bank of 4x4x4 filters with strides of 1x1x1, followed by a leaky ReLU activation function and a max pooling layer which has 2x2x2 filters and strides of 2x2x2. The number of output channels of max pooling layer starts with 64, doubling at each subsequent layer and ends up with 512. The encoder is lastly followed by two fully-connected layers to embed semantic information into latent space. The decoder is composed of 5 symmetric up-convolutional layers which are followed by ReLU activations except for the last layer with sigmoid function. Skip-connections between encoder and decoder guarantee propagation of local structures of the input 2.5D view. It should be noted that without the two fully connected layers and skip-connections, the vanilla autoencoder is unable to learn reasonable full 3D structures as the latent space is limited and the local structure is not preserved. During training, the generator is supervised supplying by ground true 3D shapes. The loss function and optimization methods are described in Section <ref>.The discriminator aims to distinguish whether the estimated 3D shapes are plausible or not. Based on conditional GAN, the discriminator takes both real reconstruction pairs and fake reconstruction pairs as input. Particularly, it consists of five 3D convolutional layers, each of which has a bank of 4x4x4 filters with strides of 2x2x2, followed by a ReLU activation function except for the last layer which is followed by a sigmoid activation function. The number of output channels of each layer is the same as that in the encoder part. Unlike the original GAN and conditional GAN, our discriminator is not designed as a binary discriminator to simply classify fake against real reconstructions. The reason is both real reconstruction pairs and fake reconstruction pairs are extremely high dimensional distributions, i.e. 2*64^3 dimensions. To naively classify it as only two categories would result in it being unable to capture geometric details of the object, and the discrimination loss is unlikely to benefit the generator through back-propagation. Instead, our discriminator is designed to output a long latent vector which represents distributions of real and fake reconstructions. Therefore, our discriminator is to distinguish the distributions of latent representations of fake and real reconstructions, while the generator is trained to make the two distributions as similar as possible. We use WGAN-GP as loss functions for our 3D-RecGAN.§.§ ObjectivesThe objective function of our 3D-RecGAN includes two main parts: an object reconstruction loss L_ae for autoencoder based generator; the objective function L_gan for conditional GAN.(1) L_ae For the generator, inspired by the work <cit.>, we use modified binary cross-entropy loss function instead of the standard version. The standard binary cross-entropy weights both false positive and false negative results equally. However, most of the voxel grid tends to be empty and the network easily gets a false positive estimation. In this regard, we impose a high penalty on false positive than false negative results. Particularly, a weight hyperparameter α is assigned to false positives, with (1-α) for false negative results, as shown in following Equation <ref>.L_ae = -αy log(y^') - (1-α ) (1-y) log(1-y^')where y is the target value in {0,1} and y^' is the estimated value in (0,1) for each voxel from the autoencoder.(2) L_gan For the discriminator, we leverage the state-of-the-art WGAN-GP loss functions. Unlike the original GAN loss function which presents an overall loss for both real and fake inputs, we separately represent the loss function L_gan^g in Equation <ref> for generating fake reconstruction pairs and L_gan^d in Equation <ref> for discriminating fake and real reconstruction pairs. Detailed definitions and derivation of the loss functions can be found in <cit.> <cit.>, but we modify them for our conditional GAN settings.L_gan^g = -𝐄[ D(y^'|x) ] L_gan^d = 𝐄[ D(y^'|x)] - 𝐄[ D(y|x) ] + λ𝐄[ (∇_ŷ D(ŷ|x)_2 - 1 )^2]where ŷ = ϵ x +(1-ϵ) y^', ϵ∼ U[0,1].λ controls the trade-off between optimizing the gradient penalty and the original objective in WGAN, x represents a voxel value, e.g.{0,1}, of an input 2.5D view, while y^' is the estimated value in (0,1) for the corresponding voxel from generator, and y is the target value in {0,1} for the same voxel.For the generator in our 3D-RecGAN network, there are two loss functions, L_ae and L_gan^g, to optimize. As we discussed in Section <ref>. Minimizing L_ae tends to learn the overall 3D shapes, whilst minimizing L_gan^g estimates more plausible 3D structures conditioned on input 2.5D views. To minimize L_gan^d is to improve the performance of discriminator to distinguish fake and real reconstruction pairs. To jointly optimize the generator, we assign weight β to L_ae, (1-β) to L_gan^g. Overall, the loss functions for generator and discriminator are as follows:L_g = β L_ae + (1-β) L_gan^g L_d = L_gan^d §.§ TrainingWe adopt an end-to-end training procedure for the whole network. To simultaneously optimize both generator and discriminator,we alternate between one gradient descent step on discriminator and then one step on generator. For the WGAN-GP, λ is set as10 for gradient penalty as in <cit.>. α ends up as 0.85 for our modified cross entropy loss function, while β is 0.05 for the joint loss function L_g. The Adam solver <cit.> is applied for both discriminator and generator with batch size of 8. The other three Adam parameters are set as default values, i.e. β_1 is 0.9, β_2 is 0.999 and ϵ is 1e-8. Learning rate is set to 0.0005 in the first epoch, decaying to 0.0001 in the following epochs. As we do not use dropout or batch normalization, the testing phase is exactly the same as training stage without reconfiguring network parameters. The whole network is trained on a single Titan X GPU from scratch.§.§ Data Synthesis For the task of 3D dense reconstruction from a single view, obtaining a large amount of training data is an obstacle. Existing real RGB-D datasets for surface reconstruction suffer from occlusions and missing data and there is no corresponding complete 3D structure for each single view. The recent work 3D-EPN <cit.> synthesizes data for 3D object completion, but their map encoding scheme is the complicated TSDF which is different from our network requirement. To tackle this issue, we use the ModelNet40 <cit.> database to generate a large amount of training and testing data with synthetically rendered depth images and the corresponding complete 3D shape ground truth. Particularly, a subset of object categories is selected for our experiments. For each category, we generate training data from around 200 CAD models in the train folder, while synthesizing testing data from around 20 CAD models in the test folder. For each CAD model, we create a virtual depth camera to scan it from 125 different angles, 5 uniformly sampled views for each of roll, pitch and yaw space. For each virtual scan, both a depth image and the corresponding complete 3D voxelized structure are generated with regard to the same camera angle. That depth image is simultaneously transformed to a partial 2.5D voxel grid using virtual camera parameters. Then a pair of partial 2.5D view and the complete 3D shape is synthesized. Overall, around 20K training pairs and 2K testing pairs are generated for each 3D object category. All data are produced in Blender.§ EVALUATIONIn this section, we evaluate our 3D-RecGAN with comparison to alternative approaches and an ablation study to fully investigate the proposed network. §.§ MetricsWe use two metrics to evaluate the performance of 3D reconstruction. The first metric is voxel Intersection-over-Union (IoU) between a predicted 3D voxel grid and its ground true voxel grid. It is formally defined as follows:IoU = ∑_ijk[I (y_ijk^'>p) * I(y_ijk) ] /∑_ijk[I ( I(y^'_ijk >p) + I(y_ijk) ) ]where I() is an indicator function, (i,j,k) is the index of a voxel in three dimensions, y^'_ijk is the predicted value at the (i,j,k) voxel, y_ijk is the ground true value at (i,j,k), and p is the threshold for voxelization. In all our experiments, p is set as 0.5. If the predicted value is over 0.5, it is more likely to be occupied from the probabilistic aspect. The higher the IoU value, the better the reconstruction of a 3D model. The second metric is the mean value of standard cross-entropy loss (CE) between a reconstructed shape and the ground true 3D model. It is formally presented as: CE =1/IJK∑_ijk[y_ijklog(y^'_ijk) + (1 - y_ijk)log(1-y^'_ijk)]where y^'_ijk and y_ijk are the same as defined in above IoU, (I, J, K) are the voxel dimension sizes of output 3D models. The lower CE value is, the better 3D prediction. The above two metrics can evaluate the overall reconstruction performance, but the reconstructed geometric details are unlikely to be well evaluated in such way. Therefore, a large number of qualitative results from reconstructed 3D models are visualized in Section <ref>.§.§ ComparisonWe compare against two alternative reconstruction methods. The first is the well-known traditional Poisson surface reconstruction <cit.> <cit.>, which is mostly used for completing surfaces on dense point clouds. The second is the state-of-the-art deep learning based approach proposed by Varley et al. in <cit.>, which is most similar to our approach in terms of input and output data encoding and the 3D completion task. It has encouraging reconstruction performance because of its two fully connected layers <cit.> in the model, but it is unable to deal with higher resolutions and it has less generality for shape completion. We also compare against the autoencoder alone in our network, i.e. without the GAN, named as 3D-RecAE for short.(1) Per-category Results. The networks are separately trained and tested on three different categories with the same network configurations. Table <ref> shows the IoU and CE results, and Figure <ref> compares qualitative results from different reconstruction approaches. (2) Multi-category Results. To study the generality, the networks are trained and tested on multiple categories without given any class labels. Table <ref> shows the IoU and CE results, and Figure <ref> shows the qualitative results.(3) Cross-category Results. To further investigate the generality, the network is trained on one category, but tested on another five different categories. Particularly, in Group 1, the network is trained on chair, tested on sofa, stool, table, toilet, and TV stand; in Group 2, the network is trained on stool, tested on chair, sofa, table, toilet, and TV stand; in Group 3, the network is trained on toilet, tested on chair, sofa, stool, table, and TV stand. Table <ref> shows the IoU andCE results; Figure <ref>, <ref> and <ref> compare qualitative cross-category reconstruction results of Group 1, Group 2 and Group 3 respectively.Overall, the above extensive experiments for per-category and multi-category object reconstruction demonstrate that our proposed 3D-RecGAN is able to complete partial 2.5D views with accurate structures and fine-grained details, outperforming the state of the art by a large margin. In addition, our 3D-RecGAN performs well in the challenging cross-category reconstruction task, which demonstrates that our novel model implicitly learns the geometric features and their correlations among different object categories. § CONCLUSIONIn this work, we proposed a novel framework 3D-RecGAN that reconstructs the full 3D structure of an object from an arbitrary depth view. By leveraging the generalization capabilities of autoencoders and generative networks, our 3D-RecGAN predicts accurate 3D structures with fine details, outperforming the traditional Poisson algorithm and the method in Varley et al.<cit.> in single-view shape completion for individual object category. We further tested our network's ability to perform reconstruction on multiple categories without providing any object class labels during training or testing, and it showed that our network is able to predict satisfactory 3D shapes. Finally, we investigated the network's reconstruction performance on unseen categories of objects. We showed that even in very challenging cases, the proposed approach can still predict plausible 3D shapes. This confirms that our network has the capability of learning general 3D latent features of the objects, rather than simply fitting a function for the training datasets. In summary, our network only requires a single depth view to recover an accurate complete 3D shape with fine details. ieee | http://arxiv.org/abs/1708.07969v1 | {
"authors": [
"Bo Yang",
"Hongkai Wen",
"Sen Wang",
"Ronald Clark",
"Andrew Markham",
"Niki Trigoni"
],
"categories": [
"cs.CV",
"cs.AI",
"cs.LG",
"cs.RO"
],
"primary_category": "cs.CV",
"published": "20170826134621",
"title": "3D Object Reconstruction from a Single Depth View with Adversarial Learning"
} |
§ INTRODUCTION VERITAS and other Imaging Air Cherenkov Telescopes (IACTs) record images of Cherenkov showers initiated by gamma rays and other types of cosmic rays. Those images are parameterized, typically assuming the images are elliptical. The parametrization of the shower is used to reconstruct the energy and sky position of the particle that initiated the air shower. However a large number of other cosmic rays are also recorded, dominated by hadron-initiated showers, which are the major source of backgrounds for the experiment. Most of the cosmic-ray showers can be removed from the analysis by using shape-based discrimination from their parameterization, but a number still remain after these gamma-ray selection cuts. IACTs are designed to detect γ rays, which point back to their point of origin. This assumption is used to subtract the remaining background. Typically, the remaining hadron showers are subtracted by selecting another region of the sky where no gamma-ray sources are expected and scaling this region to estimate the remaining background in the region of interest (ROI) for a gamma-ray source. The significance within the ROI is calculated typically by the Li&Ma 1983 equation 17 <cit.>:S = √(2)(N_ONln(1+α)N_ON/α(N_ON +N_OFF) + N_OFFln(1+α)N_OFF/N_ON +N_OFF)^1/2 where N_ON is the number of counts on an ON region of interest (ROI), N_OFF is the total number of counts in one or more background (or OFF) regions, and α is a scaling factor between events in the ON region and the OFF regions. The generalized equation for calculating α is:α = ∫ Acc_ON(x,y,t,E)dx dy dt dE/∫ Acc_OFF(x,y,t,E)dx dy dt dE where Acc is the acceptance function, defined as the probability after triggering and all off line cuts of an event being reconstructed with acertain position and energy <cit.>. The indicies ON and OFF refer to the acceptances in sampled on the ON and OFF region(s), respectively. This work focuses on the Ring Background Method (RBM) <cit.> where the background is defined as a single annulus around the ROI, but this work should be applicable to all background estimation methods available for IACTs in the literature. For the RBM method the acceptance is typically estimated using all data not in the ROI or in any background exclusion region. Regions of the FOV with gamma-ray sources and bright stars are excluded from background and acceptance estimates. More details on how acceptance is generated for the RBM, see Section 4.In the analysis of VERITAS and other IACT data, it is expected to get a distribution of statistical significances of sky map bins with mean of zero and width of unity in the absence of a VHE signal. However, it is not uncommon to see significance distributions of width greater than unity, indicating that the background is poorly estimated and the significances in the ROI are incorrect for sufficiently wide distributions. As a result, the criteria for a γ-ray detection (typically 5σ above the background) can no longer be relied upon, as the chances of a false positive are increased.This work explores the reasons for a wider significance distributions and solutions to this issue in simulations and in VERITAS data. This work is organized as follows: the next section describes and shows a simulation that was used to recreate the problem and explore different aspects of it. The following section describes a proceedure to correct for zenith gradients in the camera, followed by a description of a `modified' Li&Ma equation which takes into account systematic error of the background into the calculation. Finally we will show the results of validation of these techniques on VERITAS data. § SIMULATIONS Significance distributions calculated with the Li&Ma 17 equation were simulated to see if the observed wider significance distributions could be recreated. These were generated with the assumption that there was no significant signal in the field of view and therefore the mean rate measured in the ROI is the same as the background region(s) when scaled by α.The wider significance distributions were recreated when adding in a systematic error in α. The algorithm for the simulations are as follows:* Generate 10^4 random values for α. In this case, we assume that α is Gaussian distributed with a mean of 0.1 and width of 0.001.* Increase N_ON and N_OFF each by a random amount 10^4 times (once for each α value generated) for a time step Δ t according to: N_ON,i+1 = N_ON,i + Poisson(R_bgΔ t) N_OFF,i+1 = N_OFF,i + Poisson(R_bgΔ t/α) where N_ON,0 = N_OFF,0 = 0 and Poisson(x) is a randomly generated integer from a Poisson distribution with mean of x.* Calculate significance for each of the 10^4 sets of N_ON, N_OFF, using the mean α value of 0.1 in Equation 2.1. * Fit the distribution of the significances generated in the previous steps to a Gaussian.That width is recorded in Figure 1. * Repeat steps 2 through 4 for any number of time steps. The results shown in Figure 1 used a Δt of 1 hour and maximum timestep of 100 hours. If α is `perfect', i.e., the same α value is used in Equation 2.1 as Equation 2.4, then the width will never deviate farfrom 1.0, regardless of the number of time steps. However, if we `smear' α by adding (or subtracting) a random small amount, δα, generated from a Gaussian distribution, then the significance distribution width is ∼1.0 at small time steps, and eventually gets wider astime elapses. Figure 1 shows the results of the MC study with significance width as a function of exposure time.Based on the simulation study, a δα value randomly generated from a Gaussian with a width ∼0.001, assuming a fixed α value of 0.1, reproduces the width of the significance distribution with the standard VERITAS analysis with similar background rates. This points to some sort of systematic error in α that is not accounted for.It is possible to tighten the γ ray selection cuts to narrow the distribution, as this keeps the fluctuations small and the uncertainties in α. However this option often raises the energy threshold and reduces the total statistics in the sample, so this is not the best option for many analysis, particularly for those requiring a low energy threshold. Since α is derived from the acceptance functions, it is reasonable to look at possible factors that could possibly affect IACT acceptance functions. § ZENITH ACCEPTANCE CORRECTION As discussed in the previous section, issues related to the acceptance function is assumed to be responsible for the width of the significancedistribution seen VERITAS data. The standard VERITAS analysis packages make the assumption that the acceptance function is solely a function of ρ, the angular distance between the tracking position of the array pointing to the reconstructed event position.A zenith angle gradient was found in the data, which distorts the radial dependence of the acceptance function. Correcting the acceptance for the zenith gradient showed an improvement in the significance distribution width, bringing it closer to the expected distribution. This will be shown in the validation section. In order to qualitatively measure the acceptance gradient in the sky map, we define a new parameter called flatness, which is defined as the ratio of the acceptance in a given sky map bin to the number of events in that bin: f_i = N_i/Acc_i/<N/Acc>∼ Constant where f_i is the flatness, and N_i is the number of counts in a sky map bin and the index i denotes a particular sky map bin. Note that the ratio of counts to acceptance is divided by the mean of all bins so that flatness is always centered on 1.0. If the acceptance function accurately describes the background, then flatness should be approximately constant over the entire FOV for every sky map bin not in an a priori defined background exclusionregion. An example of a flatness map is shown in Figure 2 (left).The next step is to produce sky maps related to zenith angle. For scaling purposes, the difference of the zenith angle of the reconstructed event position and the zenith angle of the telescope tracking position is used (Figure 2, center). The contents of each bin in the zenith map and the flatness map are plotted against each other, and the Pearson correlation coefficient is used to determine the strength of any camera gradients.This part of the procedure was tested on Segue 1 for zenith angle, azimuth angle and NSB; only zenith angle showed any sort of strong correlation.After the gradient was quantified for zenith angle, we then developed a proceedure to correct for it in the data. It should be noted that the both of the other major IACTs currently in operation, MAGIC <cit.> and HESS <cit.> use acceptance functions that are depedent in zenith, so the fact that the acceptance also takes into account zenith angle in itself is not a novel idea. The method used here makes use of the Flatness/Zenith relationship (Figure 2, right). The scatter plot of Δ Zn and flatness described in the previous section is binned into a histogram. The histogram is normalized by the value at Δ Zn = 0 to better quantify the size of the effect across the camera and then is fit to a polynmial. A fourth-degree polynomial is used to remove possible second-order effects, but a linear fit most likely also acceptable. The polynomial fit is then used to re-weight the acceptance function: Acc'_i = Acc_i(1 + p_1/p_0Δ Zn + p_2/p_0Δ Zn^2 + p_3/p_0Δ Zn^3 + p_4/p_0Δ Zn^4) Where Acc_i is the acceptance in the ith RA/Dec bin, p_n is the nth coefficient from the flatness fit, and Δ Zn is theaverage reconstructed zenith angle subtracted from the average telescope tracking zenith angle. Once the zenith-corrected acceptance function,Acc'_i is obtained and the correction applied to each bin of the acceptance map, is used in Equation 1.2 to recalculate α. The new α values are then used to recalculate significance from Equation 1.1.§ MODIFIED LI&MA SIGNIFICANCE EQUATION ACCOUNTING FOR BACKGROUND SYSTEMATICS Acceptance functions are typically a derived from data and there is an intrinsic systematic uncertainty associated with it. A `Modified' version of the Li&Ma equation is needed to correctly take into account the α systematic error, which we will refer to as σ_α <cit.>. This solution presented here is analytic, but approaches have been taken by fitting using TMinuit in the literature as well <cit.>.The full derivation of the Modified Li&Ma Equation is in <cit.> along with details of its performance on simulations. It follows a maximum likelihood framework with α treated as a nuscience parameter. The Modified Li&Ma equation is: S_Mod = N_on - α N_off/|N_on - α N_off|√(S^2_LiMa(N_on,N_off,α') + (α' - α/σ_α)^2) Where S_LiMa is the Li&Ma significance calculation in Equation 1.1 and α' is the solution to the equation: 0 = α'^3 - α'^2(α - 1) - α'(α - σ_α^2N_off) - σ_α^2N_on This is a cubic equation with up to three real solutions. It's noteworthy to mention that in the case where σ_α = 0, this reduces to α' equals α, 0 and -1 and we recover Equation 1.1. If there are multiple positive and real solutions, then the solution is used that maximizes the likelihood:Λ(α') = N_onlogα' + (N_on + N_off)log(N_on + N_off/1+α') - 1/2(α' - α/σ_α)^2 It is assumed here that the main source of σ_α is the statistical uncertainty of the acceptance curves in VEGAS, one of the VERITAS standard analysis codes <cit.>. The procedure for estimating σ_α follows, which requires how this code determines α. On a run-by-run basis, an acceptance curve is calculated as a function of angular distance from the array pointing position to the reconstructed event position. That curve is then smoothed several times and is given a weight based on the number of events passing cuts in that run.The weighted acceptance curve is evaluated in each sky map position in either RA/Dec or Galactic coordinates. This process is repeated for all observation runs. α is then calculated for each position from equation 1.2, assuming E and t are constant, using an ON region described as an circular region around the region of interest (ROI) and the off region being an annulus centered on the same ROI. The process of σ_α estimation follows a similar procedure as the α calculation. The RMS from the acceptance curve is used and the same background weight is applied. The weighted RMS is then added to each sky map position, and then the process is repeated for all runs.σ_α is then calculated by error propagation of α: α = ∫ Acc_ON(x,y,t,E)dx dy dt dE/∫ Acc_OFF(x,y,t,E)dx dy dt dE = ∑ Acc_ON(x_i,y_i)/∑ Acc_OFF(x_i,y_i)σ_α = α√((∑δ Acc_ON(x_i,y_i)/∑ Acc_ON(x_i,y_i))^2 + (∑δ Acc_OFF(x_i,y_i)/∑ Acc_OFF(x_i,y_i))^2) § VALIDATION ON VERITAS DATA We test the results from these systematic studies on VERITAS data and see if they improve the significance distributions. The zenith correction and modified Li&Ma equation were tested on various data sets. Segue 1 and PKS 1424+240 were tested using soft spectral cuts since the scientific goals of both of these benefits from having a low energy threshold; IC 443 was tested with harder but extended cuts since it is an extended SNR for VERITAS. Figure 3 shows the results of the validation on PKS 1424+240. Table 1 summarizes the validation results for the target position and the best fit of the significance distributions to a Gaussian function. In all cases, the significance distribution widths decrease as the zenith correction and the modified Li&Ma are applied and the situations where both are applied the width is within 10% of unity. In the case where both are applied, the zenith correction is applied first before determining σ_α for the Modified Li&Ma equation. The zenith correction does not greatly affect the significance at the source position. Application of the modified Li&Ma equation does decrease the target position significance, and may infer that as a loss of sensitivity. But the reader should keep in mind that if the significance distribution is sufficiently wider than unity, the significance is overestimated. Instead of a loss of sensitivity, this should be interpreted as the true value of significance of the target observation. § CONCLUSIONS AND FUTURE WORK This work has addressed the systematics associated with the background weighting parameter, α, and not discussed reduction of the background rates. It is possible that other as yet unaccounted background systematics exist. Improved γ/Hadron separation would decrease the overall systematic uncertainty in the background estimation from these unaccounted for systematics and should also improve the width of significance distributions in addition to any sensitivity boost gained from a lower background rate. It is therefore encouraged to combine the methods outlined here with boosted-decision trees <cit.> or any similar methods.We show that employing the methods outlined here, the significance distributions in VERITAS and other IACT analysis can be greatly improved. The significance width of the validation data set of PKS 1424+240 is reduced from 1.43 to 1.02 and Segue 1 is improved from 1.72 to 0.99. Future work should investigate other factors that could affect camera acceptances, such as optically bright stars, night sky background and azimuth angle and how to best account for these effects. The techniques in this work and in the future in this area becomes incredibly important as targets observed by the current generation of IACTs for hundreds of hours or more and for the upcoming CTA observatory <cit.>. § ACKNOWLEDGMENTSFor the full acknowledgments, please visit https://veritas.sao.arizona.edu/.991983ApJ...272..317L Li, T.-P., & Ma, Y.-Q. 1983, apj, 272, 317 2007A A...466.1219B Berge, D., Funk, S., & Hinton, J. 2007, aap, 466, 1219Mag_Bg Prandini, E., Pedaletti, G., Da Vela, P., et al. 2015, arXiv:1509.02391 HESS_BgJ. Hinton et al., H.E.S.S. collaborationProc. Conf. Towards a Network of Atmospheric Cherenkov Detectors VII, Palaiseau, France, 2005, p. 183-1902015APh....67...70S Spengler, G. 2015, Astroparticle Physics, 67, 70Dickinson2012Dickinson, H., & Conrad, J. 2013, Astroparticle Physics, 41, 17VEGASRefCogan, P. 2008, International Cosmic Ray Conference, 3, 1385Krause2016Krause, M., Pueschel, E., & Maier, G. 2017, Astroparticle Physics, 89, 1 CTARefhttps://www.cta-observatory.org/ | http://arxiv.org/abs/1708.07444v1 | {
"authors": [
"Benjamin Zitzer",
"VERITAS Collaboration"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170824145357",
"title": "Background Systematic Studies in VERITAS Data"
} |
Robust Beamforming Techniques for Non-Orthogonal Multiple Access Systems with Bounded Channel Uncertainties Faezeh Alavi, Kanapathippillai Cumanan, Zhiguo Ding and Alister G. Burr Received: date / Accepted: date =========================================================================================================== In this letter, we propose a robust for non-orthogonal multiple access (NOMA) based multiple-input single-output (MISO) downlink systems. In , the robust power minimization problem is studied with imperfect channel state information (CSI), where the beamformers are designed by incorporating norm-bounded channel uncertainties to provide the required quality of service at each user. This robust scheme is developed based on the worst-case performance optimization framework. In terms of beamforming vectors, the original robust design is not convex and therefore, the robust beamformers cannot be obtained directly. To circumvent this non-convex issue, the original intractable problem is reformulated into a convex problem, where the non-convex constraint is converted into a linear matrix inequality (LMI) by exploiting S-Procedure. Finally, simulation results are provided to demonstrate the effectiveness of the proposed robust design.Non-orthogonal multiple access (NOMA), multiple-input single-output (MISO), robust beamforming, worst-case performance optimization § INTRODUCTIONNon-orthogonal multiple access (NOMA) is a promising multiple access technique for 5G networkswhich has the potential to address theissues associated with the exponential growth of data traffic such as spectrum scarcity and massive connectivity <cit.>. In contrast to conventional multiple access schemes, NOMA allows different users to efficiently share the same resources (i.e., time, frequency and code) at different power levels so that the user with lower channel gain is served with a higher power and vice versa. In this technique, a successive interference cancellation (SIC) approach is employed at receivers to separate multi-user signals, which significantly enhances the overall spectral efficiency. In other words, NOMA has the capability to control the interference by sharing resources while increasing system throughput with a reasonable additional complexity <cit.>.Recently, a significant amount of research has focused in studying several practical issues in NOMA scheme. In particular, beamforming designs for multiple antenna NOMA networks have received a great deal of interest in the research community due to their additional degrees of freedom and diversity gains <cit.>.A general framework for a multiple-input multiple-output (MIMO) NOMA system has been developed for both the downlink and the uplink in <cit.> whereasthe throughput maximization problem was studied for a two-user MIMO NOMA system in <cit.>. The sumrate maximization problem for a multiple-input single-output (MISO) NOMA has been investigated in <cit.> through the minorization maximization algorithm.In most of the existing work, beamforming designs have been proposed for NOMA schemes with the assumption of perfect channel state information (CSI) at the transmitter <cit.>. However,this assumption might not be always valid for practical scenariosdue to channel estimation and quantization errors <cit.>. On the other hand, channel uncertainties significantly influence the performance of theSIC based receivers as thedecoding order of the received multi-user signals is determined with respect to the users' effective channel gains. Therefore, it is important to take into account the channel uncertainties especially in the beamforming design for NOMA networks. Motivated by this practical constraint, we focus on robust beamforming design based on the worst-case performance optimization framework to tackle the norm-bounded channel uncertainties <cit.>. In <cit.>, the robust beamforming design has been developed for providing secure communication in wireless networks withimperfect CSI. By incorporating thebounded channel uncertainties, the robust sum power minimization problemis investigated in <cit.> for a downlink multicell network with the worst-case signal-to-interference-plus-noise-ratio (SINR) constraints whereas the robust weighted sum-rate maximization was studied for multicell downlink MISO systems in <cit.>. In <cit.>, a robust minimum mean square error based beamforming technique is proposed for multi-antenna relay channels with imperfect CSI between the relay and the users. In the literature, there are two types of NOMA schemes considered: I) clustering NOMA <cit.>, II) non-clustering NOMA <cit.>. In the clustering NOMA scheme, all the users in a cell are grouped into N clusters with two users in each cluster, for which a transmit beamforming vector is designed to support those two users through conventional multiuser beamforming designs. The users in each cluster are supported by a NOMA beamforming scheme. However, in the non-clustering NOMA scheme, there is no clustering and each user is supported by its own NOMA based beamforming vector. In <cit.>, the authors studied a robust NOMA scheme for the MISO channel to maximize the worst-case achievable sum rate with a total transmit power constraint.In this letter, we follow the second class of research where NOMA scheme applied between all users and there is the spectrum sharing between all users in cell. Then, we propose a robust beamforming design for NOMA-based MISO downlink systems. In particular, the robust power minimization problem is solved based on worst-case optimization framework to provide the required quality of service at each user regardless of the associated channel uncertainties. By exploitingthe originalproblem is converted into a convex one by recasting the non-convex constraints into linear matrix inequality (LMI) forms. Simulation results are provided to validate the of the robust design by comparing the performance of the robust scheme with that of the non-robust approach. The work in <cit.> also studied the worst-case based robust scheme for MISO NOMA system, however, there are main differences between our proposed scheme and the work in <cit.>. A clustering NOMA scheme is developed in <cit.> by grouping users in each cluster. In this scheme, a single beamformer is designed to transmit the signals for all users in the same cluster whereas, in this letter, the signal for each user is transmitted with a dedicated beamformer. In addition, both beamforming designs are completely different as the work in <cit.> proposes robust sum-rate maximization based design whereas this letter solves robust power minimization problem with rate constraint on each user. In terms of solutions, the work in <cit.> exploits the relationship between MSE and achievable rate and derives an equivalent non-convex problem, which is decoupled into four sub-problems and those problems are iteratively solved to realize the solution of the original problem. In this letter, the robust power minimization problem is formulated by deriving the worst-case achievable rate. The original problem formulation turns out to be non-convex and we exploit S-Procedure and semidefinite relaxation to convert it to a convex one. Hence, the work in <cit.> and the proposed work in this letter are different including problem formulation and the solution approaches.§ SYSTEM MODEL AND PROBLEM FORMULATIONWe consider NOMA-based downlink transmission where a base station (BS) sends information to K usersU_1,U_2, …, U_K.It is assumed that the BS is equipped with M antennas whereas each user consists of a single antenna.The channel coefficient vector between the BS and the k^th user U_k is denoted by 𝐡_k ∈ℂ^M × 1 (k = 1,…,K) and 𝐰_k∈ℂ^M × 1represents the corresponding beamforming vector of the k^th user U_k. The received signal at U_k is given byy_k = 𝐡_k^H 𝐰_k s_k + ∑_m≠ k𝐡_k^H 𝐰_ms_m + n_k, ∀ k, where s_k denotes the symbol intended for U_kandrepresents a zero-mean additive white Gaussian noise with variance σ_k^2. The power of the symbol s_k is assumed to be unity, i.e., 𝔼(|s_k|^2)=1. In practical scenarios, it is difficult to provide perfect CSI at the transmitter due to channel estimation and quantization errors. Therefore, we consider a robust beamforming design to overcome these channel uncertainties. In particular, we incorporate norm-bounded channel uncertainties in the design as 𝐡_k=ĥ_k +Δĥ_k,Δĥ_k_2=𝐡_k-ĥ_k_2 ≤ϵ, where ĥ_k, Δĥ_k and ϵ≥ 0 denote the estimate of 𝐡_k, the norm-bounded channel estimation error and the channel estimation error bound, respectively.In the NOMA scheme, user multiplexing is performed in the power domain and the SIC approach is employed at receivers to separate signals between different users. In this scheme, users are sortedbased on the norm of their channels,i.e., 𝐡_1_2≤𝐡_2_2≤…≤𝐡_K_2. For example, the k^th user decodes the signals intended for the users from U_1 to U_k-1 using the SIC approachwhereas the signals intended for the rest of the users (i.e., U_k+1,…,U_K) are treated as interference at the k^th user. Based on this SIC approach, the l^th user can detect and remove the k^th user's signals for 1 ≤ k < l <cit.>. Hence, the signal at the l^th user after removing the first k-1 users' signals to detect the k^th user is represented as y_l^k=𝐡_l^H 𝐰_k s_k+∑_m=1^k-1Δĥ_l 𝐰_m s_m+∑_m=k+1^K𝐡_l^H 𝐰_m s_m+n_l,∀ k, l∈{k,k+1,…,K}, where the first term is the desired signal to detect s_k andthe second term is due to imperfect CSI at the receivers duringthe SIC process.Due to the channel uncertainties,the signals intended for the users U_1,…,U_k-1 cannot be completely removed by the l^th user. The third term is the interference introduced by the signals intended to the users U_k+1,…,U_K.According to the SIC based NOMA scheme, the l^th user should be able to detect all k^th (k<l) user signals. Thus, the achievable rate of U_k can be defined as follows: R_k=log_2 (1+min_l ∈{k, k+1, …, K} SINR^k_l), where SINR^k_ldenotes the SINRof the k^th user's signal at the l^th user which can be written asSINR_l^k= 𝐡_l^H 𝐰_k𝐰_k^H 𝐡_l∑_m=1^k-1Δĥ_l^H 𝐰_m𝐰_m^H Δĥ_l+∑_m=k+1^K𝐡_l^H 𝐰_m𝐰_m^H𝐡_l+σ_l^2. For this network setup, we study robust power minimization by incorporating channel uncertainties to satisfy the required SINR at each user. This robust beamforming design is developed by considering the worst-case SINR of each user, which can be formulated asmin_𝐰_k∈ℂ^M×1∑_k=1^K𝐰_k^2_2, 𝐬.𝐭.min_Δĥ_l_2 ≤ϵ(min_l ∈{k, k+1, …, K}SINR^k_l)≥γ_k^min,∀ k,where γ_k^min=(2^R_k^min-1) is the minimum required SINR to achieve a target rate R_k^min at U_k.§ ROBUST BEAMFORMING DESIGNThe problem formulation in (<ref>) is not convex and the optimal robust beamformers cannot be obtained directly. To tackle this issue, we introduce a new matrix variable 𝐖_k=𝐰_k 𝐰_k^H and reformulate the original robust problem in (<ref>) into the following optimization framework without loss of generality: min_𝐖_k∈ℂ^M× M∑_k=1^K Tr(𝐖_k), 𝐬.𝐭.ϖ_kl, ∀ k, l=k,…,K,𝐖_k ≽0, rank(𝐖_k)=1, ∀ k, where ϖ_kl is defined in Appendix <ref>. However, the reformulated problem in (<ref>) is still not convex for two reasons; the rank-one constraint and unknown channel uncertainties, i.e., Δĥ_k, which lead to an intractable problem. The rank-one constraint in (<ref>) can be relaxed by exploiting semi-definite relaxation (SDR). To remove the unknown channel uncertainties and solve the original problem with available knowledge of imperfect CSI (error bound), we employto recast the non-convex constraints into LMIs.Lemma 1: By relaxing the rank-one constraints on 𝐖_k, the original problem in (<ref>) can be recast into the following convex problem: min_[ 𝐖_k∈ℂ^M× M,; λ_kl≥ 0 ]∑_k=1^K Tr(𝐖_k), 𝐬.𝐭. 𝐖_k ≽0, 𝐂_kl≽0, ∀ k, l=k,…,Kwhere 𝐂_kl is defined in Appendix <ref>. Please refer to Appendix <ref>.The problem in (<ref>)is a standard semidefinite(SDP) and can be efficiently solved using interior-pointThe optimal solution for the original problem in (<ref>) can be obtained through extracting the eigenvector corresponding to the maximum eigenvalue of the rank-one solution of (<ref>). Thus, the following lemma holds to show that the optimal solution to (<ref>) is rank one. Lemma 2: Provided the problem in (<ref>) is feasible, there always exists a rank-one optimal solution {𝐖^*_k}.Please refer to Appendix <ref>.§ SIMULATION RESULTSTo assess the performance of the proposed robust beamforming approach, we consider a single cell downlink transmission, where a multi-antenna BS serves single-antenna users which are uniformly distributed over the circle with a radius of 1000 meters around the BS, but no closer than d_0=100 meters.The small-scale fading of the channels is Rayleigh which represents an isotropic scattering environment. We model the large-scale fading effects as the product of path loss and shadowing fading. The log-normal shadowing is considered with standard deviation σ_0=8 dB, scaled by (d_k/d_0)^-β to incorporate the path-loss effects where d_k is the distance between U_k and the BS, measured in meters and β=3.8 is the path-loss exponent. Throughout the simulations, it is assumed that the BS is equipped with eight antennas (M = 8) and it serves three users (K = 3). The noise variance at each user is assumed to be 0.01 (i.e., σ_k^2 = 0.01) and the target rates for all users are the same.The term “Non-robust scheme" refers to the scheme where the BS has imperfect CSI without any information on the channel uncertainties and the beamforming vectors are designed based on imperfect CSI without incorporating channel uncertainty information. First, we study the impact of channel uncertainties on the required total transmit power. Fig. <ref> depicts the required total transmit power against different SINR thresholds for the robust and the non-robust NOMA schemes as well as OMA scheme with different error bounds.As seen in Fig. <ref>, the robust scheme requires more transmit power than that of the non-robust scheme. This is because the robust scheme satisfies the required SINR all the time, at the price of more transmit power at the BS whereas the non-robust scheme does not. The difference between the required transmit power for the robust and the non-robust schemes increases with error bounds. This is because incorporating all possible sets of errors in the beamforming design to satisfy high SINR thresholds requires more transmit power in the robust scheme. Moreover, as seen in , the conventional framework, orthogonal multiple access (OMA), requires more transmit power to achieve the same rate in comparison with NOMA scheme. This demonstrates that the NOMA scheme yields a better performance in terms of spectral and energy efficiencies.Next, we evaluate the performance of the proposed robust and non-robust schemes in terms of the minimum achieved SINR between users. Fig.<ref> provides cumulativefunction (CDF) and probability density function (PDF)from 1000 random sets of channels with error bounds of 0.06(ϵ = 0.06) where the SINR threshold has been set to 10 dB at each user. As evidenced by the results, the robust scheme outperforms the non-robust scheme in terms of minimum achieved SINRs. In addition, the robust scheme satisfies the SINR thresholds all the time regardless of the channel uncertainties whereasthe non-robust design fails to satisfy the minimum SINR requirements.§ CONCLUSION In this letter, we propose a robust beamforming design for the downlink of a NOMA based MISO network by taking into account the norm-bounded channel uncertainties. However, the original robust problem formulation is not convex due to the imperfect CSI. To cope with this challenge, we exploitedto reformulate the original non-convex problem into a convex optimization framework by recasting the original non-convex constraints into an LMI form. Simulation results demonstrate that the proposed robust scheme offers a better performance than the non-robust approach by satisfying the SINR requirement at each user all the time regardless of associated channel uncertainties.equationsection§ DERIVATION OF Π_KL The equivalent transformations of (<ref>) can be obtained as ϖ_kl as follows: {[ min_Δĥ_k_2 ≤ϵ(𝐡^H_k 𝐖_k 𝐡_k/∑_m=1^k-1Δĥ_k^H 𝐖_m Δĥ_k +∑_m=k+1^Kh^H_k 𝐖_m 𝐡_k +σ_k^2) ≤γ^min_k,; min_Δĥ_k+1_2 ≤ϵ(𝐡^H_k+1𝐖_k𝐡_k+1/∑_m=1^k-1Δĥ_k+1𝐖_m Δĥ_k+1+∑_m=k+1^K𝐡^H_k+1𝐖_m 𝐡_k+1+σ_k+1^2);≤γ^min_k,;⋮; min_Δĥ_K_2 ≤ϵ(𝐡^H_K 𝐖_k 𝐡_K/∑_m=1^k-1Δĥ_K 𝐖_m Δĥ_K +∑_m=k+1^K𝐡^H_K 𝐖_m 𝐡_K +σ_K^2) ≤γ^min_k, ]. ⇔min_Δĥ_l_2 ≤ϵ(𝐡^H_l 𝐖_k 𝐡_l/∑_m=1^k-1Δĥ_l^H 𝐖_m Δĥ_l +∑_m=k+1^K𝐡^H_l 𝐖_m 𝐡_l+σ_l^2) ≤γ^min_k ≜ϖ_kl,∀ k, l=k,…,K.§ PROOF OF LEMMA 1 To incorporate the channel uncertainties in the robust optimization framework, we exploit S-procedureto convert the non-convex constraint into LMI form.By applying<cit.>, the constraint (<ref>) is Δĥ_l^H 𝐈Δĥ_l -ϵ^2 ≤ 0 ⇒Δĥ_l^H (∑_m≠ k𝐖_m - 𝐖_k/γ^min_k )Δĥ_l+2Re{ĥ_l^H(∑_m=k+1^K𝐖_m -𝐖_k/ γ^min_k)Δĥ_l }+ĥ_l^H (∑_m=k+1^K𝐖_m - 𝐖_k/ γ^min_k )ĥ_l + σ_l^2 ≤ 0, Then, the constraint (<ref>) can be reformulated withas the following semidefinite constraint𝐂_kl=[[ λ_kl𝐈+ϕ_k+ν_k ϕ_k ĥ_l; ĥ_l^H ϕ_k ĥ_l^H ϕ_k ĥ_l-σ_k^2-λ_klϵ^2 ]] ≽0,whereThis completes the proof of Lemma 1. ▪ § PROOF OF LEMMA 2 To prove Lemma 2, we examine the Karush-Kuhn-Tucker (KKT) conditions of (<ref>). First, let 𝐘_k∈ℂ^M× M,and μ_kl∈ℝ_+ denote the dual variable of the constraints in (<ref>),respectively. Then, the Lagrangian dual function of (<ref>) can be written as ℒ(𝐖_k,λ_kl,𝐓_kl,μ_kl,𝐘_k)=∑_kTr(𝐖_k)-∑_kTr(𝐘_k𝐖_k)-∑_k,l Tr(𝐓_kl𝐀_1)-∑_k,l Tr[𝐓_kl𝐇^H_lϕ_k 𝐇_l]-∑_k,l Tr(𝐓_kl𝐀_2),where 𝐇_l=[𝐈𝐡_l] and 𝐀_1=([ λ_kl𝐈0; 0 -σ_k^2-λ_klϵ^2 ]), 𝐀_2 = ([ ν_k 0; 0 0 ]). The following KKT conditions hold for (<ref>)∂ℒ/∂𝐖_k=0⇒𝐘_k+𝐇_l𝐓_kl𝐇^H_l/γ^min_k= 𝐈+ ∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l +∑_j=k+1^K𝐓_jl,𝐖_k𝐘_k=0,(𝐀_1+ 𝐇^H_lϕ_k 𝐇_l+ 𝐀_2)𝐓_kl=0. We premultiply (<ref>) by 𝐖_k, i.e., 𝐖_k 𝐇_l𝐓_kl𝐇^H_l/γ^min_k=𝐖_k (𝐈 + ∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l + ∑_j=k+1^K𝐓_jl), Then, we can write the following rank relation rank(𝐖_k) =rank[𝐖_k (𝐈+∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l +∑_j=k+1^K𝐓_jl)]=rank(𝐖_k 𝐇_l𝐓_kl𝐇^H_l)≤min{rank(𝐇_l𝐓_kl𝐇^H_l), rank(𝐖_k)}.Based on (<ref>), it is required to show rank(𝐇_l𝐓_kl𝐇^H_l)≤1 if we claim rank(𝐖_k)≤1.First, we consider the following equations and Lemma 3: [𝐈0]𝐇^H_l=𝐈, [𝐈0]𝐀_1=λ_kl(𝐇^H_l-[0_M 𝐡_l]),[𝐈0]𝐀_2=ν_k(𝐇^H_l-[0_M 𝐡_l]), Lemma 3: If a block Hermitian matrixthen the main diagonal matrices 𝐁_1 and 𝐁_4 must be positive definite (PSD) matrices <cit.>.We pre-multiply [𝐈0] and post-multiply 𝐇^H_l by (<ref>), respectively, and applying the equalities in (<ref>): λ_kl(𝐇^H_l-[0_M 𝐡_l])𝐓_kl𝐇^H_l+ν_k(𝐇^H_l-[0_M 𝐡_l])𝐓_kl𝐇^H_l+ϕ_k𝐇_l𝐓_kl𝐇^H_l=0⇒(λ_kl𝐈+ϕ_k+ν_k)𝐇_l𝐓_kl𝐇^H_l=(λ_kl𝐈+ν_k)[0_M 𝐡_l]𝐓_kl𝐇^H_l By applying Lemma 3 to (<ref>), we can claim (λ_kl𝐈+ϕ_k+ν_k)≽0 and is nonsingular; thus, multiplying by a nonsingular matrix will not change the matrix rank. Thus, the following rank relation holds: rank(𝐇_l𝐓_kl𝐇^H_l)= rank((λ_kl𝐈+ν_k)[0𝐡_l]𝐓_kl𝐇^H_l)≤rank([0𝐡_l])≤ 1. This completes the proof of Lemma 2. ▪ IEEEtrans | http://arxiv.org/abs/1708.07855v1 | {
"authors": [
"Faezeh Alavi",
"Kanapathippillai Cumanan",
"Zhiguo Ding",
"Alister G. Burr"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170825182303",
"title": "Robust Beamforming Techniques for Non-Orthogonal Multiple Access Systems with Bounded Channel Uncertainties"
} |
1]Hendrik J. Weideman 1]Zachary M. Jablons 2]Jason Holmberg 3]Kiirsten Flynn 3]John Calambokidis 4]Reny B. Tyson 4]Jason B. Allen 4]Randall S. Wells 5]Krista Hupman 6]Kim Urian 1]Charles V. Stewart [1]Rensselaer Polytechnic Institute [2]WildMe [3]Cascadia Research Collective [4]Chicago Zoological Society's Sarasota Dolphin Research Program, c/o Mote Marine Laboratory [5]Massey Universityaffilsepx[6]Duke University Marine Laboratory [1]{weideh,jabloz2,stewart}@rpi.edu affilsepx[2][email protected] [3]{kflynn,calambokidis}@cascadiaresearch.org [4]{rtyson,allenjb,rwells}@mote.org [5][email protected] [6][email protected] Curvature Representation and Matching Algorithmsfor Identification of Dolphins and Whales [ December 30, 2023 ====================================================================================================empty We address the problem of identifying individual cetaceans from images showing the trailing edge of their fins.Given the trailing edge from an unknown individual, we produce a ranking of known individuals from a database.The nicks and notches along the trailing edge define an individual's unique signature.We define a representation based on integral curvature that is robust to changes in viewpoint and pose, and captures the pattern of nicks and notches in a local neighborhood at multiple scales.We explore two ranking methods that use this representation.The first uses a dynamic programming time-warping algorithm to align two representations, and interprets the alignment cost as a measure of similarity.This algorithm also exploits learned spatial weights to downweight matches from regions of unstable curvature.The second interprets the representation as a feature descriptor.Feature keypoints are defined at the local extrema of the representation.Descriptors for the set of known individuals are stored in a tree structure, which allows us to perform queries given the descriptors from an unknown trailing edge.We evaluate the top-k accuracy on two real-world datasets to demonstrate the effectiveness of the curvature representation, achieving top-1 accuracy scores of approximately 95% and 80% for bottlenose dolphins and humpback whales, respectively.§ INTRODUCTIONWe address the problem of identifying individual cetaceans from images of their fins — dorsal fins for bottlenose dolphins, and flukes for humpback whales. Fitting into the broad domain of contour-based recognition <cit.>, the fin instance recognition problem is particularly challenging, as illustrated in Figures <ref> and <ref>.Our hypothesis is that the information necessary to distinguish between the outlines of fins from distinct individuals is encoded in local measures of integral curvature <cit.>.Differential curvature measures have been applied to a variety of recognition problems, but these approaches are sensitive to noise <cit.>.Integral curvature, which produces more stable measurements, has been applied to category recognition problems like identifying leaf species <cit.>.In this paper, we propose novel combinations of integral curvature representation for extracted outline contours and two matching algorithms for identifying individuals.The first interprets the curvature representation as a sequence, and defines the similarity between two representations as the cost of warping one sequence onto the other.The second treats subsections of the representation as feature descriptors, and matches them using approximate nearest neighbors.Each takes a query image and produces a ranking of known individuals from a database.We also introduce a method for learning spatial weights that describe the relative importance of points along the trailing edge, which enables the matching algorithm to assign higher value to the most distinguishing areas of the fin contours.In cetacean research, identifying individuals observed during a survey is a fundamental part of studying populations.Traditionally, techniques such as tagging or branding are used to make identification easier.Not only do these require specialized equipment and training, but they also tend to be invasive, requiring catch-and-release of wild animals.As a less invasive alternative, photo identification requires no direct contact with the animals. Instead, researchers use images of long-lasting markings, such as nicks, notches, or scars, to track individuals over time <cit.>.Photo identification presents its own challenges. The images showing distinct individuals observed during a study need to be matched against a database containing images of known individuals.For large cetacean populations covering a large geographical area and monitored over a long time period, this can be very time-consuming when done using manual methods.Additionally, parts of the trailing edge required for identification are often occluded by water or viewpoint, requiring the use of multiple images per individual.Even when the trailing edge contours are consistently visible, direct matching between trailing edges from different images is problematic.Because the images are captured “in the wild”, animals occur in a wide variety of poses and are photographed from varying viewpoints.Even for a small change in viewpoint, out-of-plane rotations can lead to difficulty in matching nicks and notches.By presenting likely matches for a query individual to the user in order of similarity, manual identification may be substantially accelerated by reducing (on average) the number of individuals to compare per query. §.§ Time-Warping Sequence Alignment It is possible to treat the curvature representations of the trailing edges as vectors and compute their similarity using any vector norm.The start and endpoints of fins are ambiguous, and pose and viewpoint stretch some sections and foreshorten others, so we need to account for nicks and notches occurring at differing locations along the trailing edge.We use a dynamic programming time-warping algorithm <cit.> that computes the alignment cost of two representations.Rather than a one-to-one matching, as with a vector norm, this allows many points from one representation to match to one point in the other, and vice versa.This warps one representation onto the other, where the alignment cost is the sum of errors for local correspondences <cit.>. §.§ Descriptor Indexing Considering that the nicks and notches used for identification are often sparsely distributed along the trailing edge, and that points in between offer little value for identification, it seems desirable to use only the former when performing an identification.To do this, we compute local extrema in the trailing edge representation, i.e., points corresponding to regions of high curvature in the original trailing edge, and use these as feature keypoints <cit.>.Between these keypoints, we extract feature descriptors, resample them to a fixed length, and normalize using the Euclidean norm.All individuals from the database are stored in a tree-like structure, after which the most likely candidate matches for a given trailing edge are computed using the local naive Bayes nearest neighbor classifier <cit.>. §.§ ContributionsThe primary contributions of this work are that we (a) develop an integral curvature measure to represent trailing edge contours in such a way that the representation is robust to changes in viewpoint and pose that make direct comparison difficult, (b) propose integration of these measures with time-warping alignment and descriptor indexing algorithms as two approaches for ranking potential matches, (c) develop a learning algorithm to weight sections of the trailing edge contour, and (d) produce results using this representation and these algorithms on real-world datasets from active cetacea research groups to confirm their efficacy.§ RELATED WORKThe curvature computed at points along a contour is often used to represent shape information <cit.>.For differential curvature, this is defined as the change of the angle of the normal vector along the length of the contour <cit.>.This representation is sensitive to noise, however, and instead we use integral curvature.A fixed shape is placed at points along the contour, while measuring the area of the intersection of the shape with the contour <cit.>.Using integral curvature to capture shape information is a key part of the Leafsnap system <cit.>, which classifies leaves by using curvature histograms at multiple scales as shape features.We briefly compare to Leafsnap in Section <ref>.In terms of identifying dolphins from their dorsal fins, most notable is DARWIN <cit.>.The key idea behind DARWIN is to account for changes in viewpoint by computing a transformation between two fins to align them, and using the resulting sum of squared distances to define a similarity score <cit.>.As pointed out in <cit.>, a fundamental problem with this approach is that fins from distinct individuals are unlikely to align correctly, with the result that similarity scores cannot be compared reliably to produce a ranking.Another system, Finscan <cit.>, frames the problem of comparing two dorsal fins as a string matching problem <cit.>.In this setting, they compute a low-level string representation of the trailing edge curvature. Because the curvature function defined in terms of derivatives is typically noisy, they refine this representation to a high-level string representation.To compute the similarity between two string representations, they use a linear time-warping algorithm.In both DARWIN and Finscan, the extraction of the trailing edge is a semi-automatic, interactive process where the user manually corrects an initial estimate of the trailing edge.The size of our datasets, as well as our goal of scaling to real-world scenarios, makes comparison with these systems impractical.Instead, we design our representation and matching algorithms to be robust against the occasional inaccurate trailing edge extraction.In <cit.>, the authors propose a trailing edge indexing algorithm and apply it to great white sharks.After defining keypoints for feature extraction by convolving the contour with a Difference-of-Gaussian kernel, they explore the use of both the Difference-of-Gaussian norm and the descriptor from <cit.>.The descriptor indexing algorithm in our work is similar, except that we instead use the curvature representation as a feature descriptor.A notable approach to the more general problem of ranking is the triplet network <cit.>, a natural extension of the Siamese network <cit.>.A triplet network is a neural network that learns a useful representation by minimizing the distance between the representations of instances of the same class, while maximizing the distance between representations from different classes.One key advantage of this approach is that the representation does not need to be designed by hand, rather, it is learned from a large set of labeled training data.The resulting representation is then used to embed query instances into the same space, where a ranking may be computed.We briefly explored the use of triplet networks using the original trailing edge contours as well as the curvature representation, but found that our small datasets would lead to overfitting.Additionally, the parametric nature of these models makes it difficult to apply them to new datasets <cit.>, whereas we confirm in Section <ref> that our approach may be applied unchanged to an unseen dataset of the same species.§ DATASETS AND PREPROCESSING We use two real-world datasets provided by active research groups to evaluate our approach.The first dataset is provided by the Sarasota Dolphin Research Program and is illustrated in Figure <ref>.This dataset contains 10,713 images representing 401 distinct bottlenose dolphins (Tursiops truncatus).Researchers take photos of the dolphins encountered at a particular time and place, and the best images of each individual are separated into encounters.These images are cropped to the dorsal fin and added to the dataset.The second dataset is provided by the Cascadia Research Collective and shown in Figure <ref>.This dataset contains 7,173 images representing 3,572 humpback whales (Megaptera novaeangliae).Unlike the first dataset where each individual appears in multiple images per encounter, here each individual typically appears in only a single image per encounter.Given an image, a fully-convolutional neural network (FCNN) <cit.> outputs the probability that each pixel is part of the trailing edge.Anchor points are computed, and a shortest-path algorithm selects pixels based on costs determined by a combination of the FCNN and image gradients. For dorsal fins, this includes a spatial transformer network <cit.> that transforms the image such that the fin is approximately perpendicular to the image plane.§ INDIVIDUAL IDENTIFICATION§.§ Curvature Representation Given a trailing edge contour represented as an ordered set of coordinates, {(x_1, y_1), (x_2, y_2), …, (x_n, y_n)}, we wish to represent this contour such that it is robust to changes in viewpoint and pose.For this we use an integral curvature measure that captures local shape information at each point along the trailing edge.For a given point (x_i, y_i) that lies on the trailing edge, we place a circle of radius r at the point and find all points on the trailing edge that lie within this circle, i.e., 𝐩_i = {(x_j, y_j) | (x_i - x_j)^2 + (y_i - y_j)^2 ≤ r^2}.To describe a point (x_i, y_i)by its local curvature, we first orient the points 𝐩_i such that 𝐩_i(1) and 𝐩_i(n) lie on a horizontal line. The coordinates of the points in 𝐩_i are then clipped to the dimensions of an axis-aligned square with side length 2r centered at (x_i, y_i).Using the bottom side of this square as the axis for the independent variable, we use trapezoidal integration to approximate the area under the curve defined by the discrete points 𝐩_i.We define the curvature c ∈ [0, 1] at this point as the ratio of the area under the curve to the total area of the square, which implies that the curvature value for a straight line is c = 0.5.See Figure <ref> for an illustration. By computing this integral curvature measure at all points along the trailing edge, we obtain the curvature representation of the trailing edge for a single value of r.To control the extent to which we capture local and global information, we vary the radius of the circle placed at each point by choosing multiple values of r (typically four).The result is a matrix C of dimensions m × n, where m is the number of values of r that we choose and n is the number of points along the trailing edge.The scalar C_ij∈ [0, 1] is the curvature value for the ith value of r at the jth point along the trailing edge. §.§ Ranking We explore two types of methods that use the curvature representation defined in Section <ref> to produce a ranking of known individuals given a query. §.§.§ Sequence AlignmentDynamic Time-Warping. The first method for comparing representations that we explore interprets the curvature representations as temporal sequences and computes an alignment cost between them <cit.>.Given that the start and endpoints of the trailing edges are not only ambiguous, but also often under water, it is desirable to allow some degree of warping when computing correspondences.If we define 𝐜_i and 𝐜_j' as the ith and jth columns (each a vector representing the curvature values at a point) of two curvature representations C and C' of lengths m and m', respectively, then the total alignment cost c(m, m') is defined recursively as c(i, j)= d(𝐜_i, 𝐜_j') + min{c(i-1, j), c(i, j-1), c(i-1, j-1)},where d defines the distance between two points based on their curvature values at multiple scales.It is possible to use a simple vector norm, such as d(𝐜_i, 𝐜_j') = ||𝐜_i-𝐜_j'||_2.Spatial Weights. The definition above, however, treats the contribution from correspondences along the entire length of the trailing edge as equal.We know that the end of the trailing edge is often underwater for dorsal fins, and the tips often cropped for flukes.Ideally, we would thus like corresponding points from these unstable regions to contribute less towards the total alignment cost.To realize the above, we define a weight vector 𝐰, where the elements w_1, w_2, …, w_n describe the relative importance of each point along the trailing edge.In doing so, we define a more meaningful distance function asd(𝐜_i, 𝐜_j' | 𝐰) = w_iw_j||𝐜_i - 𝐜_j'||_2,where the product w_iw_j scales the contribution of each correspondence to the total alignment cost based on the relative importance of the points.Learning the Weights. To determine suitable values for the elements of 𝐰, we frame the problem as an unconstrained optimization problem where we maximize the top-k score (the fraction of times the correct individual appears in the first k entries of the ranking) over a training set.For the training set we use the images in the database, and take the images from a single encounter for each individual to be used as queries.The separation of images into database and queries is described in Section <ref>.To avoid overfitting, rather than learn all the elements of 𝐰, we reduce the number of parameters by expressing 𝐰 as a linear combination of the Bernstein polynomials <cit.> of degree n evaluated at uniformly spaced points between 0 and 1.The linear combination of Bernstein polynomials determined by coefficients 𝐜 is defined asB_n(x) = ∑_i=0^nc_ib_i,n(x),whereb_i,n(x) = n ix^i(1 - x)^n - i, i = 0, 1, …, n.These polynomials have two particular properties that are desirable for our application, namely that (a) they are positive between 0 and 1, i.e., b_i,n(x) ≥ 0 for x ∈ [0, 1], and (b) they form a partition of unity, i.e., ∑_i=0^nb_i,n(x) = 1.We use the latter to initialize the search at 𝐜 = 1, which leads to a uniform 𝐰.We set n = 10 and optimize for the coefficients 𝐜 using an open-source package for Bayesian optimization <cit.>. These coefficients 𝐜 define a polynomial f(x|𝐜) on the interval x ∈ [0, 1].After defining n uniformly-spaced points x_1, x_2, …, x_n on this interval, where n is the number of points on the trailing edge contour, we compute the entries in the spatial weight vector as w_i = f(x_i|𝐜). This leads to the weight vectors shown in Figure <ref> for the Bottlenose and Humpback datasets. §.§.§ LNBNN ClassificationSimilar to <cit.>, we use the local naive Bayes nearest neighbor (LNBNN) algorithm <cit.> to produce a ranking of known individuals <cit.>.Our work differs from <cit.> in that we use the integral curvature representation both to compute descriptors and to determine keypoints.Feature Descriptor. Instead of using the Difference-of-Gaussian norm defined in Equation 1 in <cit.> to encode local shape information, we use the curvature representation defined in Section <ref> of this work. Subsections of the curvature representation are resampled to a fixed length, normalized by the Euclidean norm, and used as feature descriptors.Feature Keypoints. Similar to <cit.>, we choose the keypoints between which to define the subsections mentioned above by resampling the curvature representation to a fixed length and choosing as keypoints the n - 2 largest local extrema of the curvature representation at each scale as well as the start and endpoints.Combinations of these keypoints yields n 2 subsections per scale, between which we extract the corresponding values from the curvature representation.LNBNN Classification. These feature descriptors are computed for all known individuals in the database, and placed in a data structure for approximate nearest neighbors using ANNOY <cit.>.We compute a score for each individual using LNBNN classification, specifically Algorithm 2 as defined in <cit.>. The benefit of using LNBNN instead of standard approximate nearest neighbors is that it considers not only the distance to the nearest descriptor from a given individual, but also the distance to the nearest descriptor from a different individual <cit.>.The difference between these is used to update the score, whichreduces the contribution from non-distinctive feature descriptors.§ EXPERIMENTS To demonstrate the effectiveness of our curvature representation and matching algorithms, we evaluate the two approaches from Section <ref> for producing a ranking of known individuals on the Bottlenose and Humpback datasets.We use the top-k score, defined as the fraction of the time the correct individual appears in the first k entries of the ranking, to evaluate time-warping (with and without learned spatial weights), as well as LNBNN using our curvature descriptor and the Difference-of-Gaussian descriptor <cit.>.In particular, we show that when using top-1 accuracy, the curvature representation outperforms the Difference-of-Gaussian descriptor by 95% to 91% on the Bottlenose dataset, and 80% to 40% on the Humpback dataset. To compare against Leafsnap <cit.>, we also construct the Histogram of Curvature over Scale from our integral curvature representation, and produce a ranking using the histogram intersection distance.We were unable to achieve good results with this, however, and we suspect that the reason is that computing a histogram over the curvature representation loses the spatial information of the nicks and notches necessary for individual identification.Additionally, because humpback whales often have uniquely identifying patterns of scarring and pigmentation on their flukes, we also compare against HotSpotter <cit.>, which uses LNBNN with SIFT descriptors <cit.>.The texture information captured by HotSpotter is complementary to the curvature of the trailing edge, and so we also evaluate combining our identification algorithms with HotSpotter.We run experiments identical to those described in the following section on two related datasets to ensure the generality of our approach. §.§ Defining Queries Bottlenose Dolphins.We randomly select m = 10 encounters for each individual, and use all the images from these encounters for the database.When an individual appears in only n encounters such that m > n, we use n-1 encounters for the database so that we have at least one query.The images from remaining encounters are used as query encounters.We also investigate the effect of varying m on the top-k accuracy in Section <ref>.Humpback Whales. The Humpback dataset typically contains only a single image per individual in each encounter and two encounters per individual.In practice, this means that most individuals are represented by one image in the database.When evaluating time-warping and HotSpotter, we use the minimum alignment cost across images in the encounter as the similarity score.For LNBNN, we stack the descriptors from all images in the encounter to build the query.We run all experiments on five random splits and report mean scores, however, there is little variance across runs. §.§ Qualitative ResultsBefore quantitatively evaluating our algorithms, we show successful and unsuccessful identifications for the Bottlenose and Humpback datasets in Figures <ref> and <ref>, respectively. In each figure, we show the pair of images (query and database) that contributes the most to the total score.We plot a minimal subset of matches such that the sum of the LNBNN scores from these matches is at least half the total score.Although the matches shown are sparse, in practice the entire length of the contour is matched, albeit with lower scores.Pairs of lines of the same color indicate the start and endpoints of the trailing edge corresponding to the matched curvature descriptor.The matches are ordered such that the strongest match is shown in red, and the weakest in purple.The sections of the trailing edge not covered by strong LNBNN matches is shown in blue.There are two main causes of misidentifications, namely (a) errors in the contour extraction that cause distinguishing features to be poorly represented in the curvature vectors, and (b)distinctions between very smooth trailing edges that are insufficiently valued by the matching algorithm.Both are amplified by significant viewpoint differences between database and query trailing edges for the correct match. §.§ Ranking Performance To evaluate ranking performance, we compare two variations of each of the algorithms from Section <ref>.Next, we describe the parameter choices for each of these algorithms.Time-Warping Alignment. When using learned spatial weights, we use the relevant 𝐰 as shown in Figure <ref> for the Bottlenose and Humpback datasets.For the Bottlenose dataset, we resample the curvature representation to 128 points and set the Sakoe-Chiba bound <cit.> in the dynamic time-warping algorithm to 8. We use scales of {0.04, 0.06, 0.08, 0.10}.For the Humpback dataset, these are set to 748, 75, and {0.02, 0.04, 0.06, 0.08}, respectively.For both datasets, we determine the radii of the circles used for integral curvature (Figure <ref>) by multiplying the scales by the maximum dimension of the fin, specifically, the height for dorsal fins, and the width for flukes.Descriptor Indexing. When doing LNBNN classification, we set the number of keypoints at which we define contour subsections to 32.The keypoints are placed along the trailing edge (which is resampled to 1024 points) at the points corresponding to the local extrema of the curvature or Difference-of-Gaussian representation, as appropriate. We set the dimension of the feature descriptors to 32.The top-k accuracy plateaus or slightly degrades on both datasets for larger dimensions.We speculate that this is due to the noisy nature of our extracted trailing edges, where resampling to a smaller feature dimension acts as a form of smoothing.The scales for the Difference-of-Gaussian descriptorsare the same as described in <cit.>.The results for the Bottlenose and Humpback datasets are shown in Figures <ref> and <ref>, respectively.With LNBNN, the curvature descriptor outperforms the Difference-of-Gaussian descriptor for both datasets.We argue that this is because of the robustness with which integral curvature can capture noisy local information — with a sufficiently large number of points representing the trailing edge, the exact coordinates of any single point have little effect on the curvature value.Evaluation. The relative performance of the two matching algorithms is different for the two datasets —it is likely that this is because of how the identifying information is distributed.For the Bottlenose dataset, where only a few distinct marks are useful for identification, the descriptor indexing approach performs better.For the Humpback dataset, where the information necessary for identification is spread along the entire length of the trailing edge, the time-warping alignment approach achieves similar results.We repeat these experiments on two smaller datasets to determine if our approach generalizes to datasets of the same (or similar) species.On a common dolphin dataset with 3744 images representing 186 individuals, we achieve 74% top-1 accuracy using time-warping and 69% using LNBNN, and on a humpback whale dataset with 1388 images representing 419 individuals, we achieve 86% top-1 accuracy using time-warping and 89% using LNBNN. §.§ Number of Encounters per Known Individual While we choose the encounters from the Bottlenose dataset randomly to evaluate our algorithms, researchers may wish to choose the best images of each known individual for the database.Figure <ref> shows the effect of increasing the number of encounters.Adding more encounters, and hence more images, to the database has several advantages. First, there are more viewpoints represented, which makes it more likely that a given query aligns well with one from the database.Second, more images per individual acts as insurance against the event where images may have distinctive parts of the trailing edge occluded.Choosing database images to maximize the information content for identification is a problem we intend to address in future work. §.§ Error Correlation In addition to evaluating the ranking performance of each algorithm separately, we also explore the possibility of combining algorithms.If the correct individual appears in the top-k entries of either of the algorithms, we consider the match to be correct.Combining our algorithms with HotSpotter <cit.> is of particular interest to us, because of their complementary nature — while our matching algorithms use the integral curvature of the trailing edge, HotSpotter uses SIFT <cit.> descriptors extracted from the interior of the fluke to describe the unique patterns of pigmentation. Figure <ref> shows that augmenting time-warping (TW+SW) with HotSpotter (HS) improves the top-1 accuracy from 80% to 89%, and augmenting LNBNN with HotSpotter improves the top-1 accuracy from 79% to 88%.§ CONCLUSIONWe introduced novel combinations of integral curvature representation and two matching algorithms for identifying individual cetaceans from their fins.This representation captures the local pattern of nicks and notches in such a way that they may be compared using either a time-warping algorithm or descriptor indexing.The effectiveness of our method is shown by computing accuracy scores on two real-world datasets, each with distinct challenges.For the Bottlenose dataset, with very little information per image,descriptor indexing outperforms time-warping because it considers not only the feature distance, but also distinctiveness.For the Humpback dataset, there are few images per individual, but many features per image.The time-warping algorithm is well-suited for this problem, because it preserves the spatial integrity of curvature along the trailing edge while exploiting learned spatial weights to emphasize matches from regions of stable curvature. As a result, the performance of the two algorithms is similar.In both cases, we demonstrate that we achieve results that can greatly accelerate the process of cetacean identification.While the focus of this paper has not been on the details of the contour extraction, a major focus of future work will be a unified algorithm that works on both species and leads to a generalization that allows rapid adaptation to new species. An important consideration will be to restrict the amount of manually-generated training data required. ieee | http://arxiv.org/abs/1708.07785v1 | {
"authors": [
"Hendrik J. Weideman",
"Zachary M. Jablons",
"Jason Holmberg",
"Kiirsten Flynn",
"John Calambokidis",
"Reny B. Tyson",
"Jason B. Allen",
"Randall S. Wells",
"Krista Hupman",
"Kim Urian",
"Charles V. Stewart"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170825154446",
"title": "Integral Curvature Representation and Matching Algorithms for Identification of Dolphins and Whales"
} |
AcceptedAsymptotic Stability ofthe Landau–Lifshitz Equation Amenda Chow August 24, 2017 ===================================================== We present new spectroscopy of the z=3.62 gravitationally lensed quasar B1422+117 from the Gemini North GMOS integral field spectrograph. We observe significant differential magnifications between the broad emission lines and the continuum, as well as across the velocity structure of the Lyman-α line. We take advantage of this differential microlensing to algebraically decompose the quasar spectrum into the absorbed broad emission line and absorbed continuum components. We use the latter to derive the intrinsic Lyα forest absorption spectrum. The proximity effect is clearly detected, with a proximity zone edge of 8.6–17.3 Mpc from the quasar, implying (perhaps intermittent) activity over at least 28 Myrs. The Lyα line profile exhibits a blue excess that is inconsistent with a symmetric fit to the unabsorbed red side. This has important implications for the use of this fitting technique in estimating the absorbed blue Lyα wings of Gunn-Peterson trough quasars. quasars: individual (B1422+231) – quasars: absorption lines – quasars: emission lines – gravitational lensing: strong – dark ages, reionization, first stars § INTRODUCTION The far ultraviolet (FUV) emission of luminous quasars is of great importance to our understanding of the evolution of the high redshift universe. The first quasars turned on in an intergalactic medium (IGM) of neutral hydrogen and may have competed with the first stars to reionize the universe <cit.>. Those same quasars are now our most powerful probe of the epoch of reionization; Gunn-Peterson (GP) troughs in the spectra of z≳ 6 quasars provide constraints on the residual neutral fraction at tail end of this process <cit.>, and the size of quasars' ionization bubble, or proximity region, probes the IGM and provides a measure of quasars' energetic contribution to its evolution <cit.>. The evolving state of the IGM at later times is then traced by the Lyman-α (Lyα) forest in therest-frame FUV spectra of quasars at z>1 (See Weinberg et al. 2003 for a review). Although a wealth of information can be extracted from the absorption of quasars' intrinsic FUV spectrum, this same absorption means that this wavelength range is very poorly characterized. An improved understanding of the intrinsic FUV emission of luminous quasars is sorely needed.Efforts to characterize the FUV spectrum of quasars are successful to a point; the use of spectroscopic samples to produce composite spectra <cit.> or principle component analysis <cit.> have given us insights into the true FUV continuum in the absorbed Lyman-α forest region. The Lyα broad emission line (BEL) itself is a greater challenge due to the wide range observed in BEL profile shapes, however progress has been made in understanding the systematics in fitting based on the unabsorbed Lyα red wing <cit.> and in reconstructing the line profile based on spectral correlations in other lines <cit.>. However due to the ubiquitous absorption in the blue wing of the Lyα line at high redshifts, there remains a fundamental uncertainty in the range of possible Lyα emission line profile shapes in high-z quasars.Gravitational lensing offers the potential to extract the elusive intrinsic FUV spectrum of high-z quasars. When a strongly-lensed quasar exhibits differential magnifications due to microlensing, it becomes possible to algebraically decompose the spectral contributions of different physical regions within the quasar <cit.>. This method has been used to extract intrinsic features within quasar spectra corresponding to particular physical processes and/or size-scales. It has also been used to extract the intrinsic broad absorption profile in the BAL quasar H1413+117 <cit.>. In this paper we apply the technique to Gemini GMOS integral field spectroscopy of the z=3.62 quasar B1422+231. We derive the intrinsic Lyman forest absorption profile along the line of sight and constrain the intrinsic Lyα line shape and FUV spectrum of this quasar. In Section <ref> we describe the observations and extraction of lensed image spectra. In Section <ref> we analyse the microlensing signatures in this new epoch of data. In Section <ref> we decompose the spectra in terms of the broad emission line region and continuum, emphasizing the physical interpretation of the decomposed regions, and we derive the intrinsic Lyman forest absorption profile. In Section <ref> we discuss the derived absorption spectrum and Lyα line shape. In Section <ref> we summarize our conclusions. § OBSERVATIONS AND SPECTRAL EXTRACTIONB1422+231 was observed on the 27th of June 2006 with the GMOS Integral Field Unit (IFU) <cit.> on the Gemini North telescope. Five 15 minute exposures were taken in one-slit mode using the B600 grating with a spectral range of 3975Å to 6575Å and a resolution of R=1688. At B1422+231's redshift of 3.62, this covers the Ovi+Lyβ through Ciii] broad emission lines. Seeing conditions were excellent, with a PSF FWHM of 06. §.§ Data Reduction and Spectral Exraction The data was reduced using the Gemini IRAF package v1.10, under IRAF v2.14.1. See <cit.> for details of the reduction of this dataset, including the correction for scattered light. Figure <ref> shows the processed data cube collapsed into a 2-D image. The excellent seeing allows us to extract the spectra of lensed images C and D with high accuracy, however images A and B, with their separation of 05 require careful deblending. <cit.> describes the method used for the extraction of spectra for each lensed image, including deblending of close image pairs.Images A, B, and C are distinct. Image D is faint but confidently detected, however it is not visible in this grey-scale stretch. Images A and B are somewhat blended, but the cores of their PSFs are resolved. Figure <ref> (upper panel) shows the extracted spectra of the four quasar images.We see distinct broad emission lines for OVI+Lyβ, Lyα, NV, OI, CII, SiIV/OIV], and CIV. § MICROLENSING SIGNATURESGravitational lensing is achromatic, and so in the absence of microlensing, differential extinction, or time-dependent spectral variation, the spectra of lensed images of a multiply-imaged quasar are identical modulo a scaling factor. Microlensing is typified by a difference in the magnification of one or more lensed images compared to the time-averaged magnification for that lensed image (which we will refer to as the macrolens magnification, or just macro-magnification).Differential microlensing is characterized by magnifications that vary between different emission regions, resulting in a change in the shape of the spectrum of the affected lensed image. Differential microlensing is sensitive to emission region size and projected location. It is frequently observed between BELs and the continuum, and often shows wavelength dependence within these features.Figure <ref> (lower panel) shows the wavelength-dependent flux ratios of lensed image A with respect to B and C, and of C with respect to B. In the absence of microlensing we expect these to be flat. The A/C ratio is indeed relatively flat, although a constant slope is apparent that may result from differential microlensing across the continuum, however this is unlikely as we would then expect features in the ratio at the locations of the BELs, yet these are minimal. A more likely explanation is differential extinction, which has a smooth wavelength dependence. In A/B and C/B there are strong features at the locations of all BELs, indicating differential microlensing between the broad emission line region and the continuum. The relative absence of such features in the A/C ratio implies that lensed image B is experiencing this differential microlensing. Figure <ref> shows the spectra of lensed images A and B in which both are scaled to have matching flux in the Lyα line core (rest frame 1216±20Å). The continuum in B appears to be experiencing a lower magnification relative to the BELs. The BELs themselves also exhibit differential magnifications in lensed image B, with the wings of the lines at lower magnification relative to the cores. We discuss this in detail in <ref>.Differential extinction between lensed images will not affect the continuum-to-BEL ratios as our measurement of the continuum strength is made at the relevant BEL wavelength range (based on power-law fits; Sect <ref>), nor will it significantly affect broad line profiles. Differential extinction only affects the continuum slope and ratios between BELs, which are not a focus of his work.In Section <ref> we use the difference in continuum-to-BEL ratios between lensed images to decompose the spectrum into intrinsic components. It's important to note that for the purpose of the decomposition of spectral components, it does not matter why this ratio varies between images. The decomposition is equally valid whether these differences are due to microlensing or intrinsic variability, as long as wavelength-dependent variations within the BEL and continuum components can be accounted for.§.§ Flux RatiosTo inspect differential microlensing properties of B1422+231 we first obtain an estimate of the unabsorbed continuum spectrum for each lensed image. In the region redward of Lyα, this can be done by fitting a power-law to the regions that are relatively uncontaminated by BELs. Fitting F_ν∝ν^α in the windows 1280–1290Å, 1350–1375Å, and 1440–1460Å yields α = -0.8±0.3. This is similar to the mean 1200–2000Å power-law index of α = -0.83 for z<1.5 AGNs measured by <cit.>. The best fit single power-law continuum is shown in figure <ref>. It is apparent that the best power-law fit redward of Lyα substantially overshoots the peaks of the Lyα forest. This may be due to from line blanketing, which at z>3 can be significant (Faucher-Giguére et al. 2008, hereafter FG08)and/or a turnover in the extreme UV (EUV) spectrum. Although quasar spectra often exhibit EUV turnovers, these are typically gradual and blueward of ∼1000–1100Å <cit.>. We can estimate the contribution of these effects by correcting the observed EUV spectrum for the average IGM absorption.To estimate the latter, we use the 2<z<4 average IGM opacity measurement of FG08. We calculate the expected continuum level in the rest-frame 1090-1150Å range, which is reasonably uncontaminated by BELs. The uncertainty in our unabsorbed continuum estimate comes from the uncertainty in the FG08 fits and the standard error of the mean in the smoothed Lyα forest. Figure <ref> shows the derived absorption-corrected spectrum in the 1090-1150Å range. The corrected EUV continuum remains significantly below the extrapolated single-index continuum. This may indicate that B1422+231 exhibits an unusually long-wavelength spectra index break, or that our estimate of line blanketing is low. To study the full range of possible continuum shapes we perform our analysis assuming both an unbroken power-law and a broken power-law fit to the opacity-corrected EUV continuum. For the latter we stitch the above α = -0.8±0.3 power law redward of Lyα to a power-law fit at 1090-1150Å, which yields α = -2.0±0.5. This is steep compared to the mean 500—1000Å power-law index of α = -1.41 measured by <cit.>, however it is within the range of EUV slopes found in that study. Additional free parameters in deriving the broken power-law continuum were the width of a Gaussian smoothing function over the break region and the break wavelength. These latter have very little effect on the resulting analysis. Figure <ref> shows the best-fit broken power-law. We use both the unbroken and broken power-law continua spectra to explore the range of spectral decompositions in Section <ref>. Table <ref> gives the flux ratios for the Lyα, OVI+Lyβ, OI, OII, OIV]+SiIV, and CIVemission lines and for the continuum. For all line ratios but Lyα, integrated, continuum-subtracted line fluxes are calculated over windows centered on rest-frame emission line wavelengths with widths given in Table <ref> . For Lyα we calculate the flux ratio in the 10Å window redward of rest-frame 1216Å and blueward of significant NV emission. Figure <ref> shows the continuum-subtracted Lyα line for lensed images A and B, along with the A/B line ratio taken after continuum subtraction. This ratio reveals clear differential microlensing across the velocity structure of the Lyα line. The line center has a microlensed magnfication ratio that is closer to the continuum ratio (also shown) compared to the red and blue wings. The NV line shows a similar behavior. Figure <ref> shows the continuum-subtracted spectrum spanning the remaining BEL species OI, CII, SiIV, and CIV, including the continuum-subtracted A/B ratio. The range of uncertainties n this ratio are also shown in Fig. <ref>, based on propagating the uncertainties due to shot noise and the continuum fit.A naive interpretation of these relative magnification ratios suggests that the low-projected-velocity core of the Lyα line may be closer in scale to the continuum emission than the higher velocity wings. Such an increase in BELR velocity with increasing size crudely suggests an accelerating outflow, and that we are seeing little orbital motion towards the base of this flow.However we must be cautious before drawing conclusions about the kinematics of the Lyα line. Magnification ratio does not always tend towards the macromodel ratio monotonically with increasing source size in the event of size-dependent differential microlensing. This is demonstrated in figure <ref>, which shows a random selection of A/B magnification ratios simulated for B1422+231 for source sizes between 0.1–2 Einstein Radii. In this simulation, Gaussian surface brightness profiles over this size range are convolved with simulated microlensing magnification maps produced as part of the GERLUMPH project <cit.>. We use the B1422+231 lens macro-model of <cit.> with an 80% smooth matter percentage, consistent with the values estimated in other lenses where the line of sight does not pass through the galactic core <cit.>. Tracks show how the expected magnification ratios change with source size for a given set of locations for images A & B on the simulated microlensing magnification maps. For a number of tracks our observed magnification ratios are consistent with the high velocity wings of Lyα arising from a smaller size scale than the low velocity core. § SPECTRAL DECOMPOSITIONGravitational microlensing grants the ability to decompose quasar spectra, separating components that are subject to different levels of microlensing.The basic approach is to express the spectra of two lensed images as linear combinations of differently-magnified components. Theseequations can then be solved for an unknown spectral component. The technique has been used to decompose microlensed from non-microlensedcomponents (e.g. the Fμ method — ), as well as extractwhat can be interpreted as intrinsic broad line, continuum components, and absorption features <cit.>. In <cit.> we describe our approach to spectral decomposition, and we refer the reader there for more detail. In short, we express the observed spectrum as a linear combination of an intrinsic continuum spectrum, F_C(λ), and broad emission line spectrum, F_L(λ). Each is subject to its own magnification factor, m_C(λ) and m_L(λ), which describe both macro- and microlensing magnifications, and which may or may not have wavelength dependence. These factors may also encompass any other differential factors between these spectral components, for example due to intrinsicvariability. We also describe a separate absorption factor for continuum and BEL, A_C(λ) and A_L(λ).In <cit.>this absorption component described the intrinsic broad absorption features in H1413+117. In the case of B1422+231 they primarily describe the Lyman forest absorption. The spectra for images A and B then become: F_A =m_C,A A_CF_C +m_L,A A_LF_LF_B =m_C,B A_CF_C +m_L,B A_LF_L Bold symbols represent potential wavelength dependence. The only known quantities in these equations are the observed spectra, F_A and F_B, however it is possible to estimate the magnified continuum spectrum for each lensed image (m_CF_C;see Sect. <ref>) and the continuum and BEL flux ratios, which we define as η_C =m_C,A/ m_C,B and η_L = m_L,A/m_L,B. We can then solve for the absorbed, magnified continuum 𝒞 and BEL ℒ spectrum for either lensed image: ℒ_A =m_L,A A_LF_L = η_L/η_L - η_C ( F_A - η_CF_B)𝒞_A =m_C,A A_CF_C = η_C/η_C - η_L ( F_A - η_LF_B) We take η_C to be the wavelength-dependent continuum ratio from the fits described in Section <ref>. As seen in Sect.<ref>, η_L is also wavelength dependent for Lyα, and so we use the continuum subtracted line ratio for this component.Figure <ref> shows the ℒ_A and 𝒞_A components derived in the region of the Lyα line from equations <ref> and <ref> assuming the best-fit continua (both broken and unbroken power-law fits). We note that the decompositions assuming an unbroken power-law fit for the continuum produce significantly negative fluxes in the line component below rest-frame ∼1160Å, and below 1145Å results in extreme fluctuations in both decompositions. We take this as an indication that there is at least some spectal index turnover in the continuum at or above ∼1160Å.§ ESTIMATING THE INTRINSIC ABSORPTION AND LINE PROFILESFollowing reionization, quasars continued to contribute a significant part of the energy budget of the IGM. All quasars at z>1 are surrounded by a region of diminished opacity due to their ionization of residual neutral hydrogen clouds in the IGM. The size-scale of this proximity zone is typically taken to be the distance from the quasar at which the quasar's ionizing flux equals the metagalactic background <cit.>. However, in practice, the duration of a quasar's activity also plays a part, and defines the far edge of the ionization front. Measuring the exact size and shape of proximity region absorption is very challenging because it overlaps the quasar's own Lyα broad emission line, and the profile of the Lyα blue wing in these luminous high-z quasars has never been observed due to this same absorption. §.§ The Intrinsic Lyman-α ForestThe derived continuum component reveals the intrinsic shape of the Lyα absorption spectrum(Fig. <ref>, lower). By dividing 𝒞_A by the continuum spectrum fit for lensed image A, m_C,A F_C we recover A_C,A, the transmission function experienced by the continuum light traveling the line-of-sight path of lensed image A (A_C,A), as shown in Figure <ref>. Certain features are clearly independent of these uncertainties:1) The onset of Lyman forest absorption is coincident with the Lyα line center, indicating no significant infalling HI contributing to the absorption.2) A region of enhanced transmission in the vicinity of the quasar is clear, and presumably results from the line-of-sight (LOS) proximity effect within the quasar's Strömgren sphere. Transmission drops off blueward of the resonant Lyα wavelength, reaching the IGM average in the observed-frame wavelength range ∼5560—5500Å. This corresponds resonant Lyα absorption at z∼3.57—3.52, or a proper distance of 8.6–17.3 Mpc assuming H_0 = 70 km s^-1Mpc^-1, Ω_M = 0.3 and Ω_Λ = 0.7. Note that this proper distance corresponds to the observable edge of the proximity zone. This is distinct from the characteristic size of this zone, typically defined as the distance at which the ionizing flux of the quasar is equal to metagalactic ionizing flux. Instead, we estimate the distance at which the opacity approaches the average IGM value at that redshift, corresponding to the point at which the ionization rate due to the quasar is small compared to that due to the metagalactic background flux. From light travel time arguments alone, this radius may be taken as a lower limit on the timescale of the quasar's activity. As such, B1422+231 appears to have been luminous, at least intermittently, for more than 28 Myrs. This activity timescale is comparible to the >34 Myr timescale for the z=3.286 quasar Q0302-003 based on its HeII proximity zone <cit.>. These authors also determine that an integrated activity period of as little as 0.17 Myrs is sufficient to ionize this zone, implying intermittent activity and a small duty cycle. To illustrate how an intermittently luminous quasar can produce a proximity effect comparible to that observed for B1422+231, we have generated a set of simulated absorption profiles using the C15 hydrodynamic simulation of <cit.>. This was scaled to reproduce the mean opacity of FG10, and includes the ionizing effect of continuously shining quasar at z=3.62 with the average quasar spectrum of <cit.> (James Bolton, private communication). B1422+231's unlensed absolute magnitude is calculated assuming α=-0.8, m_V= 16.4—16.7 for lensed image A (<cit.>; CASTLES survey[https://www.cfa.harvard.edu/castles]), and a lensing magnification of μ_A∼ 5—10; <cit.>. B1422+231 is extremely luminous, with M_V ∼ 28.6 to 29.6. To approximate the effect of intermittent activity with a duty cycle of 0.1 to 0.2, we simulate the absorption profile using a continuously-emitting quasar with M_V=-26.8, so 5 to 10 times fainter than B1422+231. Figure <ref> shows the average of 50 simulated sight-lines towards the quasar. It can be seen that this power output results in a region of decreased opacity with a similar size to the proximity zone of B1422+231. Note that this simulation is for illustrative purposes only, and is not intented to provide constraints on any model parameters. Nonetheless, the duty cycle needn't be higher than 0.1 to produce the observed proximity effect.3) Outside the proximity region the transmission spectrum rises relatively monotonically towards lower redshifts. This is as expected, and is reasonably consistent with the opacity-redshift relation of FG08. Note that this derived absorption profile uses a single opacity from FG08 (the average in the z∼3.15—3.35 range) as an input parameter in our estimate of the intrinsic UV continuum. However this is effectively just a scaling factor; the monotonic increase in transmission is a result of the decomposition and a feature of the data. The absorption profile derived assuming a broken power-law continuumis consistent with the simulated profile, while the profile derived from the unbroken power-law rises alittle too quickly. §.§ The Intrinsic Lyman-α Emission Line Profile Figure <ref> shows the derived broad emission line component over the range of the Lyα line. It's important to remember that this is component is the combination m_L,A A_L F_L: the intrinsic line profile (F_L) as affected by Lyα forest absorption over the footprint of the broad line region (A_L) and the size-dependent magnification factor for Lyα at lensed image A (m_L,A). However, even with these convoluting influences, we can derive useful contraints on the intrinsic line profile.The line connecting the peaks of the Lyα forest can be taken as a minimum for the intrinsic flux of the blue wing of the Lyα line; i.e. assuming no absorption in the regions corresponding to these peaks. In Section <ref> we saw that, based on the IGM opacity measurements of FG08, some absorption is likely even at the Lyα forest peaks for the region blueward of the Lyα line (Fig. <ref>). Average opacity, and also opacity at the forest peaks will diminish approaching the Lyα line center due to a decreasing neutral fraction inside the quasar's ionization front. We use the formalism of <cit.> to describe the relationship between opacity and distance from the quasar, assuming the range of proximity zone edge sizes (8.6—17.3 Mpc) measured in Section <ref>, and assuming that the measured edge size corresponds to the point at which opacity is within 10% of the average IGM forest opacity. This proximity region opacity is scaled to converge with the IGM opacity in the regions corresponding to the Lyα forest peaks. In this way we can correct the upper envelope of the Lyα forest to produce an “intrinsic” Lyα line profile, also shown in Figure <ref>.The shaded areas surrounding this “intrinsic” line profile and the Lyα forest peaks show the range of possible reconstructed line profiles based on propagating the uncertainties in the continuum fit (Sect. <ref>) through the decomposition. These include: uncertainties in spectral indices, normalization, continuum break wavelength, and the width of the Gaussian smoothing kernel across that break. Note that these fit uncertainties incorporate the uncertainty in the absorption correction for the FUV continuum. The uncertainty in the continuum normalization dominates over most of the blue wing of the line profile. The uncertainty in the absorption-corrected profile (red region) includes the range of proximity zone sizes measured in Section <ref>.§.§.§ Lyα Blue Excess and Implications for Reionization Studies The derived Lyα component enables us to place the first strong constraints on the intrinsic blue wing of the Lyα line profile in a luminous, high-z quasar. Understanding the Lyα profile, and in particular its potential asymmetries, is critical for reionization studies. Tracking the final stages of reionization depends heavily on measuring the depths of Gunn-Peterson (GP) troughs <cit.>, GP damping wings <cit.>, and proximity region sizes for quasars at z≳6. For these measurements the blue wing of the Lyα line must be inferred, and its intrinsic shape is a key uncertainty in derived neutral fractions. A common approach is to fit an analytic profile to the unabsorbed red wing, for example Voigt profiles <cit.>, composite Gaussians <cit.>, while a recent more sophisticated approach by <cit.> tunes a composite Gaussian to variations in longer wavelength lines. All of these approaches assume symmetry in the Lyα line profile. Another other common approach is to construct composite spectra from lower redshift quasars with unabsorbed Lyα lines <cit.>, however the largely z≲ 1 quasars necessarily used for these composites are for the most part very different in power to the extremely luminous quasars observed at z≳ 6.Lyα line profiles in low-to-medium redshift quasars do indeed exhibit asymmetries. <cit.> (KH09) analyzed Lyα profiles from archival HST spectroscopy of 78 quasars at z≲ 1.5, finding that Gaussian-fitted narrow (broad) line component have median blueshifts relative to the Lyα rest frame of ∼300 (∼400 km/s) km/s. KH09 analyze the effect of Lyα blue excess on reionization studies; they simulate z>6 GP troughs in their low-z sample and attempt to extract neutral fractions by fitting symmetric Lyα line profiles, with line centers based on the metal lines. They find that neutral fraction would be systematically underestimated if high-z quasars had similar Lyα profile asymmetries as are observed at low z. While this sample may not be representative of the typically much more luminous quasars used as reionization probes, the result advocates caution in how neutral fraction measurements are interpreted. This result also motivates us to determine whether luminous, high-z quasars have similarly asymmetric Lyα to their lower luminosity counterparts at low z.At first glance, the derived Lyα line profile for H1413+117 is inded asymmetric around the Lyα rest wavelength; it appears to have some blue excess based on the flux at the Lyα forest peaks.We assess this excess by fitting a symmetric function to the red wing of the derived profile at 1216Å <λ < 1250Å. We deliberately “over-fit” with three independent Gaussians at the Lyα line center and two Gaussians at NV. Our aim is to assess the implication of symmetric reflection of the red wing in the case of an excellent fit,and in the case of this Lyα profile we don't get an adequate fit with a double Gaussian. Note, however, that any fitted symmetric function will show a blue excess. At first we fix centers to systemic rest-frame wavelengths and fit normalizations as free parameters. The resulting best fit composite Gaussian is shown in Figure <ref>. The fit falls significantly below the peaks of the Lyman-α forest in the blue wing of the Lyα line. To check whether a blue-shifted central Lyα line wavelength can reproduce the apparent blue excess without a line asymmetry, we attempt the same fit with central wavelength also a free parameter. The fits are poorly constrained, but in general it is impossible to reproduce the blue excess in the high-velocity broad component without significantly overshooting flux in the narrow component core. This would imply an opacity that increases close to the quasar, yet we expect and only ever observe the reverse of this. A typical fit with a blue-shifted line center is also shown in Figure <ref>.We conclude that a symmetric fit falls significantly below the true line flux in the blue wing. The implied blue excess may result from gas kinematics (outflow) or contributions from other emission lines such as SiIII (1206Å), SiII (1190, 1193Å),and CIII (1175Å).The lower panel of Figure <ref> shows the fractional residuals, comparing the best-fit composite Gaussian (fixed Lyα wavelength) to two cases: 1) the envelope of the Lyα forest line peaks (the minimum blue-wing flux), and 2) the absorption corrected blue wing. The mismatch rises to at least ∼33% from line center to 10Å from the core, and the true line flux is anywhere between this and 60% higher than the fit across the blue wing. If a similar blue excess were present in a z≳ 6 GP trough quasar then it would lead to a significant underestimate of the neutral fraction. This underestimate would be most significant for neutral fraction measures that use the GP trough itself. A measurement using the GP damping wing would be less affected as this probes much closer to the Lyα core; typically within rest-frame 10Å <cit.>, where the blue excess in B1422+231 is less than 20%. Combined with the results of KH09, this clear Lyα blue excess in a powerful, high-z quasar suggests that neutral fractions may be systematically underestimated in many studies. As KH09 point out, in studies affected by this bias we may interpret any detection of a significant neutral fraction even more confidently.§ CONCLUSIONS We have presented new spectroscopy of the gravitationally lensed quasar B1422+117 using Gemini North's GMOS IFU. Extracted spectra of the four lensed images reveal that differential microlensing strongly affects the relative magnification of the broad emission lines compared to the continuum in lensed image B, with the continuum in B appearing anomalously bright compared to all other lensed images. Differential microlensing is apparent across the velocity structure of the Lyα line, in which the line center of lensed image B is more strongly microlensed; it is closer to the magnification of the continuum, while the higher velocity wings appears to be closer to the expected macro-model magnification. This does not enable trivial constraints on the kinematics due to the non-monotonic dependence of microlensing magnification on source size, however with more detailed simulations such contraints may be possible.We take advantage of the differential magnification between lensed images A and B to algebraically decompose the quasar spectrum into two components: the absorbed broad emission line spectrum and the absorbed continuum spectrum, and we use the latter to derive the intrinsic Lyman forest absorption profile. Analysis of these components leads to the following conclusions:∙The proximity zone of the B1422+117 has an outer edge at a proper distance of between 8.6–17.3 Mpc from the quasar, indicating activity over at least 28 Myrs. Note that this does not necessarily mean a period of continuous activity of this length; a duty cycle of 0.1 is more than adequate to produce a similar proximity zone.∙ The monotonic increase in IGM opacity with increasing redshift is consistent with that measured for larger quasar samples.∙ A Gaussian fit to the red wing shows that the blue wing of the reconstructed Lyα line exhibits significant excess flux relative to that fit. This indicates that caution should be taken in the interpreting neutral fraction measurements based on GP trough quasar Lyα lines. If the blue excess observed in theB1422+231's Lyα line is typical for luminous, high-z quasars then the common assumption of a symmetric line profile may have lead to significant underestimates of the neutral fraction.§ ACKNOWLEDGEMENTSWe would like to thank Michael Shull for his illuminating referee report which helped us to dramatically improve and clarify this work. We are also very grateful to James Bolton for generating tailored hydrodynamic simulations of the absorption profile. Finally, we thank the Gemini Observatory for their collaboration in acquiring and reducing these data. [Allington-Smithet al.2002]AllingtonSmith Allington-Smith, J., 2002, PASP, 114, 892 [Angoninet al.1990]Angonin90 Angonin M.-C., Vanderriest C., Remy M., Surdej J., 1990, A&A, 233, L5 [Bajtlik, Duncan & Ostriker1988]Bajtlik+88 Bajtlik, S., Duncan, R. C., Ostriker, J. P., 1988, ApJ, 327, 570 [Bateet al.2011]Bate11 Bate, N. F., Floyd, D. J. E., Webster, R. L., Wyithe, J. S. B., 2011, ApJ, 731, 71 [Becker, et al.2001]B01Becker, R. H., et al., 2001, AJ, 122, 2850 [Becker et al.2011]Becker+11 Becker, G. D., Bolton, J. S., Haehnelt, M. G., Sargent, W. L. W., 2011, MNRAS, 410, 1096 [Bolton & Haehnelt2007]BoltonHaehnelt07 Bolton, J. S., Haehnelt, M. G., 2007, MNRAS, 374, 493 [Bosman & Becker2015]BosmanBecker2015 Bosman, S. E. I., Becker, G. D., 2015, MNRAS, 452 , 1105 [Calverley et al.2011]Calverley+11 Calverley, A. P., Becker, G. D., Haehnelt, M. G., Bolton, J. S., 2011, MNRAS, 412, 2543 [Carilli et al.2010]Carilli10 Carilli, C. L., et al. 2010, ApJ, 714, 834 [Cen & Haiman2000]CenHaiman2000 Cen, R., & Haiman, Z. 2000, ApJL, 542, L75 [Cen & McDonald2002]CM02Cen, R., McDonald, P., 2002, ApJ, 570, 457 [Chartaset al.2009]Chartas09 Chartas, G., Kochanek, C. S., Dai, X., Poindexter, S., Garmire, G., 2009, ApJ, 693, 174 [Chiba et al.2005]Chiba05Chiba, M., Minezaki, T., Kashikawa, N., Kataza, H., Inoue, K. T. 2005, ApJ, 627, 53 [Daiet al.2009]Dai09 Dai, X., Kochanek, C. S., Chartas, G., Kozlowski, S., Morgan, C. W., Garmire, G., Agol, E. 2010, ApJ, 709, 278 [Donahue & Shull1987]DonahueShull87 Donahue, M., Shull, J. <., 1987, ApJ, 323, L13[Fan, et al.2002]F02Fan, X., Narayanan, V. K., Strauss, M. A., White, R. L., Becker, R. H., Pentericci, L., Rix, H.-W., 2002, AJ, 123, 1247 [Fan, et al.2006a]F06aFan, X., et al., 2006, AJ, 131, 1203 [Fan et al.2006b]F06b Fan, X., et al., 2006, AJ, 132, 117 [Faucher-Giguére et al.2008]FG08 Faucher-Giguére, C-A., Prochaska, J. X., Lidz, A., Hernquist, L., Zaldarriaga, M., 2008, ApJ, 681, 831 [2016]Finlator+16 Finlator, K., Oppenheimer, B. D., Davé, R., Zackrisson, E., Thompson, R., Huang, S., 2016, MNRAS, 459, 2299 [Giallongo et al.2015]Giallongo+15 Giallongo, E. et al., 2015, A&A, 578, 83 [Greig et al.2017]Greig+17b Greig, B., Mesinger, A., Haiman, Z., Simcoe, R. A., 2017, MNRAS, preprint (arXiv:1606.00441) [Greig et al.2016]Greig+16 Greig, B., Mesinger, A., McGreer, I. D., Gallerani, S., Haiman, Z., 2016, MNRAS, preprint (arXiv:1605.09388) [Guerraset al.2013]Guerras13 Guerras, E., Mediavilla, E., Jimenez-Vicente, J., Kochanek, C. S., Muñoz, J. A., Falco, E., Motta, V., 2013, ApJ, 764, 160 [Gunn & Peterson1965]GunnPeterson65 Gunn, J. E., Peterson, B. A., 1965, ApJ, 142, 1633 [Harris et al.2016] Harris+16 Harris D. W., et al., 2016, AJ, 151, 155 [Hook et al.2004]Hook Hook, I., Jorgensen, I., Allington-Smith, J. R., Davies, R. L., Metcalfe, N., Murowinski, R. G., Crampton, D., 2004, PASP, 116, 425 [Hutsemékerset al.2010]H10 Hutsemékers, D., Borguet, B., Sluse, D., Riaud, P., Anguita, T., 2010, A&A, 519, 103[Keetonet al.2006]Keeton06 Keeton, C. R., Burles, S., Schechter, P. L., Wambsganss, J., 2006, ApJ, 639, 1 [Kochanek2004]Kochanek04 Kochanek, C. S., 2004, ApJ, 605, 58 [Kormann, Schneider & Bartelmann1994]Kormann+94 Kormann, R., Schneider, P., Bartelmann, M., 1994, A&A, 286, 357 [Kramer & Haiman2009]KramerHaiman09 Kramer, R., Haiman, Z., MNRAS, 400, 1493 [Lee & Spergel2011]LeeSpergel11 Lee K.-G., Spergel D. N., 2011, ApJ, 734, 21 [Lidz et al.2002]L02Lidz, A., Hui, L., Zaldarriaga, M., Scoccimarro, R., 2002, ApJ, 579, 491 [Madau, Haardt & Rees1999]Madau+99 Madau, P., Haardt, F., Rees, M. J., 1999, ApJ, 514, 648 [Madau & Haardt2015]MadauHaardt15 Madau, P., Haardt, F. 2015, ApJ, 813, L8 [McGreer, Mesinger & Fan2011]McG11McGreer, I. D., Mesinger, A., Fan, Z., 2011, MNRAS, 415, 3237 [Mediavillaet al.2009]Mediavilla09 Mediavilla, E., Muñoz, J. A., Falco, E., Motta, V., Guerras, E., Canovas, H., Jean, C., Oscoz, A., Mosquera, A. M., 2009, ApJ, 706, 1451 [Mesinger & Haiman2007]MesingerHaiman07 Mesinger, A., Haiman, Z., 2007, ApJ, 660, 923 [Miralda-Escude1998]ME98Miralda-Escude, J., 1998, ApJ, 501, 15 [Mortlock et al.2011]Mortlock+11 Mortlock D. J. et al., 2011, Nature, 474, 616 [O'Dowdet al.2011]O11 O'Dowd, M., Bate, N. F., Webster, R. L., Wayth, R., Labrie, K., 2011, MNRAS, 415, 1985 [O'Dowdet al.2015]O15 O'Dowd, M., Bate, N. F., Webster, R. L., Labrie, K., Rogers, J., 2015, ApJ, 813, 62 [Patnaik et al.1999]Patnaik+99 Patnaik, A. R., Kemball, A. J., Porcas, R. W., Garrett, M. A., 1999, MNRAS, 307, L1 [Pooleyet al.2009]Pooley09 Pooley, D., Rappaport, S., Blackburne, J., Schechter, P. L., Schwab, J., Wambsganss, J., 2009, ApJ, 697, 1892 [Remy et al.1993]Remy+93 Remy, M., Surdej, J., Smette, A., Claeskens, J. -F., 1993, A&A, 278, L19 [Richardset al.2004]Richards04 Richards, G. T. et al. 2004, ApJ, 610, 679 [Schechter et al.2014]Schechter+14 Schechter, P. L., Pooley, D., Blackburne, J. A., Wambsganss, J., 2014, ApJ, 793, 96 [Schroeder, Mesinger & Haiman2013]Schroeder13 Schroeder, J., Mesinger, A., Haiman, Z., 2013, MNRAS, 428, 3058 [Shapiro & Giroux1987]ShapiroGiroux1987 Shapiro P.R., & Giroux M.L., 1987, ApJ, 321, L107 [Shull et al.2012] Shull+12 Shull J. M., Stevans M., Danforth C. W., 2012, ApJ, 752, 162 [Sluseet al.2007]Sluse07 Sluse, D., Claeskens, J.-F., Hutsemékers, D., Surdej, J. 2007, A&A, 468, 885 [Sluseet al.2011]Sluse11 Sluse, D. et al. 2011, A&A, 528, 100 [Sluseet al.2015]Sluse15 Sluse, D., Hutsemékers, D., Anguita, T., Braibant, L., Riaud, P., 2015, A&A, 582, 109 [Syphers & Shull2014] SyphersShull2014 Syphers D., Shull J. M., 2014, ApJ, 784, 42 [Stevans et al.2014] Stevans+14 Stevans M., Shull J. M., Danforth C. W., Tilton E. M., 2014, ApJ, 794, 75 [Suzuki et al.2005] Suzuki+05 Suzuki N., Tytler D., Kirkman D., O’Meara J. M., Lubin D., 2005, ApJ, 618, 592 [Suzuki2006] Suzuki06 Suzuki N., 2006, ApJS, 163, 110 [Telfer et al.2002] Telfer+02 Telfer R. C., Zheng W., Kriss G. A., Davidsen A. F., 2002, ApJ, 565, 773 [Vernardoset al.2014]gerlumph Vernardos, G., Fluke, C. J., Bate, N., F., Croton, D., 2014, ApJ, 211, 16 [Wayth, O'Dowd & Webster2005]Wayth05 Wayth, R. B., O’Dowd, M. J., Webster, R. L., 2005, MNRAS, 359, 561 [Weinberg et al.2003]Weinberg+03 Weinberg, D. H., Davé, R., Katz, N., Kollmeier, J. A., 2003, AIPC, 666, 157 [White et al.2003]W03 White, R. L., Becker, R. H., Fan, X., Strauss, M. A., 2003, AJ, 126, 1 [Wyithe & Loeb2004]WyitheLoeb04 Wyithe, J. S. B., Loeb, A., 2004, Nature, 432, 194 [Wyithe & Bolton2011]WyitheBolton11 Wyithe, J. S. B., Bolton, J. S., 2011, MNRAS, 412, 1926 [Yee & Ellington1994]YeeEllington94 Yee, H. K. C., Ellingson, E., 1994, AJ, 107, 28 | http://arxiv.org/abs/1708.07535v2 | {
"authors": [
"M. O'Dowd",
"N. F. Bate",
"R. L. Webster",
"K. Labrie",
"A. L. King",
"S-. Y. Yong"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170824194032",
"title": "The Intrinsic Far-UV Spectrum of B1422+231"
} |
The class of second order quasilinearequations: models, solutions and background of classificationO. Makarenko^†,[e-mail: <[email protected]>], A. Popov^,[e-mail: <[email protected]>], S. Skurativskyi^,[e-mail: < [email protected]>]^†Institute for Applied System Analysis NTU ”KPI”Politekhnichna st., 14, 14B,Kyiv, Ukraine, 03056^Institute of Physics and Technology NTU ”KPI”Prosp. Peremohy, 37, Kyiv, Ukraine, 03056^Subbotin institute of geophysics, Nat. Acad. of Sci. of Ukraine Bohdan Khmelnytskyi str. 63-G, Kyiv, Ukraine Abstract.The paper is concerned with the unsteady solutions to the model of mutually penetrating continua and quasilinear hyperbolic modification of the Burgers equation (QHMB). The studies were focused on the peculiar solutions of models in question.On the base ofthese models and their solutions, the ideas of second order quasilinear models classification were developed. Keyword: hyperbolic equations, attractors, multivaluedness§ INTRODUCTION Nonlinear models for phenomena and systems are the cornerstones in modern physics. The examples of these models are well known, namely the Burgers, KdV, Liouville, nonlinear Shredingerequations and etc. The derivation of analytical solutions for these equations is a challenge for scientists. Therefore, the numerical treatments of such models are developed intensively. Among the models mentioned above it is worthaccentuating the second order in time nonlinear hyperbolic differential equations <cit.> having a broad applications recently. Note that the partial solutions for these equations had been derived <cit.>.Due to the significance of considered equation and their solutions, this paper deals with the classification of these equations, definition of general expression of quasilinear models. Models' solutions obtained via the numerical modelling are analyzed in detail. We also discuss the possible ways of investigations, in particular,combining the concept of dynamical systems (attractors) and artificial intelligence methods (neural networks). The aspects related to the multivaluedness of solutions including symmetries are considered as well. § WAVE REGIMES IN MEDIA WITH OSCILLATING INCLUSIONS To begin with, let us note that the class of second order quasilinear models is not emptyand coversmany differentmodels originated from the physics and biology. In particular, consider the model of mutually penetrating continua which uses for the description of physical processes in complex media <cit.>. It turned out that this model possesses the specific wave solutions. Consider these solutions in more detail. The model we are going to deal with has the following formρ∂ ^2 u/∂ t^2= ∂σ/∂ x - m ρ∂ ^2 w /∂ t^2 , ∂ ^2 w /∂ t^2+ Φ( w - u) = 0,where ρ ismedium's density, u and w are the displacements of carrying medium and oscillator from the rest state,mρ is the density of oscillating continuum.We also usethe cubic constitutive equation for the carrying mediumσ= e_1 u_x+ e_3 u_x^3, where e_1, e_3 are theelastic moduli, and relation for applied force Φ(x)=ω ^2 x+δ x^3, where ωdenotes the natural frequency of oscillator, whereasthe parameter δ appearsdue to accounting for the cubic term in the expansion of restoring force in a power series.Thetraveling wave solutions of model (<ref>) have the following formu=U(s),w=W(s),s=x-Dt, where the parameter Dis aconstant velocity of the wave front.Inserting (<ref>) intomodel (<ref>), it is easy to see thatthe functions U and Wsatisfy the dynamical system D^2 U' = ρ^-1σ( U') - mD^2 W^',W” + Ω ^2 ( W - U) +δ D^-2(W-U)^3= 0,where Ω= ω D^-1. This system can be written in the form W' =α _1 R + α _3 R^3,U' =R,(α_1+3α_3 R^2)R'+Ω^2(W-U)+δ D^-2(W-U)^3=0, where α _1= e_1 - D^2 ρ/m ρ D^2, α _3 = e_3 /m ρ D^2 >0. Through the report we fixe_1=ρ=1,e_3=0.5,m=0.6,ω=0.9in numerical treatments. At first, consider system (<ref>) at δ=0.The detail descriptionofphase plane of dynamical system had been done in the paper <cit.>, we thus summarizethe main results only.At α_1<1 the phase plane contains three fixed points, whereas at α>1the only one fixed point (center) remains.Forα_1<0, whenD=1.2 is fixed for definiteness,a typical phase portrait is depicted in the Fig. <ref>a. In this case, all fixed points are centers surrounded by periodic orbits. There areseparatrices separated the regions with periodic and unbounded trajectories and two lines corresponding to discontinuity of system. When 0<α<1, at D=0.9 for instance, in the phase portrait (Fig. <ref>b) one can distinguish the homoclinic trajectories that go throughthe origin. The homoclinic loop can be written in the explicit form [ s - s_0= 3 /2Ωarcsin( 4α _3 R^2 - 3 + 4α _1 /√(9 - 8α _1 )) - 1/2Ω√(α _1 /1 - α _1 )×;; ln(1/R^2 + 3α _3- 4α _1 α _3 /4( α _1- α _1^2 ) +√({1/R^2 + 3α _3- 4α _1 α _3 /4( α _1- α _1^2 )}^2- α _3^2 ( 9 - 8α _1 )/16( α _1- α _1^2 )^2 ))|_R_0^2 ^R^2 ]Solution (<ref>) corresponds to the solitary wave solution with infinite support.When α_1 tends to zero, the angles between separatrices of saddle point O are growing. As a result, atα _1 = 0 we observethe transformation ofsolitary wave into the compacton, i.e.,solutions with finite support <cit.>.These orbits are described by the following expressionsU^' _s=√(3/2α _3 )sinΩ s/3,Ω s/3∈[ 0;π]0,Ω s/3∉[ 0;π] U= 0,Ω s/3∈[ -∞; 0], 3/Ω√(3/2α_3)(1-cosΩ s/3),Ω s/3∈(0; π], 6/Ω√(3/2α_3),Ω s/3∈(π; ∞). As above, there arethe periodic orbits enclosed in the homoclinic loopsandperiodic trajectories l lyingbeyond the homoclinic contour. §.§ Phase diagrams in the model with cubic nonlinearity in the equation of motion for oscillating inclusions If δ≠ 0, then system (<ref>) does not reduce to the dynamical system in the plane (R;R'). But the first integralfor(<ref>)can still be derived in the formI=μ_1/2(W-U)^4 + μ_2(W-U)^2+ f(R),where μ_1=δ D^-2, μ_2=ω^2 D^-2, f(R) = α _3^2 R^6+ 4α _1 - 3/2α _3 R^4+ ( α _1^2- α _1 )R^2. Since dI/ds=0 on the trajectories of system (<ref>), then I ≡.It is easy to see that expression (<ref>) can be used for splittingsystem(<ref>). Indeed, solving (<ref>) with respect W-U we obtain W-U=±√(-μ_2 ±√(μ_2^2-2μ_1 f(R)+2μ_1 I)/μ_1). This allows us to separate the last equation of (<ref>) from other ones. Unfortunately, the resulting equation cannot beintegrated for general case, therefore,let us consider its phase plane structure, which is equivalent to the structure oflevel curves for function (<ref>).Now consider the position and the form of homoclinic trajectories when the parameter δ is varied.Starting from the loopat δ=0 which coincides with the orbits of Fig. <ref>b, we see that increasing δ causes the attenuationofloop's size along vertical axis.Actually, the level curve consists of the closed curve (homoclinic loops) and unbounded trajectories.If δ decreases,loop's size grows, but at δ_0 the additional heterocycle connecting four new saddle points appears.The bifurcational value δ_0can be derived viaanalyzing the function (<ref>). Namely, δ_0 corresponds to the moment when different branches of(<ref>) are tangent. This happens when μ_2^2-2μ_1 f(R)=0. Thus, the condition of contact for two branches leads us to a cubic equation with respect to R^2with zero discriminant.Then δ_0=-α_3ω^4 /D^2(α_1-1)^2 or δ_0=27 α_3ω^4 /D^2α_1^2 (9-8α_1). The last value of δ_0 is not interesting because four branches of (<ref>) degenerate into two ones forming homoclinic loops.Considering the first value δ_0, we put D=0.9 and deriveδ_0=-2.24. For δ>δ_0 we have homoclinic loops placed along horizontal axis accompanied by appearingthe unbounded curvesin the upper and lower parts of diagram. When δ<δ_0, the homoclinic loops are placed in the vertical quarters of the phase plane, whereas the unbounded orbits appearat the left and at the right sides of diagram (Fig. <ref>a). Note that the profiles of the resulting solitary waves are different (Fig. <ref>b), namely, at δ<δ_0 the U profile looks like a bell-shape curve,but at δ>δ_0 it is a kink-like regime.The homoclinic orbits divide the phase plane into parts filled by closed curves corresponding to the periodic regimes.If we chooseδ=-1.5>δ_0, we getthe typical phaseportrait of system (<ref>) plotted inthe Fig. <ref>a. In the portraittwo pairs of nontrivial fixed pointsA_± (± Q;0) and B_± (0; ±ω/√(-δ)) can be distinguished. Inserting the coordinates of these points intorelation (<ref>),we obtain the values of I_1=-(α_1-1)^2/2α_3 and I_2=ω^4/2D^2|δ| which allows us to state the conditions of periodic regimes existence.For fixed δ=-1.5, I_1=-0.18 and I_2=0.27. We thus get that periodic regimes exist if I_1<I<I_2 only. Now let us chooseδ=-2.7<δ_0. In this case the homoclinic loop going through the origin is placed along vertical axis andthe phase portrait looks likeFig. <ref>b. As above, we have I_1=-0.18, but I_2=0.15. §.§ Wave dynamics of model (<ref>) To model the wave dynamics, we usedthe three level finite-difference numerical scheme for model (<ref>). Solitary waves. To consider the evolution of solitary waves, let us construct the initial data v_i, q_i, G_i, F_i for numerical simulation on the base of homoclinic contour.To do this, we integrate dynamical system (<ref>) with initial data R(0)=10^-8, Z(0)=0, s ∈ [0;L]and choose the righthomoclinic loop in the phase portrait (Fig. <ref>b).Then the profiles of W(s), U(s), and R(s) can be derived. Joining the proper arrays, we can build the profile in the form ofarch:v=U(ih)∪ U(L-ih),q=u(x+τ D)=[U(ih)+τ D R(ih)]∪[U(L-ih)+τ D R(L-ih)].The arrays G and F are formed in similar manner.Combining two arches and continuing the steady solutionsat the ends of graph, we get more complicated profile.We apply the fixed boundary conditions, i.e. u(x=0,t)=v_1, u(x=Kh,t)=v_K, where K is the length of an array. Starting from the two-arch initial data, we see (Fig. <ref>) that solitary waves move to each other, vanish during approaching, and appear with negativeamplitude and shift of phases. After collision in the zones betweenwaves some ripples are revealed.Secondary collisions of waves are watched also. Note that thesimulation of compacton solutions displays similar properties.Propagation of solitary waves at δ≠0 depends on the sign of δ. Numerical simulationsshow that thecollision of waves atδ>0 is similar to the collision atδ=0. Behavior of waves after interaction does not change essentiallywhenδ<0 and close to zero (Fig. <ref>a).But ifδ is not small,after collision the amplitude of solution is increasing in the place of soliton's intersectionandafter a while the solutionisdestroyed <cit.>.This suggests that we encounterthe unstableinteraction of solitary waves or the numerical scheme we used possesses spurious solutions. But if we takehalf spatial step and increase the schemeparameter r up to 0.8, the scenario of solitary waves collision is not changed qualitatively. Therefore, the assertion on the unstable nature of collision is more preferable.Periodic waves. To simulate the evolution of periodic waves, we take the initial profile corresponding to a periodic orbit (red curve in Fig. <ref>b) surrounding the homoclinic loop at the phase portrait of Fig. <ref>a at δ=-0.2 and D=0.9.Due to the periodicity of the problem,the boundary conditions for numerical schemeshould be modified <cit.>. The resulting profile derived after time interval 1200τhas passed is depicted in Fig. <ref>b with dark green curve. It is obvious that this wave is shifted a distance 1200τ D to the right in a self-similar manner.§ UNSTEADY SOLUTIONS TO QHMB Let us consider the evolution of localized perturbation within the framework of QHMB: τ∂ ^2 u/∂ t^2+α∂ u/∂ t+βφ(u)∂ u/∂ x=μ k(u)∂^2 u/∂ x^2+νψ (u) (∂ u/∂ x)^2+θ f(u), where τ, α, β, μ, ν, θ are constants. As an initial data for equation (<ref>)б we chose the profile u(x,0)=exp(x+a/b)+exp(x-a/b) (fig. <ref>a), where a=5, b=4.5. The evolution of this two-hump perturbation is shown in figs. <ref>b,c,d. From the analysis of these figures it follows thatthe number of oscillations increases during evolution. The oscillations have non-regular character but remain localized in spatial domain. Thus, one can call them the “pre-turbulent oscillations” or, due to the localization, the “turbulons”. Let us also note that the appearance of this type solutions was discovered in other numerical simulations of 2D and 3D problems <cit.>. § THE CLASS OF QUASILINEAR EQUATIONS AND THEIR FORMAL DESCRIPTIONThe presented examples of modelling the carrying processes manifest the variety of solutions of models considered. Analyzing these models in details, onecan be convinced that they belong to the second order quasilinear equations, i.e., the relation is a linear with respect to higher derivatives whereas it is nonlinear with respect to lower derivatives, unknown functions,and independent variables.It is obvious that this class of equations is very broad and needs to systematize.Let us start from the defining the general expression of quasilinear second order models, namely τ r(u) ∂ ^2 u/∂ t^2 +α s(u) ∂u/∂ t+βφ (u) ∂ u/∂ x = μ k(u)∂ ^2 u/∂ x^2 +νψ (u) ( ∂u/∂ x)^2 + γ h(u)∂ ^2 u/∂ x∂ t ++ξ b(u) ( ∂u/∂ t)^2+θ f(u)+ χ I(u,x,t,±Δ t, ±Δ x;...),where I is the source (integral form addmited),τ, α, β, μ, ν, θ are constants, r, s, k, ψ, h, b, f are the specified functions.Note that, depending on the coefficients and functions, class (<ref>) covers the classical linear equations (elliptic, parabolic and hyperbolic) and well-known nonlinear models (sin-Gordon, Burgers, Liouville,Hopf, hyperbolic modification for the Burgers equation, equations with blow-up solutions).For future handling the variety of models and their solutions, we need a convenient method for their identification. We propose the following descriptor for such objects: EQ( τ,α,β,μ,ν,γ,ξ,θ,χ;{ NONLINEAR_FUNCTION: r,s,φ,k,ψ,h,b,f,I}; { TITLE_OF_EQUATIONS};{TYPES_OF_SOLUTIONS}) For instance, the descriptor for QHMB can be chosen in theform EQ( 1, 1, 1, 1, 1, 0, 0, 1, 0;{ NONLINEAR_FUNCTION: φ, k,ψ, f }; { QHMB};{compactons;blow-up;oscillations}).When the system of equations is considered, the coefficients and functions in (<ref>) should be assumed asmatrices. In particular, rewrite system (<ref>) as follows∂ ^2 u/∂ t^2=ρ^-1(e_1+3 e_3(∂ u/∂ x)^2) ∂ ^2 u /∂ x^2 + m (ω^2(w-u)+δ(w-u)^3),∂ ^2 w /∂ t^2=-ω^2(w-u) - δ(w-u)^3.Thenits descriptor is EQ( ([ 1 0; 0 1 ]), 0̂,0̂, ([ ρ^-10;01 ]),0̂ ,0̂,0̂,([m0;0 -1 ]) ,0̂;{ NONLINEAR_FUNCTION: k=([ e_1+3 e_3(∂ u/∂ x)^2;0 ]), f=([ ω^2(w-u)+δ(w-u)^3; ω^2(w-u)+δ(w-u)^3 ])}; { EXAMPLE_ONE};{solitarywaves; compactones}). Remark that the construction of descriptor helps us not only to classify the equations but at the statement ofproblems. In particular, the Hopf equation can be considered as a limit case of hyperbolic Burgers equation.Let us also outline the types of solutions of such models. For generality, it is worth to mention thatcertain equations from class (<ref>) possess interesting solutions, namely, autowave solutions to the Burgers equation,singular solutions to theLiouville equation, blow-up solutions to parabolic and hyperbolic equations with nonlinear sources, solitons for the sin-Gordon equation. Among the solutions to thehyperbolic modification of the Burgers equation we encounter the packets of oscillations, compactons, autowaves. Another type of solutions is related to the multivaluedness. This means that the solutions haveseveral values in given spatial point at fixed moment of time <cit.>. In particular, the Hopf equationadmits the solutions when their profiles“overturn"and becomes multivalued. Similar behavior of wave solutions is observed in the Vakhnenko equation which admits the loop solutions. Studies of such solutions cause the introduction of new class of solutions named the “foldons”. Foldons are the multivalued autowaves. It is important that both the Hopf equation and Vakhnenko equation belong to class (<ref>) and theirmultivalued solutions have analytical expressions. Up today we have hardly any methods to treat multivalued models. In this case the group analysis methods seem to be useful. To apply group methods, the multivalued differential equations should be considered as multivalued surfaces (geometric objects). The promising approach of multivalued model studies is concerned with the asymptotic transition from the singlevalued to multivalued models.We encounter thiswhen transform the hyperbolic Burgers equation to the Hopf equation. Another way totreat the multivaluedness is the expanding the inverse scattering transform.It is worth to mention the development of novel approach dealing with the specific type ofinverse problems. The examples considered in the paper are concerned with analyzing the solutionsand their dependence on the parameters when the equations subjected to the initial and boundary data are available. Traditionally, such problems have been solved via the trial and error methods to find the self-similar solutions. But the inverse statement of the problem is possible. Indeed, we can take the functions ũ (x,t) in such a way that they satisfy equation (<ref>), when the components in(<ref>)are chosen correctly.In general case we would like to have the method when the set of functions ũ_1(x,t), ũ_2(x,t), ... , ũ_n(x,t) satisfiesequation(<ref>) subjected to proper set of initial conditions. This problem is similar to approaches in the theory of neural networks. The initial and boundary conditions are regarded as inputs of neural networks, whereas the functions ũ_1(x,t), ũ_2(x,t), ... , ũ_n(x,t) considered as potential solutions are identified as outputs of neural networks. It is obvious that for arbitrary equation with fixed initial conditions the solutionsu_1(x,t), u_2(x,t), ... , u_n(x,t) differ from the“desired” set of function ũ_1(x,t), ũ_2(x,t), ... , ũ_n(x,t). Then one canfind the values of equation's coefficients providing the small deviation Δ=ũ_i(x,t)-u_i (x,t). To realize this procedure, the iteration process of approaching to the equation's coefficients can be applied. This process is an analog of studying process in neural networks. We can use multivalued neural networks in these problems as well. § CONCLUSION Summarizing, we presented the specific solutions occurred in quasilinear models for the carrying processes in nature. The results obtained were encouraged us to develop the systematic approach to the studies of nonlinear dynamical models within the framework of quasilinear second order equations and their solutions. We outlinednew ways to analyze quasi-linear models including the multivalued cases. 00 monog Danylenko V.A., Danevych T.B.,Makarenko O.S.,Skurativskyi S.I.,Vladimirov V.A., Self-organization in nonlocal non-equilibrium media,Kyiv, Subbotin in-t of geophysics NAS of Ukraine, 2011.Makarenko_2012Makarenko A. S., Model equations and formation of structures in media with memory. Ukr. J. Phys. 57(4) (2012) 408–421.Popov_Mak_2015 Popov A.S.,Makarenko O.S., To the numerical solution of quasilinearmodification for the Burgers equation. Analysis, modelling, management.Vol.2: Kyiv, NTTU”KPI”, 2015.– P. 197-203.Mak_2015_Austria Makarenko O. S., Toward multivaluedness aspects in self-organization, complexity and computations investigations. Forth Int. Workshop on Nonlinear Dynamics and Sinchronization INDS’15, July 31, Klagenfurt, Austria, Alpen-Adria University, 2015. –Pp. 84-93nasu Danylenko V.A.,Skurativskyi S.I, Resonance regimes of the spreading of nonlinear wave fields in media with oscillating inclusions, Reports of NAS of Ukraine, 11(2008) 108–112.nonlin_dymDanylenko V.A., SkurativskyiS.I.,Travelling wave solutions ofnonlocal modelsfor media with oscillating inclusions, Nonlinear Dynamics and Systems Theory, 4(12) (2012) 365–374.dan_lodz_16 Danylenko V.A., Skurativskyi S.I.,On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua, Dynamical systems. Mechatronics and life sciences, 2 (2015) 453–460. (arXiv:1512.05226v1 [nlin.PS] 15 Dec 2015) dan_skur_ufj Danylenko V.A.,Skurativskyi S.I., SkurativskaI.A.,Asymptotic wave solutions for the model of a medium with Van Der Pol oscillators, Ukrainian Journal of Physics, 59(9) (2014) 932–938.skur_akustSkuratovskii S.I., Skuratovskaya I.A., Localized autowave solutions of the nonlinear model of complex medium, Electronic Journal ”Technical Acoustics”, 6 (2010)http: ejta.org. Hyman-Rosenau Hyman J. andRosenau Ph., Compactons: solitons with finite wavelength, Phys. Rev. Lett.,70(1993) 564. vsan_lodz2016 Vladimirov V. A. and Skurativskyi S. I., Solitarywaves inone-dimensional pre-stressed lattice and itscontinual analog, Dynamical systems. Mechatronics and life sciences, 2(2015) 531–542. (arXiv:1512.06125v1 [nlin.PS] 18 Dec 2015) Dan_Skur_2016Danylenko V.A. andSkurativskyi S.I., Peculiarities of wave dynamics in media with oscillating inclusions,International Journal of Non-Linear Mechanics, 84 (2016)31–38. | http://arxiv.org/abs/1708.08002v1 | {
"authors": [
"O. Makarenko",
"A. Popov",
"S. Skurativskyi"
],
"categories": [
"nlin.PS"
],
"primary_category": "nlin.PS",
"published": "20170826175538",
"title": "The class of second order quasilinear equations: models, solutions and background of classification"
} |
firstpage–lastpage 2016On annihilation of the relativistic electron vortex pair in collisionless plasmas S.V. Bulanov December 30, 2023 =================================================================================We present late-time spectra of eight Type Ia supernovae (SNe Ia) obtained at >200 days after peak brightness using the Gemini South and Keck telescopes. All of the SNe Ia in our sample were nearby, well separated from their host galaxy's light, and have early-time photometry and spectroscopy from the Las Cumbres Observatory (LCO). Parameters are derived from the light curves and spectra such as peak brightness, decline rate, photospheric velocity, and the widths and velocities of the forbidden nebular emission lines. We discuss the physical interpretations of these parameters for the individual SNe Ia and the sample in general, including comparisons to well-observed SNe Ia from the literature. There are possible correlations between early-time and late-time spectral features that may indicate an asymmetric explosion, so we discuss our sample of SNe within the context of models for an offset ignition and/or white dwarf collisions. A subset of our late-time spectra are uncontaminated by host emission, and we statistically evaluate our nondetections of Hα emission to limit the amount of hydrogen in these systems. Finally, we consider the late-time evolution of the iron emission lines, finding that not all of our SNe follow the established trend of a redward migration at >200 days after maximum brightness.supernovae: general§ INTRODUCTIONType Ia supernovae (SNe Ia) are a powerful standardisable candle, but their progenitor scenarios and explosion mechanisms are not yet well understood (e.g., ). Nebular-phase spectra, taken >200 days after peak brightness when the material is optically thin, are a unique diagnostic for SN Ia models — especially when combined with time-series spectra and photometry starting before peak brightness. Modern wide-field surveys have increased the number of nearby SNe Ia that are sufficiently brightfor nebular-phase spectroscopy, and the new robotic spectrographs and imagers of the Las Cumbres Observatory (LCO; ) have improved our ability to monitor SNe from early times. In this paper we present and analyse a new sample of nebular-phase spectra of eight nearby SNe Ia obtained with programs at Keck and Gemini Observatories.There are several ways that nebular spectra of SNe Ia can help to constrain the progenitor scenario and/or explosion mechanism. Once the material is optically thin, the spectrum is dominated by the broad, forbidden emission lines of the nucleosynthetic products of the thermonuclear explosion: nickel, cobalt, and iron. <cit.> use a line-emission code to model a set of SN Ia nebular spectra, measuring the masses of radioactive and stable nucleosynthetic products and intermediate-mass elements. They show that the mass ratios of these components vary continuously between SNe Ia but that all events have a similar total mass. Building on the work of <cit.>, <cit.> established that the mass of ^56Ni synthesised in the explosion can also be measured from the decline rate of [CoIII] λ5893 line flux in nebular-phase spectra. In the past, the 5893 Å feature has been suspected to have contributions from NaI D (e.g., ), but <cit.> have suggested that the sodium emission is weak compared to cobalt — although at extremely late times (∼1000 days), after the ^56Co has decayed, the NaI D line may be visible (e.g., ).Aside from the total mass of nucleosynthetic products, the nebular-phase spectrum can also help reveal its distribution within the supernova (SN) ejecta (e.g., Figure 1 of ). <cit.> suggested that an off-centre ignition followed by an asymmetric explosion causes the nebular emission lines to be redshifted or blueshifted, depending on the observer's viewing angle, and such asymmetries may be a contributing factor to the SN Ia width-luminosity correlation <cit.>. Asymmetry is also a natural outcome of the collisional scenario in which two white dwarfs collide <cit.>, creating synthesised material with a bimodal velocity distribution that is observable as double-peaked nebular lines, such as those identified by <cit.>. The leading progenitor scenarios for SNe Ia include accretion from or merger with another white dwarf (double-degenerate) or accretion from a main-sequence or red-giant star (single-degenerate). In the latter scenario, leftover hydrogen from the companion star might become visible as a narrow feature at late times after the flux of the SN subsides (e.g., ), but this has so far only been tentatively identified for one SN Ia by <cit.>. Since SNe Ia fade by ≳5 mag between their light-curve peak and 200 days later, nebular-phase spectroscopy is possible only for objects that are relatively nearby and thus reach brighter apparent peak magnitudes. Nearby SNe are also easier to monitor with a dense sampling of photometry and spectroscopy during the first few months. This is advantageous because there are several photospheric-phase characteristics of SNe Ia that correlate with nebular-phase observations in a physically meaningful way. One example is the synthesised mass of radioactive ^56Ni, M_ ^56Ni, which is directly proportional to the peak luminosity <cit.>.M_ ^56Ni can also be measured from nebular-phase spectra, as mentioned above, and is important because it is the physical origin of the width-luminosity relation by which SNe Ia are calibrated to events of standard brightness <cit.>. Another example is the rate of decline of the velocity of the SiII λ 6355 line, v̇_ Si II, during the weeks after peak brightness. In the off-centre explosion model of <cit.>, measuring a high or low v̇_ Si II depends on the observer's viewing angle and is thus also correlated with the nebular emission lines appearing to be redshifted or blueshifted, as mentioned above.In this work, we present nebular-phase spectra for eight nearby SNe Ia, obtained during several semesters of a dedicated follow-up program at Gemini Observatory South and a transient follow-up program at Keck Observatory. We also present photospheric-phase characteristics derived from the photometric and spectroscopic monitoring program at the Las Cumbres Observatory. In Section <ref> we present our sample of SNe Ia and our observations. We analyse our nebular-phase spectra in Section <ref>, first describing how we measure properties of the emission lines and then providing a comprehensive overview of the particular attributes of each individual SN Ia. In Section <ref> we interpret our observations in terms of the progenitor scenarios and explosion models for SNe Ia discussed above, and we conclude in Section <ref>. § OBSERVATIONS Our sample of SNe Ia was chosen with the following criteria: (1) nearby (<100 Mpc) and bright enough to observe at late times (i.e., B≲22 mag at ≥200 days after peak); (2) significantly offset from the host galaxy, in regions with a low background surface brightness; (3) well observed at early times by the LCO (an optical light curve that covers peak brightness, and at least two epochs of optical photospheric-phase spectra). The last criterion ensures that we have physical properties of the explosion such as the nickel mass and photospheric velocity in order to properly interpret and contextualise our nebular spectra. In Table <ref>, we list the names and coordinates of these SNe Ia, their redshift and distance based on host-galaxy observations, and the Galactic line-of-sight extinction and reddening terms that we will apply to our observations. The photospheric-phase data for our targets are described in Section <ref>. We present our nebular-phase optical spectroscopy and imaging from Gemini Observatory in <ref> and our late-time spectra from Keck Observatory in <ref>. Section <ref> provides an analysis of these data, including detailed discussions of each of our SNe Ia as individual objects. §.§ Early-Time Observations At early times, our targeted SNe were monitored with optical imaging and spectroscopy as part of the Supernova Key Project by the Las Cumbres Observatory (LCO). Typical follow-up observations for nearby SNe include imaging in the BVgri filters every 3–7 days and low-resolution spectroscopy with the FLOYDS instrument every 5–10 days. These data are all automatically processed and calibrated using in-house pipelines designed by LCO astronomers (see Appendix B of ). We display the multiband light curves from LCO in Figure <ref> and provide the photometry in Table <ref>. The photometry for SN 2015F was previously published by <cit.>. We use the Python package SNCosmo[http://sncosmo.readthedocs.org/] to fit light curves to the photometry, and the resulting peak apparent magnitudes and post-peak decline rates Δ m_15 (B) are listed in Table <ref>. Unfortunately, most of LCO's near-peak photometry of SN 2013aa was saturated, so for its light-curve fit we used V-band photometry from the Swift Optical/Ultraviolet Supernova Archive (SOUSA[V-band photometry was derived from observations with the Ultraviolet/Optical Telescope (UVOT; ) on the Swift spacecraft <cit.>. Reduced and calibrated photometry was downloaded from the SOUSA website at <http://swift.gsfc.nasa.gov/docs/swift/sne/swift_sn.html>. The reduction is based on that of <cit.>, including subtraction of the host-galaxy count rates, and uses the revised UV zeropoints and time-dependent sensitivity from <cit.>.]; ), which is shown also for reference in Figure <ref>.Given the established correlation between the SiII velocity gradient in the weeks after peak brightness (v̇_ Si II) and the nebular-phase emission-line velocity (e.g., ), we attempt to classify our SNe Ia as either “low" or “high" velocity gradient (LVG or HVG). For our SNe Ia with at least two spectra between days 0 and 14, we make a simple linear fit to the velocity of the SiII λ 6355 line from the LCO FLOYDS spectra (i.e., SNe 2012fr, 2012hr, 2013aa, and 2013dy in Table <ref>). The true decline is not linear, but this is a fine approximation for our purposes of classifying as LVG or HVG. Of these, only SN 2012hr approaches a value of v̇_ Si II that is HVG-like. Unfortunately, not all of our SNe Ia had the necessary photospheric-phase spectral coverage for this, but we can still make an educated guess at whether they are LVG or HVG-like SNe Ia. <cit.> established that LVG SNe Ia typically have v̇_ Si II < 70 km s^-1 d^-1 and v_ Si II( peak) ≲ 12,000 km s^-1, and that HVG SNe Ia typically have v̇_ Si II > 70 km s^-1 d^-1 and v_ Si II( peak) ≳ 12,000 km s^-1. Furthermore, <cit.> show that v_ Si II can be used as a proxy for v̇_ Si II (in their Figure 6). For the remaining SNe Ia in our sample, we measure v_ Si II in the spectrum nearest to peak light (0, +12, -7, and -1 days for SNe 2013cs, 2014jg, 2015F, and 2013gy, respectively; Table <ref>), and use these ranges from <cit.> to estimate the velocity-gradient classification from a single epoch alone. Of these, only SN 2013cs exhibits a velocity in the range of HVG-like SNe Ia. In Table <ref> we list the LVG and HVG determination for each of our SNe Ia, with a “comments” column describing the reasoning behind each determination. To summarize, we (or previous publications) have directly measured v̇_ Si II for SNe Ia 2012fr, 2012hr, 2013aa, and 2013dy. Of these four, only SN 2012hr has v̇_ Si II > 70 km s^-1 d^-1, but since there is a large uncertainty in our measurement of v̇_ Si II, and because SN 2012hr also exhibits v_ Si II( peak) ≲ 12,000 km s^-1 (like a LGV SN), its status as an HVG SN is not secure and we therefore list it as “HVG?". We consider the other three SNe Ia (SNe 2012fr, 2013aa, and 2013dy) securely classified as LVG, and list them as such. For the remaining four SNe Ia in our sample (SNe 2013cs, ASASSN-14jg, 2015F, and 2013gy), we must instead rely on an indirect estimate of its LVG/HVG classification using v_ Si II( peak) as a proxy for v̇_ Si II, as described in the previous paragraph. Of these, only SN 2013cs has v_ Si II( peak) ≳ 12,000 km s^-1. However, we know from SN 2012fr – which exhibited v_ Si II( peak) ≳ 12,000 km s^-1 but was clearly an LVG SN Ia – that the value of v_ Si II( peak) is not a robust proxy for the velocity gradient. We therefore list SN 2013cs as “HVG?", and furthermore list the remaining three SNe Ia (ASASSN-14jg, SN 2015F, and SN 2013gy) as “LVG?" to represent the uncertainty in using v_ Si II( peak) instead of v̇_ Si II. Since our LVG and HVG assignments are more tentative estimates than secure measurements, in this work we will refrain from drawing any strong conclusions that rely on these classifications.§.§ Nebular-Phase Spectroscopy and Imaging from Gemini ObservatoryNebular spectra for SNe Ia in the southern hemisphere were obtained using the Gemini Multi-Object Spectrograph (GMOS; ) on the Gemini South telescope on Cerro Pachon, Chile. Our GMOS configurations use the 0.75 long slit, the B600 grating with no filter and a central wavelength of 4500 Å, and the R400 grating at a central wavelength of 7500 Å with the OG515 long-pass filter to block second-order light at λ < 5200 Å. These spectra are listed in Table <ref>. We processed the data with the Gemini IRAF package: the raw two-dimensional (2D) spectra were flat-fielded, bias-subtracted, and trimmed; cosmic-ray rejection was performed; wavelength calibration was determined using the Cu-Ar comparison arc lamp; and sky-subtracted one-dimensional (1D) apertures were extracted from the processed frames. The sensitivity function was determined from standard-star observations obtained in our GMOS configuration, as near in time as possible (often the same night), and applied to flux calibrate our data. We also used the standard-star spectra to remove telluric features from our spectra. The second-epoch spectra that we obtained of SN 2012hr and SN 2013aa turned out to have such low signal-to-noise ratios (S/N) that we excluded them from our analysis, but as they are at uniquely late times (>450 days) we are publishing them to make them available to the community.We also included a sequence of ∼ 30 s gri images with each GMOS observation, which typically occurred during target acquisition. We reduced these images using the Gemini IRAF package, and calibrated the photometry using the sequence of local standards used for the LCO imaging analysis. All of our GMOS spectra were flux calibrated to match the late-time photometry, and the dates and magnitudes are listed in Table <ref>. In general, our observing requirements were relaxed to include poor weather conditions, particularly for semesters during which our program was in Gemini's Band 3; this is the cause for the variety in photometric uncertainties. §.§ Nebular-Phase Spectroscopy from Keck ObservatoryNebular spectra for SNe Ia in the northern hemisphere were obtained with the Low Resolution Imaging Spectrometer (LRIS; ) at Keck Observatory. We used the 1.0 slit rotated to the parallactic angle to minimise the effects of atmospheric dispersion (; in addition, LRIS has an atmospheric dispersion corrector). In our LRIS configuration, coverage in the blue with the 600/4000 grism extends over λ=3200–5600 Å with a dispersion of 0.63 Å pixel^-1 and a full width at half-maximum intensity (FWHM) resolution of ∼4 Å. We use the 5600 Ådichroic, and our coverage in the red with the 400/8500 grating extends over λ=5600–10,200 Åwith a dispersion of 1.16 Å pixel^-1 and a resolution of FWHM ≈ 7 Å. These spectra are listed in Table <ref>. Note that one Keck spectrum was obtained with DEIMOS; see Pan et al. (2015) for details.The data were reduced with routines written specifically for LRIS in the Carnegie Python (CarPy) package. The 2D images were flat-fielded, corrected for distortion along the y (slit) axis, wavelength calibrated with arc-lamp spectra, and cleaned of cosmic rays before extracting the 1D spectrum of the target. These spectra were flux calibrated using a sensitivity function derived from observations of a standard star obtained the same night in the same instrument configuration. The standard-star spectrum was also used to remove the telluric sky absorption features.§ ANALYSIS We begin our analysis with a visual comparison of the nebular-phase spectra in our sample that are listed in Tables <ref> and <ref>. In Figure <ref> we plot the reduced and calibrated spectra for each object individually, with two epochs for each of SN 2012fr (top left) and SN 2013dy (bottom right) plotted to show the spectral evolution at late times. In Figure <ref> we show one rest-frame nebular-phase spectrum for each of the eight SNe Ia on a single plot to facilitate a comparison and highlight the diversity in nebular features. We also plot the +309 day spectrum of the prototypically normal SN Ia 2011fe; this spectrum was obtained with the Kast Spectrograph at Lick Observatory, and previously published by <cit.>. In particular, we note that while the lines attributed to [FeIII] λ4700 and [CoIII] λ 5900 are similar in shape and velocity for most of our sample (although less similar in flux), the most diversity is seen in the red-side blend of [FeII] and [NiII].For the nebular-phase emission lines we measure three parameters: the line velocity (i.e., from the central wavelength), the FWHM, and the integrated flux. These three parameters are connected to the physical state of the ejecta in the following ways.(1) The emission line for an expanding symmetric sphere of SN Ia ejecta material should be centred at zero velocity in the rest frame. Any offset is indicative of a net flow of the material toward or away from the observer, which indicates asymmetry in the nucleosynthetic products (e.g., ). (2) The FWHM represents the total distribution in velocity of the material along the line of sight, which is related to the density profile of the material at the time of explosion and its kinetic energy (e.g., ).(3) The integrated flux is a measure of the total amount of energy escaping the nebula, and is related to the synthesised mass of ^56Ni and the physical state of the nebula (e.g., the temperature, ionisation state, positron trapping efficiency; ).We use these data as an opportunity to compare two main methods for calculating emission-line parameters: fitting a Gaussian function and direct measurement. These two methods have been used by (for example) <cit.>, <cit.>, and <cit.>. For the Gaussian fits, we follow the same procedure as did <cit.>: we subtract a pseudocontinuum created by joining the local minima on each side of the emission line, and perform a least-squares fit of a Gaussian function (with two components for the blended [FeII] and [NiII] lines) to the non-normalised, flux-calibrated spectra and bootstrap our parameter uncertainties. For the direct measurements, we first subtract the pseudocontinuum and then smooth the spectrum (specifically, we use the IDL routine POLY_SMOOTH with the Savitzky-Golay filter option and a 100 Å bin size). We then calculate the line velocity from the wavelength of the pixel at peak flux, the FWHM from the difference in wavelength between the two pixels at which the flux value is half the peak flux, and the total line flux by analytically integrating over the line. We estimate the uncertainties in these line parameters with a bootstrap method: we randomly scatter the pixel flux values by an amount proportional to their uncertainty, recalculate the line parameters in 300 trials, and take the standard deviation of the results as the parameter's error value. For the rest-frame central wavelengths of the nebular emission lines, we use [FeIII] λ4701, [CoIII] λ5891, [FeII] λ7155, and [NiII] λ7378.The resulting line parameters and their uncertainties for the Gaussian fits and the direct measurements are presented in Tables <ref> and <ref>, respectively. Note that the blended [FeII] and [NiII] lines' parameters can only be measured with the Gaussian fitting method. The two methods are compared for each measurement in Figure <ref>. From these plots we take away three main points: (1) the values deviate the most when the uncertainties are large, (2) the direct method leads to more precise measurements in general, and (3) there is a small systematic offset in the Gaussian-fit integrated fluxes. This offset was also remarked on by <cit.> (i.e., their Figure 3), and is caused by deviations in the line profile from Gaussianity. In addition to the measurement errors of these emission features, there are systematics introduced by line-blending with weaker features from other species. For example, Figure 3 of <cit.> demonstrates how models of nebular spectra indicate that the [FeIII] λ≈ 4700 Å and the [CoIII] λ≈ 5900 Å lines are reasonably pure, but that the [FeII] and [NiII] blend can be contaminated by [CaII], especially in SNe Ia that synthesise relatively little stable nucleosynthetic products. Fortunately, it appears that strong contamination from calcium would also cause the two emission lines to be quite heavily blended into almost a single feature, whereas we see a clear double peak in all our spectra (Fig. <ref>). To fully quantify the impact of line blending requires detailed modeling which we consider beyond the scope of this analysis.In the following sections we analyze our observations for each of the SNe Ia in our sample individually, and put them in context with other existing published observations where available. One of these contexts that we will frequently refer to is the off-centre explosion model of <cit.>, proposed to explain the observed relation that HVG (LVG) SNe Ia also exhibit nebular emission lines of [FeII] and [NiII] that are redshifted (blueshifted or unshifted). In this model, an off-centre ignition shifts the deflagration burning either toward or away from the observer's line of sight, imparting a blueshift or a redshift to the nebular emission lines of the nucleosynthetic products of the deflagration: [FeII] and [NiII]. After the deflagration phase, the detonation proceeds throughout the rest of the material. <cit.> find that on the side opposite to the off-centre ignition, the burned material has a shallower density distribution, and suggest that a SN Ia observed from this side will exhibit a higher SiII absorption-line velocity and velocity gradient. This is how an ignition offset away from the observer along the line of sight will result in an HVG SN Ia with redshifted nebular lines, and vice versa. In the following sections we will refer regularly to this correlation between the SiII velocity gradient and the nebular emission-line velocity, and discuss whether each of our SNe Ia exhibits characteristics that are consistent with the model of <cit.>. §.§ SN 2012fr SN 2012fr was discovered on October 27, 2012 (UT dates are used throughout this paper), two weeks before it reached peak brightness <cit.>. It was classified as a young Type Ia SN with a spectrum obtained on October 28, 2012 <cit.>. Spectra obtained three and five days later exhibited a SiII λ 6355 feature with the main photospheric component at ∼ 14,000 km s^-1 and a detached, high-velocity component at ∼ 19,000 km s^-1 (high-velocity CaII H&K was also seen; ). SN 2012fr was well offset from its host, NGC 1365, a face-on Sb-type galaxy at a distance of D=18.07±2.7 Mpc (z=0.005457; )[The redshift-independent summary statistic of 50 distances in the literature, from the NASA Extragalactic Database, NED: <https://ned.ipac.caltech.edu/>.]. As SN 2012fr was nearby, radio observations were triggered and <cit.> report a nondetection at 5.8 GHz with the Very Large Array on October 30, 2012 with an upper limit of 18 μ Jy. Pre-explosion archival Hubble Space Telescope imaging existed for the host galaxy; <cit.> report a nondetection at the location of the SN, and put an upper limit on the progenitor system brightness of M_V≳-5.9 mag, ruling out a supergiant companion star.Analyses of the evolution in the optical photometric, spectroscopic, and spectropolarimetric features of SN 2012fr are presented in the literature. <cit.> give a dense time series of optical spectra, showing that the high-velocity component of SiII λ 6355 fades by 5 days before peak brightness, and that the photospheric component is narrow and remains at v≈ 12,000 km s^-1 after peak brightness, classifying SN 2012fr as an LVG SN Ia. They also point out that while the overluminous peak brightness and slow decline of SN 2012fr resemble those of SN Ia 1991T <cit.>, spectroscopically SN 2012fr sits on the border of SN Ia subclasses (normal vs. high-velocity, and shallow-silicon vs. core-normal in the scheme of ). <cit.> present spectropolarimetry around peak brightness and use the CaII near-infrared triplet to determine that the polarisation angle of the high-velocity component is oriented at 90 from that of the low-velocity components of CaII and SiII, and that the overall continuum polarisation is low. They suggest that their observations support a delayed-detonation model over a merger model for SN 2012fr, and furthermore predict that the iron-group nebular lines will be blueshifted (based on the model of ). <cit.> present an analysis of optical and ultraviolet (UV) photometry covering the full light curve out to ∼120 days, plus a time series of optical spectroscopy to +71 days after peak brightness. They evaluate the bolometric light curve of SN 2012fr and derive the mass of ^56Ni synthesised in the explosion, M_ Ni = 0.88 ± 0.08 M_⊙, which they point out is smaller than that of SN 1991T but larger than that of the paragon of normal SNe Ia, SN 2011fe. We find that the nebular-phase spectra of SN 2012fr deviate from expectations in two ways. First, although it was securely classified as an LVG SN Ia based on its v̇_ Si II≈ 0 km s^-1, the nebular lines of [FeII] and [NiII] exhibit a redshift of ∼ 2500 km s^-1 instead of the blueshift generally seen for LVG SNe Ia <cit.>. We speculate that these redshifted nebular lines may be related to the relatively high SiII absorption-line velocity of SN 2012fr, v_ Si II( peak) ≈ 12,000 km s^-1 d^-1, which is more consistent with an HVG-like SN Ia and would correlate with redshifted nebular emission lines in the offset ignition model. Second, the velocity of the [FeIII] λ4700 line is typically observed to progress redward over time in the nebular phase, but in SN 2012fr it appears to be significantly bluer, by ∼ -1400 km s^-1, at +415 days than at +289 days (see the inset plot in Figure <ref>). Although the later epoch's [FeIII] λ4700 line has a broader, flatter shape and lower S/N than the earlier epoch — which could cause a bluer peak wavelength to be reported from our fits — we find that this blueward progression is not an artifact. In the top-left panel of Figure <ref>, the inset plot compares the iron complex for the two epochs, clearly showing how their blue-side edge broadens while the red-side edge remains at the same wavelength, which is mainly driving the shift in the peak of the line. The evolution in iron line velocities at late times are discussed further in Section <ref>. §.§ SN 2012hr SN 2012hr was discovered in an image taken on December 16, 2012 by the Backyard Observatory Supernova Search (BOSS; ) in the host galaxy ESO 121-G026 (also known as PGC 18880; host offset 2 W and 94 N), a barred spiral galaxy located at redshift z = 0.007562 <cit.>. The Carnegie Supernova Project obtained a near-infrared spectrum of SN 2012hr on December 20, 2012 and classified it as a SN Ia approximately a week before peak brightness, exhibiting MgII absorption at 12,200 km s^-1 <cit.>. SN 2012hr was also slightly overluminous (Table <ref>), similar to SN 2012fr. The similarity of SN 2012hr to SN 2012fr extends to late times, with similar shapes and velocities for their [FeIII] and [CoIII] lines, as seen in Figure <ref> and Table <ref>. As shown in Table <ref> and discussed in Section <ref>, we directly measure the SiII velocity gradient of SN 2012hr to be v̇_ Si II = 120 ± 80 km s^-1 and classify it as an HVG SN Ia. However, since the large uncertainty is consistent with LVG SNe Ia (v̇_ Si II < 70 km s^-1), and its SiII velocity (v_ Si II( peak) ≈ 11,500 km s^-1 d^-1) is lower than typically seen for HVG SNe Ia (∼ 12,000 km s^-1 d^-1), we consider this classification insecure and label it as “HVG?". In this way, it is dissimilar to SN 2012fr. In our nebular spectra of SN 2012hr we find that the [FeII] and [NiII] lines are redshifted, as expected for an HVG object, but by only a small amount (<1000 km s^-1): this is smaller than most of the HVG SNe Ia in the sample of <cit.>, but still consistent with the lower boundary (i.e., SN 2002er in their Figure 2). The relationship between photospheric and nebular line velocities is further discussed in Section <ref>. §.§ SN 2013aa SN 2013aa was discovered by the Backyard Observatory Supernova Search (BOSS) on February 13, 2013 <cit.>. SN 2013aa is 74 W and 180 S from the core of NGC 5643, a barred spiral galaxy at D ≈ 16.9 Mpc <cit.>. Follow-up observations by the Las Cumbres Observatory on February 15, 2013 spectroscopically confirmed SN 2013aa as a SN Ia near maximum light <cit.>. We securely classify SN 2013aa as an LVG SN Ia based on a direct measurement of v̇_ Si II = 50 ± 10 km s^-1 (Table <ref>), and also find that it was overluminous at peak brightness (Table <ref>).Our nebular-phase spectrum of SN 2013aa is at a much later phase (+400 days) than the other SNe Ia in our sample. This was possible because it is one of the most nearby objects, and thus brighter in apparent magnitude. At +400 days we see that the [FeIII] λ4700 line is redshifted, similar to SNe Ia at late times in general (see Section <ref> for further discussion). The emission lines of [FeII] and [NiII] are both blueshifted, consistent with expectations for an LVG SN Ia under the model of <cit.>. We also find that [CoIII] λ5900 has a broader, flatter profile in SN 2013aa than some of our other spectra, but that it is similar to our spectra of SN 2012fr and SN 2013dy at comparably late phases, >1 yr after maximum light. An interpretation of [CoIII] line-profile shapes at late nebular phases is discussed in Section <ref>. §.§ SN 2013cs SN 2013cs was first discovered by the La Silla Quest Survey (as LSQ13aiz) on May 12, 2013 in the host galaxy ESO 576-G017 <cit.>, and independently discovered on May 14, 2013 by the Catalina Real-time Transient Survey in an unfiltered Siding Spring Survey Image <cit.>. SN 2013cs was classified as an SN Ia by Yamanaka and collaborators using an optical spectrum obtained on May 15, 2013 <cit.>. This spectrum, at a phase of -10 days (i.e., before peak brightness), exhibited SiII λ6355 at a high velocity of 15,000 km s^-1 and the rarely observed CII λ 6580 absorption feature. At peak brightness the SiII velocity had decreased to 12,500 km s^-1, but this is still quite high, similar to that exhibited by HVG SNe Ia. We do not have any photospheric-phase spectra in the two weeks post-maximum, so we cannot confirm that it belongs in the HVG class, and have instead given it the insecure designation of “HVG?" in Table <ref>.Our +269 day nebular-phase spectrum of SN 2013cs is similar to that of SN 2012fr and SN 2012hr, as seen in Figure <ref>. The emission lines of [FeII] and [NiII] in SN 2013cs are redshifted by an average of v_ neb≈2600 km s^-1. This is a much higher velocity than in SN 2012hr, our only other “HVG?"-classified SN Ia, and consistent with the expectation for an HVG object in the asymmetric explosion model of <cit.> (discussed further in Section <ref>). §.§ ASASSN-14jg ASASSN-14jg was discovered by the All Sky Automated Survey for SuperNovae (ASASSN) on October 29, 2014 using the double 14 cm Cassius telescope in Cerro Tololo, Chile <cit.>. It is located 8.5 north and 12.5 east from the core of PGC 128348, a spiral galaxy at D = 59.6 Mpc. ASASSN-14jg was classified as an SN Ia with a FLOYDS spectrum obtained on the Faulkes Telescope South <cit.>. A spectrum obtained on November 6, 2014 by the ANU WiFeS SuperNovA Program (AWSNAP) showed no local Hα emission but did exhibit the NaI λλ5890, 5896absorption feature at a velocity of 8.9 ± 15.3 km s^-1 with respect to the SN host galaxy — i.e., most likely associated with the host-galaxy interstellar medium and not potential evidence of circumstellar material <cit.>. As shown in Tables <ref>, ASASSN-14jg exhibited a normal peak brightness and decline rate. We could not directly measure the velocity gradient, and instead designate ASASSN-14jg as an “LVG?" SN Ia based on its low velocity of photospheric SiII, v_ Si II( peak) ≈ 10,200 km s^-1 d^-1, as described in Section <ref> (see also Table <ref>). Our nebular-phase spectrum of ASASSN-14jg at +266 days appears to exhibit a relatively higher continuum flux on the red side, as seen in Figure <ref> where all the spectra are scaled to have equal integrated blue-side fluxes. However, the conditions at Gemini South were poor, with variable clouds on the night of the observation, making it difficult to properly flux calibrate the separately obtained blue-side and red-side spectra. Further evidence of this is the large (0.2–0.3 mag) uncertainty in our Gemini South photometry from that night (Table <ref>). The nebular-phase emission lines of [FeII] and [NiII] are significantly redshifted, with an average v_ neb≈2500 km s^-1. This is inconsistent with expectations for LVG-like SNe Ia in the model of <cit.>, which typically exhibit blueshifted nebular emission. This is discussed in further detail in Section <ref>. §.§ SN 2015F SN 2015F was discovered on March 9, 2015 in host galaxy NGC 2422 <cit.>, a peculiar, weakly barred spiral galaxy at D = 20.5 Mpc. SN 2015F is offset by 2.5 from the host-galaxy core <cit.>. PESSTO obtained an optical follow-up spectrum on March 11, 2015 that confirmed SN 2015F as an SN Ia, resembling the spectrum of SN 2002bo∼12 days before peak brightness. They further subclassify SN 2015F as at the boundary of normal and SN 1991bg-like <cit.>[See <cit.> for a description of SN 1991bg.] SN 2015F reached peak brightness on March 25, 2015 with a subluminous maximum B-band magnitude of -18.4 ± 0.8 <cit.>, and exhibited a decline rate of Δ m_B(15) ≈ 1.2 mag. While this is the largest Δ m_B(15) value in our sample of SNe, and SN 2015F does lie just below the <cit.> relation (e.g., Fig. 1 of ), the decline is significantly slower than that of a typical SN 1991bg-like SN Ia (Δ m_B(15) ≈ 1.9 mag) and more similar to that of a normal SN Ia.<cit.> find that the early-time spectra of SN 2015F exhibit strong CaII high-velocity features and an absorption feature at ∼6800 Å, which has been noted in several SN 1991bg-like SNe Ia. They attribute this absorption feature to either photospheric AlII or unburned carbon material in the outer layers of the SN ejecta. Aluminum has not been observed in other normal SNe Ia, but it has been reported in peculiar SNe like SN 2010X <cit.>.<cit.> present first-light photometry of SN 2015F from their daily-cadenced survey and identify extremely early-time emission occurring ∼1.5 days before the light curve begins to rise in earnest. Specifically, they find two 2σ detections out of 38 observations, on days -2 and -3. Statistically, one should observe 0.05×38 = 1.9 detections at 2σ in such a dataset, so these observations are not inconsistent with a nondetection. Despite this, <cit.> interpret this as the signature of ejecta interacting with a nondegenerate companion star of at least R>0.1 R_⊙. In this progenitor scenario we may see swept-up hydrogen radiating at late times, but given the underlying host-galaxy emission for SN 2015F, that is unfortunately impossible to confirm with our data. Returning to the suggestion that SN 2015F's subluminous peak brightness might make it similar to the SN 1991bg-like class, we consider whether SN 2015F is SN 1991bg-like at late times. In our +279 day optical nebular spectrum of SN 2015F, we observe that the line velocities and widths are similar to those of other normal nebular-phase SNe Ia of our sample (i.e., SN 2013aa excluded). In this way the nebular features of SN 2015F are dissimilar to those of SN 1991bg, the latter being dominated by narrow [FeIII] at a similar phase <cit.>, which was suggested by <cit.> to be from a high degree of ionisation in the nebula at late times. Our nebular-phase spectrum of SN 2015F exhibits [FeII] and [NiII] emission lines that are nearly equal in peak flux, which is unique among our sample of SNe Ia. As [FeII] represents a product of the decay of radioactive ^56Ni and any Ni at late times must have been formed stably, this observation qualitatively indicates that SN 2015F produced a relatively lower ratio of radioactive to stable nucleosynthetic products compared to other SNe Ia. This may be an indication of a higher than average metallicity for the progenitor star of SN 2015F. In this scenario, the additional neutrons from the higher abundance of heavy elements could lead to more stable nuclear burning (e.g., as investigated by ). As a final note, the nearly equal peak flux of the [FeII] and [NiII] emission lines of SN 2015F is similar to the case of SN 1991bg, although the latter were more blended. <cit.> show that model explosions with a low central density, a low mass of radioactive ^56Ni (both of which can affect the ionisation state at late times), and/or an additional boost from CaII at 7200 Å can all generate simulated spectra that match those of SN 1991bg in this respect, but it is difficult to isolate the underlying physical cause.For SN 2015F we could not directly measure the velocity gradient, and instead designate it as an “LVG?" SN Ia because it exhibited a low photospheric SiII velocity, v_ Si II( peak) ≈ 10,100 km s^-1 d^-1, as described in Section <ref> and listed in Table <ref>. The average nebular-phase emission-line velocities of [FeII] and [NiII] is v_ neb≈ 340 ± 120 km s^-1 (from Table <ref>), slightly redshifted yet still consistent with the upper bound for LVG SNe Ia from <cit.> (i.e., <1000 km s^-1, from their Figure 2).§.§ SN 2013dy SN 2013dy was discovered by the Lick Observatory Supernova Search <cit.> with the 0.76 m Katzman Automatic Imaging Telescope (KAIT) in an image taken on July 10, 2013 <cit.>. It was detected only 2.4 hr after first light (equivalent to the explosion time unless there is a dark phase, which might last many hours), making it the earliest-observed SN Ia <cit.>. SN 2013dy is located in the host galaxy NGC 7250, an irregular/diffuse spiral galaxy at D ≈ 12.4 Mpc (this is a mean of redshift- and SN-independent distance measurements[<ned.ipac.caltech.edu>] from , , and ). SN 2013dy is offset 2.3 west and 26.4 north of its host galaxy's core (∼1.76 kpc). It was spectroscopically classified as a normal SN Ia with a strong CII absorption feature at early times <cit.>.<cit.> present a thorough analysis of SN 2013dy with UV-optical-infrared photometry and spectroscopy up to +500 days post-explosion. With early-time photospheric-phase spectra, they securely classify SN 2013dy as a member of the LVG subclass. They find that the light curve declines slowly, like that of SN 1991T, but existing redshift-independent distances (based on the Tully-Fisher relation) indicate that it was not overluminous like SN 1991T, even after significant host extinction (A_V, host=0.64 mag) is accounted for <cit.>. However, they also show that the distance modulus derived directly from the light curve of SN 2013dy is significantly larger. In Table <ref>, we quote the intrinsic peak B-band magnitude of M_ peak(B) = -19.66, based on the SN-derived distance and not the Tully-Fisher relation. This is at the bright end of the <cit.> relation (e.g., Figure 13 of <cit.>).With their late-time optical spectra of SN 2013dy, <cit.> show how the [FeIII] line velocity evolves redward from ∼-2700 km s^-1 at +100 days to +1350 km s^-1 at +400 days. This is in agreement with our own measurement of -476 km s^-1 at +335 days. In the asymmetric model of <cit.>, this blueshifted nebular emission velocity is consistent with expectations for SN 2013dy, as it is a member of the LVG subclass. <cit.> find that SN 2013dy exhibits weaker [FeII] and [NiII] than the paragon SN 2011fe, as is also seen in our Figure <ref>. This leads them to infer a larger mass of ^56Ni for SN 2013dy than for SN 2011fe, which they note is consistent with the broader light curve (see paragraph above). §.§ SN 2013gy SN 2013gy was discovered by LOSS with KAIT on December 6, 2013 in the host galaxy NGC 1418 <cit.>, offset 35.08 from the nucleus <cit.>. NGC 1418 is a barred spiral galaxy at D ≈ 66 Mpc (a mean of redshift-independent distances[<ned.ipac.caltech.edu>] from , , and ). SN 2013gy was classified as a normal SN Ia with an optical spectrum obtained on December 7, 2013 <cit.>, consistent with its normal light curve. We did not have the necessary post-peak spectra to directly measure the velocity gradient for SN 2013gy, and instead designate it as an “LVG?" SN Ia because it exhibited a low photospheric SiII velocity, v_ Si II( peak) ≈ 10,700 km s^-1 d^-1, as described in Section <ref> and listed in Table <ref>.In our +275 day optical nebular spectrum, SN 2013dy appears normal and similar to the others, except that like ASASSN-14jg, it seems to have a relatively large red-side flux. In spectra of faint objects that have a flux gradient near the 5600 Å dichroic, joining the blue and red sides of an LRIS spectrum can be problematic, and in Figure <ref> we see substantial noise at 5600 Å. Co-epochal photometry (as we obtained for some objects with GMOS) can help resolve such issues, but we did not obtain LRIS photometry of SN 2013gy. However, this is not a serious issue, because our analysis does not rely on an absolute flux calibration; our measurements of line widths, velocities, and continuum-subtracted fluxes should be unaffected. We find that the lines of [FeII] and [NiII] are slightly blueshifted, with an average v_ neb≈ -100 km s^-1, consistent with expectations of LVG-like SNe Ia in the model of <cit.>, as discussed in Section <ref>.§ DISCUSSION As in the analysis above, we focus our discussion on the attributes of our individual events and provide physical interpretations for their characteristics with respect to SN Ia progenitor scenarios and explosion mechanisms. When appropriate, we compare the distribution of derived parameters for our eight events to those of larger published datasets, but otherwise we find that our sample size is too small to provide new statistical constraints on correlations involving nebular-phase spectra of SNe Ia (e.g., the relation between emission-line redshift/blueshift and the photospheric SiII velocity gradient). We hope that future analyses of the large, communally available samples of nebular-phase SN Ia spectra will incorporate our data and the derived properties, but such a study is beyond the scope of this paper. We now discuss the explosion asymmetry (Section <ref>), limits on hydrogen in the system (Section <ref>), and the redward progression of the ∼4700 Å feature (Section <ref>). §.§ Explosion Asymmetry As described briefly elsewhere in this paper, an off-centre ignition followed by an asymmetric SN Ia explosion has been proposed as the underlying physical cause ofan observed correlation between the photospheric-phase velocity gradient of SiII and the nebular-phase velocity of the emission lines of the nucleosynthetic products <cit.>. In the model of Maeda et al., the outer layers of the ejecta on the same side as the explosion are compacted, and relatively denser than the outer ejecta layers on the opposite side. A SN Ia in which the explosion is offset away from the observer (on the far side along the line of sight) exhibits photospheric SiII line velocities that evolve more rapidly because the photosphere recedes more quickly down into the slower layers of ejecta — i.e., they belong to the HVG group. A SN Ia with this orientation with respect to the observer will also exhibit redshifted nebular-phase emission lines from the nucleosynthetic products. This is reversed for SNe Ia which are observed with an offset explosion on the near side, toward the observer: an LVG SN Ia with blueshifted nebular-phase emission lines. A near-side offset also predicts a narrower light curve and fainter peak luminosity, and so has been proposed as a contributing factor to the width vs. luminosity correlation <cit.>.<cit.> initially demonstrated the relation between the photospheric SiII velocity gradient and the nebular emission line velocity in a small sample of 17 SNe Ia, and this trend was confirmed by <cit.> with a larger sample. Unfortunately, our small sample has only one tentative HVG SN Ia with a direct measurement of the SiII velocity gradient: SN 2012hr exhibits v̇_ Si II = -120±80 km s^-1 d^-1, but its large uncertainty makes it consistent with the LVG group. As discussed in Section <ref>, HVG SNe Ia typically also exhibit higher-velocity photospheric SiII near maximum light, and by using this method we also tentatively identify SN 2013cs as a potential HVG SN Ia. In the top panel of Figure <ref> we plot the photospheric-phase velocity of SiII versus the nebular-phase velocity of the [FeII] and [NiII] emission lines for all our SNe Ia, along with the sample from <cit.> for comparison. In the bottom panel of Figure<ref> we plot the SiII velocity gradient vs. the nebular line velocity for the four SNe Ia with directly measured v̇_ Si II, along with the sample from <cit.> for comparison.Our goal here is not to confirm the relation (or the proposed physical model) — our sample is too small to make an impact in this regard, even if we had direct measures of v̇_ Si II for all objects — but to put our sample of SNe Ia in context. We instead remark that we see the same general structure in this diagram as did <cit.> and <cit.>: the tentative HVG-group SNe Ia exhibit redshifted nebular-velocity emission lines associated with the nucleosynthetic products. The two notable exceptions here are SN 2012fr and ASASSN-14jg, which had redshifted nebular emission lines but were classified as LVG, but there are several caveats to mention. SN 2012fr was classified as an LVG SN Ia with directly measured v̇_ Si II, but as we can see in the top panel of Figure <ref>, the velocity of its photospheric SiII absorption line was similar to that of other HVG SNe Ia. Based on the clearly and significantly redshifted nebular lines, SN 2012fr remains a candidate for an off-centre ignition oriented away from the observer, but would be a special case in which the associated photospheric-phase SiII velocity gradient, v̇_ Si II, does not evolve as expected by the model of <cit.>. If, as suggested by <cit.>, the ejecta of SN 2012fr were stratified and had only a thin layer of partial-burning products at v>12000 km s^-1, then it was simply impossible to observe the effect of a fast-receding photosphere using the SiII absorption line in this case. ASASSN-14jg was classified as an LVG SN Ia based on its low photospheric SiII velocity, v_ Si II = 10,200 km s^-1, but the earliest spectrum was obtained 12 days after peak brightness. If it had a v̇_ Si II = 150 km s^-1 d^-1, then v_ Si II( peak) ≈ 12,000 km s^-1. Ultimately, we find that all of our observations are potentially consistent with an off-centre ignition model, although some deviations from the relation between nebular line velocity and v̇_ Si II from <cit.> are required.§.§.§ Collisional SN Ia Candidates Asymmetric explosions are also predicted by the double white dwarf collision scenario (e.g., ) in which two carbon-oxygen white dwarf stars collide. They could be two single stars in an environment of high local stellar density, or a binary system with altered trajectories from (for example) tidal interactions or the Kozai mechanism <cit.>. A tell-tale signature of the collisional model is predicted at late times if the collision is aligned with the observer's line of sight: double-peaked nebular-phase emission lines from the two regions of nucleosynthetic products with distinct velocities, as identified in [CoIII] by <cit.>. Depending on the viewing angle, a collision-triggered SN Ia could also have nebular-phase emission lines that appear relatively normal or broadened.We look for collisional candidates in our sample of SNe Ia by examining the profiles of their [CoIII] features. In Figure <ref> we plot our nebular-phase spectra in the region of [CoIII] along with the nebular-phase spectrum of SN 2003gs <cit.>, which is one of the 3 SNe Ia exhibiting double-peaked [CoIII] identified by <cit.> as a collisional candidate. SNe Ia ASASSN-14jg and 2013gy are omitted from this plot because their spectra suffer from an unfilled chip gap and poor S/N in this region, respectively. We do not find any clear indication of two velocity components for the [CoIII] λ5900 line of SNe 2012fr (at 289 days), 2012hr, 2013cs, or 2015F, but we do find this line has broadened and deviated from a Gaussian shape in SNe 2012fr (at 415 days), 2013aa, and 2013dy. The former group's spectra are all at <300 days, while the latter's were obtained >300 days after peak brightness. Although broadened features can be created by multiple velocity components, we find it more likely that the shape of this feature is simply common in nebular-phase spectra at and beyond 1 yr after peak brightness. A similar observation of a broad and flattened 5900 Å feature was made for prototypically normal SN 2011fe at >1 yr by <cit.>. They postulated that the observed emission near 5900 Å may instead be from NaI D, which is predicted by models to dominate during later nebular phases (>1 yr). None of our SNe Ia fall into the low-luminosity, fast-declining subgroup in which <cit.> find all their collisional candidates (i.e., Δ m_15(B)>1.3 mag). In fact, SNe 2012fr, 2013dy, and 2013aa were the three most intrinsically luminous of our sample. While it may not necessarily be that all collisions produce underluminous SNe Ia — the existing models are able to predict a wide range of ^56Ni masses <cit.> — there is another potential underlying physical cause of broad line profiles in SNe Ia. Overluminous SNe Ia create a higher mass of radioactive nickel, have more energy to impart to their ejecta, and can thus exhibit broader FWHM in their nebular-phase emission lines. However, in this scenario the line profiles would still retain a Gaussian shape.Although we find that some of our nebular spectra have broadened, non-Gaussian emission features near [CoIII] λ5900, none exhibits the clear double-peaked profiles seen by <cit.>, and ultimately we cannot confirm them as collisional candidates. §.§ Hydrogen at Nebular Phases In the single-degenerate model for SNe Ia, mass is accreted onto the primary white dwarf from a main-sequence or red-giant companion star. In this scenario, unaccreted hydrogen from the companion could be swept-up by the SN Ia ejecta and create an observable emission line with a velocity v≈ -1000 km s^-1 and a FWHM ≈ 22 Å (e.g., , Figure 9). Recently, <cit.> presented the results of a statistical search for hydrogen in 11 SNe Ia and found a tentative detection in one of them. Despite our selection criteria that the objects we observe be well separated from the host galaxy, a visual inspection of the 2D spectra in the region of Hα reveals that extended Hα emission from the host intersects with the trace of the SN in four of our eight objects.The four SNe Ia in apparently “clean” environments are SNe 2012fr, 2012hr, 2013aa, and 2013gy. However, for SNe 2012hr the residuals from sky-line subtraction in the 1D spectrum in the vicinity of Hα cause the S/N to be too poor to constrain the presence of Hα emission. Also, we find that SN 2013gy is projected onto a z ≈ 0.52 background galaxy exhibiting narrow emission lines of Hα, Hβ, Hγ, and [OIII], and that the Hβ line appears near to where we would look for Hα at SN 2013gy's redshift. In Figure <ref>, we show our spectra in the region of Hα for SNe 2012fr and 2013aa, in which we find no discernible emission line. To determine an upper limit on the Hα emission-line flux in our spectra, we follow an analysis similar to that of <cit.>, which in turn is based on the methodology of <cit.>, <cit.>, and <cit.>, and on the models of <cit.> and <cit.>; see also searches for hydrogen in near-infrared SN Ia spectra by <cit.>. To summarise briefly, the maximum size of an undetected emission line is modeled as a Gaussian with FWHM ≈ 22 Å and a peak intensity 3 times the RMS flux error in the region. For the flux uncertainties we use the residual between the data and the local continuum, which we estimate by applying to the data a least-squares polynomial smoothing filter with a wide window. In Figure <ref> we draw this maximum undetected Hα emission line onto our spectra, and write the peak intensity in the upper-left corner of the plot. We use the distances in Table <ref> to find that the intrinsic upper limit on the Hα intensity is ∼ (1–2) ×10^35 erg s^-1 Å^-1 for SNe 2012fr and 2013aa, which we convert to upper limits on the mass of hydrogen in the system of < (1–2) × 10^-2 M_⊙ (following the same method as, say, ). Ultimately, we find that our limits on the mass of hydrogen in the systems of SN 2012fr and SN 2013aa are quite high, and that we cannot use them to constrain the progenitor scenarios.§.§ The Redward Evolution of [FeIII] The continuous redward progression of the blueshifted peak of the [FeIII] feature at λ≈ 4700 Å was first noted by <cit.> and confirmed in a larger sample of nebular-phase SN Ia spectra by <cit.>. The latter find that the evolution in the velocity of [FeIII] begins at -4000 to -2000 km s^-1 at 100 days after peak brightness, increases by 10–20 km s^-1 d^-1, and reaches v≈0 km s^-1 at >200 days after peak brightness. They also find that the evolution appears to continue, with the line becoming redshifted. Recently, the evolution in the nebular-phase emission lines was explored using models for SN 2011fe by <cit.>. They find that a change in the dominant species contributing to the line flux from FeIII and FeII to FeI is the underlying cause of the apparent velocity shift, as the emission lines of FeI are just redward of FeIII and FeII. Additionally, <cit.> combine spectral synthesis modeling with a sample of 160 nebular-phase spectra for 27 normal SNe Ia; they find that line blending, changes in opacity, and contributions from permitted iron lines may all participate in the redward evolution of the nebular-phase Fe lines at λ≈ 4700 Å. All of our spectra were obtained >200 days past maximum brightness, and for most of our objects we have only one epoch, so we cannot confirm or investigate the above findings in the same way. Instead, in Figure <ref> we plot the [FeIII] λ4700 line velocity as a function of phase for the SNe Ia in our sample, and compare with the dataset of <cit.>. We see that all of our <300 day spectra still exhibit a blueshifted [FeIII] λ4700 line, and our sample does demonstrate the general trend that spectra at later phases exhibit higher-velocity FeIII lines — with the exception of SN 2012fr at >400 days. As presented in Section <ref> and Figure <ref>, we find that the blue-side edges of the [FeIII] λλ4700, 5300 emission lines have moved to bluer wavelengths at the later epoch, while the red-side edges match fairly well: SN 2012fr is a true outlier in terms of it's 400 day [FeIII] λ4700 line velocity. Given the work of <cit.> described in the preceding paragraph, we suspect that this is the result of a different opacity or ionisation level in the nebula of SN 2012fr compared to other SNe Ia – especially considering that SN 2012fr was a more energetic explosion, forming a larger mass of ^56Ni as described in Section <ref>. <cit.> and <cit.> also discuss how the early-time observations of SN 2012fr suggest the explosion may have been a delayed detonation or a double detonation, and the constraint of a surface layer of He could have affected the nebular-phase evolution. However, given that <cit.> find a myriad of underlying physical causes at play in generating the observed line positions in nebular spectra, and mindful of the risk of overinterpreting our analysis, we leave further evaluation to detailed models of the explosion of SN 2012fr, which are beyond the scope of this work (see, e.g., the fine treatment of SN 2011fe by ). § CONCLUSIONS We have presented new nebular-phase spectra from Gemini and Keck Observatories for eight nearby SNe Ia which were well observed at early times by the Las Cumbres Observatory. Data products for these SNe Ia were derived at early and late times, including light-curve parameters such as peak brightness and B-band decline rate, photospheric-phase spectral parameters such as the velocity of the SiII absorption line, and nebular-phase emission-line parameters such as FWHM and velocity. We provided an analysis comprised of biographies for each of our SNe Ia as individual, unique events, remarking on particular characteristics and discussing instances where predictions about the late-time qualities from the early-time behaviour were or were not observed. For example, based on its photospheric-phase SiII velocity, SN 2012fr should have exhibited blueshifted nebular emission lines, but did not. We demonstrated the diversity in nebular-phase features of SNe Ia by comparing our eight spectra with each other and with the fiducial SN Ia 2011fe, and interpreted these features with respect to various physical models for SN Ia progenitor scenarios and explosion mechanisms. None of our SNe Ia is a good candidate for the collisional scenario. Although we determined that two of our spectra were of appropriate quality to evaluate an upper limit for the mass of hydrogen from a nondegenerate companion, our results do not place strong constraints on the progenitor system. We also commented on the redward progression of the peak wavelengths of the iron emission lines at late phases and noted that SN 2012fr is again the outlier in this category for its apparent blueward shift, but that modeling of nebular ejecta would be necessary to properly interpret this peculiarity. The data released by this publication should also be useful for future conglomerate studies of the growing sample of nebular-phase spectra of SNe Ia.§ ACKNOWLEDGEMENTSWe thank our anonymous referee for a thoughtful and constructive review. This research makes use of observations from the Las Cumbres Observatory and is supported by US National Science Foundation (NSF) grant AST–1313484.We made use of Swift/UVOT data reduced by P. J. Brown and released in the Swift Optical/Ultraviolet Supernova Archive (SOUSA). SOUSA is supported by NASA's Astrophysics Data Analysis Program through grant NNX13AF35G.This work is based in part on observations and discoveries from the the Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory. We are grateful to the staff at Lick Observatory for their assistance. KAIT and its ongoing operation were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation. Research at Lick Observatory is partially supported by a generous gift from Google. We also used observations obtained at the Gemini Observatory, acquired through the Gemini Observatory Archive through programs GS-2013A-Q-10, GS-2013B-Q-48, GS-2015B-Q-62, and GS-2016A-Q-61 and processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the NSF (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). We thank the queue service observers and technical support staff at Gemini Observatory for their assistance with the acquisition and reduction of our data.This work is based in part on observations from the Low Resolution Imaging Spectrometer at the Keck-1 telescope. We are grateful to the staff at the Keck Observatory for their assistance, and we extend special thanks to those of Hawaiian ancestry on whose sacred mountain we are privileged to be guests. The W. M. Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; it was made possible by the generous financial support of the W. M. Keck Foundation. We thank Daniel Perley[http://www.astro.caltech.edu/∼dperley/programs/lpipe.html] and S. Bradley Cenko for the use of, and assistance with, their Keck LRIS imaging and spectroscopy reduction pipelines. We also thank Patrick Kelly, WeiKang Zheng, Isaac Shivvers, and Brad Tucker for their assistance with some of the observations. The supernova research of A.V.F.'s group at U.C. Berkeley is supported by Gary & Cynthia Bengier, the Richard & Rhoda Goldman Fund, the Christopher R. Redlich Fund, the TABASGO Foundation, NSF grant AST–1211916, and the Miller Institute for Basic Research in Science (U.C. Berkeley).A.V.F.'s work was conducted in part at the Aspen Center for Physics,which is supported by NSF grant PHY-1607611; he thanks the Center forits hospitality during the neutron stars workshop in June and July 2017. This research has made use of the online repository of data from the long-term Berkeley Supernova Project that is publicly available at <http://heracles.astro.berkeley.edu/sndb/>.Support for I.A. was provided by NASA through the Einstein Fellowship Program, grant PF6-170148.D.J.S. acknowledges support from NSF grant AST–1517649. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.This work has made use of the Weizmann interactive supernova data repository presented by <cit.> and accessible at <http://wiserep.weizmann.ac.il>.This work has made use of the Open Supernova Catalog that is publicly available at <https://sne.space/>, presented by <cit.>. We thank the OSC creators for their quick responses to our questions.This work has made use of the free, online, collaborative article-writing website <www.authorea.com>. We thank the staff at Authorea for their responsive assistance during the preparation of this document.apj | http://arxiv.org/abs/1708.07799v1 | {
"authors": [
"Melissa L. Graham",
"Sahana Kumar",
"Griffin Hosseinzadeh",
"Daichi Hiramatsu",
"Iair Arcavi",
"D. Andrew Howell",
"Stefano Valenti",
"David J. Sand",
"Jerod T. Parrent",
"Curtis McCully",
"Alexei V. Filippenko"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170825162444",
"title": "Nebular-Phase Spectra of Nearby Type Ia Supernovae"
} |
=-15mm =-10mm =155mm =220mm y_t m_H 11_R11 trz_tx_ty_tp_tq_tk_tu̅q̅ Elastic scattering of a quark from a color field: longitudinal momentum exchangeJamal Jalilian-Marian^1,2,3 ^1Department of Natural Sciences, Baruch College, CUNY,17 Lexington Avenue, New York, NY 10010, USA ^2CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA ^3Centre de Physique Théorique, École Polytechnique, CNRS,Université Paris-Saclay, 91128 Palaiseau, France August 24, 2017 =============================================================================================================================================================================================================================================================================================================================Perturbative QCD in the small Bjorken x limit can be formulated as aneffective theory known as the Color Glass Condensate (CGC) formalism. TheCGC formalism takes into account the dynamics of large gluon densitiesat small x and has been successfully applied to Deep InelasticScattering (DIS) and particle production in high energy hadronic andnuclear collisions in the small x kinematic region. The effectivedegrees of of freedom in CGC are Wilson lines which enter in theeffective quark (and gluon) propagators and re-sum multiple softscatterings from the small x gluon field of the target. It ishowever known that the CGC effective theory breaks down when one probesthe moderately large x (high p_t) kinematics where collinear factorizationand DGLAP evolution of parton distribution functions should bethe right framework. Here we propose a general framework whichmay allow one to eventually unify the two approaches and tocalculate pQCD cross sections in both small and large Bjorken xregions. We take the first step towards this goal by deriving anexpression for the quarkpropagator in a background field which includes scatterings fromboth small and large x modes of the gluon field of the target.We describe how this quark propagator can be used to calculateQCD structure functions F_2 and F_L at all x and thusgeneralize the dipole model of DIS. We outline this approach canalso be used to extend the so-called hybrid approach toparticle production in the forward rapidity region of high energyhadronic and nuclear collisions to all x and p_t regions andspeculate on how one may apply the same techniques to extendthe McLerran-Venugopalan effective action used in high energyheavy ion collisions to include high p_t physics. § INTRODUCTION Parton (gluon) distribution functions are known togrow very fast at small Bjorken x which would lead to a violationof the Froissart bound on growth of physical cross sections with energy.Gluon saturation was proposed <cit.> as a dynamical mechanism which cantame this growth and restore perturbative unitarity. The Color GlassCondensate (CGC) formalism <cit.> is an effective action approachto gluon saturation in QCD that includes coherent multiple scatterings fromthe gluon fields of the target, which are essential when gluon densityof the target is large, i.e. at small x.It also includes high energy effects by re-summing large logs of 1/x viathe JIMWLK/BK evolution equation <cit.> and as such it differs fromthe collinear factorization approach to particle production in pQCD whereparton distribution functions evolve according to the DGLAP evolutionequation <cit.>.The CGC formalism has been developed significantly since its inceptionboth in terms of precision (higher order corrections) as well as the range ofphysical processes considered (see <cit.> for references). Perhapsthe most important aspect of the CGCformalism is the emergence of a dynamically generated and potentially largescale, the saturation scale Q(x, A, b_t), which allows one to understanda wide range of phenomena in high energy QCD by making controlled approximationsand quantitatively reliable calculations. Despite many successes of the CGC formalism, its kinematic domain ofapplicability remains limited; at best it may be applied to processes where thecross section is dominated by small x gluons (as a rough guide one can take small x to mean x < 0.01). Recalling that in particle production in highenergy hadronic/nuclear collisions x and p_t are kinematically correlated,the dominant contribution to high p_t processes is from the not so small xregion. Therefore high p_t particle production is currently not computablein CGC formalism where one must have log 1/x ≫log p_t^2. Furthermore it'sbeen realized quite recently that CGCcalculations of higher order corrections to particle production cross sectionsare not stable and can become negative <cit.> at transversemomenta slightly higherthan the saturation scale Q_s (x). While there has beenvarious remedies proposed to cure this problem <cit.> incase of single inclusivehadron production in asymmetric collisions (such as proton-proton andproton-nucleus collisions in the forward rapidity region in the hybridapproach <cit.>),the general situation remains unsatisfactory and is likely anindication that at highp_t important physics is missing. One such deficiency is the physics ofDGLAP evolution which re-sums potentially large logs of p_t^2 in partondistribution functions in the context of collinearly factorized cross sectionsin pQCD in the leading twist approximation. This is the dominant effect at highp_t where log p_t^2 ≫log 1/x and is not included in the CGC formalism.The two approaches coincide when log p_t^2 ∼log 1/x, knownas the double log limit. Ideally one would like to have a unified formalism for particle production in QCD which reduces to collinear factorization and DGLAP evolution equationat intermediate/large x and/or p_t and to CGC and JIMWLK equation atsmall x, p_t (see <cit.> for instance). In collinear factorization/DGLAPformalism quarks and gluons and their distribution functions are the degreesof freedom used while in the small x limit it is most efficient to use(fundamental or adjoint) Wilson lines, path ordered exponentials of gluonfield which re-sum coherent multiple scatterings of a quark or gluon on asoft color field. This Wilson line is very closely connected to the quark/gluon propagator in a background color field <cit.>. One may thennaturally ask, what would be the effective degrees of freedom in a unifiedapproach? In other words, what could be the building blocks of a physical cross section in a more general approach which has both small x and large x combined?In the small x limit one makes a drastic approximation by taking the x→ 0 limit, for example, in parton splitting functions whereas in DGLAP approach the full splitting function is kept. Clearly to have any hope of a unified approach, one must relax the x→ 0 limit. As x is the ratio of longitudinal momenta or energies of the partons in the parent proton/nucleus, this means that one must consider scattering of a projectile parton not only from the small x modes of the targetproton/nucleus but also from the more energetic modes at large x. Furthermore, DGLAP is a leading twist formalism so that multiple exchangesthat involve extra powers of the hard scale are dropped. Keeping these two points in mind, we propose a new approach which may enable us toeventually "unify" the two formalisms such that one recovers the CGCformalism and JIMWLK evolution equation at small x and pQCD collinearfactorization and DGLAP evolution equation at large x. Toward this end we consider here scattering of a quark from a background colorfield which is, unlike in the small x limit, allowed to have both large and small x modes. We re-sum the multiple scatterings from the fields with small x modes to all orders while keeping only one scatteringfrom large x modes of the field. Therefore the scattered quark can in general carry any energy and transverse momentum and be deflected by large angles, unlikethe small x limit where the scattering is eikonal (energy of the quark remainsthe same) and limited to small transverse momentum exchanges. From thecalculated quarkscattering amplitude we extract the effective quark propagator in this moregeneral background field. We show how this new quark propagator can be used togeneralize the dipole model of the QCD structure functions F_2, F_Lat small x. We expect that the resulting expressions would form the startingpoint for calculating the one-loop correction to the structure functionswhich would then result in a more general evolution equation containingJIMWLK evolution at small x and DGLAP evolution at intermediate/large x.We outline how this new approach may be used to generalizethe CGC framework to include particle production at high p_t.§ MULTIPLE SCATTERING AT SMALL X: EIKONAL APPROXIMATION In this section we briefly review multiple scattering in the eikonallimit and its use in small x physics. There are already excellentreviews covering eikonal scattering <cit.> so everythingin this section is already well-known. Therefore here we use the example ofeikonal scattering to remind the reader of the approximations andassumptions made at small x, and to set up our formalism. In this section we closely follow the approach of Casalderrey-Solana and Salgado in section 4.1 of <cit.> (see also section 4 of the first reference in <cit.>). The light cone coordinates are defined asx^+ ≡t + z √(2) , x^- ≡t - z √(2) and similarly for momenta and fields. We start by considering scattering of a high energy quark moving in the positivez direction, on a high energy target (proton/nucleus) moving to the left as shown in Fig. (<ref>). The quark has momentum p^μ = (p^+ ∼√(s),p^- = 0,p_t =0)whereas the target has momentumP^μ = (P^+ = 0,P^- ∼√(s),P_t = 0) even thoughwhat's really meant in the eikonal approximation is that p^+ ≫ p^-, p_tand likewise for the target momentum. In the CGC formalism one thinks of the target as a current of color charges which hasonly one large component, J^μ_a ≃δ^μ -ρ_a. The color currentis covariantly conserved and satisfies D_μJ^μ = D_- J^- = 0If we now choose to work in the light cone gauge A^+ = 0, we get ∂_- J^- =0 which means the color current (charge) is independent of the coordinate x^-,J^- = J^- (x^+, x_t)To find the color fields generated by this color current one can solve the classical equations of motion D_μF^μν = J^ν in the light cone gauge A^+ = 0 and find the classical solution A^μ interms of the color charge ρ. The only non-zero component of the field is A^- which is independent of the coordinate x^-,A^- = A^- (x^+, x_t)so that in momentum space A^- = A^- (p^+ ∼ 0, p^-, p_t). This is important in eikonal scattering, the targetfield carries almost no plus momentum so that it can not impart any plusmomentum to the projectile. As a result the plus momentum of the projectileis conserved as we will see in detail below. Also related, the small xmomentum modes of the target (moving in the negative z direction) carrynegligible P^- momentum and therefore can not impart any sizable minus component of momentum to the projectile. So basically the only momentumexchanged in the eikonal limit is small transverse momenta. An equivalentway to understand this to consider the coupling of the gauge field A^μto the fermion current u̅ (p_f) γ_μu (p_i). In theeikonal approximation one takes p_f ≃ p_i and usingu̅ (p) γ_μu (p) ∼ p_μ we see that if the projectile quark has a large p^+ it couples only to A^- component of the gauge field in eikonal scattering. This is a general result; the quark moving along the "+" direction in its light cone frame will couple only to the "-" component of the filed in that frame. This is extremely importantand will be used extensively here.Clearly to include large x modes of the target one needs toallow exchange of longitudinal momentum which amounts toscattering of a quark not only from the small x modes of thetarget but also from the large x modes carrying a finitefraction of the target energy P^-. We will consider this inthe next section. Therefore to study scattering of a quark on atarget at small x we consider scattering (propagation) of aquark in a background field A^- which is independent ofcoordinate x^-. Each scattering costs a power of g so that ifA^- ∼ 1/g then gA^- ∼𝒪 (1)and one needs to re-sum all such scatterings. Note that we haven't madeany assumption about the x^+ dependence of the fields which in theextreme limit is usually taken to be a delta function due to Lorentzcontraction of the target. Here we will keep the x^+ dependencegeneral to allow for an extended target. To proceed it is useful todefine a light-like vector n^μ which points in the x^- directionso that n^μ = (n^+ = 0, n^- = 1, n_t = 0) and n^2 = 0 and thatn· A = 0 defines our gauge with = γ^+. Using thislight-like vector one can also writeA^-_a (x^+, x_t)= n^-S_a (x^+, x_t) so that= S = S. So basically the Lorentzindex of the field A^- is carried by the vector n^μ. To start we consider multiple scattering of a quark on the targetcolor fields one scattering at a time; momentum of the incomingquark is denoted p, while q is its outgoing momentum. To makethe approximations made clear (and so that we don't have to repeat thedetails in the next section) we will show all the details of thecalculation here. The amplitude for one scattering is shown inFig. (<ref>) where solid line denotes a quark while the dashed line denotes thesoft gluon field of the target at coordinate x_1^μ = (x_1^+, x_1t). The target is represented by point x_1 and not shown for brevity.The amplitude is i ℳ_1 =(i g) ∫ d^4 x_1e^i (q - p) x_1(q) [S (x_1)]u(p) =(i g) (2π) δ (p^+ - q^+)∫ d^2 x_1td x_1^+ e^i (q^- - p^-) x_1^+e^- i (q_t - p_t) x_1t(q)[S (x_1^+, x_1t)] u(p).Since the field S does not depend on coordinate x^- the integration over x^- is trivial and gives an overall delta function 2 πδ (p^+ - q^+) so that the plusmomentum of the quark is conserved. Including a second scattering is depicted inFig. (<ref>) and givesi ℳ_2 = (i g)^2 ∫ d^4 x_1d^4 x_2∫d^4 p_1(2 π)^4 e^i (p_1 - p) x_1 e^i (q - p_1) x_2(q) [ S (x_2) ip_1^2 + i ϵ S (x_1)]u(p).Once again one can perform the x_1^-, x_2^- integration giving two deltafunctions 2 πδ (p_1^+ - q^+)2 πδ (q^+ - p_1^+) one of which can be used to perform the p_1^+ integration setting p_1^+ = p^+ = q^+ with an overall2 πδ (p^+ - q^+) remaining. We geti ℳ_2=(i g)^2 2 πδ (p^+ - q^+) ∫ d^2 x_1td x_1^+ d^2 x_2td x_2^+∫d^2 p_1t (2 π)^2 d p_1^-(2 π)e^i (p_1^- - p^-) x_1^+e^- i (p_1t - p_t) · x_1t e^i (q^- - p_1^-) x_2^+e^- i (q_t - p_1t) · x_2t(q) [ S (x_2^+, x_2t)ip_1^2 + i ϵ S (x_1^+, x_1t)]u(p)and p_1^+ = p^+ = q^+ in the propagator. The next step is to perform the integral over p_1^-. This can be done via contour integration noticing thatin the numerator is next toand that p_1^- γ^+ = p_1^- = 0. Therefore the integration over p_1^- can be performed by closing the contour below the real axis and gives ∫d p_1^-(2 π)e^i p_1^- (x_1^+ - x_2^+)2 p^+ [p_1^- - p_1t^2 - i ϵ 2 p^+] = - i2 p^+ θ (x_2^+ - x_1^+)e^i p_1t^22 p^+ (x_1^+ - x_2^+) There are two points which we should mention here, first an incoming quark withpositive p^+ forces ordering along the direction of motion x^+, enforcedvia the theta function, and second, in the strict eikonal approximation onedisregards the exponential on the right hand side by using the fact thatp_1t p^+≪ 1. It is possible to go beyond this strict eikonallimit and keep these exponentials factors (for example see <cit.>),however we will not do that here. We then havei ℳ_2=(i g)^2 (- i ) 2 πδ (p^+ - q^+) ∫ d^2 x_1td x_1^+ d^2 x_2td x_2^+θ (x_2^+ - x_1^+) e^- i p^- x_1^+e^i q^- x_2^+∫d^2 p_1t (2 π)^2e^- i (p_1t - p_t) · x_1te^- i (q_t - p_1t) · x_2t(q) [ S (x_2^+, x_2t)i2 p^+ S (x_1^+, x_1t)]u(p) There are two further approximations we need to make before we get the finalresult, first, we note that p^- = p_t^22 p^+≪ 1 andq^- = q_t^22 q^+≪ 1 and one therefore can drop the exponentialterms involving p^-, q^- above. More importantly, there is still a p_1t term in the intermediate propagator which we also neglect since it is of the formp_1t p^+≪ 1and can be dropped (it is possible to keepthese terms, order by order, see <cit.> for example). After this laststep, one can perform the integration over transverse momentum p_1twhich gives ∫d^2 p_1t (2 π)^2e^- i p_1t· (x_1t - x_2t) = (2 π)^2 δ^2 (x_1t - x_2t)and enables one to perform one of the transverse coordinate integrations togeti ℳ_2=(i g)^2 (- i ) ( i ) 2 πδ (p^+ - q^+) ∫d x_1^+ d x_2^+θ (x_2^+ - x_1^+)∫ d^2 x_1te^- i (q_t - p_t) · x_1t (q) [ S (x_2^+, x_1t) 2 p^+ S (x_1^+, x_1t)]u(p)whereis now understood to be p_1^+γ^-. The last step is to use 2 p^+= since p_1^+ = p^+. We note that the two small x fields S are at the same transverse coordinate x_1t which was made possible due to neglecting terms p_1t p^+ in boththe exponentials and in the intermediate propagator. The second point to be kept in mind is the ordering of the scatterings which was caused by positivity of p^+ of the incoming quark and that contour integration over the propagator pole puts the intermediate quark line on shell. The approximations and steps needed to extend this procedure to include any number of scatterings from the small x fields S should now be clear; the energeticprojectile moving in a given direction will couple to the component of the gluonfield Lorentz-conjugate to that direction, for example p^+ :A^-. One neglects terms of the order p_tp^+ everywhere and that the p^+component of theprojectile momentum is conserved. The generic diagram for n soft scatterings is shown in Fig. (<ref>), The amplitude is theni ℳ_n=2 πδ (p^+ - q^+)(q)∫ d^2 x_te^- i (q_t - p_t) · x_t{ (i g)^n (- i )^n ( i )^n ∫d x_1^+ d x_2^+⋯ d x_n^+θ (x_n^+ - x_n - 1^+)⋯θ (x_2^+ - x_1^+)[ S (x_n^+, x_t)S (x_n - 1^+, x_t) ⋯ S (x_2^+, x_t)S (x_1^+, x_t)]}u(p)which after summing over all n scatterings(iℳ = ∑_n=1^∞ i ℳ_n) givesthe standard result for quark-target scattering amplitude in theeikonal limit <cit.>,i ℳ (p,q) = 2 πδ (p^+ - q^+)(q)∫ d^2 x_te^- i (q_t - p_t) · x_t [V (x_t) - 1]u(p)where V (x_t) is the Wilson line in the fundamental representation,defined as [Here we are being sloppy about path ordering vsanti-path ordering of the fields. The Wilson lines appearing in thepropagators are path ordered, i.e., the soft fields increase in theirx^+ argument from left to right, whereas the Wilson lines in theamplitudes are anti-path ordered. As introducing different symbols for path-ordered vs anti path-orderedwill result in clutter and because it will not have any bearing onour result we will ignore this distinction.] V (x_t) ≡P̂ exp{i g ∫_- ∞^+∞ d x^+ S^-_a (x^+, x_t)t_a} and S^- (x^+, x_t) = n^-S (x^+, x_t) is our small x gluon field.One can extract an effective quark propagator using the result forscattering amplitude. To do so we recall that the incomingprojectile was a quark with positive p^+ which is all we need incase of scattering of a physical particle. However, to construct aFeynman propagator one also needs to consider the case when aparticle with - p^+ is moving backward (what we have computed so far would give us the retarded and not the Feynman propagator). This ispractically trivial to implement since the only change in thederivation above is the locations of p^- poles of the intermediatepropagators which will be abovethe real axis as compared to before where the poles were below thereal axis. Thisreverses the ordering of projectile propagation in x^+ direction.It also results in a relative minus sign between the positive andnegative p^+ contributions due to the reversal of the directionof the integration contour when the contour is closed above vsbelow the real axis.Defining (recall that in calculating the amplitude one goes from right to left in a Feynman diagram while for the propagator we go from left to right) τ_F (p,q) ≡2 πδ (p^+ - q^+)∫ d^2 x_te^- i (q_t - p_t) · x_t {θ (p^+)[V (x_t) - 1] - θ (- p^+)[V^† (x_t) - 1] } the quark propagator is then given by S_F (p,q) = (2 π)^4 δ^4 (p - q)S_F^0 (p) + S_F^0 (p) τ_F (p,q) S_F^0 (q)where we have added the free propagator for completeness and the scattering amplitude is related to τ_F via i ℳ (p,q) =(q) τ_F (p,q)u (p). § MULTIPLE SCATTERING AT SMALL AND LARGE XIt should be clear from the derivation in the last sectionthat the target fields denoted by S have only small P^+ modes.Furthermore, since we are considering scattering from thesmall x fields of the target, they carry small fraction ofthe target energy P^- and therefore can not impart large P^-into the projectile. So the only momentum exchanged istransverse momentum which is 𝒪 (p_t p^+). Therefore these soft multiple scatterings can not cause a big deflection of the projectile whichcontinues to travel along a straight line in x^+ directionand remains at the same transverse coordinate x_t. This fact allows one to re-sumthe soft multiple scatterings into a Wilson line. Physical observablesin DIS and the hybrid formulation of particle production inproton-proton and proton-nucleus collisions involve multi-pointcorrelators of these Wilson lines; the most prominent one beingthe dipole scattering cross section which is the trace of twoWilson lines in the fundamental representation and satisfies theJIMWLK/BK evolution equation. In this sense Wilson lines are theeffective degrees of freedom at small x.Clearly to have any hope of merging thefull DGLAP physics, i.e. exact splitting functions, with the JIMWLK/BKevolution which takes the small x limit of the splitting functions,one needs to relax the small x approximation and include scatteringfrom the largex modes of the target. One would also need to figure out what theeffective degrees of freedom are in this more general kinematics where large x modes of the target are included. This is our main goal inthis paper. To this end we propose to go beyond scattering from smallx fields by including one general field which can carry any momentumfractions x. Such a general field will clearly depend on all4-coordinates and all of its components (subject to the gauge condition)will participate in the scattering. Therefore we consider scattering ofa quark on a target when in addition to the small x fields one "all x"field participates. Since this general field will carry large (as well as small) P^-, it can impart a large p^- momentum to theprojectile and cause a large angle deflection so that the scattered quarkmay have parametrically large transverse momentum(p^- = p_t^22 p^+) and lose some (or even all) of its p^+momentum (rapidity). We expectthat we may not have to include more than one scattering from an all xfield and that higher number of such scatterings would be suppressed bythe coupling constant. In this paper we will ignore higher number ofscatterings from the all x field but intend to come back to thispoint in the future as it may be relevant for gauge invariance at large x.§.§ A hard scattering after multiple soft scatteringsWe therefore consider scattering of a quark from an all x field which we denote A^μ = A^μ (x^+, x^-, x_t), the single scattering amplitude is shown in Fig. (<ref>). From now on wewill refer to the all x field as hard and the small x fieldsas soft so that hard or soft refers to momentum fraction x carried by a field.The result isi ℳ_1=(i g) ∫ d^4 xe^i (q - p) x(q) [(x)]u(p)=(i g) ∫ d^4 xe^i (q - p) x(q) [(x)2 p^+ ]u(p)where we have used the Dirac equation to arrive at the second line. We can now successively include more scatterings from the small xfields S. Let us consider the case that all such soft scatteringshappen before the hard scattering. Including one such soft scatteringgivesi ℳ_2 = (i g)^2 ∫ d^4 xd^4 x_1∫d^4 p_1(2 π)^4 e^i (p_1 - p) x_1 e^i (q - p_1) x(q) [(x)ip_1^2 + i ϵ S (x_1)]u(p)shown in Fig. (<ref>). As before we can perform the x_1^- integration since the soft field Sdoes not depend on it, and use the resulting delta function to setp_1^+ = p^+. Next integration over p_1^- can be done using contourintegration which gives( - i)θ (x^+ - x_1^+)12 p^+ e^i p_1t^22 p^+ (x_1^+ - x^+).We geti ℳ_2= (i g)^2 ∫ d^4 xd^2 x_1td x_1^+θ (x^+ - x_1^+) ∫d^2 p_1t (2 π)^2 e^i (q - p_1) xe^- i (p_1t - p_t) · x_1t (q) [(x) 2 p^+ S (x_1^+, x_1t)]u(p)where p_1^- = p_1t^2p^+ and p_1^+ = p^+. This procedure can now be repeated to include any number of soft scatterings which happen before the hard one. For example,inserting two soft scatterings before the hard one is shown in Fig.(<ref>) and givesi ℳ_3= (i g)^3 ∫ d^4 xd^2 x_1td x_1^+ d x_2^+θ (x^+ - x_2^+)θ (x_2^+ - x_1^+)∫d^2 p_2t (2 π)^2e^i (q - p_2) x e^- i (p_2t - p_1t) · x_1t (q) [(x) 2 p_2^+ S (x_2^+, x_1t)2 p_1^+ S (x_1^+, x_1t) ]u(p)where the soft fields are now at the same transverse coordinate as before.It should be now clear how to repeat this procedure by adding more and more soft scatterings. The amplitude for n soft scatterings followed by ahard one is shown in Fig. (<ref>),After re-summing all the contributions from n soft scatterings followed by a hard one we geti ℳ =(i g) ∫ d^4 xd^2 z_t∫d^2 k_t(2 π)^2e^i (q - k) x e^- i (k_t - p_t) · z_t (q) [(x) 2 k^+ V (z_t, x^+)]u(p).where k^+ = p^+, k^- ≡k_t^2k^+ (momentum k is on shell)and the Wilson line now extends to x^+,V (z_t, x^+) ≡P̂ exp{i g ∫_- ∞^x^+ d z^+ S^-_a (z_t, z^+)t_a} and S^- (z_t, z^+) = n^-S (z_t, z^+) as before.This has a simple interpretation; the quarkpropagates through the soft field and scatters eikonally from the target starting fromz^+ = - ∞ until the point x^+ where the hard scattering takes place. As before multiple soft scatterings do not change the transverse coordinate ofthe projectile quark and re-sum into a Wilson line.§.§ Multiple soft scatterings after a hard one In the eikonal approximation used for the soft multiple scatterings the final state quark can gain some transverse momentum but not much asp_t ≪ p^+. Furthermore its p^+ is conserved and does not change which also means it has negligibly small p^-. However, a hard scattering such as the one considered above can in principle impart large p^- and p_t to the projectile (and P^+, p_t to the target) so that after the hard scattering the quark will not necessarily propagate along its original trajectory but can be deflected by a large angle and lose some or even all of its p^+. This means we must take into account the possibility that the scattered quark isnot moving along the original (positive) z direction but can now move in an arbitrary direction with projections on all x, y, z coordinates. Equivalently the scattered quark will have momenta p^+, p^-, p_t without any restriction on their magnitude (still subject to being on shell). For example, in the extreme case of back scattering p^+ → 0 while p^- will be very large. Orin the case of scattering at right angle we will have p^+ ∼ p^- ∼ p_t.So it seems that we can not use our eikonal methods after the hard scatteringany more.However one can still our eikonal resultsif we work in the light coneframe of the scattered quark so that light cone frame of the scattered quark is rotated (in 3 dimensions) with respect to the light cone frame of the incoming quark. We define the direction of motion of the scattered quark to be the newpositive z axis and label it as z̅. The direction of propagation ofthe scattered quark z̅ will now be related to it x,y,z in the originalframe (defined by the incoming quark) via the standard rotation matrix𝒪 in 3 dimensions. ([ x̅; y̅; z̅ ])=𝒪 ([ x; y; z ])The same rotation matrix will allow us to express the momentum vectorof the scattered quark in the rotated frame in terms of quantitiesin the original frame. The rotation matrix 𝒪 will dependon azimuthal and polar angles of the scattered quark with respect to the original frame which can be related to energy and rapidity of thefinal state quark. In the new coordinate system the scattered quark is moving along positive z̅ so that it will have a large p_z̅ butno p_x̅ or p_y̅. However it will possibly haveall momentum components when expressed in terms old frame quantitiesp_x, p_y, p_z. One can then define the new light cone coordinatesx̅^+, x̅^-, x̅_t and light cone momentap̅^+, p̅^-, p̅_t in the new frame in the standard way.To facilitate this we define a new light cone vector n̅ whichprojects out the plus component in the new frame, so thatn̅·p̅ = p̅^+.The scattered quark now moves along the new plus direction x̅^+ and has only one large momentum component p̅^+. Therefore it will couple only to the minus component of the field expressed in the new coordinatesS̅^- = n̅^- S(x̅^+, x̅_t). It is important to realize that even though only the - component of the field in the new coordinate system isinvolved, the rotation matrix will generate the other Lorentz components if oneexpresses the soft field S̅^- in terms of the fields in the old coordinatesystem via S̅^μ = Λ^μ_νS^ν where the only non-zerocomponents of the Lorentz transformation matrix Λ^μ_ν are the elements of the 3-dimensional rotation sub-group.To illustrate this we consider the case when there is a hard scattering first, followed by multiple soft scatterings. The amplitude for one soft scattering after the hard one, shown in Fig. (<ref>), can be written asi ℳ_2 = (i g)^2 ∫ d^4 xd^4 x̅_1∫d^4 p̅_1(2 π)^4 e^i (p̅_1 - p) x e^i (q̅ - p̅_1) x̅_1(q̅) [S (x̅_1)i p̅_1^2 + i ϵ(x) ]u(p).All the steps we went through in section (<ref>) can now be repeated;the field S (x̅_1) is independent of x̅_1^- so that integration over x̅_1^- can be performed leading to a delta function of2 π δ (p̅_1^+ - q̅^+) which allows one to perform thep̅_1^+ integration setting p̅_1^+ = q̅^+. One then performsthe contour integration over p̅_1^ - noticing that it is next towhich puts p̅_1 on shell and gives a factor of θ (x̅_1^+ - x^+),we geti ℳ_2= (i g)^2 ∫ d^4 xd^2 x̅_1td x̅_1^+ θ (x̅_1^+ - x^+) ∫d^2 p̅_1t (2 π)^2e^i (p̅_1 - p) x e^- i (q̅_t - p̅_1t) ·x̅_1t (q̅) [S (x̅_1^+, x̅_1t) 2 p̅_1^+(x) ]u(p)withp̅_1^+ = q̅^+ and p̅_1^- = p̅_1t^22 p̅_1^+. By repeating this procedure for further soft scatterings as shown inFig. (<ref>),and re-summing them we geti ℳ =(i g) ∫ d^4 xd^2 z̅_t∫d^2 k̅_t(2 π)^2e^i (k̅ - p) x e^- i (q̅_t - k̅_t) ·z̅_t (q̅) [V (x^+, z̅_t)2 k̅^+(x) ]u(p)wherek̅_1^+ = q̅^+ and k̅_1^- = k̅_t^22 k̅_1^+ and the Wilson line is nowV (x^+, z̅_t) ≡P̂ exp{i g ∫_x^+^+∞ d z̅^+ S^-_a(z̅_t, z̅^+)t_a}Again this expression has a simple interpretation; the projectile quarkundergoes a hard scattering which changes its direction (possibly in all3 dimensions) after which it undergoes multiple soft scatterings whenexpressed in terms of coordinates in the new frame.§.§ Multiple soft scatterings before and after a hard scatteringWe now have all the ingredients to write the general expression forscattering of a projectile quark from the target including soft (small x) scatterings to all order and hard (all x) to the first order. This is shown in Fig. (<ref>) where the final state quark may now have a large q^- or q_t as measured in the incoming quark's frame (but hasonly a large momentum in z̅^+ direction in the rotated frame). The procedure is the same as above, one considers multiple scatterings from the soft fields which are independent of the "-" coordinate leading to conservation of the"+" momentum across a soft scattering, and then doing the contour integrationover the "-" component of the intermediate momentum which puts the quark line on shell. All the terms of the form p_tp^+x^+ orp̅_t p̅^+ x̅^+ are neglected which allows one todo the transverse coordinate integration over the soft fields. The only exception is if there is a "mis-match" of barred and un-barred momenta and coordinates in the exponentials which are then kept, for example exponents likep̅^-x^+ may have large components when expressed fully in termsof quantities in the old frame. The rest of details are the same as before; during soft multiple scatterings the transverse coordinate of the quarkremains the same and there is ordering in the direction of x^+ or x̅^+ which allows one to re-sum the multiple scatterings into a Wilson line. As a result we have x_1t = x_2t = ⋯ = x_it= z_t andx̅_1t = x̅_2t = ⋯ = x̅_nt= z̅_t. Thereforethe full amplitude can be re-summed and written asi ℳ = (i g) ∫ d^4 xd^2 z_t d^2 z̅_t∫d^2 k_t(2 π)^2 d^2 k̅_t(2 π)^2 e^i (k̅ - k) x e^- i (q̅_t - k̅_t)·z̅_t e^- i (k_t - p_t)· z_t (q̅) [V (x^+, z̅_t)2 k̅^+(x)2 k^+ V (z_t, x^+) ]u(p)with k^+ = p^+, k^- = k_t^22 k^+ andk̅^+ = q̅^+, k̅^- = k̅_t^22 k̅^+. As before the scattering amplitude can be written in terms of theinteraction part of the propagator viai ℳ (p,q̅) =(q̅) τ_F (p,q̅)u (p)in terms of which the full (Feynman) propagator is S_F (p,q̅) = (2 π)^4 δ^4 (p - q̅)S_F^0 (p) + S_F^0 (p) τ_hard (p,q̅) S_F^0 (q̅)with interacting part of the propagator is then given by(including the contribution where quark is moving backwardin x^+ which is needed for the Feynman propagator) τ_hard (p,q̅)≡ (i g) ∫ d^4 x ∫d^2 k_t(2 π)^2 d^2 k̅_t(2 π)^2 d^2 z_t d^2 z̅_t e^i (k̅ - k) x e^- i (q̅_t - k̅_t)·z̅_t e^- i (k_t - p_t)· z_t {θ (p^+)θ (q̅^+) V (z_t, x^+) 2 k^+(x) 2 k̅^+V (x^+, z̅_t) - θ (- p^+)θ (- q̅^+) V^† (z_t, x^+) 2 k^+(x) 2 k̅^+V^† (x^+, z̅_t)}. It is straightforward to check that this expression re-sums all thediagrams shown in Fig. (<ref>) by expandingthe Wilson lines. The dashed lines correspond to scatteringsfrom the soft fields carrying small x fractions of the targetenergy whereas the wavy line corresponds to scattering from a hard fieldcarrying arbitrary x fraction of the target energy. It describespropagation of an energetic quark along longitudinal direction z^+ andundergoing multiple soft interactions described by V (z_t, x^+) untilthe location x^μ at which point it undergoes a potentially hard(large x) scattering which changes it direction. After the hardscattering it propagates again along the new longitudinal directionz̅^+ and undergoing multiple scatterings described byV (x^+, z̅_t). The location of the hard scatteringx^μ is integrated from - ∞ to + ∞ (or from - R to R where 2 R is the diameter of the nucleus/size of the medium)so that the hard scattering can happen at any x^+, at the front,back or middle of the nucleus/medium, followed or preceded by multiplesoft scatterings. There is one more technical but essential point that we need to address;the couplingbetween the quark and gluon fields we have considered is of the form Ψ̅Ψ which is not gauge invariant. Instead one needs to use the full covariant coupling Ψ̅Ψ in the gauge invariant Lagrangian. This does not matter in the eikonal limit since replacing any of thesoft fields by the normal derivative gives a vanishingly small correction.However in our more general case it can act on the Wilson line in therotated frame and give contributions which are of the same order as considered here. Therefore to have the interacting part of the effective propagator transformcovariantly one needs to include it. To do so is trivial and amountsto replacing the hard fieldin our step by step derivation bythe covariant derivative . Therefore our expression in eq.(<ref>) is modified to τ_hard (p,q̅)≡ ∫ d^4 x ∫d^2 k_t(2 π)^2 d^2 k̅_t(2 π)^2 d^2 z_t d^2 z̅_t e^i (k̅ - k) x e^- i (q̅_t - k̅_t)·z̅_t e^- i (k_t - p_t)· z_t {θ (p^+)θ (q̅^+) V (z_t, x^+) 2 k^+ _x 2 k̅^+V (x^+, z̅_t) - θ (- p^+)θ (- q̅^+) V^† (z_t, x^+) 2 k^+ _x 2 k̅^+V^† (x^+, z̅_t)}.This is our final result for the interacting part of the quark propagator. This expression has the interesting property that it includes bothpartially and fully coherent multiple scatterings of the quarkfrom the target and as such generalizes the fully coherent multiplescattering in eikonal approximation. It is very interesting to consider the soft limit of this expression, i.e. the limit when the all x fieldcarriesa small x fraction of the target energy. To do so we need to replace(x^+, x^-, x_t)→S(x^+, x_t) and take allbarred quantities to be un-barred, for example,→ and k̅→ k. It is easy then to show eq. (<ref>) vanishes in this limit due to action of the covariant derivative on a Wilson lines under a parallel transport along the direction of motion (n· D_xV (x^+, x_t) = 0).This property of eq. (<ref>) suggests that to have a quarkpropagator which can be used for construction of physical quantities such as structure functions F_2 and F_L valid for any x one shouldinclude both contributions from purely eikonal scattering as given by eq.(<ref>) and hard scattering given by eq. (<ref>).Therefore we take the most general interaction part of the effectivepropagator to be τ_all x≡τ_F + τ_hard which is shown symbolically in Fig. (<ref>) and where the thick solid lines denote a Wilson line representing multiple soft scatterings from the target to all orders.It should be noted that while the projectile quark scatters from bothlow and high x modes of the target color fields, we have not included interactions between the different x modes. In this sense the small and large x fields are non-interacting with respect to each other. Forinstance it should be possible for a soft field to couple to a hard fieldas shown in Fig. (<ref>) Apreliminary study shows that the first coupling ofthe soft field to the hard field forces the two fields to be at the sameposition in coordinate space. Contributions of diagrams like this arecurrently under investigation and will be reported elsewhere. § DISCUSSION AND SUMMARY We have constructed an effective quark propagator in the background of color fields which can carry both small and large x energyfractions of the target from which the quark scatters. This generalizesthe known results for the quark propagator in the background ofa color field carrying small x fraction of the target energy based on eikonal approximation. The constructed effective propagator can be useful in many ways; first, it includes the possibility that a projectile quark can scatter from themore energetic (large x) partons of the target and get deflected bya large angle (high p_t). This may already give significant contributionsto single inclusive particle production in high energy collisions <cit.>when one uses the hybridformalism. It may also lead to significant final state momentum anisotropy at high p_t if one couples it to realistic nuclear geometries thatinclude fluctuations in the target nucleus/medium <cit.>.Furthermore, one can use this propagator to calculate the QCD structurefunctions F_2 and F_L following the procedure suggested in <cit.>. This would generalize the standard dipole picture of DIS at small x by including the contributions of the large x gluons in the target. This work is in progress and will be reported elsewhere <cit.>.To proceed further, one can use this method to calculate the effectivegluon propagator in a similar fashion. One can then generalize thehybrid formalism for particle production in asymmetric high energyproton-proton and proton-nucleus collisions and extend the validityof the hybrid formalism to high p_t, keeping in mind that high p_t in this context is equivalent to large x. Hence it would includecontributions of both fully and partially coherent multiple scatterings.It will be interesting to see if one can use our results to generalizeand extend the McLerran-Venugopalan (small x) effectiveaction <cit.> to include contributions of large x gluons.If so one would then be able to apply the MV model to not only low <cit.> but also high p_t particle production in high energy heavy ion collisions.This would allow us toinvestigate and quantify the contributions of early time dynamics tojet energy loss in Quark-Gluon Plasma formed in high energy heavy ioncollisions.Perhaps most significantly, we hope the physical quantities computed using this effective propagator will serve as the Leading Orderexpressions which can then be used for deriving Leading Log evolutionequations which would have DGLAP and JIMWLK evolution equations inappropriate limits <cit.>.§ ACKNOWLEDGMENTSWe acknowledge support by the DOE Office of Nuclear Physics through Grant No. DE-FG02-09ER41620 and by the Idex Paris-Saclaythough a Jean d'Alembert grant. We would like to thank N. Armesto,F. Gelis, E. Iancu, C. Lorcé, C. Marquet, A.H. Mueller, S. Munier,B. Pire, C. Salgado, G. Soyez and B. Xiao for critical questions,illuminating discussions and helpful suggestions. 99glr L. V. Gribov, E. M. Levin and M. G. Ryskin,Phys. Rept.100, 1 (1983). doi:10.1016/0370-1573(83)90022-4mq A. H. Mueller and J. w. Qiu,Nucl. Phys. B268, 427 (1986). doi:10.1016/0550-3213(86)90164-1mv L.D. McLerran and R. Venugopalan, Phys. Rev.D 49, 2233 (1994);ibid. 3352 (1994);ibid. 50, 2225 (1994).jimwlk J. Jalilian-Marian, A. Kovner, L. D. McLerran and H. Weigert,Phys. Rev.D55, 5414 (1997); J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert,Nucl. Phys.B504, 415 (1997), Phys. Rev.D59, 014014 (1999), Phys. Rev.D59, 014015 (1999), Phys. Rev.D59, 034007 (1999),A. Kovner, J. G. Milhano and H. Weigert, Phys. Rev.D62, 114005 (2000); A. Kovner and J. G. Milhano,Phys. Rev.D61, 014012 (2000); E. Iancu, A. Leonidov and L. D. McLerran,Nucl. Phys.A692, 583 (2001), Phys. Lett.B510, 133 (2001);E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran,Nucl. Phys.A703, 489 (2002).bk I. Balitsky, Nucl. Phys.B 463, 99 (1996);Yu.V. Kovchegov, Phys. Rev.D 61, 074018 (2000).DGLAPG. Altarelli and G. Parisi, Nucl. Phys.B 126, 298 (1977); V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys.15, 438 (1972);ibid. 675 (1972); Yu. Dokshitzer, Sov. Phys. JETP46, 641 (1977). review-cgcF. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan,Ann. Rev. Nucl. Part. Sci.60, 463 (2010) doi:10.1146/annurev.nucl.010909.083629 [arXiv:1002.0333 [hep-ph]];J. Jalilian-Marian and Y. V. Kovchegov,Prog. Part. Nucl. Phys.56, 104 (2006) doi:10.1016/j.ppnp.2005.07.002 [hep-ph/0505052]. H. Weigert,Prog. Part. Nucl. Phys.55, 461 (2005) doi:10.1016/j.ppnp.2005.01.029 [hep-ph/0501087].cgc-negativeG. A. Chirilli, B. W. Xiao and F. Yuan,Phys. Rev. D86, 054005 (2012) doi:10.1103/PhysRevD.86.054005 [arXiv:1203.6139 [hep-ph]]; B. W. Xiao and F. Yuan,arXiv:1407.6314 [hep-ph]; Z. B. Kang, I. Vitev and H. Xing,Phys. Rev. Lett.113, 062002 (2014) doi:10.1103/PhysRevLett.113.062002 [arXiv:1403.5221 [hep-ph]];E. Iancu, A. H. Mueller and D. N. Triantafyllopoulos,JHEP1612, 041 (2016) doi:10.1007/JHEP12(2016)041 [arXiv:1608.05293 [hep-ph]]; A. M. Stasto and D. Zaslavsky,Int. J. Mod. Phys. A31, no. 24, 1630039 (2016) doi:10.1142/S0217751X16300398 [arXiv:1608.02285 [hep-ph]].hybrid A. Dumitru and J. Jalilian-Marian,Phys. Rev. Lett.89, 022301 (2002) doi:10.1103/PhysRevLett.89.022301 [hep-ph/0204028], Phys. Lett. B547, 15 (2002) doi:10.1016/S0370-2693(02)02709-0 [hep-ph/0111357];A. Dumitru, A. Hayashigaki and J. Jalilian-Marian,Nucl. Phys. A765, 464 (2006) doi:10.1016/j.nuclphysa.2005.11.014 [hep-ph/0506308],Nucl. Phys. A770, 57 (2006) doi:10.1016/j.nuclphysa.2006.02.009 [hep-ph/0512129].balI. Balitsky and A. Tarasov,JHEP1606, 164 (2016) doi:10.1007/JHEP06(2016)164 [arXiv:1603.06548 [hep-ph]],JHEP1510, 017 (2015) doi:10.1007/JHEP10(2015)017 [arXiv:1505.02151 [hep-ph]].fgA. J. Baltz, F. Gelis, L. D. McLerran and A. Peshier,Nucl. Phys. A695, 395 (2001) doi:10.1016/S0375-9474(01)01109-5 [nucl-th/0101024];F. Gelis and A. Peshier,Nucl. Phys. A697, 879 (2002) doi:10.1016/S0375-9474(01)01264-7 [hep-ph/0107142].eik-reviewA. Kovner and U. A. Wiedemann,In Hwa, R.C. (ed.) et al.: Quark gluon plasma 192-248 doi:10.1142/9789812795533-0004 [hep-ph/0304151]; J. Casalderrey-Solana and C. A. Salgado,Acta Phys. Polon. B38, 3731 (2007) [arXiv:0712.3443 [hep-ph]].b-eik T. Altinoluk and A. Dumitru,Phys. Rev. D94, no. 7, 074032 (2016) doi:10.1103/PhysRevD.94.074032 [arXiv:1512.00279 [hep-ph]]; T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso,JHEP1601, 114 (2016) doi:10.1007/JHEP01(2016)114 [arXiv:1505.01400 [hep-ph]]; T. Altinoluk, N. Armesto, G. Beuf, M. Martínez and C. A. Salgado,JHEP1407, 068 (2014) doi:10.1007/JHEP07(2014)068 [arXiv:1404.2219 [hep-ph]].adjjm2 A. Dumitru and J. Jalilian-Marian,Phys. Rev. Lett.89, 022301 (2002) doi:10.1103/PhysRevLett.89.022301 [hep-ph/0204028]. jjm-dAJ. Jalilian-Marian,Nucl. Phys. A748, 664 (2005) doi:10.1016/j.nuclphysa.2004.12.001 [nucl-th/0402080].bsrv T. Lappi, B. Schenke, S. Schlichting and R. Venugopalan,JHEP1601, 061 (2016) doi:10.1007/JHEP01(2016)061 [arXiv:1509.03499 [hep-ph]]. mv-f2L. D. McLerran and R. Venugopalan,Phys. Rev. D59, 094002 (1999) doi:10.1103/PhysRevD.59.094002 [hep-ph/9809427].jjm-f2 J. Jalilian-Marian, work in progress.jv S. Jeon and R. Venugopalan,Phys. Rev. D71, 125003 (2005) doi:10.1103/PhysRevD.71.125003 [hep-ph/0503219]; A. Dumitru, J. Jalilian-Marian and E. Petreska,Phys. Rev. D84, 014018 (2011) doi:10.1103/PhysRevD.84.014018 [arXiv:1105.4155 [hep-ph]]. kmw A. Kovner, L. D. McLerran and H. Weigert,Phys. Rev. D52, 6231 (1995) doi:10.1103/PhysRevD.52.6231 [hep-ph/9502289]; A. Kovner, L. D. McLerran and H. Weigert,Phys. Rev. D52, 3809 (1995) doi:10.1103/PhysRevD.52.3809 [hep-ph/9505320]. | http://arxiv.org/abs/1708.07533v1 | {
"authors": [
"Jamal Jalilian-Marian"
],
"categories": [
"hep-ph",
"nucl-ex",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170824193321",
"title": "Elastic scattering of a quark from a color field: longitudinal momentum exchange"
} |
]Towards the map of quantum gravity^*Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland ^†Institute for Theoretical Physics, University of Wrocław, Pl. Borna 9, 50-204 Wrocław, Poland In this paper we point out some possible links between different approaches to quantum gravity and theories ofthe Planck scale physics. In particular, connections between Loop Quantum Gravity, Causal DynamicalTriangulations, Hořava-Lifshitz gravity, Asymptotic Safety scenario, Quantum Graphity, deformations ofrelativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantumdeformations of the Hypersurface Deformations Algebra and Poincaré algebra, nonlinear structure of phasespace, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present anattempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantumgravity. The analysis is performed in the spirit of a mind map, which represents the architectural approachto the studied theory, being a natural way to describe the properties of a complex system. We hope that theconstructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity asa particular complex system and serve as a useful guide for the researchers. [ Jakub Mielczarek^* and Tomasz Trześniewski^*,†December 30, 2023 ===================================================§ INTRODUCTION For around 100 years[The importance of quantum effects in gravity was noted by Einstein alreadyin 1916 <cit.>.] of constantly branching research, the landscape of quantum gravityinvestigations has become very broad and diverse. Nowadays it contains many scattered building blocks –conceptual dilemmas and conjectures, novel mathematical tools, models at the nascent stage of developmentand theories that are already quite extended. The usual perspective is that we have distinct, competingapproaches to the construction of quantum gravity as well as the modification of classical generalrelativity (assuming that the latter is a useful intermediate step), which arose partly due to thescarcity of experimental guidelines. They can be more or less sorted in several ways, e.g. some formulationsof the quantum theory are classified as canonical and some as covariant, depending on the appliedprocedure of quantization. One possible realization of such a taxonomy of (modified) classical andquantum gravity has been depicted in Fig. <ref> but it should be understood as an outline rather than a precise diagram. In the proposed scheme we roughly dividequantum gravity approaches and the more general theoretical frameworks according to whether they are built around the assumed fundamentaldegrees of freedom or represent some type of the effective description (at least at their currentstage of development). There is also included a spectrum of potential classical approximations to the theory, consisting of general relativity and its various modifications,which are extensively reviewed in e.g. <cit.>. Certain frameworks can be seen as reductions or developments of others, sometimes due to the synergy of different branches of research, but for simplicity we show only a few links (and we do not include different versions of quantum cosmology). In general, we have restricted here to the approachesthat are either most popular or especially interesting in the context of issues that will be consideredin this paper. However, simple taxonomy can not help to address the problem of consistency or contradictionbetween various existing models. On the other hand, as we will argue in the next sections, some sort of convergence in physical predictionsof various frameworks has recently started to become apparent. If a lucky coincidence is excluded, this mayindicate significant underlying connections. Moreover, there is an interesting idea to consider(see especially Sec. <ref> and references therein) that (quantum) gravity is actually, or at least can be treated as, a complexsystem, which exhibits emergent phenomena. Its fundamental degrees of freedom might behave in avery different way than the gravitational field at some effective level. Then individual models could bepotentially applied to different layers of the theory and in this sense become unified. In order to exploreall of these possibilities we should employ a more architectural way of thinking, by which we mean tryingto uncover the hypothetical structure of the still incomplete theory and to understand its internal functions.This contrasts with what may be called the engineering approach, in which independent groupsof researchers keep working on their rival theories and which has dominated quantum gravity over the years.In our opinion, only an appropriate combination of both architecture and engineering is able to bring usto the destination point. The architecture considered in this paper is not a very sophisticated one but every story has its beginning.In order to represent the structure and phenomena of the discussed theory we will apply the idea of a mindmap. The nodes of such a graph (playing the role of bricks) will denote different models of quantum gravityas well as their features and known results. The links (which are like mortar) will tell us which of theseelements are directly interrelated. The arrow of a link will not mean an implication but show a directionin which the graph should be read. In Fig. <ref> we present a piece of the architecture that we are going to study here.The initial node in our map is “quantum gravity”. From there one can pass to the concept of“phases of gravity”. Indeed, gravity is a system with a huge (perhaps infinite)number of interacting degrees of freedom. Systems of this kind, occurringin the physics of complex systems, generically exhibit such emergent phenomenaas the nontrivial phase structure. Various phases can arise,depending on the surrounding environment. The particular feature distinguishingdifferent phases is a “symmetry” – our second building block. Ananalogy that can be recalled here is the difference between the solid and liquid statesof water. The symmetries of the liquid state are the invariance under rotations and translations,while in the solid state these symmetries are partially broken due to the formation of the crystal structure. The notion of symmetry can be expressed in the algebraic terms by introducingthe generators of symmetry transformations. For example, the generators of rotations form the𝔰𝔬(3) algebra. This leads us to the third block of our construction, namely the“Deformed Hypersurface Deformation Algebra” (DHDA). Let us explain how does it arise.Classical theory of gravity – General Relativity – tells us that physicsdoes not depend on the choice of a coordinate system. Such a property is a symmetryof the theory, which is known as general covariance. The generators of this symmetry(the scalar and vector constraints) form the first class algebra, the so-called Hypersurface Deformation Algebra (HDA). On the other hand, when quantum gravityeffects are taken into account, the structure of HDA may become deformed, so that it isreplaced by a certain DHDA. HDA encodes the symmetry of any pseudo-Riemannian manifold but each manifoldlocally reduces to Minkowski spacetime, whose symmetries are described by thePoincaré algebra. The Poincaré algebra is, therefore, a sub-structure of HDA.This generalizes to the case of an arbitrary DHDA, which is expected to reduce to thecorresponding deformation of the Poincaré algebra. Various versions of such aquantum deformed Poincaré algebra have been considered in the context of quantum gravity.It shows how the small scale structure of spacetime is modified by the quantumgravity effects. Another simple way in which these effects could manifest is a changein the effective dimensionality of spacetime. The latter phenomenon has been studied indifferent approaches to quantum gravity by investigating the variability (as a function of scale)of the dimension of spacetime, usually defined as the Hausdorff or spectral dimension.We will focus here on the latter, which quantifies the spectral properties of a given fieldconfiguration and explicitly depends on the form of the (deformed) Poincaré algebraand the associated Laplace operator. In the mathematical sense, the spectral dimensionbelongs to the “spectral properties of field configurations”. In the next sections we will gather the ingredients necessary to built a more detailedrealization of the architecture introduced above, which will be presented in Sec. <ref>.It should be stressed that in this realization we restrict to a particular selection of approachesfrom Fig. <ref>, especially the ones that we are most familiar with. The obtainedconstruction will be hopefully extended in the sequel to this paper, where we would liketo include other aspects of quantum gravity. Here we focus on understanding therelationships between: phases of gravity, the running dimension of spacetime, deformationsof relativistic symmetries and the nonlinear structure of phase space.§ PHASES OF GRAVITYThere is growing evidence (see e.g. <cit.>) that thegravitational field can exist in different phases (i.e. macroscopic states). This is actually not surprising,since fields as well as systems of particles usually form various phases, invariant under different sets ofsymmetries and determined by the initial conditions and interaction with the environment. The so-called analogue gravity models <cit.> have shown many interesting similarities between gravity and condensed matter physics. Furthermore, it is becomingapparent that the quantum gravitational phenomena might also be explained in terms of the quantum many-body systems, which naturallyreflects their non-trivial phase properties <cit.>. In the case of gravity, the first indication of thenontrivial phase structure came from the numerical results of Euclidean Dynamical Triangulations (EDT),formulated within the path integral quantization approach. EDT facilitate the computationsby discretizing spacetime into simplices, with the help of the Regge calculus, and making the Wickrotation to the Euclidean domain. Consequently, the quantum gravitational field can be described bya statistical ensemble and studied using the Monte Carlo simulations, which allow to find the equilibriumconfigurations for different values of the coupling constants. An equilibrium is equivalent to a classical path,while thermal fluctuations correspond to quantum fluctuations around it. The (2+1)-dimensional EDT were the first to predict two distinct phases of gravity,separated by a first order phase transition <cit.>:the so-called branched polymer phase and the crumpled phase.The investigations were later extended to 3+1 dimensions, showing that the abovephase structure is preserved in this case <cit.>. Gravitational phases canusually be distinguished by the effective dimensionality of spacetime (which will bediscussed in more detail in Sec. <ref>). In particular, in the crumpledphase of EDT the Hausdorff dimension behaves as d_H →∞,while in the branched polymer phase we have d_H = 2 and the (constant) spectraldimension d_S = 4/3. EDT in the standard formulation does not exhibit a phase with“extended” four dimensional spacetime, which would describe a semiclassical solutionof the theory. It has recently been verified <cit.> that such a phase does not emerge even after introducing a nontrivial measure in the path integral. Further information about the presumed phase structure of gravity comes from simulationsof Causal Dynamical Triangulations (CDT) <cit.>.This improved approach is constructed by imposing a causal structure onconfigurations of EDT. The causality condition is realized owing to the introduction of a preferredfoliation of spacetime, which may seem to be somewhat restrictive. However, it has also beenargued <cit.> that the preferred foliation is not a necessary ingredient but only aconvenient choice and the results of CDT are indeed sufficiently generic. One of the remarkableconsequences of CDT is appearance of the “extended” four dimensional phase.In total, three main phases, called A, B and C, have been observed. An averagespacetime configuration in the phase C is a well extended blob characterized by the Hausdorffdimension d_H ≈ 4. Moreover, most of the vertices of such a triangulationhave relatively small valence, while the maximal valence in a spatial slice grows proportionally to its volume <cit.>. The notion of geometry emerges naturally via a global coordinate system thatcan be introduced to parametrize the graph representing a given triangulation. The graph corresponding to the phase B is lacking the latterproperty since in this case the typical valence of vertices is too large. According to the numerical results of 4dCDT, this phase is characterized by the Hausdorff and spectral dimensions tending to infinity,d_H →∞, d_S →∞. Therefore, the phase B is often perceivedas a counterpart of the crumpled phase in 4d EDT. On the other hand, the phase A sharessome properties of the branched polymer phase. However, a detailed study of this phase hasnot been performed so far. The phases C and A are separated by a first order transition line <cit.>,while the transition between C and B is of the second order <cit.>. The orderof the A-B transition is unknown. The C-A, C-B and A-B transition lines meet at thetriple point. Furthermore, recent results in 4d CDT suggest that there exists an additional phaseinside the phase C, called the bifurcation phase <cit.>, which seems toexhibit the phenomenon of the metric signature change <cit.>. Further analysis is requiredto understand the nature of this new region on the phase diagram.It should also be stressed here that different types of the order parameter can be applied in order to discern thephases and transitions between them. Each order parameter is sensitive to certain particular features (including symmetries)of a given field configuration. The A,B,C phases in CDT are detected using the heuristically introduced“average geometry" parameter. However, in order to observe the bifurcation phase, which isa subphase of the phase C, a more sophisticated parameter has to be chosen.Therefore, the number of perceived distinct phases depends on what type of the order parameter is considered. Interestingly, it has been shown <cit.> that three phasessimilar to those present in 4d CDT appear also in the so-called Quantum Graphity <cit.>model of the Planck scale physics. In this case the (extended/semiclassical) phase C is realized at theminimum of energy and is the most stable configuration (i.e. a vacuum). The phases A and B can bereached by departing from the minimal energy. Furthermore, the transition betweenthe high temperature non-geometric phase and low temperature geometric phase sharesproperties of the transition between the crumpled phase B and the geometric phase C in CDT.In this context the term geometrogenesis has been coined and possible observationalrelevance of such gravitational phase transitions has been studied <cit.>.One can say that the B-C transition in CDT provides a concrete realization of thegeometrogenesis discussed in the context of Quantum Graphity. Cf. a recent study <cit.>, which also explores the relation between geometrogenesis and the ultralocal limit of gravity (we will discuss the latter in Sec. <ref>). Transitions between different phases of gravity in principle may be of the first, second or higher order.Among them, the second order transitions deserve a special attention. The reason is that atsuch a transition the correlation function of a given order parameter diverges and thetheory becomes scale invariant. Consequently, field configurations in the continuum limit(of discretization) should be described by a conformal quantum field theory. This concerns eitherthe full spacetime geometry or only its spatial part. Therefore, it is clear that GR, which does not satisfy the conformal invariance, cannotdescribe the classical state of gravitational field at a second order transition point (or line, as in the case ofCDT). We may suppose that GR is just a (semi)classical theory of the extended phase of gravity,while at the phase transitions, or in other phases, the fundamental quantum theory of gravityreduces to certain effective theories that are different from GR. Let us compare this possibility with thephase structure of water. The fundamental quantum Hamiltonian of water is the same for every phase butvarious effective descriptions are needed in individual phases. In particular, the Clapeyron equationprovides a sufficient effective model in the low-density gaseous state. On the other hand, the liquidstate is well modeled as a non-compressive fluid, characterized by the Navier-Stokes equation.Both equations may be called the equations of state, as the relations between variables that are satisfiedin a given phase. Analogously, perhaps the Einstein equation is an equation of state that is valid onlyin the extended phase of gravity. Then different equations of state would be required as the properdescription of gravity in other regions of phase space. Coming back to the second order phase transition, a choice of the proper equation of state in such a casedepends on whether it is full spacetime that is conformally invariant or only its spatial slices. Forthe full conformal invariance one might consider e.g. the Weyl gravity[The actionof the Weyl gravity is S = 1/16π G∫ d^4x √(-g) C_μναβ C^μναβ, where C_μναβ denotes the Weyl tensor. The Weyl tensor is invariant under aconformal transformation g_μν→ g'_μν≡Ω^2(x) g_μν.].Therefore, a classical theory of gravity different than GR, such as the Weyl gravity (investigated so far dueto the purely theoretical reasons), can potentially arise as an effective description of the gravitationalfield at the second order transition. On the other hand, when the conformal invariance holds only at thelevel of spatial hypersurfaces, the so-called Cotton tensor will be more adequate to consider. Thelatter situation occurs in the Hořava-Lifshitz approach to quantum gravity <cit.>.In this context it is worth to mention that, as analyzed in <cit.>, the Hořava-Lifshitztheory and its precursor – the Lifshitz scalar, have the phase structure analogous to the one observedin CDT for the “average geometry" order parameter. The existence of various phases of gravity has been also predicted within Group FieldTheory (GFT) <cit.>, Loop Quantum Gravity and Spin Foam models. Worth mentioninghere is that, although strictly related, these three approaches are characterized by different Hilbertspaces and dynamics. A GFT model defined on the group U(1)^× 3 was studiedwith the help of the Functional Renormalization Group and found to have the nontrivial phase structure,containing the Gaussian and non-Gaussian fixed points <cit.>. Furthermore,the emergence of bosonic condensate phases of GFT, which are probably associated with the vacuum states different from the Fock vacuum, has been observed and interpretedin the cosmological terms, cf. <cit.> and references therein. Such GFT condensates are another potential realization of the geometrogenesis mentioned above. Meanwhile, the phase diagram of Spin Foam modelswas tentatively derived in <cit.>. An indication of distinct phases in LQG can also be noticedwithin investigations of its vacuum. In the recently proposed new formulation of LQG<cit.> there appears a new vacuum state, the so-called BF-vacuum |0⟩_BF.Such a ground state (given by a spin network with a large number of vertices) has the constant curvatureand is a sort of the condensate state, similar to the ones considered in GFT. This contrasts with theAshtekar-Lewandowski vacuum |0⟩_AL, for which no single node of a spin network exist.The two vacuum states correspond to two inequivalent representations of the LQG algebra of canonicalvariables. Therefore, they are possibly associated with different phases of the gravitational field.Finally, the Random Tensor Models, which are the higher dimensional extension of the RandomMatrix Models used to study 2D quantum gravity and can also be seen as a simple version of GFT, indicate the non-trivial phasestructure via their observed critical behavior (in the large N expansion) <cit.>. Further discussion and examplesof the many-body aspects of quantum gravity and the related non-trivial phase structure can be foundin e.g. <cit.>.§ ASYMPTOTIC SAFETY AND UV FIXED POINT There is much more to say about the second order phase tradition in gravity. Namely, this type of a transition is associated with existence of the critical point(or critical line). The criticality leads to the hypothesis of asymptotic safety<cit.>, first proposed by Weinberg (see below). General Relativity is perturbatively nonrenormalizable, which is caused by the dimensionful character ofthe Newton coupling constant. In a perturbative expansion of the probability amplitudes there occuradditional UV divergences, which require a regularization by adding the appropriate counterterms to theHamiltonian. Consequently, new coupling constants (that multiply counterterms) appear in the quantumtheory and have to be fixed in order to make predictions. Since values of these couplings are a prioriunknown, the theory is lacking the predictive power. While GR is perturbatively nonrenormalizable, there is still a possibility that it is nonperturbativelyrenormalizable. This is precisely the idea of asymptotic safety. Namely, the conjecture put forward byWeinberg <cit.> is that if there exists a non-Gaussian UV fixed point in the RenormalizationGroup (RG) flow and it can be reached by trajectories belonging to a finite dimensional critical surface,then only a finite number of couplings (equal to the dimension of the critical surface) has to befixed to cancel the divergences of GR. Investigation of such a scenario has attractedsignificant attention in recent years <cit.>. Analysis of various effective quantum actions of GR (containing higher order derivatives)supports the asymptotic safety conjecture by indicating existence of a critical point (nontrivial fixedpoint) for the considered actions. However, in order to prove the conjecture, all allowedeffective actions should be taken into account. If the conjecture is correct, then there would be no formalnecessity to introduce new degrees of freedom for gravity at the Planck scale and GR could be quantizedjust as it is. Furthermore, in the latter case there would be no interpretation of GR as an effective theory,valid only at the energy scales below the Planck energy, as one may expect from its perturbativenonrenormalizability. Nevertheless, it is important to stress that asymptoticsafety does not rule out that GR actually is an effective theory. It is still possible that morefundamental degrees of freedom have to be introduced due to the underlying physical reasons. Such a situation happensin the theory of hydrodynamics, which despite being simultaneously perturbatively nonrenormalizable andasymptotically safe, does not provide the accurate description of liquid water at the atomic scale or near the phase transitions. Searching for the critical behavior of the gravitational field gains, therefore, an additional motivation.In fact, the CDT results discussed in the previous section seem to support the asymptoticsafety conjecture. This is due to the presence of the second order transition line.Preliminary results of the RG analysis in CDT <cit.> indicate that thetriple point, which ends the second order C-B transition line, is probably relatedto the UV critical point via the continuum limit. However, then there may appear the anisotropic scalingbetween spatial and temporal directions, as suggested in <cit.>.In such a situation, RG trajectories would “flow” from approximately the centre of the phase Ctowards the second order transition line and finally converge at the triple point.Further numerical simulations have to be performed in order to verify these findings. The type of anisotropy that is needed to obtain a UV fixed point in CDT is exactly the oneassumed in the Hořava-Lifshitz gravity (HLG) approach to quantum gravity. HLG is ageneralization of GR whose action (in the UV) is invariant under the anisotropic scaling of spacetime coordinates<cit.>x→ bx ,t → b^z t ,with an arbitrary constant b and the critical exponent z. The standard GR can be recovered for z = 1, while forz = 3, at the so-called Lifshitz point of the RG flow,the theory in 3+1 dimensions (similar to a Lifshitz scalar field theory) becomes power-counting renormalizable. This and otherchoices of the critical exponent will be discussed in more detail in the next section. An extremely promising possibility is that the UV fixed point appearing in theasymptotic safety scenario, the triple point on the CDT phase diagram and theLifshitz point (z = 3) of HLG are actually the same critical point.Besides the arguments presented above, further support for such a case comesfrom the results for the spectral dimension of spacetime. § SPECTRAL DIMENSION AND PHASES OF GRAVITYA method to characterize the structure of quantum spacetime that has been widely usedin recent years are calculations of the effective number of spacetime dimensions, especiallyapplying the notion of the spectral dimension. This particular definition of the dimensionemploys the fact that the return probability of diffusion (random walk) to the same pointstrongly depends on the dimensionality of a manifold on which it is considered. In particular,in Euclidean space ℝ^d the averaged (over the whole space) return probabilityscales as P(σ) ∝σ^-d/2, where σ is the auxilliary diffusion time.By analogy with the scaling of P(σ) in Euclidean space,the spectral dimension of a manifold is introduced asd_S := -2 ∂log P(σ)/∂logσ ,so that for Euclidean space we have simply d_S = d. Since the above definition concerns Riemannian manifolds, spacetime first has to be Wick-rotated. Moreover, it should be mentioned that the expression (<ref>) does nottake into account the effects of topology or curvature. The scaling of P(σ) is a probe of departure from the Euclidean geometry, which isassociated with the spectral properties of a given manifold. In order to determine this quantity one considers a fictitious diffusion process, described bythe heatequation∂/∂σ K( x, y; σ) = Δ_x K( x, y; σ) ,where Δ_x denotes the appropriate Laplace operatorand the initial condition K( x, y; σ) = δ^(d)( x - y)is assumed (when the manifold is flat). The heat kernel K( x, y; σ), which is the solution to (<ref>), allowsto calculate the average return probabilityP(σ) := ∫ dxK( x, x; σ) = ∫ dμ(p)e^σΔ_p ,where Δ_p is the momentum representation of Δ_x and μ(p)an invariant measure on momentum space. For quantum spacetimesit is required that at large diffusion times (probing large scales) the classical value of the dimensionis recovered. On the other hand, for small times the smallscale structure of spacetime is explored, corresponding to the UV limit of the theory, which may lead to the nontrivial behavior of the dimension. Indeed, running of the spectral dimension as a function of scale has been found insuch approaches to quantum gravity and models of quantum spacetime as: CDT <cit.>,Hořava-Lifshitz gravity <cit.>, asymptotic safety scenario <cit.>, nonlocal quantum gravity <cit.>,broadly understood loop quantum gravity <cit.> (so far, only at the kinematical level),causal sets <cit.>, multifractionalspacetimes <cit.>, noncommutative spacetimes <cit.>and spacetimes characterized by a deformed hypersurface deformation algebra <cit.> (see the next section). In almost all of thecases the obtained results show a dimensional reduction in the UV limit, usually tod_S ≈ 2, which is the value that makes gravity power-counting renormalizable. This is also in agreement with what is expected at the UV fixed point in the asymptotic safety conjecture<cit.>. Other results may be associated with different phases of the theory. In general, spacetimes exhibiting the phenomenon of the running dimension are called multiscale. Various relevant aspects of thedimensionality of spacetime are discussed e.g. in <cit.> and within a recent review of multifractional spacetimes <cit.>. In the case of CDT, each of the four phases (taking into account the bifurcation phase) ischaracterized by a different scale dependence of the spectral dimension.All phases and transitions between themhave been collected on a graph in Fig. <ref> together with the corresponding IR and UV limits of the spectral dimension. As it was already mentioned in Sec. <ref>, in the non-geometric phase B the UVvalue of the spectral dimension tends to infinity. This is a consequence of the extremely highconnectivity between different points of space. It takes only a few Planck steps to go from onepoint to any other since for small diffusion times (for small number of steps) there is always ahuge number of neighboring points. Spacetime is effectively very high dimensional, which is reflectedin the spectral dimension. On the other hand, it is also easy to return in several steps to thestarting point. Therefore, while the spectral dimension is sharply peaked at small diffusion times,it quickly falls to zero for larger times, as it was indicated by simulations performed in CDT.This shows that the crumpled spacetime configuration in the IR looks like a single space-time point. Spacetime in the extended phase C has acompletely different structure, which can be interpreted as semiclassical. At large scales, i.e. in the IR limit, thespectral dimension saturates at d_S ≈ 4, recovering (when a contribution of the assumed compacttopology is subtracted) the value from the classical theory. In the seminal paper <cit.> itwas also discovered that the dimension monotonically decreases with scale and small scales (the UV limit) arecharacterized by the value d_S ≈ 2. Based on the requirement of consistency, one may speculatethat the same behavior occurs at the triple point on the phase diagram but this presumption has to be verifiedby further numerical studies. Furthermore, as more extensive simulations have recently shown <cit.>,the UV limit of the spectral dimension actually depends on a location within the phase C. In particular, it hasbeen found that the value d_S ≈ 2 is being measured in a region lying deep inside the phase C.However, if one approaches the transition line to the phase A, the spectral dimension decreases tod_S ≈3/2, which probably reflects the renormalization group flow. Finally, the spectral dimension in the phase A has not yet been a subject of systematicstudies. In the numerical simulations spacetime in this phase appears to behave as a sequence of causallydisconnected spatial slices. A preliminary analysis of the spectral dimension indicates thatat large scales it becomes d_S →4/3 <cit.>, which is the sameas for the branched polymer phase in EDT. Meanwhile, in HLG (the Hořava-Lifshitz gravity) it has been found that the spectraldimension depends on the value of the (running) critical exponent z and is given by <cit.>d_S = 1 + D/z ,where D denotes the topological dimension of space, which we fix here as D = 3 (i.e. we consider (3+1)-dimensional spacetime). It is clearthat the classical case with d_S = 4 is correctly recovered for z → 1. In turn,at the Lifshitz point z → 3 the spectral dimension reduces to d_S = 2.The latter result coincides with the UVlimit of d_S(σ) in the central region of the phase C in CDT. Indeed, at least in the case of 2+1 dimensions, it has been shownto high accuracy <cit.> that HLG can reproduce the spectral dimension in the above region on the CDT phase diagramfor the range of scales lying between the UV and IR (although this is not enough to prove that the small scale structure of spacetime inthe two theories is identical <cit.>). Moreover, the case of d_S = 3/2, which occurs in 3+1 dimensions in bothCDT and EDT, in the Hořava-Lifshitz gravity is obtained for z = 6. Analysis of the theory with this value of z has not been carriedout so far. Furthermore, in Hořava-Lifshitz gravity the value of spectral dimension corresponding to the branched polymer case (or CDTphase A) is obtained for z = 9.There are two more special situations in HLG that should be explored, namely z → 0 and z →∞. In particular,analysis of the scaling properties of the gravitational action leads to the conclusion that the case z → 0 realizes the so-calledultralocal limit <cit.> of gravity (see also Sec. <ref>), in which spacetime splits into a congruence of causallyindependent worldlines. This is a consequence of suppressing the spatial derivatives in the action, so that only the kinetic and cosmologicalconstant terms remain. According to the formula (<ref>), the value of the spectral dimension in such a limit will tend to infinity,d_S →∞. The result may seem counterintuitive since in the ultralocal state of spacetime the spectral dimension is expectedto reduce to d_S = 1, corresponding to the remaining single direction. The possible explanation is found by considering the phase B ofCDT, which can be seen as a particular realization of the ultralocal state, characterized by the anisotropic scaling z → 0 (see thenext paragraph). As we already discussed, in the phase B the dimension at small scales becomes d_S →∞ but at sufficientlylarge scales spacetime is effectively a point. Comparing the ultralocal state and the phase B of CDT one can suspect that they are closely related. Indeed, it has been shown<cit.> that when the phase B is approached from the phase C, the ratio of lengths of spacelike and timelike simpliciallinks is decreasing. Such an effect is associated with the collapse of lightcones into worldlines, which is exactly what happens in the ultralocallimit. The ultralocality of the phase B is, however, partially obscured due to the constraints that are imposed at the level of numerical simulations.The crucial assumption of CDT is that the (compact) topology of spatial slices is preserved, which is necessary to ensure that the causality inquantum spacetime is not violated. However, this constraint also prevents the gravitational configuration from achieving the “extended” ultralocal state, inwhich all points of space evolve as disconnected universes. Another restriction, introduced for computational reasons, is that the total numberof simplices (or the average of it) is fixed, being controlled by the cosmological constant, which plays the role of a chemical potential in the statistical ensemble. Due tothe combined effect of the above constraints in CDT, an attempted simulation of the ultralocal state leads to a pointlike universe pierced by a timelikestraw (the latter may or may not have the physical meaning). The dimensionality of space in the IR is correctly zero, whichresults from the crumpled structure in the UV. The only possible real difference with respect to the ultralocal state in HLG is the behavior of the time direction. Depending on whetherthe straw observed in numerical simulations is interpreted as a physical feature or a numerical artifact, the time directionstretches a single dimension or disappears, respectively. Let us also mention that since the phase Bcorresponds to the anisotropic scaling z = 0, while the phase C_dS to z = 1, we conjecture that the (sub-)phase C_b should correspond to thescaling that interpolates between these two values of z. However, the properties of the phase C_b have not yet been completely studied. Coming back to the Hořava-Lifshitz gravity, the last case to consider is the critical exponent z →∞,for which the spectral dimension d_S → 1. While the dimension behaves here as it can be expected forthe ultralocal state, field configurations are not characterized by vanishing of the spatial derivatives. It is quite theopposite, since the action invariant under the z →∞ scaling contains derivatives of the infinitely highorder, possibly describing an extremely strongly correlated configuration. Physical properties of the case z →∞deserve a separate detailed analysis, which still has to be carried out. Concluding the discussion, we can tentatively consider that the Hořava-Lifshitz gravity and Causal Dynamical Triangulationsare connected by the network of relations depicted in Fig. <ref>.In this section we actually restricted mainly to the above two approaches to quantum gravity. An outline of the more inclusive map of quantumgravity that is based on predictions for the dimensionality of spacetime (using also other notions besides the spectral dimension) hasrecently been proposed in <cit.>.§ DEFORMATIONS OF THE HYPERSURFACE DEFORMATION ALGEBRAThe symmetry of General Relativity is general covariance. In the Hamiltonian frameworkit is embedded in the structure of the algebra of constraints, the so-calledHypersurface Deformation Algebra (HDA). This algebra (more precisely, it is an algebroid <cit.>)is of the first class and therefore the constraints are given the interpretation of symmetry generators.In the caseof classical GR transformations generated by the constraints coincide with the Liederivatives. However, the evidence gathered in recent years suggests that this propertymay be spoiled when the Planck scale physics is taken into account. Namely, the effects of quantum gravity can lead to certain deformations (i.e. modifications of the brackets) of the hypersurfacedeformation algebra. The ensuing Deformed Hypersurface Deformation Algebra (DHDA)remains of the first class and the constraints still act as the generators of gaugetransformations. Nevertheless, apart from certain limits, the transformations will differ from theLie derivatives. Consequently, the spacetime metric will no longer be a covariant object, which suggests that itdoes not play such a significant role here as in the classical theory. The deformations of thistype have been derived through an analysis of the effective algebra of quantumconstraints in loop quantum gravity. For both spherically symmetric configurations andcosmological perturbations the obtained algebra has the brackets <cit.>{D[N^a_1], D[N^a_2]} =D[N_1^b ∂_b N^a_2 - N_2^b ∂_b N^a_1] ,{S^Q[N], D[N^a]} =-S^Q[ N^b ∂_b N] ,{S^Q[N_1], S^Q[N_2]} =D [s Ωg^ab(N_1 ∂_b N_2 - N_2 ∂_b N_1)],where Ω is the deformation factor (affecting only the last bracket), g^ab denotes the spatial metric ands the spacetime metric signature, which is s = 1 in the Lorentzian case and s = -1 in the Euclidean one.The superscript Q means that the scalar constraints S^Q[N] are themselves quantum deformed(with respect to their classical counterparts). In the case of cosmological perturbations it has been found that thedeformation factor takes the following form <cit.>Ω = cos(2γμ̅k̅) = 1 - 2 ρ/ρ_c∈ [-1,1] ,with the critical energy density ρ_c = 3/(8π G γΔ) ∼ρ_ Pl, depending on theImmirzi parameter γ and the minimal surface area Δ.Similarly, for the spherically symmetric configurations we have Ω = cos(2δ K_φ). In bothcases the expression for Ω is a cosine function of the extrinsic curvature. Such an effectoriginates from the so-called holonomy corrections, which are the result of replacing the Ashtekarconnection by the corresponding holonomies[The holonomy of the Ashtekar connection along acontour C is given by the path-ordered exponential h_ C :=P e^∫_ C A.].However, the inverse volume corrections are to be expected as well, which will modify theexpression (<ref>) <cit.>. It has also been shown<cit.> that substituting a function f(p) in the place of the kinetic termp^2 in the field Hamiltonian leads to the DHDA with the deformationfactorΩ = 1/2d^2f(p)/dp^2 .In particular, when the polymer quantization is applied to the canonical momentum via the functionf(p) = sin^2(λ p)/λ^2(as it is done in loop quantum cosmology), one obtains Ω = cos(2λ p). λ is herethe scale of polymerization, introduced in such a way that the classical symmetries are recovered for λ→ 0. This explainsthe origin of the cosine form of deformations observed in the above mentioned effective models of LQG. The graph presented in Fig. <ref> contains different cases of the DHDA (<ref>) and their relations with some other aspects of quantum gravity.Depending on the value of Ω, several physical scenarios are realized.In particular, for Ω→ 1 we recover classical GR with the Lorentzian metric signature. For Ω→ -1 classical GR is recovered as well butwith the Euclidean signature, while the case of Ω→ 0 is equivalent to theultralocal limit, discussed in the previous section. It is worth to add here that the ultralocal state can also be obtainedin classical GR, by taking the strong coupling limit G →∞ <cit.>.Furthermore, in the sense of the BKL conjecture <cit.>, theultralocality is a general prediction when evolution of the gravitational field towards a singularity is considered. In this contextit is associated with the concept of asymptotic silence <cit.>. As can be seen from (<ref>), the type of DHDA obtained in loop quantum cosmology (LQC)leads to the effect of the metric signature change. When energy density of the matter content of universe reachesthe value ρ = ρ_c/2, then Ω changes its sign from positive to negative, which has theinterpretation of going from the Lorentzian to Euclidean signature. This curious phenomenon has been asubject of several investigations in recent years <cit.>. Furthermore, theultralocal state that emerges at ρ = ρ_c/2 has been studied in <cit.>. An essentialfeature of the Ω-deformation in LQC, pointed out in <cit.>, is that it is actuallyassociated with the non-Riemannian geometry of spacetime. Further understanding on this small scale structurein the regime where deformations are significant may be achieved through an analysis of the correspondingdeformations of the Poincaré algebra. This issue is the topic of the next section.§ CONNECTION BETWEEN DHDA AND THE DEFORMED POINCARÉ ALGEBRA The hypersurface deformation algebra (HDA) describes symmetries of an arbitrarypseudo-Riemannian manifold and the simplest case of such a manifold is naturally the Minkowskispacetime. The isometries of Minkowski spacetime form the Poincaré algebra,which can also be obtained in the limit of linear hypersurface deformations in the corresponding HDA.Accordingly, any modification of the standard HDA is expected to affect the structure ofthe Poincaré algebra. In particular, this is predicted to be the case in LQG, wherethe effective algebra of constraints (which is equivalent to HDA) is deformed, as we have already discussed. It has been argued that the deformed Poincaré algebra corresponding to the DHDA (<ref>)has the following form <cit.>:{J_a,J_b } = ϵ_abc J^c ,{J_a,K_b } = ϵ_abc K^c ,{K_a,K_b } =-s_effϵ_abc J^c ,{J_a,P_b } = ϵ_abc P^c ,{J_a,P_0 } =0 ,{K_a,P_b } = δ_ab P_0 ,{K_a,P_0 } =s_eff P_a ,{P_a,P_b } =0 ,{P_a,P_0 } =0 ,with the deformed signature factor s_eff≡ s Ω̃, which is a function of the algebra generators and becomes the metric signature s_eff = s in the limits Ω̃→± 1. So far, the form ofΩ̃ has not been determined in general but some attempts have beenmade. In particular, by requiring the algebra to satisfy the Jacobi identities and Ω̃ to be a separable function of P_0^2 and P_i^2 (which is a convenient Ansatz), one obtains thefollowing expression <cit.>:Ω̃ = P^2_0 - α/P^2_i - α ,where α∈ℝ is a free parameter associated with the energy scale ofthe deformation. In such a case the Euclidean mass Casimir element (with the properclassical limit) can be constructed asC^E = P_0^2 +P^2/1 - α^-1 P^2and the simplest possible guess concerning the form of the d'Alambert operatoron momentum space is Δ_p := - C^E. Substituting the latter into (<ref>), weare able to calculate the spectral dimension of spacetime equipped with the deformed symmetry algebra (<ref>-<ref>). It has been found<cit.> that the dimension reduces from d_S = 4at large scales to d_S = 1 at short scales, reflecting theultralocality arising in the limit Ω̃→ 0. The connection between HDA and the Poincaré algebra is a crucial element inthe construction of our map of quantum gravity. When generalized to the deformedcase, it allows to relate (see the end of this section) the two classes of, previously separated, approaches to the Planckscale physics: LQG and models of spacetime with deformed relativistic symmetries. This is illustrated in Fig. <ref>. In particular, let us focus on the DHDA given by (<ref>). In theΩ→ 0 limit it becomes an ultralocal algebra. By the restrictionto the case of linear hypersurface deformations, this algebra reduces to the so-called Carroll algebra.So far, limited amount of research has been directed toward this structure. However, as we saw,it may actually reflect the quantum gravitational effects. The Carroll algebra has actually been introduced at the classical level, where it isobtained by taking the limit of vanishing speed of light, c → 0, of the Poincaré algebra<cit.>. This qualitatively overlaps with the observation made forDHDA, where the effective speed of propagation is found to be v_eff = √(Ω)and vanish for Ω→ 0 <cit.>. The general form of DHDA defined by the brackets (<ref>) may lead to various deformations of the Poincaréalgebra. In particular, in <cit.> it was argued that the DHDA that was previously discussed in<cit.> possibly corresponds to the famous κ-Poincaré algebra <cit.>.The latter algebra constitutes the archetypal example of quantum deformations of spacetime symmetries,which naturally appear in the semiclassical frameworks known as Doubly Special Relativity <cit.> andRelative Locality <cit.> (see below). Moreover, the form of deformations of both the Poincaré algebraand the corresponding HDA has been derived for certain multifractional spacetimes <cit.>. The resultsin these cases turn out to differ from (<ref>). Meanwhile, the reconstruction of the appropriate DHDA in RelativeLocality is an open task. An important role will probably be played there by the nonlinear momentum space, or nonlinearphase space, which is associated with the deformed relativistic symmetries.§ NONLINEAR STRUCTURE OF PHASE SPACE Deformations of relativistic symmetries discussed in the previous section areclosely related to another concept appearing in the context of quantum gravity: momentum space or, more generally,phase space with the nontrivial geometry or topology. The origins of this idea go back to M. Born, who argued thatmomentum space and the space of particle positions,i.e. spacetime, in quantum physics are connected by the reciprocity symmetry. Therefore,in the regime of full quantum gravity, curved spacetime should be complementedby momentum space with similarly nontrivial Riemannian geometry <cit.>.Later it was observed <cit.> that non-vanishing curvature ofmomentum space leads to the noncommutativity of spacetime coordinates.This intuition was subsequently confirmed in the mathematical formalism ofquantum groups, i.e. nontrivial Hopf algebras, which give a natural descriptionof the deformed relativistic symmetries <cit.>. The most interesting caseof such deformed symmetries is the κ-Poincaré (Hopf) algebra<cit.>. This algebra acts covariantly onthe noncommutative κ-Minkowski space <cit.>, while themomentum space corresponding to the latter is the AN(3) Lie group,which as a manifold is equivalent to half of de Sitter space <cit.>. The above mathematical structures have been utilized <cit.> in the framework known as Doubly(or deformed) Special Relativity, which is the family of models that attempt to representthe semiclassical regime of quantum gravity characterized by the existence of twoinvariant scales, given by the speed of light and Planck mass. These two scalesdetermine the geometry of flat spacetime and curved momentum space. Doublyspecial relativity has recently been recast and generalized into the so-calledRelative Locality approach <cit.>, whoseunderlying principle is that (at sufficiently high energies) only the full phase spaceis an absolute physical entity, while its decomposition into spacetime and momentumspace depends on the choice of an observer. More specifically, the structure of spacetime is inferredfrom the dynamics of particles in (curved) momentum space. This leads to the relativityof locality of events in spacetime, i.e. physical events which have the same spacetimecoordinates in the frame of a certain observer are not coincidental according toobservers that are distant from the former. If we want to extend the above conceptto the generally covariant case, both spacetime and momentum space should be allowed to havethe nontrivial geometry. In order to study such a scenario there have been attempts to construct theaction principle <cit.> or the Hamilton geometry <cit.> for particles having the nontrivial phase space. On the morefundamental level, there have been proposed <cit.>a unified description of the (dynamical) phase space geometry, which involves thesymplectic form, the generalized metric (combining the metrics on spacetime andmomentum space) and the locality metric (which encodes the pairing betweenspacetime and momentum space), satisfying the appropriate compatibility conditions.Using such a formulation the relative locality approach can also be introduced in theframework of string theory, where it is realized as a generalization to the meta-string theory. On the other hand, the nontrivial geometry of phase space has also been consideredfrom the perspective of the candidate full quantum gravity theories (and not justmodels of quantum spacetime). For example, curved momentum space can in principle arisewithin the Group Field Theory approach. At theelementary level of this framework, spacetime is replaced by a quantumfield defined on the background of several copies of a certain Lie group(associated with relativistic symmetries), which is also acurved manifold and whosetangent space is given by the related Liealgebra. A particle excitation of the field corresponds toa single quantum of discrete space(time), while different quantabecome connected via the combinatorially nonlocal field interactions. Continuousspacetime is expected to emerge only in the semiclassical limit and therea group field theory should reduce to a certaineffective field theory (describing perturbations around the classical solution) <cit.>. It is reasonable to suppose thatthe latter field will perceive some effective Lie group as configuration spaceand the (dual) Lie algebra as momentum space. However, one may also envisagethe opposite situation and then, at the mathematical level, thedifference between a given group field theoryand the theory covariantunder the action of the appropriate Hopf algebra will be thecombinatorial structure of the former. Indeed, it has been tentatively shown <cit.> that ascalar field theory on κ-Minkowski space is equivalent to the effective,semiclassical approximation of a Poincaré group field theory. Another avenuethat potentially leads to nontrivial phase space is the already mentioned polymer quantization scheme,commonly applied within LQG, which can result in the circular (periodic) momentum space of a quantum mechanical system <cit.>. One might also wonder whether the concept of nontrivial phase spaceshould be extended to the domain of field theory, so that it is the phase space of valuesof a given field that has some nonlinear structure (rather than just the phase space on whicha field is defined). In <cit.> (see also <cit.>) the authors of this paper proposed the conjecture that ordinaryfield theories are actually the low energy limit of theories whose phase spacesof field values are not affine spaces but manifolds with nontrivial geometry or topology.This has been called the Nonlinear Field Space Theory (NFST). As a prototypicalexample of such a framework we considered a scalar field theory whose phase space at every point of spacetime, or for every Fourier mode, hasthe symplectic geometry of a sphere (while the background spacetime is assumed to beMinkowski space). Performing quantization of the field we then obtain a number ofpredictions typical to the quantum gravity models. Moreover, if the assumed phase space iscompact, as it is the case for a sphere, the so-called principle of finiteness <cit.> is automatically imposed ona given field, which automatically resolves the ubiquitous problem of UV divergences.In <cit.> we also preliminarily investigated the application of such a scalar field as thematter content of universe in the standard cosmological model, which turned out to leadto different interesting results. Worth discussing is a further extension of the nonlinear phase space framework to the area of quantum gravity.In this context let us take a look at the case of LQG. The starting point of the latter approach is the formalism ofAshtekar variables, in which the connection A and densitized triad E are the canonical fields, whose valuesbelong to the 𝔰u(2) algebra <cit.>. The phase space of classical GR parametrizedin terms of the Ashtekar variables is still affine. However, passing to LQG, A is subject to the exponentiation and forms a holonomy, being an element of a compact group SU(2) <cit.>. Meanwhile,the fluxes constructed with the use of E remain elements of the 𝔰u(2) algebra, which is isomorphicto ℝ^3. Therefore, similarly as we observed above in the context of Group Field Theory,one can say that the phase space of LQG is partially curved. On the other hand, in Sec. <ref> it was mentioned that the opposite setup for holonomies and fluxes is considered in the recentlyproposed alternative formulation of LQG <cit.>. Namely, E is then subject to the exponentiation,while fluxes of A remain elements of 𝔰u(2). The natural next step one could consider is to exponentiateboth E and A, so that the phase space per link of a spin network becomes Γ =SU(2) × SU(2),which is a compact manifold. The idea of generalizing LQG to such a case was discussed in <cit.>.While, so far, it has been mostly analyzed only in the (2+1)-dimensional theory, significant progresstowards the extension of this framework to 3+1 dimensions has recently been achieved <cit.>.§ SYNTHESIS: THE FIRST MAPIn this section we make the first attempt to collect the results discussed aboveinto a single map, representing relations between different approaches to quantum gravity.The map in Fig. <ref> has been constructed using the architectural structure exemplified in Fig. <ref>.Its core elements are:* Quantum deformations of HDA andthe corresponding deformations of the Poincaré algebra. The bridge between them allows to relate modelsof quantum spacetime with the candidate theories of quantum gravity. * Various phases of the gravitational field, invariant under different symmetries,which are described by the (deformed) symmetry algebras. * The phase transition lines, meeting at the triple/quadruple point. This point is possibly relatedto a UV fixed point of the RG flow, which is also supposed to coincide with the Lifshitz point of HLG.The asymptotic safety conjecture is expected to be realized there.* Nonlinear phase spaces of particles and fields, which can lead to deformations of the relativistic symmetries. * The ultralocal state, predicted to occur in such approaches to quantum gravity as the effective LQG, CDT andHořava-Lifshitz gravity. § SUMMARY AND DISCUSSION In this paper, in the spirit of the architectural approach to a physical theory,we carried out our first attempt to draw a mind map of quantumgravity. It is based on the most up to date results, obtained within variousapproaches to the Planck scale physics. As several researchers have observed (see Sec. <ref>), there are strong reasons to suppose that quantum gravity belongs to the realm of complex systems.A crucial indication is that the gravitational field is a system with a huge (perhaps infinite) number of degreesof freedom, which are subject to strongly non-linear interactions. Systems of thiskind exhibit various emergent phenomena, such as the nontrivial phase structure, containingphase transitions. Consequently, presumably there is no single effective description of states ofquantum gravity but rather several “equations of state”, corresponding todifferent phases. Nonlinearity implies the existence of various underlying mechanisms,each deserving a separate analysis and explanation with the help of adequatevariables and a set of concepts. Similarly, the functioning of a human braincannot be described using a single equation. Rather than that, a huge network of biochemicalprocesses, composed of numerous contributing reactions, has to be considered. The concept of a network, or graph, occurs naturally in the descriptionof complex systems. It consists of two essential levels: structural andfunctional. The first one means that a graph serves as a method forcapturing relations between elementary constituents of the complex system(e.g. connections between neurons in a brain) in the visual language.Meanwhile, at the second level a graph is a way of representing differentoperating processes and emergent phenomena in the system(e.g. epileptic seizures happening in a brain). While the first (structural) point of view is extensively used in the investigations ofquantum theory of gravity (e.g. spin networks, Regge calculus, tensor networks), thesecond one (functional) has had only a residual appearance in the quantumgravity research so far. This paper constitutes the first serious attempt to introduce thefunctional network approach into the domain of quantum gravity. In our discussion we have considereda simple architecture, combining such issues as the gravitational phase structure, running dimensionof spacetime, asymptotic safety of the theory as well as deformations of relativistic symmetries andnonlinear phase space structures. The obtained construction indicates certainmissing points, which lead to a number of open questions listed in the Appendix.Among the most important observations that contributed to the structure of the presented mapwe should mention: * Different phases of quantum gravity will have different effective descriptions(equations of state). * There may be a direct connection between the UV fixed point in theasymptotic safety scenario, the Lifshitz point in the Hořava-Lifshitz gravityand triple (quadruple) point on the CDT phase diagram. * A variety of approaches to quantum gravity predict the dimensional reductionof spacetime at small scales. * A given deformed hypersurface deformation algebra in the linear limitleads to the corresponding deformation of the Poincaré algebra. * The nonlinear structure of phase space is associated with deformationsof general and special relativistic symmetries. * The ultralocal (silent) state probably plays an important role in quantum gravity.Quantum gravity is a fascinating jigsaw puzzle. Many of its elements are definitelystill missing. The only way to find out which elements we need to look for is to puttogether the pieces that are already available. This has been the most important motivation behindour proposal, which deserves a further systematic extension, including additionalmodels and aspects of quantum gravity.This work is supported by the Iuventus Plus grant No. 0302/IP3/2015/73 from the Polish Ministryof Science and Higher Education. TT was additionally supported by the National Science CentrePoland, project 2014/13/B/ST2/04043.§ APPENDIX – OPEN PROBLEMS Here we collect the main open problems that emerged in the course of our investigations:* What is the behavior of the spectral dimension at the triple point on the phase diagram of CDT?* In the continuum, is the CDT triple point a Lifshitz point with z = 3?* Is this point also a nontrivial UV fixed point?* What is the order of the A-B phase transition in CDT?* Is the BKL conjecture realized in the phase B of CDT?* What is the nature of the Hořava-Lifshitz theory with the critical exponent z →∞?* What are the properties of the Hořava-Lifshitz theory with z = 6, which leads to d_S = 3/2?* What are the properties of the Hořava-Lifshitz theory with z = 9, which leads to d_S = 4/3?* What is the (class of) DHDA corresponding to the κ-Poincaré algebra?* Is the loop-deformed Poincaré algebra associated with certain nonlinear structure of phase space? Finding answers to the above questions will be a significant guidance in the further extension of themap of quantum gravity.§ REFERENCES 99 Rovelli:2000aw C. Rovelli,gr-qc/0006061. Blagojevic:2012gn M. Blagojević and F. W. Hehl,Imperial College Press, London 2012 [arXiv:1210.3775 [gr-qc]]. Clifton:2012my T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis,Phys. Rept. 513, 1 (2012) [arXiv:1106.2476 [astro-ph.CO]]. Smolin:1996ca L. Smolin,Lect. Notes Phys. 461, 184 (1996) [gr-qc/9505022]. Stephens:2001sd G. J. Stephens and B. L. Hu,Int. J. Theor. Phys. 40, 2183 (2001) [gr-qc/0102052]. Bojowald:2015qw M. Bojowald,Rep. Prog. Phys. 78, 023901 (2015) [arXiv:1501.04899 [gr-qc]]. Mielczarek:2014aka J. Mielczarek,Adv. in High Energy Phys. 2017, 4015145 (2017) [arXiv:1404.0228 [gr-qc]]. Barcelo:2005ay C. Barceló, S. Liberati and M. Visser,Liv. Rev. Rel. 8, 12 (2005) [gr-qc/0505065]. Oriti:2017twl D. Oriti,in “Many-body Approaches at Different Scales: A Tribute to Norman H. March on the Occasion of his 90th Birthday,” eds. G. G. N. Angilella and C. Amovilli, Springer, New York 2018 [arXiv:1710.02807 [gr-qc]]. Ambjorn:1991rx J. Ambjørn, D. V. Boulatov, A. Krzywicki and S. Varsted,Phys. Lett. B 276, 432 (1992). Bialas:1996wu P. Białas, Z. Burda, A. Krzywicki and B. Petersson,Nucl. Phys. B 472, 293 (1996) [hep-lat/9601024].Coumbe:2015em D. Coumbe and J. Laiho,JHEP 1504, 028 (2015) [arXiv:1401.3299 [hep-th]]. Ambjorn:2005qm J. Ambjørn, J. Jurkiewicz and R. Loll,Phys. Rev. Lett. 93, 131301 (2004) [hep-th/0505154]. Ambjorn:2012jv J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll,Phys. Rept. 519, 127 (2012) [arXiv:1203.3591 [hep-th]]. Ambjorn:2018ty J. Ambjørn, J. Gizbert-Studnicki, A. Görlich, J. Jurkiewicz and D. Németh,arXiv:1802.10434 [hep-th]. Jordan:2013cn S. Jordan and R. Loll,Phys. Lett. B 724, 155 (2013) [arXiv:1305.4582 [hep-th]]. Ambjorn:2017ns J. Ambjørn, D. N. Coumbe, J. Gizbert-Studnicki, A. Görlich and J. Jurkiewicz,Phys. Rev. D 95, 124029 (2017) [arXiv:1704.04373 [hep-lat]]. Ambjorn:2012ij J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll,Phys. Rev. D 85, 124044 (2012) [arXiv:1205.1229 [hep-th]]. Ambjorn:2011cg J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll,Phys. Rev. Lett. 107, 211303 (2011) [arXiv:1108.3932 [hep-th]].Ambjorn:2015qja J. Ambjørn, D. N. Coumbe, J. Gizbert-Studnicki and J. Jurkiewicz,JHEP 1508, 033 (2015) [arXiv:1503.08580 [hep-th]]. Wilkinson:2014zga S. A. Wilkinson and A. D. Greentree,Phys. Rev. D 90, 124003 (2014) [arXiv:1409.2557 [gr-qc]]. Wilkinson:2015fja S. A. Wilkinson and A. D. Greentree,Phys. Rev. D 92, 084007 (2015) [arXiv:1506.07588 [gr-qc]]. Konopka:2008hp T. Konopka, F. Markopoulou and S. Severini,Phys. Rev. D 77, 104029 (2008) [arXiv:0801.0861 [hep-th]]. Magueijo:2006fu J. Magueijo, L. Smolin and C. R. Contaldi,Class. Quant. Grav. 24, 3691 (2007) [astro-ph/0611695]. Mandrysz:2018us M. Mandrysz and J. Mielczarek,arXiv:1804.10793 [gr-qc]. Horava:2009uw P. Hořava,Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]]. Ambjorn:2010hu J. Ambjørn, A. Görlich, S. Jordan, J. Jurkiewicz and R. Loll,Phys. Lett. B 690, 413 (2010) [arXiv:1002.3298 [hep-th]]. Oriti:2014uga D. Oriti,in “Loop Quantum Gravity: The First 30 Years,” eds. A. Ashtekar and J. Pullin, World Scientific, Singapore 2017 [arXiv:1408.7112 [gr-qc]]. Benedetti:2014qsa D. Benedetti, J. Ben Geloun and D. Oriti,JHEP 1503, 084 (2015) [arXiv:1411.3180 [hep-th]]. Gielen:2016qw S. Gielen and L. Sindoni,SIGMA 12, 082 (2016) [arXiv:1602.08104 [gr-qc]]. Oriti:2016acw D. Oriti,Comptes Rendus Physique 18, 235 (2017) [arXiv:1612.09521 [gr-qc]]. Delcamp:2016dqo C. Delcamp and B. Dittrich,Class. Quant. Grav. 34, 225006 (2017) arXiv:1612.04506 [gr-qc]. Dittrich:2014wpa B. Dittrich and M. Geiller,Class. Quant. Grav. 32, 112001 (2015) [arXiv:1401.6441 [gr-qc]]. Bonzom:2011zz V. Bonzom, R. Gurau, A. Riello and V. Rivasseau,Nucl. Phys. B 853, 174 (2011) [arXiv:1105.3122 [hep-th]]. Bonzom:2014ts V. Bonzom, R. Gurau, J. P. Ryan and A. Tanasa,JHEP 1409, 051 (2014) [arXiv:1404.7517 [hep-th]]. Oriti:2018tym D. Oriti,arXiv:1803.02577 [physics.hist-ph].WeinbergAS S. Weinberg,in A. Zichichi (ed.), Understanding the Fundamental Constituents of Matter. The Subnuclear Series 14, 1-52 (1978). Percacci:2007sz R. Percacci, in D. Oriti (ed.), Approaches to quantum gravity: Towards a New Understanding of Space, Time and Matter 111-128, Cambridge 2009 [arXiv:0709.3851 [hep-th]]. Reuter:2012id M. Reuter and F. Saueressig,New J. Phys. 14, 055022 (2012) [arXiv:1202.2274 [hep-th]]. Ambjorn:2014gsa J. Ambjørn, A. Görlich, J. Jurkiewicz, A. Kreienbuehl and R. Loll,Class. Quant. Grav. 31, 165003 (2014) [arXiv:1405.4585 [hep-th]]. Ambjorn:2005db J. Ambjørn, J. Jurkiewicz and R. Loll,Phys. Rev. Lett. 95, 171301 (2005) [hep-th/0505113]. Coumbe:2014noa D. N. Coumbe and J. Jurkiewicz,JHEP 1503, 151 (2015) [arXiv:1411.7712 [hep-th]].Horava:2009if P. Hořava,Phys. Rev. Lett. 102, 161301 (2009) [arXiv:0902.3657 [hep-th]]. Lauscher:2005fy O. Lauscher and M. Reuter, JHEP 0510, 050 (2005) [hep-th/0508202]. Modesto:2012sy L. Modesto, Phys. Rev. D 86, 044005 (2012) [arXiv:1107.2403 [hep-th]]. Modesto:2009fm L. Modesto, Class. Quant. Grav. 26, 242002 (2009) [arXiv:0812.2214 [gr-qc]]. Calcagni:2014cza G. Calcagni, D. Oriti and J. Thürigen,Phys. Rev. D 91, 084047 (2015) [arXiv:1412.8390 [hep-th]]. Carlip:2016dy S. Carlip, Class. Quant. Grav. 32, 232001 (2015) [arXiv:1506.08775 [gr-qc]]. Belenchia:2016ss A. Belenchia, D. M. T. Benincasa, A. Marcianò and L. Modesto, Phys. Rev. D 93, 044017 (2016) [arXiv:1507.00330 [gr-qc]]. Calcagni:2012qn G. Calcagni, Phys. Rev. D 86, 044021 (2012) [arXiv:1204.2550 [hep-th]]. Benedetti:2009fe D. Benedetti, Phys. Rev. Lett. 102, 111303 (2009) [arXiv:0811.1396 [hep-th]]. Arzano:2014de M. Arzano and T. Trześniewski, Phys. Rev. D 89, 124024 (2014)[arXiv:1404.4762 [hep-th]]. Arzano:2017ua M. Arzano and F. Nettel, Phys. Lett. B 767, 236 (2017)[arXiv:1611.10343 [hep-th]]. Mielczarek:2016zfz J. Mielczarek and T. Trześniewski,Phys. Rev. D 96, 024012 (2017) [arXiv:1612.03894 [hep-th]]. Carlip:2017dy S. Carlip, Class. Quant. Grav. 34, 193001 (2017) [arXiv:1705.05417 [gr-qc]]. Calcagni:2017mw G. Calcagni, JHEP 1703, 138 (2017); 1706, 020 (2017) [arXiv:1612.05632 [hep-th]]. Sotiriou:2011ss T. P. Sotiriou, M. Visser and S. Weinfurtner,Phys. Rev. Lett. 107, 131303 (2011) [arXiv:1105.5646 [gr-qc]]. Calcagni:2013pn G. Calcagni, A. Eichhorn and F. Saueressig, Phys. Rev. D 87, 124028 (2013)[arXiv:1304.7247 [hep-th]]. Isham:1976 C. J. Isham,Proc. Roy. Soc. Lond. A 351, 209 (1976). Calcagni:2016as G. Calcagni, Eur. Phys. J. C 76, 181 (2016); 76, 459 (2016) [arXiv:1602.01470 [hep-th]]. Bojowald:2016hgh M. Bojowald, S. Brahma, U. Büyükçam and F. D'Ambrosio,Phys. Rev. D 94, 104032 (2016) [arXiv:1610.08355 [gr-qc]]. Bojowald:2011aa M. Bojowald and G. M. Paily,Phys. Rev. D 86, 104018 (2012) [arXiv:1112.1899 [gr-qc]]. Bojowald:2012ux M. Bojowald and G. M. Paily,Phys. Rev. D 87, 044044 (2013) [arXiv:1212.4773 [gr-qc]]. Cailleteau:2011kr T. Cailleteau, J. Mielczarek, A. Barrau and J. Grain,Class. Quant. Grav. 29, 095010 (2012) [arXiv:1111.3535 [gr-qc]]. Cailleteau:2013kqa T. Cailleteau, L. Linsefors and A. Barrau,Class. Quant. Grav. 31, 125011 (2014) [arXiv:1307.5238 [gr-qc]]. Bojowald:2014zla M. Bojowald,Front. in Phys. 3, 33 (2015) [arXiv:1409.3157 [gr-qc]].Belinsky:1970ew V. A. Belinsky, I. M. Khalatnikov and E. M. Lifshitz,Adv. Phys. 19, 525 (1970). Belinsky:1982pk V. A. Belinsky, I. M. Khalatnikov and E. M. Lifshitz,Adv. Phys. 31, 639 (1982). Andersson:2004wp L. Andersson, H. van Elst, W. C. Lim and C. Uggla,Phys. Rev. Lett. 94, 051101 (2005) [gr-qc/0402051]. Mielczarek:2012pf J. Mielczarek,Springer Proc. Phys. 157, 555 (2014) [arXiv:1207.4657 [gr-qc]]. Bojowald:2015gra M. Bojowald and J. Mielczarek,JCAP 1508, 052 (2015) [arXiv:1503.09154 [gr-qc]]. Mielczarek:2012tn J. Mielczarek,AIP Conf. Proc. 1514, 81 (2012) [arXiv:1212.3527 [gr-qc]]. Mielczarek:2013rva J. Mielczarek,Europhys. Lett. 108, 40003 (2014) [arXiv:1304.2208 [gr-qc]]. Levy-Leblond:1965J.-M. Lévy-Leblond, Ann. I. H. P.: Phys. Theor. 3, 1 (1965). Amelino-Camelia:2016gfx G. Amelino-Camelia, M. M. da Silva, M. Ronco, L. Cesarini and O. M. Lecian,Phys. Rev. D 95, 024028 (2017) [arXiv:1605.00497 [gr-qc]]. Lukierski:1991qa J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoi, Phys. Lett. B 264, 331 (1991). Lukierski:1992ny J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B 293, 344 (1992). Magueijo:2001cr J. Magueijo and L. Smolin,Phys. Rev. Lett. 88, 190403 (2002) [hep-th/0112090]. Amelino:2011py G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Gen. Relativ. Gravit. 43, 2547 (2011) [arXiv:1101.0931 [hep-th]]. Calcagni:2017ds G. Calcagni and M. Ronco, Phys. Rev. D 95, 045001 (2017) [arXiv:1608.01667 [hep-th]]. Born:1938ay M. Born, Proc. R. Soc. Lond. A 165, 291 (1938). Snyder:1947qe H. S. Snyder, Phys. Rev. 71, 38 (1947). Majid:1995fy S. Majid,“Foundations of Quantum Group Theory,”Cambridge University Press 1995. Majid:1994by S. Majid and H. Ruegg, Phys. Lett. B 334, 348 (1994) [hep-th/9405107]. Kowalski:2002dy J. Kowalski-Glikman, Phys. Lett. B 547, 291 (2002) [hep-th/0207279]. Kowalski:2003de J. Kowalski-Glikman and S. Nowak, Class. Quant. Grav. 20, 4799 (2003) [hep-th/0304101]. Amelino:2002re G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002) [gr-qc/0012051]. Amelino:2002dy G. Amelino-Camelia, Nature 418, 34 (2002) [gr-qc/0207049]. Girelli:2005dy F. Girelli, E. R. Livine and D. Oriti, Nucl. Phys. B 708, 411 (2005) [gr-qc/0406100]. Amelino:2011re G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84, 084010 (2011) [1106.0313 [hep-th]]. Cianfrani:2014ge F. Cianfrani, J. Kowalski-Glikman and G. Rosati, Phys. Rev. D 89, 044039 (2014) [arXiv:1401.2057 [gr-qc]]. Barcaroli:2015hs L. Barcaroli, L. K. Brunkhorst, G. Gubitosi, N. Loret and C. Pfeifer,Phys. Rev. D 92, 084053 (2015) [arXiv:1507.00922 [gr-qc]]. Freidel:2014be L. Freidel, R. G. Leigh and D. Minic, Phys. Lett. B 730, 302 (2014) [arXiv:1307.7080 [hep-th]]. Freidel:2015me L. Freidel, R. G. Leigh and D. Minic, JHEP 06, 006 (2015) [arXiv:1502.08005 [hep-th]].Oriti:2012ty D. Oriti, in “Foundations of Space and Time: Reflections on Quantum Gravity,” eds.J. Murugan, A. Weltmann and G. F. R. Ellis, Cambridge University Press 2012[arXiv:1110.5606 [hep-th]]. Girelli:2010fs F. Girelli, E. R. Livine and D. Oriti, Phys. Rev. D 81, 024015 (2010) [arXiv:0903.3475 [gr-qc]]. Bojowald:2011ge M. Bojowald and A. Kempf, Phys. Rev. D 86, 085017 (2012) [arXiv:1112.0994 [hep-th]]. Mielczarek:2016ty J. Mielczarek and T. Trześniewski, Phys. Lett. B 759, 424 (2016) [arXiv:1601.04515 [hep-th]]. Mielczarek:2016fe J. Mielczarek, Universe 3, 29 (2017) [arXiv:1612.04355 [hep-th]]. Trzesniewski:2017fe T. Trześniewski, Acta Phys. Pol. B Proc. Suppl. 10, 329 (2017) [arXiv:1701.06865 [hep-th]]. Bilski:2017gic J. Bilski, S. Brahma, A. Marcianò and J. Mielczarek,arXiv:1708.03207 [hep-th]. Born:1934fy M. Born and L. Infeld, Proc. Roy. Soc. A 144, 425 (1934). Mielczarek:2017ny J. Mielczarek and T. Trześniewski, Phys. Rev. D 96, 043522 (2017) [arXiv:1704.01934 [gr-qc]]. Ashtekar:1986yd A. Ashtekar,Phys. Rev. Lett. 57, 2244 (1986).Rovelli:1989za C. Rovelli and L. Smolin,Nucl. Phys. B 331, 80 (1990). Bahr:2015bra B. Bahr, B. Dittrich and M. Geiller,arXiv:1506.08571 [gr-qc].Rovelli:2015fwa C. Rovelli and F. Vidotto,Phys. Rev. D 91, 084037 (2015) [arXiv:1502.00278 [gr-qc]]. Dittrich:2017nmq B. Dittrich,JHEP 1705, 123 (2017) [arXiv:1701.02037 [hep-th]].Dittrich:2016typ B. Dittrich and M. Geiller,New J. Phys. 19, 013003 (2017) [arXiv:1604.05195 [hep-th]].Riello:2017iti A. Riello,Phys. Rev. D 97, 025003 (2018) [arXiv:1706.07811 [hep-th]]. | http://arxiv.org/abs/1708.07445v3 | {
"authors": [
"Jakub Mielczarek",
"Tomasz Trześniewski"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170824150006",
"title": "Towards the map of quantum gravity"
} |
Sales Forecast in E-commerce using Convolutional Neural NetworkKui Zhao College of Computer Science Zhejiang University Hangzhou, China [email protected] Can Wang College of Computer Science Zhejiang University Hangzhou, China [email protected]: December 30, 2023/ Accepted: December 30, 2023 ==========================================================================================================================================================================================================================================================fancyshortarticlesinglecolumn§ INTRODUCTION Current theoretical formulations and computational algorithms for time domain electromagnetic problems are well-developed for applications that range from transient spectroscopy <cit.> to telemetry involving high speed spacecrafts <cit.>. Under conditions relevant to micro-photonics in which the interest in regimes of wavelengths can be small or large compared to the characteristic dimensions of the scatterers or the need to have accurate values of the phases and amplitudes of the field near boundaries, current methodologies encounter challenges.This paper addresses the time domain electromagnetic problem of the scattering of electromagnetic pulses by working directly in terms of the components of the electric field using a non-singular surface integral formulation in the frequency domain.Fourier transform is then used to give the complete space-time behavior. This approach retains the advantages of reduction in spatial dimension of surface integral methods, and because of the non-singular nature of the surface integrals, it has the added ability to handle surface geometric intricacies that often arise in micro-photonics problems in which certain length scales in the problem many be small compared to the wavelength or where there is the need to obtain accurate results for field values on or near surfaces.The use of the Fourier transform to give the time evolution also avoids possible instabilities that can arise with march on time algorithms. Examples of the space-time dependence of scattering of an incident wave pulse by conducting and dielectric scatterers are provided to illustrate the transition between wave and geometric optics.The structure of the field near the surfaces is used to demonstrate field focusing and multiple scattering effects.§ BACKGROUND One established approach to time domain electromagnetics is based on extending the surface integral formulations, the electric field integral equation (EFIE) or magnetic field integral equation (MFIE) method, to the time domain using a march on time method, see for example, <cit.>, <cit.> and <cit.> for reviews.This widely used approach involves solving surface integral equations for the surface currents. The electric and magnetic field are then obtained by post-processing the surface current values. In present formulations, the surface integral equations that need to be solved for the induced surface current densities contain singular kernels that originate from the Green's function. This mathematical feature that does not have a physical basis means a loss of precision in the calculation of field values close to boundaries between different media.Furthermore, march on time algorithms can also lose precision as time progresses.Another approach to time domain electromagnetics is the finite difference time domain (FDTD) method of Yee <cit.> where the space- and time-dependent Maxwell's equations are discretized in the 3D space variables and time stepping is used to track the space-time evolution from given initial conditions. Although the FDTD algorithm is simple conceptually, there are a number of technical issues that require care in implementation.Convergence constraints impose limitations on the step sizes in time and space.Numerical dispersion effects associated with the relative orientation of the spatial grid and the direction of propagation can arise and unphysical reflections can occur at the boundaries (that conform to the stepwise nature of the grid) between regions of the different grid densities. If the problem domain is infinite, an outer boundary needs to be constructed with suitable boundary conditions to satisfy the radiation condition at infinity so as to avoid unphysical reflections back into the solution domain <cit.>.Earlier works on pulses have been based on extending the analytic Mie theory for scattering by a sphere <cit.> and as such are not readily applicable to consider problems involving general scatterers.§ THEORYThe present theoretical development is motivated by the Mie theory <cit.> in which the solutions of Maxwell's equations in the frequency domain are constructed from 2 scalar Debye potentials that obey the Helmholtz equation.Although the Debye potentials can be found by solving boundary integral equations, they are not suitable for numerical implementation because the boundary conditions on the electric, E and magnetic, H fields involve the first and second derivatives of these scalar potentials.The following is a summary of the field only, non-singular boundary integral formulation of the solution of Maxwell's equation in the frequency domain. Detailed development and numerical implementation issues have been given in the literature <cit.>.For a vector field that satisfies wave equation∇^2 E + k^2 E = 0the divergence free condition on E can be written as2 ∇·E≡∇^2 (x·E) + k^2 (x·E) = 0whereby the scalar function (x·E), where x is the position vector, also satisfies the scalar wave equation or the Helmholtz equation.Therefore the solution of Maxwell's equation for the scattered electric field, E can be expressed as the simultaneous solution of 4 coupled scalar Helmholtz equations of the general form∇^2 p_i(x) + k^2 p_i(x) = 0, i = 1 .. 4 where p_i(x) denotes (x·E) or one of the 3 Cartesian components of E. These 4 equations are coupled by the usual boundary conditions on the normal and tangential components of E at material boundaries. The scalar function (x·E) is the result of the application of the angular momentum operator on one of the Debye potentials <cit.>. Writing the solution of Eq. <ref> as surface integral equations furnishes 3 relations between the 6 unknowns: E_α and ∂ E_α / ∂ n, (α =x,y,z), where ∂/∂ n ≡n·∇ and n is the outward unit normal of the surface, S of the solution domain. The boundary integral solution of Eq. <ref> for the quantity (x·E) provides one more relation between E_α and ∂ E_α / ∂ n since: ∂ (x·E) / ∂ n = n·E + x·∂E / ∂ n. The electromagnetic boundary conditions on the continuity of the tangential components of E provide the remaining 2 equations to determine E and ∂E / ∂ n completely. At the surface of a perfect electrical conductor, the tangential component of the incident field must be canceled by that of the scattered field.The boundary integral solution of Eq. <ref> for the scattered field is based on Green's Second Identity that gives a relation between p_i(x) and its normal derivative ∂p_i/∂n at points x and x_0 on the boundary, S.All singularities associated with the Green's function G ≡ G(x,x_0) = exp(ikr) / r, as r ≡ |x - x_0|→ 0 can be removed analytically by using the following non-singular formulation of the boundary integral equations <cit.>∫_S^[p_i (x) - p_i (x_0) g(x) - ∂p_i(x_0)/∂nf(x)] ∂G/∂n dS(x) =∫_S^G [ ∂p_i (x)/∂n - p_i (x_0) ∂g (x)/∂n - ∂p_i(x_0)/∂n∂f (x)/∂n ]dS(x). The requirement on the functions f(x) and g(x) is that they satisfy the Helmholtz equation and the following conditions at the point x = x_0 on surface, S: <cit.>.f(x_0) = 0n·∇ f(x_0) = 1 g(x_0) = 1n·∇ g(x_0) = 0Therefore, if p_i (or ∂p_i/∂n) is given, then Eq. <ref> can be solved for ∂p_i/∂n (or p_i) in a straightforward manner. The reason is that for f(x) and g(x) that obey the above conditions, Eq. <ref>, the terms that multiply G and ∂ G/∂ n vanish at the same rate as the rate of divergence of G or ∂ G/∂ n as x→x_0 and consequently both integrals have non-singular integrands and can thus be evaluated accurately by quadrature, see <cit.> for details.Consequently, this approach affords higher numerical precision with fewer degrees of freedom and confers numerically robustness that enables, in particular, the accurate calculation of field values on or near boundaries without adverse numerical issues. Note that the solid angle at x_0 that occurs in the traditional boundary integral equation has also been eliminated in Eq. <ref>. Suitable choices for f(x) and g(x) can be found in <cit.>. Similar coupled equations also hold for the magnetic field H, see <cit.> for details.Furthermore, the present field only formulation is not affected by the so called zero frequency catastrophe as k → 0 that limits the accuracy of the familiar electric and magnetic field integral equations for the surface current density <cit.>.As to be expected, the system of equations to be solved are simpler for perfect electrical conductor (PEC) scatterers <cit.> than for dielectric scatterers <cit.>, but the underlying physical concepts are the same. § RESULTS A demonstration of our field only approach for small and large dimensionless wave number, ka where a is a characteristic length scale of the problem, is now given.To this end we consider variations of the magnitude of the scattered field around:a) a perfect electrical conductor (PEC) sphere of radius, a,b) an axisymmetric bowl shaped PEC object obtained by rotating the parametric closed curve in the body coordinates, see Fig. <ref>(X, Z) = a ( 2 sinθ, βsin^2 θ + γ [cosθ - 1]),0 ≤θ < 2 π,about the Z-axis.The bowl rim has radius, 2a and the inner concave bottom of the bowl can be fitted to a parabola Z = X^2/(4f) where the focal length, f = R/2 is related to the radius of curvature, R = 4a/(2β-γ) at the inner center of the bowl.Here we choose β = 0.6 and γ = 0.5 that gives f = 2.86 a and R = 5.71 a as shown in Fig.<ref>. c) an axisymmetric thin nano-rod with length, L and maximum cross-section diameter, b whose surface is defined by rotating an analytic curve about the long axis <cit.>. The aspect ratio of the rod is L/b = 10. The cases of the nano-rod being a PEC and a dielectric are studied.In Fig. <ref> we show the scattered field amplitudes by perfect electrical conductors (PEC): a sphere and a bowl, excited by an incident plane wave that propagates in the positive z-direction and with E polarized in the y-direction with wave number, k. The results shown in the yz-plane demonstrate the transition from the electrostatic limit at small wave numbers, ka, to approaching geometric or physical optics as ka increases.Note also the high induced local field strengths associated with the corona effect that is evident around the high curvature rim of the bowl in the long wavelength (small ka) limit.To demonstrate time dependent effects, we consider the scattering of an incident plane wave pulse: E^inc = (0, E^inc_y, 0) that is polarized in the y-direction and propagates in the positive z-direction with the following form in a window of width, w E^inc_y = {[0, -4 π N_c ≤τ< -2 π N_c; sin(τ) exp (-α |τ|), -2 π N_c ≤τ≤ 2 π N_c; 0, 2 π N_c < τ≤ 4 π N_c; ].where τ≡ k_0 (z-vt).Examples of such pulse functions that comprise of 2N_c oscillatory cycles and their Fourier components calculated using a 128 point discrete Fourier transform are given in Fig. <ref> for different values of N_c and α <cit.>.Since the Fourier amplitudes are small at high wave numbers, using say the first 20 Fourier components is generally sufficient to give an accurate representation of the pulse.For each of these Fourier components of the incident pulse we solve for E and (x·E) at the corresponding value of k and then the time domain behavior can be found by inverse discrete fast Fourier transform <cit.>. As in earlier work on the scattering of acoustic pulses <cit.>, we set the total window width of the incident pulse to be w = 20.1a, of which the central oscillatory portion has width w/2.So for a pulse with 2N_c oscillatory cycles over the width w/2, the wave number, k_0 in Eq. <ref> is determined by the relation (2N_c)λ_0 ≡ (2N_c) (2π/k_0) = w/2, giving k_0 a = 8π N_c/20.1. The results are presented in such a way, that the pulse just reaches the bottom of the scatterer at the first frame. To account for this, each frequency component of the incoming wave must be multiplied with a phase factor e^iβ with β=-k_ma[w/(4a)-1] and k_m the m^th wave number of the Fourier spectrum. In Fig. <ref> we show both the total and scattered E_y electric field amplitudes as well as the electric field vectors excited by the plane wave pulse given by Eq. <ref>. Results in the yz-plane around a PEC sphere are shown at selected time steps out of a total of 128 steps (1002 nodes and 500 quadratic elements were used). The field strengths are indicated by the color scale and their strengths and directions by arrows. These results have been validated against available analytic solutions <cit.>.In Fig. <ref>, we show the magnitudes of the total field excited by a broader incident plane wave pulse given by Eq. <ref> with N_c = 1, α = 0.5 and hence k_0 a = 8π/20.1 in the neighborhood of a PEC bowl with 1002 nodes and 500 quadratic elements. Here the high field strengths associated with the corona effect are more evident around the high curvature rim of the bowl although the maximum field amplitude is only about half that in Fig. <ref>.In Fig. <ref> we show the total amplitudes and field vectors of E excited by the same incident plane wave pulse as in Fig. <ref> in the neighborhood of a PEC bowl with rim radius 2a as given in Eq. <ref>.The electric field strengths are indicated by the color scale and the field strengths and directions by arrows. The focusing effects due to the curvature of the bowl are clearly evident with regions of high field amplitudes located near the position of the focal point discussed above.The focusing effect increases the field amplitude by up to about 50%.Complex structures in the vector E are evident in the enlarged sub-figures near the concave surface of the bowl. To demonstrate the ability of the present field only formulation to handle multiple scattering effects in which spatial configurations of the scatterer can cause significant field enhancements, we consider the scattering by the combined effect of a PEC bowl that has a rim radius, 2a with a sphere of diameter a, that is located near the focal point of the bowl. The bowl and the sphere each have 362 nodes and 180 quadratic elements. In Fig. <ref>, we show the space-time variation of the total field amplitudes and directions, for an incident pulse given by Eq. <ref> with N_c = 2, α = 0.1 and k_0 a = 16π/20.1 – the same as that in Fig. <ref>. Here we see that this configuration of a bowl with a sphere placed near its focus can enhance the local field amplitude by about a factor of 3, twice that achieved by a bowl on its own. The complex field structures in the region between the bowl and the sphere are shown in the enlarged sub-figures.The scattering from an object with high aspect ratio is tackled next, both for a PEC and a dielectric object, a task which is rather challenging for conventional methods.In Fig. <ref> and Fig. <ref>, we compare the values of the E_y and E_z components of the total electrical field excited by the incident pulse given in Eq. <ref> by a PEC and by a dielectic nano-rod with refractive index n_p=3 <cit.>, the surrounding medium has unit refractive index.The y-component of the electric field, E_y is shown on the left and the z-component, E_z on the right at three time instances in both figures.In both cases, the long axis of the nano-rod is oriented at an angle of 45^∘ to the propagation direction of the incident pulse.The surface of the axial symmetric nano-rod is defined by rotating an analytic closed curve about the long axis <cit.>.The rod has length, L and maximum cross-section diameter, b at the middle of the rod, with L/b = 10.In both figures, the incident pulse is defined by Eq. <ref> with N_c=2 and α = 0.1.This corresponds to, k_0L = 32π/20.1 and k_0b = 3.2π/20.1 so that the length of the nano-rod is approximately the same wavelength, λ_0 ≡ 2π/k_0 of the incident pulse and the width of the nano-rod is about λ_0/10.As expected, the PEC nano-rod scatters much more strongly than its dielectric counterpart.This is evident by comparing the snapshots at the same time step in Fig. <ref> and Fig. <ref>.At frame 31, when nano-rod is near the center of the pulse, the scattering due to the PEC nano-rod is larger and consequently the perturbation of the total field is more severe.Towards the tail of the pulse at frame 53, the scattered field around the PEC is again much more pronounced than that of the dielectric nano-rod. After the pulse has passed, the scattered field from the dielectric nano-rod is essentially zero, whereas that due to the PEC is still creating a scattering effect which slowly dies out.This effect is even more obvious in the supplementary Visualizations 5 and 6).As a check of our numerical implementation, the scattering goes to zero smoothly, as expected, when the nano-rod is made transparent by taking the limit of its refractive index to be the same as that of the surrounding medium, n_p→ 1.§ CONCLUSIONSWe have leveraged on the direct nature and numerical stability of the field only formulation of the solution of Maxwell's equations in the frequency domain <cit.> to extend it to find time domain solutions using the Fourier transform.The field only formulation solves directly for the field components instead of first finding the surface current and then computing the field by post-processing. It is therefore particularly suitable for situations in which one is interested in the field amplitudes, phases and directions, especially near material boundaries - often desired in the design of microphotonic devices or in surface enhanced Raman spectroscopy.In contrast, because of the singular nature of the surface integrals for the surface currents, the post processing step to find field components from the surface currents becomes unstable numerically as the field point becomes close to a boundary. Although only results for the field in the yz-plane of symmetry are given for simpler visualization purposes, the complex 3D structure of the field around the scatterers are readily available. Although the finite difference time domain (FDTD) method also solves directly for the field components, it is necessary to discretise the 3D spatial domain as opposed to only to focus on 3D boundaries in surface integral formulations. This becomes especially beneficial for spatial regions with different characteristic length scales such as near the rim of the PEC bowl and around the sharp ends of the nano-rods in our examples. The 3D discretization problem can be challenging for conventional methods. Furthermore, when the natural orientation of the scatterer is at an angle to the direction of field propagation, care is needed in the construction of the 3D spatial mesh to avoid spurious dispersion effects <cit.>. The present non-singular boundary integral formulation <cit.> for the solutions of the Helmholtz equation for the field components, is unaffected by numerical instabilities at the low frequency regime or in the regime when the characteristic length scale of the problem is small compare to the wavelength. In this low frequency or long wavelength regime, the conventional boundary integral equation approach will exhibit numerically troublesome near-singular behavior sometimes referred to as the zero frequency catastrophe that limits the precision that can be attained in calculations that involve first having to find induced surface currents <cit.>. In contrast, the present approach is numerically robust for wavelengths that span the electrostatic to approaching the geometric optics regimes without the need to modify the algorithm.The absence of singularities in the integral equations also means numerical instabilities do not arise in multiple scattering effects due to scatterers in close proximity or in the evaluation of field values close to material boundaries. And in the absence of singular kernels, it is easy to use quadratic surface elements with consistent quadratic interpolation of the continuous integrands <cit.>, as we have done so in our numerical examples, to evaluate the surface integrals. This means higher order precision can be obtained with fewer number of nodes compared to using planar surface elements For a fixed set of scatterers, the boundary integral equations only need to be solved once from which the space-time variation of the total field for different incident pulses can be found by inverse Fourier transform.In summary, the present approach involves solving non-singular scalar surface integral equations directly for the electric field components on the boundary of scatterers. In contrast, the conventional surface integral equation approach involves solving integral equations with singular kernels for surface currents. Although there are well developed methods to deal with such issues in the conventional surface integral approach, the integral equations become numerically ill-conditioned when the characteristic length scales of the scatterers are small compared to the wavelength. Since the electric field is obtained from the surface current by post processing, the presence of singularities in the integral equations means that the precision of field values at or near surfaces is compromised. The present approach avoids such issues by having to solve 4 scalar Helmholtz equations by a non-singular integral equation formulation. Thus, in handling the time evolution by Fourier transform, the robustness of our method at all frequencies is a distinct advantage.Compared to the finite difference time domain (FDTD) method, the ability to achieve a reduction in dimension by only having to solve the problem on the boundary of scatterers also removes the challenge of having to discretize a complex3D domain when there are vastly different characteristic length scales and spatial directions in a problem. In the FDTD approach, changing incident field for the same set of scatterers, requires repeating the solution process with each new incident field. In contrast, with the present approach, once the frequency domain problem has been found for a fixed set of scatterers, the solution can be used to find the response to different incident fields by simply taking the inverse Fourier transform. This work is supported by the Australian Research Council (ARC) through a Discovery Project Grant (DP170100376) to DYCC and a Discovery Early Career Researcher Award (DE150100169) to QS. Main | http://arxiv.org/abs/1708.07934v4 | {
"authors": [
"Evert Klaseboer",
"Qiang Sun",
"Derek Y. C. Chan"
],
"categories": [
"physics.optics",
"physics.class-ph",
"physics.comp-ph"
],
"primary_category": "physics.optics",
"published": "20170826053753",
"title": "Field-only integral equation method for time domain scattering of electromagnetic pulses"
} |
APS/123-QED Institute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, JapanDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Hongo, Tokyo 113-8656, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan National Institute for Materials Science (NIMS), Tsukuba 305-0047, JapanNara Institute of Science and Technology (NAIST), 89165, Takayama, Ikoma, Nara 630-0192, JapanInstitute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, JapanInstitute for Solid State Physics, University of Tokyo, Chiba 277-8581, JapanInstitute for Solid State Physics, University of Tokyo, Chiba 277-8581, JapanInstitute for Solid State Physics, University of Tokyo, Chiba 277-8581, JapanResearch Institute for Energy Conservation, National institute of Advance Industrial Science and Technology (AIST), Umezono 1-1-1, Tsukuba 305-8568, JapanDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Hongo, Tokyo 113-8656, JapanRIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, JapanNara Institute of Science and Technology (NAIST), 89165, Takayama, Ikoma, Nara 630-0192, JapanDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Hongo, Tokyo 113-8656, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, JapanDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Hongo, Tokyo 113-8656, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, JapanInstitute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, Japan The spin states of Co^3+ ions in perovskite-type LaCoO_3, governed by complex interplay between the electron-lattice interactions and the strong electron correlations, still remain controversial due to the lack of experimental techniques which can detect directly. In this letter, we revealed the tensile-strain dependence of spin states, i. e. the ratio of the high- and low-spin states, in epitaxial thin films and a bulk crystal of LaCoO_3 via resonant inelastic soft x-ray scattering. The tensile-strain as small as 1.0% was found to realize different spin states from that in the bulk. 78.70.En, 78.70.Dm, 36.40.Mr, 73.43.FjTensile-Strain Dependent Spin States in Epitaxial LaCoO_3 Thin FilmsH. Wadati December 30, 2023 ======================================================================Charge, spin, and orbital degrees of freedom activated by strong electron correlation realize emergent physical phenomena such as superconductivity, metal-insulator transition, and charge ordering <cit.>. Perovskite-type cobalt oxide LaCoO_3 is one of the most intriguing materials due to its versatile electron degrees of freedom. Since the spin state of LaCoO_3 is sensitive to the crystal field, it shows a spin crossover from the low-spin (LS) to high-spin (HS) state with increasing temperature <cit.>. In addition, the spin states can be modified by external stimuli such as a magnetic field and pressure. For example, the possibility of an excitonic insulating state under a high magnetic field was theoretically proposed in the bulk crystal and has also attracted considerable interest these days <cit.>. Epitaxial strain can also influence the spin states as observed for ferromagnetism in LaCoO_3 thin films at lower temperatures (≲ 85 K) <cit.>.The crystal structure of bulk LaCoO_3 is a rhombohedral perovskite-type with corner-sharing CoO_6 octahedra, in which the valence of Co is 3+ with 3d^6 configuration. The spin state of LaCoO_3 is considered to take LS state with e_g^0t_2g^6, intermediate-spin (IS) state with e_g^1t_2g^5, or HS state with e_g^2t_2g^4. The population of HS state gradually increases as the temperature increases from around 100 K and the HS state becomes dominant above 500 K <cit.>. Although the spin state and its temperature dependence of LaCoO_3 bulk have already been studied by using various experimental and theoretical approaches <cit.>, the spin states have not been completely determined yet. Especially, the spin states in the intermediate temperature region remain controversial, i. e. the mixed HS/LS state or the IS state. On the other hand, in ferromagnetic LaCoO_3 thin films, it is indicated that the ferromagnetism originates from the spin-state, orbital, and spin orderings <cit.>. From the resonant x-ray diffraction, modulation vector q = (1/4, -1/4, 1/4)_pc (the suffix pc stands for the pseudo-cubic setting) and q = (1/6, 1/6, 1/6)_pc were observed in the thin films grown on LSAT(110) and LSAT(111) substrates, respectively [LSAT: (LaAlO_3)_0.3(SrAl_0.5Ta_0.5O_3)_0.7]. Then, the spin states are considered to depend on the tensile-strain. In these thin films, some possible models of spin state orderings were reported in Ref. <cit.>. The proposed relationships between the strain and the estimated ratio of spin states per unit cell are shown in TABLE I.In previous studies, the spin states were conjectured from the orderings of 3d electrons on the basis of resonant x-ray diffraction <cit.>. However, direct observations of the electronic structures are needed to clarify the spin states. Since it is difficult to determine the HS/LS ratio by conventional methods such as x-ray absorption spectroscopy (XAS), we performed resonant inelastic soft x-ray scattering (RIXS) by using Co 2p → 3d → 2p process (L-edge). The RIXS is one of the most powerful techniques to clarify the spin states by observing the d-d excitations <cit.>.A higher-energy-resolution (a few hundreds of meV) is required to distinguish the d-d excitations from the elastic scattering <cit.>.The LaCoO_3 epitaxial thin films with the thickness of 30 nm were fabricated on LSAT(110) and LSAT(111) substrates by pulsed laser deposition technique. Details for the syntheses and the characterizations were described on Ref. <cit.>. The lattice constant of LSAT substrate is 3.868 Å, while that of LaCoO_3 bulk is 3.804 Å<cit.>, meaning that the LaCoO_3 epitaxial thin films grown on LSAT substrates are distorted by tensile-strain. The tensile-strains from LSAT(110) and LSAT(111) substrates are 1.0% and 0.5%, respectively (strains are defined as the ratio of the cubic root of the unit cell volume) <cit.>. The experiments of XAS and RIXS were performed at BL07LSU HORNET, SPring-8 <cit.>. Before RIXS measurements, we obtained the XAS spectra at 300 K by total electron yield (TEY) in order to determine the excitation energy of RIXS. The RIXS measurements were performed at 40 K and 300 K by using soft x-ray from 770 eV to 810 eV (Co L_3,2 edge). In the range of the x-ray energy, energy resolution Δ E ∼ 300 meV, which was determined via full width of half maximum (FWHM) of the elastic peak. The charge coupled device (CCD) detector was set at 90^∘ relative to the incident x-ray with horizontal polarization to suppress the elastic scattering (see inset in Fig. 1(b)).First, we measured Co L_3,2 XAS by TEY mode of LaCoO_3 thin films at 300 K. The spectra are shown in Fig. 1(a). The peaks around 779 eV and 794 eV correspond to the Co L_3 and L_2 edges, respectively. Although the magnitude of the tensile-strain and modulation vector q are different between LSAT(110) and LSAT(111), the spectral shapes are quite similar. To investigate the detailed electronic structures from d-d excitations, we performed RIXS measurements. As the excitation energy of RIXS, we selected the energy of A: L_3-2.9 eV, B: L_3-1.5eV, C: L_3-1.0 eV, D: L_3, E: L_3+1.7 eV, and F: L_2. The Co L_3,2 RIXS spectra of the LaCoO_3 thin films are shown in Fig. 1(b). These spectra are normalized by the intensity of the highest peak. In this figure, the thick solid line shows the center of elastic scattering peaks and the arrows indicate the fluorescence ones. Other peaks from 0 eV to 4 eV correspond to the d-d excitations and the peak around 5.0 eV may be a charge-transfer (CT) excitation. In the spectra of A and B, the peaks of d-d excitations are clearly observed. We find that the d-d excitations are obviously different between the thin films on LSAT(110) with 1.0% tensile-strain and LSAT(111) with 0.5% tensile-strain; this indicates that the spin states of LaCoO_3 change drastically according to the magnitude of the tensile-strain. On the other hand, in the spectra excited by higher energy (C, D, E, and F), the peaks of d-d excitations are not clear because of the larger fluorescence. Then, we selected the excitation energy A: L_3-2.9 eV and compare with the RIXS spectra of bulk single crystal in order to discuss the relationship between the spin states and the epitaxial strain. In the analysis, we carried out impurity Anderson model calculations in O_h and D_2h local symmetry. Figure 2(a) show the schematic figures of the crystal structures in bulk and thin film on LSAT(110). The symmetry of bulk is nearly O_h, while that of the thin film on LSAT(110) and LSAT(111) are D_2h and D_3d, respectively. In the film on LSAT(111), the strain is so small that we consider its symmetry as O_h. Therefore, we performed the calculations in O_h symmetry for the LaCoO_3 bulk and the thin film on LSAT(111), while in D_2h symmetry for the thin film on LSAT(110). As shown in Fig. 2(b), we consider the [CoO_6]^9- cluster model with the scaling factors δ and β in order to describe the crystal distortion, i.e. shrinkage along [010] axis and elongation along [001] axis. In the present model, we consider the Co 3d orbitals of a central Co atom (core hole site) and appropriate linear combinations of Co 3d orbitals on neighboring sites.The anisotropic effect due to the lattice mismatch of substrate is also taken into account through the anisotropic hybridization and the crystal field for Co 3d states, where the local symmetry around the Co ion is approximately treated as D_2h. As shown in Fig. 2(c), the energy diagram in D_2h symmetry is different from that in O_h symmetry, where 10Dq is defined as the crystal field splitting between e_g and t_2g orbitals. In D_2h symmetry, the two energy levels (e_g and t_2g) are split into five energy levels by reducing the symmetry, indicating that the new spin states can be realized. In our calculations, the charge-transfer energy from the O 2p ligand band to the upper Hubbard band and that from the non-local band to the upper Hubbard band are defined as Δ and Δ^*, respectively. The Coulomb interaction between Co 3d states (U_dd) and that between Co 3d and 2p core-hole states (-U_dc) were set as U_dd=5.6 eV and U_dc=7.0 eV, respectively. The meaning of these parameters were defined as in Ref. <cit.>. The Slater integrals and the spin-orbit coupling constant are calculated by Cowan's Hartree-Fock program <cit.> and then the Slater integrals are rescaled by 85%, as usual. (pdσ) and (pdπ) are the transfer integrals between Co 3d and O 2p orbitals and we used the parametrization given in Harrison's rule as (pdσ) ∝ d^-3.5 <cit.>. (d denotes the atomic distance.) In our calculations, the parameters were set as TABLE II. The spin crossover (HS ↔ LS) occur between 10Dq = 1.38 and 1.40 eV in the O_h symmetry. The value of δ and β in the case of the D_2h symmetry were obtained from lattice parameters <cit.>. Note that the calculations were performed in the wide range of parameters (10Dq: 0 - 4.00 eV, β: 0 - 0.040, δ: 0 - 0.040, and (pdσ): -3.00 - -1.00 eV) and there is no parameter region where the IS state is the most stable.The experimental RIXS spectra excited with A: L_3-2.9 eV (776.5 eV) measured at 40 K and 300 K are shown in comparison to the theoretical ones with several electronic states in Fig. 3. There are two kinds of strong peaks in the energy range between 0 and 2 eV in all three samples.The peaks observed around 0.3 eV can be assigned to the excitation originated form the HS states. On the other hand, the peaks around 1.3 eV observed in LaCoO_3/LSAT(111) and bulk correspond to the LS ground states. As the temperature increasing, the behavior that the peak intensities at 0.3 eV increase and those at 1.3 eV decrease is consistent with the fact that population of the HS state increases with increasing temperature. However, peaks which appear at 1.0 eV in LaCoO_3/LSAT(110) can be explained by neither the LS nor HS with O_h symmetry. As seen in the theoretical spectra, the peak for HS state is shifted to 1.0 eV by lowering the symmetry from O_h to D_2h.It indicates that the spin state of LaCoO_3/LSAT(110) consists of the HS states with different local symmetries, i.e. the mixture of O_h and D_2h symmetries. To compare the present results with the previous study by resonant x-ray diffraction, we estimate the ratio of spin states by the theoretical spectra of various spin states. Since the theoretical spectra are normalized by the intensity of the theoretical XAS at A: L_3-2.9 eV, we can reproduce the mixed spin states by linear combinations of the spectra and estimate the ratio of the spin states by the peak intensity in each spin state. As shown in Fig. 4(a), HS(O_h) : HS(D_2h) = 1 : 1 is in good agreement with the experimental spectrum of LaCoO_3/LSAT(110), indicating that the CoO_6 octahedra in O_h and D_2h local symmetry coexist as the ratio of 1 : 1. Since the spatial modulation of spin state is fourfold periodicity, one plausible model is that the spin states are aligned in order of O_h - D_2h(d_yz) - O_h - D_2h(d_zx), where the HS states with D_2h have two different orbital states. In this study, we did not find the theoretical spectra of IS states which reproduce experimental results and thus could not confirm the spin state model of IS : HS = 3 : 1.In conclusion, the spin states of LaCoO_3 change from LS to HS according to the magnitude of the epitaxial strain, which could not be detected by conventional XAS but could be clarified by RIXS measurements. The LS ground state in bulk fully changes to HS states by tensile-strain as small as 1.0%. Especially in the thin film on LSAT(110), we observed the tensile-strain induced unique spin state, i. e. HS(O_h) : HS(D_2h) = 1 : 1. Since the spin state is not stabilized by temperature or hydrostatic pressure in bulk, applying the tensile-strain is an important factor for realizing novel spin states.The authors thank T. Arima and K. Yamamoto for productive discussions. The present work was performed under the approvals of SPring-8 Japan Synchrotron Radiation Research Institute (Proposal No. 2016A7501, No. 2016B7515).99 Imada M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 79, 1039 (1998).Haverkort M. W. Haverkort, Z. Hu, J. C. Cezar, T. Burnus, H. Hartmann, M. Reuther, C. Zobel, T. Lorenz, A. Tanaka, N. B. Brookes, H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Phys. Rev. Lett. 97, 176405 (2006).Kunes1 J. Kuneš, J. Phys.: Condens. Matter 27, 333201 (2015).Sotnikov A. Sotnikov and J. Kuneš, Sci. Rep. 6, 30510 (2016).Nasu J. Nasu, T. Watanabe, M. Naka, and S. Ishihara, Phys. Rev. B 93, 205136 (2016).Fuchs1 D. Fuchs, C. Pinta, T. Schwarz, P. Schweiss, P. Nagel, S. Schuppler, R. Schneider, M. Merz, G. Roth, and H. v. Löhneysen, Phys. Rev. B 75, 144402 (2007).Fuchs2 D. Fuchs, E. Arac, C. Pinta, S. Schuppler, R. Schneider, and H. v. Löhneysen, Phys. Rev. B 77, 014434 (2008).Freeland J. W. Freeland, J. X. Ma, and J. Shi, Appl. Phys. Lett. 93, 212501 (2008).Mehta V. V. Mehta, M. Liberati, F. J. Wong, R. V. Chopdekar, E. Arenholz, and Y. Suzuki, J. Appl. Phys. 105, 07E503 (2009).Rata A. D. Rata, A. Herklotz, L. Schultz, and K. Dörr, Eur. Phys. J. B 76, 215 (2010).Fujioka1 J. Fujioka, Y. Yamasaki, H. Nakao, R. Kumai, Y. Murakami, M. Nakamura, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 111, 027206 (2013).Fujioka2 J. Fujioka, Y. Yamasaki, A. Doi, H. Nakao, R. Kumai, Y. Murakami, M. Nakamura, M. Kawasaki, T. Arima, and Y. Tokura, Phys. Rev. B 92, 195115 (2015).Yamasaki Y. Yamasaki, J. Fujioka, H. Nakao, J. Okamoto, T. Sudayama, Y. Murakami, M. Nakamura, M. Kawasaki, T. Arima, and Y. Tokura, J. Phys. Soc. Jpn. 85, 023704 (2016).Abbate M. Abbate, J. C. Fuggle, A. Fujimori, L. H. Tjeng, C. T. Chen, R. Potze, G. A. Sawatzky, H. Eisaki, and S. Uchida, Phys. Rev. B 47, 16124 (1993).Korotin M. A. Korotin, S. Yu. Ezhov, I. V. Solovyev, and V. I. Anisimov, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. B 54, 5309 (1996).Magnuson M. Magnuson, S. M. Butorin, C. Såthe, J. Nordgren, and P. Ravindran, Europhys. Lett. 68, 289 (2004).Podlesnyak A. Podlesnyak, S. Streule, J. Mesot, M. Medarde, E. Pomjakushina, K. Conder, A. Tanaka, M. W. Haverkort, and D. I. Khomskii, Phys. Rev. Lett. 97, 247208 (2006).Knizek K. Knížek, Z. Jirák, J. Hejtmánek, P. Novák, and W. Ku, Phys. Rev. B 79, 014430 (2009).Kunes2 J. Kuneš, and V. Křápek Phys. Rev. Lett. 106, 256401 (2011).Hariki A. Hariki, A. Yamanaka, and T. Uozumi, J. Phys. Soc. Jpn. 84, 073706 (2015).Kotani A. Kotani and S. Shin, Rev. Mod. Phys. 73, 203 (2001).Chiuzbaian S. G. Chiuzbǎian, G. Ghiringhelli, C. Dallera, M. Grioni, P. Amann, X. Wang, L. Braicovich, and L. Patthey, Phys. Rev. Lett. 95, 197402 (2005).Ament L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, Rev. Mod. Phys. 83, 705 (2011).Kobayashi Y. Kobayashi, T. Mitsunaga, G. Fujinawa, T. Arii, M. Suetake, K. Asai, and J. Harada, J. Phys. Soc. Jpn. 69, 3468 (2000).Harada Y. Harada, M. Kobayashi, H. Niwa, Y. Senba, H. Ohashi, T. Tokushima, Y. Horikawa, S. Shin, and M. Oshima, Rev. Sci. Instrum. 83, 013116 (2012).Taguchi M. Taguchi, A. Chainani, K. Horiba, Y. Takata, M. Yabashi, K. Tamasaku, Y. Nishino, D. Miwa, T. Ishikawa, T. Takeuchi, K. Yamamoto, M. Matsunami, S. Shin, T. Yokoya, E. Ikenaga, K. Kobayashi, T. Mochiku, K. Hirata, J. Hori, K. Ishii, F. Nakamura, and T. Suzuki, Phys. Rev. Lett. 95, 177002 (2005).Cowan R. D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, 1981). Harrison W. A. Harrison,Electronic Structure and the Properties of Solids (W. H. Freeman and Co., San Francisco, 1980) Appendix. | http://arxiv.org/abs/1708.07529v2 | {
"authors": [
"Y. Yokoyama",
"Y. Yamasaki",
"M. Taguchi",
"Y. Hirata",
"K. Takubo",
"J. Miyawaki",
"Y. Harada",
"D. Asakura",
"J. Fujioka",
"M. Nakamura",
"H. Daimon",
"M. Kawasaki",
"Y. Tokura",
"H. Wadati"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170824191313",
"title": "Tensile-Strain Dependent Spin States in Epitaxial LaCoO$_3$ Thin Films"
} |
BMS in higher space-time dimensions and Non-relativistic BMS. < g r a p h i c s >Author: Delmastro, D.G. Supervisors:Batlle, C. and Gomis, J.June 2017 ============================================================================================= .9 BMS in higher space-time dimensions and Non-relativistic BMS. by Delmastro, D.G. under the supervision of Batlle, C. and Gomis, J. Submitted to the department of physics in partialfulfilment of the requirements for the degree of Master of science in Astrophysics, Particle Physics and Cosmology at the Universitat de Barcelona. [1]email: [email protected] [2]email: [email protected] [3]email: [email protected] CHAPTER: ACKNOWLEDGMENTS.emptyNeedless to say, nothing in this document – or in my career as a whole – would have been possible without the continuous support of my family. I will forever be grateful to them.I am also thankful to friends and colleagues, with whom I have shared many priceless conversations and unforgettable experiences, and who not only have helped shape my scientific view of the world but also made the process enjoyable.In addition, I am also in debt to the many professors I have had the pleasure to interact with and from whom I have learnt invaluable things, both from the Universidad Autónoma de Madrid, where I studied my Bachelor's Degree, and from the Universitat de Barcelona, where I studied my Master's Degree. I would also like to thank the very attentive staff from both universities.The professors that supervised my final projects in both cycles deserve a special mention. I owe a lot to Antonio González-Arroyo España, from Madrid, and to Carles Batlle Arnau and to Joaquím Gomis Torné, from Barcelona. Last but not least, I am also in debt to the coordinator of the Master's Degree, Bartomeu Fiol Núñez. I have refrained from making other names explicit for fear of forgetting someone. I am sure the relevant people will know I am talking about them. CHAPTER: OUTLINE. tocchapterOutline.The Bondi-Metzner-Sachs (BMS) Group is the group of asymptotic isometries of asymptotically flat space-times. This group is given by the semi-direct product of the Lorentz Group with the infinite-dimensional group of the so-called super-translations, which constitutes an abelian sub-group.The isometries of a strictly flat manifold are, as is well-known, Poincaré. If one admits a non-trivial geometry, but such that it becomes Minkowski in some of its asymptotic directions, then one may try to find the transformations that leave the asymptotic form of the metric invariant. If the metric is asymptotically flat, the most natural result one would expect is that the asymptotic isometries are Poincaré. < g r a p h i c s >Indeed, this was the programme that BMS undertook, as a part of their study of gravitational waves, in the early sixties <cit.>. In an attempt to rederive the Poincaré Group as the group of isometries of a manifold that is asymptotically flat, these authors discovered, much to their surprise, that the actual group of asymptotic isometries does indeed contain Poincaré, but it contains much more: it is in fact infinite-dimensional. The new set of symmetries, today called super-translations, can be regarded as a generalisation of the standard translations. We will review the details of the BMS Group in Chapter <ref>.The new physics discovered by these authors soon transcended by far their original intents, to the point that we nowadays consider it to be a just a part of a much larger and richer structure; for a recent review, see the lecture notes <cit.>. For example, there is compelling evidence that the BMS Group is an essential ingredient of quantum gravitational scattering <cit.>, black holes and the information paradox <cit.>, flat holography <cit.>, soft theorems <cit.>, the memory effect <cit.>, etc.The last two of these topics, the soft theorems and the memory effect, have a major role in the study of the BMS Group. The soft theorems describe the universal behaviour of scattering amplitudes that include massless particles with vanishing momentum. These soft particles can be understood as the Goldstone bosons of spontaneously broken large gauge symmetries; in particular, a broken BMS symmetry gives rise to soft gravitons. On the other hand, the memory effect is the permanent displacement of two detectors as the consequence of passing-by gravitational waves; in particular, the relative positions and clock times of the detectors before and after the radiation transit differ by a super-translation. As stressed by Strominger and collaborators <cit.>, the memory effect, the soft theorems, and the concept of asymptotic symmetries are the three corners of an infra-red triangle – a triality that is present in any theory with non-trivial infra-red dynamics.1.0 [thick,blue,<->,>=stealth] (1,0) – (3,0); [thick,blue,<->,>=stealth] (3.5, 0.866) – (2.5, 2.598); [thick,blue,<->,>=stealth] (0.5, 0.866) –(1.5, 2.598);at (0,.2) [align=center] Soft Theorem; at (4,.2) [align=center] Memory Effect; at (2,3.3) [align=center] Asymptotic Symmetry; This triangular equivalence interweaves these three seemingly unrelated topics and makes them interdependent. This is relevant to our work for the following reason: on the one hand, it has recently been argued that the memory effect vanishes for d>3, where d is the number of space dimensions <cit.>. On the other hand, the soft theorems exist for any number of space-time dimensions <cit.>. This puts the third corner, the asymptotic symmetries, in an uncertain position: to what extent should we expect a non-trivial BMS Group in higher dimensions? This apparent contradiction has recently been addressed by Strominger et al <cit.>. According to these authors, the classical analysis of asymptotic symmetries (e.g. <cit.>, etc.) imposed unnecessarily strong conditions for asymptotic flatness, thus eliminating the degrees of freedom corresponding to super-translations. If one slightly relaxes these conditions, the whole family of super-translations emerges, and the paradox concerning the triangle is resolved.This provides the main motivation for the first part of our thesis (Chapter <ref>): we will argue that one can construct the algebra of the BMS Group in any number of space-time dimensions, by a different method than the standard approach: we will not consider the gravitational problem, but we will consider instead the canonical construction of the 𝔟𝔪𝔰 algebra <cit.>. This construction is based on the study of the symmetries of the equations of motion of a free field in flat space-time. As is well-known, the set of point symmetries of such equations contains Poincaré transformations; but, as we will show, it is much larger than that, and it contains, in particular, super-translations. The algebra of these symmetries coincides with the one of the gravitational problem, which is one more piece of evidence that suggests that the relevance of the BMS Group goes beyond the gravitational problem.The existence of a canonical BMS Group for any number of space-time dimensions means that either we should expect a non-trivial gravitational BMS for any d, or that the agreement between the canonical and gravitational BMS in d=2,3 is just a fortuitous coincidence. Of course, the first of these two options is not only more attractive, but also much more natural. The second part of this thesis (Chapter <ref>) concerns the problem of a non-relativistic BMS Group. One could define this group as the set of isometries of asymptotically flat Newtonian space-times, in the sense of Cartan <cit.>. Even though some interesting results in this area have been obtained (e.g., <cit.>), the overall picture is much less clear than in the relativistic case. In order to construct the non-relativistic analogue of 𝔟𝔪𝔰 there are several approaches one could take. The simplest one is to consider a suitable İnönü-Wigner contraction of the corresponding relativistic 𝔟𝔪𝔰 algebra, in the same sense that the Galilei algebra can be obtained by contracting the Poincaré one. An alternative to obtain a possible non-relativistic 𝔟𝔪𝔰 algebra is to mimic the aforementioned canonical construction, but using a scalar field with Galilean space-time symmetries instead of a relativistic field. Finally, a third alternative is, naturally, to follow the same steps that led to the original 𝔟𝔪𝔰 algebra, but in a non-relativistic setting, i.e., one could try to study the set of isometries of asymptotically flat Newtonian space-times. In this work, we will only consider the first two possibilities.It bears mentioning that some partial results concerning the non-relativistic 𝔟𝔪𝔰_4 algebra have already been published, cf. <cit.>. In this thesis we extend the results therein and contextualise them by comparing them to the relativistic algebra.CHAPTER: THE BMS GROUP. In this chapter we will provide a brief but hopefully self-contained introduction to the BMS Group. A more detailed discussion can be found in <cit.>.The problem that BMS addressed is the following: given a certain metric g_μν(x) that describes the geometry of an asymptotically flat (lorentzian) manifold,g_μν(x)η_μν,we want to find the vector fields ξ such that they leave the asymptotic form of the metric invariant,ℒ_ξ g_μν(x)0,where ℒ denotes a Lie derivative (see e.g. <cit.>). Of course, to make this problem well-defined we must properly specify what we mean by the asymptotic limit x∼∞. To make the discussion above more precise, let us consider a flat manifold with cartesian coordinates x^μ=(x^0,…,x^3), and let us define the radial coordinate r and the retarded time u_+ asr √(x_1^2+x_2^2+x_3^2)u_+x^0-rThe coordinate u_+ is the parameter that appears in the solution to Maxwell's equations in the Lorenz gauge <cit.>, ∂^2 A^μ= j^μ, viz.A^μ(x) =∫Δ_ret(x-y) j^μ(y) dy=∫1/2πδ((x_0-y_0)^2-( x- y)^2)Θ(y^0-x^0) j^μ(y) dy=∫j^μ(x^0-| x- y|, y)/4π| x- y| d yand as such, it has the following interpretation: if there is a perturbation in the source at a time u_+, then a detector placed at a distance r will receive the signal at a time x^0. In other words, the world-line of a massless particle is of the form u_+=const. If we emit massless particles radially away from the origin, at a time u_+, then the sphere they reach at null infinity, r→∞, is called the celestial sphere 𝒞𝒮^2. We let z^A, with A=1,2, parametrise this sphere. Some common choices for these coordinates are: spherical coordinates, z^A=(θ,φ), defined byx^1=rsinθcosφ, x^2=rsinθsinφ, x^3=rcosθand the stereographic (complex) coordinates z^A=(z,z̅), defined byz= x^1+i x^2/r+x^3,z̅= x^1-i x^2/r+x^3 We will ultimately be interested in a possible extension of the upcoming discussion to higher dimensional space-times. Therefore, the parametrisation in terms of angles seems to be more convenient than the stereographic one. We will come back to this point later on; for now, we keep the coordinates z^A unspecified. The coordinates u_+,r,z^A are called retarded Bondi coordinates. The set of all celestial spheres,ℐ^+={(u_+,z^A): u_+∈ℝ, z^A∈𝒞𝒮^2}is called future null infinity, and it is topologically a cone ℝ× S^2.There exists an analogous construction using advanced Bondi coordinates, where one introduces an advanced time u_- x^0+r. In the context of electrodynamics, this coordinate appears in the advanced propagator Δ_adv, and is relevant to, for example, the Wheeler-Feynman absorber theory <cit.>. As in the retarded case, one defines past celestial spheres, which foliate past null infinity ℐ^-. All massless particles originate in ℐ^- and end up in ℐ^+. On the other hand, massive particles originate in past timelike infinity i^-, and end up in future timelike infinity i^+. Finally, spatial infinity is denoted by i^0 (see Fig. <ref>). 1.0In Bondi coordinates, the Minkowski metric readsds^2=-du^2_±∓2dr du_±+r^2γ_ABdz^Adz^Bwhere γ_AB is the metric on the unit sphere. In spherical coordinates, γ=(1,sin^2θ), and in stereographic coordinates, γ_zz̅=γ_z̅z=2/(1+zz̅)^2.In cartesian coordinates, the Killing vectors of this manifold are given byξ=a^μ∂_μ+Ω^μνx_ν∂_μ,with a^μ,Ω^μν∈ℝ a set of constants, with Ω^μν=-Ω^νμ. In Bondi coordinates, these Killing vectors are given by a more complicated and not particularly illuminating expression (see e.g. <cit.>). In the particular case of r→∞, that is, at ℐ^±, translations correspond tou→ u+α(z^A)where, in spherical coordinates,α(θ,φ)=a^0+a^1sinθcosφ+a^2sinθsinφ+a^3cosθ For future reference, one should note that these are just the spherical harmonics Y_0^0,Y_1^m, with m=0,±1. In stereographic coordinates,α(z,z̅)∼linear combinations of 1,z,z^2,z̅,z̅^2 On the other hand, (homogeneous) Lorentz transformations correspond to conformal transformations on the celestial sphere, given byz →ω_11z+ω_12/ω_21z+ω_22withω=ω(Ω)∈SL(2,ℂ)u → (1+δ(z,z̅))ufor a certain function δ (whose form shall not concern us here).The infinitesimal generators M_μν of these transformations are given by <cit.>M_12 =z∂_z-z̅∂_zM_13 =-i/2((z^2+1)∂_z+(z̅^2+1)∂_z̅)M_23 =-1/2((z^2-1)∂_z-(z̅^2-1)∂_z̅)M_01 =+i/2((z^2-1)∂_z+(z̅^2-1)∂_z̅)-iuz+z̅/1+zz̅ ∂_uM_02 =+1/2((z^2+1)∂_z-(z̅^2+1)∂_z̅)-iui(z̅-z)/1+zz̅ ∂_uM_03 =-i(z∂_z+z̅∂_z)-iu1-zz̅/1+zz̅ ∂_u These vector fields satisfy the Lorentz algebra. As a matter of fact, the angular part can be written as linear combination of the generators of the Witt algebra <cit.>, l_n=-z^n+1∂_z,l̅_n=-z̅^n+1∂_z̅, with n=0,±1. On the other hand, in spherical coordinates the Lorentz generators are given byM_12 =-i∂_φM_13 =icosφ ∂_θ-iθsinφ ∂_φM_23 =isinφ ∂_θ+iθcosφ ∂_φM_01 =icosφcosθ ∂_θ-isinφθ ∂_φ-iucosφsinθ ∂_uM_02 =isinφcosθ ∂_θ+icosφθ ∂_φ-iusinφsinθ ∂_uM_03 =isinθ ∂_θ+iucosθ ∂_u Upon Lie bracketing, one verifies that these Killing vectors satisfy the Poincaré algebra,[P^μ, P^ν] =0[P_α, M^μν] =2iδ^[μ_α P^ν][M_αβ,M^μν] =4iδ^[μ_[α M^ν]_β]Let us now move on to the non-flat case. With our previous definitions at hand, we we can now properly define what it means for a manifold to be asymptotically flat at ℐ^±: it means that it admits local coordinates u_±,r,z^A, such that, as r→∞, the metric takes the form (<ref>) up to terms subleading in r. Apart from the metric becoming flat at null infinity, g_μν(ℐ^±)=η_μν, we also assume it to be smooth there. One can prove <cit.> that this is equivalent to assuming that the components of the metric admit a well-defined 1/r expansion around r→∞ (however, see <cit.>).A much more technical and elegant definition of asymptotic flatness can be found in <cit.> and references therein. This definition is based on the existence of a particular Penrose compactification of the manifold and, as such, it is much more geometrical and coordinate-independent than ours. As discussed in <cit.>, the Penrose definition is not easily extended to higher (odd) dimensions, due to a potential non-analiticity of the metric at null infinity. On the other hand, the extension to higher dimensions of the Bondi definition in terms of an explicit 1/r expansion is in principle much more direct. This latter definition will be enough for our purposes.For concreteness, we will now focus on the asymptotic future ℐ^+, so we will drop the `+' label on the retarded time. The corresponding analysis at ℐ^- is analogous.Following Bondi, we take u to be a null coordinate, so that g^uu=0, and we take the angular coordinates z^A to be constant along null rays, so that g^uA=0. Imposing that the two-spheres spanned by z^A have area 4π r^2, we get g_AB=r^4γ_AB. These conditions fix the diffeomorphism invariance of the metric, and are usually known as the Bondi gauge conditions. One should note that some of these conditions are sometimes modified in the literature; for example, Penrose <cit.> replaces the metric on the sphere, γ_AB by one conformally equivalent to it, with an arbitrary conformal factor (see <cit.> for a comparison of the Bondi-Sachs and the Penrose approaches). In any case, we shall stick to the Bondi conditions in what follows. After fixing the gauge, the metric takes the so-called Bondi-Sachs form <cit.>:ds^2=-V/re^2βdu^2-2 e^2βdr du+r^2h_AB(dz^A-1/2 U^Adu)(dz^B-1/2 U^Bdu)where we have defined h_AB r^-2g_AB, so that the Bondi area gauge condition reads h_AB=γ_AB. The indices on the sphere A,B,… are lowered and raised with the metric h_AB and its inverse h^ABh_BC=δ^A_C. We also denote the covariant derivatives with respect to h_AB by the letter D_A.Asymptotic flatness, in Bondi coordinates, meanslim_r→∞β =lim_r→∞U^A=0 lim_r→∞V/r =1 lim_r→∞h_AB =γ_ABIf, as before, we denote the covariant derivative with respect to h_AB by D_A, and we let ƪ_A be the covariant derivative with respect to γ_AB, then the asymptotic condition h_AB→γ_AB implies that D_A→ƪ_A. The precise correspondence isD_Av^B=ƪ_A v^B+1/2rγ^BF(ƪ_A c_FE+ƪ_Ec_FA-ƪ_Fc_AE)v^E+𝒪(r^-2)for an arbitrary vector field v^B. Here, c_AB=c_AB(z) is defined as the subleading part of the metric on the sphere:h_ABγ_AB+c_AB/r+𝒪(r^-2) The rest of the components of the metric must have a well-defined falloff too. There is a certain freedom in the choice of these falloffs, and certain consistency conditions that determine some coefficients in terms of the rest. The (physically motivated) choice for the subleading behaviour of the metric made by Bondi, van der Burg, Metzner and Sachs wasV =r-2M+𝒪(r^-1) β =-1/32c^ABc_AB/r^2+𝒪(r^-3)U^A =-ƪ_B c^AB/2r^2+2L^A/r^3+𝒪(r^-4)where M,L are arbitrary functions defined on ℐ^+; M=M(u,z) is called the mass aspect, and L^A=L^A(u,z) is called the angular momentum aspect. We are now in position to tackle the problem of characterising the isometries of asymptotically flat space-times; the set of these diffeomorphisms is called the BMS Group. These are, by definition, the diffeomorphisms that preserve the Bondi gauge conditions as well as the metric falloffs (<ref>). Preservation of the gauge conditionsg_rr=g_rA=0, h_AB=γ_ABrequiresℒ_ξ g_rr=ℒ_ξ g_rA=ℒ_ξ h_ABh^AB=0while preservation of the metric falloffs requiresℒ_ξ g_ur=𝒪(r^-2),ℒ_ξ g_uA=𝒪(1),ℒ_ξ g_AB=𝒪(r),ℒ_ξ g_uu=𝒪(r^-1)The general solution to the first set of equations (<ref>) isξ(r,u,z)=α(u,z)∂_u+f^A(u,z)∂_A+ξ^r∂_r+𝒪(r^-1)where α,f^A are a pair of arbitrary functions, and where the actual form of ξ^r and of the 𝒪(r^-1) term shall not concern us here.With this, the first equation in (<ref>) implies thatα(u,z)=α(z)+1/2∫_0^uƪ_Bf^B(u',z) du'while the second equation implies that f^B(u,z)=f^B(z). The third equation is equivalent to requiring f^B to bea conformal Killing vector of the sphere,ƪ_(Af_B)-1/2γ_ABƪ_Cf^C=0while the fourth equation is automatically satisfied. The infinitesimal generators of the BMS Group thus read <cit.>ξ(ℐ^+)=[α(z)+1/2u ƪ_Bf^B(z)]∂_u+f^A(z)∂_Awhere α(z) is an arbitrary function, and f^A is a conformal Killing vector of the sphere.The Killing vectors with α(z)≡ 0 correspond to local conformal transformations on the celestial spheres. In complex coordinates, the conformal Killing equation becomes the Cauchy-Riemann equations, which means that f^z (resp. f^z̅) is given by an arbitrary holomorphic (resp. anti-holomorphic) function on the Riemann sphere S^2=ℂ∪{∞}. One can always write such a function asf^z∂_z=∑_n∈ℤ f_n l_n,with l_n=-z^n+1∂_zand a similar relation for f^z̅. If f_n≠ 0 for n<-1, then the Killing vector is singular at z→ 0, while if f_n≠0 for n>1, then it is singular at z^-1→ 0. If we restrict ourselves to non-singular functions (so that the global transformation is well-defined) we recover the Lorentz generators (<ref>), that is, the values n=0,±1 (for more details, see <cit.>). It has recently been argued <cit.> that there is no reason to require global well-definedness. If we abandon such condition, the general set of generators is l_n,l̅_n, for n∈ℕ. The new transformations are called super-rotations. We will not be interested in them in this thesis.for d>3, and due to Liouville's theorem <cit.>, the solutions to the conformal Killing equation are just the Lorentz generators, with no super-rotations. A simplified argument for the augmented group of conformal diffeomorphisms on the d-1=1,2 spheres is the following <cit.>: let n=d-1 be the dimension of the celestial spheres S^n. Then the conformal Killing equation has n unknowns f^A, and 1/2(n-1)(n+2)independent equations. For n=1 we get no constraints, so the general solution depends on one arbitrary function. For n=2, the two equations for the two unknowns are the Cauchy-Riemann equations, so the general solution depends on two arbitrary functions. For n> 2, the system is over-determined, and the general solution has a finite number of real parameters: 𝔠𝔬𝔫𝔣(S^n)<∞. Indeed, 𝔠𝔬𝔫𝔣(S^n) coincides with the orientation-preserving part of Spin(1,d), which is the double cover of the Lorentz Group Möb(n)≅SO(1,d).On the other hand, the vector fields with f^A(z)≡ 0 are called super-translations, and they include standard translations for a specific choice of the function α. Indeed, if we letα(θ,ϕ)=a^0+a^1sinθcosφ+a^2sinθsinφ+a^3cosθwe recover equation (<ref>), i.e., we find the expected isometries of Minkowski. The surprising fact is that α(z) can be an arbitrary function, which means that the group of isometries is now infinite-dimensional.one could argue that Minkowski is itself an asymptotically flat space-time, and therefore its group of asymptotic symmetries includes super-translations as well. In this sense, there would be no enlargement of the group of symmetries when one goes from flat space-time to asymptotically flat space-times: in both cases the group of symmetries is BMS. This is not really the case: the formal definition of an asymptotic symmetry is one such that ℒ_ξ g_μν≈ 0 “as good as possible”. In flat space-time, the best approximation to the Killing equation is the Killing equation itself, ℒ_ξη_μν≡ 0, whose solutions are exactly Poincaré. In non-flat space-times, in general the Killing equation doesn't hold anywhere except possibly at ℐ^±. There, the solutions include an arbitrary function α, and there is no general way (or reason) to single out the particular case (<ref>). If we happen to find translations, they must come hand in hand with super-translations.Following Sachs <cit.> (in spherical coordinates) and Barnich andTroessaert <cit.> (in stereographic coordinates), we may expand the function α(z) in spherical harmonics Y(θ,φ), or as a Laurent seriesα(z) =∑_ℓ∈ℕ∑_m=-ℓ^+ℓα^ℓ_m Y_ℓ^m(θ,φ)=∑_n,n̅∈ℤα_nn̅z^nz̅^n̅which leads to the (countable) infinite-dimensional family of super-translations, generated byP_ℓ^m(ℐ^+)Y_ℓ^m∂_uP_n,n̅(ℐ^+)z^nz̅^n̅∂_uDenoting by M_μν the generators of (homogeneous) Lorentz transformations, one may show that the generators of the BMS Group satisfy the algebra[M_23,P_ℓ^m] =i/2√((ℓ+m) (ℓ-m+1)) P_ℓ^m-1-i/2√((ℓ-m) (ℓ+m+1)) P_ℓ^m+1[M_31,P_ℓ^m] =1/2√((ℓ-m) (ℓ+m+1)) P^m+1_ℓ+1/2√((ℓ+m) (ℓ-m+1)) P^m-1_ℓ[M_12,P_ℓ^m] =mP^m_ℓ [M_01,P_ℓ^m] =ℓ +2/2[ √((ℓ-m) (ℓ-m-1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m+1+√( (ℓ +m)(ℓ+m -1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-1]+ℓ -1/2[√((ℓ +m+1) (ℓ +m+2)/(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^m+1+√((ℓ-m+1)(ℓ-m+2) /(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^m-1][M_02,P_ℓ^m] =i(ℓ +2)/2[√((ℓ-m) (ℓ-m-1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m+1-√( (ℓ +m)(ℓ+m -1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-1]+i(ℓ -1)/2[√((ℓ +m+1) (ℓ +m+2)/(2 ℓ +1) (2 ℓ +3))P_ℓ+1^m+1-√((ℓ-m+1)(ℓ-m+2) /(2 ℓ +1) (2 ℓ +3))P_ℓ+1^m-1][M_03,P_ℓ^m] =i (ℓ +2) √((ℓ -m) (ℓ +m)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-i (ℓ -1)√((ℓ -m+1) (ℓ +m+1)/(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^mor, in stereographic coordinates,[l_n,l_n'] =(n-n')l_n+n'[l̅_n̅,l̅_n̅'] =(n̅-n̅')l̅_n̅+n̅' [l_p,P_n,n̅] =(12(p+1)-n)P_n+p,n̅[l̅_p,P_n,n̅] =(12(p+1)-n̅)P_n,n̅+pThis is the 𝔟𝔪𝔰_4 algebra. In stereographic coordinates, the result is much more compact and transparent, but it has the disadvantage that its generalisation to higher dimensions is not straightforward. We shall henceforth focus on the parametrisation in terms of spherical coordinates. One of the most prominent features of the 𝔟𝔪𝔰_4 algebra is that it contains, as one would expect, a Poincaré sub-algebra. Indeed, if we restrict ourselves to the super-translations with ℓ=0,1, then the algebra above is easily seen to contain a closed sub-algebra, because the coefficients of the terms with ℓ→ℓ+1 are proportional to (ℓ-1), which vanishes at ℓ=1. Under the identificationP^1 =P^1_1+P^1̅_1/i√(2)P^2 =P_1^1-P_1^1̅/√(2)P^3 =P^0_1 inverse⟺ P_1^1̅ =-P^2+iP^1/√(2)P_1^1 =+P^2+iP^1/√(2)P^0_1 =P^3one may check that the ℓ=0,1 sub-set of the 𝔟𝔪𝔰_4 algebra coincides with the Poincaré algebra. This in turns means that there is a well-defined notion of energy and momentum for asymptotically flat manifolds; this is the Bondi momentum p_B <cit.>. the Killing equation that we used to determine the super-translations is homogeneous, which means that the normalisation of P_ℓ^m is in principle arbitrary. Here we have followed the standard convention P_ℓ^m=Y_ℓ^m∂_u, which we shall call the gravitational normalisation. Had we included some ℓ-dependent coefficient in front of the spherical harmonics, the actual value of the structure constants of the algebra (<ref>) would have been different. In the next chapter we will rederive the 𝔟𝔪𝔰 algebra in a different context, and there we will find that the most natural normalisation for the super-translations is different; we will call it the canonical normalisation. These two normalisations will be related through an ℓ-dependent rescaling. To extend the discussion above to higher-dimensional space-times, we would have to make a careful analysis of the falloff conditions, and repeat the calculation of the Killing vector fields. From amore pragmatic and perhaps naïve point of view, we could simply anticipate that the solution takes the same form as in the four dimensional case,ξ(ℐ^+)?=[α(z)+u/d-1ƪ_Bf^B(z)]∂_u+f^A(z)∂_Awhere α(z) is an arbitrary function, and f is a conformal Killing vector on S^d-1,ƪ_(Af_B)-1/d-1γ_AB ƪ_Cf^C=0 It has recently been argued <cit.> that in d>3 the most natural falloff conditions for the metric are actually too strong to admit a non-trivial BMS Group. In such a case, the only admissible solutions to the BMS problem would be Poincaré, that is, the Killing vectors ξ would correspond to Lorentz transformations and standard translations. The case is not closed yet though: <cit.> counter-argues that these conditions are unnecessarily strong, and that the BMS Group exists and is non-trivial in any number of space-time dimensions.In any case – be it physically realistic or not – we can dismiss these arguments and impose whatever falloff conditions we may need in order for ξ(ℐ^+)=α(z) to be a valid asymptotic isometry.In this case, the calculation of the 𝔟𝔪𝔰_d+1 algebra becomes straightforward: we just need to expand the super-translations in (higher dimensional) spherical harmonics, Y_ℓ, where ℓ=(ℓ_1,ℓ_2,…,ℓ_d-1) is a multi-index:P_ℓ(ℐ^+) Y_ℓ ∂_uand calculate the different Lie brackets among the P_ℓ,M_μν. We will come back to this problem, from a different perspective, in Chapter <ref>. There we will see that the 𝔟𝔪𝔰 algebra is a natural construction in any number of space-time dimensions, which suggests that, irrespective of its applicability to the gravitational problem, the BMS Group exists and is non-trivial for any d.To close this chapter, it is worthwhile mentioning that the group that we have been studying is BMS^+, corresponding to the symmetries that act on future null infinity ℐ^+. The analogous group acting on past null infinity ℐ^- is denoted by BMS^-. Both groups admit extensions in the sense of super-rotations (and even an extension to include conformal transformations <cit.>). Moreover, it has recently been conjectured <cit.> that the so-called diagonal subgroup of BMS^+×BMS^- is the group of symmetries of (both classical and quantum) gravitational scattering, giving rise to an infinite number of conservation laws. This subgroup is obtained by a certain identification of future-acting to past-acting super-translations, α^+(z)=α^-(z). We refer the reader to <cit.> for more details.CHAPTER: CANONICAL REALISATION OF BMS. In this chapter we willstudy a subset of the symmetries of the equations of motion of a relativistic free field ϕ of arbitrary spin in flat space-time. This object can be regarded as a classical field, or as a quantum operator; for definiteness, here we will only consider the first case, though our main conclusions will hold in the second case as well.Let the equations of motion of ϕ:ℝ^d+1→ℂ^sbe𝒟ϕ=0for 𝒟 a certain Poincaré invariant linear differential operator. By Poincaré invariance we mean that 𝒟 commutes with translations and Lorentz transformations:[𝒟,-i∂_μ]=[𝒟,ix_[ν∂_μ]+spin]=0where “spin” corresponds to a finite-dimensional matrix that generates the representation of the Lorentz Group under which ϕ transforms. In simple terms, Poincaré invariance means that 𝒟 does not depend on x, and that its spin indices are contracted in a Lorentz-invariant way. Typical examples include the Klein-Gordon equation, the Dirac equation or the Maxwell equations.What we want to do is to find the on-shell symmetries of 𝒟, that is, the differential operators Q such that[𝒟,Q]∝𝒟 It is clear that if -i∂_μ is a symmetry of 𝒟, then so is f(-i∂_μ) for any function f. Standard translations are recovered from f by taking it to be the identity. We will see that, for a certain family of functions f, the associated differential operators f(-i∂_μ) will correspond to super-translations, and they will therefore furnish a representation of the 𝔟𝔪𝔰 algebra. The point of this construction is that it is qualitatively independent of the number of space-time dimensions, so it provides a natural extension of the 𝔟𝔪𝔰_4 algebra to any d.instead of considering ϕ to be a free field, we could take it to be an interacting field such that the interactions vanish at large times t→±∞. In fact, this is the usual setting considered in interacting quantum field theories, where fields are asymptotically free (e.g., the LSZ formalism <cit.> or the more technical Haag-Ruelle scattering theory <cit.>, the Gell-Mann and Low theorem <cit.>, etc.). This would lead to two groups of symmetries, at t→±∞, and one could perhaps draw an analogy with BMS^±, acting on ℐ^±.The equation 𝒟ϕ=0, being free, can be solved most easily in momentum space. We will frame our discussion directly in terms of a,a^*, that is, the Fourier modes of ϕ. From now on, no explicit reference to the underlying field ϕ shall be made (we will come back to it in Chapter <ref>).Let us therefore introduces a pair of canonical variables a_k⃗,σ,a_k⃗,σ^*, with algebra{a_k⃗,σ,a_k⃗',σ'} =0{a_k⃗,σ,a^*_k⃗',σ'} =δ_σσ'δ_k⃗k⃗'where {·,·} denotes a Dirac bracket.Here, k⃗∈ℝ^d is a continuous index – the momentum – and σ∈ℕ is a discrete index – the spin. On the other hand, δ_σσ' is the Kronecker delta, while δ_k⃗k⃗' is the Dirac delta on the mass hyperboloid,δ_k⃗k⃗' (2π)^d2√(k⃗^2+m^2) δ(k⃗-k⃗') Given this pair of canonical variables, one may realise the Poincaré algebra through the standard construction,P_μ ∑_σ∫ a^*_k⃗,σk_μ a_k⃗,σ ḳ M_0it P_i+∑_σ∫ a^*_k⃗,σℳ_0i a_k⃗,σ ḳ+spin M_ij ∑_σ∫ a^*_k⃗,σℳ_ij a_k⃗,σ ḳ+spinwhere ḳ is the invariant measure on the mass hyperboloid,ḳ1/(2π)^ddk⃗/2√(k⃗^2+m^2)and ℳ_μν are the differential operatorsℳ_0iik_0∂_i ℳ_ij2ik_[i∂_j] The differential operators ℳ_μν satisfy the Lorentz algebra,[ ℳ_αβ,ℳ^μν]=4iδ^[μ_[αℳ^ν]_β]while the functions P_μ,M_μν satisfy the Poincaré algebra,{ P^μ, P^ν} =0 { P_α, M^μν} =2iδ^[μ_α P^ν] { M_αβ,M^μν} =4iδ^[μ_[α M^ν]_β] if we regard ℳ_μν as a vector field instead of a differential operator, then it is straightforward to check that it satisfies the Killing equation, that is, ℳ_μν generates the isometries of the mass hyperboloid which, as a manifold (Euclidean AdS_d), is maximally symmetric. This corresponds to the well-known isomorphism Iso(EAdS_d)=SO(1,d).If we think of the functions a,a^* as the Fourier modes of a free field, then the objects P_μ, M_μν are actually the Noether charges corresponding to space-time symmetries. In particular, P_μ are the generators of space-time translations, and they are conserved. In fact, for free theories one may actually construct an infinite number of conserved charges,Q_f=∑_σ∫ a^*_k⃗,σf(k⃗) a_k⃗,σ ḳwhere f is an arbitrary function defined on the mass hyperboloid. For each f we will have a charge which, albeit perhaps non-local, is conserved Q̇_f=0.The standard Poincaré algebra is obtained by taking f(k⃗)= k^μ, in which case the conserved charge Q_f coincides with the generators of translations, P^μ. The generalisation of the Poincaré algebra to the 𝔟𝔪𝔰 algebra is obtained by extending the functions k^μ into the larger set χ_ℓ^λ(k⃗),k^μ → mχ^λ_ℓ(k⃗)P^μ → P^λ_ℓ m∑_σ∫ a^*_k⃗,σχ_ℓ^λ(k⃗) a_k⃗,σ ḳfor a certain set of (dimensionless) functions χ^λ_ℓ defined on the mass hyperboloid. Here, ℓ=(ℓ_1,ℓ_2,…,ℓ_d-1)∈ℤ^d-1 is a certain multi-index whose precise role will be clarified below; while λ∈ℝ is a real parameter. The set of indices (λ,ℓ) generalise the vector index μ.We will call the symmetries generated by the conserved charges P_ℓ^λ super-translations. As long as the set of functions {χ^λ_ℓ} contains k^μ for certain values of (λ,ℓ), the corresponding conserved charges will agree with the generators of standard translations. Furthermore, if the functions {M_μν,P_ℓ^λ} satisfy a closed algebra, then we will have succeeded in finding a generalisation of the Poincaré algebra. The new algebra, the 𝔟𝔪𝔰 algebra, will be the sought-after infinite-dimensional generalisation of the Poincaré algebra. In what follows, we will describe how the functions χ^λ_ℓ are constructed. The functions P^λ_ℓ clearly commute among themselves, while the functions M_μν satisfy the Lorentz algebra. The 𝔟𝔪𝔰 algebra is obtained by extending the relation { P_α, M^μν}=2iδ^[μ_α P^ν] into{ P^λ_ℓ, M_μν}=∑_α∈ℤ^d-1c^α_μνP^λ_ℓ+αfor a certain set of structure constants c^α_μν=c^α_μν(λ,ℓ). Equivalently, we can also write the relation above asℳ_μνχ^λ_ℓ=∑_α∈ℤ^d-1c^α_μν χ^λ_ℓ+α From this equation we see that what we wantis essentially to construct an infinite-dimensional representation of the Lorentz algebra. Therefore, we will define the functions χ^λ_ℓ as the eigenvectors of the Casimir operator of the Lorentz algebra,Δ1/2ℳ_μνℳ^μνsuch thatΔχ^λ_ℓ=λχ^λ_ℓ For one thing, we note that the functions k^μ themselves satisfy Δ k^μ=d k^μ, which nicely fits in our picture: we are trying to generalise the functions k^μ into a larger set, so it is reassuring to see that they indeed satisfy the equation we are using to define χ^λ_ℓ.In order to better understand the equation above, it is useful to point out that Δ is actually the Laplace-Beltrami operator on the mass hyperboloid k^2+m^2=0. In other words, given the flat metric ds^2=-dk_0^2+d k^2 and the embedding k^0→ +√(k⃗^2+m^2), the induced metric is ds^2=g_ijdk^idk^j, whereg_ij=δ_ij-k^ik^j/k⃗^2+m^2inverse⟺ g^ij=δ_ij+k^i k^j/m^2with determinant g=(g_ij)=m^2/k⃗^2+m^2.With this, the induced Laplacian reads1/m^2Δ=1/√(g)∂_i(√(g)g^ij∂_j)=g^ij∂_i∂_j+d/m^2 k^i∂_iwhich agrees with (<ref>). For completeness, we also mention that the Ricci scalar of this manifold is R=-d(d-1)/m^2 and the Christoffel symbols are Γ^k_ij=-g_ijk^k/m^2. With this, we will define our functions χ^λ_ℓ to be the eigenvectors of Δ, that is, the solutions ofΔχ^λ_ℓ=λχ^λ_ℓ The operator Δ is formally self-adjoint, and has an empty point spectrum <cit.>. In what follows we will be interested in the continuous spectrum of Δ and, in particular, in the singular value λ=d, as we shall discuss later on. For this particular value of λ, the equation above becomes invariant under conformal transformations g_ij→Ω^2 g_ij, χ→Ω^(2-d)/2χ.The equation (<ref>) does not determine χ^λ_ℓ uniquely, because the eigenvalues λ are degenerate; this is where the labels ℓ come into play. In order to fix the functions χ^λ_ℓ, we will simultaneously diagonalise the set of commuting operators {Δ, ℳ_12,Δ_S^n}, where Δ_S^n is the Laplacian of the unit n-sphere,Δ_S^n=∑_n>i>jℳ_ijℳ^ijwhose eigenvalues are of the form -ℓ_n(ℓ_n+n-1), with ℓ_n∈ℕ a non-negative integer (see the appendix <ref>).With this, our functions χ^λ_ℓ are defined through{Δχ^λ_ℓ =λχ^λ_ℓ ℳ_12χ^λ_ℓ =ℓ_1χ^λ_ℓ Δ_S^nχ^λ_ℓ =-ℓ_n(ℓ_n+n-1)χ^λ_ℓ n=2,3,…,d-1 .with |ℓ_1|≤ℓ_2≤ℓ_3≤⋯≤ℓ_d-1. The last of these labels, ℓ_d-1, plays the role of “total angular momentum”, so we will sometimes denote it by Lℓ_d-1.The equations (<ref>), together with univaluedness and finiteness, are enough to uniquely determine the functions χ^λ_ℓ. We will solve them next. These equations are separable in spherical coordinates (ρ,θ):k^1 =ρsinθ_d-1sinθ_d-2⋯sinθ_2sinθ_1k^2 =ρsinθ_d-1sinθ_d-2⋯sinθ_2cosθ_1k^3 =ρsinθ_d-1sinθ_d-2⋯cosθ_2⋯k^d-1 =ρsinθ_d-1cosθ_d-2k^d =ρcosθ_d-1 In these coordinates, the Laplacian reads1/m^2Δ=(1+ρ^2/m^2)∂_ρ^2+(d-1/ρ+d ρ/m^2)∂_ρ+1/ρ^2Δ_S^d-1where Δ_S^d-1 is the Laplacian on the (d-1)-sphere. The equationsℳ_12χ^λ_ℓ =ℓ_1χ^λ_ℓ Δ_S^nχ^λ_ℓ =-ℓ_n(ℓ_n+n-1)χ^λ_ℓ n=2,3,…,d-1determine the angular part of χ(ρ,θ) to be the spherical harmonics Y_ℓ,χ^λ_ℓ(ρ,θ)=f(ρ)Y_ℓ(θ),while the radial equation, as given by(Δ-λ)χ(ρ,θ)=0reads(1+z^2)f”+(d-1/z+d z)f'-(L(L+d-2)/z^2+λ)f=0where z=ρ/m, and a prime denotes differentiation with respect to z.Throughthe change of variables f(z)=z^L F(-z^2), this equation adopts the standard hypergeometric form <cit.>z(zd/dz+α)(zd/dz+β)F=zd/dz(zd/dz+γ-1)Fwhereα =1/4(+√((d+1)^2+4 (λ -d))+d-1+2 L) β =1/4(-√((d+1)^2+4 (λ -d))+d-1+2 L) γ =d/2+LThe solution to the radial equation thus readsf^λ_L(ρ) =c_1 (ρ/m)^L_2F_1[d+L+Λ-1/2,L-Λ/2;d/2+L;-ρ ^2/m^2]+c_2(m/ρ)^d+L-2_2F_1[Λ-L+1/2,2-d-Λ-L/2;2-d/2-L;-ρ ^2/m^2]where _2F_1 is the Gaussian hypergeometric function, andΛ1/2 (√((d+1)^2+4 (λ-d))-d+1)inverse⟺λ=Λ (Λ+d-1)As L≥ 0, the solution corresponding to c_2 is singular at ρ→ 0, so we will only keep the one corresponding to c_1 (which we set to c_1≡ 1, which we dub the canonical normalisation). For λ<0 the function f_L^λ(ρ) decays as ρ→∞, while for λ≥0 it does not. From now on we will focus on the second case.Using _2F_1(α,β;γ;z)=1+𝒪(z), together with <cit.>_2F_1(α ,β ;γ ;-z) =z^-αΓ (γ )Γ (β -α )/Γ (β ) Γ (γ -α ) _2F_1(α ,α -γ +1;α -β +1;-1/z)+z^-βΓ (γ )Γ (α -β ) /Γ (α ) Γ (γ -β ) _2F_1(β ,β -γ +1;β-α +1;-1/z)we see that the asymptotic behaviour of f_L^λ(ρ) isf^λ_L(ρ)=(ρ/m)^L+⋯ ρ≪ m (d+2 Λ-3)!/(d+2 Λ-4)!!(d+2 L-2)!!/(d+L+Λ-2)!(ρ/m)^Λ+⋯ ρ≫ mFrom this expression we see that for λ<d the function f^λ_L(ρ) grows sub-linearly with ρ, while for λ>d the growth is super-linear. The edge case λ=d is of particular interest: for example, for λ=d, the lowest modes L=0 and L=1 reproduce the functions k^μ, because mf^d_0=√(ρ^2+m^2) (see <cit.>) and mf^d_1=ρ:mχ_00⋯0^d ∼√( k^2+m^2)mχ_ℓ_1ℓ_2⋯1^d ∼linear combinations of k^iand therefore k^μ∈{χ_ℓ^λ}, as required. The higher-order modes L≥ 2 have a more complicated expression, but they are alllinear in ρ for large ρ. This is an essential property of the radial function f_L^d(ρ), because it guarantees that the integral that defines P_ℓ^d converges as long as that that defines P^μ does. We note that if λ<d, the integral is still convergent, so the admissible range for λ is 0≤λ≤ d, which is equivalent to 0≤Λ≤ 1. If we wish to pick a larger value of λ, we must impose a faster decay rate for the Fourier modes a_k⃗ (for arbitrary λ, we need | k|^Λ |a|^2∈ L^1(ℝ^d,ḳ)).the second argument of f_L^λ in the hypergeometric function (<ref>) is 1/2(L-Λ). Therefore, if Λ is an integer, then this argument vanishes for L=Λ, in which case the radial function becomes f=ρ^L. This in turns means that the mode χ_ℓ^λ, for L=Λ∈ℕ, is a (harmonic) polynomial in k of order L, that is, χ∼linear combinations of k^i_1k^i_2⋯ k^i_Λ. Now that we have characterised the modes χ_ℓ^λ, we can go back to our initial intention of enlarging the Poincaré algebra into the 𝔟𝔪𝔰 algebra. To this end, let us note that ℳ_μν commutes with (Δ-λ), and therefore(Δ-λ)ℳ_μνχ_ℓ^λ=0which implies that ℳ_μνχ_ℓ^λ must be a linear combination of the modes themselves:ℳ_μνχ^λ_ℓ=∑_α∈ℤ^d-1c^α_μν(λ,ℓ) χ^λ_ℓ+αwhere c^α_μν are a set of constants. This is precisely the relation we were after, (<ref>). we have now completed the construction of the modes χ, and showed that they satisfy a closed algebra of the form [ℳ,χ]=χ. A crucial ingredient of this construction was the fact thatthe differential operator (Δ-λ) commutes with ℳ_μν. In fact, it possible to prove that (Δ-λ) is the most general second order differential operator that commutes with ℳ_μν, so our construction is in a sense unique: if we were to insist that the modes χhave to be given by a differential equation, then this equation has to be (Δ-λ)χ=0.The coefficients c_μν^α(λ,ℓ) are the structure constants of the λ-extended 𝔟𝔪𝔰_d+1 algebra:{ P_ℓ^λ, P_ℓ'^λ'} =0 { M_αβ,M^μν} =4iδ^[μ_[α M^ν]_β] { P^λ_ℓ, M_μν} =∑_α∈ℤ^d-1c^α_μν(λ,ℓ) P^λ_ℓ+α We note that this algebra is uncountable infinite-dimensional, its elements being labelled by the integers ℓ∈ℤ^d-1 and the real number Λ∈[0,1] (or a larger interval if we impose a faster decay rate on the Fourier modes). It contains the standard (gravitational) algebra, corresponding to the value λ=d, which itself contains the Poincaré algebra, corresponding toall the integer values of |ℓ_1|≤ℓ_2≤…ℓ_d-1=1:λ𝔟𝔪𝔰_d+1⊃𝔟𝔪𝔰_d+1⊃𝔦𝔰𝔬(1,d)of dimensions 2^ℵ_0> ℵ_0>1/2 (d+1) (d+2).The algebra that is relevant to the gravitational problem is the one corresponding to λ=d, since it contains a Poincaré sub-algebra generated by the super-translations with L=0,1 (see Fig. <ref>).1.0 A rather straightforward method to calculate the structure constants c^α_μν(λ,ℓ) is to use the orthogonality of the spherical harmonics:c_μν^α(λ,ℓ)=1/f^λ_ℓ_d-1+α_d-1(ρ)∫ Y^*_ℓ+α(θ) ℳ_μνχ^λ_ℓ(ρ,θ) dΩ_d-1which is in fact independent of ρ, so we can take ρ→ 0 or any other convenient value. In the appendix <ref> we quote the coefficients c_μν^α(λ,ℓ) for arbitrary λ, for d=3,4,5.As the differential operators ℳ_μν have one unit of angular momentum (à la Wigner-Eckart), the indices α_i take values in α_i∈{0,±1}. One should also note that, as the angular momentum generators only act on the angular part of the modes, the spatial coefficients c_ij^α are only non-zero if α_d-1=0 (so that L→ L). On the other hand, the boost operators include a radial derivative ∂_ρ, so they do change the value of L. This means that c^α_0i will be non-zero for α_d-1=±1 (see Fig. <ref>). Another interesting fact about the coefficients c^α_μν is that, when α_d-1=+1, they are proportional to (ℓ_d-1(ℓ_d-1+d-1)-λ). Therefore, if λ=Λ (Λ+d-1) with Λ∈ℕ, then c^α_μν(λ,ℓ)|^α_d-1=1_ℓ_d-1=Λ vanishes, which means that we get a closed sub-algebra, of dimension(SO(1,d))+∑_ℓ_d-1=0^Λ ∑_ℓ_d-2=0^ℓ_d-1⋯∑_ℓ_2=0^ℓ_3 ∑_ℓ_1=-ℓ_2^ℓ_21=1/2 d (d+1)+(d+Λ-2)!/(d-1)! Λ!(d+2 Λ-1) a simple argument to prove that when L=Λ∈ℕ the action of the Lorentz operators do not raise the value of L (that is, that the coefficient corresponding to L→ L+1 vanishes) is to realise that for L=Λ, the mode χ_ℓ^λ is given by a linear combination of monomials of the form k^i_1k^i_2⋯ k^i_Λ, and therefore the derivatives ℳ_0i=ik_0∂_i cannot raise the power of ρ=| k| (it can only lower it or leave it as is). Therefore, we cannot have a term with χ_L+1∼ρ^L+1 in the expansion of ℳ_0iχ_ℓ.In the particular case λ=d, that is, Λ=1, the second term of (<ref>) becomes d+1, which is the number of space-time translations in a manifold of d spatial dimensions. This is the Poincaré sub-algebra. For Λ=0 we get 𝔰𝔬(1,d)×𝔲(1). For other integer values of Λ, we get other closed sub-algebras (see Fig. <ref>), whose relevance and physical interpretation is not clear yet. For non-integer values of Λ, there are no closed sub-algebras.In any case, equation (<ref>) proves that our construction does indeed lead to an infinite-dimensional generalisation of the Poincaré algebra. We will next discuss how this construction is related to the gravitational 𝔟𝔪𝔰 algebra. The correspondence between the canonical 𝔟𝔪𝔰 algebra and the gravitational one is most clear in what we will call the massless limit of the former, to be discussed below. Let us therefore discuss this limit, together with the opposite limit, the non-relativistic limit. These two limits correspond to taking ρ≫ m and ρ≪ m, respectively (see Fig. <ref>).Let us begin with the massless limitm→ 0. In this limit, the mass hyperboloid becomes singular, and the Laplacian blows up. One may nevertheless expand the Laplacian, the zero-modes, and the 𝔟𝔪𝔰 algebra in power series in m, and keep the leading terms. The result isΔ̃ lim_m→ 0Δ=ρ^2∂_ρ^2+d ρ∂_ρ ℳ̃_0i lim_m→ 0ℳ_0i=iρ∂_i ℳ̃_ij lim_m→ 0ℳ_ij=2ik_[i∂_j] χ̃_ℓ^λ lim_m→ 0m^Λχ_ℓ^λ=(d+2 Λ-3)!/(d+2 Λ-4)!!(d+2 L-2)!!/(d+L+Λ-2)! ρ^Λ Y_ℓ(θ)where we have used the limit ρ≫ m as given by (<ref>). From here on, a tilde ·̃ over an object means that it corresponds to the massless limit of the same.These modes satisfy the same algebra as the standard modes, with the exact same coefficients, c̃^α_μν=c^α_μν. As the radial function f^λ_L(ρ) is a power function instead of a hypergeometric one, the calculation of the structure constants c^α_μν(λ,ℓ) is particularly simple in this limit. If we take the particular case λ=d, the massless modes becomeχ̃_ℓ^d=(d-1)!/(d-2)!!(d+2 L-2)!!/(d+L-1)! ρ Y_ℓ(θ),to be compared with the asymptotic Killing fields, ξ(ℐ^+), as given by (<ref>). The L-dependent prefactor accounts for the different normalisation between the canonical and gravitational normalisations. We can now consider the opposite limit, the non-relativistic limit. In the limit m→∞, the mass hyperboloid becomes flat, and the Laplace operator becomesΔ̂ m^2lim_m→∞1/m^2Δ=m^2∂_i^2which is, once again the Casimir operator, but now of the homogeneous Galilei Group instead of the Lorentz Group. The Galilei Group is generated byℳ̂_0imlim_m→ 01/mℳ_0i=im∂_i ℳ̂_ij lim_m→ 0ℳ_ij=2ik_[i∂_j]such that Δ̂=ℳ̂_0iℳ̂^0i. Here and in the remainder of this document, we will place a hat ·̂ over an object to indicate that it corresponds to the non-relativistic counterpart of the corresponding relativistic object.In this limit, the modes χ̂_ℓ^λ becomeχ̂_ℓ(ρ,θ) 1/m^Llim_m→∞m^Lχ_ℓ^λ=(ρ/m)^LY_ℓ(θ)as given by the ρ≪ m limit of our previous f^λ_L(ρ), cf. (<ref>). In this limit, the modes become independent of λ. This was to be expected, because1/m^2Δχ^λ_ℓ=λ/m^2χ^λ_ℓ0,which is independent of λ, and therefore so are non-relativistic modes (which satisfy Δ̂χ̂=0, being the non-relativistic limit of Δχ=λχ). This is consistent with the fact that the functions k^i satisfy Δ̂k^i=0, where Δ̂ is the flat Laplacian. Therefore, our modes χ̂_ℓ contain the functions k^i for L=1 (the case L=0 plays the role of the central charge of the Bargmann group, χ̂_0=const., as we shall discuss in the following chapter). Unlike in the massless case, and due to the factor of m in the non-relativistic limit of the boost generators (cf. (<ref>)), here the algebra is not the same as in the relativistic case. If we writeℳ̂_μνχ̂_ℓ=∑_αĉ_μν^α(ℓ)χ̂_ℓ+αthen the new structure constants ĉ_μν are given byĉ^α_ij =c^α_ij ĉ^α_0i = c^α_0i α_d-1=-10α_d-1=+1 We call this new algebra the 𝔫𝔯𝔟𝔪𝔰 algebra, as it corresponds to the non-relativistic limit of the canonical realisation of the 𝔟𝔪𝔰 algebra. For consistency, one should realise this non-relativistic algebra in terms of a scalar field with a Galilean symmetry instead of a relativistic field. We postpone this construction to the next chapter. There we will also consider the non-relativistic limit in more abstract terms, that is, as an İnönü-Wigner contraction of the abstract 𝔟𝔪𝔰 algebra. CHAPTER: NON-RELATIVISTIC BMS. In this chapter we will address the problem of a non-relativistic BMS Group. To define this group, we could try to repeat the analysis of Chapter <ref> but in the context of Newtonian space-times, characterised by a contravariant degenerate spatial metric h^μν andcovariant vector τ_μ; see for example <cit.> and references therein. We will not do this here; instead, we will construct a possible candidate for the 𝔫𝔯𝔟𝔪𝔰 algebra by performing an İnönü-Wigner contraction of the relativistic 𝔟𝔪𝔰 algebra (in the same spirit as the Bargmann algebra can be obtained by contracting the Poincaré one <cit.>); and we will then provide an explicit realisation of the contracted algebra, by means of a free field with Galilean space-time symmetries, mimicking the construction of Chapter <ref>. This chapter is essentially a generalisation of the results of <cit.> to an arbitrary number of space-time dimensions.§ THE ALGEBRA 𝔫𝔯𝔟𝔪𝔰 AS A CONTRACTION OF 𝔟𝔪𝔰. The 𝔟𝔪𝔰 algebra is the semi-direct sum of the Lorentz algebra with the generators of the super-translations, which form an infinite-dimensional abelian sub-algebra. In a similar fashion, the 𝔫𝔯𝔟𝔪𝔰 algebra will be given by the semi-direct sum of the Bargmann algebra with the generators of super-translations.The canonical realisation of the λ-extended 𝔟𝔪𝔰_d+1 algebra in terms of the Fourier modes ofrelativistic free field leads to thealgebra (<ref>){ P_ℓ^λ, P_ℓ'^λ'} =0 { M_αβ,M^μν} =4iδ^[μ_[α M^ν]_β] { P^λ_ℓ, M_μν} =∑ c^α_μν(λ,ℓ) P^λ_ℓ+αwhere M_μνare thegenerators of Lorentz transformations, and P_ℓ^λ are the generators of super-translations. To simplify the discussion, we shall henceforth set λ≡ d, and omit this label altogether.As discussed in the previous chapter, for this value of λ the algebra above contains a Poincaré sub-algebra. This sub-algebra is spanned by M_μν,P_ℓ, with ℓ_d-1=0,1. These operators form a closed sub-algebra of 𝔟𝔪𝔰_d+1 because the structure constants c^α_μν(ℓ) vanish when α_d-1=+1 and ℓ_d-1≥ 1.In the particular case d=3, the algebra above agrees with the algebra of the gravitational 𝔟𝔪𝔰 (<ref>) (modulo an ℓ_2-dependent prefactor, resulting from the different normalisation for the generators of super-translations). We have argued that the correspondence should hold for any number of space dimensions d.We now proceed to perform the contraction of this algebra in order to obtain its non-relativistic analogue. In order to accommodate the central extension of the Galilei algebra, we consider the direct product of the BMS Group with U(1), with generator Z, and introduce the following transformation: H ω(P_0+ Z) Ẑ 1/ω(P_0- Z) M̂_0i 1/ωM_0i M̂_ijM_ij inverse⟺ P_0 =1/2ωH+ωẐZ =1/2ωH-ωẐM_0i =ωM̂_0iM_ij =M̂_ijtogether withP̂_ℓω^f(L)P_ℓinverse⟺ P_ℓ=ω^-f(L)P̂_ℓ,L≥ 1,where L=ℓ_d-1 and f(L) is an unspecified function, and ω is a dimensionless parameter which we shall take ω→∞ at the end.In the limit ω→∞, the centrally-extended relativistic algebra (<ref>) becomes[M̂_ij,M̂_mn] =4iδ_[i[mĴ_n]j][M̂_ij,M̂_0m] =2iδ_m[iM̂_j]0[M̂_0i,M̂_0j] =0[P̂_ℓ,H] =0[M̂_ij,H] =0[M̂_0i,H] =iP̂_i[P̂_ℓ,P̂_ℓ'] =0[M̂_ij,P̂_ℓ] =∑ĉ^α_ijP̂_ℓ+α[M̂_0i,P̂_ℓ] =∑ĉ^α_0iP̂_ℓ+αwhere ĉ^α_ij c^α_ij andĉ^α_0ilim_ω→∞ω^f(L)-f(L')-1 c^α_0i In order to have a non-trivial ω→∞ limit, we must havef(L)-f(L')-1≡0for some L'. Moreover, the structure constants c^α_0i are non-zero only if |L'-L|=1. This means that the only non-trivial contractions are the ones that satisfyf(L)=f(0)± L Furthermore, in order to obtain the Bargmann algebra as a sub-algebra,for L=1 we should recover the standard contraction P̂_i=P_i, so that f(1)=0. With this,f(L)=±(L-1) The conclusion of this discussion is that if we restrict ourselves to scalings of the form P̂_ℓ=ω^f(L)P_ℓ then there are only two non-trivial contractions of the 𝔟𝔪𝔰_d+1 algebra, corresponding to either sign in f(L)=±(L-1), which we will call 𝔫𝔯𝔟𝔪𝔰^(↓) (corresponding to the plus sign) and 𝔫𝔯𝔟𝔪𝔰^(↑) (corresponding to the negative sign). In either case, the structure constants are given byf^(↓)(L)=L-1ĉ^α_ij =c^α_ij ĉ^α_0i =c^α_0i α_d-1=-10α_d-1=+1 f^(↑)(L)=1-Lĉ^α_ij =c^α_ij ĉ^α_0i =0α_d-1=-1 c^α_0i α_d-1=+1the first of these algebras is the exact same algebra we obtained by the non-relativistic limit ρ≪ m of the modes χ_ℓ (cf. (<ref>)). This was actually to be expected, because the factors of ω in the abstract contraction (<ref>), (<ref>) are the same as the factors of m in the explicit limit (<ref>), (<ref>). The limits ω→∞ and m→∞ are formally identical. The second algebra is new.The two possible algebras above contain time and space translations, rotations, boosts, super-translations, and a central charge, and they both contain a Bargmann sub-algebra (see Fig.<ref>).1.0It is important to note that in the case of the 𝔫𝔯𝔟𝔪𝔰^(↓) contraction, the boost operators lower the value of L to L-1. This is in stark contrast with the relativistic case, where we have simultaneous contributions from L-1 and L+1. Therefore, unlike in the relativistic case, here the algebra resulting from the `↓' contraction contains an infinite number of finite-dimensional sub-algebras, obtained by considering all the super-translation generators with 1≤ L≤Λ for given Λ∈ℕ; the dimension of these sub-algebras is given by (<ref>) (plus one, due to the central charge). All these sub-algebras contain a Bargmann sub-algebra, and the associated matrices ĉ^α_ij,ĉ^α_0i provide an infinite number of finite-dimensional representations of the homogeneous Galilei group (finite-dimensional indecomposable representationsof homegeneous Galilei have been studied in <cit.>). On the other hand, in the case of the 𝔫𝔯𝔟𝔪𝔰^(↑) algebra, the boost operators raise the value of L to L+1, which means that the onlyfinite-dimensional sub-algebra of 𝔫𝔯𝔟𝔪𝔰^(↑) is the Bargmann algebra (see Fig.<ref>). 1.0In the following section we will construct an explicit realisation of the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra corresponding to the plus sign contraction, and we will also discuss the possibility of adding dilatations and expansions. § CANONICAL REALISATION OF THE 𝔫𝔯𝔟𝔪𝔰 ALGEBRA. We now proceed to constructthe canonical realisation of the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra. The construction will be analogous to the canonical realisation of 𝔟𝔪𝔰_d+1 from the previous chapter: we want to study the symmetries of galilean equations of motion of a free fields. As before, we will frame our discussion directly in terms of the Fourier modes of the non-relativistic field, a,a^*, in terms of which one may realise the Bargmann algebra throughẐ ∫ a^*( k)ma( k) d k/(2π)^dH ∫ a^*( k) k^2/2m a( k) d k/(2π)^d P̂_i ∫ a^*( k)k_i a( k) d k/(2π)^d M̂_0itP̂_i+∫ a^*( k)ℳ̂_0i a( k) d k/(2π)^d M̂_ij ∫ a^*_k⃗ℳ̂_ij a_k⃗ d k/(2π)^dwhere Ẑ is the central charge which generates U(1) rotations (here, and due to a lack of Coleman-Mandula <cit.>, external symmetries and internal symmetries mix in a non-trivial way). The differential operators ℳ̂_0i,ℳ̂_ij are given by (<ref>):ℳ̂_0i =im∂_i ℳ̂_ij =2ik_[i∂_j]and they satisfy the (homogeneous) Galilei algebra[ℳ̂_ij,ℳ̂_mn] =4iδ_[i[mℳ̂_n]j][ℳ̂_ij,ℳ̂_0m] =2iδ_m[iℳ̂_j]0[ℳ̂_0i,ℳ̂_0j] =0On the other hand, the Noether charges satisfy the the Bargmann algebra,{M̂_ij,M̂_mn} =4iδ_[i[mM̂_n]j] {M̂_ij,M̂_0m} =2iδ_m[iM̂_j]0 {M̂_0i,M̂_0j} =0 {P̂_ℓ,H} =0 {M̂_ij,H} =0 {M̂_0i,H} =iP̂_i {P̂_i,P̂_j} =0 {M̂_ij,P̂_m} =2iδ_m[iP̂_j] {M̂_0i,P̂_m} =iδ_imẐWhat we want to do is to generalise the last column of this algebra, by introducing a set of functions χ̂_ℓ in place of the standard momentum,k^i → mχ̂_ℓ(k⃗) P̂^i →P̂_ℓ m∫ a^*_k⃗χ̂_ℓ(k⃗) a_k⃗ d k/(2π)^dsuch that k_i∈{χ̂_ℓ} for some values of ℓ, thus extending the Bargmann algebra into the 𝔫𝔯𝔟𝔪𝔰 algebra.As before, let us consider the quadratic Casimir operator of the homogeneous Galilei group, Δ̂ℳ̂_0iℳ̂^0i=m^2∂_i^2which coincides with the Laplace-Beltrami operator on ℝ^d (flat space). With this, we will define our modes as the zero-modes of Δ̂:Δ̂χ̂_ℓ( k)=0together with the same angular dependence as in the relativistic case (cf. (<ref>)). As a consistency check, we note that, once again, the momentum k^i satisfies Δ̂k^i=0, so the family of zero-modes {χ̂} will contain the functions k^i as a subset. The general solution to the equation above isχ̂_ℓ(ρ,θ)=f_L(ρ)Y_ℓ(θ)where ρ,θ are the spherical components of k_i, and Y_ℓ are the spherical harmonics. On the other hand, f_L is given by the solution tof”+d-1/zf'-L(L+d-2)/z^2f=0where z=ρ/m, and a prime denotes differentiation with respect to z. The solution to this equation isf_L(r)=c_1(ρ/m)^L+c_2(m/ρ)^L+d-2 As before, we set c_2≡0 to avoid the singular behaviour at ρ→ 0, and set c_1≡ 1. With this,χ̂_ℓ(ρ,θ)=(ρ/m)^L Y_ℓ(θ) We see that the L=0 mode corresponds to the central charge (being momentum-independent), while the L=1 modes agree with the spherical components of k, so that, as expected, the family {χ̂_ℓ} contains the functions k_i as a special subcase. Comparing (<ref>) with the equation (<ref>), we see that the non-relativistic modes agree with the non-relativistic limit of the relativistic modes from the previous chapter, as the notation suggests. Therefore, they satisfy the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebraℳ̂_μνχ̂_ℓ=∑_αĉ_μν^α(ℓ)χ̂_ℓ+αwhere the structure constants are given by (<ref>):ĉ^α_ij =c^α_ij ĉ^α_0i = c^α_0i α_d-1=-10α_d-1=+1With this, we define the generators of super-translationsasP̂_ℓ m∫ a^*_ kχ̂_ℓ( k) a_ k d k/(2π)^dOne should note that, unlike the relativistic case, here the existence of P̂_ℓ is not guaranteed by the existence of P̂_i, because the non-relativistic modes χ̂ scale as ρ^L for large ρ instead of linearly with ρ. Therefore, a_ k being square-integrable is not enough for the integral defining P̂_ℓ to converge; we must impose the stronger condition that | k|^L |a_ k|^2 is integrable for all L∈ℕ. If we only require for a finite number of super-translations to exist, those corresponding to 0≤ L≤Λ for a certain integer Λ, then we must impose that | k|^L |a_ k|^2 is integrable for all 0≤ L≤Λ. This doesn't interfere with the closedness of the sub-algebra, because the structure constants ĉ_μν^α only connect super-translations of order L among themselves and with super-translations of order L-1 (see the left diagram of Fig. <ref>).In any case, using (<ref>) we see that the functions P̂_ℓ satisfy the algebra{P̂_ℓ,M̂_μν}=∑_α∈ℤ^d-1ĉ^α_μν(ℓ) P̂_ℓ+αwhich, together with the first two columns of (<ref>), agrees with the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra (<ref>). In principle, it may possible to construct a realisation of 𝔫𝔯𝔟𝔪𝔰_4^(↑) by using the second solution to the radial equation, f_L∼ρ^2-L-d, but the fact that these functions are singular at the origin implies that the Fourier modes a_ k must go to zero faster than any polynomial if we want the integral that defines P̂_ℓ to converge. We will not consider this possibility any further here.we have chosen the definition of χ̂ to be Δ̂χ̂≡ 0 in order to match the non-relativistic limit of the relativistic modes (cf. (<ref>)). We could have worked with a more general equation (Δ̂-λ̂)χ̂=0 for an arbitrary real constant λ̂>0. The solutions {χ̂^λ̂_ℓ} satisfy a more general algebra than 𝔫𝔯𝔟𝔪𝔰^(↓), but such that it reduces to the it when we take λ̂≡ 0. In order to obtain this generalised algebra by contracting the λ𝔟𝔪𝔰 algebra, one must take the ω,λ→∞ limit of the latter, while keeping λ̂≡λ/ω fixed. We state without proof that the radial equation for general λ̂ reads f^λ̂_L(z)=z^L _0F_1(;d/2+L;-λ̂/4 z^2) where _0F_1 is the confluent hypergeometric function, and that the structure constants for general λ̂ are ĉ^α_μν(λ̂,ℓ)=c^α_μν(λ̂+ℓ_d-1(ℓ_d-1+d-1),ℓ). We will not study this extension of 𝔫𝔯𝔟𝔪𝔰 here. In what follows, we will restrict ourselves to the case λ̂≡0, that is, the regular 𝔫𝔯𝔟𝔪𝔰^(↓) algebra, where the modes are given by χ̂_ℓ( k)=(ρ/m)^LY_ℓ(θ).The algebra 𝔫𝔯𝔟𝔪𝔰_d+1^(↓) has a very important property, one that is not present in the relativistic λ𝔟𝔪𝔰_d+1 case: here, the modes χ̂( k) are actually homogeneous polynomials in k^i, of degree L. This means that the symmetries generated by P̂_ℓ are local when acting on the field ϕ, meaning that δ_ℓϕ(x)={P̂_ℓ,ϕ(x)}=χ̂_ℓ(-i∂)ϕ(x)where χ̂_ℓ is a harmonic polynomial of degree L. In other words, χ̂_ℓ(-i∂) is nothing but a polynomial in ∂_i:χ̂_ℓ(-i∂)=∑_|α|=L𝒞^α∂_αfor a certain set of coefficients 𝒞^α. Differential operators of this form (and generalisations thereof), in the context of symmetries of partial differential equations, have been studied extensively in the literature; see for example <cit.>. Furthermore, the fact that the polynomials χ̂_ℓ( k) are homogeneous and of degree L implies that they satisfy𝒟χ̂_ℓ=Lχ̂_ℓwhere 𝒟 is the homogeneity operator, 𝒟 k^i∂_i=ρ∂_ρ. This means that if we define the dilatation operator asD 2tH+i∫ a^*_ k𝒟 a_ k d k/(2π)^dthen the generators of super-translations satisfy{D,P̂_ℓ}=iL P̂_ℓwhich extends the 𝔫𝔯𝔟𝔪𝔰 algebra to include dilatations, giving rise to a “Weyl-𝔫𝔯𝔟𝔪𝔰”. We note that the main obstruction of this extension to the relativistic algebra is the presence of a non-zero mass m, which introduces a length scale to the theory (so that it is not invariant under dilatations). From a more pragmatic point of view, the energy k_0=√( k^2+m^2) is not a homogeneous polynomial, so it is not an eigenvector of 𝒟 (unless we take m≡ 0).Once we realise that the algebra admits dilatations, it becomes natural to ask ourselves about the action of the expansion operator (or Schrödinger conformal transformations), given byC-t^2H+tD+m/2∫ a^*_ kΔ̂a_ k d k/(2π)^dand which satisfies <cit.>{D,C}=2iC,{H,C}=iD,{C,P̂_i}=iM̂_0i, etc. Using[Δ̂,χ̂_ℓ] =Δ̂χ̂_ℓ+2(∂_iχ̂_ℓ)∂/∂ k^i=2/im[∑_αĉ_0i^α χ̂_ℓ+α]∂/∂ k^iwe obtain{C,P̂_ℓ}= itL P̂_ℓ-im∫ a^*_ k[∑_αĉ_0i^α χ̂_ℓ+α]∂/∂ k^i a_ k d k/(2π)^d The r.h.s. of (<ref>) is not an element of 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1. This means that if we attempt to extend the algebra to include C, the resulting algebra is not closed, which seems to preclude a possible “Schrödinger-𝔫𝔯𝔟𝔪𝔰”. In any case, the r.h.s. (<ref>) is the bracket of two conserved quantities, which means that it is conserved as well. Indeed, a straightforward calculation confirms that[∂/∂ t+{·,H}]{C,P̂_ℓ}=0 This means that {C,P̂_ℓ}, despite not being an element of 𝔫𝔯𝔟𝔪𝔰, is a conserved operator, i.e., it generates a symmetry of the equations of motion. In the particular case L=1, this commutator agrees with the generators of boosts, {C,P̂_i}=iM̂_0i. It is tempting to let {C,P̂_ℓ} define a new kind of generator of symmetries, which would generalise the standard generators of boosts; we could dub these objects super-boosts. It will be interesting to explore their relation with the relativistic super-rotations. CHAPTER: CONCLUSIONS AND OUTLOOK. tocchapterConclusions and Outlook. This completes our study of the BMS Group. Here we summarise the essential points and highlight those topics that require a further analysis.§.§.§ The BMS Group. In Chapter <ref> we reviewed this group in its original context, the problem of characterising the asymptotic isometries of asymptotically flat space-times. We saw that, while in the strictly flat case the isometries are generated by a finite number of vector fields, once we relax the notion of isometry into an asymptotic isometry, the number of generators becomes infinite. In the first case, these vector fields generate the Poincaré Group, and in the second case they generate the BMS Group, which contains Poincaré as a sub-group. Moreover, we saw that one may further extend the symmetry algebra by including the infinite-dimensional set of super-rotations.Finally, we mentioned that the generalisation of this procedure to higher dimensional manifolds is non-trivial, and there are several opposing views in the literature. The very existence of super-translations depends on the particular falloff rate for the component of the metric, and it is not clear how strong these falloffs should be in arbitrary dimensions. Even more so in the case of super-rotations, whose existence depends on whether the conformal Killing equation on the sphere admits or not an infinite number of solutions; and, as we have mentioned, that only happens in d=2,3. Therefore, unless we alter the definition of super-rotations in higher dimensions, one would not expect to find them in d>3. §.§.§ Canonical realisation of BMS. In Chapter <ref> we generalised the canonical construction of the 𝔟𝔪𝔰_4 algebra to an arbitrary number of space-time dimensions, and we argued that this result suggests a non-trivial gravitational 𝔟𝔪𝔰_d+1 for all d. We also generalised this algebra to include an additional label, λ, which makes the resulting algebra uncountable infinite-dimensional. For some particular values of λ, the corresponding algebra contains finite-dimensional sub-algebras, such as Poincaré for λ=d.We have not addressed the problem of super-rotations in the canonical construction. One may presume that, like in the gravitational case, these symmetries are only present in d=2,3. These dimensions require a case-by-case analysis.Another unexplored aspect of the canonical construction is the fact that it admits the extension in terms of the parameter λ. The particular value λ=d was relevant to the gravitational problem, but it would be nice to find a physical system whose group of symmetries is λ𝔟𝔪𝔰, with arbitrary λ. Finally, we would like to mention that in massless theories the group of external symmetries usually gets enhanced to the Conformal Group, which includes conformal transformations and dilatations together with the standard Poincaré transformations. In our canonical construction, it may be possible to do the same in the case of massless super-translations, that is, to consider a conformal BMS, perhaps related to the gravitational conformal BMS constructed in <cit.>.§.§.§ Non-relativistic BMS. In Chapter <ref> we studied some possible analogues of 𝔟𝔪𝔰 for non-relativistic systems, both by means of a contraction of the abstract relativistic algebra and by the canonical method. We saw that, in a sense, there are only two admissible contractions, and that one of them explicitly arises in the study of the symmetries of Galilei-invariant equations of motion for free fields.Unlike in the relativistic case, the 𝔫𝔯𝔟𝔪𝔰 super-translations are local: they act as polynomials in the differential operator ∂_i. Being homogeneous, these polynomials have a simple behaviour under the action of the dilatation operator, which means that the 𝔫𝔯𝔟𝔪𝔰 algebra admits an extension that includes dilatations. However, it does not seem to admit conformal transformations, which means that we cannot use the non-relativistic super-translations to extend the Schrödinger algebra. The existence of super-rotations in the non-relativistic setting might prove essential in the construction of a possible non-relativistic conformal NRBMS.Finally, it will be interesting to study the asymptotic symmetries of asymptotically flat Newtonian space-times and to check whether the algebra coincides with the one of the two 𝔫𝔯𝔟𝔪𝔰 algebras we have constructed. In other words, it will be nice to have an explicit verification of the correspondence between the canonical NRBMS Group and the gravitational one, provided the latter actually exists. CHAPTER: SPHERICAL HARMONICS.Here we quote the form of the spherical harmonics on S^d-1. A much more detailed discussion can be found in <cit.>.The metric on the sphere can be defined recursively throughds_1^2 =dθ_1^2 ds_n^2 =dθ_n^2+sin^2θ_nds_n-1^2where θ_1∈[0,2π) and θ_n∈[0,π). The volume element isdΩ_1 =dθ_1 dΩ_n=sin^n-1θ_n dθ_n dΩ_n-1 On the other hand, the Laplacian readsΔ_S^1 =∂^2/∂θ_1^2 Δ_S^n =1/sin^n-1θ_n∂/∂θ_n[sin^n-1θ_n∂/∂θ_n]+1/sin^2θ_nΔ_S^n-1 We define the spherical harmonics through[∂/∂θ_1-iℓ_1]Y_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1) =0 [Δ_S^n+ℓ_n(ℓ_n+n-1)]Y_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1) =0where n=1,2,…,d-1.The first equation is solved byY_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1)∝e^iθ_1ℓ_1where the constant of proportionality is an arbitrary function of θ_2,…,θ_d-1. Univaluedness of Y implies that ℓ_1∈ℤ is an integer. The rest of equations are solved byY_ℓ_1ℓ_2⋯ℓ_n(θ_1,θ_2,…,θ_n)=Y_ℓ_1ℓ_2⋯ℓ_n-1(θ_1,θ_2,…,θ_n-1)𝒥(θ_n)where 𝒥(θ) satisfies[1/sin^n-1θ∂/∂θ[sin^n-1θ∂/∂θ]-ℓ_n-1(ℓ_n-1+n-2)/sin^2θ +ℓ_n(ℓ_n+n-1)]𝒥(θ)=0If we set 𝒥(θ)=sin^ℓ_n-1θ y(cosθ), and make the change of variables x≡cosθ, this equation becomes(1-x^2) y”(x)-(2 μ+1)x y'(x)+ν (ν+2 μ) y(x)=0where μℓ_n-1+(n-1)/2 and νℓ_n-ℓ_n-1. The solution to this differential equation isy(x)= c_1 C_ν^(μ)(x)+c_2 (1-x^2)^1/4 (1-2 μ) Q_ν+μ-1/2^1/2-μ(x)where C_ν^(μ),Q_ν^μ are the Gegenbauer function and the Legendre function of the second kind <cit.>: C_ν^(μ)(x) Γ(ν+2μ)/Γ(2μ)Γ(ν+1)_2F_1(-ν,ν+2μ;μ+1/2;1-z/2)Q_ν^μ(x) e^iμππ^1/2Γ(μ+ν+1)(x^2-1)^μ/2/2^ν+1x^μ+ν+1 ··_2F̃_1(μ+ν+2/2,μ+ν+1/2;ν+3/2;x^-2)where F̃ denotes the regularised hypergeometric function. Finiteness at x=±1 requires c_2≡ 0, and ν to be a non-negative integer. By induction, ℓ_n∈ℕ, with |ℓ_1|≤ℓ_2≤ℓ_3…≤ℓ_n. The series for C_μ^(μ) now terminates; the Gegenbauer functions become polynomials:C_ν^(μ)(x)= ∑_j=0^⌊ν/2⌋(-1)^jΓ(ν+μ+j)/Γ(μ)j!(ν-2j)!(2x)^ν-2jWith this,y_ℓ_n,ℓ_n-1^(n)(x)∝ C_ℓ_n-ℓ_n-1^(ℓ_n-1+(n-1)/2)(x)∝(d/dx)^ℓ_n-1C_ℓ_n^((n-1)/2)(x)If we normalise the functions to∫_-1^+1 (1-x^2)^n/2+ℓ_n-1-1[y_ℓ_n,ℓ_n-1^(n)(x)]^2 dx=1we gety_ℓ_n,ℓ_n-1^(n)(x) (-1)^ℓ_n-1 2^ℓ_n-1+n/2-1Γ(n-12+ℓ_n-1) ··√((ℓ_n+n-1/2) Γ (ℓ_n-ℓ_n-1+1)π Γ (ℓ_n+ℓ_n-1+n-1)) C_ℓ_n-ℓ_n-1^(ℓ_n-1+(n-1)/2)(x)where we have included the conventional Condon-Shortley phase (-1)^ℓ_n-1.With this, our spherical harmonics are given by the recursive formulaY_ℓ_1(θ_1) =1/√(2π) e^iθ_1ℓ_1Y_ℓ_1ℓ_2…ℓ_n(θ_1,θ_2,⋯,θ_n) =sin^ℓ_n-1θ_n y_ℓ_n,ℓ_n-1^(n)(cosθ_n)Y_ℓ_1ℓ_2⋯ℓ_n-1(θ_1,θ_2,…,θ_n-1)and they have been normalised to∫ Y^*_ℓ_1ℓ_2⋯ℓ_n(θ)Y_ℓ'_1ℓ'_2⋯ℓ'_n(θ) dΩ_n=δ_ℓ_1ℓ'_1δ_ℓ_2ℓ'_2⋯δ_ℓ_nℓ'_n left=2cm,right=2cm CHAPTER: STRUCTURE CONSTANTS. In this appendix we quote the value of the structure constants c^α_μν(λ,ℓ) for d=3,4,5.For d=3,c_01^±+ =1/2 (ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_01^±- =1/2√(2 ℓ_2+1/2 ℓ_2-1 (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1)) c_02^±+ =±i/2(ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±- =±i/2√(2 ℓ_2+1/2 ℓ_2-1(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1)) c_03^0+ =-i (ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0- =i √(2 ℓ_2+1/2 ℓ_2-1(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_12^00 =ℓ_1c_13^±0 =-1/2√((ℓ_2∓ℓ_1) (ℓ_2±ℓ_1+1))c_23^±0 =∓i/2√((ℓ_2∓ℓ_1) (ℓ_2±ℓ_1+1))For d=4,c_01^±++ =-1/8 (ℓ_3 (ℓ_3+3)-λ )··√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2) )c_01^±-+ =-1/8 (ℓ_3 (ℓ_3+3)-λ )··√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )c_01^±+- =-1/2√(ℓ_3+1/ℓ_3 (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_01^±– =-1/2√(ℓ_3+1/ℓ_3(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1)) c_02^±++ =∓i/8 (ℓ_3 (ℓ_3+3)-λ )··√(1/(ℓ_3+1) (ℓ_3+2)^3 (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±-+ =∓i/8 (ℓ_3 (ℓ_3+3)-λ )··√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+1) (ℓ_3-ℓ_2+2)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )c_02^±+- =∓i/2√(ℓ_3+1/ℓ_3(ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±– =∓i/2√(ℓ_3+1/ℓ_3(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_03^0++ =+i/4 (ℓ_3 (ℓ_3+3)-λ )··√(1/ (ℓ_3+1) (ℓ_3+2)^3 (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0-+ =-i/4 (ℓ_3 (ℓ_3+3)-λ )·√(1/(ℓ_3+1) (ℓ_3+2)^3 (ℓ_3-ℓ_2+1)(ℓ_3-ℓ_2+2) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_03^0+- =+i √(ℓ_3+1/ℓ_3(ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0– =-i √(ℓ_3+1/ℓ_3(ℓ_2+ℓ_3) (ℓ_2+ℓ_3+1)/(1-2 ℓ_2) (2 ℓ_2+2-1)(ℓ_1-ℓ_2) (ℓ_1+ℓ_2))c_04^00+ =-i/4 (ℓ_3 (ℓ_3+3)-λ )√(/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))c_04^00- =i √(ℓ_3+1/ℓ_3(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))c_14^±+0 =1/2√( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_14^±-0 =1/2√( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_24^±+0 =±i/2√( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_24^±-0 =±i/2√( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_34^0+0 =-i √( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_34^0-0 =+i √( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))For d=5,c^±+++_01 =1/4 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3)(2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2)(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c^±+-+_01 =1/4 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4-ℓ_3+1) (ℓ_4-ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1)(ℓ_2±ℓ_1+2))c^±-++_01 =1/4 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c^±–+_01 =1/4 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c^±+–_01 =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c^±++-_01 =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c^±-+-_01 =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2±ℓ_1) (ℓ_2±ℓ_1-1))c^±—_01 =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_02^±+++ =±i/4 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±-++ =±i/4(ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_02^±–+ =±i/4(ℓ_4 (ℓ_4+4)-λ )··√(1/ (2 ℓ_4+3)(2 ℓ_4+5)^3(ℓ_4-ℓ_3+1)(ℓ_4-ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_02^±+-+ =±i/4(ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3)(2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±— =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_02^±+– =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±++- =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_02^±-+- =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_03^0+++ =-i/2(ℓ_4 (ℓ_4+4)-λ )··√(1/ (2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0-++ =+i/2 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2+1) (2 ℓ_2-1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_03^0+-+ =-i/2(ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0–+ =+i/2 (ℓ_4 (ℓ_4+4)-λ )··√(1/ (2 ℓ_4+3)(2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_03^0++- =-i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0-+- =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2+1) (2 ℓ_2-1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_03^0+– =-i/2√(2 ℓ_4+3/2 ℓ_4+1 (ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_03^0— =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1)) c_04^00++ =+i/2 (ℓ_4 (ℓ_4+4)-λ )··√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))c_04^00-+ =-i/2(ℓ_4 (ℓ_4+4)-λ )··√(1/ (2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))c_04^00– =-i/2√(2 ℓ_4+3/2 ℓ_4+1 (ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))c_04^00+- =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))c_05^000+ =-i (ℓ_4 (ℓ_4+4)-λ )√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+3))c_05^000- =+i √(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4+ℓ_3+2))c_15^±++0 =-1/4√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_15^±–0 =-1/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_15^±-+0 =-1/4√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )c_15^±+-0 =-1/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_25^±++0 =∓i/4√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_25^±–0 =∓i/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_25^±-+0 =∓i/4√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))c_25^±+-0 =∓i/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))c_35^0++0 =+i/2√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_35^0-+0 =-i/2√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_35^0+-0 =+i/2√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))c_35^0–0 =-i/2√( (ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))c_45^00+0 =-i/2√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))c_45^00-0 =+i/2√( (ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))ieeetr | http://arxiv.org/abs/1708.07564v1 | {
"authors": [
"D. G. Delmastro"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170824214033",
"title": "BMS in higher space-time dimensions and Non-relativistic BMS"
} |
V>c<height 4./imgs/BICICCV-015 emptySynthesising Wider Field Images from Narrow-Field Retinal Video Acquired Using a Low-Cost Direct Ophthalmoscope (Arclight) Attached to a SmartphoneKeylor Daniel Chaves Viquez^a Ognjen Arandjelović^† a Andrew Blaikie^bIn Ae Hwang^b^aSchool of Computer Science^bSchool of Medicine University of St AndrewsScotland, United Kingdom ^† December 30, 2023 ====================================================================================================================================================================================================================Access to low cost retinal imaging devices in low and middle income countries is limited, compromising progress in preventing needless blindness. The Arclight is a recently developed low-cost solar powered direct ophthalmoscope which can be attached to the camera of a smartphone to acquire retinal images and video. However, the acquired data is inherently limited by the optics of direct ophthalmoscopy, resulting in a narrow field of view with associated corneal reflections, limiting its usefulness. In this work we describe the first fully automatic method utilizing videos acquired using the Arclight attached to a mobile phone camera to create wider view, higher quality still images comparable with images obtained using much more expensive and bulky dedicated traditional retinal cameras. § INTRODUCTIONIt is estimated that some 285 million people in the world visually impaired, with the majority of cases potentially preventable or treatable if diagnosed promptly <cit.>. Unfortunately the burden of disease is greatest in low and middle income countries (East Asia, Latin America, and Africa) where access to diagnostic devices presents a major challenge <cit.>. Observation of the retina by an eye care professional using an ophthalmoscope or through acquiring a video or photograph allows screening for key blinding diseases such as diabetic retinopathy, glaucoma and retinopathy of prematurity <cit.>.However, one of the key barriers to fundal screening in low and middle income countries (LMICs) emerges from the lack of access to affordable retinal imaging equipment <cit.>. The Arclight direct ophthalmoscope has been recently developed as a means of addressing this challenge <cit.>. The device, despite its simplified design and low cost, has been shown to be as effective as more expensive traditional devices <cit.>. In addition, the Arclight can be attached to the camera of a mobile phone and acquire a digital recording of the fundus for the purposes of telemedicine. However, the captured images are intrinsically limited to a narrow field of view by the optics of direct ophthalmoscopy. Other limiting factors relating to the optics, sensor, and compression software of mobile phone cameras can further reduce the quality of the acquired data in comparison to more expensive dedicated traditional retinal cameras <cit.>.§ CONTEXT AND PRACTICAL CHALLENGESTraditionally, retinal images (see Figure <ref>) are acquired using expensive and bulky tabletop units operated by trained technicians in a hospital setting<cit.>. Consequently such devices are impractical for use in low income areas where resources to buy and maintain traditional cameras are limited and robustness and portability is key for allowing the sharing and movement between inhospitable sites <cit.>.However, the availability of low cost yet powerful smartphones in LMICs has led to the development of `m-health' <cit.> whereby mobile phones can play several roles: contacting and reminding patients to take medicines and attend appointments, transferring money to pay for health care, storing guidelines and diagnostic algorithms, acquiring and archiving audio and visual information from patients, analysing data, utilizisng GPS tracking, and transferring information wirelessly.These approaches <cit.> have quickly been adopted in low resource health care settings where the portability and connectivity of smartphones <cit.> has made them particularly valuable in telemedicine <cit.>, making it possible to reach patients and communities that currently receive suboptimal care due to geographical and financial barriers <cit.>. §.§ The Arclight deviceThe Arclight is a highly portable (dimensions: 110mm ×26mm ×9mm, mass: 18g) low-cost ( 5 if purchased in high volume[See <https://iapb.standardlist.org/get-arclight/>]) ophthalmoscope that has been developed in response to the need for affordable diagnostic eye care tools in LMICs <cit.>. Uniquely the device is illuminated by an LED allowing it to be powered by a slim lithium battery that is charged by an integrated solar panel reducing the need for expensive and hard to find consumables. Employing a small LED allows simplification of the design of the device reducing manufacturing costs and its overall size, see Figure <ref><cit.>. The novel features of the Arclight make it ideal for use in mobile clinics and remote low resource settings with limited access to power. Consequently over 10000 devices have been distributed, enabling thousands of health care workers worldwide to perform fundal examinations for the first time <cit.>.Studies have shown that the Arclight is just as accurate for screening for signs of diabetic retinopathy and glaucoma as more expensive traditional devices when used manually by an eye care professional <cit.>. However, one of the key inherent limitations of any direct ophthalmoscopy device is the narrow retinal field of view. Single still images are typically of limited utility but a sweeping video of a greater area of the fundus may offer greater clinical information, benefiting a telemedicine approach. Still, transferring video files is challenging in remote areas of a low income country where fast and reliable broadband internet may not be available. Synthesising a single wider field still image from multiple frames of a video fused together is a promising solution for overcoming many of the limitations of the optics of direct ophthalmoscopy as well as the data-heavy nature of video.§ PROPOSED METHODIn this section we describe a novel method for processing narrow field retinal video acquired using the Arclight ophthalmoscope while attached to an Apple iPhone 6s. By fusing the information contained in different frames of a video we synthesise a wider field image of higher quality, which is easier to interpret clinically and more compact to transmit and share. We start with an overview of the method and then follow up with detailed descriptions of each of the processing steps. §.§ OverviewAs we noted earlier, the quality and useful information content of different frames in a video varies greatly. Hence, in order to ensure robustness, our algorithm starts by automatically identifying the highest quality frame. Tracking, and thereafter incremental image stitching, commences both in forward and backward directions in time. Each new frame is processed in order to remove confounding appearance content, processed to extract salient information, registered with the growing synthetic image (initially the single frame which which tracking begun), and stitched with it. §.§ Detection of the region of interestA crucial step in the pipeline outlined in the previous section concerns the detection of the region of interest (ROI) i.e. the part of the image which corresponds to the fundus and contains salient appearance content: the blood vessels and the optic disc. The parts of the frame which are outside of it are of no use for us and correspond to the sclera (the white, outer part of the eyeball), eyelids, etc.As the frames in Figure <ref> illustrate, the region of the interest is elliptical in shape, and bright. However, colour in the image can not be used reliably as a discriminative feature for segmentation as significant variation can be observed across different individuals (e.g. there are differences between individuals of Caucasian and African descent). Variation in location, ellipse shape and orientation, as well as scale should also be noted as significant challenges.Though the region of interest is elliptical in shape, observe that there is no sharp delineation, prohibiting the use of edge detection based approaches. Instead, we propose a two step method. In the first step, simple binary segmentation is performed using the well-known method first introduced by Otsu <cit.>. Here the frame is treated as a greyscale image, see Figure <ref>. Then the contour of the largest region used to find the best elliptical fit. We achieve this using a genetic algorithm based approach. Our methodology is inspired by the work of Conn and Arandjelović who used ellipse fitting as the first stage in the task of localizing and segmenting ancient coins in real-world images <cit.>.An ellipse embedded within a 2-dimensional Euclidean space is parameterized by five parameters. This can be readily seen from the implicit equation:x^2 + k_xy x y + k_yy y^2 + k_x x + k_y y + k = 0.The choice of the parameterization is crucial in making the most of ourgenetic search strategy. In particular, the parameterization should be such that when encoded as a `chromosome' (in the context of a genetic algorithm), operations such as crossover, mutations, and others, are likely to effect an improvement in the fitness of a hypothesis. With this goal in mind, we parameterize an ellipse (hypothesis) using five points on its circumference and, following Conn and Arandjelović <cit.>, use a short, non-binary chromosome comprising the coordinates of these points. As argued by Conn and Arandjelović, by enforcing the indivisibility of coordinate values we ensure that the constraint imposed by two circumference points is retained during evolutionary operations, thereby achieving a higher chance of greater generational fitness improvement.The fitness of a specific hypothesis is evaluated in a straightforward fashion. From chromosomes formed by concatenating the coordinates of five points on an ellipse's circumference –(x_a,y_a), (x_b,y_b), (x_c,y_c), (x_d,y_d), and (x_e,y_e)– the parameters in (<ref>) can be obtained by solving a simple linear equation:[ [ x_a^2 x_a y_a y_a^2 x_a y_a 1; x_b^2 x_b y_b y_b^2 x_b y_b 1; x_c^2 x_c y_c y_c^2 x_c y_c 1; x_d^2 x_d y_d y_d^2 x_d y_d 1; x_e^2 x_e y_e y_e^2 x_e y_e 1; ]] [[1; k_xy; k_yy;k_x;k_y;k;]] =[[ 0; 0; 0; 0; 0; ]]. From this parameterization, the fitness of a hypothesis can be readily quantified by (i) uniformly sampling the circumference of the corresponding ellipse, and (ii) computing the number of samples which are located on edge pixels of the edge image. Equispaced samples along the circumference of the corresponding ellipse are readily generated using the canonical form, i.e. using the major and minor radii (a and b) of the ellipse, its centre (x_0, y_0), and the rotation angle θ relative to the coordinate system:a = -√(2 ψψ_1)/k_xy^2 - 4k_yy and b = -√(2 ψψ_2)/k_xy^2 - 4k_yy,x_0=2k_yy k_x- k_xy k_y/k_xy^2 - 4k_yy and y_0=2 k_y- k_xy k_x/k_xy^2 - 4k_yy,where:ψ=k_y^2+ k_yyk_x^2 - k_xyk_x k_y + (k_xy^2 - 4k_yy)k, ψ_1=1+k_yy + √((1-k_yy)^2+k_xy^2), ψ_2=1+k_yy - √((1-k_yy)^2+k_xy^2).Then, if there are n_s samples, and the samples are { (x_1,y_1),(x_2,y_2),…,(x_n_s,y_n_s)}, the fitness becomes:ϕ = ∑_i=1^n_s E(x_i,y_i) / n_s,where E(x,y) is the value of the pixel (x,y) in the edge image 𝐄. §.§ Glare detection and removalAs mentioned earlier, one of the challenges with the input videos is the presence of significant glare. It is of critical importance that the areas of the fundus occluded by glare are detected so as not to confound the tracking process, as well as so that the relevant regions can be filled-in by exploiting information from other frames in which they are not occluded.Glare areas are for practical purposes of unpredictable shape, often but not universally present, and their location varies. All of these characteristics make their detection difficult. Another challenge which emerges in our specific application concerns the appearance of the optic disk which is usually brighter than the rest of the fundus, and could be mistaken for glare. Yet, the appearance of the optic disk is a crucial structure of interest to ophthalmologists, used to diagnose glaucoma and a number of other conditions <cit.>.Our approach to detecting glare is inspired by the work of Lange <cit.> on images of the uterine cervix. Broadly speaking, we detect glare areas as those regions of the image which exhibit high contrast or saturation, and are approximately white. Formally, the criterion is based on the following local measure g(x,y):g(x,y)=min(R_x,y,G_x,y,B_x,y) - min(R_x,y,G_x,y,B_x,y)/max(R_x,y,G_x,y,B_x,y)where R_x,y, G_x,y, and B_x,y are respectively the red, green, and blue colour components of the pixel at the locus (x,y). After computing g(x,y) for all image loci the corresponding feature image is smoothed using the Alternating Sequential Filter (a sequence of morphological closing and opening operations), and the binary mask 𝐌 computed with M_x,y = 1 iff G_x,y exceeds a predefined threshold, and M_x,y = 0 otherwise. Examples are shown in Figure <ref>. §.§ Salient information extractionHaving removed confounding image content (glare, background, etc.) the processed frames are ready for the main part of our algorithms with concerns the tracking of salient structures present in the fundus, and the subsequent stitching. However, notwithstanding the significant processing described thus far, these tasks are still far from straightforward. For example, note that neither simple appearance nor any of the commonly used local features can be used for tracking (or, equivalently, registration). Raw appearance changes greatly due to global illumination changes, as well as local changes caused by the inherent optics of the eye. Local features – such as appearance based SIFT <cit.> and SURF <cit.>, or edge based ones <cit.>– can also not be reliably detected as the appearance of the fundus does not contain corner-like loci.Our approach is specifically targeted at the type of appearance content of interest in fundal images, namely blood vessels. These are line like features which form complex and characteristic structures, and by virtue of this allow different frames to be mutually registered.We extract characteristic vessel based features using the method introduced by Frangi et al. <cit.>. Their so-called vesselness filter, first proposed for the use on 3D MRI data but later successfully employed in a number of different applications <cit.>, extracts tubular structures from an image. For a 2D image consider the two eigenvalues λ_1 and λ_2 of the Hessian matrix computed at a certain image locus and at a particular scale. Without loss of generality let us also assume that |λ_1| ≤ |λ_2|. The two key values used to quantify how tubular the local structure at this scale is are ℛ_𝒜 = |λ_1|/|λ_2| and 𝒮 = √(λ_1^2 + λ_1^2). The former of these measures the degree of local 'blobiness'. If the local appearance is blob-like, the Hessian is approximately isotropic and |λ_1|≈|λ_2| making ℛ_𝒜 close to 1. For a tubular structure ℛ_𝒜 should be small. On the other hand, 𝒮 ensures that there is sufficient local information content at all: in nearly uniform regions, both eigenvalues of the corresponding Hessian will have small values. For a particular scale of image analysis s, the two measures, ℛ_𝒜 and 𝒮, are then unified into a single vesselness measure:𝒱(s) =0 if λ_2 > 0(1-e^-ℛ_ℬ/2β^2) × (1-e^-𝒮/2c^2) otherwise,where β and c are the parameters that control the sensitivity of the filter to ℛ_𝒜 and 𝒮. Finally, if an image is analyzed across scales from s_min to s_max, the vesselness of a particular image locus can be computed as the maximal vesselness across the range:𝒱_0 = max_s_min≤ s ≤ s_max𝒱(s)We empirically set the parameter values to β= 0.75 and c = 15, and process frames at scales s=3,4,5. Examples of typical results are shown in Figure <ref>.§.§.§ Algorithm initializationRecall that our algorithm starts the tracking (and thus the stitching process) from an automatically frame detected as the most reliable one i.e. one that contains sufficient salient information content to facilitate robust registration with the subsequent frames, and free of motion or out of focus blur. The vessel extraction method we just described is used to this end. In particular, following the computation of the vesselness image, the richness of salient information content is quantified by computing the entropy of this image. Input frames which do not contain many blood vessels, or which are corrupted by blur, will not produce many vessel detections, and thus result in low entropy vesselness images. Therefore, the frame with the highest entropy of the corresponding vesselness image is selected as the starting frame from which tracking commences, and as the initial synthetic image which is subsequently extended and improved though merging with newly processed frames. §.§ Image registrationRecall that our algorithm incrementally expands the synthetic image of the fundus by expanding it by (stitching with) newly processed frames. The first step in the stitching process concerns the registration of a frame with the synthetic image. This is achieved by using the extracted vessel feature images. Specifically, we adopt the use of normalized cross-correlation as the hypothesis matching criterion <cit.>, and exhaustively (though in a coarse to fine fashion, to speed up the process) explore the space of different scaling and translation parameters. The combination of the parameters which results in the highest normalized cross-correlation is accepted as the correct hypothesis but only if matching goodness exceeds that of the second best hypothesis (the second highest local normalized cross-correlation maximum) by a significant enough margin (herein we used a factor of 1.5). If this condition is not satisfied (which can happen, for example, when the newly acquired frame does not contain enough information to allow confident registration) the frame is discarded as unreliable. §.§ Incremental stitchingHaving registered a new frame with the growing synthetic image, the last step of our algorithm concerns the merging of the two images and with it the expansion of the synthetic result. There are several factors which make the process difficult. Firstly and as we noted before, there may be a significant illumination discrepancy between the two images. Moreover, a gradual darkening of the region of interest in the proximity of its boundary can be readily observed; note that this is a feature of the imaging process, rather than an artefact introduced by any of the steps of our algorithm. Lastly, the regions corresponding to glare areas, void of useful information, should be taken into account.Ideally, the result of the merging process should not only produce a seamless output image, spatially expanded (in general), but also leverage information content from both input images to increase the quality of information in the overlapping areas too. To achieve this while at the same addressing the challenges summarized above, we adopt an adaptive weighting strategy whereby the relative contributions of the two input images are adjusted depending on the particular locus under the consideration. To motivate out approach intuitively, the idea is to use preferentially data from an image in which a particular location is more central (relative to the region of interest) i.e. where the optical setup of mini ophthalmoscopies and the eye itself, produce better quality data (higher signal to noise ratio). We formalize this through the use of the distance transform <cit.>. Specifically, after a linear illumination transform of brightness which normalizes global changes, for each pixel in an image we compute its approximate distance from the boundaries of the region of interest (the outer boundary and, if present, the boundary of the glare region). Then, when combining two corresponding pixels their relative contributions are computed by considering the inverse value of their distances from the boundaries in the original images, i.e. the output value of a RGB channel c_x,y becomes:c_x,y= w_f c^f_x,y + w_s c^s_x,y/w_f w_swhere c^f_x,y is the value of the pixel in a new frame and c^s_x,y in the synthetic image, and:w_f = 1/d^f_x,y w_s = 1/d^s_x,ywhere d^f_x,y and d^s_x,y are the corresponding distance transform values (see Figure <ref>). The process can be therefore described as a form of adaptive linear fusion <cit.>.§ EVALUATIONThe data corpus we used in this work was collected from healthy volunteers. The acquisition process was entirely unaffected and unguided by us, guaranteeing realistic video input as would be collected in the field. Thus, we found that the sequences were highly variable in length, lasting from approximately 20 up to 90 seconds. The majority of the sequences start with a distant view of the eye (and hence little useful fundal information), and progressively close in onto the areas of actual interest such as blood vessels and the optic disc. Major motion and out of focus blur can be observed in this early phase. Once a section of the retina is clear the acquisition becomes more controlled, exhibiting purposeful motion in an effort to capture as much of the fundus as possible. Only a small section can be seen at any given moment due to the narrow field of view of direct ophthalmoscopy devices. Throughout this process blur continues to pose problems, with the retina easily going out of focus, drastic changes in the direction of motion occurring, and the imaging stopping unexpectedly. Considering the nature of our data set, it was not possible to obtain objective ground truth high quality images using a traditional ophthalmoscope. Hence, at this stage we limited ourselves to a qualitative assessment of the results. Representative examples of synthetic images and the effects of specific challenges are shown in Figures <ref> and <ref>.§ SUMMARY AND CONCLUSIONSThe Arclight is a recently developed low-cost direct ophthalmoscope that offers great potential for the prevention of blindness amongst those living in low and middle income countries. However, currently this potential is limited by the inherent optical limitations of direct ophthalmoscopy, most notably the narrow field of view. In this paper we described the first method that uses this low quality data to create a synthetic, higher quality, wider view image comparable to one acquired using expensive and bulky traditional retinal cameras.ieee | http://arxiv.org/abs/1708.07977v1 | {
"authors": [
"Keylor Daniel Chaves Viquez",
"Ognjen Arandjelovic",
"Andrew Blaikie",
"In Ae Hwang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170826142821",
"title": "Synthesising Wider Field Images from Narrow-Field Retinal Video Acquired Using a Low-Cost Direct Ophthalmoscope (Arclight) Attached to a Smartphone"
} |
Low Cost, Open-Source Testbed to Enable Full-Sized Automated Vehicle Research Austin Costley Chase Kunz Ryan Gerdes Rajniaknt Sharma ============================================================================= * State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, Zhejiang, China. * Channing Division of Network Medicine, Brigham and Women's Hospital and Harvard Medical School, Boston, MA 02115. * Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Boston, MA 02115. * School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287. * Department of Electrical Engineering, Princeton University, Princeton, NJ 08544.()AbstractCentrality is widely recognized as one of the most critical measures to provide insight in the structure and function of complex networks. While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different layers. This framework can leverage prior knowledge about the interplay among layers to better characterize multicentrality for varying scenarios. Two interesting cases are presented to illustrate how to model multilayer influence by choosing appropriate functions of interlayer influence and design algorithms to calculate eigenvector multicentrality. This framework is applied to analyze several empirical multilayer networks, and the results corroborate that it can quantify the influence among layers and multicentrality of nodes effectively.§ INTRODUCTIONCentrality quantifiesthe importance of nodes in a graph, and has been widely studied to understand the structure and function of complex networks<cit.>. For example, it can be utilized toidentify the most influential person in an online social network<cit.>, the most crucial artery in transport congestion<cit.>, or the most important financial institution in the global economy<cit.>. Over 30 differentcentrality measures (e.g., degree centrality, betweenness centrality, closeness centrality,eigenvector centrality, and control centrality) have been examinedin the literature<cit.>. Among these, eigenvector centrality, defined as the leading eigenvector of the adjacency matrix of a graph, has receivedincreasing attention<cit.>. It is worth noting that PageRank, a variant ofeigenvector centrality, is the primary algorithm used in Google's search engine<cit.>.Notably, most previous studies have focused on eigenvector centrality in a single-layer network, in which all nodes/links are assumed to be of the same type (centrality-homogeneous). As revealed recently<cit.>, many practical complex systems, ranging from the Internet to airline networks, have multiple typesof nodes and/orlinks between nodes. Multilayer networks, which consist of multiple layers of nodes with intra- and interlayer links, can be used to model suchcomplex systems. Figure <ref> shows two examples of multilayer networks (see Supplementary Fig. 1 for more examples). Simply aggregating a multilayer network into a single-layer one would obviously lead to a miscalculation of centrality. Recent work on eigenvector-like centrality in multilayer networks eitherassigned constant weights to predetermine interlayer influence (which can be regarded as the gain or loss of the interplay strength between two layers<cit.>), or focused on a special case of multilayer networks, i.e., the so-called multiplex networks (where all layers share the same set of nodes, andinterlayer links only exist between counterpart nodes)<cit.>. It is of significant interest to develop a framework for studying eigenvector-like centrality in general multilayer networks, hereafterreferred to as eigenvector multicentrality. In this study, we introduce a tensor-based framework that enables the quantification of the relationship between interlayer influence and eigenvector multicentrality. It is challenging to compute eigenvector multicentrality of nodes in such a framework sinceinterlayer influenceand eigenvector multicentrality are interdependent. We prove the existence and uniqueness of eigenvector multicentrality for given appropriate forms of interlayer influence. We also design efficient algorithms to calculate it for two interesting scenarios. This framework offers a novel approach formodeling and quantifying the interlayer interactions in multilayer networks, providing a systematic way of characterizing eigenvector multicentrality. Experimental results based on several real-world multilayer networks corroborate our analytical results.§ RESULTS §.§ A tensor-based framework for studying eigenvector multicentrality. In the calculation of eigenvector centrality of nodes in a single-layer network, a directed link to a node can be viewed asa vote of support. Each node fairly propagates its entire centrality score to its neighbors recursively.The eigenvector centrality of a node is defined as the scores that it gathersfrom its neighbors after appropriate normalization in the steady state. Formally, the vector consisting of the eigenvector centrality of all nodes is defined as the leading left eigenvector of the adjacency matrix associated with the single-layer network. We generalize this definition to multilayer networks by taking into account interlayer influence among layers. Specifically, a multilayer network is modeled as ℳ=(ℒ,ℰ), where ℒ={L_α;α=1,2,⋯, K} is a collection of graphs L_α=(V_α, E_α) representing layers in ℳ;V_α={v_1,α,v_2,α,⋯ v_n_α,α} is the set of nodes, where n_α denotes the number of nodes, and E_α is the set of intralayer links in layer α; ℰ={E_αβ⊆ V_α× V_β; α,β =1,2,⋯,K (αβ) } contains the interlayer links in ℳ<cit.>. To avoid confusion, we use Latin letters {i,j,⋯} to indicate nodes and Greek letters {α,β,⋯} to indicate layers. Tensors provide a general mathematical tool to describe high-dimensional objects<cit.>, and notably, tensors have been employed to study multilayer networks<cit.>. For example, a fourth-order tensor M_jβ^iα, called the adjacency tensor, is used to encode a directed, weighted link from node i in layer α to node j in layer β (see Supplementary Text I for further details on tensorial representations). We further introduce the influence tensor W^α_β, which is a second-order tensor measuring the interlayer influence from layer α to layer β. In our framework, the influence tensor W may be treated as a constant tensor when quantitative knowledge is available. The interaction tensor is defined as H^iα_jβ=W^α_β M_jβ^iα, encoding the interaction from node i in layer α to node j in layer β. Intuitively, when the influence fromlayer α to layer β is greater than one, the centrality scores propagating along the links from layer α tolayer β will be magnified,and vice versa. The second-order tensor Φ_iα is defined as the solution to the following tensorial equation:H_jβ^iαΦ_iα=λ_βΦ_jβ,where λ_β is a coefficient related to layer β, and the Einstein notation<cit.> is adopted here (see Supplementary Text I for further details). Because Φ is a eigenvector-like centrality, it will hereafter be referred toas eigenvector multicentrality, and Φ_iα represents the eigenvector multicentrality score of node i in layer α. Inthe calculation ofeigenvector multicentrality, after each node propagatesits entire multicentrality score to neighbors, the scores from layer α to layer β will be multiplied by the influence coefficient W^α_β.Hence, we can obtain the eigenvector multicentrality via appropriate normalization in the steady state. Notice that the normalizing coefficient λ_β may be different for different layers in a multilayer network.This differs from the eigenvector centrality in a single-layer network, where all nodes share a common normalizing coefficient λ_1 (namely, the leading eigenvalue of the adjacency matrix).Many existing models can be incorporated in our eigenvector multicentrality frameworkby choosing the influence tensor W appropriately. For instance, in an author-document heterogeneous network, unweighted interlayer links connect documents to their authors. A directed unweighted intralayer link exists between two documents if one document refers to the other and the undirected weighted intralayer link between two authors represents their social tie. The multicentrality in such an author-document network was defined as the leading eigenvector of a stochastic matrix, which encodes the probabilities that a random surfer moves along intra- and interlayer linksin a combined random walk process<cit.>. Clearly, this model can be incorporated in our framework by setting the intra- and interlayer influence weights to predetermined constants. Further, a definition of multicentrality in multiplex networks has been proposed in Ref. 20, considering the influence from counterpart nodes in other layers by importing the influence matrix Q=(q_αβ)∈ℝ^K× K. The influence matrix is non-negative, and q_αβ measures the influence of layer β on layer α. One can calculate eigenvector-like multicentrality of a multiplex network once Q has been obtained.Observe that in Ref. 20 the influence matrix Q is predetermined and given as in a lower-dimensional form (i.e., a matrix), whereas in our framework the influence tensor W depends on the eigenvector multicentrality and hence they are interdependent. Moreover, interconnected multilayer networks have been proposed to predict diffusive and congestion processes<cit.>, where the undirected unweighted interlayer links connect nodes to their counterparts in other layers.Eigenvector centrality in these networksis a special case of our framework where the interlayer influence is equal to one. §.§ Leveraging prior knowledge about interlayer interactions.Quantifying the influence tensor W is important to calculate multicentrality in our framework. As expected, the influence tensor W is typically a function of the adjacency tensor M and the multicentrality Φ, rather than being a constant. In practice, precisely predetermining the influence tensor is often infeasible. One advantage of our general framework lies in the leveraging of prior knowledge about the influence tensor W in diverse applications and thecalculation of multicentrality even when W and Φ are interdependent.Consider a typical scenario in which all nodes are centrality-homogeneous and layers are heterogeneous in a multilayer network. In such a scenario, the multicentrality scores of all nodes are comparable, and can be represented by a vector C∈ℝ^N, where N is the number of all nodes. We call the eigenvector multicentrality in such a scenarioglobal multicentrality.We normalize the multicentrality such that the vector C is defined over an N-dimensional simplex. For example, in a multilayer network consisting of web pages on different subjects (see the empirical results for further details), we may need to compare the multicentrality scores of two web pages on different subjects. Clearly, the multicentrality score propagates differently along interlayer links and along intralayer links, owing to differences in the popularity of different subjects.Here, we assume that the importance of layer α is a function of the multicentrality of all nodes in layer α, denoted by f(Φ_:α), where the colon “:" indicates all elements of a given dimension<cit.>.Notably, the function f(·) describing the layer importance is application-dependent. The function f could be, for example, the L^1-norm f(Φ_:α)=Φ_:α_1, which means the aggregated multicentrality scores of nodes in layer α, or it could be f(Φ_:α)=Φ_:α_1/n_α, which denotes the average multicentrality score over nodes in layer α. There are a variety of ways to define the influence tensor. In this article, we defineW^α_β=f(Φ_:α)/f(Φ_:β).That is to say, the interlayer influence between two layers depends on their relative layer importance. It is clear that there isno gain or loss for links between nodes in the same layer,because the interlayer influence W^α_α (α=1,2,⋯,K) is equal to one. Forlinks from a node in a more important layer, there is a gain in the multicentrality, and vice versa. Then the interaction tensor can be written asH^iα_jβ=f(Φ_:α)/f(Φ_:β)· M_jβ^iα. Further, we prove that λ_α=λ_1, ∀α∈{1,2,⋯, K}, and Eq. <ref> reduces toH_jβ^iαΦ_iα=λ_1Φ_jβ,where λ_1 is the leading eigenvalue of the interaction tensor H, and is irrelevant to β in this scenario. Notably, λ_1 is also the leading eigenvalue of the adjacency tensor M, indicating that the interaction tensor H maintains the leading eigenvalue of the adjacency tensor M (see Supplementary Text V for further details). From Eq. <ref>, we can see how multicentrality score propagates in a multilayer network. Considering that a node i in layer α links to another node j in layer β, node i will propagate its multicentrality score to node j scaled by an influence coefficient W^α_β, which could be a gain (W^α_β>1), a loss (W^α_β<1), or even (W^α_β=1), and the multicentrality Φ_jβ comprises the scores that node j in layer β gathers in the steady state. The existence and uniqueness of the multicentrality Φ are proved in Supplementary Text V.For a given function f, we can calculate the global multicentrality using the compressed power iteration method introduced in the Materials and Methods section. We consider another interesting scenario in which nodes in different layers areheterogeneous, and thus are not comparable. For example, in a heterogeneous network consisting of authors and papers, it is not meaningful to compare the multicentrality of an author with that of a paper. Because the multicentrality of nodes in different layers may have varying implications, we can only calculate the local multicentrality of nodes in each layer while taking into account the interlayer influence, where local multicentrality means that the nodes in each (local) layer are centrality-homogeneous. In such a scenario, the multicentrality score of a node cannot simply propagate along interlayer links to other layers. We measure the local multicentrality of nodes in each layer by defining the influence tensorin the framework asW^α_β=∑_i,j=1^NM^jβ_iαΦ_jβ/∑_i,j=1^NM^iα_jβΦ_iα.Notice that the denominator ∑_i,j=1^NM^iα_jβΦ_iα in W^α_β is a normalizing constant, which is the sum of scores propagating along interlayer links from layer α to layer β.Moreover, the sum of interactions from nodes in layer α to nodes in layer β is given by the numerator ∑_i,j=1^NM^jβ_iαΦ_jβ, which is the sum of scores propagating from layer β to layer α. Therefore, by defining the influence tensor in Eq. <ref>, we assume that the score flow going out of one layer will be returned to the layer. In such a way, we can calculate the local multicentrality of nodes in each layer independently while the interlayer influence is taken into account. The detailed proofs of the existence and uniqueness of Φ in local multicentrality are provided in Supplementary Text VI. We can also calculate the local multicentrality Φ numerically using the compressed power iteration method. Because the prior knowledge is network-specific and application-dependent, we present two interesting scenarios, for global multicentrality and local multicentrality respectively, to illustrate how to leverage prior knowledge to find W and compute the multicentrality of nodes. The PageRank algorithm has been now widely utilized in social, transportation, biology and information network analysis for link prediction, recommendation, etc.<cit.>. Note that PageRank centrality is a variant of eigenvector centrality. In PageRank centrality, each node distributes its PageRank score to its neighbors along outgoing links on an equal footing, and a node's PageRank score is defined as the sum of scores that it gathers from its neighbors in the steady state. Our eigenvector multicentrality framework can be easily carried over to characterize PageRank multicentrality (see Supplementary Text IV for further details about PageRank multicentrality).§.§ Multicentrality in empirical networks. We firstconsider a dataset from Wikipedia consisting of 4,604 web pages (see the Materials and Methods section for further details about this dataset and how it has been obtained). The web pages are divided by Wikipedia into 15 subjects, including art, business studies, citizenship, countries, design and technology, everyday life, geography, history, IT, Language and literature, mathematics, music, people, religion and science, and one web page belongs solely to one subject. We build a multilayer network by placing web pages of the same subject in the same layer, and establishing a directed link between two web pages if there is a hyperlink between them (a subnetwork is shown in Fig. 1A). For comparison, we also build a single-layer network by aggregating all web pages in all layers (see Supplementary Fig. 4 for further details). We select 4,128 web pages from these, in order to guarantee that the network is connected and all nodes have at least one out-degree and one in-degree.We measure the PageRank multicentrality of web pages in the constructed multilayer network. Moreover, because web pages in different subjects are centrality-homogeneous, global PageRank multicentrality is used. We consider three common forms for layer importance f(·): f_1(Φ_:α)=ln(1+N·|Φ_:α|_1/n_α), f_2(Φ_:α)=|Φ_:α|_∞ and f_3(Φ_:α)=|Φ_:α|_1/n_α. Table 1 shows the results of global PageRank multicentrality (f=f_1) in the Wikipedia multilayer network and PageRank centrality in the aggregated Wikipedia network, where the digits in parenthesis indicate the differences between these two rankings (see Supplementary Tables 1-3 for further details).Note that the entry “United States” has the largest multicentrality score, becauseit has the largest number of incoming links. Furthermore, theweb pages linked to it have high multicentrality scores. The entry “Europe” in the layer “Geography” has many enhanced links fromnodes in the layer “Countries”, because the average PageRank multicentrality score of nodes in “Countries” is higher than that in “Geography”. “France” haslarge numbers ofincoming links from entries in the layers “Art”, “Music” and other subjects; scores along these incoming links, however, will be diminished since “Art” and “Music” are less important than “Countries”. Note that the entry “Television” is significantly promoted, because the layer importance of “Design and Technology” is relatively low.Global multicentrality can effectively quantify the influence between layers even when we have limited prior knowledge, rather than aggregating all nodes without considering interlayer influence, as shown in many previous methods.Eigenvector multicentrality is an effective predictor for searching for the most important node in multilayer networks, which is also validated by the results obtained from the page views (PVs) in Wikispeedia. Wikispeedia is a human-computation game<cit.>, in which users are requested to navigate from a given web page to atarget one by only clicking on Wikipedia links.We collect all completed navigation paths and obtain the PVs of each web page from Wikispeedia. For all entries, we calculate their PageRank multicentrality scores, PageRank centrality and degree centrality scores, and compare them to their PVsin Wikispeedia (see Supplementary Tables 4-8 for further details).The results are shown in Figs. <ref> and <ref>. We also list the average PageRank multicentrality score of each layer (subjects) in Wikipedia (see Supplementary Tables 9-11 for further details). The Spearman rank correlationcoefficients show that PageRank multicentrality outperforms PageRank centrality and degree centrality in the aggregated network.Next, we consider a transportation network consisting of airports and air routes between them. We first consider 450 airports in Europe<cit.> (see the Materials and Methods section for details about this dataset and how it has been obtained), and we focus on three main airlines.We then build a multiplex network with three layers (airlines) and450 nodes (airports) in each layer as shown in Fig. <ref>B, where the dotted lines are interlayer links between airports and their counterparts in other layers. We measure the global PageRank multicentrality in the multiplex network using three forms of layer importance: f_1(Φ_:α)=e^|Φ_:α|_1/n_α-1, f_2(Φ_:α)=|Φ_:α|_1/n_α and f_3(Φ_:α)=ln(1+N·|Φ_:α|_1/n_α). Then, the PageRank multicentrality score of each airport is obtained by assembling the multicentrality scores of all its counterparts in all layers. For comparison, we also build a single-layer network, called the aggregated network, by combining the same airports in the three layers. We further introduce the versatility, a good predictor for diffusive and congestion processes in multilayer networks<cit.>, which is a special case of our framework when settingall componentsof the influence tensor W to one. We focus on the coverage ρ(t), a suitable proxy for the exploration efficiency of the network<cit.>, defined as the average fraction of distinct nodes being visited up to time t regardless of the layer,assuming that a walker starts from a certain node in the network (see the Materials and Methods section for further details about the coverage). We investigate whethermulticentrality helps understand the role that a node plays in dynamical scenarios. To this end, we compute the Spearman correlation coefficient between the ranking of the airports by multicentrality and that by the coverage at time t of a hypothetical epidemic spreading process that starts from a certain airport. For comparison, we also compute baselines such as the rankings by versatility and PageRank centrality in the aggregated network (see Supplementary Tables 12-14 for further details). We calculate the Spearman correlation coefficients for these five methods at each time step, and the results are shown in Fig. <ref>, where the time ranges from t=1 to t=4,000. It is shown that the three multicentrality measures achieve higher accuracy (theircorrelation coefficients exceed 0.947) in the steady state (t≥3,000). We then perform a similar analysis on an airline network from the U.S., which contains the airlines flying from the U.S. on Jan 3, 2008 (with data provided by the American Statistical Association Sections on Statistical Computing, <http://stat-computing.org/>).We build a multiplex network with 20 layers and284 nodes in each layer, and a similar conclusion can be drawn (see Supplementary Table 15 for further details).Another real-world example we consider is a social network, where we apply our framework to ane-mail network constructed from a large European research institution with 1,005 nodes (individuals) and 42 layers (departments), on which we consider an epidemic spreading process.The simulation results indicate that the nodes with higher eigenvector multicentrality play a more important role in theepidemic spreading process (see Supplementary Text VII for details).§ DISCUSSIONAs shown in recent work on eigenvector centrality (and its variants)<cit.>, it is of significant interest to build a framework for studying eigenvector-like centrality in multilayer networks. The existing studies, however, assumed empirical influence coefficients or relied on specific types of multilayer networks. Here we develop a general framework for studying eigenvector multicentrality in multilayer networks, which enables the quantification of the impact of interlayer influence on eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentrality propagates among different layers. Further, this framework can easily leverage prior knowledge about the interplay among layers to characterize eigenvector multicentrality for varying scenarios. As the interlayer influence and multicentrality of nodes are interdependent, they are jointly solved using a compressed power iteration method. Furthermore, we formulate and analyze PageRank multicentrality for practical applications within the proposed framework. We also perform theoretical analyses to prove the existence and uniqueness of the solutions in Supplementary Text V-VI, which allows us to calculate global and local multicentrality in any strongly connected multilayer networks.For an arbitrary multilayer network, we treat a dead end (a node with no outgoing links) the same as if it had outgoing links to all nodes, and introduce a damping factor to guarantee the existence and uniqueness of the solution. The results fromempirical networks demonstrate that our general framework can effectively quantify the interlayer influence, and eigenvector multicentrality is a good measure to identify important nodes from both structural and dynamical perspectives. Thus,multicentrality aids in understanding and predicting the behaviors of dynamic processes by leveraging network structure, and describes the structure-function relationship of multilayer networks well. We believe that the concept of multicentrality has the potential to enable a deep understanding of the structure and function of multilayer networks. Because the real-world scenarios of multilayer networks vary, a key step is to find appropriate forms of interlayer influence for theoretical analyses. Here weconsider two interesting scenarios, for global multicentrality and local multicentrality respectively. We believe that our proposed tensor-based framework can be applied tomore empirical networks in variousscenarios, including social networks, transportation networks, biological networks, etc. § MATERIALS AND METHODS §.§ Numerical solution.The crux of the proposed framework is to solve the tensorial equationH_jβ^iαΦ_iα=λ_βΦ_jβ,where the solution Φ isthe multicentrality tensor. To obtain the numerical solution, we first flatten the adjacency tensor M into a matrix, i.e., we represent the fourth-order tensor M∈ℝ^N× N× K× K as a matrix M∈ℝ^NK× NK, where M denotes the lower-dimensional form of the tensor M. Then we vectorize the second-order tensor Φ∈ℝ^N× K into a supravector Φ∈ℝ^NK and denote by W the matrix form of the influence tensor W (see Supplementary Text II for further details oftensor decomposition). Further, we denote by Λ=[λ_1,λ_2,⋯,λ_K]^⊤∈ℝ^Ka vector encoding thenormalizing coefficient in each layer. Thus, we obtain the matrix equation(W⊙M)^⊤·Φ=Λ⊙Φ,where ⊙ denotes the Khatri-Rao product and W=W(M,Φ) isa function of the matrix M and the multicentrality Φ. For the numerical solution, we propose a compressed power iteration method, whose iteration scheme is as follows:Φ^(k+1)=Φ^(k)+D^(k)[(W⊙M)^⊤(k)⊙Ω^(k)-E_N]Φ^(k),where Ω^(k)=[ω^(k)_1,ω^(k)_2,⋯,ω^(k)_K]∈ℝ^K is the normalizing vector. Denoting B^(k)=(W⊙M)^⊤(k)⊙Ω^(k)-E_N, we can write the iteration scheme asΦ^(k+1)=Φ^(k)+D^(k)B^(k)Φ^(k),where D^(k)∈ℝ^N× N is a diagonal matrix related to B^(k), and D^(k) compresses the induced infinity norm of the matrix B^(k) such that the L^1-norm of each row in D^(k)B^(k) is strictly less than one.Specifically, we let ω^(k)_γ=Φ_γ^(k)_1^-1. With regard to the global multicentrality, the vector Λ contains equivalent elements, i.e., Λ=[λ_1,λ_1,⋯,λ_1]^⊤. Thus we have Ω^(k)=[ω^(k)_1,ω^(k)_1,⋯,ω^(k)_1]^⊤, where ω^(k)_1=Φ^(k)_1^-1. Further, we can specify the diagonal matrix D^(k) to compress the induced infinity norm of matrix B^(k), whereD^(k) is not unique in practice. For example, we could takeD^(k)=( diag{(B^(k)+E_N)·1_N× 1})^-1. Then for each Φ^(k)>0 (i.e., all the components in Φ^(k) are positive), the matrix [E_N+D^(k)B^(k)] is strictly diagonal dominant with positive elements. Hence, the iterations in Eq. <ref> converges to a unique solution<cit.> and this solution satisfies the matrix Eq. <ref> (see Supplementary Text VIII for further details). §.§ Multilayer network ofWikipedia.The Wikipedia datasetcontains 4,604 entries and 119,882 hyperlinks<cit.> (data provided by the Stanford Network Analysis Project, <http://snap.stanford.edu/index.html>). To ensure connectivity, we select the entries that have at least one out-degree and one in-degree. Then 4,128 entries and 113,441 links are obtained and these entries are divided into 15 subjects by Wikipedia. Resultantly, we can build a multilayer network with N=4,128 nodes and K=15 layers, where each layer contains the entries of a single subject. The adjacency tensor M ∈ℝ^N× N× K× K encodes the directed and unweighted links of the multilayer network and the influence tensor W ∈ℝ^K× K encodes the interlayer influence between any two layers. We measure the interlayer influenceW^α_β=f(Φ_:α)/f(Φ_:β),where f(Φ_:α) indicates the layer importance of layer α. Here, we consider three forms of layer importance: f_1(Φ_:α)=ln(1+N·|Φ_:α|_1/n_α), f_2(Φ_:α)=|Φ_:α|_∞ and f_3(Φ_:α)=|Φ_:α|_1/n_α. Further, we obtain the interaction tensor H ∈ℝ^N× N× K× K as H^iα_jβ=W^α_β M_jβ^iα. Following the construction of the interaction tensor H, we then solve the tensorial equationH_jβ^iαΦ_iα=λ_1Φ_jβusing the compressed iteration method. Finally, the multicentrality tensor Φ is in the space ℝ^N× K, andΦ_iα represents the multicentrality ofnode i inlayer α. §.§ Multilayer network of the European airlines.The European airline network contains450 airports in Europe and the air routes for 37 airlines (see Ref. 34 for more details about this dataset). For each airline, we can build a network with N=450 nodes and a set of links representing routes betweenairports. We select those airlines with the number of air routes greater than N/2, such that the average degree for each nodein the constructed network is at least one.In this way, we obtain three main airlines: Ryanair, Lufthansa and Easyjet. We thus have a multiplex network with K=3 layers and 450 nodes in each layer. Then, we interconnectthe same airport across layers, obtaining a three-layer multiplex network. The adjacency tensor M is in the space ℝ^N× N× K× K. In the context of themultiplex network, M_jα^iα encodes the undirected and unweighted intralayer links in layer α, while M_jβ^jα=1 encodes the undirected and unweighted interlayer link for node j between layer α and layer β. For the interlayer influence, we again consider three forms of the layer importance: f_1(Φ_:α)=e^|Φ_:α|_1/n_α-1, f_2(Φ_:α)=|Φ_:α|_1/n_α and f_3(Φ_:α)=ln(1+N·|Φ_:α|_1/n_α). After obtaining the influence tensor W viaEq. <ref>, we have the interaction tensor H^iα_jβ=W^α_β M_jβ^iα in the space ℝ^N× N× K× K. Finally, we solve the tensorial Eq. <ref> using the compressed iteration method and obtain the multicentrality tensor Φ_iα.With respect to the coverage ρ(t), we haveρ(t)=1-1/N^2∑^N_i,j=1δ_i,j(0)exp[-P_j(0)ℙE_i^⊤],where δ_i,j(0)=0 for j=i, and δ_i,j(0)=1 otherwise. Here, P_j(0) represents the supravector of probabilities at time t=0 (assuming that the walker starts at node j) and the matrix ℙ indicates the probability of reaching each node through any path of length 1,2,⋯,t+1. Furthermore, E_i=(e_i,e_i,⋯,e_i) is the supravector in which e_i is the ith canonical row vector (see Ref. 35 for details about the derivation of Eq. <ref>).§ SUPPLEMENTARY INFORMATION §.§ Supplementary Text§.§.§ I. Tensorial representationsWe first introduce a mathematical formulation of multilayer networks<cit.>, in which tensorial representations are adopted. The column vector e(s)=(0,…,0,1,0,…,0)^⊤ is a canonical vector in the vector space ℝ^N, in which the sth component is equal to one and others are zeros. In a network with N nodes, to describe the relations between any two nodes, the Kronecker product ⊗ is adopted. Note that E(st)=e(s)⊗e^⊤(t) is a second-order canonical tensor in the space ℝ^N× N. Given the intensity u_st of the link between nodes s and t, the tensor U∈ℝ^N× N encodes all linksin the network, whereU=∑_s,t=1^Nu_stE(st).Next, we follow the tensorial notation in which a^i (i=1,2,…,N) represents the ith component of a contravariant vector (column vector) a∈ℝ^N while a_j(j=1,2,…,N) represents the jth component of the corresponding covariant vector (row vector) a^⊤∈ℝ^N. We use (s) to indicate the sth vector, and (st) to denote the (st)-th tensor. Further,i and j are used to represent the corresponding components of a certain vector or tensor. The adjacency tensor U can be written as a 1-covariant and 1-contravariant tensorU_j^i=∑_s,t=1^Nu_ste^i(s)e_j(t)=∑_s,t=1^Nu_stE^i_j(st),where E^i_j(st) indicates the component of E(st) in the ith row and jth column. In multilayer networks, there may be various intra- and interlayer links. We further introduce the second-order adjacency tensor C^i_j(θϕ) (θ,ϕ=1,2,…,K)to describe them. To avoid confusion, Latin lettersindicate nodes in networks and Greek letters representlayers in the multilayer networks, the same as described in the article. Clearly, C^i_j(θθ) encodes the intralayer links inlayer θ, and C^i_j(θϕ) encodes the interlayer links from layer θ to layerϕ. Further, u_st(θϕ) indicates the intensity from node s inlayer θ to node t inlayer ϕ. Because of the varying influence between layers,we use a canonical basis of layers. The vector e^⊤(θ)∈ℝ^K(θ=1,2,…,K) is the θth covariant canonical vectorand e^α(θ) represents the αth component of e^⊤(θ), while e_β(θ) denotes the βth component of the corresponding contravariant canonical vector e_θ. The set of all second-order tensors E^α_β(θϕ)=e^α(θ)e_β(ϕ) forms the canonical basis of the space ℝ^K× K. The multilayer adjacency tensor M^iα_jβ can be represented by the tensor product of C^i_j(θϕ) and E^α_β(θϕ): M^iα_jβ= ∑_θ,ϕ=1^K C^i_j(θϕ)E^α_β(θϕ)= ∑_θ,ϕ=1^K∑_s,t=1^Nu_st(θϕ)π^iα_jβ(stθϕ),where the fourth-order tensor π^iα_jβ(stθϕ)=e^i(s)e_j(t)e^α(θ)e_β(ϕ) forms the canonical basis in the space ℝ^N× N× K× K.Introducingw_αβ to indicate the interlayer influence fromlayer α tolayer β, the tensor W∈ℝ^K× K encodes the interlayer influence between any two layers inmultilayer networks, where W can be written as the 1-covariant and 1-contravariant tensorW^α_β=∑_θ,ϕ=1^K w_αβe^α(θ)e_β(ϕ)=∑_θ,ϕ=1^K w_αβE^α_β(θϕ).Finally, we introduce the fourth-order tensor H, called the interaction tensor,in terms of thetensor productH^iα_jβ= W^α_β· M^iα_jβ= ∑_θ,ϕ=1^K∑_s,t=1^Nu_st(θϕ)w_αβπ^iα_jβ(stθϕ),which encodes the interactions between any two nodes in the multilayer networks.To simplify the notation, the Einstein summation convention isadopted. Repeated indices in a term, where one index is a subscript and the other is a superscript, denote the summationover all indices, which is also calleda contraction in tensorial operations. For example, we use the convention in the following equations:A_i^i =Σ_i=1^NA_i^i,A_j^i B^j_i =Σ_i=1^NΣ_j=1^NA_j^i B^j_i,A_jβ^iα B_iα =Σ_i=1^NΣ_α=1^K A_jβ^iα B_iα.This convention can be adopted for the product of any number of tensors with arbitrary order.§.§.§ II. Tensor decompositionTensors can be decomposed into lower-dimensional objects like matrices and supravectors<cit.>. We first flatten the adjacency tensor M into a matrix, i.e., we represent the fourth-order tensor M∈ℝ^N× N× K× K by a matrix M∈ℝ^NK× NK. The overline here allows us to distinguish a tensor from its lower-dimensional form, such as a matrix. Thus, the matrix M is a block matrixM= [ (@,20pt,20pt)c.c.c.cc.c.c.cM_11 M_12 ⋯ M_1K M_21 M_22 ⋯ M_2K ⋮ ⋮ ⋱ ⋮ M_K1 M_K2 ⋯ M_KK ],where the block M_αβ∈ℝ^N× N encodes the links fromlayer α to layer β.We can vectorize the second-order tensor Φ_iα∈ℝ^N× K into a supravector Φ∈ℝ^NK:Φ= [ (@,20pt,20pt)cc.c.c.cΦ_1 Φ_2 ⋮ Φ_K ],where Φ_α∈ℝ^N encodes the multicentrality of nodes inlayer α.Denoting the matrix form of the influence tensor W by W∈ℝ^K× K, we haveW= [ (@,20pt,20pt)c.c.c.cc.c.c.c w_11w_12 ⋯w_1Kw_21w_22 ⋯w_2K ⋮ ⋮ ⋱ ⋮w_K1w_K2 ⋯w_KK ],where element w_αβ quantifies the interlayer influence from layer α tolayer β. Using these decompositions, the tensorial equationH_jβ^iαΦ_iα=λ_βΦ_jβbecomes amatrix equation [ (@,20pt,20pt)c.c.c.cc.c.c.c w_11·M_11w_12·M_12 ⋯w_1K·M_1Kw_21·M_21w_22·M_22 ⋯w_2K·M_2K ⋮ ⋮ ⋱ ⋮w_K1·M_K1w_K2·M_K2 ⋯w_KK·M_KK ]^⊤·[ (@,20pt,20pt)cc.c.c.cΦ_1 Φ_2 ⋮ Φ_K ]= [ (@,20pt,20pt)cc.c.c.cλ_1·Φ_1 λ_2·Φ_2 ⋮ λ_K·Φ_K ]. We further define the vector Λ=[λ_1,λ_2,⋯,λ_K]^⊤∈ℝ^K. Then the Eq. <ref> can be written as(W⊙M)^⊤·Φ=Λ⊙Φ,where ⊙ denotes the Khatri-Rao product.In the calculation of the global multicentrality in multilayer networks, we takew_αβ=f(Φ_α)/f(Φ_β),and we prove that there exists a unique normalized supravector Φ satisfying(W⊙M)^⊤·Φ=λ_1·Φ,where λ_α=λ_1 (α=1,2,⋯,K).In the calculation of the local multicentrality in multilayer networks, we takew_αβ=∑_i,j=1^NM^jβ_iαΦ_jβ/∑_i,j=1^NM^iα_jβΦ_iα,and we also prove that there existsa unique normalized supravector Φ satisfying(W⊙M)^⊤·Φ=Λ⊙Φ. It is possible that different layers have different numbers of nodes, resulting in different tensor size. To maintain the unified tensor framework, we need to expand the tensor M and Φ to a specific size by adding zeros for nodes which are not in the current layer. Thus, after tensor decomposition, many rows and the corresponding columns only have zero elements in the matrix M, and the corresponding multicentrality Φ_iα are also zeros. To reduce the computational complexity in practice, we can omit these rows and columns in the matrix M and the corresponding elements in the supravector Φ.§.§.§ III. PageRank centrality in a single-layer networkConsider a graph G=(V,E), where V={v_1,v_2,⋯,v_N} is the set of nodes and E is the set ofdirected links. We have the adjacency matrix A=(a_ij), wherea_ij=1, if (i,j)∈ E0, otherwise.Further, we construct a stochastic matrix T=(t_ij), which is the transposition of the adjacency matrix A after normalization, i.e.,t_ij= 1/k^out_j, if (j,i)∈ E0, otherwisewhere k^out_j is the out-degreeof node j. The PageRankcentrality C∈ℝ^N<cit.> is the normalized leading right eigenvector of T, which corresponds to the leading eigenvalue λ_1=1, i.e., T· C=1· C. Next, we will briefly prove the existence and uniqueness of C. We first assume that the graph G is strongly connected.Clearly, the summation of each column of T is equal to one, and T has a left eigenvector 1_1× N, whose components are all equal to one, corresponding to the case of eigenvalue λ=1.Since ∑_i|t_ij|=1, all eigenvalues satisfy |λ_j|≤1 according to Gershgorin's circle theorem<cit.>. Furthermore,the matrix T is irreducible since the graph G is strongly connected. Hence, the leading eigenvalue λ_1 of T is uniqueaccording to the Perron-Frobinus theorem<cit.>, leading to the fact that the normalized leading eigenvector C is unique.When the graph G is not strongly connected, the following two steps can be employed to guaranteethe existence and uniqueness of the PageRank centrality. At the first step, find all dead ends (nodes with no outgoing links) in the graph G, and add outgoing links from them to all nodes. Since we add outgoing links between a dead end and all nodes in the graph, such an operation will not change the relative centrality of other nodes<cit.>. At the second step, introduce a damping factor d ranging between zero and one<cit.>, andthe corresponding stochastic matrix T_d becomesT_d=d· T+1-d/N·1_N× N.This makesT_d irreducible, and thus the leading eigenvalue λ_1 of T is unique<cit.>.The PageRank centrality can be interpreted by two models: the flow model<cit.> and the random walk model<cit.>. In the flow model,the PageRank score is regarded as a kind of negotiable flow. At each iteration, the current flow (score) of each node is distributed through its outgoing links on an equal footing to its neighboring nodes. In the steady state, the flow that each node gains is its PageRank score. In the random walk model, we consider that a random walkerstarts at a certain node and randomly chooses an outgoing link to walk iteratively. In this context, the PageRank score of each node is the probability of thewalker resting on the node in the steady state.§.§.§ IV. PageRank multicentrality in multilayer networksIn multilayer networks, we define the stochastic tensor T_jβ^iα=M_jβ^iα/∑_j,βM_jβ^iα such that v^jβT_jβ^iα=v^iα, where v is the tensor with all components equal to one. Further, we define the interaction tensor as H_jβ^iα=W^α_β· T_jβ^iα. It follows from Eq. <ref> that the leading eigentensor of H, denoted by Φ_iα, isthe PageRank multicentrality of node i in layer α, i.e.,H_jβ^iαΦ_iα=1·Φ_jβ.Also, the influence tensor W can be a function of the tensor T and centrality Φ. It is noteworthy that in PageRank multicentrality, the leading eigenvalue λ_1 of the interaction tensor is equal to one in accordance with PageRank centrality in a single-layer network. As before, a damping factor d ∈ [0,1] should be introduced to guarantee the existence and uniqueness of global and local PageRank multicentrality when the multilayer networks are not strongly connected (see Text III of the SI above for further details).Similarly, to characterize global PageRank multicentrality, we can choose the influence tensor as W^α_β=f(Φ_:α)/f(Φ_:β). In particular, after a node in layer α equally distributes its multicentrality score to its neighbors, the scores propagating along intralayer links stay unchanged, while the scores propagating along interlayer links to nodes in layer β will be scaled by a factor f(Φ_:α)/f(Φ_:β). In this case, we prove that the leading eigenvalue of H is equal to 1 (see Text V of the SI below for further details about the theoretic analysis). For clarification, we providea toy example in Supplementary Fig. <ref>. In local PageRank multicentrality, we specify the influence tensor asW^α_β=∑_i,j=1^NT^jβ_iαΦ_jβ/∑_i,j=1^NT^iα_jβΦ_iα.Likewise, we prove that the leading eigenvalue of H remains one (see Text VI of the SI below for the proof). When a node in layer α propagates its multicentrality score to nodes in layer β, the interactions depend on the scores from nodes in layer βto nodes inlayer α. In other words, the scores leaving from layer α will be returned to layer α. We provide an illustration of local PageRank multicentrality in Supplementary Fig. <ref>. In the random walk model, the interactiontensor H^iα_jβ encodes the transition probability of node i in layer α leaving for any node j in layer β. We can interpret the local multicentrality by a random walk process over a multilayer network as follows. If a walker starts at a certain node in layer α, she will walk along the intralayer links normally according to the transition probability. However, once the walkerleaves fromlayer α to any other layer along the interlayer links, she will not rest on nodes in other layers until she goes back to layer α. The local PageRank multicentrality score of each node in layer α is the probability of such a walker resting on the node in the steady state.§.§.§ V. Theoretical demonstration of global multicentrality Theorem 1: Given a strongly connectedmultilayer network ℳ and a function f:ℝ^N_+→ℝ_+, we denote the eigenvector centrality in the corresponding single-layer network by Θ. Then, a unique global eigenvector multicentrality Φ exists if and only if, for ∀ Θ_:α, Θ_:β (α,β=1,2,⋯,K), there exists only one y_αβ∈ (0,+∞) such that f(y_αβ·Θ_:β)=1/y_αβ· f(Θ_:α).Proof of Theorem 1. (Sufficiency.) First, we will prove the existence of Φ. Because the multilayer network is strongly connected,there is a unique normalized leading eigentensor Θ (i.e., the eigenvector centrality in the corresponding single-layer network), satisfyingM_jβ^iαΘ_iα=λ_1Θ_jβ,which we have stated in Text III. Then for Θ_:1∈ℝ^N and Θ_:γ∈ℝ^N (γ =1,2,⋯,K), there exists only one y_1γ∈ (0,+∞) such that f(y_1γ·Θ_:γ)=1/y_1γ· f(Θ_:1). Let Φ_:γ=y_1γ·Θ_:γ (γ=1,2,⋯,K). We have∑_α=1^K∑_i=1^NH^iα_jβΦ_iα= ∑_α=1^K∑_i=1^N(w_αβ M^iα_jβ) · (y_1α·Θ_iα)= y_1β∑_α=1^K∑_i=1^N M^iα_jβΘ_iα= λ_1Φ_jβ. Then, we will prove the uniqueness of Φ by contradiction. Suppose there exist two tensors Φ' and Φ” such that ∑_i,αΦ'_iα=∑_i,αΦ”_iα andH_jβ^iαΦ'_iα=λ_1Φ'_jβ,H_jβ^iαΦ”_iα=λ_1Φ”_jβ.For Φ'_iα, we have∑_i,αH_jβ^iαΦ'_iα =λ_1·Φ'_jβ ∑_i,αf(Φ'_:α)/f(Φ'_:β)M_jβ^iαΦ'_iα =λ_1·Φ'_jβ ∑_i,αM_jβ^iα· f(Φ'_:α)Φ'_iα =λ_1· f(Φ'_:β)Φ'_jβ.Similarly for Φ”_:α, wehave∑_i,αM_jβ^iα· f(Φ”_:α)Φ”_iα =λ_1· f(Φ”_:β)Φ”_jβ.Due to the uniqueness of the leading eigentensor Θ of M, we have f(Φ'_:α)·Φ'_iα=f(Φ”_:α)·Φ”_iα=k·Θ_:α, andΦ'=Φ”. This is in contradiction with our hypothesis. Thus, the solution Φ is unique.(Necessity.) We proceed to prove the necessity of Theorem by contradiction. Consider a strongly connectedmultilayer network ℳ and a function f:ℝ^N_+→ℝ_+. Suppose that a unique global eigenvector multicentrality Φ exists. Further, we suppose that ∃ Θ_:μ,Θ_:ν, and either of the following two conditions is satisfied: 1) ∄ y_μν such that f(y_μν·Θ_:ν)=1/y_μν· f(Θ_:μ); 2)∃ y'_μν,y”_μν∈ (0,+∞) such that f(y'_μν·Θ_:ν)=1/y'_μν· f(Θ_μ) and f(y”_μν·Θ_ν)=1/y”_μν· f(Θ_μ). By hypothesis, there isa unique global eigenvector multicentrality Φ∈ℝ^N× K for the tensor M, such thatH_jβ^iαΦ_iα=λ_1Φ_jβ,i.e.,∑_i,αH_jβ^iαΦ_iα =λ_1·Φ_jβ ∑_i,αf(Φ_:α)/f(Φ_:β)M_jβ^iαΦ_iα =λ_1·Φ_jβ ∑_i,αM_jβ^iα· f(Φ_:α)Φ_iα =λ_1· f(Φ_:β)Φ_jβ.Then we have f(Φ_:μ)·Φ_:μ=k·Θ_:μf(Φ_:ν)·Φ_:ν=k·Θ_:ν.Further, there exist y_μμ>0 and y_μν>0 such that y_μμ/y_μν is a constant, and the following two equations hold: Φ_:μ=y_μμ·Θ_:μ Φ_:ν=y_μν·Θ_:ν.Without loss of generality, we let y_μμ=1, i.e., Φ_:μ=Θ_:μ. Thus, there exists only one y_μν>0 such thatf(Φ_:μ)=f(Φ_:ν)· y_μν=k,i.e.,f(y_μν·Θ_:ν)=1/y_μν· f(Θ_:μ),which is in contradiction with the two conditions of the hypothesis. ▪ Note that Φ_:α represents the eigenvector multicentrality scores of nodes in layer α, Θ_:β represents the eigenvector centrality scores of nodes in layer β, and Φ_:β=y_αβ·Θ_:β represents the eigenvector multicentrality of nodes in layer β. Thus, y_αβ is a scale factor comparing the eigenvector multicentrality in a multilayer network with the eigenvector centrality in the corresponding single-layer network. If y_αβ<1, the eigenvector multicentrality score of nodes in layer β is relatively less important than the eigenvector centrality score in the corresponding single-layer network when taking layer α as a reference (Φ_:α=Θ_:α), and vice versa. Hence, it is necessary to guarantee the existence anduniqueness of y_αβ for any Θ_:α and Θ_:β as the theorem states. In fact, the condition in the theorem requires that the slope of the function fhas a lower bound. So, if the function f is a nonnegative monotonic increasing function such as the L^p-norm, the condition in the theorem is satisfied since the slope ofamonotonic increasing function is always positive. In addition, from the proof above, we can see that λ_1 is the leading eigenvalue of the tensor M, satisfyingM_jβ^iαΘ_iα=λ_1Θ_jβ.Also, λ_1 is the leading eigenvalue of the tensor H, satisfyingH_jβ^iαΦ_iα=λ_1Φ_jβ. In the global PageRank multicentrality, the leading eigenvalue of the tensor M is equal to one. Clearly, the leading eigenvalue of the tensor H is also equal to one.§.§.§ VI. Theoretical demonstration of local multicentrality Theorem 2: Given a strongly connected multilayer network ℳ and influence matrix W described byw_αβ=∑_i,j=1^NM^jβ_iαΦ_jβ/∑_i,j=1^NM^iα_jβΦ_iα,thelocal eigenvector multicentrality Φ has a unique normalized solution to the equationH_jβ^iαΦ_iα=λ_βΦ_jβ. Proof of Theorem 2. Weflatten the interaction tensor H into a matrix H∈ℝ^NK× NK, and vectorize the second-order tensor Φ∈ℝ^N× K into a vector Φ∈ℝ^NK. We will show thatH·Φ is a contraction mapping of Φ. To this end, we first suppose that the adjacency tensor M satisfies v^jβM_jβ^iα=v^iα, where v is the tensor with all components equal to one, i.e., the case of local PageRank multicentrality. Consider two different vectors Φ'_γ and Φ”_γ. For any given Φ_η (η≠γ), we supposeΦ'(γ)= [ (@,20pt,20pt)cc.c.c.c.c.cΦ_1 Φ_2 ⋮ Φ'_γ ⋮ Φ_M ]∈ℝ^N,Φ”(γ)= [ (@,20pt,20pt)cc.c.c.c.c.cΦ_1 Φ_2 ⋮ Φ”_γ ⋮ Φ_M ]∈ℝ^N.Then|H·Φ'(γ)-H·Φ”(γ)|= |M_γγΦ'_γ+∑_η≠γM_ηγ·Φ'_η_1/M_γη·Φ'_γ_1·M_γη·Φ'_γ-M_γγΦ”_γ-∑_η≠γM_ηγ·Φ”_η_1/M_γη·Φ”_γ_1·M_γη·Φ”_γ| ⩽ p_γ· |Φ'_γ-Φ”_γ|,where p_γ is given byp_γ=max_i,k∑_jM_γγ(i,j)-M_γγ(k,j)_1<1.Thus, for any positive vectors Φ' and Φ”,|H·Φ'-H·Φ”|⩽max_γ{p_γ}· |Φ'-Φ”|.This implies that the function H·Φ is a contraction mapping of Φ. According to the Banach fixed-point theorem<cit.>, H admits a unique fixed-point Φ, satisfyingH·Φ=Φ,from which we find that the eigenvalue of H_jβ^iα remains one in the local PageRank multicentrality analysis. For the general case, we can construct an analogous contraction mapping. ▪§.§.§ VII. Epidemic spreading inmultilayer networks We first introduce epidemic spreading process in a single-layer network. There are many models describing epidemicspreading process, such as the susceptible-infected (SI) model, the susceptible-infected-susceptible (SIS) model, the susceptible-infected-recovered (SIR) model and so on<cit.>. Here we take the SI model as an illustration.In an undirected single-layersocial network, nodes denote individuals and links indicate their social relations. We study the spreading of a disease in such a social network. Each node of the network has onlytwo states, “susceptible” (S) or “infected” (I). A susceptible node is a temporarily healthy node, which can be infected by infected nodes, while an infected node will stay in the infected state. At each step, an infected node will infect the susceptible nodes connected to it with probability μ, which is called the transition rate. Clearly, if a susceptible node has more infected neighbors, it will have a higher probability of being infected. Since the infected nodes will notrecover, all nodes will eventuallybe infectedif the network is strongly connected and μ>0.In multilayer networks, nodes lie in different layers. The transition rate can be different along intralayer links and interlayer links. Here we consider a datasetfrom a large European research institute<cit.> (data provided by the Stanford Network Analysis Project, <http://snap.stanford.edu/index.html>). There are 1,005 individuals, each belonging to exactly one of 42 departments at the research institute. A multilayer network is built withK=42 layers indicating departments, and N=1,005 nodes representing individuals. There is an undirected link between two nodes if either one of them has sent at least one email to the other, and the connections are encoded by the adjacency tensor M ∈ℝ^N× N× K× K. We then consider an epidemic spreading process in this multilayer network. Considering a certain node as the infection source, we set the intralayer transition rate to 0.05 and the interlayer transition rate to 0.03 at eachstep<cit.>.Since we already have prior knowledge about interlayer influence, the influence tensor W ∈ℝ^K× K is given as follows:W^α_β=0.05, α=β0.03, α≠β. Then, we obtain the interaction tensor H ∈ℝ^N× N× K× K, and the multicentrality tensor Φ∈ℝ^N× K by solving the tensorial equationH_jβ^iαΦ_iα=λ_1Φ_jβ.We simulate the spreading process in this multilayer network with only one infected node in the initial state. At eachstep, the infected node/nodeswill infect its/their susceptible neighboring nodes at the transition rate, and the rates are different between intralayer links and interlayer links as we mentioned above. All nodes will be infected as long as there is a path between any two nodes in the multilayer network. We focus on the spreading coverage, i.e., the percentage of the infected nodes after each step. The results are shown in Supplementary Fig. <ref>, indicating that the nodes with higher eigenvector multicentrality play a more important role in the process of epidemic spreading in a multilayer network.§.§.§ VIII. Theoretical demonstration of the numerical solution Here we prove that the iteration scheme in the Materials and Methods section,Φ^(k+1)=Φ^(k)+D^(k)[(W⊙M)^⊤(k)⊙Ω^(k)-E_N]Φ^(k),converges to the solution to the matrix equation(W⊙M)^⊤·Φ=Λ⊙Φ. Proof. We first consider the global multicentrality. The iteration scheme Eq. <ref> becomesΦ^(k+1)=Φ^(k)+D^(k)[(W⊙M)^⊤(k)/λ_1^(k)-E_N]Φ^(k),and the matrix Eq. <ref> becomes(W⊙M)^⊤·Φ=λ_1⊙Φ.In Text V of the SI above, we have proved that there exists a unique solution to Eq. <ref>. We denote the unique solution by Φ^* and the corresponding eigenvalue by λ_1^*. The convergence of the iteration scheme Eq. <ref> has been demonstrated in the Materials and Methods section. We denote the limit of Φ^(k) by Φ^(∞). The primary task is to prove thatΦ^(∞) is equal to Φ^*. Clearly,Φ^(∞) =Φ^(∞)+D^(∞)[(W⊙M)^⊤(∞)/λ_1^(∞)-E_N]Φ^(∞)0 =D^(∞)[(W⊙M)^⊤(∞)/λ_1^(∞)-E_N]Φ^(∞).As the matrix D^(∞) is a diagonal matrix with non-zero diagonal elements, it is invertible. Thus[(W⊙M)^⊤(∞)/λ_1^(∞)-E_N]Φ^(∞) =0(W⊙M)^⊤(∞)/λ_1^(∞)·Φ^(∞) =Φ^(∞).Therefore, we have(W⊙M)^⊤(∞)·Φ^(∞)=λ_1^(∞)·Φ^(∞).Note that Φ^(∞) satisfies Eq. <ref> and has a unique solution Φ^*. Thus,we have Φ^(∞)=Φ^* and λ_1^(∞)=λ_1^*. Similarly, we can prove the convergence of the iteration scheme for local multicentrality. ▪ §.§ Supplementary Figures §.§ Supplementary Tables § ACKNOWLEDGEMENTS.We thank Manlio De Domenico for sharing the processed dataset of the European airline network and the details for calculating the versatility. This work wassupported by the National Natural Science Foundation of China (Grant No. 61731004) and the Zhejiang Natural Science Foundation (Grant No. LR16F020001).This work was supported in part by the U.S. Army Research Office under Grant W911NF-16-1-0448 and DTRA under Grant HDTRA1-13-1-0029.§ AUTHOR CONTRIBUTIONS.S.H., J.C., J.Z., and H.V.P. conceived and designed the project. M.W. conducted all the theoretical analysis and empirical calculations. All authors analyzed the results. M.W., S.H., J.Z., and J.C. wrote the manuscript. Y.-Y.L., J.Z., H.V.P., and Y.S. edited the manuscript. §.§ Competing Interests.The authors declare no competing financial interests.§ REFERENCES | http://arxiv.org/abs/1708.07763v3 | {
"authors": [
"Mincheng Wu",
"Shibo He",
"Yongtao Zhang",
"Jiming Chen",
"Youxian Sun",
"Yang-Yu Liu",
"Junshan Zhang",
"H. Vincent Poor"
],
"categories": [
"physics.soc-ph",
"cs.SI"
],
"primary_category": "physics.soc-ph",
"published": "20170825144736",
"title": "A Tensor-Based Framework for Studying Eigenvector Multicentrality in Multilayer Networks"
} |
[pages=1-last]SANE2017-30.pdf | http://arxiv.org/abs/1708.07920v1 | {
"authors": [
"Hidetoshi Furukawa"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170826024409",
"title": "Deep Learning for Target Classification from SAR Imagery: Data Augmentation and Translation Invariance"
} |
http://arxiv.org/abs/1708.07856v3 | {
"authors": [
"Jason Iaconis",
"Chunxiao Liu",
"Gábor B. Halász",
"Leon Balents"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170825182915",
"title": "Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice"
} |
|
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Bauman Moscow State Technical University, 5 stroenie 1 2-ya Baumanstaya st., Moscow, 105005, Russia In this paper, we present the project of the heliogyro solar sail unit for deployment of CubeSat constellation and satellite deorbiting. The ballistic calculations show that constellation deployment period can vary from 0.18 years for 450km initial orbit and 2 CubeSats up to 1.4 years for 650km initial orbit and 8 CubeSats. We also describe the structural and electrical design of the unit and consider aspects of its integration into a standard CubeSat frame. Nanosatellite aerobrake maneuvering device Nikita V. Goncharov December 30, 2023 ==========================================V. MELNIKOVA ET AL. § INTRODUCTIONThe rising century of small satellites creates more challenging tasks for them. One of the development trends of small satellites is thecreation and deployment of CubeSat <cit.> constellations to implement some distributed activities: remote sensing, including Earth observation and climate monitoring <cit.>, Sun observation, and space weather <cit.>, communications <cit.>, ionospheric research <cit.>, and other activities. Several projects have already been announced, some of them have started space operations: QB50[<https://www.qb50.eu/>] <cit.>, Planet Labs[<https://www.planet.com/>] <cit.>, Spire[<https://spire.com/>] <cit.>, and others are coming <cit.>.In the case of deploying of satellites at the same time from the same launcher, it needs a propulsion system to deploy a constellation. Another way is to launch CubeSats in different time during several months from the same launching satellite (e.g., as Planet Labs do on ISS). The disadvantage of launch from the ISS is the growing period of constellation formation and a considerable atmospheric drag which reduce the operational time of the satellites. The advantage of this option is a possible reduction of launch failures, so there is a trade-off between potential risk value and an operational lifetime.During the AeroCube4 mission[<https://directory.eoportal.org/web/eoportal/satellite-missions/a/aerocube-4>] <cit.>, the satellite successfully implemented several orbital maneuvers using atmospheric drag forces, but the problem of constellation deployment was not solved. There are several studies with the similar method for constellation deployment using atmospheric drag forces <cit.>. It is also possible to utilize this technology for spacecraft de-orbiting – as it is already implemented now[<http://www.sstl.co.uk/Blog/September-2013/The-De-Orbit-Sail-that-will-bring-TechDemoSat-1-ou>] <cit.>.Deployment of several CubeSats on the orbit using drag forces can be accomplished within several months. The duration of deployment depends on orbit altitude and drag area. We should notice that any active CubeSat propulsion system will reduce the deployment time to the days and even hours <cit.>, and this is the significant disadvantage of any design of the satellite equipment which utilizes atmospheric drag force for passive formation deployment.In this paper, we proposed a method for formation of satellite constellation using solar sail technology. A solar sail uses light pressure to create thrust without consumption of propellant. There were several successful experiments with solar sails on CubeSats, namely Nanosail-D2 <cit.> and Lightsail-1 <cit.>. For the low Earth orbit, light radiation pressure is lower than atmospheric drag <cit.>. However, the similar effect of air drag on the sail can be used to deploy a satellite constellation at altitudes lower than 550 km.One of the types of solar sails is a Heliogyro – rotary solar sail <cit.>. Recent progress in the field of Heliogyro solar sails <cit.> allows us to utilize this technology for practical space applications. § THE CONSTELLATION DEPLOYMENT ALGORITHMThe operating principle is a changing of the atmospheric drag area. By utilizing of Heliogyro-like solar sail, it is possible to roll-out and roll-in the sail blades onto the bobbins.Let us consider the problem of separation of N satellites from the common initial position on the orbit. For every satellite, we know the geometrical parameters of its surface and its weight. All spacecraft should be dispersed evenly throughout the orbit. The final orbit has to be close to the starting orbit. At the initial time t=0, all satellites remain approximately at the same point of the orbit. The satellite number 1 deploys its solar sail, which results in changes of parameters of its orbit. After this, the phase mismatch Δϕ_1 between satellite 1 and the other satellites will start to increase. Next, after a particular time, satellite 1 folds the sail, and the next satellite 2 deploys its sail. Then the satellite 2 appears on the orbit which minimizes the relative movement of both spacecraft, keeping a constant angle Δϕ_0 between the satellites, and the sail on the satellite 2 folds up. This procedure continues up to the satellite N-1. The last satellite should also deploy its sail for a particular time Δ T_N to achieve the final orbit in which phase mismatch with all other satellites will remain approximately constant.This constellation deployment procedure can be described as the following algorithm.The particular values of Δ T_k may be acquired by a multidimensional optimization. The optimization condition is the fastest time of formation deployment T_N. The optimization parameters are the time of start of deployment for each sail (t_k) and the period for which the particular solar sail will be opened (Δ T_k). § MODELING OF CONSTELLATION DEPLOYMENTLet us consider the Earth-centered inertial equatorial coordinate system for the formulation of differential equations of the movement of the center of mass of spacecraft.These are the general assumptions we used for the differential equations of motion: * all external forces on the spacecraft acts to its center of masses; there is no external torque from gravitational and atmospheric forces; * we neglect gravitation of the Moon, Sun, and other planets; * the atmosphere of the Earth does not rotate; * the solar sail maintained its initial orientation during the calculation time, which is typical for a rotary solar sail, stabilized by rotation. The differential equations of motion of the spacecraft have the following simple form:d^2𝐗/dt^2 = 1/m(𝐅_g+𝐅_a+𝐅_s),Where: 𝐗 – vector of coordinates of the satellite; m – the weight of satellite; 𝐅_g - vector of the gravitation force of the Earth, in which model we take into account only a second zonal harmonic; 𝐅_a - vector of an atmospheric drag force; 𝐅_s - vector of a solar radiation pressure force. The atmospheric drag pressure calculated by existing mathematical models (Eq. (<ref>)) is very uncertain and depends on many factors defined by mission profile (attitude, solar activity index, Earth penumbra effects, sail orientation, and its shape).We used these conservative assumptions for the model of the atmosphere: * solar activity index F_10.7 = 100 sfu for the 2016…2022 as NOAA prognoses[<http://services.swpc.noaa.gov/ images/solar-cycle-10-cm-radio-flux.gif>]; * flow is normal to sail surface; * atmospheric drag coefficient C_x = 2.2. We used the GOST R 25645.166-2004 atmospheric model <cit.>. We neglected the effects of Earth penumbra. The comparison between GOST atmosphere and MISI-E-90 model[<http://omniweb.gsfc.nasa.gov/vitmo/msis_vitmo.html>] <cit.> is presented in the Fig. <ref>. The comparison was made in GMAT 2015a software package[<http://gmatcentral.org/display/GW/2015/11/02/Announcing+GMAT+R2015a>].The solar radiation pressure was assumed to be 0.9·10^-6 Pa, and the sail is flat and entirely specularly reflective. The solar radiation flux was also assumed to be perpendicular to the sail surface (cannonball model). Currently, we are investigating the improved model of solar radiation pressure based on the modification of the Generalized Sail Model <cit.>.Initial orbit inclination 51.6 (ISS orbit inclination), longitude of the ascending node 0, the argument of periapsis 0, eccentricity 0.0001, true anomaly 0, a various initial height of the orbit. Satellite parameters The weight of each nanosatellite is always equal to 3kg. This assumption corresponds to the average weight of the 3U CubeSat satellite. The initial height of the orbit changed from 450 km to 650 km with step 50 km, the drag area of solar sail changed from 0.135 m^2 to 1.01 m^2, and the number of satellites was assumed to be 2, 4, or 8. Optimization Starting from the maximum possible time span, which is equal to 1000 seconds, we looked for the value of time of deployment of next solar sail t_i using golden section method. After this, the software calculated values of T_i also using golden section method. The optimization stopped when the deficiency of the calculated and predicted phase separation angles became less than proposed accuracy level.Fig. <ref> shows that the phase angle between spacecraft is stabilized and becomes constant after the separation. Thus the constellation is formed, confirming the feasibility of the concept. The divergences at the end of simulation time in the figure depend on the accuracy of the calculation, and it is possible to reduce them with the increase of the accuracy of the simulation. As shown in the figure, with the increase of an altitude of the initial orbit the average density of the atmosphere decreases, and the time of constellation formation increases.All carried calculations showed that cumulative decrease of an altitude of the orbit does not exceed 15 km. The decline of satellite lifetime is almost equal to the duration of constellation deployment for low attitude orbit (450 km) and become negligible for altitudes higher approximately 550 km (Fig. <ref> and <ref>). § SOLAR SAIL DEPLOYMENT The nanosatellite should spin around some axis perpendicular to the sail surface to provide stiffness to the sail. In this case, sail stiffness has to be enough to deploy sail and to withstand atmospheric drag pressure and solar pressure. According to the results from BMSTU-Sail deployment simulations <cit.>, there is a relationship between initial spinning velocity and the required sail deployment velocity. This relationship can be obtained by the numerical simulations of the sail deployment process considering it as a tether <cit.>.The duration of sail deployment may differ from hundreds of seconds to the thousands of seconds or some hours, but in all cases, the deployment time is negligible compared with nanosatellite lifetime (months or years). The solar radiation pressure and drag pressure affect the value of spinning velocity because they deform the shape of the solar sail. We introduced the angle α between sail tip point before deformation, central body and sail tip position after deformation as a numerical measure of such deformation (Fig. <ref>).For normal sail operation, α has to be as low as possible to prevent the decrease of the effective area of the solar sail.For this paper, we used condition α≤10which provides effective sail area about 95% from its overall area. Simulations of sail deformations indicated that at α<20, sail deformed shape is close to the straight line (Fig. <ref>). In that case, one can derive the equation for α from equilibrium equation of torques from combined atmospheric and solar pressure, and centrifugal forces:α = arctanq/( 2/3 m_sail + 2m_tip) Ω^2 ,Where: α – deformation angle, radians (Fig. <ref>); q – combined atmospheric drag and solar pressure on one meter of blade length, N/m; m_sail – weight of one sail blade, kg; m_tip – sail blade tip mass; Ω – spinning velocity of satellite with deployed sail, radians/s. Comparing the exact solution for sail deformation, the error extimation for Eq. (<ref>) is about 1 that is negligible.We can find the relationship between spinning velocity of deployed sail from Eq. (<ref>) and the required initial spinning velocity before deployment using the principle of conservation of angular momentum:Ω_0 = Ω( 1 + Y_sail/Y_CubeSat),Where: Ω_0 – required initial spinning velocity; Y_sail – inertia moment of deployed sail about spinning axis; Y_CubeSat – inertia moment of CubeSat about spinning axis before deploying.Fig. <ref> represents the results of calculations of initial spinning velocity for different flight altitudes.We conducted the calculations of the spin-up time considering the standard magnetic attitude control system[<http://www.cubesatshop.com/product/cubetorquer-and-cubecoil/>] and the known control laws <cit.>, Fig. <ref>. For the range of the possible sail surface area and altitudes of the orbit, the required initial angular velocity varies from 0.2 to 4.0 rad/s. The commercially available CubeSat magnetic torques can provide this spin-up. Taking into account the various constraints on the CubeSat, such as power and link budget, the spin-up duration can last up to the several days at maximum. This period is much lower comparing with nanosatellite lifetime and the length of constellation deployment time by utilizing of proposed in this paper technology.§ SOLAR SAIL UNIT§.§ Operation The ground station guides the deployment of a constellation.The guidance consists of commands for opening and closing of a solar sail for every nanosatellite at the different time. We choose the following separation of operational functions between nanosatellite bus and a solar sail unit: * nanosatellite provides power supply; * nanosatellite provides commands from ground station for deploying/folding of the sail using the standard digital interface; * nanosatellite provides spinning with the necessary angular velocity and necessary attitude, e.g., by using of magnetic coils; * the sailing unit deploys or folds the sail; * the sailing unit provides telemetry to the nanosatellite which is relaying it to the ground station. Solar sail unit is operating in the hibernation mode at the time when deploying/folding of the sail does not need. §.§ General design Design requirements The main requirements for the unit layout are the following: * Dimensions of all of the PCBs with electronic components has to be 96 x 90 mm; * Height of the unit should be minimized to provide additional space for a payload, but enough to be connected with the other parts of the CubeSat using electrical interfaces; * Width of the sail should be as large as possible to minimize the length of the sail; * Bobbin's structure should be designed in two versions: one made from aluminum with a milling machine, and the other should be polyamide; * It has to be possible to make several structural elements with a 3D printer; * All of the designed components has to be durable and reliable enough to withstand the loads during the launch and subsequent flight.UnitFig. <ref> shows the dimensions of the unit. The layout of the unit constrained by the sail width which depends on the distance between the coupling nuts (Fig. <ref>). The location of the bobbins is symmetrical relative to the spinning axis of the satellite; they should be close to the center of the unit mass. The fixture of the bobbins to the main plate is eight screws locked with nuts and washers. The materials of all the fittings are non-magnetic stainless steel. The PCB has micro motion sensors mounted on a gear axis behind the motor wheel, and there are specific holes in the wheel which help to register the spinning velocity of the bobbins (Fig. <ref>).BobbinA bobbin body design has two different variants: one for manufacturing on a 3D printer, and the other for manufacturing on a milling machine. At Fig. <ref> one can see a bobbin with a polyamide body consisting of two parts and the middle parting plane in between. The top of the body is fixed to the bottom with four screws. The axis and the disks preventing the sail from contacting the bearings while deploying or folding are printed as one.The plastic gears are mounted on printed polyamide bushes. We decided not to make a thread from polyamide because it is flexible and can damage the sail film. The other variant of the bobbin is the variant in which the bobbin body is made of duralumin (Fig. <ref>). The body parts are fixed together with three bolts.Comparison of these two variants showed that the second one seems to be more reliable and durable. However, the 3D printing technology allows making things appropriate for use in space. Firstly, the polyamide structure is much lighter than an aluminum one, and there can be an extra payload on a board of the CubeSat.Secondly, it is much easier and cheaper in production comparing traditional milling technology. Thirdly, the polyamide does not dissolve during the flight, and finally, it proved to be durable and hard enough to sustain the loads during the launch and detachment. There is only one problem with polyamide – it should be metalized to prevent charging and ESD effects in the space environment. This problem can be solved by using specific paintings on the outer surfaces of bobbin body.§.§ Integration There are several requirements to the sailing unit for its integration into the CubeSat: * The outer dimensions of the unit are to be no larger than 90mm x 96mm x 37mm; * It should provide as much as possible of free space to the CubeSat; * It should afford the possibility of its placement in the middle of CubeSat stack as well as on the periphery including by-passing of electrical interfaces; * There must be cutouts for the sail in the CubeSat skeleton which provide the 1mm clearance between the sail and the skeleton; * It should include the special adapter which closes the gap between the sail and the CubeSat skeleton (Fig. <ref>). It is proposed to attach this adapter to the solar panels with glue after the unit integration. The example of solar sail unit integration to 3U CubeSat is shown in Fig. <ref>. For this paper, we used the standard CubeSat components from CubeSatShop[<http://www.cubesatshop.com/faq/>], however, it is proposed that this unit can be adapted to any other CubeSat-class satellite. Electrical interfaces in different CubeSats have similar principles, although practical implementation differs. At the current stage, we selected the ISIS interface, which includes two board-to-board connectors.To provide the solar sail unit functionality, we used I2C for main and reserve information buses, and one power feeder (3.0 to 5.0 V).§ CONTROL SYSTEMThe solar sailing unit is equipped with its avionics which uses the CubeSat's electricity supply, data, and command link with the ground station through CubeSat’s transceiver. The solar sail unit acts as a slave device on the I2C bus. The avionics has to be reliable; it should be tolerant to one permanent fault. It was proposed to utilize theRussian components of industrial class primarily. It should work on the broad allowable range of input/output voltage levels.Avionics consist from DC-DC converter, microcontrollers,MEMS sensors, FLASH memory, input/output ports, etc.The DC-DC converter provides stabilized power buses (3,3 V & 5,0 V) for unit operation in the broad range of input voltage. There is input filter (capacitance) and protection (TVS diode) at the input power line from CubeSat. It consists of several DC-DC devices based on KP1156EY chip (STC SIT) and is constructed by the cascade principle. The first step-up works in the voltage range from 3 to 5 V and transform it to the stable 5 V. This line provides power for some failsafe subsystems, for the second stage of DC-DC and other components. The second stage of the DC-DC converter is a step-down converter. It provides the stable voltage of 3.3 V for main integrated circuits. This scheme ensures the operation of a unit at input voltage range from 3 to 5 V. Each stage of the DC-DC has hot redundancy, and there is protection from current overload.Two I2C interfaces provide data and command handling. There is a bidirectional optical isolation on the entrance of the interface for more reliability. The I2C bus usually reserved on a board of CubeSat that is why these parts have no redundancy.Internal electronic equipment operates on another two separate I2C buses.We selected the industrial-class version of Russian microcontroller K1986BE92QI by Milandr based on the ARM Cortex-M3 architecture as a core of the avionics. Failure of the microcontroller is critical for all the system, that is why we use the scheme of cold redundancy. This scheme repairs itself because integrated circuits without power are not sensitive to SEL. For the successful operation of this scheme, it is necessary to provide a health check of microcontrollers using their software and provide switching of the power supply of microcontroller in a case of failure. These problems we solve using the watchdog MAX6369 by Maxim; JK-trigger produced in the CIS and the Solid State Relay PRAC37S by Proton. The microcontrollers have to provide impulses to MAX6369 every 5 seconds, generated by the software and hardware health check. In the case of failure, the watchdog gives an impulse to the T-trigger. T-trigger switches to the opposite state and starts power supply for the second microcontroller while powering off the first. T-triggers are also redundant. In the worst conditions, in the case of failure of the T-trigger, there is a possibility of both microcontrollers powered on at the same time. There are two communication lines between microcontrollers for software arbitration mechanism in such case.The unit uses the SPI and 1-wire buses for communication with other devices inside the unit. Data storage is a FLASH memory AT45DB011D by Atmel with the SPI interface. There are two MEMS 3-Axis gyroscope and the 3-Axis accelerometer MPU-9250 by InvenSense which provide a measurement of a spinning precession.The deployment of the sail is provided by two bipolar stepper motors, as in the project BMSTU-Sail. Stepper motors are the most suitable for usage in the space due to the lack of brushes and possibility of control of spinning angle without feedback. In each coil, we used the bipolar stepper motor AM1524 by Faulhaber. To provide self-retardation, we used the reduction drive with a gear ratio more than 40. Motors are less capable of operation in the low-temperature(from -20C) that's why they may need heating. For this purpose, there are two resistors connected with the thermal sensors.Control of the motor is performed with the use of the dual H-bridge. They has wide temperature range: from - 60C to +125C. We use four digital signals from microcontroller's output ports to control sail deploying or stowing.The tolerance to one failure is provided by the redundancy only of a driver because the probability of a motor hardware failure is negligible. The second H-bridge chip is connected parallel to the primary chip and stays in the cold reserve. All chips have independent power switches.The detailed analysis and experiments have shown that to provide full redundancy it needs to include the separate relays in each of four circuits from K1128KT4BR output to the stepper motor coils.The operational tests of the primary and reserve chips and switching between them have been carried out on the breadboard. The drive has shown good fault tolerance on the accidents such as the excess of the maximum current, break of several operating wires, break up from the wires to the motor.Two photosensors AOT137A1 from Optron control the sail deployment process. These sensors check if the bobbin is rotating or not. Sensors are located opposite to the cogwheels connected to shafts and detect the presence or absence of the windows in gear wheels. If they do not provide any changes in data at the specific time, the primary driver of this motor is considered failed, and the reserve driver turns on. A pair of sensors is necessary for ensuring the tolerance to one failure of the photosensors.There are JTAG, UART and USB interfaces for debugging and programming.The PCB design has to satisfy these requirements: coupling nuts have to be isolated from electrical circuits; bobbins bodies metallization has to be connected with a metallization circuit on PCB; electrical components have to be glued to PCB with thermally conductive glue; the area near high-power electrical components has to be covered with thermally conductive glue. All electric schematic is developed in Dip Trace EDA/CAD software for creating schematic diagrams and printed circuit boards.§ QUALIFICATION To prove the design of solar sail unit we need an extensive on-ground testing: electrical and functional testing, tests of software, tests of deployment of sail, vibration testing, climatic, thermal vacuum, and other tests. To carry out these test, we need the working mock-up that have to be similar to flight unit.The ground qualification will be conducted in the Bauman MSTU. The partial space qualification of two-blade solar sail technology will be carried out during BMSTU-Sail space experiment <cit.>.We also proposed a mission which will involve three CubeSats with the solar sail. Each nanosatellite will be equipped with a spectrophotometer to carry out the observation of the Sun. One of the satellites should always be on the illuminated side of the orbit to ensure the continuous Sun observation. Thus, taking into account that all three nanosatellites will be launched from the same rocket booster at the same time, we need a proposed solar sailing device to create and to maintain the constellation. This project started in 2017 as a research collaboration between Bauman Moscow State Technical University and the P.N. Lebedev Physical Institute of Russian Academy of Science, Moscow, Russia, and currently is in the preliminary phase. § CONCLUSION In this work, we showed the feasibility and technological details of a method of the deployment of nanosatellite constellations using solar sails. Solar sails, especially heliogyro-like, are still unproven to be reliable for space operations, but the calculation results showed that it is possible to deploy and maintain the constellation of small CubeSat class satellites on the particular range of orbit altitudes.The advantage of this deployment method is the fact that this approach is faster than passive phase separation, and may be cost-effective relative to traditional space propulsion systems. The other advantage of the described device is a selected type of solar sail – it is possible to roll back the solar sail blade onto the bobbin for future use. However, this possibility is still unproven to be achievable in space flight. It is planned to deploy and roll back the similar sail during the BMSTU-Sail Space Experiment. The disadvantage of using a heliogyro-like solar sail is the fact that for several payloads, e.g., Earth remote sensing payload, it may be impossible to operate since the satellite should rotate around some axis.This unit can increase the effectiveness of CubeSat satellite constellations for real-time space weather, Earth observation, improving life quality of people all over the world. § ACKNOWLEDGMENTAuthors would like to thank Department “Aerospace Systems (SM-2)” of Bauman MSTU for granting access to their computer cluster, department “Technologies of rocket engineering (SM-12)” of Bauman MSTU and A.N. Korolev from this department for help with polyamide 3D printing.The research was performed at Bauman Moscow State Technical University with the financial support of the Ministry of Education and Science of the Russian Federation under the Federal Target Program "Research and development on priority directions of the scientific and technological complex of Russia for 2014-2020". Agreement # 14.574.21.0146 (unique identifier RFMEFI57417X0146). apa | http://arxiv.org/abs/1708.07417v2 | {
"authors": [
"Valeriia Melnikova",
"Alexander Borovikov",
"Maksim Koretskii",
"Yuliya Smirnova",
"Ekaterina Timakova",
"Zhaokai Yu",
"Arseniy Kuznetsov",
"Kirill Frolov",
"Stepan Tenenbaum",
"Dmitriy Rachkin",
"Oleg Kotsur",
"Nikolay Nerovny",
"Vera Mayorova",
"Anton Grigorjev",
"Nikita Goncharov"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170824135315",
"title": "Nanosatellite aerobrake maneuvering device"
} |
Department of Chemistry, Stanford University, Stanford CA 94305, USAThe variational principle for conformational dynamics has enabled the systematic construction of Markov state models through the optimization of hyperparameters by approximating the transfer operator. In this note we discuss why lag time of the operator being approximated must be held constant in the variational approach.MSM lag time cannot be used for variational model selection Vijay S. Pande December 30, 2023 ===========================================================Markov state models (MSMs) are a powerful master equation framework for the analysis of molecular dynamics (MD) datasets that involve a complete partition of the conformational space into disjoint states. <cit.> By representing each frame of a MD dataset as its state label, the populations of and conditional pairwise transition probabilities between the states can be counted, leading to thermodynamic and kinetic information about the system, respectively. This information is represented by a transition matrix, which contains all the information necessary to propagate the system forward in time. The transition matrix is the discrete-time approximation to the transfer operator 𝒯(τ), which is characterized by its lag time τ. The transfer operator propagates the system, represented by a normalized probability density u_t(x), forward by a time step of τ, and admits a decomposition into eigenfunctions and eigenvalues (see Ref. Bowman_Book14, Ch. 3),𝒯(τ) ∘ u_t(x) = u_t+τ(x), 𝒯(τ) ∘ψ_i = λ_i ψ_i.The eigenvalues λ_i are real and numbered in decreasing order. The unique highest eigenvalue λ_1=1 corresponds to the stationary distribution, and the subsequent eigenvalue/eigenfunction pairs represent dynamical processes in the time series. Importantly, the timescale of each process can only be retrieved with knowledge of the lag time at which the operator was defined using the equation, t_i = -τ/logλ_i. Choosing a lag time at which the system is Markovian depends on what type of system is being modeled. At a long enough lag time for the system to be approximated as a Markov process, intrastate transitions occur much more quickly than interstate transitions. The appropriate lag time depends on the system of study: for protein folding, 50 ns might be appropriate; for electron dynamics, a suitable lag time might be on the order of femtoseconds. If a system is Markovian at a lag time τ (if the intrastate transitions occur more quickly than τ), then the system will be Markovian at all lag times greater than τ and the timescales of the subprocesses will be constant for all Markovian lag times. This idea has motivated the use of implied timescales plots to choose a lag time. <cit.> Lag times after which the timescales “level out” are assumed to be Markovian, and usually the shortest such time is chosen for the most temporal resolution.In practice, we usually do not know the true eigenfunctions ψ in (<ref>) and instead need to guess them. For a MSM, this means choosing how to divide phase space into disjoint states. Until recently, choosing how to define the states occupied by a dynamic system represented a bottleneck in MSM methods development, and heuristic, hand-selected states were common. However, the derivation of a variational principle for conformational dynamics by <cit.> in 2013 opened the door for a systematic approach to choosing the states of a system.Our guess, or ansatz eigenfunctions, ψ̂_i will admit corresponding eigenvalues λ̂_i. Using our ansatz, we can state the variational principle derived by Noé and Nüske, <cit.> GMRQ≡∑_i=1^mλ̂_̂î≤∑_i=1^mλ_i, where GMRQ stands for generalized matrix Rayleigh quotient, which is the form of the approximator when the first m eigenfunctions are estimated simultaneously. By recalling the relation of the eigenvalues and operator lag time τ to the system timescales in (<ref>), we see that the variational principle establishes an upper bound on the timescales of the slowest m processes in the dynamical system. In practical cases, the variational bound can be exceeded due to statistically undersampled processes; therefore, the GMRQ must be evaluated under cross-validation as described in Ref. McGibbon_JChemPhys15a.When we variationally choose a set of eigenfunctions, we can only compare them if we are trying to approximate the same transfer operator. Therefore, the lag time τ must not be changed when the ansatz is changed, and cannot be variationally optimized using the GMRQ—instead, it must be determined using such techniques as implied timescale plots. In contrast, all transformation and dimensionality reduction choices leading up to the state decomposition are ideal hyperparameters to optimize using the GMRQ. This might include: * RMSD cutoffs for geometric clustering;* internal coordinate choices such as dihedral angles or contacts pairs, including which angles and pairs to include, and any transformations thereof; * internal parameters for time-structure based independent component analysis (tICA) such as tICA lag time, number of components retained, and any transformations of these components;* clustering algorithm and number of clusters;but, as discussed above, cannot include: * the operator lag time, or* the number of timescales scored.These choices are illustrated in Fig. <ref>. For protein folding, we refer the reader to Ref. Husic_JChemPhys16 for a systematic study of these choices in the context of the VAC. In practice, we recommend starting with reasonable parameters for the system of study and choosing a valid lag time. Then, at the chosen lag time, perform a hyperparameter search for any of the state decomposition choices listed above. This two step process can then be repeated, alternating lag time validation using fixed hyperparameters with hyperparameter searches for a fixed lag time.Perhaps the most natural way to understandthe separate treatment of the lag time is to consider its true role in kinetic model building.While previous MSM approaches have treated it effectively like a hyperparameter (e.g., choosing a lag time based on flattening of implied timescales), in actuality, this approach is fundamentally philosophically incorrect. The lag time must be chosen a priori by the researcher, as it directly reflects the resolution of interest to study. Given a method which can directly identify the relevant degrees of freedom, choosing a lag time of picoseconds would bring water dynamics into the state space, vs. nanoseconds for backbone and side chain dynamics or microseconds for longer time slow scale rearrangements. For this reason, it simply doesn’t make sense to let the model choose the lag time and instead one must have the protocol choose the best model given a pre-chosen set lag time.MSMs are just one example of the general set of models to which the VAC applies. The popular tICA framework <cit.> can also be variationally optimized. When using tICA as an intermediate step in MSM construction, the tICA lag time may be varied and optimized. However, in the case where the tICA model is the entity being evaluated, the tICA lag time and number of components scored must be held constant in order to ensure that the same operator is being approximated. Additional extensions of the VAC can be found in Ref. Wu_Arxiv17. We also refer the interested reader to Ref. McGibbon_JChemPhys15b, which presents continuous-time Markov processes that do not have lag times. We would also like to note that the VAC is not a panacea: the slowest dynamical processes are often assumed, but not guaranteed, to be the processes of interest, and it is important to verify this for each analysis. We anticipate this note will help guide hyperparameter optimization when using VAC. The open-source software Osprey <cit.> has been designed for this type of optimization and is available on <msmbuilder.org>. The authors are grateful to Matt Harrigan, Carlos Hernández, Jade Shi, Anton Sinitskiy, Nate Stanely, and Muneeb Sultan for discussion and manuscript feedback. We acknowledge the National Institutes of Health under No. NIH R01-GM62868 for funding. V.S.P. is a consultant & SAB member of Schrodinger, LLC and Globavir, sits on the Board of Directors of Apeel Inc, Freenome Inc, Omada Health, Patient Ping, Rigetti Computing, and is a General Partner at Andreessen Horowitz. 10 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Bowman, Pande, and Noé(2014)]Bowman_Book14 author author G. R. Bowman, author V. S. Pande, and author F. Noé(publisher Springer, year 2014)NoStop [Swope, Pitera, and Suits(2004)]Swope_JPCB04a author author W. C. Swope, author J. W. Pitera, and author F. Suits, 10.1021/jp037421y journal journal J. Phys. Chem. B volume 108, pages 6571 (year 2004)NoStop [Noé and Nüske(2013)]Noe_MMS13 author author F. Noé and author F. Nüske, 10.1137/110858616 journal journal Multiscale Model. Simul. volume 11, pages 635 (year 2013)NoStop [McGibbon and Pande(2015a)]McGibbon_JChemPhys15a author author R. T. McGibbon and author V. S. Pande, http://scitation.aip.org/content/aip/journal/jcp/142/12/10.1063/1.4916292 journal journal J. Chem. Phys. volume 142, eid 124105 (year 2015a)NoStop [Husic et al.(2016)Husic, McGibbon, Sultan, and Pande]Husic_JChemPhys16 author author B. E. Husic, author R. T. McGibbon, author M. M. Sultan,andauthor V. S. Pande, http://scitation.aip.org/content/aip/journal/jcp/145/19/10.1063/1.4967809;jsessionid=Td617tvPkXmj4MqYBi1JOe8C.x-aip-live-02 journal journal J. Chem. Phys. volume 145, eid 194103 (year 2016)NoStop [Pérez-Hernández et al.(2013)Pérez-Hernández, Paul, Giorgino, De Fabritiis, and Noé]PerezHernandez_JChemPhys13 author author G. Pérez-Hernández, author F. Paul, author T. Giorgino, author G. De Fabritiis,and author F. Noé, http://dx.doi.org/10.1063/1.4811489 journal journal J. Chem. Phys. volume 139, eid 015102 (year 2013)NoStop [Schwantes and Pande(2015)]Schwantes_JCTC15 author author C. R. Schwantes and author V. S. Pande, 10.1021/ct5007357 journal journal J. Chem. Theory Comput. volume 11, pages 600 (year 2015)NoStop [Wu and Noé(2017)]Wu_Arxiv17 author author H. Wu and author F. Noé,@noopjournal journal arXiv preprint arXiv:1707.04659(year 2017)NoStop [McGibbon and Pande(2015b)]McGibbon_JChemPhys15b author author R. T. McGibbon and author V. S. Pande, http://dx.doi.org/10.1063/1.4926516 journal journal J. Chem. Phys. volume 143, eid 034109 (year 2015b)NoStop [McGibbon et al.(2016)McGibbon, Hernández, Harrigan, Kearnes, Sultan, Jastrzebski, Husic, and Pande]McGibbon_JOSS16 author author R. T. McGibbon, author C. X. Hernández, author M. P. Harrigan, author S. Kearnes, author M. M. Sultan, author S. Jastrzebski, author B. E. Husic,and author V. S. Pande, http://dx.doi.org/10.21105/joss.00034 journal journal J. Open Source Software volume 1 (year 2016)NoStop | http://arxiv.org/abs/1708.08120v1 | {
"authors": [
"Brooke E. Husic",
"Vijay S. Pande"
],
"categories": [
"q-bio.BM",
"physics.bio-ph",
"physics.chem-ph"
],
"primary_category": "q-bio.BM",
"published": "20170827182624",
"title": "MSM lag time cannot be used for variational model selection"
} |
A Machine Learning Approach For Identifying Patients with Mild Traumatic Brain Injury Using Diffusion MRI Modeling Shervin Minaee^1,*, Yao Wang^1, Sohae Chung^2, Xiuyuan Wang^2, Els Fieremans^2, Steven Flanagan^3, Joseph Rath^3, Yvonne W. Lui^2,^1Electrical and Computer Engineering Department, New York University ^2Department of Radiology, New York University ^3Department of Rehabilitation Medicine, New York University ^*Correspondence to Shervin Minaee: [email protected] December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================ Purpose. While diffusion MRI has been extremely promising in the study of MTBI, identifying patients with recent MTBI remains a challenge <cit.>. The literature is mixed with regard to localizing injury in these patients, however, gray matter such as the thalamus and white matter including the corpus callosum and frontal deep white matter have been repeatedly implicated as areas at high risk for injury. The purpose of this study is to develop a machine learning framework to classify MTBI patients and controls using features derived from multi-shell diffusion MRI in the thalamus, frontal white matter and corpus callosum. Method. This IRB-approved, prospective study includes 69 MTBI subjects between 18 and 64 years old, within 1 month of MTBI as defined by the American College of Rehabilitation Medicine (ACRM) criteria for head injury and 40 healthy age and sex-matched controls. Imaging was performed on a 3.0 Tesla Siemens Trio (Erlangen, Germany) magnet including multi-shell diffusion MRI at 5 b-values (250, 1000, 1500, 2000, 2500 s/mm2) in a total of 136 directions using multiband 2 at isotropic 2.5mm image resolution. Denoising, Gibbs correction, geometric and eddy current distortion and motion correction was performed. In addition to Mean Diffusion (MD), Fractional Anisotropy (FA) and Mean Kurtosis (MK), additional modeled white matter tract integrity (WMTI) metrics were calculated and parametric maps generated for: 1) axonal water fraction (AWF), 2) intra-axonal diffusivity (“D”_“axon”, diffusivity within axons), 3) extra-axonal axial and 4) extra-axonal radial diffusivities (“De”_“par”and “De”_“per”, diffusion parallel and perpendicular to the axonal tracts in the extra-axonal space). The relationship between WMTI metrics and tissue microstructure has been studied in several animal validation works, specifically demyelination following acute injury and remyelination in the chronic phase and also applied in vivo to characterize normal and abnormal neural development in the central nervous system (CNS).Two approaches were attempted differing in feature representation. In the first approach, mean values of above 7 MR metrics in five regions (thalamus, prefrontal white-matter, corpus callosum (CC) Body, CC-Genu, and CC-Splenium) were used in the feature representation along with two demographic features (age and sex) and four neurocognitive tests (Stroop, Symbol Digit Modalities Test (SDMT), California Verbal Learning Test (CVLT) and Fatigue Severity Scale (FSS)). Agreedy forward feature selection approach was used to choose the best feature subset upon which a support vector machine achieves the highest cross-validation accuracy. In the second approach, 16x16 patches are extracted from brain slices through the areas of interest and all the training patches in the MTBI patients and control subjects are separately clustered to learn the most representative visual patterns (called “visual” words). Each 3D dataset is then shown as a histogram of these words, known as the Bag of Words (BoW) representation <cit.>. The block-diagram of this approach is shown in Fig. 1. The same feature selection approach was used to derive a best feature subset for the support vector machine classifier <cit.>.Results. Using approach 1, we found the best single metric cross-validation classification accuracy of 72% using AWF in CC-Body. The best feature subset chosen by the greedy feature selection had an accuracy of 80% with 8 features (De_par in thalamus, De_par and DA in pre-frontal white matter, FA in CC-Genu, AWF and De_per in CC-Body, Stroop and SDMT).BoW approach achieved further improvement in accuracy to 85.5%. In this approach, the raw representation is 206 dimensional, including 20 words for each of 5 MR metrics (AWF, DA, De_par, FA, MD) in each of 2 brain regions (thalamus and corpus collasum), and 6 demographic and neurocognitive features, and the optimum subset contains 8 features (which includes age, and words from AWF, De_parand FA in thalamus, and also DA, De_par, FA and MDin corpus callosum). Fig. 2 shows example BoW histograms.We can see MTBI and control subjects have clear differences in frequency of some visual words, for example along the right side of the histograms.Conclusions. Here we show the application of two approaches to multi-feature analysis using advanced diffusion MRI for classification of patients with mTBI compared with controls: first employing greedy forward feature selection and support vector machine algorithms and second exploring a novel approach in which computer vision and machine learning techniques are used to learn useful feature representation. Our early results show that feature learning approach achieves 5.5% gain over mean value features. These visual features can also be used for long-term outcome prediction of mTBI patients <cit.>. § 1615ACKNOWLEDGMENTWe would like to thanks Siyun Wang for her help during this project.1 yvonne X Wu, II Kirov, O Gonen, Y Ge, RI Grossman, YW Lui, “MR imaging applications in mild traumatic brain injury: an imaging update”, Radiology, 279(3), 693-707, 2016. bow J Yang, YG Jiang, AG Hauptmann, CW Ngo, “Evaluating bag-of-visual-words representations in scene classification', Proceedings of the international workshop on Workshop on multimedia information retrieval, ACM, 2007. svm WS Noble, “What is a support vector machine?”, Nature biotechnology 24.12: 1565-1567, 2006. sherv S Minaee, Y Wang, and YW. Lui. “Prediction of longterm outcome of neuropsychological tests of MTBI patients using imaging features”, Signal Processing in Medicine and Biology Symposium, IEEE, 2013. | http://arxiv.org/abs/1708.09000v1 | {
"authors": [
"Shervin Minaee",
"Yao Wang",
"Sohae Chung",
"Xiuyuan Wang",
"Els Fieremans",
"Steven Flanagan",
"Joseph Rath",
"Yvonne W. Lui"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170827201428",
"title": "A Machine Learning Approach For Identifying Patients with Mild Traumatic Brain Injury Using Diffusion MRI Modeling"
} |
Aeronautics and Astronautics, University of Washington, Seattle, WA, USA, 98195-2400Equally contributed first author Aeronautics and Astronautics, University of Washington, Seattle, WA, USA, [email protected] Aeronautics and Astronautics, University of Washington, Seattle, WA, USA, 98195-2400Inspired by the quantum spin Hall effect shown by topological insulators, we propose a plate structure that can be used to demonstrate the pseudo-spin Hall effect for flexural waves. The system consists of a thin plate with periodically arranged resonators mounted on its top surface. We extend a technique based on the plane wave expansion method to identify a double Dirac cone emerging due to the zone-folding in frequency band structures. This particular design allows us to move the double Dirac cone to a lower frequency than the resonating frequency of local resonators. We then manipulate the pattern of local resonators to open subwavelength Bragg band gaps that are topologically distinct. Building on this method, we verify numerically that a waveguide at an interface between two topologically distinct resonating plate structures can be used for guiding low-frequency, spin-dependent one-way flexural waves along a desired path with bends. 45.70.-n 05.45.-a 46.40.CdSubwavelength and directional control of flexural waves in zone-folding induced topological plates Jinkyu Yang December 30, 2023 ====================================================================================================§ INTRODUCTIONA topological insulator has emerged as a new state of matter in condensed matter physics. This is a special type of insulator that conducts electricity only on its boundary. Here topology is relevant because one can predict the boundary properties of these finite materials (i.e., finite-sized lattices) solely by knowing the bulk properties of infinite materials (i.e., infinitely large lattices). Topological framework provides an elegant way to categorize the bulk properties in terms of a topological invariant, and thus, one expects a topological protection and a degree of robustness for the boundary properties <cit.>.It is recent that this whole framework dealing with the flow of electrons has evolved further and influenced other areas such as photonics<cit.> and acoustics <cit.>. It has also propelled new design paradigm for artificial mechanical structures, so-called topological mechanical metamaterials, to achieve unconventional static (zero frequency) <cit.> and dynamic (non-zero frequency) <cit.> responses. In particular, tailoring non-zero frequency responses, i.e., elastic waves in structures, on topological grounds shows tremendous potential to be used for energy harvesting, sensing, and impact mitigation purposes <cit.>.One of the most unique topological effects is the quantum spin Hall effect, the underlying phenomenon shown by topological insulators <cit.>. These systems are passive in the sense that they do not require any external field, but still possess directional boundary states. This is due to the presence of Kramers partners, i.e., two opposite spins of electron, which travel in the opposite directions on their boundaries, thereby keeping the time reversal symmetry intact. Although mechanical counterparts, being bosonic systems, do not possess these intrinsic spins, one can carefully design the system to have two pseudo-spins by imposing certain symmetries in the lattice, and thus realize the pseudo-spin Hall effect <cit.>. While previous studies have successfully reported the feasibility of the pseudo-spin Hall effect in mechanical settings, in this study, we focus on the feasibility of the same in less-explored continuum structures such as plates. One of the approaches that has been recently applied in plate structures is a so-called zone-folding technique <cit.>, in which one rather considers a larger unit cell than an irreducible one in a hexagonal lattice arrangement, so that the frequency band structure folds onto itself, creating a double Dirac cone at the Γ point. Based on the same, Brendel et al. <cit.> and Yu et al. <cit.> showed that purely geometric manipulation of holes can invoke topological effects in plates. However, these topological effects have been restricted to high-frequency wave modes. Therefore, in this research, we ask the question: How can one invoke the pseudo-spin Hall effect at low frequencies for a given plate dimension? It is important because of several reasons, including (1) the low-frequency plate modes, such as flexural modes, carry a large amount of energy, and manipulating them could lead to relevant engineering applications, and (2) these lower modes generally require bigger lattice patterns of holes on conventional plates due to the Bragg condition, and thus, an improved way of controlling the low-frequency wave modes can relax the current stringent size limitations. Therefore, it would be a significant advancement to the current research trend if one can demonstrate low-frequency pseudo-spin Hall effect in a continuum mechanical structure such as plates, which are ubiquitous in many engineering disciplines. To address the aforementioned challenges, we propose a topological plate system that consists of a thin plate with periodically arranged local resonators mounted on its top surface. This locally resonant (LR) plate is a reminiscence of sonic crystals <cit.>. Pal et al. <cit.> proposed such a structure for realizing the elastic analogue of quantum valley Hall effect. Building on the similar methodology, in this research, we employ a technique based on the combination of the classical plate theory and the plane wave expansion (PWE) method <cit.>, which enables fast and efficient calculation of the wave dispersion relation. Furthermore, we integrate into this scheme the zone-folding technique to create a double Dirac cone for flexural wave modes. As a result, we report that the double Dirac cone can be formed in low-frequency regimes by tuning the resonating frequency of the resonators. In this way, we can acquaint a subwavelength characteristic of the proposed plate system, i.e., the lattice size of the LR plate being smaller than the wavelengths in the bare plate at operating frequencies. We then show that a purely geometric manipulation of the local resonator pattern results in the opening of a topologically trivial and non-trivial subwavelength Bragg band gaps around the double Dirac cone. Building on these findings, we verify numerically—by using the finite element method (FEM)—that a waveguide created at the interface of topologically distinct LR plates can guide low-frequency flexural waves along a designed path. Moreover, it shows a unique spin-depenedent one-way propagation characteristic. Unlike the traditional plate-based waveguides studied in the past <cit.>, we show that this LR topological plate system has potential to guide one-way flexural waves along a path with multiple bends—generally challenging in topologically trivial waveguides.The structure of this manuscript is as follows: in section II, we describe the design of the topological plate. In section III, we present the PWE method to calculate dispersion relation. In section IV, we show the zone-folding of bands and create a double Dirac cone in a subwavelength regime. In section V, we show the formation of a band gap around the double Dirac cone by perturbing the pattern of resonators on the LR plate. This facilitates the system to transition from a topologically trivial state to a non-trivial state. In section VI, we employ the FEM to show the existence of two local modes, each designated by a pseudo-spin (clockwise or counterclockwise), at the interface of topologically trivial and non-trivial lattices. In section VII, we demonstrate the feasibility of guiding low-frequency flexural wave modes along a path with bends and having a spin-dependent one-way propagation characteristic. In section VIII, we conclude this manuscript.§ DESCRIPTION OF THE LOCALLY RESONANT TOPOLOGICAL PLATEOur system consists of a thin plate on which multiple local resonators are attached to form a lattice arrangement (Fig. <ref>). The rhombus-shaped unit cell is of length a and consists of six resonators in a hexagonal arrangement (Fig. <ref>a). Each resonator is at a distance R and rotationally symmetric from the center of the unit cell, showing the C_6 symmetry. a⃗_1 and a⃗_2 are the lattice vectors. We model the resonators as cylindrical heads attached to the plate with a thin neck (Fig. <ref>b). In order to invoke the topological effects in the system, we will only vary radius R, keeping the C_6 symmetry intact in this unit cell.As a substrate material, we choose an aluminum plate (E=77.6 GPa, ρ=2,730 kg/m^3, ν=0.352) of thickness h=1 mm with the unit cell of size a=45 mm. The resonator neck is made of acrylic plastic (E_neck=3.2 GPa) with h_neck=5 mm and d_neck=2 mm, whereas the resonator head is made of tungsten(ρ_head=19,260 kg/m^3) with h_head=14 mm and d_head=9 mm. The aforementioned material properties are based on nominal values of standard materials.§ CALCULATION OF THE UNIT CELL DISPERSIONWe first evaluate dispersion characteristics of the unit cell design with six resonators for variable R. For fast calculations, we rely on the PWE method. To this end, we simplify the resonator design with a lumped mass (m=πρ_head d_head^2 h_head/4) connected to the plate with a linear spring (β=π E_neck d_neck^2/4h_neck). In this process, we neglect the bending motion of the resonators and only consider their motion in the out-of-plane z-direction. The bending (and other modes) of resonators, though taken into account in full-scale models in later sections, do not affect the topological phenomenon in our system as those frequencies can be separated from the Dirac point we will be investigating and have minimal coupling with out-of-plane wave modes (see Appendix A). We have h ≪ a, therefore the plate can be assumed to be thin <cit.>, and the transverse motion of the plate can be calculated as per the classical plate theory (i.e., Kirchhoff-Love theory) <cit.>. Following the approaches taken by Pal et al. <cit.>, Xiao et al. <cit.>, and Torrent et al. <cit.>, governing equations for the time-harmonic vibration of the unit cell with angular frequency ω can be written asD∇^4 w(r) -ω^2 ρ h w(r)=-β∑_α [w(R_α)-w̃(R_α)]δ(r-R_α) ,-ω^2 m w̃(R_α)= β [w(R_α)-w̃(R_α)],where D=Eh^3/12(1-ν^2) represents the flexural rigidity of the plate, r=(x,y) denotes the generalized coordinate of the plate, w(r) represents the transverse displacement of the plate, and w̃(R_α) represents the displacement of the resonating masses attached at points R_α. We have α=1,2,...,6 for six different resonating masses per unit cell and δ(r-R_α) is a delta function in two dimensions. We introduce the following non-dimensional angular frequencyΩ = ωa^2 √(ρ h/D). Also, the mass of the resonator can be normalized asγ = m/ρ A_c h,where A_c=√(3) a^2/2 is the area of a unit cell. We can write the normalized resonance frequency of a resonator as Ω_r =a^2 √((β/m) ρ h/D).Employing the PWE method, we write the displacement of the plate for a Bloch wave vector K as a superposition of multiple plane waves such thatw(r)=∑_G W(G)e^-i(K+G)·r, where W(G) is a plane wave coefficient and G denotes the reciprocal lattice vector given by G=pb_1 +qb_2, in which p and q are integers, and b_1 and b_2 are the basis vectors of the reciprocal lattice. We truncate the summation with respect to G by choosing both p and q as -M,-(M-1),...,0,...,(M-1), M. Therefore, the reciprocal space is a N× N finite grid with N=2M+1. The displacement of the plate at the locations where the resonators are attached can be simply deduced from Eq. (<ref>) asw(R_α) = ∑_G W(G)e^-i(K+G)·R_α. SubstitutingEq. (<ref>) and Eq. (<ref>) into Eq. (<ref>)D ∑_G'K+G'^4 W(G')e^-i(K+G')·r - ω^2 ρ h ∑_G' W(G')e^-i(K+G')·r= β∑_α[w̃(R_α)- ∑_G' W(G') e^-i (K+G')·R_α]δ(r-R_α). Multiplying both sides with e^i(K+G)·r, we obtain∑_G'[D K+G'^4-ω^2 ρ h ] W(G') e^-i(G'-G)·r =β∑_α e^i(K+G)·r[w̃(R_α) - ∑_G' W(G') e^-i (K+G')·R_α]δ(r-R_α).Taking the area integral over the entire unit cell of area A_c leads to∑_G'[D K+G'^4 - ω^2 ρ h ] W(G') ∬_A_c e^-i(G'-G)·rdr^2 = β∑_α[w̃(R_α) - ∑_G' W(G') e^-i (K+G')·R_α] ∬_A_c e^i(K+G)·rδ(r-R_α) dr^2. We now use the following relations∬_A_c e^-i(G'-G)·rdr^2=A_c, if G=G'0, otherwise ∬_A_c f(r) δ(r-R_α) dr^2=f(R_α)to obtainA_c [D K+G^4 - ω^2 ρ h ] W(G)= β∑_α[w̃(R_α) - ∑_G' W(G') e^-i (K+G')·R_α] e^i(K+G)·R_α. Using the Bloch's theorem for the resonators, we write w̃(R_α) =w̃(0_α) e^-i K·R_α, where w̃(0_α) represents the Bloch displacement of the resonator (indexed with α) at the reference unit cell. Thus, we deduce [ a^4 K+G^4 -Ω^2] W(G) =γΩ_r^2∑_αe^i G·R_α[w̃(0_α)- ∑_G' W(G') e^-i G'·R_α]. Similarly, we simplify the second governing Eq. (<ref>) (for α=1,2,...,6) as-Ω^2 w̃(0_α) = Ω_r^2 [ ∑_G W(G) e^-i G·R_α-w̃(0_α) ]. Given the N × N size of the reciprocal space, we arrange Eq. (<ref>) and Eq. (<ref>) in the form of an eigenvalue problem to solve for Ω at a specific Bloch wave vector K and obtain the dispersion relation. Note that we multiply Eq. (<ref>) with γ to make the matrices Hermitian. Therefore, we have[ P_11 P_12; P_21 P_22 ]{[W(G); w̃(0_α) ]} = Ω^2 [ Q_11 Q_12; Q_21 Q_22 ]{[W(G); w̃(0_α) ]}withP_11 = a^4 [ K+G_1^4 0 ⋯ 0; 0 K+G_2^4 ⋯ 0; ⋮ ⋮ ⋱ ⋮; 0 ⋯ 0 K+G_N^2^4 ] + γΩ_r^2 exp{ i [ G_1; G_2; ⋮; G_N^2 ][ R_1 R_2 ⋯ R_6 ]}exp{-i [ R_1; R_2; ⋮; R_6 ][ G_1 G_2 ⋯ G_N^2 ]},P_12 = P_21^†=-γΩ_r^2 exp{ i [ G_1; G_2; ⋮; G_N^2 ][ R_1 R_2 ⋯ R_6 ]},P_22 = γΩ_r^2 I_6,Q_11 = I_N^2,Q_12 = Q_21^† = 0_{N^2,6},Q_22 = γI_6,where `exp', †, I, and 0 represent the exponential function, conjugate transformation, the identity matrix, and the null matrix, respectively. We choose N=7 for further calculations.§ BAND FOLDING AND SUBWAVELENGTH UNIT DESIGNBy using the aforementioned technique, we calculate the dispersion relation for the hexagonal arrangement of resonators, i.e., R=a/3 (see Fig. <ref>a) and plot in Fig. <ref>a. Torrent et al. <cit.> showed the existence of a single Dirac cone in such a system. Building on this finding, in the current study, we create a double Dirac cone (two Dirac cone dispersion curves superimposed) with the frequency f_d = 0.94 kHz at the Γ point. This is possible because we have chosen a bigger unit cell consisting of six resonators instead of two (compare the unit cells in Fig. <ref>b of different colors and corresponding Brillouin zones below). This results in the dispersion curves folded onto a smaller Brillouin zone <cit.>. Note that the physics is the same in both representations and it is simply a mathematical zone-folding of bands. However, achieving a double Dirac cone—which is a key ingredient of the spin Hall systems—guides us to realize topological effects by manipulating the geometrical configuration of the larger unit cell (to be further discussed in the Section V). In Fig. <ref>a, we also mark the resonating frequency f_r of the local resonator. It equals f_r=(1/2π) √(β/m)=1.72 kHz. It is important to realize that f_d ≤ f_r, as thoroughly investigated by Torrent et al. <cit.>. Therefore, the resonator design can be used as a tuning knob to push the Dirac frequency further down in the dispersion relation. In the same figure, we also plot the dispersion relation for a bare plate (i.e., the identical plate as the substrate described in Section II, but without local resonators attached). This is to compare the wavelength (λ_d) of flexural wave in the bare plate if excited at the Dirac frequency. This is indicated by the star marker on the dispersion curve. For the chosen set of design parameters, λ_d is approximately 2.3 times longer than the length of the large unit cell (i.e., a), and 4 times longer than the size of the small, irreducible unit cell (a/√(3)). Figure <ref>b shows the relative sizes of the unit cells compared to this wavelength, indicating subwavelength units of the LR plate. Therefore, as the topological effects will be seen around the Dirac frequency, this opens up new pathways to controlling large-wavelengths flexural waves by using a relatively small substrate. Again, by further reducing the resonant frequency, it is possible to shift the Dirac point to even lower frequency regime, thereby making the plate design deep-subwavelength. However, practical challenges in designing such a system can limit the same, e.g., due to heavy resonating masses and soft neck structures.§ BAND INVERSION AND TOPOLOGY We now vary the radius R and see its effects on the wave dispersion in the system. For R<a/3, as shown in Fig. <ref>a, there emerges a band gap near the Dirac frequency. We call it a subwavelength Bragg band gap because it lies in the subwavelength regimes as discussed above but emerges due to the change in translational periodicity of the resonators. By keeping the C_6 symmetry intact, two modes on each side (lower or higher side) of the gap are degenerate at the Γ point. Seen in the insets are the corresponding degenerate mode shapes of the plate at the Γ point, which are obtained by the PWE method described in Section III. Here, the lower frequency modes are of p-type (p_1 and p_2 as shown in the bottom panel of Fig. <ref>a), and the higher frequency modes are d-type (d_1 and d_2, upper panel in Fig. <ref>a) as per the analogy to electronic orbital shapes. For R>a/3, however, the band gap still exists, but its topological characteristic is different from the earlier case. As shown in Fig. <ref>b, the degenerate modes are flipped, i.e., d-type modes are at the lower frequency compared to p-type modes. This band inversion as we vary R around R=a/3indicates a typical topological transition in the system. Again, the validity of this result based on the lumped mass model is verified and discussed in Appendix A in comparison with the FEM (using COMSOL Multiphysics), which takes into account all geometrical features in the resonator design.The presence of degenerate modes around the band gap has important implications in realizing pseudo-spin Hall effect. One can take linear combinations of these modes and construct two alternate modes, i.e., pseudo-spin modes, without changing the physics of the system. Let p_±=p_1± i p_2 and d_±=d_1± i d_2 represent such spin modes for these degenerate points. The sign in the middle determines if these are rotating clockwise or counterclockwise. We can interpret the dispersion near the Γ point in terms of the pair of spins by projecting the eigenstates onto the spin basis {p_±,d_±}. Therefore, the effective Hamiltonian of the system around the Γ point reduces to the one for Cd/Te/HgTe/CdTe quantum well <cit.> and would resemble a mechanical pseudo-spin Hall system. One can show that the bands have non-zero spin Chern number for the case with R>a/3, hence, proving it to be topologically non-trivial <cit.>. § EMERGENCE OF TOPOLOGICAL INTERFACE STATENow that we verified the feasibility of the double Dirac cone formation and the band inversion in the unit-cell level, we move to the investigation of wave guiding characteristics in multi-cell configurations. To account for more complicated geometry and boundary conditions in such a multi-cell setting, we resort to the FEM henceforth. According to the bulk-boundary correspondence of topology <cit.>, we expect distinct behaviors on the boundaries of topologically trivial and non-trivial lattices. One way to observe it clearly is to have topologically distinct lattices placed adjacently and investigate their connecting interface for a non-trivial local response. To this end, we take a supercell, which consists of both topologically trivial (R=0.8a/3) and non-trivial (R=1.1a/3) lattices, 10 units of each placed as one strip (Fig. <ref>a). The periodic boundary condition is introduced in the direction of another lattice vector (at 60^∘ from the horizontal). In this way, such a system provides a quick way to calculate vibration responses at the interface and monitor their propagation along the periodic direction. In Fig. <ref>b, we plot the eigenfrequencies of the supercell as a function of wave number in the periodic direction. The presence of two modes inside the band gap (shown in purple and yellow colors) is especially striking, since those have the following non-trivial properties.First, they represent two types of pseudo-spin modes localized at the interface: one rotates clockwise, while the other rotates counterclockwise. Second, both have opposite group velocities at a given frequency. Figure <ref>c shows the respective mode shapes corresponding to points S_1 and S_2 in Fig. <ref>b, which are excited at 0.91 kHz. Opposite spins and group velocities of these modes can be verified by looking at the harmonic evolution of these modes (see Supplementary Movie 1). We also plot in-plane time-averaged mechanical energy flux (I_j=-σ_ijv_j, where σ_ij and v_j are stress tensor and velocity vector, respectively) over a harmonic cycle as black arrows. This further confirms the spin nature of these flexural modes in the LR plate.There is a small frequency gap at the Γ point for these spin modes. The absence of topological interface modes indicates the absence of topological protection, and that suggests these pseudo-spin modes are not topologically protected in the full frequency band gap. This is because, in our system, the protection is guaranteed by the C_6 symmetry, which we break by introducing a sharp interface between topologically trivial and non-trivial lattices, and thereby resulting in an avoided crossing at the Γ point. Nevertheless, we will show in the next section that these modes can still be used to build robust and directional waveguides. The remedy to reduce the gap at the Γ point is to minimize the effect of the C_6 symmetry breaking at the interface. This can be done in several ways, including (1) by choosing the radii of trivial and non-trivial configurations as close as possible, or (2) by constructing a graded interface between two topologically distinct lattices (see Appendix B). This therefore leads to a `greater' degree of protection of the topological spin-modes.§ DIRECTIONAL WAVEGUIDES In the previous section, we have shown that the mechanical spin Hall effect enables us to have two pseudo-spins at one frequency but in the opposite directions. In order to demonstrate how this property can be used to build unconventional waveguides on plates, we combine topologically trivial and nontrivial LR plates to form a 2D structure.Figure <ref> shows the waveguide (three linear segments with two bends) along the interface of two types of lattices. We give a forced excitation in the z-direction at the center of the plate (indicated by the star symbol) in such a way that we selectively excite spin modes. This could be done, for example, by choosing multiple points in the vicinity but with a phase difference in their forcing. Insets show the three excitation points in a nontrivial unit cell (i.e., with R=1.1a/3) where the spin is predominantly d-type. We extract the phase information from the spin modes in the supercell analysis done earlier for 0.91 kHz (see the resonators in red and blue colors, representing out-of-phase oscillations in Fig. <ref>c). We apply this phased excitation as F_1=Fexp(iω t), F_2=Fexp(iω t+2π/3), and F_3=Fexp(iω t+π). Note that the phase differences in the three excitation points are not equally spaced, but show π/3 and 2π/3 differences between the neighboring ones (i.e., π/3 between F_2 and F_3, and 2π/3 between F_1 and F_2). This excitation tactic induces clockwise spin in Fig. <ref>a and counterclockwise spin in Fig. <ref>b. We enforce low-reflecting boundary conditions on the plate and perform harmonic analysis using the FEM. We confirm the unique features of this topological waveguide. The clockwise spin mode propagates to the left (Fig. <ref>a) and the counterclockwise spin propagates only to the right (Fig. <ref>b). These spin waves propagate robustly along the waveguide interface in a way that even though there are sharp bends, there is no back-scattering and the spins remain intact (see Supplementary Movie 2).These simulation results imply that by using the pseudo-spin Hall effect induced in this LR plate structure, we can guide flexural waves in a selected path and direction without resorting to the breakage of time-reversal symmetry. That is, without using any active components, we can achieve directional control of low-frequency flexural waves simply by creating a topological boundary and exciting the host medium strategically via a phased excitation. It is important to note that robustness of these spin waves shown along a waveguide with sharp bends does not imply that these are also robust against any other types of `defects' along the waveguide as shown in acoustics recently <cit.>. Though our elastic LR plate structure demands a thorough stand-alone study on this subject in future, we have explored exemplary cases here, in which certain defects along the waveguides can(not) back scatter these spin waves (Appendix C). § CONCLUSIONSWe have proposed a locally resonant plate structure to demonstrate the pseudo-spin hall effect for directional control of flexural waves. We show that the resonator design can be simplified with a lumped mass model and solved by employing the plane wave expansion method. This method enables us to efficiently investigate the key design parameters responsible for forming a double Dirac cone at a lower frequency than the resonating frequency of the local resonators.Keeping the C_6 symmetry intact, we perturb the unit cell and show an opening of subwavelength Bragg band gaps and the corresponding band-inversion process. This provides us with two topologically distinct lattice configurations. When these lattices are placed adjacently, we show the existence of two pseudo-spin modes traveling in the opposite directions along the interface. This unique feature is used to build topological waveguides with multiple bends and robustly guiding the spin-dependent flexural waves in a selected direction.The finding could be useful in designing compact and robust one-way channels for guiding low-frequency flexural waves in applications such as energy harvesting, sensing, and impact mitigation. Future studies include the optimization of the locally resonant plate configurations by using the proposed numerical techniques, as well as the experimental verification of the waveguiding effects, which will be reported by the authors' future publications. We gratefully acknowledge fruitful discussions with Krishanu Roychowdhury (Cornell University), Rui Zhu (Beijing Institute of Technology), Raj Kumar Pal (Georgia Institute of Technology), Simon Yves (CNRS), Romain Fleury (EPFL), Cheng He (Nanjing University), Zhiwang Zhang (Nanjing University), and Panayotis Kevrekidis (University of Massachusetts, Amherst). We are grateful for the support from NSF (CAREER-1553202 and EFRI-1741685).99Hasan2010 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). Qi2011 X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). Lu2014 L. Lu, J. D. Joannopoulos, and M. Soljac̆ić, Nat. Photonics 8, 821 (2014). Xiao2015 M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, Nat. Phys. 11, 240 (2015). Yang2015 Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Phys. Rev. Lett. 114, 114301 (2015). Lu2016 J. Lu, C. Qiu, M. Ke, and Z. Liu, Phys. Rev. Lett. 116, 093901 (2016). He2016 C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, Nat. Phys. 12, 1124 (2016).Mei2016 J. Mei, Z. Chen, and Y. Wu, Sci. Rep. 6, 32752 (2016). Fleury2016R. Fleury, A. B. Khanikaev, and A. Alù, Nat. Commun. 7, 11744 (2016). Zhang2017 Z. Zhang, Q. Wei, Y. Cheng, T. Zhang, D. Wu, and X. Liu, Phys. Rev. Lett. 118, 084303 (2017). Yves2017 S. Yves, R. Fleury, F. Lemoult, M. Fink, and G. Lerosey, New J. Phys. 19, 075003 (2017). Kane2013 C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39 (2014). Paulose2014J. Paulose, B. G. Chen, and V. Vitelli, Nat. Phys. 11, 153 (2015). Rocklin2016 D. Z. Rocklin, B. G. G. Chen, M. Falk, V. Vitelli, and T. C. Lubensky, Phys. Rev. Lett. 116, 135503 (2016). Stenull2016 O. Stenull, C. L. Kane, and T. C. Lubensky, Phys. Rev. Lett. 117, 068001 (2016). Bilal2017 O. R. Bilal, R. Süsstrunk, C. Daraio, and S. D. Huber, Adv. Mater. 29, 1700540 (2017). Huber2015 R. Süsstrunk and S. D. Huber, Science 349, 47 (2015). Nash2015 L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Proc. Natl. Acad. Sci. 112, 14495 (2015). Wang2015 P. Wang, L. Lu, and K. Bertoldi, Phys. Rev. Lett. 115, 104302(2015). Mousavi2015 S. H. Mousavi, A. B. Khanikaev, and Z. Wang, Nat. Commun. 6, 8682 (2015). Kariyado2015 T. Kariyado and Y. Hatsugai, Sci. Rep. 5, 18107 (2015). YWang2015Y.-T. Wang, P.-G. Luan, and S. Zhang, New J. Phys. 17, 073031 (2015). Pal2016a R. K. Pal, M. Schaeffer, and M. Ruzzene, J. Appl. Phys. 119, 084305 (2016). Pal2016b R. K. Pal and M. Ruzzene, New J. Phys. 19, 25001 (2016). Vila2017 J. Vila, R. K. Pal, and M. Ruzzene, Phys. Rev. B 96, 134307 (2017). Brendel2017Valley C. Brendel, V. Peano, O. J. Painter, and F. Marquardt, Proc. Natl. Acad. Sci. 114, E3390 (2017). Brendel2017SpinC. Brendel, V. Peano, O. Painter, and F. Marquardt, Phys. Rev. B 97, 020102(R) (2017). Yu2017 S.-Y. Yu, C. He, Z. Wang, F.-K. Liu, X.-C. Sun, Z. Li, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, arXiv:1707.04901 (2017). Salerno2016G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto, Phys. Rev. B 93, 085105 (2016). Weinstein2014 C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Proc. Natl. Acad. Sci. 111(24), 8759 (2014). Chaunsali2016 R. Chaunsali, F. Li, and J. Yang, Sci. Rep. 6, 30662 (2016). Chaunsali2017 R. Chaunsali, E. Kim, A. Thakkar, P. G. Kevrekidis, and J. Yang, Phys. Rev. Lett. 119, 024301 (2017). Prodan2017 E. Prodan, K. Dobiszewski, A. Kanwal, J. Palmieri, and C. Prodan, Nat. Commun. 8, 14587 (2017). Huber2016 S. D. Huber, Nat. Phys. 12, 621 (2016). Kane2005 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). Bernevig2006 B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006). Wu2015 L.-H. Wu and X. Hu, Phys. Rev. Lett. 114, 223901 (2015). Liu2000 Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, Science 289, 1734 (2000). Yu2006 D.-L. Yu, G. Wang, Y.-Z. Liu, J. Wen, and J. Qiu, Chinese Phys. 15, 0266 (2006). Sigalas1994 M. M. Sigalas and E. N. Economou, J. Appl. Phys. 75, 2845 (1994). Xiao2012 Y. Xiao, J. Wen, and X. Wen, J. Phys. D. Appl. Phys. 45, 195401 (2012).Hsiao2007 F. L. Hsiao, A. Khelif, H. Moubchir, A. Choujaa, C. C. Chen, and V. Laude, Phys. Rev. E 76, 056601 (2007). Pennec2008 Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur, and A. C. Hladky-Hennion, Phys. Rev. B 78, 104105 (2008). Pennec2009 Y. Pennec, B. Djafari-Rouhani, H. Larabi, A. Akjouj, J. N. Gillet, J. O. Vasseur, and G. Thabet, Phys. Rev. B 80, 144302 (2009). Wu2009 T. C. Wu, T. T. Wu, and J. C. Hsu, Phys. Rev. B 79, 104306 (2009). Oudich2010 M. Oudich, M. B. Assouar, and Z. Hou, Appl. Phys. Lett. 97, 193503 (2010). Casadei2012 F. Casadei, T. Delpero, A. Bergamini, P. Ermanni, and M. Ruzzene, J. Appl. Phys. 112, 064902 (2012). Torrent2013 D. Torrent, D. Mayou, and J. Sánchez-Dehesa, Phys. Rev. B 87, 115143 (2013). XWang2015 X. P. Wang, P. Jiang, T. N. Chen, and J. Zhu, AIP Adv. 5, 107141 (2015). Jiang2015 P. Jiang, X. P. Wang, T. N. Chen, and J. Zhu, J. Appl. Phys. 117, 154301 (2015). Baboly2016 M. Ghasemi Baboly, A. Raza, J. Brady, C. M. Reinke, Z. C. Leseman, and I. El-Kady, Appl. Phys. Lett. 109, 183504 (2016). Chen2017 Y. Chen, G. Hu, and G. Huang, J. Mech. Phys. Solids 105, 179 (2017).Fahy2007 F. Fahy and P. Gardonio, Sound and Structural Vibration: Radiation, Transmission and Response(Academic Press, Oxford, 2007) Second Edition.Deng2017 Y. Deng, H. Ge, Y. Tian, M. Lu, and Y. Jing, Phys. Rev. B 96, 184305 (2017).§ COMPARISON BETWEEN THE LUMPED MASS MODEL (PWE) AND FULL-SCALE MODEL (FEM)We corroborate the results obtained earlier based on the lumped mass model now by using the FEM, in which we account for all geometrical features described in Fig. <ref>b. As shown in Fig. <ref>, we clearly see an excellent agreement of the PWE method (red curves) with the FEM (green circles). It should be also noted that the p- and d-type degenerate mode shapes obtained through the lumped mass model comply with those obtained by the full-scale model (compare the inset images between Fig. <ref> and Fig. <ref>). In the dispersion relation obtained by the FEM, however, we observe that previously neglected shear-horizontal (SH_0) and shear (S_0) plate modes do appear for a thin plate. In our analysis, it is reasonable to ignore such modes and focus solely on anti-symmetric (A_0) flexural modes for the transverse source excitation at low frequencies as it was already demonstrated by full-scale simulations in Section VI and Section VII. We also see nearly flat dispersion curves due to the other modes of local resonators, which were not accounted for in the lumped mass model. Nevertheless, these modes have minimal coupling with the out-of-plane vibration of the plate, and these are away from the double Dirac cone. Therefore, the band-inversion process is not affected by them and it is reasonable to neglect them in the PWE method. § GRADED INTERFACE BETWEEN TWO TOPOLOGICALLY DISTINCT LATTICESHere we verify the scheme of reducing the gap observed at the Γ point in Fig. <ref>b. The gap emerges due to the breakage of the C_6 symmetry at the interface. Therefore, we minimize the effect of symmetry breakage by modifying the interface. Figure <ref>a shows a supercell, in which a topologically non-trivial lattice (R=1.1a/3) smoothly transitions to a topologically trivial lattice (R=0.8a/3) via four lattices with a gradient in their radii, i.e., R_1=1.05a/3, R_2=1.02a/3, R_3=0.95a/3, and R_4=0.9a/3. The gap reduction is confirmed in Fig. <ref>b. It is about 3 times lesser than the one observed in Fig. <ref>b. Spin nature of these interface modes is also intact as shown in Fig. <ref>c.§ THE PRESENCE OF DEFECTS ALONG THE WAVEGUIDEWe further examine the robustness of spin-dependent one-way flexural waves against certain defects present at the topological interface as shown in Fig. <ref>. The first defect, i.e., Defect 1, is introduced by removing four local resonators from one of topologically trivial unit cells along the interface (see Fig. <ref>a). We then excite a point on the left most end of the waveguide (shown by star marker) with low-reflecting boundary conditions and perform harmonic analysis for the same frequency as that in Fig. <ref>. We observe that the presence of this defect does not have any noticeable effect in terms of scattering the spin mode. The mode continues to propagate from the left to the right with a counterclockwise spin (the same as in Fig. <ref>b) and maintains similar modal amplitude before and after the defect location. Therefore, this defect would qualify as a non spin-mixing defect <cit.>. =1.0em We take another defect, i.e., Defect 2, in which we remove all six local resonators from the same unit cell (see Fig. <ref>b). Thus this defect is `stronger' than Defect 1. Under the same excitation and boundary conditions as those in the previous case, we observe that this defect affects the spin mode more drastically. In Fig. <ref>b, we see that the modal amplitude is almost negligible on the right side of the defect because of strong back-scattering. The defect causes the spin modes to mix. Consequently, the rightward-propagating counterclockwise spin mode is converted to leftward-propagating clockwise spin mode. This therefore represents a case of spin-mixing defect <cit.>. | http://arxiv.org/abs/1708.07994v2 | {
"authors": [
"Rajesh Chaunsali",
"Chun-Wei Chen",
"Jinkyu Yang"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170826163245",
"title": "Subwavelength and directional control of flexural waves in zone-folding induced topological plates"
} |
§ INTRODUCTIONParton distribution functions (PDFs) together with parton level matrix elements allow for a very accurate description of “hard” events in hadron-hadron and hadron-electron collisions. The bulk of such analysis is carried out within the framework of collinear factorization <cit.>. However, there exist classes of multi-scale processes where the use of more general schemes is of advantage. An example of such a process is a high-energy or low x limit of hard processes with s ≫ M^2 ≫Λ_QCD^2 where x = M^2/s. In such a scenario it is necessary to resum terms enhanced by logarithms ln 1/x to all orders in the α_s, which is achieved by BFKL evolution equation <cit.>. The resulting formalism called high-energy or k_T factorization <cit.> provides a factorization of such cross-sections into a TMD coefficient or “impact factor” and an “unintegrated” gluon density. Unfortunately, the high-energy factorization framework has important limitations that makes it cumbersome to apply it directly to an arbitrary process. For example it takes into account only gluon densities which is not acceptable for some quark initiated processes. What is more, since it is valid only in the low x region (x≲10^-2) by construction it is restricted to exclusive observables such that x of both gluons is fixed; e.g. description of processes involving fragmentation function, requiring integration over full x range, rises problems. In this contribution we summarize our efforts <cit.> to formulate a prescription to accommodate the advantages of both collinear and high-energy factorizations. In particular, our goal is to construct evolution equation (or rather a system of equations) for unintegrated parton distributions (with k_T-dependence) that would have the following properties: (i) it should resum the low x logarithms, (ii) has smooth continuation to the large x region, (iii) include both quarks and gluons, and (iv) reproduce the correct collinear limit given by DGLAP. To this end we extend the Catani-Hautmann (CH) <cit.> and Curci-Furmanski-Petronzio (CFP) <cit.> formalisms, which allows us for calculation of k_T-dependent splitting functions. For the moment we concentrate only on the real part of the quark splitting functions, and in the second step we use the newly calculated P_gq splitting to construct a low x evolution equation incorporating quark contributions.§ TMD SPLITTING FUNCTIONSSo far we have calculated the real emission parts of the quark TMD splitting functions (P_qg, P_gq, P_qq). The method used for the calculation is based on the two-particle-irreducible (2PI) expansion of refs. <cit.> and <cit.>. The details of the method can be found in <cit.> where the computations were performed. In this contribution we only highlight two crucial ingredients used in the calculation. The first are projection operators used to obtain factorization by decoupling the 2PI kernels in the momentum and helicity space. These projectors are generalization of the ones introduced in <cit.> accounting for the more general kinematics (featuring off-shell momenta on the incoming legs of the 2PI kernels):ℙ_g, in^ μν = k_⊥^μ k_⊥^ν/^2,ℙ_q, in = yp/2,where the corresponding kinematics is shown in Fig. <ref> and the involved momenta are parametrized as follows:k^μ = y p^μ + k_⊥^μ,q^μ = x p^μ + q_⊥^μ + q^2+^2/2x p· n n^μ,p'=k-q.The second new element is the construction of an appropriate vertex that can be used in the presence of off-shell particles such that the obtained results are gauge invariant. We quote here only one of the vertices that is needed for calculation of the P_gq splitting:Γ_g^*q^*q^μ (q, k, p') = igt^a (γ^μ - p^μ/p· qk).The remaining vertices and explanations on how we construct them can be found in <cit.>. The prescription to compute the TMD splitting function P̃_ij^(0) follows from the ladder expansion of refs. <cit.> and reads:K̂_ij(z, ^2/μ^2, ϵ)= z ∫d^2 + 2 ϵ/2(2π)^4+2ϵ∫ d q^2ℙ_j, in⊗K̂_ij^(0)(q, k) ⊗ℙ_i, out _P̃_ij^(0)(z, , , ϵ) Θ(μ_F^2+q^2),where K̂_ij^(0) is the first order expansion of the 2PI kernel of <cit.>, and ℙ_j, in and ℙ_i, out are the projection operators discussed above. As an example, in Fig. <ref>, we show the diagram representing the first order expansion of K̂_gq^(0) which is the only diagram necessary to compute the NLO P_gq^(0) TMD splitting function in the presented approach. Using eq. (<ref>) and integrating the angular dependence we obtain the followingangular averaged TMD splitting functions:P_qg^(0)(z, ^2/^2, ϵ)=T_R (^2/^2 + z(1-z) ^2)^2 [ z^2+(1-z)^2 + 4z^2(1-z)^2 ^2/^2], P_gq^(0)(z, ^2/^2, ϵ)= C_F [ 2^2/z |^2-(1-z)^2 ^2|- ^2 (^2(2-z) + ^2 z(1-z^2))/(^2 + z(1-z)^2)^2 + ϵ z ^2 (^2 + (1-z)^2 ^2)/(^2 + z(1-z)^2)^2], P_qq^(0)(z, ^2/^2, ϵ) =C_F (^2/^2 + z(1-z)^2) [ ^2 + (1-z^2)^2/(1-z)|^2 - (1-z)^2 ^2| + z^2 ^2 -z(1-z)(1 - 3z + z^2)^2 + (1-z)^2ϵ (^2 + z^2 ^2)/(1-z)(^2 +z(1-z)^2)],where =- z and z=x/y.The P_qg^(0) splitting function have been obtain previously <cit.> and we reproduce this result. On the other hand the results for P_gq^(0) and P_qq^(0) are new.§ EVOLUTION EQUATION WITH QUARKSOur starting point is the leading order BFKL equation which describes evolution in ln 1/xforthe dipole amplitude in the momentum space:F(x,^2) = F^0(x,^2) + α_s∫_x^1dz/z∫d^2'/π'^2[ F(x/z,| q+'|^2)-θ(^2-'^2) F(x/z,^2)]where α_s=C_Aα_s/π. This form is particularly usefulto promote the BFKL equation to a system of equations for quarks and gluons. We will incorporate a contribution from quarks in the following form: α_s/2π∫_x^1dz/z∫d^2'/π'^2 P_gq(z,',) Q(x/z,| q+'|^2), where Q(x/z,| q+'|^2) is a distribution of quarks.Next we introduce a resolution scale μ allowing us to decompose the kernel of the gluonic part of (<ref>) into a resolved real emission part with '^2>μ^2 and the unresolved part with '^2<μ^2. Additionally, since the integral over ' in the quark contribution is divergent it needs to be regulated. In the following we achieve this through introducing the same cut-off μ as used for the gluonic part. Technically this is obtained through including a theta function θ('^2-μ^2) in the quark term: [Note that, in the gluon case, this additional scale is just for technical convenience as 1/'^2 is regularized by the virtual contribution, in the case of quarks, μ scale is really needed for regularizing the corresponding expression.]F(x,^2) = F^0(x,^2) +α_s∫_x^1dz/z∫d^2'/π'^2 F(x/z,| q+'|^2)θ('^2-μ^2)+α_s∫_x^1dz/z∫d^2'/π'^2[ F(x/z,| q+'|^2)θ(μ^2- '^2)- θ(^2-'^2) F(x/z,^2)]+α_s/2π∫_x^1dz∫d^2'/π'^2P_gq(z,',) Q(x/z,|'+|^2)θ('^2-μ^2) .Equation (<ref>) is in a suitable form to perform resummation of the virtual and unresolved emissions by going to the Mellin space (x→ω→ x). The detailed steps of this resummation are presented in ref.<cit.> in the following we only present the final result:F(x,^2) = F̃^0(x,^2)+ α_s/2 π∫_x^1 dz/z∫_μ^2d^2 '/π'^2 [ Δ_R(z, ^2, μ^2)( 2 C_AF(x/z,|+'|^2) + C_F Q(x/z,|+'|^2) )-∫_z^1 dz_1/z_1Δ_R(z_1, ^2, μ^2)[ P̃'_gq(z/z_1, , )z/z_1] Q(x/z,|+'|^2) ].From the above expression we can see that the 1/'^2 singularity of the quark term is now regularized by the Regge formfactor Δ_R(z,^2,μ^2)≡exp(-α_sln1/zln^2/μ^2) in a direct analogy with the gluonic term.We note that the above resummation can be in a straight forward manner extended <cit.> to the situation where the gluon density is large and therefore subject to a nonlinear evolution equation, taking into account saturation effects <cit.>.As a last step we check the stability of the obtained result (<ref>) with respect to the cut-off μ. To this end we perform a convolution of the low z and fine z parts of the P_gq kernel with the quark density <cit.>. This leads us to the conclusion that as μ^2→0 the cutoff dependence gets weaker, see Fig. <ref>. § CONCLUSIONS We have generalized a framework of Catani and Hautmann and used it to calculate the real emission k_⊥-dependent P_qq, P_gq and P_qg splitting functions. These splitting functions were then used to construct evolution equation for gluons, receiving contribution from quarks. We have demonstrated that the singularity of the P_gq kernel can be regularized by the means of the same cut-off as in case of BFKL kernel. Furthermore, resumming the combined contribution of the virtual part and the small ' real part of the BFKL kernel to all orders in the strong coupling, one finds that both the pure gluonic contribution to the evolution equation as well as the quark induced term are finite if we send this cut-off to zero.In particular we have demonstrated, via performing one iteration of the kernels, that the equation has a realistic chance to be stable against variation of the cutoff parameter, since after one iteration the result stabilizes.To perform a fully consistent study of the complete system of k_T-dependent evolution equations,we need to calculate the virtual contributions to the quark-to-quark splitting function in the framework of k_T-factorization.Additionally, to demonstrate the completeness of our framework, the P_gg splitting function should be recalculated using the presented approach. utphys_spires | http://arxiv.org/abs/1708.08057v1 | {
"authors": [
"A. Kusina"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170827065420",
"title": "TMD splitting functions and the corresponding evolution equation"
} |
alpha Sales Forecast in E-commerce using Convolutional Neural NetworkKui Zhao College of Computer Science Zhejiang University Hangzhou, China [email protected] Can Wang College of Computer Science Zhejiang University Hangzhou, China [email protected]: December 30, 2023/ Accepted: December 30, 2023 ========================================================================================================================================================================================================================================================== In this paper we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it?More precisely, let PConf_n(S) be the space of ordered n-tuple of distinct points in S.Let f_n(S): PConf_n+1(S) →PConf_n(S) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n). We classify all continuous sections of f_n up to homotopy by proving the following.1. If S=ℝ^2 and n>3, any section of f_n(S) is either “adding a point at infinity" or “adding a point near x_k". (We define these two terms in Section 2.1; whether we can define “adding a point near x_k" or “adding a point at infinity" depends in a delicate way on properties of S. )2. If S=S^2 a 2-sphere and n>4, any section of f_n(S) is “adding a point near x_k"; if S=S^2 and n=2, the bundle f_n(S) does not have a section. (We define this term in Section 3.2)3. If S=S_g a surface of genus g>1 and for n>1, we give an easy proof of <cit.> that the bundle f_n(S) does not have a section.§ INTRODUCTIONLet M be a manifold. There is a natural geometric question: How can we continuously introduce a new point on M for any collection of n distinct points on M? We denote by PConf_n(M) the pure configuration space parametrizing ordered n-tuple of distinct points on M. Let f_n(M): PConf_n+1(M) →PConf_n(M) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n). There is a natural action of permutation group Σ_n on PConf_n(M) by permuting the n points. Permutation group Σ_n acts on the fiber bundle f_n(M) as well. Thus we get a new fiber bundle F_n(M): PConf_n+1(M)/Σ_n →PConf_n(M)/Σ_n, given by F_n(x_0,{x_1,… ,x_n}):={x_1,… ,x_n}. The quotient PConf_n(M)/Σ_n=:Conf_n(M) is called the configuration space parametrizing unordered n-tuple of distinct points on M. In this article, we will study the existence and uniqueness of sections off_n(M) and F_n(M) when M is a surface.The study of sections of configuration spaces of open manifolds goes back to the work of McDuff and Segal <cit.> <cit.>. They introduce a point “at infinity", which allows them to prove homological stability for configuration spaces. For closed manifolds, the possibility of adding a point depends on the topology of the manifold. For a manifold M with a nowhere vanishing vector field, Cantero and Palmer <cit.>, Berrick, Cohen, Wong and Wu <cit.> introduced another way to add a new point by adding a point infinitesimally near an old point using the vector field. This allows Ellenberg and Wiltshire-Gordon <cit.> to improve eventual polynomiality to immediate polynomiality of the betti numbers of PConf_n(M) for some closed manifolds M. The following figures illustrate adding a point “at infinity" and adding a point infinitesimally near an old point on the plane ℝ^2.We call a section s of f_n(ℝ^2) (resp. f_n(S^2)) “adding a point near x_k" if s is homotopic to an element in the collection of sections Add_n,k(ℝ^2) (resp. Add_n,k(S^2)). Informally, we assign x_0 at a sufficiently small distance to x_k along some nonvanishing vector field. See Figure <ref> for a demonstration of “adding a point near x_k". Notice that there are infinitely many homotopy classes of sections in Add_n,k(ℝ^2) and Add_n,k(S^2) and they are classified by a kind of twists or sections of a circle bundle. See Section 2.1 and Section 3.2 for formal definitions of Add_n,k(ℝ^2) and Add_n,k(S^2) respectively.We call a section s of f_n(ℝ^2) “adding a point at infinity" if s is homotopic to an element in the collection of sections Add_n,∞(ℝ^2); see Figure <ref>. Informally, we consider ℝ^2 as S^2 missing a point ∞, we can assign x_0 at a sufficiently small distance to ∞ along some nonvanishing vector field. See Section 2.1 for a formal definition of Add_n,∞(ℝ^2). Let S_g be a surface of genus g and S^2 be the 2-sphere. In this paper, we will classify the sections of the fiber bundle f_n(S) for 3 cases: ℝ^2, S^2 and S_g when g>1. Here by section we mean continuous section. The following holds:(1) If S=ℝ^2 and n>3, any section of f_n(S) is either “adding a point at infinity" or “adding a point near x_k" for some 1≤ k≤ n.(2) If S=S^2 and n=2, the bundle f_n(S) does not have a section. If S=S^2 and n>4, any section of f_n(S) is “adding a point near x_k" for some 1≤ k≤ n.(3) If S=S_g a surface of genus g>1 and for n>1, the bundle f_n(S) does not have a section.For unordered case, we have the following corollary.The following holds:(1) If S=ℝ^2 and n>3, any section of F_n(S) is “adding a point at infinity";(2) If S=S^2 and n>4 or n=2,3, the bundle F_n(S) does not have a section. We discuss the exceptional cases when n=3 for S=S^2 in Section <ref>. Our method does not work for the case n=4 but <cit.> proved that F_4(S^2) does not have sections. The g=1 case seems to be more complicated to analyze, therefore we do not pursue here. Notice that the construction “adding a point near x_k" works for the torus as well; see Section 2.1. It is classical that f_n(ℝ^2) admits a section. In <cit.>, Fadell showed that when n>2, the bundle f_n(S^2) admits a section. The unordered case for S=ℝ^2, i.e. (1) of Corollary <ref> has been proved by <cit.> and <cit.>. In <cit.>, they prove the case (2) of Corollary <ref>, and even stronger, they deal with the multi-section problems. All the previous proofs make use of the braid relation and the presentations of braid groups and do not imply (1) and (2) in Theorem <ref>. Our main novelty is to use the characterization of lantern relation in analyzing the canonical reduction systems. The canonical reduction system uses the Thurston classification of isotopy classes of diffeomorphisms of surfaces. This idea originated from <cit.>. The ordered case for S=S_g of g>1, i.e. (3) of Theorem <ref> has been proved by <cit.>. Their proof makes heavily use of the presentations of surface braid group. We give a simpler proof using the cohomology of surface braid group and a classification theorem in <cit.>.The structure of the paper * In Section 2, we introduce the construction and the main tool we use: canonical reduction system.* In Section 3, we reduce Theorem <ref>(1) to a more algebraic statement Theorem <ref>.* In Section 4, we prove Theorem <ref>, which is the main work of this paper.* In Section 5, we prove Theorem <ref>(2) by reducing it to Theorem <ref>(1).* In Section 6, we prove Theorem <ref>(3) by a classification of maps between configuration spaces of surfaces in <cit.>.* In Section 7, we ask further questions.Acknowledgements The author would like to thank Kevin Casto for telling her about the construction of adding a nearby point and thank Nir Gadish for discussion. She would also like to thank Paolo Bellingeri, Dan Margalit,Cihan Bahran and an anonymous referee for pointing out many references of previous works on braid groups and typos. Finally, she would like to thank her advisor Benson Farb for his extensive comments and for his invaluable support from start to finish. § THE CONSTRUCTION AND BACKGROUND ON CANONICAL REDUCTION SYSTEMSLet S be a surface and let PConf_n(S) the pure configuration space be the space of ordered n-tuple of distinct points on S. The natural embedding PConf_n(S)⊂ S^n gives the topology on PConf_n(S). Let f_n(S): PConf_n+1(S) →PConf_n(S) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n).There is a natural action of permutation group Σ_n on PConf_n(S) by permuting the n points. Thus the quotient space Conf_n(S) is the space of unordered n-tuple of distinct points in S. Permutation group Σ_n acts on the fiber bundle f_n(S) as well. Let F_n(S): PConf_n+1(S)/Σ_n →PConf_n(S)/Σ_n be the map given by F_n(x_0,{x_1,… ,x_n}):={x_1,… ,x_n}. The subject of this section is to classify the sections of the fiber bundles f_n(ℝ^2) and F_n(ℝ^2). §.§ Constructing sectionsIn this subsection we give constructions of sections of the fiber bundle f_n(ℝ^2). There are two cases: “adding a point near x_k" and “adding a point at infinity". These constructions originate from Berrick, Cohen, Wong and Wu <cit.>, but the idea appeared in <cit.>.Case 1: adding a point near x_k. DefinePConf_n,k(ℝ^2)={(v_k,x_1,...,x_n)|x_1,...,x_nbenpoints on ℝ^2andv_kbe a unit vector atx_k}.This is the total space of a circle bundle by forgetting the vector v_kS^1→PConf_n,k(ℝ^2)→PConf_n(ℝ^2).Equip ℝ^2 with the Euclidean metric. Setϵ(x_1,...,x_n)=1/2min_1≤ i≠ j≤ n{d(x_i,x_j)}.By the definition of ϵ(x_1,...,x_n), setting x_0 to be the image of the v_k-flow at time ϵ(x_1,...,x_n) from x_k gives a map:em_n,k(ℝ^2):PConf_n,k(ℝ^2)↪PConf_n+1(ℝ^2). Composing a continuous section s:PConf_n(ℝ^2)→PConf_n,k(ℝ^2) of the fiber bundle (<ref>) with em_n,k(ℝ^2) gives a section of the fiber bundle f_n(ℝ^2).We denote by Add_n,k(ℝ^2) the collection of sections of f_n(ℝ^2) consisting of compositions of a section of (<ref>) with em_n,k(ℝ^2).Notice that there are infinitely many homotopy classes of sections in Add_n,k(ℝ^2) and they are in one-to-one correspondence with the homotopy classes of sections of (<ref>).Case 2: adding a point at infinity. Let us call the north pole of a 2-sphere the point at infinity ∞. Then ℝ^2≅ S^2-∞ through the stereographic projection. DefinePConf_n,∞(ℝ^2)={(v_∞,x_1,...,x_n)|x_1,...,x_nbenpoints on ℝ andv_∞ be a unit vector at ∞}. This is the total space of a circle bundle by forgetting the vector S^1→PConf_n,∞(ℝ^2)→PConf_n(ℝ^2). Equip S^2 with the spherical metric; i.e. the metric that is induced from the standard embedding S^2⊂ℝ^3. Setϵ(x_1,...,x_n)=1/2min_1≤ i≤ n{d(x_i,∞)}.By the definition of ϵ(x_1,...,x_n), setting x_0 to be the image of the v_∞-flow at time ϵ from ∞ gives a map:em_n,∞(ℝ^2):PConf_n,∞(ℝ^2)↪PConf_n+1(ℝ^2). Composing a continuous section s:PConf_n(ℝ^2)→PConf_n,∞(ℝ^2) of the fiber bundle (<ref>) with em_n,∞(ℝ^2) gives a section of the fiber bundle f_n(ℝ^2).We denote by Add_n,∞(ℝ^2) the collection of sections of f_n(ℝ^2) consisting of compositions of a section of (<ref>) with em_n,∞(ℝ^2).Notice that there are infinitely many homotopy classes of sections in Add_n,∞(ℝ^2) and they are in one-to-one correspondence with the homotopy classes of sections of (<ref>). §.§ BackgroundIn this subsection we discuss some properties of canonical reduction systems and the lantern relation. Let S=S_g,p^b be a surface with b boundary components and p punctures. Let Mod(S) (reps. PMod(S)) be the mapping class group (resp. pure mapping class group) of S, i.e. the group of isotopy classes of orientation-preserving diffeomorphisms of S fixing the boundary components pointwise and punctures as a set (resp. pointwise).By “simple closed curves", we often mean isotopy class of simple closed curves, e.g. by “preserve a simple closed curve", we mean preserve the isotopy class of a curve. Thurston's classification of elements of Mod(S) is a very powerful tool to study mapping class groups. We call a mapping class f∈Mod(S) reducible if a power of f fixes a nonperipheral simple closed curve. Each nontrivial element f∈Mod(S) is of exactly one of the following types: periodic, reducible, pseudo-Anosov. See <cit.> and <cit.> for more details. We now give the definition of canonical reduction system. A reduction system of a reducible mapping class h in Mod(S) is a set of disjoint nonperipheral curves that h fixes as a set up to isotopy. A reduction system is maximal if it is maximal with respect to inclusion of reduction systems for h. The canonical reduction system CRS(h) is the intersection of all maximal reduction systems of h. For a reducible element f, there exists n such that f^n fixes each element in CRS(f) and after cutting out CRS(f), the restriction of f^n on each component is either periodic or pseudo-Anosov. See <cit.>. Now we mention three properties of the canonical reduction systems that will be used later.CRS(h^n)=CRS(h) for any n.This is classical; see<cit.>.For a curve a on a surface S, denote by T_a the Dehn twist about a. For two curves a,b on a surface S, let i(a,b) be the geometric intersection number of a and b. For two sets of curves P and T, we say that S and T intersect if there exist a∈ P and b∈ T such that i(a,b)≠ 0. Notice that two sets of curves intersecting does not mean that they have a common element.Let h be a reducible mapping class in Mod(S). If {γ} and CRS(h) intersect, then no power of h fixes γ. Suppose that h^n fixes γ. Therefore γ belongs to a maximal reduction system M. By definition, CRS(h)⊂ M. However γ intersects some curve in CRS(f); this contradicts the fact that M is a set of disjoint curves.Suppose that h,f∈ Mod(S) and fh=hf. Then CRS(h) and CRS(f) do not intersect.By conjugation, we have that CRS(hfh^-1)=h(CRS(f)). Since hfh^-1=f, we get that CRS(f)=h(CRS(f)). Therefore h fixes the whole set CRS(f). A power of h fixes all curves in CRS(f). By Proposition <ref>, curves in CRS(h) do not intersect curves in CRS(f). Now, we introduce a remarkable relation for Mod(S) that will be used in the proof.There is an orientation-preserving embedding of S_0,4⊂ S and let x,y,z,b_1,b_2,b_3,b_4 be simple closed curves in S_0,4 that are arranged as the curves shown in the following figure.In Mod(S) we have the relationT_xT_yT_z=T_b_1T_b_2T_b_3T_b_4.This is classical; see <cit.>. § AN ALGEBRAIC RESULT AND HOW IT IMPLIES (1) OF THEOREM <REF>In this section we give an algebraic result about the braid groups and prove how it implies (1) of Theorem <ref>. PConf_n(ℝ^2) and PConf_n+1(ℝ^2) are both K(π,1) spaces. This can be seen by induction on n and taking the long exact sequence of homotopy groups of the fiber bundle f_n(ℝ^2). Therefore, the homotopy classes of sections of f_n(ℝ^2) only depend on the homomorphisms of the fundamental groups. Let PB_n=π_1(PConf_n(ℝ^2)) and let F_n be a free group of n generators. The fundamental groups of the fiber bundle f_n(S) gives us the following short exact sequence, i.e. the Fadell-Neuwirth short exact sequence:1→ F_n→ PB_n+1 PB_n→ 1. Let D_n be the disk with n punctures {x_1,...,x_n} and D_n+1 be the disk with n+1 punctures {x_0, x_1,...,x_n}and the forget map forgets the point x_0. We view PB_n and PB_n+1 as mapping class groups as the following:PB_n=PMod(D_n) and PB_n+1=PMod(D_n+1).A simple closed curve a on D_n separates D_n into two parts: the outside of a, i.e. the component containing the boundary of D_n and the inside of a, i.e. the one not containing the boundary of D_n. We say that a surrounds x_k if x_k∈ the inside of a. The following algebraic result on the splittings of the exact sequence (<ref>) is a key ingredient in the proof of Theorem <ref>.Suppose that we have a section s: PB_n→ PB_n+1. Then the image s(PB_n) either preserves a simple closed curve c surrounding points {x_1,...,x_n}, or preserves a simple closed curve c surrounding {x_i,x_0} for some i∈{1,2,...,n}. The rest of this subsection focuses on how Theorem <ref> implies part (1) of Theorem <ref>. Let c be a curve inside D_n+1 surrounding k points. Let D_k^l be a disk with k punctures and l open disks removed. We call the boundary of the l disks the small boundary components and the original boundary of D the big boundary component. See the following figure for a geometric explanation.Let PB_k,l:=PMod(D_k^l) be the pure mapping class group of D_k^l. The difference between punctures and boundary components is that the Dehn twist about a puncture is trivial but the Dehn twist about a boundary component is nontrivial. The following proposition describes the centralizer of T_c. Denote the centralizer of T_c by C_PB_n+1(T_c). C_PB_n+1(T_c) satisfies the following exact sequence1→ℤ PB_k× PB_n+1-k,1→ C_PB_n+1(T_c)→ 1where k is the number of points that c surround. This is classical. The centralizer of T_c is the subgroup of Mod(D_n) that fixes c. The curve c separates D_n into two components: C_1 that contains the boundary and C_2 that does not contain the boundary. Since C_1 and C_2 are not homeomorphic, we have that C_PB_n+1(T_c) only contains elements that preserve C_1 and C_2. Therefore, our statement holds. Now we are ready to prove (1) of Theorem <ref>.Let g:PConf_n(ℝ^2)→PConf_n+1(ℝ^2) be a section of the fiber bundle f_n(ℝ^2). Let s=g_*: PB_n→ PB_n+1 be the induced map on the fundamental groups of g. By Theorem <ref>, the image s(PB_n) preserves a curve c that either surrounds 2 points or n points. Therefore, s(PB_n) is in the centralizer of T_c in PB_n+1 by the fact that fT_cf^-1=T_f(c).Case 1: when c surrounds {x_0,x_k}. PB_2≅ℤ, which is generated by the Dehn twist about the boundary component. From Proposition <ref> we have1→ℤℤ× PB_n-1,1→ C_PB_n+1(T_c)→ 1.Therefore C_PB_n+1(T_c)≅ PB_n-1,1. The inclusion PB_n-1,1↪ PB_n+1 is induced by gluing a 2-punctured disk inside the small boundary of D_n-1^1.On the other hand, we have thatπ_1(PConf_n,k(ℝ^2))=PB_n-1,1.The fundamental groups of the fiber bundle (<ref>) is the following exact sequence:1→ℤ PB_n-1,1→ PB_n→ 1.Here T_d is the Dehn twist about the small boundary component. The embedding em_n,k: PConf_n,k(ℝ^2)↪PConf_n+1(ℝ^2) induces a homomorphism on the fundamental group em_n,k*:PB_n-1,1→ PB_n+1. On the mapping class group level, since T_d in PB_n-1,1 is mapped to the Dehn twist about a curve surrounding {x_0,x_k}, we know that em_n,k* is also induced by gluing a 2-punctured disk inside the small boundary of D_n-1^1. The theorem holds.Case 2: when c surrounds {x_1,...,x_n}. Since PB_1,1≅ℤ×ℤ, which is generated by the Dehn twists about the two boundaries, we have the following exact sequence:1→ℤℤ×ℤ× PB_n→ C_PB_n+1(T_c)→ 1.On the mapping class group level, PB_n× PB_1,1→ C_PB_n+1(T_c)→ PB_n+1 is induced by gluing D_1^1 outside the big boundary component of D_n. Therefore ℤ× PB_n≅ C_PB_n+1(T_c) and the generator of ℤ is mapped to T_cT_b^-1 where b is the big boundary of D_n+1.On the other hand, we have thatπ_1(PConf_n,∞(ℝ^2))=ℤ× PB_n.The embedding em_n,∞: PConf_n,∞(ℝ^2)↪PConf_n+1(ℝ^2) induces em_n, ∞*:ℤ× PB_n→ PB_n+1 on the fundamental groups. On the level of mapping class groups, since ℤ maps to T_cT_b^-1, we know that em_n,∞* is induced by the embedding of D_n in D_n+1 and maps the generator of ℤ to T_cT_b^-1. Therefore, em_n,∞* is induced by gluing D_1^1 outside the big boundary component of D_n as well. Our theorem holds. The classification of the sections of the fiber bundle f_n(S) is not entirely the same as the classification of the splittings of the exact sequence (<ref>). There is an subtlety coming from the choice of base point in the fundamental groups. Therefore, we classify the splittings of the exact sequence (<ref>) up to conjugacy. In Theorem <ref>, all the choices of c is coming from a conjugacy by an element F_n; thus they decide the same sections. § THE PROOF OF THEOREM <REF>Throughout the section we prove Theorem <ref>, which implies Theorem <ref>(1). The strategy of the proof is the following. We assume that there exists a section s:PB_n→ PB_n+1, i.e. f_n(ℝ^2)_*∘ s=id. The strategy is that we first determine s(T_a) for any simple closed curve a on D_n. We first prove that the lift s(T_a) is always a multi-twist about at most two curves on D_n+1; these two curves or one curve are either trivial or isotopic to a after forgetting the point x_0. This is done by using a result of McCarthy on centralizer of pseudo-Anosov element and lantern relation. We find a generating set of PB_n consisting of Dehn twists about curves bounding two points. We then argue depending on whether s(T_a) is a multi-twist on two curves or a single twist. The main tool of this part is Proposition <ref>, characterizing lantern relation that we deduce from Thurston's construction. §.§ Step 1: constrain the image of s(T_c) for a simple closed curve c The following proposition characterizes intersection number 2 of two curves and will be used many times in the proof.Let i(a,b)≠ 0. Then T_aT_b is a multitwist if and only if i(a,b)=2.This result was previously obtained by Margalit <cit.> and Hamidi-Tehrani <cit.>. We give a different proof using Thurston's construction; see e.g. <cit.>. There is a subspace T of S that a,b fills, i.e. the tubular neighborhood of a∪ b. Let ⟨ T_a,T_b⟩ be the group generated by T_a and T_b in Mod(T). Thurston's theorem says that when a,b fill, there is a representation ρ: ⟨ T_a,T_b⟩→PSL(2,ℝ) such that T_a→[ 1 -i(a,b); 0 1 ] and T_b→[10; i(a,b)1 ].ρ(h) is parabolic if and only if h is reducible on T. We know that ρ(T_aT_b)= [ 1 -i(a,b); 0 1 ][10; i(a,b)1 ]=[ 1-i(a,b)^2-i(a,b); i(a,b)1 ]Since Trace(ρ(T_aT_b))=2-i(a,b)^2, we know that T_aT_b is reducible on T if and only if i(a,b)=2. By the lantern relation, we know that T_aT_b is a multitwist when i(a,b)=2. The following lemma determines s(T_a) for any simple closed curve a on D_n.Let a be a simple closed curve on D_n, then s(T_a) can only be one of the following three cases:(1) It can be a Dehn twist T_a' about a curve a' on D_n+1 such that after forgetting x_0, we have a'=a.(2) It can be a multitwist T_a'T_c^m (i.e. a product of twists on disjoint curves) about two curves a' and c on D_n+1 for m∈ℤ, where c surrounds 2 points {x_0,x_k} and after forgetting x_0, we have that a'=a.(3) It can be T_a'(T_a'T_a”^-1)^n, where a' and a” are disjoint on D_n+1such that after forgetting x_0, we have that a'=a”=a.We start with a the proof of the claim:After forgetting x_0, any element of CRS(s(T_a)) is either a or surrounding one puncture (trivial). The centralizer of T_a contains T_b when a,b are disjoint curves. By injectivity of s, the centralizer of s(T_a) contains a copy of ℤ^2 when n>3. By <cit.> that the centralizer of a pseudo-Anosov element is virtually cyclic, we know that s(T_a) is not pseudo-Anosov. The injectivity of s also implies that s(T_a) is not a torsion element. Therefore we have that s(T_a) is reducible under Thurston's classification of mapping classes. Assume there exists b'∈CRS(s(T_a)) such that after forgetting x_0, we have that b is not trivial and b≠ a. Since b'∈CRS(s(T_a)), we have that a power of s(T_a) fixes b'. Also a power of any mapping class that commutes with s(T_a) fixes b as well. We break our discussion into the following two cases. * Case 1: If i(a,b)≠ 0, then no power of T_a fixes b. However we also have some power of s(T_a) fixes b'. This is a contradiction.* Case 2: If i(a,b)=0 but b≠ a, then there exists a curve c such that i(c,b)≠ 0 but i(c,a)=0. Since s(T_c) commutes with s(T_a), we know that s(T_c) preserves CRS(s(T_a)). However i(b,c)≠ 0, which shows that no power of T_c preserve b. This contradicts the fact that a power ofs(T_c) preserves b'. By the disjointness of curves in canonical reduction system, we have that CRS(s(T_a)) contains at most 2 curves. We break the rest of the proof into 3 cases. * Case 1: CRS(s(T_a)) only contains one curve a'. It depends on the location of x_0, only one side of a' will contain x_0, which means only that side could s(T_a) could possibly be not identity. If a' surrounds x_0 and a surrounds more than 2 points, there is a curve b inside of a containing 2 points, therefore s(T_a) fixes CRS(s(T_b)) inside of a', which means s(T_a) does not acts as pseudo-Anosov inside a'. This proves that a power of s(T_a) is the identity on the inside. Therefore a power of s(T_a) is a power of the T_a'. Since s(T_a) is a lift of T_a, we have that a power of s(T_a)T_a'^-1 is the identity. Therefore s(T_a)=T_a'.If a' surrounds x_0 and a surrounds 2 points, we position a as in the Figure <ref>.By the lantern relation, we haveT_cT_dT_b=T_aT_e.Since T_b,T_c,T_d,T_e commutes with T_a, we have that s(T_b),s(T_c),s(T_d),s(T_e) fix a' and therefore is identity on the component that a' stay. Since s(T_a) is a product of s(T_b),s(T_c),s(T_d),s(T_e), we know that s(T_a) is also identity in the interior of a'. Therefore s(T_a)=T_a'. The case when a' does not surround x_0 is similar. * Case 2:CRS(s(T_a)) only contains two curves a' and c such that c surrounds 2 points {x_0,x_k}.On both the exterior of a' and the interior of a' pinching curve c, we know that s(T_a) is identity. Therefore we know that s(T_a) is the multi-twist on a',c. * Case 3:CRS(s(T_a)) only contains two curves a' and a” such that after forgetting x_0, both curves a' and a” become a.On both the interior of a' and the exterior of a”, we know that s(T_a) is identity. Therefore we know that s(T_a) is the multi-twist on a',a”. In the following argument, we will use small letters like a,b,c,... to represent simple closed curves on D_n and small letters with a prime or double primes like a',a”,b', ... to represent the canonical reduction systems of s(T_a), s(T_b),.... If we have two curves in CRS(s(T_a)), we use a' and a”.§.§ Step 2: the case of adding points at infinityOn D_n, we call a simple closed curve surrounding 2 points by a basic simple closed curve. The following lemma gives one condition for s(PB_n) to preserve a simple closed curve surrounding {x_1,...,x_n}.If the canonical reduction system of any basic simple closed curve does not contain a curve surrounding x_0, then s(PB_n) preserves a simple closed curve surrounding {x_1,...,x_n}. Suppose that there exists a simple closed curve a such that CRS(a) contains a curve surrounding x_0. We call a the innermost if a surrounds k points and the canonical reduction systems of all curves surrounding k-1 points does not contains a curve surrounding x_0. Take an innermost curve a such that a surrounds k points in D_n. By the assumption of Lemma <ref>, we have that k>2. There are three cases according to Lemma <ref>.* Case 1: CRS(a)={a'} such that after forgetting x_0, we have a'=a. We take b and c inside D_n as in the following figure, we have the lantern relation T_bT_c=T_eT_aT_d^-1.Because b,c,d,e surround less points than k and a is the innermost curve, we know that CRS(s(T_b)), CRS(s(T_c)), CRS(s(T_d)) and CRS(s(T_e)) each only contains one curve not surrounding x_0 and we denote them by b',c',d',e'. Since T_e, T_a and T_d commute with each other, their canonical reduction systems would be disjoint. By Lemma <ref>, we know that s(T_eT_aT_d^-1) is also a multitwist. Therefore by Lemma <ref>, we know that i(b',c')=2. However CRS(T_b'T_c') does not contain a' because a' surround x_0 but b',c' do not. This is a contradiction.* Case 2: CRS(a)={a',a”} such that a” surrounds 2 points {x_0,x_k} and after forgetting x_0, we have that a'=a. * Case 3: CRS(a)={a',a”} such that after forgetting x_0, we have that a'=a”=a.Case 2 and 3 can be proved using a similar argument as in Case 1. We construct the same lantern relation and use the fact that a is the innermost curve to reach a contradiction. Therefore if the canonical reduction systems of all basic simple closed curves do not contain a curve surrounding x_0, then the canonical reduction systems of any curve does not surround x_0. This is true for the center element of PB_n as well. Let c be the boundary curve of D_n. Then CRS(c)={c'} does not contain x_0. However, all Dehn twists commute with T_c which preserves c'.Now we introduce a generating set for PB_n. Consider the n-punctured disk D_n in Figure <ref>. Let L be a segment below all the other points. Let L_1,...,L_n be segments connecting x_1,...,x_n to the segment L. Figure <ref> is the corresponding figure for D_n+1. For {i_1,...,i_k} a subset of {1,...,n}, let a_i_1i_2...i_k be the boundary curve of the tubular neighborhood of L∪∪_m=1^kL_i_k. Denote by A_i_1i_2...i_k the Dehn twist about a_i_1i_2...i_k.For {i_1,...,i_k} a subset of {0,1,...,n}, let b_i_1i_2...i_k be the boundary curve of the tubular neighborhood of L∪∪_m=1^kL_i_k. Denote by B_i_1i_2...i_k the Dehn twist aboutb_i_1i_2...i_k. See Figure <ref> for an example of a curve representing a_124.The following proposition describes a generating set of the group PB_n. There is a generating set of PB_n consisting of all the Dehn twists about the basic curves a_ij for 1≤ i<j≤ n. This is classical and can be prove it by induction on the exact sequence1→ F_k→ PB_k+1→ PB_k→ 1.This generating set is given by Artin; e.g. see <cit.>§.§ Step 3: Finishing the proof of Theorem <ref>In this subsection, we prove Theorem <ref>. We break the proof into several cases. By Lemma <ref>, we only need to consider the case that there exists at least one basic simple closed curve a such that some element of CRS(a) surrounds x_0. We break our discussion into the following four cases.Case 1: The canonical reduction systems of any basic curves only contain one curve a' and a' surrounds x_0.Let a,b,c,d be the curves in Figure <ref>. We have the lantern relation T_aT_bT_c=T_d.Since T_c and T_d commute, s(T_c)=T_c' and s(T_d)=T_d' also commute. Therefore s(T_a)s(T_b)=T_c'^-1T_d' is a multitwist by Lemma <ref>. By Lemma <ref>, we know that i(b',a')=2 as in Figure <ref>. Suppose that b' does not surround x_0. By the lantern relation, T_a'T_b'=T_d'T_e'T_c'^-1. Since s(T_d) and s(T_c) are commuting multicurves, s(T_d)s(T_c)^-1 is multicurve as well. Since a,b,c are basic curves, we know that s(T_d)=T_d'T_e' and s(T_c)=T_c'. By the same reason, we have that s(T_f)=T_f' in Figure <ref> and <ref>.In the following, we prove that s(PB_n) preserves b_01. Under Notation <ref> for PB_n, we have a=a_12, b=a_23, c=a_13 and d=a_123. We also have that s(A_12)=B_012, s(A_23)=B_23 and s(A_13)=B_013. Since A_ij generates PB_n, all we need to show is that s(A_ij) preserves b_01. Since CRS(d) contains b_01, any curve disjoint from d preserves b_01. We only need to consider the curves that intersect with d. Without loss of generality, we only need to show that s(A_14), s(A_24) and s(A_34) preserve b_01. By the assumption of Case 1, we only need to show that the CRS(a_14), CRS(a_24) and CRS(a_34) are disjoint from b_01. Since i(a_12,a_34)=0, we have that CRS(a_12) is disjoint from CRS(a_34), which means that CRS(a_34) is disjoint from b_01.Since s(T_f)=T_f' in Figure <ref> and Figure <ref>, CRS(a_24) is also disjoint from b_01. We have the following lantern relation:A_13A_34A_14=A_134. The image of relation under lift s is:B_013B_34s(A_14)=s(A_134). A_134 commutes with A_13 and A_34, thus CRS(a_134) is disjoint from b_013 and b_34. The only possible curves are b_01 and b_0134. If s(A_134)=B_0134, we have another lantern relation in D_n+1:B_013B_34B_014=B_0134B_01. This proves that s(A_14)=B_014B_01^-1 preserving b_01=e'. If CRS(a_134) contains b_01, we also have that s(A_14) preserves b_01=e'. The case when b' surrounds x_0 follows from the same argument. Case 2: There exists a basic simple curve a such that CRS(a) has two curves and both are isotopic to a after forgetting x_0.Let b,c,d,e be curves in Figure <ref>. We have the lantern relation T_bT_cT_d=T_eT_a.Since b,c,d,e are disjoint from a, we have that CRS(b), CRS(c), CRS(d) and CRS(e) are disjoint from {a',a”}. Therefore s(T_b)=T_b', s(T_c)=T_c', s(T_d)=T_d' and s(T_e)=T_e' as in Figure <ref>. But we also have the lantern relation T_b'T_c'T_d'=T_e'T_a'. Thus s(T_a)=T_a'. This contradicts the assumption of Case 2.Case 3: There exists a basic simple curve a such that CRS(a) has two curves a',a” such that a' is isotopic to a and a” is trivial after forgetting x_0, and a' surrounds a”. We arrange a to be a_12 and a',a” to be b_01,b_012. Then we have s(A_12)=B_012B_01^k for k≠ 0 by Lemma <ref>. Without loss of generality, we only need to show that CRS(a_13) and CRS(a_23) are disjoint from b_01. First of all, we have the following lantern relation:A_123A_34A_124=A_12A_1234. Since all of the curves above are disjoint from a_12, their canonical reduction systems are disjoint from a_12'=b_01 and a_12”=b_012. We have the lantern relation:B_0123B_34B_0124=B_012B_01234.Since s(A_12)=B_012B_01^k, there exists at least one other curve in a_123, a_34, a_124, a_1234, whose canonical reduction system contains b_01. We break our discussion into the following four subcases depending on whether b_01 is an element in CRS(A_1234), CRS(A_123),CRS(A_124) orCRS(A_34), respectively.Subcase 1 and 2: In the first two cases, it is clear that CRS(a_13) and CRS(a_23) are disjoint from b_01 because a_13 and a_23 are disjoint from a_123 and a_1234.Subcase 3: By i(a_14,a_124)=0, we have that b_014∈CRS(a_14) and b_01 does not intersect CRS(a_14). Since i(a_23,a_14)=0, we have that b_014 does not intersect CRS(a_23). Suppose CRS(a_23) contains another curve z that is trivial after forgetting x_0. Since a_23 is disjoint from a_123 and a_14, we have that z has to be disjoint from b_014 and b_0123. The only possibility is that z=b_01. Because of the disjointness of a_123 and a_12, we have that s(A_123) preserves CRS(a_12). This shows that s(A_123) preserves b_01. We have a lantern relationA_12A_23A_13=A_123After applying the homomorphism s to the above relation, all of the above element except s(A_13) preserves b_01. Therefore s(A_13) fixes b_01.Subcase 4:Since i(a_234,a_34)=0, we have that b_234∈CRS(a_234). Since i(a_123,a_12)=0, we have that b_0123∈CRS(a_123). Therefore b_23∈CRS(a_23) and CRS(a_23) may contain another curve z that is trivial after forgetting x_0. However a_23 is disjoint from a_123 and a_234, which implies that z is disjoint from b_0123 and b_234. Therefore z can only be b_01. By the same argument as Subcase 3, we know that A_13 also fixes b_01. Case 4: There exists a basic simple curve a such that CRS(a) has two curves a',a” such that a' is isotopic to a and a” is trivial after forgetting x_0, and a' does not surround a”. Let a',a” be positioned into the following Figure <ref> such that a'=b_34 and a”=b_01. If a curve c is disjoint from a_34, then s(T_c) preserves b_01. Therefore without loss of generality, we only need to show that s(A_23) and s(A_13) preserve b_01.Since i(a_12,a_34)=0, we have that b_012∈CRS(a_12). Since i(a_124,a_12)=0, we have that b_0124∈CRS(a_124). PossiblyCRS(a_124) contains another curve z that is trivial after forgetting x_0. However b_24∈CRS(a_24) because b_234∈CRS(a_234) and i(a_234,a_24)=0. Therefore, z is disjoint from b_24 and b_012, which means z=b_01. By the same reason, we can prove that b_0123∈CRS(a_123) and s(A_123) preserves b_01. We have the following lantern relation.A_123A_34A_124=A_12A_1234.Since A_34=B_34B_01^k for nonzero k, therefore the canonical reduction system of one of the curves in the relation (<ref>) contains b_01. The rest of the discussion is similar to Case 3 by doing a case study.§.§ The proof of (1) of Corollary <ref> Let B_n=π_1(PConf_n(ℝ^2)/Σ_n) and B_n,1=π_1(PConf_n+1(ℝ^2)/Σ_n). The fiber bundle F_n(S) gives the first line of the following commutative diagram:1[r]PB_n+1[r][d]^f_n(ℝ^2)_* B_n,1[r][d]^F_n(ℝ^2)_* Σ_n [r][d]^= 11[r] PB_n[r]B_n[r] Σ_n[r]1. Every splitting of F_n(ℝ^2)_* induces a splitting of f_n(ℝ^2)_*. Therefore, we only need to study the extension of a splitting of f_n(ℝ^2)_* to a splitting of F_n(ℝ^2)_*. Let ϕ:B_n→ B_n,1 be a splitting of F_n(ℝ^2)_*. Let x∈ PB_n and e∈ B_n. We have that ϕ(exe^-1)=ϕ(e)ϕ(x)ϕ(e)^-1.Denote by C_e the conjugation of e on PB_n. Therefore, we have the following diagram:PB_n[r]^C_e[d]^ϕ|_PB_nPB_n[d]^ϕ|_PB_nPB_n+1[r]^C_ϕ(e) PB_n+1. By Theorem <ref>, there are two possibilities of ϕ|_PB_n: (1) ϕ fixes a simple closed curve c surrounding {x_k,x_0}(2) ϕ fixes a simple closed curve c surrounding {x_1,...,x_n}. We claim that ϕ|_PB_n fixes a simple closed curve surrounding {x_1,...,x_n}. To prove this claim, we assume the opposite that ϕ|_PB_nfixes a simple closed curve c surrounding {x_k,x_0}. There exists an element e∈ B_n such that e permutes punctures k and j≠ k. Since c is the only curve that ϕ(PB_n) fixes, we have that c is the only curve that ϕ(C_e(PB_n))=PB_n fixes, which contradicts that C_ϕ(e)(ϕ(PB_n)) also fixes ϕ(e)(c) surrounding {x_j,x_0}. Therefore, ϕ|_PB_n fixes a simple closed curve surrounds {x_1,...,x_n}. In this case, the section is adding a point at infinity.§ THE CASE WHEN S IS THE 2-SPHERE S^2In this subsection we give a construction of sections of the fiber bundle f_n(S^2). §.§ Nonexistence of a continuous section for n=2We prove a more general result on the sections of the fiber bundle f_n(S^2) for n=2. Let S^2k be 2k-dimensional sphere for k>0 integer.Let x_1,x_2 be two distinct points in S^2k. The following is classical; see <cit.>.The following fiber bundleS^2k-{x_1,x_2}→ PConf_3(S^2k) PConf_2(S^2k)does not have a continuous section. Suppose that there is a continuous map s:PConf_2(S^2k)→PConf_3(S^2k) such that f_2(S^2k)∘ s=identity. Then after post-composing with a forgetful map to the last coordinate, we obtain a map f: PConf_2(S^2k)→ S^2k. We denote by p_i:PConf_2(S^2k)→ S^2k the projection to the ith component. Letg_i:PConf_2(S^2k) PConf_2(S^2k)⊂ S^2k× S^2k. Let ⊂ S^2k× S^2k be the diagonal subspace in the product. Let []∈ H^2k(S^2k× S^2k,ℚ) be the Poincaré dual of . By the Thom isomorphism, there is an exact sequence for the computation of cohomology:0→ℚ H^2k(S^2k× S^2k,ℚ)→ H^2k(S^2k× S^2k-;ℚ)→ 0.Let c∈ H^2k(S^2k;ℚ) be the fundamental class and c_i=p^*_i(c). The image of diagonal is the Thom class c_1+c_2. Therefore in H^2k(S^2k× S^2k-;ℚ), we have c_1+c_2=0. This means that H^2k(S^2k× S^2k-;ℚ)=ℚc_1. Suppose that f^*(x)=kc_1 for k an integer. Therefore we will have g_i^*([])=kc_1+c_i. Since the image of g_i misses the diagonal , we have that g_1^*([])=kc_1+c_1=0 and g_2^*([])=kc_1+c_2=kc_1-c_1=0. Since c_1 is a generator of H^2k(S^2k× S^2k-;ℚ), we have that k+1=0 and k-1=0. These two formulas cannot be satisfied at the same time. §.§ Constructing sections when n>2DefinePConf_n,k(S^2)={(v_1,x_1,...,x_n)|x_1,...,x_nbenpoints inS^2andv_kbe a unit vector atx_k}.This is the total space of a circle bundle by forgetting the vector:S^1→PConf_n,k(S^2)→PConf_n(S^2).For n>2, the fiber bundle (<ref>) is a trivial bundle.S^1-bundle is classified by Euler class, i.e. a second cohomology class of the base. We investigate H^2(PConf_n(S^2);ℤ) first. There is a graded-commutative ℚ-algebra [G_ij] defined in <cit.>, where the degree of the generators G_ij is 1. By Totaro <cit.>, there is a spectral sequence E_2^p,q=H^p((S^2)^n;ℚ)[G_ij]^q converging to H^*(PConf_n(S^2);ℚ). Since we only compute H^2, the differential involved is d_2:E_2^0,1=H^0(S^2;ℚ)[G_ij]→ E_2^2,0=H^2(S^2;ℚ). Let [_ij]∈ H^2(S^2;ℚ) be the Poincaré dual of _ij⊂ S^2. By <cit.>, the differential d_2(G_ij)=[_ij]. Let p_i:(S^2)^n→ S^2 be the projection to the ith coordinate and [S^2]∈ H^2(S^2;ℤ) be the generator of H^2(S^2;ℤ). Therefore we have thatH^2(PConf_n(S^2);ℤ)=⊕_i=1^nℤp_i^*[S^2]/(p_i^*[S^2]+p_j^*[S^2])≅ℤ/2,which is generated by p_k^*[S^2] and we have that 2p_k^*[S^2]=0. The circle bundle (<ref>) is induced from the circle bundle S^1→PConf_1,1(S^2)→ S^2by the projection to the kth coordinate. The bundle (<ref>) is the unit tangent bundle over S^2. Since the Euler characteristic of S^2 is 2, the Euler class of (<ref>) is eu=2[S^2]∈ H^2(S^2;ℤ). Therefore the Euler class of (<ref>) is p_k^*[eu]=2p_k^*[S^2]=0∈ H^2(PConf_n(S^2);ℤ). Equip S^2 with the spherical metric; i.e. the metric that is induced from the standard embedding S^2⊂ℝ^3. Let ϵ(x_1,...,x_n)=1/2min_1≤ i<j≤ n{d(x_i,x_j)}.Set x_0 to be the image of the v_k-flow at time ϵ from x_k; that isem_n,k(S^2):PConf_n,k(S^2)↪PConf_n+1(S^2)Composing a continuous section s:PConf_n(S^2)→PConf_n,k(S^2) of the fiber bundle (<ref>) with em_n,k(S^2) gives a section of the fiber bundle f_n(S^2). We denote by Add_n,k(S^2) the collection of sections of f_n(S^2) consisting of compositions of a section of (<ref>) with em_n,k(S^2).Notice that there are infinitely many homotopy classes of sections in Add_n,k(S^2) and they are classified by sections of (<ref>).A special section for n=3.Since there is a unique Mobius transformation ϕ(x_1,x_2,x_3) that transforms (0,1,∞) to any ordered three points (x_1,x_2,x_3). we have that PConf_3(S^2)PSL(2,ℂ).We can assign any new point x_0=ϕ(x_1,x_2,x_3))(a) such that a≠ 0,1,∞. §.§ The proof of (2) of Theorem <ref>In this subsection we prove (2) of Theorem <ref>. Let S_0,n a sphere with n punctures. Let Diff(S_0,n) be the orientation-preserving diffeomorphism group of S_0,n fixing the n punctures pointwise. While the following is surely known to experts, we could not find this statement or a proof in the literature. I am thus incluing it for completeness. We believe that it follows from Earle-Eells <cit.> in the punctured case.For n>2, we have that BDiff(S_0,n)≅ K( PMod(S_0,n),1)We only need to prove that the homotopy group π_k(Diff(S_0,n))=0 for k>0. For n=0, by Smale <cit.>, Diff(S^2)≃SO(3). By fiber bundleDiff(S_0,n+1)→Diff(S_0,n)→ S_0,n,we deduce that Diff(S_0,1)≃SO(2) and Diff(S_0,2)≃SO(2). The long exact sequence of homotopy groups of the fiber bundle (<ref>) is 1→π_1(Diff(S_0,3))→π_1( Diff(S_0,2))→π_1(S_0,2)→PMod_0,3→PMod_0,2→ 1.However we know that PMod_0,3=1 (see <cit.>), we get that π_1(Diff(S_0,3))=0 and also π_i(Diff(S_0,3))=0 for i>1. The other cases are the same.Let PB_n(S^2)=π_1(PConf_n(S^2)). Now we are ready to prove (2) of Theorem <ref>.Let S_0,n+1→UDiff(S_0,n+1)BDiff(S_0,n+1)be the universal S_0,n+1-bundle in the sense that any S^2 bundle with n+1 sections S_0,n+1→ E→ Bis the pullback from u_n+1 by a continuous map f:B→BDiff(S_0,n+1). By Proposition <ref>, BDiff(S_0,n+1)≅ K(PMod(S_0,n+1),1). This means that UDiff(S_0,n+1) is also a K(π,1)-space. Therefore S_0,n+1-bundles are determined by their monodromy representations and the sections of an S_0,n+1-bundle are also determined by the maps on fundamental groups. A splitting of the following exact sequence gives us a section of the fiber bundle f_n(S^2).1→ F_n→ PB_n+2(S^2) PB_n+1(S^2)→ 1. We have the following diagram:S_0,n+1[r][d] PConf_n+1(S_0,1)[r]^f_n(S_0,1)[d]PConf_n(S_0,1)[d]S_0,n+1[r]PConf_n+2(S^2)[r]^f_n+1(S^2)[d]^p_n+1 PConf_n+1(S^2)[d]^p_n+1S^2[r] S^2. By the long exact sequence of homotopy groups of the fiber bundle PConf_n(S_0,1)→PConf_n+1(S^2)→ S^2,we have that PB_n+1(S^2)=PB_n/Z where Z denotes the center of PB_n and is generated by the Dehn twist about the boundary of D_n; see <cit.>. Therefore a section of f_n(S_0,1) induced from a section of f_n+1(S^2) satisfies that f_n(S_0,1)_* maps the center to the center.Since S_0,1≈ℝ^2, the section problem for f_n(S_0,1) has been fully discussed in Section 2. Every section of f_n+1(S^2) induces a section of f_n(ℝ^2), thus we could use the classification of sections of f_n(ℝ^2) to study the sections of f_n+1(S^2). Let s:PB_n+1(S^2)→ PB_n+2(S^2) be a splitting of f_n+1(S^2)_* such that f_n+1(S^2)_*∘ s=id. By (1) of Theorem <ref>, we break the discussion into the following two cases according to the sections of f_n(S_0,1). Case 1: the section of f_n(S_0,1) is adding a point near x_k. In this case, s(PB_n+1(S^2)) fixes a curve c around {x_0,x_k}. Then the image lies in the stabilizer of c. The stabilizer of c in PMod(S_0,n+2) is PMod(D_n)≅ PB_n. The boundary of D_n is c surrounding {x_1,...,x_k-1,x_k+1,...,x_n+1}. On the other hand by Proposition <ref> the circle bunldeS^1→PConf_n,k(S^2)→PConf_n(S^2)is trivial, we have that π_1(PConf_n+1,k(S^2))≅ℤ×π_1(PConf_n+1(S^2))≅ℤ× PB_n+1(S^2)≅ PB_n.The last isomorphism is coming from the splitting of the following exact sequence; see <cit.>.1→ Z→ PB_n→ PB_n/Z→ 1. Since the ℤ component of π_1(PConf_n+1,k(S^2)) is mapped to the Dehn twist about a curve d surrounding {x_0,x_k}, it means that d also surrounds {x_1,...,x_k-1,x_k+1,...,x_n+1}. Therefore we have that f_n+1(S^2) is adding a point near x_k.Case 2: the section of f_n(S_0,1) is adding a point near ∞. In this case, s(PB_n+1(S^2)) fixes a curve c around {x_1,...,x_n}. Then the image lies in the stabilizer of c. The stabilizer of c in Mod(S_0,n+2) is PMod(D_n)≅ PB_n. The boundary of D_n is c surrounding {x_1,...,x_n}. On the other hand by Proposition <ref> the circle bunldeS^1→PConf_n+1,n+1(S^2)→PConf_n+1(S^2).is trivial, we have that π_1(PConf_n+1,n+1(S^2))≅ℤ×π_1(PConf_n+1(S^2))≅ℤ× PB_n+1(S^2)≅ PB_n.Since the ℤ component of π_1(PConf_n+1,k(S^2)) is mapped to the Dehn twist about a curve d surrounding {x_0,x_n+1}, it means that d also surrounds {x_1,...,x_n}. Therefore we have that f_n+1(S^2) is adding a point near x_n+1.§.§ The unordered caseBy the same argument as the proof of (1) of Corollary <ref>, we show that none of the sections of f_n(S^2): PConf_n+1(S^2)→PConf_n(S^2)can be extended to a section of F_n(S^2): PConf_n+1(S^2)/Σ_n→PConf_n(S^2)/Σ_n. §.§ The exceptional casesFor the special cases n=3, we have the following classification.There is a unique section for the fiber bundle f_3(S^2) up to homotopy. There is no section for the bundle F_3(S^2). By Proposition <ref>, we have that BDiff(S_0,3)≅ K(PMod(S_0,3),1). Since PMod(S_0,3)=1, the classifying space BDiff(S_0,3) is contractible. Therefore every S^2-bundle with 3 sections is a trivial bundle. Thus f_3(S^2) is a trivial bundle. Therefore, a section of f_3(S^2) is determined by a map PConf_3(S^2)→ S_0,3. Since S_0,3≅ K(F_2,1), a map PConf_3(S^2)→ S_0,3 up to homotopy is determined by Hom(PB_3(S^2),F_2) up to conjugation. However PB_3(S^2)=PB_2/Z=1 implying that Hom(PB_3(S^2),F_2)=1. Therefore, there is a unique section up to homotopy. For the unordered case F_3(S^2), let Mod(S_0,3,1) be the mapping class group of S^2 fixing a set of 3 points and a set of 1 point. There is an exact sequence1→PMod(S_0,4)→Mod(S_0,3,1)→Σ_3 → 1.Since π_1(PConf_3(S^2)/Σ_3)≅Σ_3 and π_1(S_0,3)≅PMod(S_0,4), we have that Mod(S_0,3,1)=π_1(PConf_4(S^2)/Σ_3). Therefore, the section of F_3(S^2) is the determined by the splittings of the exact sequence (<ref>).Let Diff(S_0,3,1) be the orientation-preserving diffeomorphism group of S^2 fixing a set of 3 points and a set of 1 point. By definition there is a map ρ: Diff(S_0,3,1)→Mod(S_0,3,1) which induces isomorphism on π_0. A version of the Nielsen Realisation Theorem (e.g. <cit.> and <cit.>) tells us that a finite subgroup of Mod(S_0,3,1) has a lift to Diff(S_0,3,1). However every finite subgroup of Diff(S_0,3,1) is cyclic because Diff(S_0,3,1) fixes a point. Therefore every finite subgroup of Mod(S_0,3,1) is cyclic. Since Σ_3 is noncyclic, (<ref>) does not split.For the special cases n=4, we have the following classification.The sections of fiber bundle f_4(S^2) correspond to the splittings of the exact sequence1→ F_3 → PB_5(S^2) F_2→ 1up to conjugation. We have the following Birman exact sequence; see <cit.>.1→π_1(S_0,3)→ PB_4(S^2) PB_3(S^2)→ 1.Since PB_3(S^2)=1, we have that PB_4(S^2)=π_1(S_0,3)≅ F_2. By Proposition <ref>, the sections of f_4(S^2) is determined by the splittings of the following Birman exact sequence up to conjugaction.1→π_1(S_0,4)→ PB_5(S^2) PB_4(S^2)→ 1.§ THE CASE WHEN S=S_G A CLOSED SURFACE OF GENUS G>1In this section, we prove Theorem <ref>(3).Let S_g^n be the product of n copies of S_g. There is a natural embedding PConf_n(S_g)⊂ S_g^n. Let p_i:PConf_n(S_g)→ S_g be the projection onto the ith component. Denote by _ij≈ S_g^n-1⊂ S_g^n the ijth diagonal subspace of S_g^n; i.e., _ij consists of points in S_g^n such that the ith and jth coordinates are equal. Let H_i:=p_i^*H^1(S_g;ℚ) and let [S_g] be the fundamental class in H^2(S_g;ℚ). Now, we display the computation of H^*(PConf_n(S_g);ℚ) from <cit.>. (1) For g>1 and n>0,H^1( PConf_n(S_g);ℚ)≅ H^1(S_g^n;ℚ)≅⊕_i=1^nH_i.(2)We have an exact sequence1→⊕_1≤ i<j≤ nℚ[G_ij]H^2(S_g^n;ℚ)≅⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j H^2(PConf_n(S_g);ℚ),where ϕ(G_ij)=[_ij]∈ H^2(S_g^n;ℚ) is the Poincaré dual of the diagonal _ij.See <cit.>. Let {a_k,b_k}_k=1^g be a symplectic basis for H^1(S_g;ℚ). For 1≤ i,j ≤ m, we denote M_i,j=∑_k=1^n p_i^*a_k ⊗ p_j^*b_k-p_i^*b_k⊗ p_j^*a_k. The diagonal element [_ij]=p_i^*[S_g]+p_j^*[S_g]+M_ij∈⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j≅ H^2(S_g^n;ℚ). See <cit.>.The following lemma is the classification of homomorphisms π_1(PConf_n(S_g))→π_1(S_g) from <cit.>.Let g>1 and n>0. Let R: π_1(PConf_n(S_g))→π_1(S_g) be a homomorphism. The followings hold:(1)If R is surjective, then R=A∘ p_i* for some i and A an automorphism of π_1(S_g). (2)If Image(R) is not a cyclic group, the homomorphism π_1(PConf_n(S_g))→π_1(S_g) factors through p_i* for some i. See <cit.>.Now, we are ready to prove (3) of Theorem <ref>.Suppose that there is a map s:PConf_n(S_g)→PConf_n+1(S_g) such that f_n(S_g)∘ s=identity. Then after post-composing with a forgetful map of the last coordinate, we obtain a map f: PConf_n(S_g)→ S_g. We denoteg_i:PConf_n(S_g) PConf_2(S_g)⊂ S_g× S_gLet ⊂ S_g× S_g be the diagonal subspace and []∈ H^2(S_g× S_g;ℚ) be the Poincaré dual of . Let f^*:H^1(S_g)→ H^1(PConf_n(S_g)) and f_*:π_1(PConf_n(S_g))→π_1(S_g) be the induced map on cohomology and the fundamental groups. By Lemma <ref>, either f_* factors though a forgetful map p_i* or Image(f_*)≅ℤ. We break the proof into two cases according to the image of f_*.Case 1: Image(f_*)≅ℤ. There are two subcases:(1) If f^*=0, then g_i^*([])=p_i^*[S_g]≠ 0. This contradicts the fact that the image of g_i misses .(2) If f^*≠ 0, then Imf^*≅ℤ because f_* has image ℤ on the fundamental groups. We assume that there exists a symplectic basis {a_k,b_k}_k=1^g for H^1(S_g;ℚ) such that f^*(a_i)=0 for any i≠ 1 and f^*(b_i)=0 for any i. Let f^*(a_1)=(x_1,x_2,...,x_n)≠ 0∈⊕_i=1^nH_i≅ H^1(PConf_n(S_g);ℚ). Assume without loss of generality that x_1≠ 0. Therefore for k≠ 1 by Lemma <ref>, we have thatg_k^*([])=p_k^*[S_g]+∑_i=1,i≠ k^nx_i⌣ p_k^*b_1∈⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j≅ H^2(S_g^n;ℚ).The coordinate x_1⊗ p_k^*b_1 is not zero, therefore g_k^*([])≠ 0. This contradicts the fact that the image of g_i misses .Case 2: f_* factors though the forgetful map p_i*. Without loss of generality, we assume that i=1. We have thatg_2^*([])=f^*[S_g]+p_2^*[S_g]+∑_k f^*a_k ⌣ p_2^*b_k-f^*b_k⌣ p_2^*a_k.Since Image(f^*)⊂Image(p_1^*), we have that g_2^*([]) only has nonzero terms in ℚG_12⊕ H_1⊗ H_2. The fact that g_2 missesimpliesf^*[S_g]+p_2^*[S_g]+∑_k f^*a_k ⊗ p_2^*b_k-f^*b_k⊗ p_2^*a_k=λ ([_12])∈ℚp_1^*[S_g]⊕ℚp_2^*[S_g]⊕ H_1⊗ H_2. The coefficient of p_2^*[S_g] tells us that λ=1. Therefore we have that f^*[S_g]=p_1^*[S_g] and ∑_k (f^*a_k-p_1^*a_k) ⊗ p_2^*b_k-(f^*b_k-p_1^*b_k)⊗ p_2^*a_k=0∈ H_1⊗ H_2By the property of tensor product, we know that f^*a_k-p_1^*a_k=0 and f^*b_k-p_1^*b_k=0. However in this case, if we look at the map g_1:PConf_n(S_g)S_g× S_g. We have that g_1^*([])=f^*[S_g]+p_1^*[S_g]+∑_k f^*a_k ⌣ p_1^*b_k-f^*b_k⌣ p_1^*a_k=2p_1^*[S_g]-2g p_1^*[S_g]=(2-2g) p_1^*[S_g]≠ 0.This contradicts the fact that the image of g_1 misses .§ FURTHER QUESTIONSIn this section we list a few further questions. Let m,n be two positive integers. Let (x_1,...,x_n)∈PConf_n(S) for any manifold S. Let the permutation group Σ_m acts on PConf_n+m(S) by permuting the last m points. We have the following fiber bundle:PConf_m(S-{x_1,...,x_n})/Σ_m→PConf_n+m(S)/Σ_mPConf_n(S).Here denote by f_n+m,n(S) the forgetful map that forgets the first n points. A section of the fiber bundle (<ref>) is called a multi-section.Classify the continuous sections of the fiber bundle (<ref>) up to homotopy for S a surface. Classify the continuous sections of the fiber bundle (<ref>) up to homotopy for any manifold S.California Institute of Technology Department of Mathematics Pasadena, CA 91125, USAE-mail: [email protected] | http://arxiv.org/abs/1708.07921v3 | {
"authors": [
"Lei Chen"
],
"categories": [
"math.GT"
],
"primary_category": "math.GT",
"published": "20170826025002",
"title": "Section problems for configuration spaces of surfaces"
} |
.pdf, .jpg, .pnp, .bmp, .fig y all #1 #1 #1 #16.50in-0.50in 0in0in 9.00in.equationequation equation(.equation)argument .argument | http://arxiv.org/abs/1708.07857v1 | {
"authors": [
"Ioannis S. Stamatiou"
],
"categories": [
"math.NA",
"60H10, 60H35, 65C20, 65C30, 65J15, 65L20"
],
"primary_category": "math.NA",
"published": "20170825184055",
"title": "A note on pathwise stability and positivity of nonlinear stochastic differential equations"
} |
Mining the Demographics of Political Sentiment from Twitter Using Learning from Label Proportions Ehsan Mohammady Ardehaly Department of Computer ScienceIllinois Institute of TechnologyChicago, Illinois 60616Email: [email protected] Aron Culotta Department of Computer ScienceIllinois Institute of TechnologyChicago, Illinois 60616Email: [email protected] 30, 2023 ===================================================================================================================================================================================================================================================================================== Opinion mining and demographic attribute inference have many applications in social science. In this paper, we propose models to infer daily joint probabilities of multiple latent attributes from Twitter data, such as political sentiment and demographic attributes. Since it is costly and time-consuming to annotate data for traditional supervised classification, we instead propose scalable Learning from Label Proportions (LLP) models for demographic and opinion inference using U.S. Census, national and state political polls, and Cook partisan voting index as population level data. InLLP classification settings, the training data is divided into a set of unlabeled bags, where only the label distribution in of each bag is known, removing the requirement of instance-level annotations. Our proposed LLP model, Weighted Label Regularization (WLR), provides a scalable generalization of prior work on label regularization to support weights for samples inside bags, which is applicable in this setting where bags are arranged hierarchically (e.g., county-level bags are nested inside of state-level bags). We apply our model to Twitter data collected in the year leading up to the 2016 U.S. presidential election, producing estimates of the relationships among political sentiment and demographics over time and place. We find that our approach closely tracks traditional polling data stratified by demographic category, resulting in error reductions of 28-44% over baseline approaches. We also provide descriptive evaluations showing how the model may be used to estimate interactions among many variables and to identify linguistic temporal variation, capabilities which are typically not feasible using traditional polling methods. § INTRODUCTIONRecent research has demonstrated the feasibility of estimating quantities of public interest from online social network data, with applications to health <cit.>, politics <cit.> and marketing <cit.>. However, practitioners are often more interested in investigating the interactions among sets of variables, rather than estimating the trend of a single variable. For example, health researchers may want to know not only what the influenza rate is, but also how it is distributed among demographic groups. Similarly, in politics observers may want to know not only which candidate has stronger support from the electorate, but also how that support varies by geography, income, and race/ethnicity.This type of analysis poses significant challenges to internet-based systems because many of the variables of interest (e.g., demographics) are not publicly observable. Thus, one must build a separate classification model for each variable, for example classifying the demographics of a social media user based on linguistic and social evidence. Traditional supervised approaches to this problem suffer from two primary limitations: (1) it is costly and time-consuming to annotate data for each variable of interest for training and validation; and (2) in streaming settings models quickly becoming outdated due to rapidly changing linguistic patterns. For example, Figure <ref> shows the association of the term `#hillary2016' on Twitter with various class labels (described in more detail below). We can see that this term was highly indicative of some demographic classes (e.g. college graduates) for almost five months and faded after that. Thus, models need to be robust to rapidly shifting distributions in the data.To address these challenges, in this paper we propose an approach based on Learning from Label Proportions (LLP) <cit.>. Unlike traditional supervised learning, LLP models do not require instance-level annotations for training. Instead, in LLP the training data consist of bags of instances annotated with label proportions – e.g., a collection of 1,000 users, of which 80% are expected to be male. LLP models are appealing in this domain because there are many pre-existing data sources that can provide approximate label proportions. For example, by combining county-level demographics with geolocated tweets, we can associate bags of users with expected demographic distributions. To deal with data drift, we retrain the LLP models daily, which is possible because no additional labeled data is required.In this paper, we develop an LLP approach to estimate the relationship between political sentiment and demographics during the 2016 U.S. presidential election. We collect 88M geo-tagged tweets posted in the year leading up to election day and group them into 424 counties per day. We use U.S. Census county population data as the expected demographic label proportions. We also use national and state polls (Clinton vs. Trump) and Cook partisan voting index (PVI)[<http://cookpolitical.com//>] as expected state level label proportions for political sentiment classification. Using these data, we ultimately fit LLP models to classify tweets along seven dimensions: political sentiment (pro-Clinton or pro-Trump), race/ethnicity, education, income, gender, native or foreign born, and health insurance coverage status.To do so, we propose a new LLP training algorithm, called weighted label regularization, which is appropriate for settings in which bags are organized hierarchically — e.g., county bags are nested inside of state bags, which are in turn nested inside of a nation-wide bag. The approach combines ideas from label regularization (<cit.>) and ridge regression into a scalable model that can be retrained frequently to maintain model freshness. For each day, we retrain all seven LLP models, then calculate conditional probabilities, such as P(pro-Clinton|College Graduate). For quantitative evaluation, we compare our estimates to CNN/ORC[<http://orcinternational.com/news-category/cnn-poll/>] polls that stratify results by demographics. Additionally, we compare our results on election day with exit polls[A poll of voters taken after they have exited the polling stations.] and the final election results. Our proposed approach produces estimates that closely align with the polls, reducing error by 28-44% over competing baselines on average across all demographic variables.An additional advantage of our approach is that we can stratify our estimates using combinations of variables, which is often impractical with polling data due to small sample sizes. For example, our estimates of P(pro-Clinton|No College and Income < $50K) show a strong decline in the months prior to the election, in line with journalistic reports of the weakness of Clinton's support among this group. We also provide some qualitative analysis of how linguistic patterns change over time with respect to political sentiment.The paper is organized as follows. In Section <ref>, we review related work on internet-based tracking methods and demographic inference, and in Section <ref> we describe the data collected for the experiments. Section <ref> provides background onLLP models and introduces our proposed approach. In Section <ref> we present our experimental results; Section <ref> concludes and provides discussion of limitations and future work. * RQ1 Can a model that trained on population level data be used to infer conditional probabilities of latent attributes? We found that LLP models can successfully train to deduce hidden attributes, which can then be used to estimate joint probability attributes and then the conditional probability can be computed (e.g.P(Democratic | Florida, white, male, 11/6/2016)).* RQ2 What is the effect of population level data? Previous works show that US Census demographic data can be used as population-level data for demographic attribute inference. For political sentiment inference, we propose three methods based on national polls, state polls, and PVI, and in the evaluation step, we found that using only national polls with PVI has the best result. That can be in part because of inaccuracy of some state polls (especially for Florida and Midwest).* RQ3 Can temporal dynamic of linguistic features be detected? The coefficients of our daily trained models can be used to show temporal changes of terms for one year. We found that while some unigrams (e.g. `drinking' and `God') have stable indication over a year,some words (e.g. `university' and 'Hillary') change from one class to another(possibly multiple times) over time. That can be useful as the linguistic perspective of social and political science analysis.§ RELATED WORK Analyzing temporal dynamic of social media is investigated in many recent works.Abel et al. (2010) study temporal dynamic in Twitter for personalized recommendation <cit.>. For example, their model can detect new users who become interested in a new topic. Yang et al. (2011) develop a spectral clustering algorithm to address how some hashtag's popularity grows and fades overtime <cit.>. Traditional supervised learning is widely being used in previous works <cit.>; however, annotations can be costly to obtain in a timely fashion. Also, many attributes such as political affiliation are hard to interpret, and in a temporal environment such as Twitter, the annotated users became outdated soon.Other work has jointly modeled demographic variables in social networks <cit.>, though again this relies on user-level annotations.One possible approach to resolving old annotated data is domain adaption.Li et al. (2015) propose the Naive Bayes approach with Expectation Maximization (EM) to predict a new disaster based on labeled data available from Twitter for past catastrophes <cit.>. The advantage of their model is that they do not need to annotate labeled data for the current disaster with unsupervised domain adaptation. Imran et al. (2016) propose a domain adaptation model for disaster classification <cit.>. They show that the labels from the previous crisis are useful when the source and the target events are the same types (e.g. earthquakes).They also indicate that cross-language domain adaptation works better when two languages are similar (e.g. Italian and Spanish). However, these methods still base on labeled data on source domain.Domain adaptation also can be applied to Learning from Label Proportions <cit.>. In this approach, the modeltrained on the domain of origin isused to transfer to the new domain.For social media with temporal dynamic, the model that was fitted previously (e.g. last month),can be transferred to another time (e.g. now) with self-training. The main problem of self-training is scalability and sensitivity to hyperparameters, and it can degrade adaptation to the temporal dynamic of social media. An attractive alternative is training LLP classifiers with a sliding temporal window. In this case, we do not need domain adaptation, and we can smooth the output of the classifier with moving average to make it more robust to noise. While LLP has a satisfactory result for manyclassification applications in social science<cit.>, fraud detection <cit.>,and computer vision <cit.>, to the best of our knowledge, it has not been applied to time series tasks.The main challenges to using LLP on time series are scalability and robustness to noisy environment of social media (e.g. Twitter). Prior work has proposed an exhaustive greedy bag selection algorithm to deal with noise <cit.>. While this method has accurate result on some domains, it is not scalable and cannot apply to time series environments. Therefore, a scalable model is required to use in this area.In this work, we develop a scalable LLP model by using a sliding window for training and estimate the conditional probability between different latent attributes. To improve robustness against inherent noise in social media, we apply moving average instead of using exhaustive bag selection algorithms. In this paper, we propose the Weighted Label Regularization (WLR) model with several key differences from Label Regularization (LR) for LLP settings. With these contributions, WLR can efficiently be applied to time series data over Twitter with millions of samples. The differences between the two models are as following: * LR uses softmax, but WLR uses logistic function (because we need only binary classification).* LR assumes all unlabeled samples have the same weight, while WLR supports weighted samples.* In WLR model, feature vectors are the average of features of sub bags, and as a result, training is significantly faster than LR,making it feasible to apply to big data.Using temporal models on Twitter for event detection has many applications. Culotta (2010) uses text regression model to find a correlation between term frequencyof daily tweets with CDC statistics of influenza, and train a document classifier to identify flu <cit.>. Sakaki et al. (2010) propose a probabilistic model for earthquake detection from time-series tweets and information diffusion related to a real-time event <cit.>. Benhardus et al. (2013) investigate methods for trend detection from streaming Twitter data by identifying term frequency of n-grams <cit.>. § DATAFor purposes of this study, we collect both individual level data (from social media)and population level data. These data are used to train the LLP models and to make inferences for different social media activity. This section describes the detail of our data collection.[Replication code and data will be made available upon publication.] §.§ Twitter dataTo understand temporal dynamics in social media, we use the Twitter Streaming API to collect a random sample of geo-located tweets in the United States for roughly one year (Oct 20, 2015, to Nov 7, 2016). We use reverse geocoding to find the originating U.S. county based on the geo-tagged attribute of tweets, and remove tweets which reverse geocoding is failed. After this process, 88M tweets remains with 1.3B tokens and 9M terms. To reduce model complexity, we use only the top 17.5K of the most common unique unigrams that roughly appear in at least 5K tweets.Then, we create daily bags for each county. Since less populous counties usually lean toward the Republican Party and removing them can introduce bias towards the Democratic Party, to reduce variance, we collapse counties with fewer than 40 tweets per day in each state together. As a result, we totally create 424 county bags (including collapsed bags) per day. §.§ Population-level dataWe do not have any Twitter labeled data, and the aim of LLP is to use population-level data as a light supervision to predict individual level data. As a result, we need to collect aggregated data as described in this section. Even though it isgenerally accepted that social media users are not a representative sample ofpopulation, an advantage of LLP algorithms is that they are often robust to slight mispecifications of bag proportions <cit.>.§.§.§ US CensusFor demographic attributes, we collect the latest (2014) county statistics from the U.S. Census. For purposes of this study, we only considered binary classification for 6 attributes: race (white or non-white), education (college graduate or non-college graduate), income (under $50K household income or $50K or more), gender (male or female), nativity (native or foreign born),and health insurance (insured or uninsured).§.§.§ PollsTo estimate temporal dynamics of political sentiment, we use averaged poll data from the “Real Clear Politics”[<http://www.realclearpolitics.com/>]website. This site reports the moving average of polls from highly graded pollsters for both national and state level, and we use their daily average estimates for Clinton vs.Trump as population-level data for political sentiment classification.§.§.§ Cook partisan voting index (PVI)Cook Political Report periodically reports this index as an estimate of how strongly a state leans toward major parties. For example, the PVI of Florida is “R+2” for 2014, that means Florida tipped 2% more than national average toward Republican Party. We use the latest index (2014[<https://en.wikipedia.org/wiki/Cook_partisan_voting_index>]) as an additional aggregation level estimate (described in more detail below). §.§ Data qualityThe advantage of the above data is that we can easily associate bags of tweets with label proportions obtained from pre-existing census and polling data. Of course, this data, while convenient, is far from perfect. First, there is selection bias from the fact that Twitter users are not representative of the overall population. Second, census statistics and polling data are themselves only approximations, and so any errors in them will propagate to the trained model. Third, relying on geolocated data presents further challenges to sample size and quality. Despite these challenges, there is considerable evidence in prior work that LLP models are quite robust to noisy label proportions and biased data <cit.>. This is in part due to the "softness" of the training objective, which accommodates mislabeled instances, and in part due to the fact that the model contains intercept terms that can account for some of the selection bias. (E.g., if younger users are overrepresented in Twitter, the intercept for the age classifier adjusts for this.)§ MODELSIn this section, we investigate a linear model and propose a non-linear model for learning from label proportion (LLP). Both of these models are scalable and robust for big data settings. §.§ Linear model We begin with a baseline model that uses ridge regression for LLP. Let T_i ∈ T indicate a set of tweets assigned to bag i, where tweet j is represented by a d-dimensional term frequency vector x_ij∈ℝ^d. The linear model averages the feature vectors for each user in bag i, and minimizes the mean squared error between the true and predicted label proportions. Let X̅_i = ∑_j x_ij/|T_i| be the average feature vector for each user in bag i, and let ỹ_̃ĩ be the known label proportion for bag i (e.g., the proportion of males in county i).The linear model is simply the dot product between the average feature vector and the model parameters θ∈ℝ^d: h_i = X̅_i^TθThe θ parameters are optimized to minimize mean-squared error with L2 regularization:θ^* ←argmin_θ1/|T|∑_i (ỹ_i - h_i)^2 + λ/2||θ||^2where λ controls the regularization strength. While this linear model is conceptually simple, recent research has found that it produces accuracy comparable to traditional supervised models on social media tasks<cit.>. The main advantage of ridge regression for LLP is that it only needs term frequency per bag, without using individual features of samples. This can significantly speed up the training time with accuracy competitive with supervised models such as logistic regression. As a result, it can apply for big data or streaming data, and only the average of features per bag need to be stored in memory.We use this model for demographic attribute prediction as follows. For each day, we use the mean of features for the prior week's tweets per bag to create feature matrix X (each rowis the average term frequency of one county), and our target variable (ỹ) is the normalized population for the corresponding demographic attribute. Since all our demographic attributes are binary, we just need to train ridge regression for one of the classes. For example, for gender classification we compute the proportion of men in each county as the ỹ vector, and train ridge regression for (X, y) to optimize θ.To predict the class label for an individual tweet with feature vector x,we estimate the probability that sample x is Male as x^Tθ (truncated between 0 and 1). As a result, if x^Tθ > .5, we classify this sample as a male, otherwise as female. Also, we use same L2 regularization strength λ for our all experiments, and we find that the results are not very sensitive to this parameter.Because of reported high accuracy and scalability of ridge regression <cit.>, we use it as a state of the art baseline model. However, because our population level data for political sentiment classification is at the state level, and our bags are at the county level, we use the label proportion of a state as the corresponding county-level label proportion. Also, we use the sample rate for each county based on the number of samples in that county. We combine polls and PVI index as described in Section <ref> to assign label proportions to bags. We call the resulting model Ridge-NP. §.§ Non-linear modelLabel regularization <cit.> is a semi-supervised non-linear model (with logistic hypothesis) and is similar to logistic regression for the supervised part. For the semi-supervised part, the model tries to minimize the cross-entropy between the given label proportionand posterior probability estimate of unlabeled data. The original experiments using label regularization assumed that there is a set of labeled data and only one bag of unlabeled data with known label proportion. However, subsequent work has extended the model to multiple unlabeled bags and without any labeled data (i.e., LLP settings) <cit.>.Scaling label regularization is challenging because at training time it must iterate over each individual instance (tweet) to compute the gradient. Therefore, its training time is much slower than the ridge regression model in the previous section, which only considers a single average feature vectors per bag. We omit label regularization from our experiments below because it is not scalable and requires prohibitively long training time for the millions of training instances in our data; it is also quite sensitive to hyper-parameters. This is the motivation for the present work. We propose a lightweight generalization of label regularization that can use term frequency of county bags. Thus, the training time is much faster than label regularization, allowing us to scale to the current problem domain. We named this scalable model weighted label regularization (WLR).Let X_u,i, w_u,i, and h_u, i be the term frequency vector, number of tweets, and the hypothesis for county u in state i, and ỹ_̃ĩ be theknown label proportion for state i. We define h_u, i same as logistic regression, i.e.h_u, i = σ(X_u, i^Tθ)where θ is the model parameter and σ is the logistic function. We define h̅_̅i̅ as weighted average of h_u, i:h̅_̅i̅ = ∑_u w_u,ih_u, i/∑_u w_u,iThus, in weighted label regularization, we estimate the state-level proportions as a weighted average of the predicted county-level proportions. Similar to label regularization, we use cross-entropy (H) as the error function:J(θ)=∑_i H(ỹ_̃ĩ, h̅_̅i̅) + λ/2 ||θ||^2 = -∑_i (ỹ_̃ĩlogh̅_̅i̅ + (1 - ỹ_̃ĩ) log(1 - h̅_̅i̅)) + λ/2 ||θ||^2where λ is the L2 regularization strength. Our experimental results (Table <ref>) show that the model is not very sensitive to λ. We set λ = .01 for all our other experiments. We also need the gradient of the cost function to apply the gradient descent algorithm. To do that, we use the gradient of logistic function, i.e.∂/∂θσ(f) = σ(f)(1 - σ(f)) ∂/∂θ f Now, the gradient of the cross-entropy part of the cost function is:= -∑_i (ỹ_̃ĩ∂/∂θlogh̅_̅i̅ + (1 - ỹ_̃ĩ) ∂/∂θlog(1 - h̅_̅i̅)) = -∑_i (ỹ_̃ĩ/h̅_̅i̅∂h̅_̅i̅/∂θ - 1 - ỹ_̃ĩ/1 - h̅_̅i̅∂h̅_̅i̅/∂θ) =∑_i h̅_̅i̅ - ỹ_̃ĩ/h̅_̅i̅(1 - h̅_̅i̅)∂h̅_̅i̅/∂θ=∑_i h̅_̅i̅ - ỹ_̃ĩ/h̅_̅i̅(1 - h̅_̅i̅)∂/∂θ∑_u w_u,ih_u, i/∑_u w_u,i=∑_i h̅_̅i̅ - ỹ_̃ĩ/h̅_̅i̅(1 - h̅_̅i̅)∑_u w_u,i∑_u w_u,ih_u, i(1 - h_u, i) X_u, i=∑_u,iw_u, ih_u, i(1 - h_u, i)(h̅_̅i̅ - ỹ_̃ĩ)/h̅_̅i̅(1 - h̅_̅i̅) ∑_u w_u, i X_u, i Finally, any gradient descent algorithm can be used to find parameters. (Although the objective is non-convex,convex optimization has been shown to work well in prior LLPstudies <cit.>.) In this study, we use the L-BFGS algorithm <cit.> to find coefficient θ that minimizes the cost function. To apply WLR for political sentiment training, same as demographic training, we use the average of features for last week per county, and group counties together to create state bags. We also need state label proportions (ỹ_̃ĩ). Because some states are polled more frequently than others, we consider three strategies to assign the label proportion for each state bag for training, summarized in Table <ref> and described below.§.§.§ WLR-NP In this approach, we use the average national poll plus PVI index for all states to assign a label proportion to each state. For example, on Nov 1, 2016, according to the Real Clear Politics site, Clinton polled at 47.5%, and Trump polled at 45.3%. Since we use binary classification, we do not consider third-party candidates. As a result, we estimate the proportion of positive Democratic sentiment as 47.5 / (47.5 + 45.3) = 51.2% at the national level. We use PVI to generate label proportions for each state. For example, Florida has PVI `R+2', so we assign the label proportion for positive Democratic sentiment in Florida on Nov 1, 2016 as 51.2% - 2%= 49.2%. §.§.§ WLR-SNIn this method, rather than using PVI, we restrict the training data to the normalized state polls for states with a poll available on the corresponding day, removing states without polls from the training data for that day. §.§.§ WLR-SNRSimilar to the prior method, we use normalized state polls when available. We additionally augment this data using the PVI method above for other states that do not have a poll available on a given day.§.§ Training stepsTo apply the proposed models, several preprocessing steps are required. For higher performance, these actions can be sped up by pre-computing steps (such as reverse geocoding and storing the daily average of features for county bags). The primary steps of training at day d are as follows: * Add tweets from last week to the training set. Formally, we select all tweets in [d - 7, d] to create the training set.* Use reverse geocoding to create county bags for training data.* Tokenize tweets; we remove mentions and URLs and maintain hashtags and description field.* Compute the average feature vector for each county bag.* Finally, train LLP models to find model parameters. We use the same hyper-parameters (i.e., random initialization, number of BFGS iterations, and L2 regularization strength) for all experiments. The reason we retrain every day is to ensure that the model coefficients best reflect the most recent data distribution. While census demographics do not change much, those who participate in political discussions on Twitter on a given day can change rapidly over time, as do the topics that they discuss. Retraining allows the model to capture these latest trends.§.§ Estimating conditional probabilitiesIn this section, we describe how to infer joint probability (and therefore conditional probability) distributions for different classes for day d, using our trained models. Let B be a boundary (e.g. state, county, city) that we are interested in (in this study we use only state boundaries), and T_B,d is the set of all sampled tweets originating from this boundary at day d. This set can be quickly populated by reverse geocoding. For the sake of simplicity, suppose we are interested in estimating P(Democratic, male|B, d). We propose two methods for this estimation.Hard-voting: In this approach, we compute the number of tweets classified as both `Democratic' and `Male' in T_B,d, and divide that by the number of total tweets in T_B,d. More formally, let θ_D and θ_M be model parameters for `Democratic' and `Male' class. We estimate the joint probability P(D, M|B, d) as follows: |{x ∈ T_B,d | P(D|x,θ_D) > .5 ∧ P(M|x,θ_M) > .5 }|/|T_B,d| Soft-voting: In this method, instead of computing the majority vote for both classes, we compute the average of P(D,M|x, θ_D, θ_M) assuming `Democratic' and `Male' classes are independent (since they are computed independently). More formally, we estimate P(D, M|B, d) as:1/|T_B,d|∑_x ∈ T_B,d P(D|x,θ_D)P(M|x,θ_M) In practice, according to Figure <ref>, we find that for regional attributes (e.g. `Midwest', `Florida') soft-voting works better, and for demographic attributes (e.g. `White') using the weighted average of soft-voting and hard-voting (75% soft, 25% hard) has the best result, and we use this method for our experiments in the next section. Once we have computed the joint probability, we then use the chain rule of probability to compute the desired conditionals; e.g., P(D | M, B, d) = P(D, M | B, d)/P(M | B, d).§ RESULTSWe compare the estimates produced by our models both with tracking polls in the year prior to the election and also with exit polls the day of the election. We investigate three research questions: * RQ1 Can a model trained on population level data produce accurate estimates of the political sentiment of demographic groups over time? * RQ2 What is the relative impact of the different methods of assigning label proportions to bags (i.e., methods WLR-NP, WLR-SN, and WLR-SNR above)?* RQ3 How do certain terms change over time with respect to their association with demographics and political sentiment? In the first experiment, we estimate conditional probabilities of political sentiment given demographic classes at the national level by computing the weighted average of the state-level estimates. We compare these estimates with the demographic breakdown of polls from CNN/ORC. There were 11 CNN/ORC polls conducted during this time, with five regions (US, Midwest, Northeast, South, and West), and ten demographic breakdown attributes. Table <ref> shows the mean absolute error (MAE) between model prediction and CNN/ORC result. The last column in this table shows the average margin of error (MOE) of polls. For all but four demographic classes, WLR-NP has an error rate less than the margin of error. The largest error belongs to race classes, which is in line with previous work <cit.>. According to this table, WLR-NP is more accurate than WLR-SNP. Also, WLR-SN and Ridge-NP have an error rate above the margin of error.The lowest error in Table <ref> belongs to `College grad' and 'Under $50K' for demographic breakdowns. Figure <ref> plots the daily probability of being Democratic (smoothed with 14 days moving average) given college graduate, compared to CNN/ORC lowestand highest margin of error. According to this plot, WLR-NP is close to the WLR-SNP method, and except for one poll, it is within the margin of CNN/ORC poll error. Furthermore, it shows that WLR-SN has the highest error. Figure <ref> plots the same pattern for `Under $50K' class and shows that WLR-NP has the lowest error; except for one poll,it is in the margin of CNN/ORC poll error. Finally, Figure <ref> shows the probability of voting Democratic given both education and income level. Here, we do not have access to polls reporting this combination of variables, in part because small sample sizes make these difficult to estimate using traditional polling methods. However, we note a significant drop in Democratic support among people with low income and low education levels in late August/early September prior to the election.In the next experiment, we use the exit poll results and compare them with our model predictions generated one day before the election date. Table <ref> compares our models with exit polls. Again, WLR-NP has the lowest error rate, and `Non-white' class has the highest error. According to this table, `$50K or more', 'White non-college graduate', and`Native' classes have the lowest error. To show the sensitivity of model parameters, we run our best model (WLR-NP) with different L2 regularization strength (λ). Table <ref> shows the error rate of WLR-NP with various model parameters. According to this table, the error rate is stable for small values of λ. This result shows that the model is not very sensitive to L2 regularization.While our primary goal is not to predict election results, as an additional validation measure we also compare our predictions (at one day before election day) to election results. Table <ref> compares our prediction of being Democratic with the election result for battleground states. While all models have a very similar average error rate, WLR-NP incorrectly predicts the winner of only 5 states (Colorado, Iowa, Michigan, Pennsylvania, Virginia), the fewest among all approaches. These results suggest that models based on state polls (WLR-SN and WLR-SNP) have a poor prediction. This may in part because of inaccurate polls in some states (notably Florida and the Midwest).Figure <ref> plots the effect of weighted average between soft-voting and hard-voting. This plot shows the error (MAE) of WLR-NP using different weighted averages between soft-voting and hard-voting. The leftmost of this plot shows using only hard-voting (0% soft-voting), and the rightmost illustrates the error of using only (100%) soft-voting. In this plot we divide CNN/ORC polls to regional (e.g. `Midwest', `Florida') and demographic (e.g. `White') attributes, and according to this plot for regional attributes (i.e. CNN/ORC regional and election result) soft-voting works better. However, for demographic attributes (i.e. CNN/ORC demographics and exit polls) using the weighted average of soft-voting and hard-voting (75% soft, 25% hard) has the best result. These results answer our first research question by indicating that the joint probability (and therefore conditional probability) of different latent attributes can be estimated by models trained on population-level data, and we evaluate our models with CNN/ORC polls, exit polls, and election result. Our models show that while some demographic attributes (such as race) are hard to predict, some characteristics such as income and college graduation are easier to predict, and as a result, it affects the accuracy of the conditional probabilities.To answer our second research question, we present evidence that WLR-NP by using national polls and PVI index has the best result, and WLR-SN has the worst result due to overfitting state polls. We believe that is because of noise in state polls in some regions. Further studies are required to investigate the source of inaccuracy in these states.To answer our third research question, we select WLR-NP as the best model for political affiliation prediction and report how using a term can change over time. Figure <ref> reports changes of weights (normalized to unit vector) for term `Hillary' for different classes (smoothed by 30 days moving average). According to this plot, except January, the unigram `Hillary' has a growing indication for `Democratic' class. But, for demographic attributes, its sign changes over time. For example, the term is a weak indicator of `native born' class before July, and after that becomes indicative of the `foreign born' class. In addition, according to this plot all demographic weights converge to near zero at nomination time, that can be in part because all classes use this term at that point.On the other hand, according to Figure <ref>, the term `#trump' has stable indication over one year. Before April, it is almost neutral for all demographic classes and a weak indicator for `Democratic' class. After April, its indication grows over time to become strongly indicative of `Republican', `Non-college graduate' and `under $50k' classes. Mildly positive coefficients are found for `White', `Uninsured', `Native', and `Male' classes.Finally, Figure <ref> plots weights of some terms for both race and political sentiment classes to show how the unigram indication changes over time (all weights are smoothed with 30 days moving average). For each term, its weight starts from `x' mark(on Nov 1, 2015) to `o' mark (on Nov 7, 2016). There are four quartiles in this plot. We select unigrams with the highest indication changes, and some terms (i.e. `drinking', `#healthcare', `God', and`check') keep in one quartile for entire year. For example, the term `drinking' is an indicator for both `Democratic' and `White' classes for the whole year. That is in part because according to 2013 national survey on drug use and health from U.S. Department of Health and Human Services[http://www.samhsa.gov/data/sites/default/files/ NSDUHresultsPDFWHTML2013/Web/NSDUHresults2013.pdf], white Americans use more alcohol than other races/ethnicities, and the rate of alcohol consumption increases with increasing levels of education (which correlates with Democratic political affiliation).Finally, some terms (i.e. `international' and `university') have a solid indication for race classes, but multiple indications for political classes. That is in part because of more temporal dynamics for political classes in election season. Also, the term `#trump' starts from almost neutral and leads to `Republican' class over time, and becomes a weak indicator for `white' class. § CONCLUSIONS AND FUTURE WORKIn conclusion, we found that the population-level data can be used to mine conditional probability of demographic and opinion attributes from Twitter. Our first contribution is scalability compared to previous works. Our proposed model, weighted label regularization,is a scalable generalization of label regularization that can apply to domains where bags of users are grouped into smaller sub bags. The training time of this model is significantly faster than label regularization because it does not require individual features of users in sub bags, relying instead on the average of feature values and the number of samples in each sub bag. This difference makes weighted label regularization applicable to the data size in this domain.Our second contribution is to investigate one step beyond the classification task by estimating joint and conditional probabilities between different latent attributes. Also, this method is more robust to noise by applying moving average instead of exhausting bag selection algorithms. This process benefits social scientist to understand the opinion of different demographic populations.Finally, our experimental results show that using national polls with PVI has the lowest error and some state polls appear to be inaccurate. This method, in turn, can be utilized as a supplement to polls to discover public opinion for election candidates.In the future, we will investigate new models to track opinion and public health in domains with high temporal dynamics and propose more scalable and accurate models to adapt to these dynamics.§ ACKNOWLEDGMENT This research was funded in part by the National Science Foundation under grants #IIS-1526674 and #IIS-1618244. IEEEtran | http://arxiv.org/abs/1708.08000v1 | {
"authors": [
"Ehsan Mohammady Ardehaly",
"Aron Culotta"
],
"categories": [
"cs.SI"
],
"primary_category": "cs.SI",
"published": "20170826174820",
"title": "Mining the Demographics of Political Sentiment from Twitter Using Learning from Label Proportions"
} |
ChisioKucukkececi et ali-Vis Research Group, Computer Eng. Dept.,Bilkent University, Ankara 06800, Turkey <http://www.cs.bilkent.edu.tr/ ivis/chisio>Chisio: A Compound GraphEditing and Layout Framework Cihan KucukkececiUgur DogrusozTo whom correspondenceshould be addressed ([email protected])Esat BelviranliAlptug Dilek December 30, 2023 ================================================================================================================================= We introduce a new free, open-source compound graph editing and layout framework named , based on the Eclipse Graph Editing Framework (GEF) and written in Java.can be used as a finished graph editor with its easy-to-use graphical interface. The framework has an architecture suitable for easy customization of the tool for end-user's specific needs as well.comes with a variety of graph layout algorithms, most supporting compound structures and non-uniform node dimensions. Furthermore, new algorithms are straightforward to add, makingan ideal test environment for layout algorithm developers.§ INTRODUCTIONAs graphical user interfaces have improved, and more state-of-the-art software tools have incorporated visual functions, interactive graph editing andlayout facilities have become important components of software systems. Overthe years, an abundant number of such tools have been made available forconsumption both commercially and academically <cit.>. However, only a few, if any, non-commercial systems seem to address compound structures ofgraphs both in regards to editing and layout capabilities.should fillan important gap in this field. can be used as a finished graph editor and layout tool. The tool has a user-friendly graphical interface for interactive editing of compound graphs, accepting graphs in GraphML format <cit.>. It also features a numberof graph layout algorithms from spring embedders to Sugiyama algorithm for hierarchical graphs. In addition, many aspects of the tool, from graphobject UIs to menus to persistency operations may be easily customized forspecific applications. There has been a great deal of work done on general graph layout <cit.> but considerably less on layout of compound graphs, which is a notion that has been in use to represent more complex types of relations or varying levels of abstractions in data <cit.>. The limited work done on compound graph layout has mostly focused on layout of hierarchical graphs <cit.>, where underlying relational information is assumed to be under a certain hierarchy.provides an implementation of a relatively recent spring embedder based layout algorithm for undirected compound graphs, with arbitrarily deep nesting relations <cit.>. In addition, some other well-known layout methods have been extended to support compound structures in . Finally,hosts an implementation of a new spring embedder based circular layout algorithm for clustered graphs. § GRAPH AND DRAWING MODEL OF was built on top of the Eclipse Graph Editing Framework (GEF) <cit.>. GEF assumes that you have a model that you would like to display and edit graphically. The controllers bridge the view and model (Figure <ref>). Each controller is responsible both for mapping the model to its view, and for making changes to the model. The controller also observes the model, and updates the view to reflect changes in the model's state. Controllers are the objects with which the user interacts. The compound graphs are modeled inusing this framework and the concept of graph managers described earlier (Figure <ref>). A compound node manages a list of children graph objects. A dedicated one is set as the root of the nesting hierarchy. Through recursive use of compound nodes as child objects, an arbitrary level of nesting can be created. Nodes, edges and compound nodes inall have distinct properties and UIs, which can be changed by its graphical user interface or through programming. Each node is drawn as a rectangle, ellipse or triangle. Edges can be drawn in a variety of styles. Compound nodes are always drawn with a rectangle, where the name text is displayed on its bottom margin and its geometry is auto-calculated using the geometry of its contents to tightly bound its contents plus user-defined child graph margins. Figure <ref> shows the basics of drawing compound graphs in .§ CHISIO AS AN EDITOR can be used as a finished graph editing and layout tool <cit.>. Graphs created with other tools (and saved in GraphML format <cit.>) and loaded intoor graphs created incan be edited and laid out interactively. The tool features standard graph editing facilities such as zoom, scroll, add or remove graph objects, move, and resize. Object property and layout options dialogs are provided to modify existing graph object properties and layout options, respectively. In addition, printing or saving the current drawing as a static image and persistent storage facilities are supported. Furthermore, a highlight mechanism is provided to emphasize subgraphs of user's interest. Figure <ref> shows a sample screen-shot of . Drawing in <ref> is a partial drawing produced byfor a biological pathway. § CUSTOMIZING CHISIO One can customizefor their specific needs by adding new node/edge types or by modifying existing nodes/edges (UI and attributes) <cit.>. In addition, you may customize the menus to add new functionality as well as modifying node and edge menus and property inspectors. Furthermore,is designed for easy integration of new layout algorithms. Layout researchers will especially findto be useful for implementation and testing of their new methods. A sample application based on Chisio is(Chisio BioPAX Editor) (<http://sourceforge.net/projects/chibe/>). The tool features user-friendly display, viewing, and editing of pathway models represented by the BioPAX format (<http://biopax.org>). Pathway views are rendered in a feature-rich format, and may be laid out automatically (Figure <ref>). Among other things, visualization of experimental data overlaid on pathway views is supported in . §.§ Layout Architecture in Chisio The basis inlayout is constructed through an abstract layout class namedand an associated l-structure, classes , , , and . Here anmaintains and manages a list of instances, which in turn maintains a list of s and s. Theclass converts the givenmodel into a generic l-structure used only for layout purposes that is destroyed at the end of layout. Individual layout algorithms extend theclass and run on this l-structure by overriding predefined methods. When the layout is finished, the geometry information of the l-level is transferred to model (Figure <ref>).§ LAYOUT ALGORITHMS IN CHISIO provides implementations of the following layout methods. §.§ CoSE Layout CoSE (Compound graph Spring Embedder) layout is an algorithm specifically designed for compound graphs <cit.>. It has been based on the traditional force-directed layout scheme with extensions to handle multi-level nesting, varying node sizes, and possibly other application-specific constraints. An expanded node and its associated nested graph are represented as a single entity, similar to a “cart”, which can move freely in orthogonal directions (no rotations allowed). Multiple levels of nesting is modeled with smaller carts on top of larger ones. Figure <ref> shows an example drawing produced by this layout algorithm.§.§ Cluster Layout Cluster Layout is designed to place the nodes that belong to the same group or cluster (say to refer to molecules with similar functionality or network devices in a LAN) near each other without the use of compound nodes (Figure <ref>). It uses CoSE layout as a subroutine to do that.§.§ CiSE Layout A popular way to draw clustered graphs is in a circular fashion. In other words, a circle of appropriate size is created for each cluster and the nodes in that cluster are placed around this circle trying to minimize edge crossings. Circular layout algorithms <cit.> address the issue of placing nodes of a cluster nicely around a circle, minimizing the number of edge crossing and total edge lengths. However most of these algorithms do not address the issue of how the cluster graph should be laid out (i.e. how the individual circles should be placed with respect to each other to minimize crossings and lengths of inter-cluster edges). The ones that do, can not handle non-tree structures nicely. CiSE layout algorithm does not place any constraints on the structure of the cluster graph and tries to satisfy the following specific requirements: * Nodes in the same cluster should be placed around the same circle spaced as evenly as possible. * Unclustered nodes should not be placed on or overlapping any circles, but they might have neighbor nodes placed around a circle. * The number of (intra-cluster) edge crossings between the nodes in the same cluster should be as small as possible. * The number of (inter-cluster) edge crossings between the nodes in different clusters or unclustered nodes should be as small as possible. The layout algorithm is based on the spring embedder, extending it for properly grouping nodes in the same cluster and placing them around a circle by extra constraints. The use of extra constraints is implemented by introducing additional properties to the physical model used by the spring embedder, trying to obey the basic laws of physics. Each cluster/circle is represented by a meta-node of circular shape, on which sits a round-shaped track on the periphery. The physical entities for each member node of a cluster is assumed to be either fixed (restrained to its current location on the owner circle) or flexible to move around the track on which they sit as needed by the different steps of the algorithm. In addition, we assume a dynamic center of gravity in the middle of the bounding rectangle of the current drawing, towards which an attractive force is assumed by unclustered nodes and cluster nodes (all nodes except member nodes of a cluster). Following are the main steps of the algorithm: * Step 1: In this step, we run a basic circular layout <cit.> for each cluster. * Step 2: Next we layout the cluster graph, where each node in the graph represent a cluster with the dimensions produced in Step 1, using a basic spring embedder. * Step 3: Here, our aim is to reposition/rotate circles according to the location of their out-nodes and inter-cluster edges incident on these nodes. However, nodes on the circles are not allowed to move individually. They are assumed to be pinned down to their circles. After this step, a draft layout of the whole graph is obtained. * Step 4: In this final polishing step, we obtain the final layout of the graph by trying to reduce inter-cluster edge crossing. The main difference between this step and Step 3 is that we allow nodes on circles to move on their parent circle (as well as moving with them). features a draft-implementation of this algorithm and Figure <ref> shows a sample drawing produced by this algorithm.§.§ Others also provides implementations for a regular spring embedder <cit.>, a hierarchical layout method <cit.>, and a circular layout algorithm <cit.>. § CONCLUSION In this paper we have presented a new open-source, free compound graph editing and layout tool, distributed under Eclipse Public License.version 1 is available at <http://www.cs.bilkent.edu.tr/ ivis/chisio.html>. abbrv | http://arxiv.org/abs/1708.07762v1 | {
"authors": [
"Cihan Kucukkececi",
"Ugur Dogrusoz",
"Esat Belviranli",
"Alptug Dilek"
],
"categories": [
"cs.HC"
],
"primary_category": "cs.HC",
"published": "20170825143607",
"title": "Chisio: A Compound Graph Editing and Layout Framework"
} |
^1Department of Physics and Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA^2Department of Physics, Southeast University, Nanjing 210096, China We study theoretically the topological surface states (TSSs) and the possible surface Andreev bound states (SABSs) of Cu_xBi_2Se_3 which is known to be a topological insulator at x=0. The superconductivity (SC) pairing of this compound is assumed to have the broken spin-rotation symmetry, similar to that of the A-phase of ^3He as suggested by recent nuclear-magnetic resonance experiments. For both spheroidal and corrugated cylindrical Fermi surfaces with the hexagonal warping terms, we show that the bulk SC gap is rather anisotropic; the minimum of the gap is negligibly small as comparing to the maximum of the gap. This would make the fully-gapped pairing effectively nodal. For a clean system, our results indicate the bulk of this compound to be a topological superconductor with the SABSs appearing inside the bulk SC gap. The zero-energy SABSs which are Majorana fermions, together with the TSSs not gapped by the pairing, produce a zero-energy peak in the surface density of states (SDOS). The SABSs are expected to be stable against short-range nonmagnetic impurities, and the local SDOS is calculated around a nonmagnetic impurity. The relevance of our results to experiments is discussed.Nematic superconductivity in Cu_xBi_2Se_3: The surface Andreev bound states Lei Hao^1,2 and C. S. Ting^1 December 30, 2023 =========================================================================== § INTRODUCTION Cu_xBi_2Se_3, the first superconductor (SC) realized in a doped topological insulator, has attracted great interest since its discovery <cit.>. After early gathering of experimental and theoretical evidences trying to imply the superconductivity (SC) pairing in this compound to be topologically nontrivial <cit.>, several later experiments seem to conclude that the actual pairing symmetry is the conventional s-wave and topologically trivial <cit.>. However, initiated by a recent nuclear magnetic resonance (NMR) experiment <cit.>, a new surge of researches on this material and its several variants (e.g., Sr_xBi_2Se_3 <cit.>, Nb_xBi_2Se_3 <cit.>, and Tl_xBi_2Se_3 <cit.>) revived the possibility that the SCphase realized in this group of materials is topologically nontrivial. In the NMR experiment, a prominent in-plane uniaxial anisotropy in the SC order parameter is observed in the Knight shift measurement <cit.>. The above twofold in-plane rotational symmetry is confirmed further by field-angle dependent specific heat measurement <cit.>. These new experiments indicate that the SC pairing in Cu_xBi_2Se_3 might be the first example of a pairing breaking spontaneously the spin rotation symmetry of the parent material, from the threefold rotational symmetry of the normal phase to the uniaxial twofold in-plane symmetry <cit.>. Amazingly, the new NMR experiment is claimed to be explainable by an odd-parity pairing proposed earlier <cit.>. The alluded pairing, with a salient in-plane anisotropy, has also been called a nematic SC <cit.>. For a simplified model with spheroidal Fermi surface, this SC pairing is known to be equivalent to the A phase of ^3Heand has two bulk nodes <cit.>. However, by including in the model the terms responsible for hexagonal warping of the Fermi surface, this SC was argued to have a full pairing gap <cit.>. If confirmed, this could be the first three-dimensional topological superconductor with a fully-gapped bulk ever discovered. However, a hallmark of the topological SC is the presence of surface Andreev bound states (SABSs) within the bulk SC gap, and its robustness against nonmagnetic impurities and interactions. In early studies in terms of simplified models based on spheroidal Fermi surface without the hexagonal warping terms, the SC is known to support novel SABSs which are flat along one direction [i.e., (k_x,0)] and linearly dispersive in the perpendicular direction [i.e., (0,k_y)] <cit.>. In addition, the topological surface states (TSSs) are well defined at the chemical potential and well separated from the bulk states<cit.>. The SC pairing would not open a gap in the TSSs and thus the TSSs coexist with the SABSs <cit.>. If the hexagonal warping term is incorporated, then on one hand the two bulk nodes would be gapped out <cit.>, whereas on the other hand the TSSs will remain gapless <cit.>. In addition, the odd-parity topological SC pairing should still supports SABSs, independent of the model parameters <cit.>. As a result, it seems natural to expect the existence of the nontrivial spectral features related to the two types of surface states.Previous experiments, on the other hand, have made conflicting statements on the surface states in the SC phase. In several point contact spectroscopy (PCS) studies, a pronounced zero-bias peak appeared and was ascribed to the existence of SABSs <cit.>. Later, a scanning tunneling spectroscopy (STS) study reported a standard BCS-like spectrum <cit.>. Moreover, a detailed PCS experiment arrived at the same conclusion that no SABSs existed <cit.>. However, recent experiments clearly indicated that the Fermi surface of the Cu_xBi_2Se_3 compound changes from spheroidal to corrugated-cylindrical surfaces as the doping x increases <cit.>. As a result and without detailed investigations, there exist works <cit.> arguing that the absence of SABSs in the odd-parity (e.g., nematic) SC state of Cu_xBi_2Se_3 is consistent with a (corrugated) cylindrical Fermi surface. Inspired by the above experimental observations and theoretical arguments, in this work we explore whether the corrugated cylindrical Fermi surface would or would not support the SABSs in the Cu_xBi_2Se_3 compound with the nematic SC pairing proposed for explaining the recent experiments <cit.>. Using the band parameters which fit qualitatively the experimental Fermi surfaces and the TSSs, we find that the SC is fully gapped, and the bulk gap is rather anisotropic. The minima of the bulk SC gap is smaller than the maximum of the bulk gap by two to three orders of magnitude. The bulk quasiparticle spectrum, while in principle is fully gapped, appears to be nodal-like from the point of view of measurements. The SABSs are shown to exist in the clean system for both spheroidal and corrugated cylindrical Fermi surfaces, and the zero-energy Majorana bound state is a characteristic of the SABSs. In addition, we verify the stability of the SABSs against short-range nonmagnetic impurities, both for uniformly distributed bulk impurities and for dilute impurities doped only to the surface layer. As to whether the SABSs could be observed experimentally may depend on the condition of the sample surfaces. For instance, the excessive magnetic Cu^2+ (3d^9) ions or Cu (3d^104S^1) atoms on the surfaces could very much suppress the SABSs. For clean and perfect surface, the SABSs should be detectable. Recently, the possible existence of surface states in superconducting Sr_xBi_2Se_3 is inferred from the Shubnikov-de Hass oscillation measurement <cit.>. In combination with the results of the present work, it is therefore highly desirable to reexamine the existence of the SABSs in Cu_xBi_2Se_3 as a crucial test for the relevancy of the proposed SC pairing <cit.>.§ MODEL We consider a two-orbital tight-binging model for the low-energy degrees of freedom of the materialH_0(𝐤) = ϵ(𝐤)I_4+M(𝐤)Γ_5+B_0c_z(𝐤)Γ_4 +A_0[c_y(𝐤)Γ_1 -c_x(𝐤)Γ_2]+R_1d_1(𝐤)Γ_3+R_2d_2(𝐤)Γ_4.The basis vector is taken as ϕ^†_𝐤=[a^†_𝐤↑, a^†_𝐤↓, b^†_𝐤↑, b^†_𝐤↓], where the two orbitals (a and b) are mainly from the two p_z orbitals on the top and bottom Se layers of each Bi_2Se_3 quintuple unit <cit.>. I_4 is the 4×4 unit matrix. Γ_1=σ_3⊗ s_1, Γ_2=σ_3⊗ s_2, Γ_3=σ_3⊗ s_3, Γ_4=-σ_2⊗ s_0, and Γ_5=σ_1⊗ s_0 <cit.>. s_i and σ_i (i=1,2,3) are Pauli matrices for the spin and orbital degrees of freedom. The parity operator is defined as P=σ_1⊗ s_0 <cit.>. The above model was obtained previously based on symmetry and comparison with an existing 𝐤·𝐩 model <cit.>. The lattice of Bi_2Se_3 and Cu_xBi_2Se_3, which belong to the D_3d^5 space group, is mapped to a hexagonal lattice in the tight-binding model. The in-plane (labeled as the xy plane) and out-of-plane (labeled as the z direction) lattice parameters, a and c, are taken as a=4.14 Å and 3c=28.64 Å <cit.>. ϵ(𝐤)=C_0+2C_1[1-cos(𝐤·δ_4)] +4/3C_2[3-cos(𝐤·δ_1)-cos(𝐤·δ_2) -cos(𝐤·δ_3)]. M(𝐤) is obtained from ϵ(𝐤) by making the substitutions C_i→ M_i (i=0,1,2). c_x(𝐤)=1/√(3) [sin(𝐤·δ_1)-sin(𝐤·δ_2)], c_y(𝐤)=1/3[sin(𝐤·δ_1)+sin(𝐤·δ_2) -2sin(𝐤·δ_3)], and c_z(𝐤)=sin(𝐤·δ_4). Finally, d_1(𝐤)=-8/3√(3)[sin(𝐤·𝐚_1)+sin(𝐤·𝐚_2) +sin(𝐤·𝐚_3)] and d_2(𝐤)=-8[sin(𝐤·δ_1) +sin(𝐤·δ_2)+sin(𝐤·δ_3)]. Here, the four nearest-neighboring bond vectors of the hexagonal lattice are δ_1=(√(3)/2a, 1/2a, 0), δ_2=(-√(3)/2a, 1/2a, 0), δ_3=(0, -a, 0), and δ_4=(0, 0, c). The three in-plane second-nearest-neighboring bond vectors in d_1(𝐤) are 𝐚_1=δ_1-δ_2, 𝐚_2=δ_2-δ_3, and 𝐚_3=δ_3-δ_1. The last two terms in H_0(𝐤) induce hexagonal warping of the Fermi surface and the topological surface states (TSSs)<cit.>. Before doping with copper, the Fermi surface of Bi_2Se_3 is spheroidal. After intercalating copper to inter-quintuple-layer positions, the material becomes more two-dimensional. According to the experiments <cit.>, the Fermi surface for certain Cu_xBi_2Se_3 becomes (corrugated) cylindrical, although the details of the evolution are still unclear. On the other hand, a common feature of the normal phase of superconducting Cu_xBi_2Se_3 is that the TSSs are well defined and coexist with the Fermi surface <cit.>. In this work, we consider three sets of parameters shown in Table I. The Fermi surface contours (on the k_y=0 plane) and the surface spectral functions for the three sets of parameters are shown in Figure 1. The surface spectral functions are calculated in terms of the iterative Green's function method <cit.>, for the upper xy surface of a sample that can be regarded as consisting of an infinite number of layers. The parameters are chosen here to fit qualitatively three different shapes of the Fermi surfaces and the coexisting TSSs. The second corrugated cylindrical Fermi surface is less corrugated compared to the first corrugated cylindrical Fermi surface. Therefore, the material described by the `Cylindrical 2' is more two-dimensional than the material described by the `Cylindrical 1'. With the three typical sets of parameters, we can study the qualitative evolution of the property of a pairing as the Fermi surface turn from spheroidal to corrugated cylindrical and then becomes even more two-dimensional.Note that, these parameter sets are chosen to reflect the evolution of the Fermi surface and the coexistence with the TSSs, which are most crucial for the low-energy physics in the superconducting phase. A completely two-dimensional model with zero hopping along the z direction is unsuitable because it cannot give the TSSs observed in experiments <cit.>. In addition, the relative magnitudes of the various parameters in Table I are in agreement with the set of parameters obtained previously by fitting the first-principle band structures for Bi_2Se_3 <cit.>. By increasing the value of R_1 artificially (e.g., to 2 eV) and keeping other parameters unchanged, the topology of the Fermi surface and the coexisting TSSs can still be retained qualitatively. We will discuss the effects of increasing R_1 at the end of Section V-B. § PSEUDOSPIN BASIS The full model contains the complete information of the phase, but it is hard to work with analytically. On the other hand, it is the states close to the Fermi level that are most important to the superconducting phase. By introducing the pseudospin basis, the full model containing both of the two bands of the model in the normal phase can be projected to a simplified model containing only the band contributing to the Fermi surface<cit.>. By making this reduction, the low-energy properties of the superconducting phase, in particular the gap structure of the bulk quasiparticle spectrum and the SABSs, can be understood more easily. Here, we follow the approach of Yip, which was originally applied to a simplified version of the model, to construct the pseudospin basis for our tight-binding model <cit.>. This method makes use of the time-reversal symmetry (T) and inversion symmetry (P) of the model, which lead to the Kramers degeneracy of each state. The two pseudospin bases for each Kramers doublet are thus required to be related by the combined action of PT operation. Throughout this work, we assume the chemical potential to lie within the conduction band. The eigenbasis can be constructed by first diagonalizing the model in the spin subspace and then in the orbital subspace. One basis can be taken as|𝐤,α'⟩=1/D̃_𝐤N_𝐤[ Ẽ_𝐤; M̃_-(𝐤) ][ A_0c_+(𝐤);D_-(𝐤) ],where the first and second vectors are separately spinors in the subspaces of the original orbital and spin degree of freedom. For notational simplicity, here and later in this work we will use the following abbreviations c_±(𝐤)=c_y(𝐤)± i c_x(𝐤), M̃_±(𝐤)=M(𝐤)± i[B_0c_z(𝐤)+R_2d_2(𝐤)], D_𝐤=√(A_0^2[c_x^2(𝐤)+c_y^2(𝐤)]+R_1^2d_1^2(𝐤)), E_𝐤=√(|M̃_±(𝐤)|^2+D_𝐤^2), Ẽ_𝐤=E_𝐤+D_𝐤, N_𝐤=√(2E_𝐤Ẽ_𝐤), D_±(𝐤)=D_𝐤± R_1d_1(𝐤), D̃_𝐤=√(2D_𝐤D_-(𝐤)). The other pseudospin basis is related to the one listed above by symmetry|𝐤,β'⟩=PT|𝐤,α'⟩=1/D̃_𝐤N_𝐤[ M̃_+(𝐤); Ẽ_𝐤 ][ -D_-(𝐤); A_0c_-(𝐤) ],In order for the pseudospin basis to have the correct P and T symmetries in the whole BZ, we fix the wave vectors in |𝐤,α'⟩ and |𝐤,β'⟩ to lie on the northern hemisphere (k_z>0). States on the southern hemisphere are obtained by symmetry, namely |-𝐤,α'⟩=P|𝐤,α'⟩ and |-𝐤,β'⟩=P|𝐤,β'⟩=T|𝐤,α'⟩.The naive choice of the pseudospin basis defined above are not guaranteed to have the correct rotational property of the original model. As a result, they may not be the suitable basis set for studying the symmetry of a specific pairing channel. As has been shown in Ref.<cit.>, a good set of the pseudospin basis can be constructed as a linear combination of |𝐤,α'⟩ and |𝐤,β'⟩ that make the magnetic moment operator expressed under this basis to have the proper transformation property under rotation <cit.>. For the model defined by Eq.(1), the magnetic moment is a linear combination of 𝐬 and σ_1𝐬. Here, following the method in Ref.<cit.>, we choose to focus on the component m_1α=g_1ασ_0+σ_1/2s_α of the magnetic moment <cit.>. α=x,y,z, and g_1x=g_1y=g_1p are assumed. In the space of {|𝐤,α'⟩, |𝐤,β'⟩}, m_1z has the following matrix formm_1z(𝐤)/g_1z|W_𝐤|^2=[ cosθ_𝐤 ie^i(φ_𝐤+2ϕ_𝐤)sinθ_𝐤; -ie^-i(φ_𝐤+2ϕ_𝐤)sinθ_𝐤-cosθ_𝐤 ],where the three phase factors, φ_𝐤, ϕ_𝐤, and θ_𝐤, are defined byc_+(𝐤)=i√(c_x^2(𝐤)+c_y^2(𝐤))e^-iφ_𝐤=ic(𝐤)e^-iφ_𝐤, W_𝐤=Ẽ_𝐤+M̃_+(𝐤)/√(2)N_𝐤=|W_𝐤|e^iϕ_𝐤, R_1d_1(𝐤)+iA_0c(𝐤)=D_𝐤e^iθ_𝐤.m_1z(𝐤) in the basis of{|𝐤,α'⟩, |𝐤,β'⟩} clearly does not have the desired form of the z-component of an axial vector. The new basis |𝐤,α⟩ and |𝐤,β⟩ are constructed such that m_1z(𝐤) is proportional to the z-component of the Pauli matrix, namely they are the eigenbasis of m_1z <cit.>. We take|𝐤,α⟩=h(𝐤)[(1+cosθ_𝐤)|𝐤,α'⟩ -ie^-i(φ_𝐤+2ϕ_𝐤)sinθ_𝐤|𝐤,β'⟩].From |𝐤,β⟩=PT|𝐤,α⟩, we get the other basis|𝐤,β⟩=h^∗(𝐤)[(1+cosθ_𝐤)|𝐤,β'⟩ -ie^i(φ_𝐤+2ϕ_𝐤)sinθ_𝐤|𝐤,α'⟩].Normalization of the eigenbasis requires|h(𝐤)|^2=1/2(1+cosθ_𝐤)=1/4cos^2θ_𝐤/2.In this basis, we have m_1z(𝐤)=g_1z|W_𝐤|^2ρ_z, where ρ_z is the conventional z-component of the Pauli matrices. The x-component of 𝐦_1 in the new basis has a purely off-diagonal form with the two off-diagonal elements[m_1x(𝐤)]_αβ/g_1p|W_𝐤|^2=[m_1x(𝐤)]^∗_βα/g_1p|W_𝐤|^2= [2ih^∗(𝐤)cosθ_𝐤/2e^i(φ_𝐤+ϕ_𝐤)]^2.Takingh(𝐤)=i/2cosθ_𝐤/2e^i(φ_𝐤+ϕ_𝐤),we havem_1x(𝐤)=g_1p|W_𝐤|^2ρ_x, m_1y(𝐤)=g_1p|W_𝐤|^2ρ_y,where ρ_x and ρ_y are the conventional x-component and y-component of the Pauli matrices. Therefore, we have shown that the new basis {|𝐤,α⟩, |𝐤,β⟩} defined by Eqs.(8), (9), and (12) can ensure the correct transformation property of the magnetic moment operator, and are thus proper choices in discussing symmetry properties of the system. This basis, employed in the present work, isshown <cit.> to coincide with the so-called manifestly covariant Bloch basis introduced by Fu <cit.>.§ PAIRING AND GAP STRUCTURE OF THE BULK QUASIPARTICLE SPECTRUMIn the Nambu basis, ψ^†_𝐤=[ϕ^†_𝐤,(ϕ_-𝐤)^T], and denoting the pairing term generically as Δ(𝐤), the model for a bulk superconducting topological insulator is written asĤ = 1/2∑_𝐤ψ^†_𝐤[H_0(𝐤)-μ I_4Δ(𝐤);-Δ^∗(-𝐤) μ I_4-H^∗_0(-𝐤) ]ψ_𝐤= 1/2∑_𝐤ψ^†_𝐤H(𝐤)ψ_𝐤,where μ is the chemical potential. The 1/2 factor accounts for the particle-hole redundancy introduced by the Nambu representation. The two-fold in-plane rotation symmetry in the Knight shift and field-angle dependent specific heat experiments indicate that the pairing must belong to a multi-dimensional representation of the symmetry group. Because the pairing order parameter for a one-dimensional representation should necessarily respect the three-fold rotational symmetry of the D_3d^5 space group. Presently, most of attention has been paid to the two-dimensional E_u representation of the D_3d^5 space group. One set of the two bases for the E_u representation is Δ_4a(𝐤)=iΔ_aσ_2⊗ s_0 and Δ_4b(𝐤)=Δ_bσ_2⊗ s_3, with Δ_a and Δ_b the pairing amplitudes. For a simplified model without the hexagonal warping terms (i.e., R_1=R_2=0), both of the two components are known to lead to bulk spectrum with point nodes <cit.>. However, it was shown by Fu that the bulk nodes for Δ_4a(𝐤) are gapped out by including the hexagonal warping term proportional to R_1 of Eq.(1) <cit.>. It seems that the fully gapped Δ_4a(𝐤) provides a most natural explanation to the Knight shift and field-angle dependent specific heat experiments. Here, we study more carefully the excitation gap of the bulk quasiparticle spectrum. Since only the conduction band contribute to the Fermi surface, the gap structure of the quasiparticle spectrum is understood more easily from the low-energy effective model obtained by projecting the full model defined by Eq.(15) to the conduction band <cit.>. The dispersion of the conduction band is ϵ(𝐤)+E_𝐤. The projection is thus achieved by replacing H_0(𝐤)-μ I_4 with (ϵ(𝐤)+E_𝐤-μ)I_2 and transforming the pairing term expressed in the spin-orbital basis to the pseudospin basis derived in the last section. For an arbitrary pairing denoted as Δ(𝐤) in the original spin-orbital basis, its expression in the new pseudospin basis of the conduction band isΔ̃(𝐤)=U^†_𝐤ΔU^∗_-𝐤,where the transformation matrix is U_𝐤=[|𝐤,α⟩,|𝐤,β⟩].For Δ_4a(𝐤)=iΔ_aσ_2⊗ s_0, we have <cit.>Δ̃_4a(𝐤) = Δ_0/E_𝐤[-R_1d_1(𝐤)ρ_1 -(B_0c_z(𝐤)+R_2d_2(𝐤))ρ_2 +A_0c_y(𝐤)ρ_3]iρ_2.On the Fermi surface, ϵ(𝐤)+E_𝐤-μ=0, the quasiparticle spectrum is determined only by the pairing termE(𝐤)=±|det[Δ̃_4a(𝐤)]| =±|Δ_a|√(1-M^2(𝐤)+A_0^2c_x^2(𝐤)/E_𝐤^2).Up to slight hexagonal warping induced by terms proportional to R_1 and R_2, both M(𝐤) and E_𝐤 are approximately symmetrical in the k_xk_y plane. As a result of the c_x(𝐤) term in Eq.(18), the size of superconducting gap is smaller along the k_y=0 contour of the Fermi surface than that along the k_x=0 contour of the Fermi surface. Therefore, the bulk energy spectrum for Δ_4a(𝐤) has a strong anisotropy between the k_x direction and the k_y direction.Let us focus on the contour of the Fermi surface on the k_y=0 plane, where the minimum of the superconducting gap is attained. Eq.(18) is written asE(𝐤)=±|Δ_a|/μ-ϵ_𝐤√(R_1^2d_1^2(𝐤)+B_0^2c^2_z(𝐤)),where ϵ(𝐤)+E_𝐤-μ=0 has been used. For clarity, we have d_1(𝐤)≃ (k_xa)^3 for k_y=0 and k_x small, and c_z(𝐤)=sin(k_zc). Notice that for both spheroidal and corrugated cylindrical Fermi surfaces, including those shown in Fig.1, d_1(𝐤) and c_z(𝐤) do not attain zero simultaneously. While for R_1=0 there are point nodes on the k_y=0 Fermi surface contour determined by c_z(𝐤)=0, a finite R_1 removes all these nodes <cit.>. One exception is a spheroidal Fermi surface with a point 𝐤=(0,0,π) on it, which marks the transition between a spheroidal Fermi surface and a corrugated cylindrical Fermi surface. For practical purpose, however, we will ignore this special case and so the bulk spectrum of Δ_4a(𝐤) is always fully gapped for R_10. On the other hand, k_xa on the Fermi surface is small for actual materials. The size of the gap for c_z(𝐤)=0 and k_y=0, which grows like (k_xa)^3 for small k_x, is actually much smaller than Δ_a. In all cases studied, μ-ϵ(𝐤)=E_𝐤 has only a small variation on the Fermi surface. Therefore, the numerator of Eq.(19) determines the qualitative behavior of the superconducting gap. For simplicity and without losing generality, we focus on the k_x≥0 and k_z≥0 portion of the k_y=0 Fermi surface contours shown in Figs.1(a, c, and e). Each point on the chosen portion of the Fermi surface contour can then by labeled by a unique k_x. For the spheroidal Fermi surface shown in Fig.1(a), as we go along the Fermi surface contour from (0,0,k_z1) (k_z1c≃0.29π) to (k_x1,0,0) (k_x1a≃0.077π), c_z(𝐤) decreases monotonously and we have |B_0c_z(0,0,k_z1)|≫|R_1d_1(k_x1,0,0)|. The size of the gap is thus expected to decrease monotonously as we go along the contour from (0,0,k_z1) to (k_x1,0,0). For the two cases with corrugated cylindrical Fermi surfaces, the Fermi surface contour is bounded by two points (k_xi,0,π) and (k_xf,0,0). k_xia≃0.039π (k_xia≃0.074π) and k_xfa≃0.103π (k_xfa≃0.103π) for Fig.1(c) [Fig.1(e)]. As we increase k_x from k_xi to k_xf, k_z changes from π to 0. As a result, we have a nonmonotonous variation of c_z(𝐤), which first increases towards 1 as k_z approaches π/2 and then decreases to 0 afterwards. Because we have |R_1d_1(k_x,0,k_z)/B_0|≪1 along the Fermi surface contour, we expect to get a nonmonotonous variation of the superconducting gap for corrugated cylindrical Fermi surfaces, which first increases and then decreases, with two minima at (k_xi,0,π) and (k_xf,0,0). In comparison to the case for Fig.1(a), the number of gap minima is doubled when the Fermi surface evolves from spheroidal to corrugated cylindrical.To have a more quantitative understanding on the evolution of the superconducting gap explained above, we plot in Fig.2 simultaneously three functions f_1=4√(R_1^2d_1^2(𝐤)+B_0^2c^2_z(𝐤)), f_2=4(μ-ϵ_𝐤)=4E_𝐤, and f_3=f_1/f_2=|E(𝐤)/Δ_a|, on the k_z≥0 and k_x≥0 portion of the k_y=0 Fermi surface contours. The states are labeled uniquely in terms of the value of k_x. The value of f_3 gives the magnitude of the bulk gap, normalized by the pairing amplitude. For all three sets of parameters considered, the minimal values of the bulk gap are much smaller than the corresponding maximum values. For the experimental transition temperature of about 3.8 Kelvin, the pairing amplitude (i.e., Δ_a) is of the order 1 meV. The minimal value of the bulk gap is two to three orders of magnitude smaller than the pairing amplitude. It is also interesting to notice that, when the smaller Fermi momentum of the corrugated cylindrical Fermi surface along k_x [e.g., k_xia≃0.039π for Fig.2(b)] is smaller than the Fermi momentum of the spheroidal Fermi surface along k_x [e.g., k_x1a≃0.077π for Fig.2(a)], the minimum superconducting gap for the corrugated cylindrical Fermi surface can be smaller than the minimum superconducting gap for the spheroidal Fermi surface. From a practical point of view, and for both spheroidal and corrugated cylindrical Fermi surfaces, the minimum of the superconducting gap acts effectively as point node of the bulk spectrum. The above picture holds as long as R_1 is not extremely (e.g., two to three orders of magnitude) larger than the value used. For the parameter set of `Cylindrical 2' in table I, a two orders of magnitude larger R_1 (i.e., 20 eV) is needed to increase the minimum of the bulk gap along k_x to the same order of magnitude to that along k_y. A further tenfold enhancement in R_1 is required to achieve the same increase in the bulk gap for the parameter set of `Cylindrical 1' in table I. These large values of R_1 are not only inconsistent with the magnitudes of other parameters but also will distort strongly the Fermi surface and thus deviate qualitatively from experiments. Therefore, the bulk spectrum for Δ_4a(𝐤) should be nodal-like for realistic parameters. For Δ_4b(𝐤)=Δ_bσ_2⊗ s_3, the effective pairing isΔ̃_4b(𝐤) = Δ_b/E_𝐤[(B_0c_z(𝐤)+R_2d_2(𝐤))ρ_1 -R_1d_1(𝐤)ρ_2 -A_0c_x(𝐤)ρ_3]iρ_2.The minimum of the superconducting gap lies along the k_x=0 plane, where d_1(𝐤)=c_x(𝐤)=0. Along the intersection contour of the Fermi surface with the k_x=0 plane, E_𝐤=μ-ϵ(𝐤) is a smooth function of the wave vector. The variation of the gap is thus determined by B_0c_z(𝐤)+R_2d_2(𝐤). For R_2=0, we reproduce the known result that Δ_4b(𝐤) has bulk point nodes determined by c_z(𝐤)=k_x=0. The number of the point nodes is two (four) for spheroidal (corrugated cylindrical) Fermi surface. For R_20, the above point nodes are gapped out, with the magnitude of the gap proportional to |Δ_bR_2d_2(0,k_Fy,0)|/E_𝐤 [and also |Δ_bR_2d_2(0,k'_Fy,π)|/E_𝐤 for corrugated cylindrical Fermi surface], where k_Fy (k'_Fy) is the k_y component of the Fermi momentum. The original point nodes do not simply vanish. Instead, they are tilted away from the (0,k_y,0) axis [and also the (0,k_y,π) axis for the case with corrugated cylindrical Fermi surface] into the k_yk_z plane. If we have |R_2/B_0|≫1, a fully-gapped bulk spectrum can be obtained. However, for realistic parameters, the point nodes are still present. For the parameters considered in Table I, the point nodes of Δ_4b(𝐤) are in fact still very close to the point nodes for R_2=0. § SURFACE ANDREEV BOUND STATESSince Δ_4b(𝐤) is nodal for practical model parameters, we will focus on the fully-gapped Δ_4a in what follows. For spheroidal Fermi surface, Δ_4a was known to support a peculiar surface Andreev bound states (SABSs) on the xy surface of a sample, which is (almost) flat along the k_x direction of the surface BZ <cit.>. It was argued in later works that, when the Fermi surface becomes two-dimensional-like with copper intercalation, the SABSs for Δ_4a would disappear <cit.>. This conclusion is natural if the Fermi surface is purely cylindrical with no dispersion along k_z, because the existence of SABSs on the xy surface is associated with a sign change in the pairing term upon reflection from the surface, which requires on one hand a finite dispersion along k_z and on the other hand a pairing component that changes sign with the reversal of k_z.However, there seems to be no reason why the Fermi surface can turn from three-dimensional to completely two-dimensional with copper intercalation. On one hand, one experiment reports corrugated cylindrical rather than completely cylindrical Fermi surface <cit.>. On the other hand, Cu_xBi_2Se_3 is known to be superconducting in a wide range of x values <cit.>. It is natural to expect that the Fermi surface evolves continuously from spheroidal to corrugated cylindrical as x increases. Finally, a completely two-dimensional Fermi surface is inconsistent with the existence of the TSSs, observed experimentally <cit.>. Therefore, compared to completely cylindrical Fermi surface, corrugated cylindrical Fermi surfaces with different degrees of corrugation [e.g., those shown in Fig.1] are better descriptions of the actual Fermi surface. For these Fermi surfaces, the bulk gap for the Δ_4a pairing has minima along the k_x direction, which is explained above and illustrated in Fig.2. Since the conduction band has a finite dispersion along k_z, and the gap is dominated by the c_z(𝐤) term which is odd in k_z, it seems natural to expect the prevalent existence of SABSs. As shown below, this is true. We first give some analytical analysis, which is then followed by numerical results. Finally, we study the stability of the SABSs against surface and bulk nonmagnetic impurities.§.§ Analytical analysis for a simplified modelTo gain a qualitative understanding of the SABSs for a Fermi surface in the shape of a spheroid or a corrugated cylinder, we ignore the hexagonal warping in the Fermi surface and consider a band with the following dispersionξ_𝐤=k_x^2+k_y^2/2m_1^∗-t_zcos k_z-μ,where m_1^∗ is the effective mass in the k_xk_y plane, and the dispersion along k_z is determined by t_z. ħ=1 is assumed. m_1^∗ is assumed to be small so that the Fermi momenta along directions in the k_xk_y plane are small, i.e. √(2m_1^∗μ) is small (m_1^∗>0 and μ>0). The Fermi surface is spheroidal (corrugated cylindrical) when |t_z/μ|≥1 (0<|t_z/μ|<1). For the pairing term, it is convenient to replace the factor E_𝐤 with a constant. Eq.(17) is thus reduced toΔ̃_4a(𝐤) = Δ̃_0[-R_1d_1(𝐤)ρ_1 -(B_0c_z(𝐤)+R_2d_2(𝐤))ρ_2 +A_0c_y(𝐤)ρ_3]iρ_2,where Δ̃_0 is the pairing amplitude divided by the constant representing E_𝐤. There are several available approaches that we can use to derive the SABSs. Here, we follow the approach of mapping the surface problem by an equivalent junction problem. Namely, we consider a junction at z=0 between the surfaces of two bulk samples; one is extended from z=0 to ∞, and the otheris from z=0 to -∞. The two bulk samples are both described by Eq.(21). The problem of scattering off the surface is mapped to a sign change in the components of the pairing term odd in k_z. The z<0 part of the junction is described simply by Eqs.(21) and (22). The z>0 part of the junction is described by Eq.(21) and Eq.(22) with the sign of the term proportional to c_z(𝐤) reversed. The model on either side of the junction is a 4×4 model with two 2×2 block diagonals for the bare bands and two 2×2 off-diagonal blocks representing the pairing term.To proceed, we adopt the quasiclassical approximation to the ansatz for the wave function of the SABSs, to separate the fast and slow degrees of freedom. The 4×4 eigenvector of the SABSs is thus taken asφ(𝐤_F,𝐫)=[ u(𝐫); v(𝐫) ]= e^i𝐤_F·𝐫[ f(𝐫); g(𝐫) ].In the same spirit, we expand the bulk band around the Fermi momentum 𝐤_F asξ_𝐤≃𝐯_F(𝐤_F)·(-i∇-𝐤_F),where the Fermi velocity is defined as𝐯_F(𝐤_F)=∇_𝐤ξ_𝐤|_𝐤=𝐤_F.We assume a perfect junction in which the translational invariance within the junction plane is preserved. The problem is thus reduced to a one-dimensional scattering problem along the z direction. That is, the dependencies in the x and y coordinates occur only through the exponential pre-factor of Eq.(23). f and g depend only on z. Consistent with this assumption on the wave function, we replace in the pairing term the k_x and k_y components of the wave vectors with k_Fx and k_Fy. The c_z(𝐤) term is then expanded to linear order of k_z-k_Fz.The interface localized states are solved by imposing the following boundary conditions to the wave functionφ(𝐤_F,x,y,z=0^+)=φ(𝐤_F,x,y,z=0^-), φ(𝐤_F,x,y,z=-∞)=φ(𝐤_F,x,y,z=+∞)=0.0^+ and 0^- are positive and negative infinitesimals. Focusing on the direction of k_Fy=0, where the minima of the superconducting gap are attained, we indeed find solutions satisfying the above boundary conditions, with energies|E(k_Fx,0,k_Fz)|=|Δ̃_0R_1d_1(k_Fx,0,k_Fz)|.From the discussions on the bulk superconducting gap in the previous section, the energy of the above bound states are well below the bulk gap in a large part of the bulk gap. Therefore, they are well-defined in-gap states.§.§ Numerical results for clean system Inspired by the existing experiments, we consider the three sets of parameters in Table I, which result separately a spheroidal Fermi surface and two corrugated cylindrical Fermi surfaces with different degrees of corrugation. Besides the shape of the Fermi surface, these parameters allow the simultaneous presence of bulk conduction band and the topological surface states (TSSs) at the Fermi level. As shown in Figure 3 are the surface spectral functions for the three parameter sets, for clean systems. The surface spectral function for a wave vector in the surface BZ are defined as summation over the imaginary part of the particle Green's function, which are obtained in terms of standard iterative Green's function method <cit.>. From Fig.3, the SABSs exist for both spheroidal and corrugated cylindrical Fermi surfaces. One essential feature is the existence of a nearly flat band of Andreev bound states along k_x at the center of the SC gap. The magnitude of the bulk gap depends sensitively on the model parameters and can be vanishingly small for the parameters with (corrugated) cylindrical Fermi surface, consistent with the previous section. A qualitative difference from previous results obtained for simplified models without the hexagonal warping terms is that <cit.>, the SABSs are not exactly flat along (k_x,0), which becomes increasingly clear as the size of the (larger) gap minimum increases from Fig.1(a) to Fig.1(c) and Fig.1(e).The surface spectral functions can be probed by ARPES <cit.>. The integrated surface spectral function, the surface density of states (SDOS), can be probed by tunneling spectroscopy <cit.>. In Fig.4, we have shown the SDOS together with the corresponding bulk density of states (BDOS). A common characteristic of the results for all three parameter sets is the appearance of a prominent zero-energy peak corresponding to the mid-gap Majorana bound states plus a continuum of low-energy states filling up the bulk superconducting gap. As is explained in Sec.IV, the minimum of the bulk gap scales linearly with R_1. We have made test calculations by increasing R_1 artificially to 2 eV and 20 eV, for the third set of parameters (`Cylindrical 2' in Table I). The minimal size of the bulk gap along k_x increases linearly with R_1, and the BDOS becomes increasingly U shaped with a flat bottom of zero DOS. The SABSs, on the other hand, persist and traverse the bulk gap for all parameters considered. The zero-energy TSSs also persist. As a result, the SDOS is still featured by the existence of in-gap states with a peak at or close to zero energy.§.§ Stability against nonmagnetic impuritiesHaving verified the presence of SABSs and seen their peculiar dispersions in clean systems, we proceed to test their stability against various imperfections. From a practical point of view relevant to Cu_xBi_2Se_3, it is plausible to focus on the effect of short-range nonmagnetic impurities. We study the effect of the nonmagnetic impurities at two levels, impurities uniformly distributed in the whole sample and separate point-like impurities situating on the surface.Firstly, we consider the effect of impurities distributed uniformly throughout the whole sample. We first obtain the self-energy correction to the bulk Green's functions in terms of the self-consistent T-matrix approximation <cit.>. We consider the simplest case of uniformly distributed short-range nonmagnetic impurities, V(𝐫)=V_0δ(𝐫-𝐫_0). In this case, the self-energy is 𝐤-independent and is determined by a set of three self-consistent equations: (1) G(𝐤,ω)=[ω+iη-H(𝐤)-Σ(ω)]^-1, (2) Σ(ω)=n_imp[T(ω)-Ṽ], and (3) T(ω)=[I_8-Ṽ/N∑_𝐤G(𝐤,ω)]^-1Ṽ. Here, n_imp is the concentration of the nonmagnetic impurity, Ṽ=V_0τ_3⊗σ_0⊗ s_0, 𝐤 denotes a wave vector in the 3D BZ, and N is the number of wave vectors in the 3D BZ. After obtaining the self-energy Σ(ω) from the above self-consistency loop, we add it as an energy correction to H(𝐤) and obtain the surface Green's function in terms of the iterative Green's function method <cit.>. The resulting surface Green's function is then the proper Green's function for the surface layer in the presence of short-range nonmagnetic impurities uniformly distributed in the bulk. The bulk density of states (BDOS) and surface density of states (SDOS) obtained by this method are as shown in Figure 5. Three concentrations of the impurities (n_imp=0.001, 0.01 and 0.02) are considered. As the bulk superconducting gap of the cases with corrugated cylindrical (spheroidal) Fermi surface is filled up for n_imp=0.01 (n_imp=0.02), the fine structures in the SDOS beyond E=0 disappear, but the zero-energy surface states are still quite robust and manifest as a single zero-energy peak in the SDOS. Secondly, we consider the effect of individual point-like impurities on the surface of an otherwise clean sample. This is achieved by keeping the bulk of the material clean, and adding impurities only to the surface layer in a manner that different impurities are far away from each other. For this case, we study the changes in the surface Green's functions for the clean system induced by the surface impurities. The effect of the impurities is taken in to account in terms of the T-matrix approximation <cit.>. For a single short-range nonmagnetic impurity, V(𝐫)=V_0δ(𝐫-𝐫_0), the T-matrix is 𝐤-independentT(ω)=[I_8-Ṽ/N_xy∑_𝐤G_0(𝐤,ω)]^-1Ṽ,where N_xy is the number of unit cells (wave vectors) in the xy plane (surface BZ), G_0(𝐤,ω) is the retarded surface Green's function obtained in terms of the iterative Green's function method for a clean system <cit.>. Ṽ=V_0τ_3⊗σ_0⊗ s_0 is the impurity potential in the Nambu space. In terms of the unperturbed Green's function G_0(𝐤,ω) and the T-matrix, the perturbed Green's function is obtained in term of the T-matrix approximation asG(𝐫,𝐫',ω)=G_0(𝐫-𝐫',ω)+G_0(𝐫-𝐫_0,ω)T(ω)G_0(𝐫_0-𝐫',ω),where G_0(𝐫,ω) is the Fourier transformation of G_0(𝐤,ω). The local Green's function at 𝐫 under the influence of the nonmagnetic impurity at 𝐫_0 is defined as G(𝐫,𝐫,ω). We consider the effect of a strong unitary impurity and take V_0=1000 eV. The results of the local density of states (SLDOS) at the impurity site and its six nearest-neighbor sites for the three sets of parameters in Table I are shown in Fig. 6. The SLDOS on the nearest-neighbor site 𝐫-𝐫_0=δ_α is equal to the SLDOS on the nearest-neighbor site 𝐫-𝐫_0=-δ_α (α=1,2,3), so only results for three of the six nearest-neighbor sites are shown. Albeit quantitative differences from the SDOS on Fig.4 and Fig.5, the in-gap surface states persist. On the other hand, the SLDOS for 𝐫-𝐫_0=δ_1 and 𝐫-𝐫_0=δ_2 are identical but are different from the SLDOS for 𝐫-𝐫_0=δ_3, which is consistent with the two-fold anisotropy of the superconducting pairing between the x and y directions. It is also noted that the zero-energy surface states are also very robust at and near the impurity site for the three types of Fermi surfaces (see Fig. 6).§ CONCLUSIONIn conclusion, extensive analyses for the bulk and surface spectra have been made for the Δ_4a(𝐤) nematic pairing, suggested to be the correct SC pairing for the Cu_xBi_2Se_3 compound based on recent Knight shift and field-angle dependent specific heat measurements<cit.>. The purpose of the present work is to explore the consequences deduced from this type of pairing, taking into account the evolution of the Fermi surface from spheroidal to corrugated cylindrical <cit.>. We show that Cu_xBi_2Se_3 with Δ_4a(𝐤) pairing should be a topological superconductor with topological surface states and the surface Andreev bound states (SABSs), even if the Fermi surface has changed from spheroidal <cit.> to the corrugated cylindrical case studied in the present work. The bulk SC spectrum, while fully gapped, show prominent twofold anisotropy with vanishingly small gap minima along k_x. One of the essential features of the SABSs is the exhibition of the zero-energy Majorana bound states regardless the shape of the Fermi surface. The SABSs are shown to be robust against short-range bulk and surface nonmagnetic impurities. This is consistent with recent works which show that the surface states of class DIII topological superconductor are stable against weak disorder and interaction <cit.>. On the other hand, there are experimental controversies on the existence of SABSs <cit.>. As to whether the SABSs could be observed experimentally may depend critically on the condition of the sample surfaces. For instance, the excessive magnetic Cu^2+ (3d^9) ions or Cu (3d^104S^1) atoms on the surfaces could very much suppress the SABSs. For clean and perfect surface, on the other hand, the SABSs should be detectable. The present pairing model, an odd-parity spin-triplet pairing, yields a very anisotropic bulk SC gap while the STM experiment detected an almost isotropic s-wave like bulk gap<cit.>. It appears that the present nematic pairing model, which has successfully explained recent Knight shift and field-angle dependent specific heat measurements, are apparently having difficulty to account for other experimental measurements like the SC density of states <cit.>. Therefore it is necessary to develop a revised pairing model which includes the essential physics as discussed here and also being able to explain other experiments. This will definitely constitute a challenging topic for future study. L.H. thanks Ting-Kuo Lee and Sungkit Yip for many helpful discussions. This work was supported in part by the Robert A. Welch Foundation under Grant No. E-1146 and the Texas Center for Superconductivity at the University of Houston. L.H. is also supported by NSFC.11204035 and would also like to acknowledge the support from the China Scholarship Council. Part of the calculations were performed in the Center for Advanced Computing and Data Systems in University of Houston.hor10 Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett. 104, 057001 (2010). wray10 L. Andrew Wray, Su-Yang Xu, Yuqi Xia, Yew San Hor, Dong Qian, Alexei V. Fedorov, Hsin Lin, Arun Bansil, Robert J. Cava and M. Zahid Hasan, Nature Phys. 1762 (2010). sasaki11 Satoshi Sasaki, M. Kriener, Kouji Segawa, Keiji Yada, Yukio Tanaka, Masatoshi Sato, and Yoichi Ando, Phys. Rev. Lett. 107, 217001 (2011).koren11 G. Koren, T. Kirzhner, E. Lahoud, K. B. Chashka, and A. Kanigel, Phys. Rev. B 84, 224521 (2011).kirzhner12 T. Kirzhner, E. Lahoud, K. B. Chaska, Z. Salman, and A. Kanigel, Phys. Rev. B 86, 064517 (2012).kriener11 M. Kriener, Kouji Segawa, Zhi Ren, Satoshi Sasaki, and Yoichi Ando, Phys. Rev. Lett. 106, 127004 (2011).das11 Pradip Das, Yusuke Suzuki, Masashi Tachiki, and Kazuo Kadowaki, Phys. Rev. B 83, 220513(R) (2011).bay12 T.V. Bay, T. Naka, Y. K. Huang, H. Luigjes, M. S. Golden, and A. de Visser, Phys. Rev. Lett. 108, 057001 (2012). levy12 N. Levy, T. Zhang, J. Ha, F. Sharifi, A. A. Talin, Y. Kuk, and J. A. Stroscio, Phys. Rev. Lett. 110, 117001 (2013). peng13 Haibing Peng, Debtanu De, Bing Lv, Fengyan Wei, and Ching-Wu Chu, Phys. Rev. B 88, 024515 (2013). matano16 K. Matano, M. Kriener, K. Segawa, Y. Ando, and Guo-qing Zheng, Nat. Phys. 12, 852 (2016).fu16 Liang Fu, Nat. Phys. 12, 822 (2016). zhao14 Lukas Zhao, Haiming Deng, Inna Korzhovska, Milan Begliarbekov, Zhiyi Chen, Erick Andrade, Ethan Rosenthal, Abhay Pasupathy, Vadim Oganesyan, Lia Krusin-Elbaum, Nature Commun. 6, 8279 (2014). pan16 Y. Pan, A. M. Nikitin, G. K. Araizi, Y. K. Huang, Y. Matsushita, T. Naka, and A. de Visser, Sci. Rep. 6, 28632 (2016). du17 Guan Du, YuFeng Li, J. Schneeloch, R. D. Zhong, GenDa Gu, Huan Yang, Hai Lin, and Hai-Hu Wen, Sci. China-Phys. Mech. Astron. 60, 037411 (2017) qiu15 Y. Qiu, K. N. Sanders, J. Dai, J. E. Medvedeva, W. Wu, P. Ghaemi, T. Vojta, and Y. S. Hor, arXiv:1512.03519. asaba17 Tomoya Asaba, B. J. Lawson, Colin Tinsman, Lu Chen, Paul Corbae, Gang Li, Y. Qiu, Y. S. Hor, Liang Fu, and Lu Li, Phys. Rev. X 7, 011009 (2017). lawson16 B. J. Lawson, Paul Corbae, Gang Li, Fan Yu, Tomoya Asaba, Colin Tinsman, Y. Qiu, J. E. Medvedeva, Y. S. Hor, and Lu Li, Phys. Rev. B 94, 041114(R) (2016).wang16 Zhiwei Wang, A. A. Taskin, Tobias Frölich, Markus Braden, and Yoichi Ando, Chem. Mater. 28, 779 (2016). yonezawa17 Shingo Yonezawa, Kengo Tajiri, Suguru Nakata, Yuki Nagai, ZhiweiWang, Kouji Segawa, Yoichi Ando, and Yoshiteru Maeno, Nature Phys. 13, 123 (2017). fu10 Liang Fu and Erez Berg, Phys. Rev. Lett. 105, 097001 (2010).fu14 Liang Fu, Phys. Rev. B 90, 100509 (2014).yip13 S.-K. Yip, Phys. Rev. B 87, 104505 (2013). hao14 Lei Hao and Ting-Kuo Lee, J. Phys.: Condens. Matter 27, 105701 (2015); ibid arXiv:1407.3329v2.hao11 Lei Hao and T. K. Lee, Phys. Rev. B 83, 134516 (2011). yamakage12 Ai Yamakage, Keiji Yada, Masatoshi Sato, and Yukio Tanaka, Phys. Rev. B 85, 180509(R) (2012). hao15 Lei Hao and Jun Wang, J. Phys.: Condens. Matter 27, 255701 (2015).lahoud13 E. Lahoud, E. Maniv, M. S. Petrushevsky, M. Naamneh, A. Ribak, S. Wiedmann, L. Petaccia, Z. Salman, K. B. Chashka, Y. Dagan, and A. Kanigel, Phys. Rev. B 88, 195107 (2013). mizushima14 Takeshi Mizushima, Ai Yamakage, Masatoshi Sato, and Yukio Tanaka, Phys. Rev. B 90, 184516 (2014).liu15 Zhongheng Liu, Xiong Yao, Jifeng Shao, Ming Zuo, Li Pi, Shun Tan, Changjin Zhang, and Yuheng Zhang, J. Am. Chem. Soc. 137, 10512 (2015). nagai15 Yuki Nagai, Phys. Rev. B 91, 060502(R) (2015).liu10 Chao-Xing Liu, Xiao-Liang Qi, Haijun Zhang, Xi Dai, Zhong Fang, and Shou-Cheng Zhang, Phys. Rev. B 82, 045122 (2010). fu09 Liang Fu, Phys. Rev. Lett. 103, 266801 (2009). zhang09 Haijun Zhang, Chao-Xing Liu, Xiao-Liang Qi, Xi Dai, Zhong Fang, and Shou-Cheng Zhang, Nature Phys. 5, 438 (2009). wang10 Qiang-Hua Wang, Da Wang, and Fu-Chun Zhang, Phys. Rev. B 81, 035104 (2010). fu07 Liang Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).hao13 Lei Hao, Gui-Ling Wang, Ting-Kuo Lee, Jun Wang, Wei-Feng Tsai, and Yong-Hong Yang, Phys. Rev. B 89, 214505 (2014).acparametersP. Larson, V. A. Greanya, W. C. Tonjes, Rong Liu, S. D. Mahanti, C. G. Olson,Phys. Rev. B 65, 085108 (2002).yip14 Sungkit Yip, Annu. Rev. Cond. Matter Physics 5, 15 (2014). yip16 Sungkit Yip, arXiv:1609.04152. zocher13 Björn Zocher and Bernd Rosenow, Phys. Rev. B 87, 155138 (2013). hashimoto13 Tatsuki Hashimoto, Keiji Yada, Ai Yamakage, Masatoshi Sato, and Yukio Tanaka, J. Phys. Soc. Jpn. 82, 044704 (2013). nagai14 Yuki Nagai, Hiroki Nakamura, and Masahiko Machida, J. Phys. Soc. Jpn. 83, 053705 (2014). takami14 Shota Takami, Keiji Yada, Ai Yamakage, Masatoshi Sato, and Yukio Tanaka, J. Phys. Soc. Jpn. 83, 064705 (2014).fu15 Liang Fu, Phys. Rev. Lett. 115, 026401 (2015). kozii15 Vladyslav Kozii and Liang Fu, Phys. Rev. Lett. 115, 207002 (2015). venderbos16a Jörn W. F. Venderbos, Vladyslav Kozii, and Liang Fu, Phys. Rev. B 94, 094522 (2016). venderbos16b Jörn W. F. Venderbos, Vladyslav Kozii, and Liang Fu, Phys. Rev. B 94, 180504(R) (2016).note1 Qualitatively the same effective pairing can be obtained in terms of a simplified method of deriving the low-energy effective model <cit.>. The only change is to substitute μ+M(𝐤)-ϵ(𝐤) for E_𝐤 in the denominator of Eq.(17). The simplified method thus gives the same picture about the nature of the pairing, at the expense of less quantitative accuracy <cit.>.balatsky06 A. V. Balatsky, I. Vekhter, and Jian-Xin Zhu, Rev. Mod. Phys. 78, 373 (2006).foster14 Matthew S. Foster, Hong-Yi Xie, and Yang-Zhi Chou, Phys. Rev. B 89, 155140 (2014). xie15 Hong-Yi Xie, Yang-Zhi Chou, and Matthew S. Foster, Phys. Rev. B 91, 024203 (2015). | http://arxiv.org/abs/1708.07800v1 | {
"authors": [
"Lei Hao",
"C. S. Ting"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170825163019",
"title": "Nematic superconductivity in Cu$_{x}$Bi$_{2}$Se$_{3}$: The surface Andreev bound states"
} |
kimuniv-m,kimuniv-n]Un-Gi Jong kimuniv-m]Chol-Jun Yucor [email protected] [cor]Corresponding author kimuniv-m]Gum-Chol Ri impp]Andrew P. McMahon impch]Nicholas M. Harrison impp]Piers R. F. Barnes imp]Aron Walshcor [email protected] [kimuniv-m]Department of Computational Materials Design, Faculty of Materials Science, Kim Il Sung University, Ryongnam-Dong, Taesong District, Pyongyang, Democratic People's Republic of Korea [kimuniv-n]Natural Science Centre, Kim Il Sung University, Ryongnam-Dong, Taesong District, Pyongyang, Democratic People's Republic of Korea [imp]Department of Materials, Imperial College London, London SW7 2AZ, United Kingdom [impp]Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom [impch]Department of Chemistry, Imperial College London, London SW7 2AZ, United Kingdom The use of methylammonium (MA) lead halide perovskites CH3NH3PbX3 (X=I, Br, Cl) in perovskite solar cells (PSCs) has made great progress in performance efficiency during recent years. However, the rapid decomposition of MAPbI3 in humid environments hinders outdoor application of PSCs, and thus, a comprehensive understanding of the degradation mechanism is required. To do this, we investigate the effect of water intercalation and hydration of the decomposition and ion migration of CH3NH3PbX3 using first-principles calculations. We find that water interacts with PbX6 and MA through hydrogen bonding, and the former interaction enhances gradually, while the latter hardly changes when going from X=I to Br and to Cl. Thermodynamic calculations indicate that water exothermically intercalates into the perovskite, while the water intercalated and monohydrated compounds are stable with respect to decomposition. More importantly, the water intercalation greatly reduces the activation energies for vacancy-mediated ion migration, which become higher going from X=I to Br and to Cl. Our work indicates that hydration of halide perovskites must be avoided to prevent the degradation of PSCs upon moisture exposure. Influence of water intercalation and hydration on chemical decomposition and ion transport in methylammonium lead halide perovskites [====================================================================================================================================§ INTRODUCTION Perovskite solar cells (PSCs) using methylammonium lead halide perovskites MAPbX3 (MA = methylammonium cation CH3NH3+; X = halogen anion I-, Br- or Cl-) as light harvesters have opened up a new stage in the field of photovoltaics.The rapid rise of power conversion efficiency of PSCs from the initial 3.81% reported by Kojima et al. in 2009 <cit.> to over 20% <cit.> within a few years is quite remarkable, as compared with other types of solar cells <cit.>. Moreover, much low cost of cell fabrication by chemical methods and much abundance of raw materials have attracted great attention, inspiring people with hope that solar power will become competitive with fossil fuels in the electricity market in the near future. In spite of a great advance in research and development of PSCs during the past years, there still remain some obstacles to prevent commercialization of PSCs, such as toxicity of lead <cit.> and mostly poor material stability of these halide perovskites <cit.>. First-principles calculations revealed that MAPbI3, the most widely used hybrid perovskite halide in PSCs, may be decomposed exothermally <cit.> and such intrinsic instability can be cured by mixing halogen ions <cit.>.On the other hand, moisture has been reported to play a critical role as an extrinsic factor in the degradation of PSC performance <cit.>. Although controlled humidity condition or a small amount of H2O has positive effects on PSC performance such as enhancement of the reconstruction in perovskite film formation and triggering nucleation and crystallization of perovskite phase <cit.>, H2O molecule is known to damage the integrity of perovskite films. Some researchers asserted that MAPbI3 may readily hydrolyze in the presence of water due to deprotonation of CH3NH3+ by H2O, resulting in degradation products such as CH3NH2, HI and PbI2 <cit.>.On the contrary, different degradation reactions in perovskite films under humidity condition have been reported. Huang et al. <cit.> focused on the interaction between H2O and MAPbI3 and suggested that degradation of MAPbI3 under ambient condition may produce PbCO3 and α-PbO rather than PbI2. Some reports found that the formation of monohydrated CH3NH3PbI3·H2O or dehydrated (CH3NH3)4PbI6·2H2O intermediate phase is the initial step in the perovskite decomposition process under 80% humidity exposure <cit.>. It was found subsequently that these intermediate phases are unstable in ambient condition, so that further decomposition into the final products could occur <cit.>. As a consequence, the decomposition of MAPbI3 in humid air is a rather complicated process and their reaction processes or mechanisms are in an active controversy. To unveil the mechanism of MAPbI3 decomposition upon humidity exposure, theoretical simulations based on the density functional theory (DFT) have been performed, focused on H2O adsorption on MAPbI3 surfaces <cit.>. However, comprehensive understanding of water-assisted decomposition of MAPbI3 is not yet fully established, and is urgent to facilitate materials engineering for enhanced material stability.In this work, we investigate the influence of water intercalation and hydration on the decomposition of MAPbX3 (X = I, Br, Cl) by performing first-principles calculations. The crystalline and atomistic structures of water intercalated phases, as we denoted MAPbX3_H2O, and monohydrated phases MAPbX3·H2O are explored carefully. These phases can be regarded as the intermediate ones on the process of water-assisted decomposition of MAPbX3. The intercalation energies of a water molecule into the pseudo-cubic phases and the decomposition energies of the water intercalated and monohydrated phases into PbX2, CH3NH3X and H2O are calculated to draw a meaningful conclusion about the stability. Finally we consider vacancy-mediated diffusion of an X- anion, MA+ cation, and H2O molecule in pristine, water intercalated and monohydrated phases, since such migrations seem to give important implication of the material stability <cit.>. Pb^2+ migration is excluded due to high formation energy of the Pb vacancy, V_Pb.§ COMPUTATIONAL METHODS All of the pimary DFT calculations were carried out using the pseudopotential plane wave method as implemented in Quantum-ESPRESSO package <cit.>. We used the ultrasoft pseudopotentials provided in the package[The pseudopotentials C.pbe-n-rrkjus_psl.0.1.UPF, H.pbe-rrkjus_psl.0.1.UPF, N.pbe-n-rrkjus_psl.0.1.UPF, Pb.pbe-dn-rrkjus_psl.0.2.2.UPF, and I(Br,Cl).pbe-n-rrkjus_psl.0.2.UPF were used.], where the valence electronic configurations of atoms are H–1s^1, C–2s^22p^2, N–2s^22p^3, Cl–3s^23p^5, Br–4s^24p^5, I–5s^25p^5, and Pb–5d^106s^26p^2. The exchange-correlation interaction between the valence electrons was estimated using the Perdew-Burke-Ernzerhof (PBE) <cit.> form within the generalized gradient approximation, which is augmented by dispersive van der Waals interaction (vdW-DF-OB86), already shown to be important for calculations of perovskite halides <cit.>. The structures of MAPbX3 and water intercalated phases were assumed to be pseudo-cubic. Unit cells containing one formula unit (f.u.) for these phases and two formula units for monohydrated phase were used for structural optimization and decomposition energetics. For diffusion process, (2×2×2) supercells for pseudo-cubic phase and (2×2×1) supercells for monohydrated phases, including 120 and 96 atoms, were used. The cutoff energy for plane-wave basis set is as high as 40 Ry and k-points are (2×2×2) with Monkhorst-Pack method, which guarantee the total energy accuracy as 5 meV per unit cell. Atomic positions were fully relaxed until the forces converge to 5×10^-5 Ry/Bohr. The activation energies for migrations were calculated using the climbing image nudged elastic band (NEB) method <cit.>.The structural optimizations of pseudo-cubic MAPbX3 phases produced lattice constants of 6.330, 5.949, and 5.682 Å for X=I, Br, and Cl, which are in good agreement with the experimental values <cit.> within 1% relative errors. A water molecule was put into the interstitial space of these optimized cubic phases, which were re-optimized. The supercell shape becomes triclinic following optimization. We have also performed optimization of the monohydrated phases MAPbX3·H2O with monoclinic crystalline lattice and experimentally identified atomic positions <cit.>. For the case of MAPbI3·H2O, the determined lattice constants a=10.460 Å, b=4.630 Å, c=11.100 Å, and β=101.50^∘ agree well with the experimental results <cit.>. It is worth noting that the MAPbI3 crystal has a 3-dimensional structure with corner-sharing PbI6 octahedra, while the monohydrated phase MAPbI3·H2O is characterized by a 1-dimensional edge-sharing PbI6 structure. The optimized atomistic structures of water intercalated phase MAPbI3_H2O and monohydrated phase MAPbI3·H2O are shown in Figure <ref>.It should be emphasized that the total energies per formula unit of MAPbX3_H2O are typically 0.3 eV higher than those of MAPbX3·H2O, indicating that the water intercalated phase would be an intermediate phase in a transformation to the monohydrated phase.In order to compare the free energies of formation for pristine and hydrated phases, additional calculations of the harmonic phonon density of states were calculated using the codes Phonopy<cit.> and VASP (PBEsol functional).<cit.> The Gibbs free energy of formation (G_solid) is calculated from the sum of the DFT internal energy (U), the vibrational free energy (F_vib), and the configurational entropy (S_conf). G_solid = U + F_vib - TS_conf S_conf takes account of the configurational entropy (rotational activity) of the MA^+ ion from statistical mechanics, whichcannot be extracted directly from static DFT calculations.<cit.>To complement the solid-state calculations, the free energy of water vapour was estimated froma standard ideal gas expression. § RESULTS§.§ Water-perovskite interaction For the water intercalated phase MAPbI3_H2O, the intercalated water molecule, with organic molecular MA+ ion, resides in the large interstitial space formed by a huge framework consisting of Pb and I atoms of inorganic PbI6 octahedra (for X=Br and Cl, similar structures are observed). It is bonded with H atoms of NH3 moiety of MA and I atoms of PbI6; the bond length between the O atom of water and the H atom of NH3 is ∼1.63 Å, and those between the H atoms of water and the I atoms of PbI6 are 2.43 and 2.54 Å. In Figure <ref>(b) for the monohydrated phase MAPbI3·H2O, the corresponding bond lengths are observed to be slightly longer, i.e., 1.78, 2.83, and 2.92 Å. These indicate that the interactions between water molecule and inorganic PbI6 as well as organic MA are through hydrogen bonding, as already pointed out in the previous works <cit.>. When going from I to Br and to Cl, the (H2O)O-H(MA) bond length increases a little for the water intercalated phases or hardly change for the monohydrated phases, whereas the (H2O)H-X(PbX6) bond lengths decrease distinctly (see Table <ref>).On consideration that water intercalation causes a volume expansion, we estimate a relatively volume expansion rate as r_vol=(V-V_0)/V_0×100%, where V is the volume of water intercalated or monohydrated phase and V_0 the volume of pseudo-cubic MAPbX3 phase per formula unit. This gradually increases going from X=I to Br and to Cl. Consequently, it can be deduced that decreasing the atomic number of halogen component induces an enhancement of the interaction between water and inorganic PbX6 matrix through hydrogen bonding, while maintaining those interactions between water and the MA ion, resulting in the contraction of the interstitial space and the volume, which might cause difficulty of water intercalation and ion migration. To estimate the ease of water intercalation, we calculated the intercalation energy by E_int=E_MAPbX3·H2O-(E_MAPbX3+E_H2O) where E_MAPbX3·H2O, E_MAPbX3, and E_H2O are the total energies of the water intercalated or monohydrated bulk, pseudo-cubic bulk unit cells per formula unit, and an isolated water molecule, respectively. The intercalation energies in the water intercalated phases were calculated to be -0.53, -0.43, and -0.36 eV for X=I, Br, and Cl, which are smaller in magnitude than those in the monohydrated phases as -0.87, -0.78, and -0.69 eV, as presented in Table <ref>. Therefore, it can be said that formation of the monohydrated phases is easier than formation of water intercalated phases. When going from X=I to Br and to Cl, the magnitude of water intercalation energy decreases, indicating that water intercalation becomes more difficult as the atomic number of halogen component decreases.This can be interpreted with the analysis of the interaction between water and perovskite, and the structural property. It should be noted that, although negative intercalation energies imply the exothermic process of water intercalation, a certain amount of energy (kinetic barrier) could be required for water molecule to intercalate into the perovskite halides, as penetration of water into the perovskite iodide surface <cit.>. We further considered decomposition of water intercalated and monohydrated phases by calculating the decomposition energy, E_dec=(E_PbX2+E_MAX+E_H2O)-E_MAPbX3·H2O where E_PbX2 and E_MAX are the total energies of crystalline PbX2 (space group: P3̄m1) and MAX (space group: Fm3̄m), respectively. As shown in Table <ref>, the decomposition energies were calculated to be positive, indicating that the decomposition is an endothermic process. At this moment, it is worth to compare with those for the pristine perovskite halides without water. The decomposition of MAPbI3 → MAI + PbI2 is exothermic due to the negative decomposition energy of -0.06 eV, whereas those of MAPbBr3 and MAPbCl3 are also endothermic with the positive decomposition energies of 0.10 and 0.23 eV <cit.>. Interestingly, the decomposition energy in magnitude is in the reverse order to the intercalation energy going from X=I to Br and to Cl. This indicates that water could intercalate more readily into the halide perovskite but the formed water-included compounds would be more resistant to decomposition into their relevant components as decreasing the atomic number of halogen component.To consider finite-temperature effects, additional calculations were performed to compute the Gibbs free energy of each phase (seeFigure <ref>).Formation of the hydrated compound is found to be favourable at lower temperatures due to the large energy gain from creation of the hydrate versus a state containing MAPbI_3 and water vapour. The calculations confirm the results stated earlier, but also predict that at higher temperaturesthe pristine MAPI_3 phase becomes stabilized by its vibrational entropy.The inclusion of configuration entropy associated with the rotational freedom of the MA^+ ion acts to further stabilize the material against hydration. §.§ Dynamics of water incorporation and charged defects We turn our attention to how the ions and water molecule diffuse inside the water intercalated or monohydrated perovskites, trying to find out their role in material instability. It is well known that diffusion of ions in crystalline solid is associated with point defects such as a site vacancy and/or interstitial. While such migrations of ions or defects can provide explicit explanations for the performance of PSC device such as ionic conduction, hysteresis, and field-switchable photovoltaic effect <cit.>, these might have important implications for material stability. It was established that, although as in other inorganic perovskite oxides several types of point defects could be formed in the hybrid perovskite halides, including vacancies (V_MA, V_Pb, V_X), interstitials (MA_i, Pb_i, X_i), cation substitutions (MA_Pb, Pb_MA) and antisite substitutions (MA_X, Pb_X, X_MA, X_Pb), vacancies except V_Pb have the lowest formation energies, while others are unstable both energetically and kinetically <cit.>. In this work we thus considered only vacancies V_MA and V_X in the pristine, water-intercalated, and monohydrated phases, which could support vacancy-mediated ionic diffusion. For migration of the water molecule inside the water intercalated or monohydrated phases, water vacancy V_H_2O is formed and allowed to migrate.For vacancy-mediated ion migration in the pristine and water intercalated MAPbX3 phases, we follow the three vacancy transport mechanisms established in the previous works <cit.>, where vacancies are allowed to conventionally hop between neighbouring equivalent sites. According to these mechanisms, X- at a corner site of PbX6 octahedron migrates along the octahedron edge towards a vacancy in another corner site, and MA+ hops into a neighbouring vacant cage formed by the inorganic scaffold. Water molecule migrates along the similar path to MA+ case. For the cases of monohydrated phase, we have devised plausible paths for each of the three defects, and pick out one that has the lowest activation energy, as discussed below in detail. Figure <ref> shows the schematic view of vacancy-mediated ion and molecule migration paths. Special attention was paid to obtaining the well-converged structures of the start and end point configurations with structural relaxations with convergence criteria as 0.01 eV/Å atomic forces. The activation energies for these vacancy-mediated migrations are summarised in Table <ref>.To see whether our computational models and parameters could give reasonable results for the ionic migrations, the pseudo-cubic MAPbI3 was first tested. As listed in Table <ref>, the activation energies for I- and MA+ migrations were calculated to be 0.55 and 1.18 eV, respectively, which are comparable with 0.58 and 0.84 eV reported in ref. <cit.>, and 0.32-0.45 and 0.55-0.89 eV in ref. <cit.>, but higher than 0.16 and 0.46 eV in ref. <cit.>. With respect to the crystalline lattice, we used the pseudo-cubic lattice as in the work in ref. <cit.>, while Haruyama et al. <cit.> and Azpiroz et al. <cit.> used the tetragonal lattice. An exchange-correlation (XC) functional including dispersion (vdW) interactions was used in our work and the work in ref. <cit.>, whereas PBEsol and PBE without vdW correction were used in ref. <cit.> and ref. <cit.>, respectively. Therefore, the slight discrepancies might be associated with the different crystalline lattices and XC functionals without vdW correction, not with the supercell size. Most important, the activation energy for I- migration is lower than that for MA+ migration in all the above-mentioned works, convincing that the results obtained in this work can be used to find out the influence of water on ion diffusion.For I- migration in the monohydrated phase MAPbI3·H2O, which is structurally characterized by 1-dimensional edge sharing PbI6 octahedra connected in [010] direction, we devised four migration pathways along the three octahedron edges in different directions and across the space between separated octahedra. The lowest activation energy was found for the migration along the edge in [010] direction, while the highest value over 2 eV was found for the one across the space, implying this pathway less likely. Figure <ref> shows the I- migration pathways along the octahedron edge in the three kinds of phases and the corresponding activation energy profile. It is found that, when water intercalates into the perovskite, the activation energy decreases, indicating more facile diffusion of I- ion upon water intercalation. Meanwhile, the activation energy in the monohydrated phase is higher than in the water intercalated phase but still lower than in the pristine phase. This demonstrates that the intercalated water molecule enhances diffusion of ions in hybrid perovskite halides, facilitating the formation of hydrated phases. However, once the hydrated phase is formed, diffusion of ions becomes a little harder. Similar arguments hold for MAPbBr3 and MAPbCl3. An MA+ ion in the monohydrated phase was enforced to migrate along the almost straight pathway in [010] direction since there is no channel in [100] direction due to the wall formed by PbI6 octahedra and the channel in [001] direction has much longer distance. It should be noted that MA+ cation in the water intercalated phase is allowed to diffuse equally both in [100] and [010] directions but not allowed to move in [001] direction due to the presence of water molecule on the path. Figure <ref> shows the intermediate states during MA+ migrations in the pristine, water-intercalated, and monohydrated MAPbI3 phases and the corresponding energy profile. We can see distortions of PbI6 octahedra during migration, much more clearly in the case of the water intercalated phase due to the rather strong interaction between PbI6 and methylammonium, which may cause difficult MA+ ion migration. Relatively high activation energies for MA+ ion migrations were found for the pristine (1.18 eV) and monohydrated phases (1.14 eV), while low activation energy of 0.38 eV was found for the water intercalated phase. This indicates that inclusion of water in the perovskite halides reduces the activation energy for MA+ ion migration as in the case of halogen ion migration. As discussed above, the volume expansion rate of the water intercalated phase is larger than that of the monohydrated phase, and thus, water intercalation can induce much more space expansion, resulting in the enhancement of MA+ ion diffusion. When compared to I- migration, MA+ migration can be said to be more difficult due to higher activation energy in agreement with the previous works <cit.>, in which the bottleneck comprising four I- ions and the high level of orientational motion of the MA+ ion were pointed out to be the reasons for such hard migration. Finally, a H2O molecule was allowed to migrate in the same direction as MA+ ion in the water intercalated and monohydrated phases, as represented in Figure <ref>. The activation energy in the water intercalated phase was calculated to be 0.28 eV, which is low enough to diffuse inside the bulk crystal and form the hydrated phase. As in the cases of ion migrations, this is lower than the one in the monohydrated phase. At this stage, it is worth to compare with the initial process of water penetration into MAPbI3 surface. From the first-principles calculations <cit.>, it was found that, when water molecules are brought into contact with MAPbI3 surface, the water molecule in the inside region is 0.2∼0.3 eV more stable than the one in the outside region, and thus, the water molecule is strongly driven to diffuse into the inside of MAPbI3. The activation barrier for this process was calculated to be 0.31 eV <cit.> or 0.27 eV <cit.> at low coverage of water and 0.82 eV <cit.> at high coverage. These are comparable to the barrier of water diffusion within the bulk crystal in this work, which can be regarded as a continuation of the water penetration, indicating easy formation of water intercalated and further hydrated phase. As can be seen in Table <ref>, water molecule can migrate more easily than MA+ ion, which might be due to larger molecular size of MA and its stronger interaction with environmental components, but more toughly than X- ion in both phases. On the other hand, the activation barrier of water migration in the monohydrated phase is higher than the one in the water intercalated phase, as in the cases of X- and MA+ ion migrations. When the atomic number of halogen component decreases from X=I to Br and to Cl, the activation barriers for ions and water molecule migrations increases monotonically, as shown in Table <ref>.In fact, when changing from I with larger ionic radius of 2.2 Å to Br with smaller ionic radius of 1.96 Å and to Cl with further smaller ionic radius of 1.81 Å, the lattice spacing and PbX6-MA bonding shrink while maintaining the intramolecular MA spacing <cit.>. This results in the enhancement of Pb-X interaction and makes passages of ions and water molecule more difficult. Similar arguments hold for water intercalated and monohydrated phases. The increase of activation barrier for ion migrations going from X=I to Cl describes enhancement of material stability when mixed I atom with Br or Cl atom, and is coincident with aforementioned decomposition energetics.§ CONCLUSIONTo understand the degradation mechanism of PSCs upon exposure of MAPbI3 to moisture, we have investigated the influence of water intercalation and hydration on decomposition and ion migrations of MAPbX3 (X=I, Br, Cl) by first-principles calculations. The crystalline lattices and atomistic structures of water intercalated MAPbX3 phases were suggested and optimized, together with those of monohydrated phases MAPbX3·H2O. It was found that water interacts with PbX6 and MA in these compounds through hydrogen bond, and the former interaction enhances gradually while the latter hardly change when going from X=I to Br and to Cl. The calculated results for water intercalation energies and decomposition energies indicate that water can exothermically intercalate into the hybrid perovskite halides, while the water intercalated and monohydrated compounds are stable with respect to decomposition. More importantly, the hydrogen bond interaction induced by water greatly affects the vacancy-mediated ion migrations, for which the activation barrier decreases upon the water inclusion inside the perovskite halides. The activation energies for ion and water molecule migrations become higher as going from X=I to Br and to Cl. These results clarify that degradation of the PSCs upon moisture exposure originates from the formation of water intercalated and further hydrated compounds and then decomposition of these compounds, which provides the idea to prevent this degradation at the atomic level.§ ACKNOWLEDGMENTSThis work was supported partially by the State Committee of Science and Technology, Democratic People's Republic of Korea, under the state project “Design of Innovative Functional Materials for Energy and Environmental Application” (No. 2016-20). The work in the UK was supported by the Royal Society and the Leverhulme Trust, and the Imperial College High Performance Computing Service. A.P.M. was supported by a studentship from the Centre for Doctoral Training in Theory and Simulation of Materials at Imperial College London, funded by the EPSRC under grant EP/G036888.The calculations have been carried out on the HP Blade System C7000 (HP BL460c) that is owned and managed by Faculty of Materials Science, Kim Il Sung University.§ NOTESThe authors declare no competing financial interest. rsc | http://arxiv.org/abs/1708.07608v3 | {
"authors": [
"Un-Gi Jong",
"Chol-Jun Yu",
"Gum-Chol Ri",
"Andrew P. McMahon",
"Nicholas M. Harrison",
"Piers R. F. Barnes",
"Aron Walsh"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170825040147",
"title": "Influence of water intercalation and hydration on chemical decomposition and ion transport in methylammonium lead halide perovskites"
} |
#1 #1#1#1 | http://arxiv.org/abs/1708.07851v2 | {
"authors": [
"Manuel Hohmann",
"Andreas Schärer"
],
"categories": [
"gr-qc",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170825181512",
"title": "Post-Newtonian parameters $γ$ and $β$ of scalar-tensor gravity for a homogeneous gravitating sphere"
} |
http://arxiv.org/abs/1708.08051v2 | {
"authors": [
"Marek Rogatko",
"Karol. I. Wysokinski"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170827044955",
"title": "Two interacting current model of holographic Dirac fluid in graphene"
} |
|
0000-0002-7867-7674]Robert O. Chancia Department of Physics, University of Idaho, Moscow ID 83844-0903 Department of Physics, University of Idaho, Moscow ID 83844-0903 Astronomy Department, Wellesley College, Wellesley MA 02481Robert O. Chancia [email protected] The η ring is one of the narrow rings of Uranus, consisting of a dense core that is 1-2 km wide and a diffuse outer sheet spanning about 40 km. Its dense core lies just exterior to the 3:2 Inner Lindblad Resonance of the small moon Cressida.We fit the η ring radius residuals and longitudes from a complete set of both ground-based and Voyager stellar and radio occultations of the Uranian rings spanning 1977-2002. We find variations in the radial position of the η ring that are likely generated by this resonance, and take the form of a 3-lobed structure rotating at an angular rate equal to the mean motion of the moon Cressida. The amplitude of these radial oscillations is 0.667±0.113 km, which is consistent with the expected shape due to the perturbations from Cressida. The magnitude of these variations provides the first measurement of the mass and density of the moon Cressida (m=2.5±0.4×10^17 kg and ρ=0.86±0.16 g/cm^3) or, indeed, any of Uranus' small inner moons. A better grasp of inner Uranian satellite masses will provide another clue to the composition, dynamical stability, and history of Uranus' tightly packed system of small moons.§ INTRODUCTIONIn March of 1977, <cit.>, <cit.>, and <cit.> discovered nine narrow rings around the planet Uranus by measuring the light blocked by each ring before and after Uranus occulted the star SAO158687. Since then, the Uranian rings have been studied extensively with ground based stellar occultations <cit.>. Occultations provide very precise radial locations of the rings at different longitudes in their orbits around Uranus. <cit.> found that the main rings of Uranus consist of six measurably eccentric rings (6, 5, 4, α, β, and ϵ) and three nearly circular rings (η, γ, and δ).In the past, measurements of the η ring's radius have not shown the ring to be anything but circular. The η ring also features a broad low optical depth sheet extending approximately 40 km exterior to its narrow core <cit.>.During the Voyager 2 flyby of Uranus <cit.> discovered ten new small inner moons, but no one has ever measured their masses or densities. Nine of the moons orbit within a radial range of 20,000 km, making the group one of the most tightly packed systems of interacting satellites in our solar system. <cit.> estimated the masses of the inner moons assuming densities equal to that of the larger moon Miranda <cit.> and shapes estimated with photometry <cit.>, but stated that at least some of Uranus' small inner moons are significantly less massive than these estimates. The lifetime of this system is highly sensitive to the masses of the individual satellites <cit.>. In fact, prior to the knowledge of the even less stable moon Cupid <cit.>, <cit.> showed that Desdemona could collide with either Cressida or Juliet within the next 4-100 million years, depending on the masses of the satellites involved. The discovery of the dusty ν and μ rings <cit.>, near the orbits of Portia/Rosalind and Mab respectively, hints at the possibility of an evolving inner ring-moon system dominated by accretion <cit.>. <cit.> also argue that anomalies in Mab's orbital motion may be explained by a ring-moon system that is undergoing re-accretion after a recent catastrophic disruption.Here we investigate a complete set of Uranian η ring occultation observations spanning their discovery in 1977 to 2002.We find that the η ring's radii exhibit a 3-lobed structure rotating around Uranus at the mean motion of the moon Cressida. We argue that this structure is a result of the η ring's close proximity to Cressida's 3:2 inner Lindblad resonance (ILR). One of the maxima in the ring's radius aligns with Cressida, as expected for the stable ring structure located exterior to the resonant radius. The measured radial amplitude of this ring structure and its distance from the resonance allow us to estimate Cressida's mass, and thus obtain the first gravity-based mass measurement of any inner Uranian moon.We have only been able to find three previous mentions of the Cressida 3:2 ILR and its association with the η ring. <cit.> identified the most relevant resonances in the Uranian ring-moon system and made a case for Cordelia and Ophelia shepherding the outermost ϵ ring through torques generated by the Lindblad resonances located appropriately on the ring's inner and outer edges <cit.>. They also note single resonances that could be perturbing the γ and δ rings. Finally, they state: “The only isolated first-order satellite resonances which fall near any of the remaining rings are located interior to the η ring." <cit.> list both the Cressida 3:2 and the Cordelia 13:12 resonances, located at a=47171.6±0.3 km and a=47173.0±0.3 km respectively. These resonances fall 3-5 km interior to the η ring. They calculate the widths of both resonances to be ∼ 1 km and dismiss the possibility that either resonances is perturbing the η ring. <cit.> later marked the location of the Cressida 3:2 ILR in their figure displaying a radial scan of a high phase image of the Uranian rings acquired by Voyager 2. Subsequently, <cit.> noted that this resonance needs to be re-examined using updated satellite parameters. At the time, with a smaller data set, there was no detection of either an m=3 or an m=13 mode in the η ring, nor any other modes due to resonances with known satellites having observed effects on any of the other previously noted rings' edges <cit.>. Thus, it was only sensible to dismiss these resonances, and it is reasonable that they have not been of interest since. We are only able to make this discovery now because we have a larger set of occultation data extending from 1977 through 2002.We present the data used in this analysis in Section 2, and describe our ring particle streamline model and our mode detection methods in Section 3. In Section 4, we report the parameters of our fit to the η ring and calculate the mass and density of Cressida. Finally in Section 5, we discuss potential implications for the dynamical stability of the tightly packed system of inner Uranian moons and the possible composition of Cressida.§ OBSERVATIONAL DATAThe observational data used for this analysis consist of 49 individual occultation observations of the η ring. In the appendix, Table <ref> contains each occultation's ring intercept time, inertial longitude, and mid-radius determined using a simple square-well model for profile fitting, developed by <cit.> and used in later orbit determinations of the Uranian rings <cit.>. Of these 49 observations, 46 are Earth-based stellar occultations, two are Voyager 2 Radio Science Subsystem (RSS) radio occultations, and one is a Voyager 2 Photopolarimeter Subsystem (PPS) stellar occultation. Several of the observations are ingress and egress pairs from the same occultation of Uranus and its rings.For each Earth-based occultation, an instrument recorded the brightness of the background star as a function of time. As the Earth moves relative to Uranus the rings can block the star's light, leaving each ring's mark as a sharp decrease in the recorded brightness of the star for some amount of time related to the width of the ring. Typically the observations were detected with an InSb photometer in the 2.2 μm band, using the K filter, where Uranus is fainter than the rings. Most observations provided limited information about the radial structure within the rings, and here we are making use only of the estimate of the radius of the mid-point of each ring occultation profile. Interested readers should see <cit.> for a review of stellar occultation studies of the solar system and <cit.> for a review of this observation method specific to the rings of Uranus.To identify possible Uranus occultation opportunities <cit.> compared positions of Uranus to stellar positions in the Smithsonian Astrophysical Observatory (SAO) catalog. Once the rings were discovered, it became more appropriate to utilize dimmer stars that are bright in the 2.2 μm band. Thus, <cit.> searched for stars on photographic plates containing star fields ahead of Uranus and created a list of ideal future occultation observations. Additional lists of this type were compiled by <cit.>, <cit.>, <cit.>, and <cit.>. The Voyager 2 PPS stellar occultation only detected the η ring on egress <cit.>. In the case of the Voyager 2 RSS occultations, the RSS instrument illuminated the rings at 3.6 cm and 13 cm wavelengths in the direction of Earth once beyond the ring plane. Stations on Earth detected the diffracted signal and relative phase change, to later be reconstructed into high-resolution radial optical depth profiles after the removal of diffraction effects<cit.>. Presently, ground based occultation opportunities are rare because Uranus has passed out of the dense Milky Way background, drastically reducing the density of appropriate background stars. The rings are also no longer as open to our view from Earth as they were in the 1980s because the apparent aspect of the ring plane as viewed from Earth changes over time. § RING PARTICLE STREAMLINE MODEL AND FITTING METHODThe procedure used here follows that of <cit.> for the Uranian rings, more recently employed by <cit.>, <cit.>, and <cit.> for analyses of Saturn's non-circular narrow rings, gaps, and edges. After taking account any inclination relative to the equatorial plane, the majority of narrow rings are well-fit by simple precessing Keplarian ellipses whose radii are described by:r(λ,t)=a(1-e^2)/1+ecos f,where the true anomaly f=λ-ϖ_0-ϖ̇(t-t_0). Here, the radius of the ring will vary with longitude λ and time t, where a and e are the ring's semi-major axis and eccentricty, ϖ_0 is the ring's longitude of periapsis at the time t_0, and ϖ̇ is the ring's apsidal precession rate. We can approximate a nearly circular (e≃ 0) ring's radii as r≃ a(1-ecos f). Additionally, several rings are found to contain forced radial oscillations and in a few cases there are even rings whose structure is dominated by free normal mode oscillations. In these cases, the structures are distinct from circles or ellipses and their radii are described by:r(λ,t)≃ a-A_mcos(mθ),where θ=λ-Ω_p(t-t_0)-δ_m, following the formalism of <cit.> and <cit.>. Here, the systematic radial oscillations of the rings form a m-lobed figure rotating around their planet at a pattern speed Ω_p with a radial amplitude A_m and phase δ_m. We show some exaggerated models of m-lobed ring streamlines, resulting from both free normal modes and Lindblad resonances, in Figure <ref>. While individual particles follow normal elliptical orbits, described by Equation <ref>, the ring as a whole consists of streamlines with m azimuthally symmetric radial minima and maxima rotating around the planet with the frequencyΩ_p≃(m-1)n+ϖ̇_sec/m.Here, the mean motion n and apsidal precession rate ϖ̇_sec are functions of the semi-major axis a of the ring, and m can be any positive or negative integer. If we consider the case of m=1 we find that Ω_p=ϖ̇_sec, A_1=ae, and δ_1=ϖ_0, so that r is equivalent to the approximation of Equation <ref> above. In the case of a free normal mode oscillation, the pattern speed will be equal to the expected pattern speed obtained by evaluating Equation <ref> at the semi-major axis of the ring. However, if the ring is perturbed by a satellite through a first-order Lindblad resonance, then the ring structure will have a forced pattern speed matching the mean motion of the perturbing satellite n_s=Ω_p and will differ from the expected pattern speed slightly based on the ring's separation from the exact radius of the resonance |a-a_res|. The ring is perturbed by the satellite due to the near commensurate ratio of the ring particles' orbital periods and the period of the perturbing satellite. As such, first-order Lindblad resonances are defined by |m|:|m-1|, where for every |m| orbits of the ring particle, there are |m-1| orbits of the corresponding satellite. In the majority of cases, the perturbing satellite lies at a larger semi-major axis than the ring (a_s>a). The relevant resonances in this case are called inner Lindblad resonances (ILR) and are assigned positive values of m. In the rare case of a satellite located interior to the rings it is possible to have both ILR and outer Lindblad resonances (OLR) at locations within the rings, allowing for negative values of m.The condition for a first-order Lindblad resonance is that the resonant argument:φ=m(λ-λ_s)-(λ-ϖ)is constant in time. Here λ and λ_s refer to the longitudes of a ring particle and the satellite respectively and ϖ is the longitude of periapsis of the ring particle. If we consider a conjunction of the ring particle and the satellite (λ-λ_s=0) occurring when the ring particle is also located at its longitude of periapsis (λ-ϖ=0), then the condition that φ is constant implies that all future conjunctions will occur when the ring particle is near periapsis. In general, this means that the ring particle will always be in the same phase of its orbit when it passes longitudinally close to the satellite. This allows the perturbing satellite to force the eccentricity and periapsis locations of streamlines located near the resonance. In Figure <ref> we show a cartoon model of the resulting streamlines surrounding a 3:2 ILR in the co-rotating frame of the perturbing satellite. Interior (exterior) to the resonant radius, marked with the dashed line, the streamlines are stable when oriented such that one of the three periapses (apoapses) is aligned with the satellite.In short, our procedure is a search for patterns in the varying mid-radii measurements of the rings. Each ring occultation observation provides the ring's radius at a particular longitude and time. To search for patterns in each ring we need the observed parameters, an m value to test, and the resulting expected pattern speeds for that m value. For each test of m, we compute the expected patten speed for the semi-major axis of the ring using Equation <ref> and create an array of 100,000 pattern speeds, evenly spaced in increments of 0.00001^∘/day, surrounding the expected pattern speed. Using each pattern speed we calculate mθ, for every ring observations' longitude λ and time t, using an initial epoch time t_0 of UTC 1977 MAR 10 20:00:00.00. We can then compute the observed ring radii r vs. mθ mod 360^∘ and fit the data to a single sinusoid. The resulting fit parameters are a, A_m, and δ_m, allowing us to compute model values of r using Equation <ref>. We compute the RMS deviation of the observed radii and the model radii for each m's 100,000 test pattern speeds and look for a RMS minimum to identify the best fitting pattern speed. We first checked our algorithms by searching for known structures in the Uranian rings. In several rings (6, 5, 4, α, β, and ϵ) we can easily detect RMS deviations that drop to nearly zero (sub-km) with the proper pattern speed and m input. These are the rings that largely follow classical Keplerian ellipses (m=1) and whose pattern speeds equal the rings' apsidal precession rate, Ω_p=ϖ̇_sec. The η, γ, and δ rings are nearly circular and their residuals are relatively larger when fit with a low amplitude m=1 ellipse.We are also able to identify the known m=2 structure of the δ ring and the combination of m=0 and m=1 for the γ ring <cit.>.We decided to identify the strongest resonances in the Uranian rings to have a better idea of the resonantly forced modes that are the most likely to be detected. To quantify the `strength' of the resonances in the system we chose to compare the expected forced radial amplitude on rings near each of the possible resonances in the main ring system. We use Equation 10.22 from Chapter 10 of <cit.>,A_m=2α a^2(m_s/m_p)|f_d|/3(j-1)|a-a_res|where A_m is the forced radial amplitude of a ring particle in the vicinity of a Lindblad resonance <cit.>. This amplitude is a function of the ratio of the perturbing satellite and central planet masses m_s/m_p, the radial separation of the ring and the resonance |a-a_res|, the ratio of the ring and satellite semi-major axes α=a/a_s, and the Laplace factor f_d, that depends on j, the integer coefficient of the satellites longitude in the resonant argument, which is equivalent to m in the case of a first-order Lindblad resonance. As shown in Figure 10.10 of <cit.>, 2α|f_d|/j-1 varies between 1.5 and 1.6, depending on j. Note that Equation <ref> isn't necessarily applicable for all cases. If |a-a_res| is smaller than the resonance half-width, then A_m calculated using Equation <ref> is not a good estimation of the radial amplitude produced by the resonance because in this regime neighboring streamlines will cross and collisional dissipation cannot be ignored. In Figure <ref> we display the forced amplitude on all 10 rings (inner and outer edges for the ϵ ring) due to all possible first-order Lindblad resonances of all Uranian moons out to Perdita. For the estimated mass of each moon, we use the radius measurements of <cit.> and <cit.> and consider a range of densities from 0.5 to 1.3 g/cm^3. In the left half of Figure <ref> all resonances mentioned by <cit.> are apparent in addition to a previously unexplored 2:1 ILR with Portia in the proximity of the 6 ring. In the right side of Figure <ref> we compare the amplitudes of the strongest resonances over a range of moon densities. The fainter patches in the left side of Figure <ref> are due to resonances inducing much weaker amplitudes due to their large distance from the rings. Despite the separation in semi-major axis of the η ring from the Cressida 3:2 ILR the η ring is expected to be the most perturbed of all the Uranian rings in this framework. The next largest expected amplitudes are the Cordelia 24:25 OLR and the Ophelia 14:13 ILR that are thought to play a roll in shepherding the ϵ ring. If this is a realistic estimation of the strength of the resonances in the system, in the future we may be able to detect the m=-24 mode on the inner edge of the ϵ ring, which was previously detected by <cit.> with occultation data and by <cit.> with images showing the ring's longitudinal brightness variations. Detecting the ϵ ring edge modes will first require determining the ring's edge positions and the removal of the larger amplitude m=1 normal mode which dominates its structure. Our analysis of these ring residuals as well as those for the other rings, whose structure is dominated by previously known normal modes, is ongoing and will be presented in a subsequent publication.§ RESULTSlr[t] η ring m=3 best fit0ptParameter Final fit and scaled errorsa (km) 47176.447±0.086 A_3 (km) 0.667±0.113 δ_3 (^∘) 58.81±6.12 Ω_p (^∘/day) 776.58208±0.00169 n_Cressida (^∘/day) 776.582789±0.000059a n_Cressida (^∘/day) 776.582414±0.000022b n_Cressida (^∘/day) 776.582447±0.000022c χ^2 13.861 χ^2/ν 0.308 σ/ν (km) 0.555 N 49 # of parameters 4 Listed on top are the four fit parameters and their formal 1-σ errors resulting from our final fit, where we have assumed an error of 0.555 km for each of the observed radii of the η ring. We also list three published mean motions of Cressida for comparison with our pattern speed. The chi-squared and reduced chi-squared below are from the initial fit assuming an error of 1 km for each radii. The unscaled errors of the parameters in the initial fit are roughly double the scaled errors from the final fit, in which we have used the standard deviation per degree of freedom as a universal error in the observed radii. The degrees of freedom ν=N-# of fit parameters. aFrom <cit.> bFrom <cit.> cFrom <cit.> After searching mode values from m=-25 to 25 of all the rings, the strongest new feature we've found is an m=3 structure of the η ring consistent with the expectations discussed above. In Figure <ref> we show the shallow minimum in RMS for our η ring m=3 fits. The top plot shows the RMS deviations of the model radii from the observed radii at each pattern speed for m=3, zoomed in on the minimum. Listed are the best fitting pattern speed, the semi-major axis of the Cressida 3:2 ILR, and the expected pattern speed for an m=3 normal mode marked by the dashed line. Note that the best fitting pattern speed and the expected pattern speed for the semi-major axis of the η ring are offset because this is not a normal mode oscillation, but is instead the effect of a resonance with a satellite whose perturbations force the pattern speed to match the satellite's mean motion. We further refine our best fit solution and formal errors by applying the best fit parameters (a, A_m, δ_m, and Ω_p) as a set of starting parameters for MPFIT, a non-linear least squares fitting IDL function <cit.>. We've initially assumed an uncertainty of 1 km in each of the 49 observed radii of the η ring, but found a reduced chi-squared of 0.308<<1. We fit again to obtain the listed errors using the standard deviation per degree of freedom (σ/ν) as a rescaled uncertainty in our observed radii which better represents the error of these data. The bottom plot shows the best fitting model radius curve on top of the observed radial separations from the fit semi-major axis of the ring, Δ r=r-a. We've listed the final fit parameters and chi-squared analysis in Table <ref>.The best fitting pattern speed for this mode, 776.58208 ±0.00169 ^∘/day, is strikingly close to the published mean motion of Cressida, the fourth moon from Uranus. Most recently <cit.> listed Cressida's mean motion as 776.582789±0.000059^∘/day. All three of the measurements of Cressida's mean motion listed in Table <ref> are well within the uncertainty of our detected pattern speed, supporting the proposed connection between this m=3 structure of the η ring and Cressida. To solidify that the m=3 structure is real and is a result of perturbations from Cressida, we have inspected the alignment of the structure with Cressida. In this case, the η ring (a=47176.447) is located exterior to the resonance (a_res=47171.51), and the dynamical model predicts that one of the three outer radial extents should track the motion of Cressida. That is, as the m=3 structure and Cressida both rotate around Uranus at n_Cressida≃Ω_p one of the apoapses is constantly aligned with Cressida. This can be confirmed by noting that the m=3 structure has a phase offset δ_3=58.81±6.12^∘ (this is the longitude of one of the 3 periapsis), which is roughly 60^∘ offset from Cressida's longitude (359.50^∘) at the epoch of the fit. We show this alignment more precisely in Figure <ref>, where we have determined the offset of each occultation scan longitude relative to Cressida's longitude at the observation time, |m|(λ-λ_Cressida). The apoapse of the phase-wrapped structure lags the longitude of Cressida by only 6±11^∘ (Cressida's longitude is 0^∘ and the fit sinusoid's largest radial excursion occurs at 354^∘). This suggests that the perturbations on the η ring are due to its proximity to the 3:2 ILR with Cressida. cccccc[t] Mass and Density of Cressida0ptA_3 (km) Radius (km) a (km) a_res (km) m_Cressida (kg) ρ_Cressida (g cm^-3)0.667±0.113 41±2 47176.447±0.086 47171.51±0.03 2.5±0.4×10^17 0.86±0.16We list the variables needed to solve for the mass of Cressida using Equation <ref>. For the calculation of m_Cressida we used GM_Uranus=5793951.3±4.4 km^3 s^-2 from <cit.> and G=6.67408±31×10^-11 m^3 kg^-1 s^-2 from <http://physics.nist.gov/cgi-bin/cuu/Value?bg>. Also note 2α|f_d|/j-1≃1.545 when j=m=3 for the case of the Cressida 3:2 ILR. The listed radius needed to calculate the density of Cressida comes from Voyager 2 photometry <cit.>. Perhaps the most significant result of this work, shown in Table <ref>, is a determination of Cressida's mass using Equation <ref>. Given A_3=0.667±0.113 km we find m_Cressida=2.5±0.4×10^17 kg. We use the effective radius for Cressida of 41±2 km from <cit.> to calculate a density of 0.86±0.16 g/cm^3 for Cressida. For our purposes, the η ring is outside the width of Cressida's 3:2 ILR and the resulting estimation of A_m is reasonable, but we note that this is not necessarily the case for all of the other rings and resonances. Curious readers should note, to test the applicability of Equation <ref>, we've calculated a resonance half-width of ∼3.5 km for Cressida's 3:2 ILR using Equation 10.23 from <cit.> along with our newly determined mass of Cressida. The other relevant variable inputs can be found in Tables <ref> and <ref>. This half-width is less than the 5 km separation of the resonance and ring, confirming we are justified in using Equation <ref>. Note that the ∼1 km resonance half-width quoted in the introduction was estimated by <cit.> and results from an approximation of the resonance half-width equation as well as a different satellite mass. § DISCUSSIONSince the Voyager 2 flyby of Uranus in 1986, several dynamicists have explored the stability of the inner Uranian moons. The moons Bianca, Cressida, Desdemona, Juliet, Portia, Rosalind, Cupid, Belinda, and Perdita are members of the most tightly packed system of moons in our solar system. Nicknamed the `Portia group' for their largest member, these satellites are thought to be unstable on short timescales compared to the age of the solar system. The stability of the Portia group is known to be highly sensitive to the masses of the individual satellites <cit.>, which are not well constrained. In fact, the mass we provide for Cressida is the first direct measurement of an inner Uranian satellite's mass. Past simulations <cit.> have relied on treating a range of possible masses for the inner Uranian satellites and suggest that Cressida will cross orbits with Desdemona in under 10^6 years <cit.>, given our mass density. Incorporation of our mass for Cressida should further constrain the timescale of satellite orbit crossing (collisions) and allow a future work to determine the masses of some of the other satellites through their resonant interactions. Strictly speaking, our density measurement does not necessarily represent a common density of the inner moons. However, a lower average satellite density will generally result in collisions occurring in the more distant future. <cit.> and <cit.> detected a possible water ice absorption feature in Hubble Space Telescope near-infrared photometry of the largest inner moon Puck. Combining this with the previously mentioned size estimates has formed the presumption that Cressida and the other inner Uranian moons are likely composed of mostly water ice with at least a veneer or contamination of dark material to explain their low albedo and flat gray spectra. The range in densities of the larger Uranian moons, determined from mass <cit.> and radius <cit.> measurements, have provided a presumed upper limit on the densities of the inner moons, usually with reference to the least dense major moon Miranda (1.214±0.109 g/cm^3)[<https://ssd.jpl.nasa.gov/?sat_phys_par>]. In Figure <ref> we plot our average density of Cressida versus radius along with other satellites in our solar system, after <cit.>. Cressida is about 50% denser than the inner icy moons of Saturn with comparable radii. It may be that Cressida, and the Uranian rings/moons in general, have either a lower porosity than these Saturnian analog or they have higher amounts of non-icy contaminants, as inferred by <cit.>. The contamination of denser and darker material may not be as high as previously expected, but it is substantial regardless.This analysis shows that there is still information about Uranus' rings and moonsfound in historical and ground based data. Still, the best means of obtaining the Uranian moon masses and compositions, determining the ultimate fate of the Portia group, and understanding the intricate structure of the rings is of course a Uranus orbiter mission.We would like to thank Phil Nicholson for his insights regarding ring occultation observations and both Phil Nicholson and Pierre-Yves Longaretti for several fruitful discussions concerning the forced radial amplitudes of ring particles orbiting near Lindblad resonances. We'd also like to thank our anonymous reviewer for helpful suggestions and comments, ultimately improving the clarity of this work. This work was supported by the NASA Solar System Workings program grant NNX15AH45G.Included below are the occultation observation data we used in this analysis of the η ring. The precise numbers for the ring's position are derived from an analysis of the entire Uranian ring data set, including re-determined pole position (Pole right ascention =77.3105814^∘ and declination =15.1697826^∘), standard gravitational parameter (GM=5.793956433×10^6 km^3s^-2), gravitational harmonics (J_2=3.340656×10^-3 and J_4=-3.148536×10^-5), and time offsets. The numbers therefore can deviate slightly from previously published values. We list each observation ID, observing location, ring plane intercept time of the relevant electromagnetic wave observed, detected mid-radius of the η ring, m=3 fit residuals, longitude of the observation, longitude of Cressida at this time, and reference to publications including the observation. Longitudes are measured in the prograde direction from the ascending node of Uranus' equator on the Earth's equator of the J2000 epoch. <cit.> have included all observations from 1977-1986 in their most recent fit, but more recent observations are unpublished. cccccccc η ring occultation observation geometry0ptID Observing Location Star Name Catalog ID t_ ring intercept (UTC) r (km) r-r_ fit (km) λ (^∘)Reference1 Kuiper Airborne Obs. U0 Hipparcos 71567 1977 MAR 10 17:48:26.95 47177.352 0.346 36.68 <cit.>2 Kuiper Airborne Observatory U0 Hipparcos 71567 1977 MAR 10 19:20:03.28 47176.465 0.673 153.51 <cit.>3 Cerro Las Campanas Obs. U5 UCAC2 25775788 1978 APR 10 03:00:16.19 47178.519 2.107 46.24 <cit.>4 Cerro Las Campanas Obs. U5 UCAC2 25775788 1978 APR 10 03:50:16.64 47177.359 0.535 143.71 <cit.>5 Cerro Tololo Interamerican Obs. U12 UCAC2 25096598 1980 AUG 15 22:20:37.70 47176.235 -0.087 22.05 <cit.>6 European Southern Obs. 1-m U12 UCAC2 25096598 1980 AUG 15 19:24:18.97 47175.681 -0.476 171.95 <cit.>7 European Southern Obs. 1-m U12 UCAC2 25096598 1980 AUG 15 22:20:36.26 47176.330 0.013 22.18 <cit.>8 Anglo-Australian Telescope U13 Hipparcos 77434 1981 APR 26 16:45:45.26 47176.726 -0.341 26.90 <cit.>9 Anglo-Australian Telescope U13 Hipparcos 77434 1981 APR 26 17:53:41.63 47176.877 -0.092 163.55 <cit.>10 European Southern Obs. 2-m U14 Hipparcos 79085 1982 APR 22 00:27:11.11 47175.767 -0.184 164.12 <cit.>11 Cerro Las Campanas Obs. U14 Hipparcos 79085 1982 APR 21 23:07:45.20 47175.548 -0.314 24.83 <cit.>12 Cerro Las Campanas Obs. U14 Hipparcos 79085 1982 APR 22 00:27:11.32 47174.793 -1.158 164.09 <cit.>13 Tenerife U14 Hipparcos 79085 1982 APR 21 23:09:03.74 47176.060 -0.027 35.09 <cit.>14 Cerro Tololo Interamerican Obs. U14 Hipparcos 79085 1982 APR 21 23:07:38.90 47175.718 -0.143 24.70 <cit.>15 Cerro Tololo Interamerican Obs. U14 Hipparcos 79085 1982 APR 22 00:27:09.91 47175.595 -0.353 164.23 <cit.>16 Mt. Stromlo U15 UCAC2 23648038 1982 MAY 01 13:53:02.53 47176.249 0.275 27.01 <cit.>17 Mt. Stromlo U15 UCAC2 23648038 1982 MAY 01 15:01:45.77 47176.542 -0.119 162.76 <cit.>18 Mt. Palomar U16 UCAC2 23892052 1982 JUN 04 02:50:34.77 47175.991 -0.012 32.37 <cit.>19 Mt. Palomar U16 UCAC2 23892052 1982 JUN 04 03:50:47.86 47176.435 -0.464 156.73 <cit.>20 South African Astronomical Obs. U17 Hipparcos 80841 1983 MAR 24 22:05:13.10 47176.847 -0.238 35.29 <cit.>21 Cerro Tololo Interamerican Obs. U23 UCAC2 22735323 1985 MAY 04 02:28:59.92 47176.178 -0.003 318.94 <cit.>22 Cerro Tololo Interamerican Obs. U23 UCAC2 22735323 1985 MAY 04 03:30:35.20 47176.885 0.774 229.93 <cit.>23 McDonald Obs. U23 UCAC2 22735323 1985 MAY 04 03:37:42.72 47175.702 -0.133 221.66 <cit.>24 Cerro Tololo Interamerican Obs. U25 UCAC2 22734194 1985 MAY 24 05:23:43.95 47176.565 0.747 316.13 <cit.>25 Cerro Tololo Interamerican Obs. U25 UCAC2 22734194 1985 MAY 24 06:10:43.14 47176.151 0.340 233.85 <cit.>26 McDonald Obs. U25 UCAC2 22734194 1985 MAY 24 05:22:06.85 47175.644 -0.154 326.24 <cit.>27 McDonald Obs. U25 UCAC2 22734194 1985 MAY 24 06:17:47.01 47175.721 -0.124 223.28 <cit.>28 Mt. Palomar U25 UCAC2 22734194 1985 MAY 24 05:23:00.51 47175.853 0.059 326.26 <cit.>29 Mt. Palomar U25 UCAC2 22734194 1985 MAY 24 06:18:47.48 47175.497 -0.360 223.09 <cit.>30 IRTF U28 UCAC2 22517254 1986 APR 26 10:58:06.57 47176.433 -0.282 333.24 <cit.>31 IRTF U28 UCAC2 22517254 1986 APR 26 12:34:39.87 47175.910 -0.632 215.97 <cit.>32 IRTF U1052 UCAC2 22296665 1988 MAY 12 10:56:55.69 47175.431 -0.563 293.09 <cit.>33 IRTF U1052 UCAC2 22296665 1988 MAY 12 11:26:09.19 47176.094 -0.584 256.32 <cit.>34 IRTF U83 UCAC2 22564036 1991 JUN 25 10:10:59.40 47176.545 -0.530 326.15 <cit.>35 IRTF U83 UCAC2 22564036 1991 JUN 25 10:59:59.96 47176.530 -0.581 224.37 Unpublished36 IRTF U84 UCAC2 22563790 1991 JUN 28 07:47:36.36 47176.741 -0.197 306.15 Unpublished37 IRTF U84 UCAC2 22563790 1991 JUN 28 08:19:53.10 47175.300 -0.507 243.98 Unpublished38 Cerro Tololo Interamerican Obs. U9539 UCAC2 23016546 1993 JUN 30 04:57:33.67 47176.082 0.063 351.62 Unpublished39 Cerro Tololo Interamerican Obs. U9539 UCAC2 23016546 1993 JUN 30 05:54:55.13 47177.216 0.256 199.09 Unpublished40 South African Astronomical Obs. U134 UCAC2 23509999 1995 SEP 09 15:29:49.47 47175.948 0.126 31.36 Unpublished41 South African Astronomical Obs. U134 UCAC2 23509999 1995 SEP 09 16:57:52.04 47177.201 0.292 161.04 Unpublished42 IRTF U137 UCAC3 141-413386 1996 MAR 16 11:59:43.68 47176.716 0.936 178.95 Unpublished43 IRTF U137 UCAC3 141-413386 1996 MAR 16 12:43:23.55 47177.408 0.372 13.13 Unpublished44 Mt. Palomar U138 UCAC2 24243463 1996 APR 10 09:27:44.07 47177.050 0.139 356.19 Unpublished45 Mt. Palomar U0201 UCAC2 27214859 2002 JUL 29 07:20:09.62 47177.709 0.829 74.55 Unpublished46 Mt. Palomar U0201 UCAC2 27214859 2002 JUL 29 07:29:17.80 47175.005 -0.807 117.00 Unpublished47 Voyager 2 - RSS1986 JAN 24 19:50:59.23 47176.817 0.080 343.08 <cit.>48 Voyager 2 - RSS1986 JAN 24 22:44:28.11 47176.557 0.375 197.13 <cit.>49 Voyager 2 - PPS β Per Hipparcos 14576 1986 JAN 24 19:36:54.98 47176.041 0.132 110.96 <cit.>The precise numbers for the ring's position are derived from an analysis of the entire Uranian ring data set, including re-determined pole position, GM, J_2, J_4, and time offsets. We used an epoch time, t_0, of UTC 1977 MAR 10 20:00:00.00 for all fits of this data set. The times, t, listed in this table refer to the exact time of the ray intercept in the ring plane for each occultation observation. The ring radii and longitudes are those observed at these times, where longitudes are measured in the prograde direction from the ascending node of Uranus' equator on the Earth's equator of the J2000 epoch. The residuals show separation of each observations' radii and the m=3 model radii. aasjournal | http://arxiv.org/abs/1708.07566v1 | {
"authors": [
"Robert O. Chancia",
"Matthew M. Hedman",
"Richard G. French"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170824221225",
"title": "Weighing Uranus' moon Cressida with the $η$ ring"
} |
calc,fit itemsep=1pt,topsep=2pt|#1argmax_#1 #1argmin_#1 #1Dir(#1) X x (<ref>)1Eq <ref> Ø#1O(#1)P EFxw θ β𝒢 ℬ ℬ^* ℳ^^*BN𝒟 ℒ 𝒳𝒴̨$̨̨̨𝒯̨𝒰̨ℳ̨ ̨N̨B̨ ̨ĄǪD̨Ę ̨P̨^̨#̨1̨ ̨Ą#̨1̨ ̨D̨Ę ̨Ąn̨D̨Ęn̨⊆̨(̨<̨r̨ęf̨>̨)̨1̨Ęq̨ ̨<̨r̨ęf̨>̨Ø̨#̨1̨Ǫ(̨#̨1̨)̨.̨p̨d̨f̨,̨.̨p̨n̨g̨,̨.̨g̨įf̨,̨.̨j̨p̨g̨.̨.̨/̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨<̨c̨įt̨.̨>̨ ̨ ̨Ąc̨c̨ųr̨ąt̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨B̨N̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ųs̨įn̨g̨ ̨H̨D̨P̨M̨ąc̨h̨įn̨ę ̨L̨ęąr̨n̨įn̨g̨ ̨F̨r̨ąn̨ç̨ǫįs̨ ̨P̨ęt̨įt̨j̨ęąn̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨F̨ąc̨ųl̨t̨y̨ ̨ǫf̨ ̨Įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨T̨ęc̨h̨n̨ǫl̨ǫg̨y̨,̨ ̨M̨ǫn̨ąs̨h̨ ̨Ųn̨įv̨ęr̨s̨įt̨y̨ ̨f̨r̨ąn̨c̨ǫįs̨.̨p̨ęt̨įt̨j̨ęąn̨@̨m̨ǫn̨ąs̨h̨.̨ęd̨ų ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨W̨r̨ąy̨ ̨B̨ųn̨t̨įn̨ę ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨F̨ąc̨ųl̨t̨y̨ ̨ǫf̨ ̨Įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨T̨ęc̨h̨n̨ǫl̨ǫg̨y̨,̨ ̨M̨ǫn̨ąs̨h̨ ̨Ųn̨įv̨ęr̨s̨įt̨y̨ ̨w̨r̨ąy̨.̨b̨ųn̨t̨įn̨ę@̨m̨ǫn̨ąs̨h̨.̨ęd̨ų ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨G̨ęǫf̨f̨r̨ęy̨ ̨Į.̨ ̨W̨ęb̨b̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨F̨ąc̨ųl̨t̨y̨ ̨ǫf̨ ̨Įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨T̨ęc̨h̨n̨ǫl̨ǫg̨y̨,̨ ̨M̨ǫn̨ąs̨h̨ ̨Ųn̨įv̨ęr̨s̨įt̨y̨ ̨g̨ęǫf̨f̨.̨w̨ęb̨b̨@̨m̨ǫn̨ąs̨h̨.̨ęd̨ų ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨N̨ąy̨y̨ąr̨ ̨Z̨ąįd̨į ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨F̨ąc̨ųl̨t̨y̨ ̨ǫf̨ ̨Įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨T̨ęc̨h̨n̨ǫl̨ǫg̨y̨,̨ ̨M̨ǫn̨ąs̨h̨ ̨Ųn̨įv̨ęr̨s̨įt̨y̨ ̨n̨ąy̨y̨ąr̨.̨z̨ąįd̨į@̨m̨ǫn̨ąs̨h̨.̨ęd̨ųĄc̨c̨ųr̨ąt̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ųs̨įn̨g̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ęs̨ ̨ ̨ ̨ ̨ ̨F̨r̨ąn̨ç̨ǫįs̨ ̨P̨ęt̨įt̨j̨ęąn̨ ̨ ̨W̨r̨ąy̨ ̨B̨ųn̨t̨įn̨ę ̨ ̨G̨ęǫf̨f̨r̨ęy̨ ̨Į.̨ ̨W̨ęb̨b̨ ̨ ̨N̨ąy̨y̨ąr̨ ̨Z̨ąįd̨į ̨ ̨ ̨ ̨ ̨R̨ęc̨ęįv̨ęd̨:̨ ̨d̨ąt̨ę ̨/̨ ̨Ąc̨c̨ęp̨t̨ęd̨:̨ ̨d̨ąt̨ę ̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨=̨ ̨ ̨ ̨T̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨įn̨t̨r̨ǫd̨ųc̨ęs̨ ̨ą ̨n̨ǫv̨ęl̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨m̨ęt̨h̨ǫd̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨t̨ąb̨l̨ęs̨ ̨ǫf̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨(̨B̨N̨C̨s̨)̨,̨ ̨ųs̨įn̨g̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ęs̨ ̨(̨H̨D̨P̨s̨)̨.̨ ̨T̨h̨ę ̨m̨ąįn̨ ̨r̨ęs̨ųl̨t̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨įs̨ ̨t̨ǫ ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨įm̨p̨r̨ǫv̨ęd̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ąl̨l̨ǫw̨s̨ ̨B̨N̨C̨s̨ ̨t̨ǫ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨l̨ęąd̨įn̨g̨ ̨l̨ęąr̨n̨įn̨g̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨R̨ąn̨d̨ǫm̨ ̨F̨ǫr̨ęs̨t̨ ̨f̨ǫr̨ ̨b̨ǫt̨h̨ ̨0̨-̨1̨ ̨l̨ǫs̨s̨ ̨ąn̨d̨ ̨R̨M̨S̨Ę,̨ ̨ąl̨b̨ęįt̨ ̨j̨ųs̨t̨ ̨ǫn̨ ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨ąt̨ąs̨ęt̨s̨.̨ ̨ ̨ ̨ ̨Ąs̨ ̨d̨ąt̨ą ̨ąs̨s̨ęt̨s̨ ̨b̨ęc̨ǫm̨ę ̨l̨ąr̨g̨ęr̨,̨ ̨ęn̨t̨ęr̨įn̨g̨ ̨t̨h̨ę ̨h̨y̨p̨ęd̨ ̨w̨ǫr̨l̨d̨ ̨ǫf̨ ̨“̨b̨įg̨”̨,̨ ̨ęf̨f̨įc̨įęn̨t̨ ̨ąc̨c̨ųr̨ąt̨ę ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨r̨ęq̨ųįr̨ęs̨ ̨t̨h̨r̨ęę ̨m̨ąįn̨ ̨ęl̨ęm̨ęn̨t̨s̨:̨ ̨(̨1̨)̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨w̨įt̨h̨ ̨l̨ǫw̨-̨b̨įąs̨ ̨t̨h̨ąt̨ ̨c̨ąn̨ ̨c̨ąp̨t̨ųr̨ę ̨t̨h̨ę ̨f̨įn̨ę-̨d̨ęt̨ąįl̨ ̨ǫf̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨(̨2̨)̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨l̨ęąr̨n̨ęr̨s̨ ̨t̨h̨ąt̨ ̨c̨ąn̨ ̨l̨ęąr̨n̨ ̨f̨r̨ǫm̨ ̨d̨ąt̨ą ̨w̨įt̨h̨ǫųt̨ ̨h̨ąv̨įn̨g̨ ̨t̨ǫ ̨h̨ǫl̨d̨ ̨įt̨ ̨ąl̨l̨ ̨įn̨ ̨m̨ąįn̨ ̨m̨ęm̨ǫr̨y̨ ̨ąn̨d̨ ̨(̨3̨)̨ ̨m̨ǫd̨ęl̨s̨ ̨t̨h̨ąt̨ ̨c̨ąn̨ ̨c̨l̨ąs̨s̨įf̨y̨ ̨n̨ęw̨ ̨d̨ąt̨ą ̨v̨ęr̨y̨ ̨ęf̨f̨įc̨įęn̨t̨l̨y̨.̨ ̨ ̨ ̨T̨h̨ę ̨l̨ąt̨ęs̨t̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨(̨B̨N̨C̨s̨)̨ ̨s̨ąt̨įs̨f̨y̨ ̨t̨h̨ęs̨ę ̨r̨ęq̨ųįr̨ęm̨ęn̨t̨s̨.̨ ̨T̨h̨ęįr̨ ̨b̨įąs̨ ̨c̨ąn̨ ̨b̨ę ̨c̨ǫn̨t̨r̨ǫl̨l̨ęd̨ ̨ęąs̨įl̨y̨ ̨b̨y̨ ̨įn̨c̨r̨ęąs̨įn̨g̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨n̨ǫd̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨g̨r̨ąp̨h̨.̨ ̨T̨h̨ęįr̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨c̨ąn̨ ̨b̨ę ̨l̨ęąr̨n̨ęd̨ ̨ǫųt̨ ̨ǫf̨ ̨c̨ǫr̨ę ̨w̨įt̨h̨ ̨ą ̨l̨įm̨įt̨ęd̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąs̨s̨ęs̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨d̨ąt̨ą.̨ ̨H̨ǫw̨ęv̨ęr̨,̨ ̨ąs̨ ̨t̨h̨ę ̨b̨įąs̨ ̨įs̨ ̨m̨ąd̨ę ̨l̨ǫw̨ęr̨ ̨t̨ǫ ̨ąc̨c̨ųr̨ąt̨ęl̨y̨ ̨m̨ǫd̨ęl̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨t̨ąs̨k̨s̨,̨ ̨s̨ǫ ̨įs̨ ̨t̨h̨ę ̨ąc̨c̨ųr̨ąc̨y̨ ̨ǫf̨ ̨t̨h̨ęįr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨'̨ ̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨ąs̨ ̨ęąc̨h̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨įs̨ ̨ęs̨t̨įm̨ąt̨ęd̨ ̨f̨r̨ǫm̨ ̨ęv̨ęr̨ ̨d̨ęc̨r̨ęąs̨įn̨g̨ ̨q̨ųąn̨t̨įt̨įęs̨ ̨ǫf̨ ̨d̨ąt̨ą.̨ ̨Įn̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨,̨ ̨w̨ę ̨įn̨t̨r̨ǫd̨ųc̨ę ̨t̨h̨ę ̨ųs̨ę ̨ǫf̨ ̨H̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨P̨r̨ǫc̨ęs̨s̨ęs̨ ̨f̨ǫr̨ ̨ąc̨c̨ųr̨ąt̨ę ̨B̨N̨C̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ęv̨ęn̨ ̨w̨įt̨h̨ ̨l̨ǫw̨ęr̨ ̨b̨įąs̨.̨ ̨ ̨W̨ę ̨c̨ǫn̨d̨ųc̨t̨ ̨ąn̨ ̨ęx̨t̨ęn̨s̨įv̨ę ̨s̨ęt̨ ̨ǫf̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨ǫn̨ ̨6̨8̨ ̨s̨t̨ąn̨d̨ąr̨d̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨ąn̨d̨ ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨ę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨r̨ęs̨ųl̨t̨įn̨g̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨p̨ęr̨f̨ǫr̨m̨ ̨v̨ęr̨y̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ęl̨y̨ ̨w̨įt̨h̨ ̨R̨ąn̨d̨ǫm̨ ̨F̨ǫr̨ęs̨t̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨p̨r̨ęd̨įc̨t̨įǫn̨,̨ ̨w̨h̨įl̨ę ̨k̨ęęp̨įn̨g̨ ̨t̨h̨ę ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨c̨ąp̨ąb̨įl̨įt̨y̨ ̨ąn̨d̨ ̨s̨ųp̨ęr̨įǫr̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨t̨įm̨ę.̨ ̨ ̨ ̨§̨ ̨ĮN̨T̨R̨ǪD̨ŲC̨T̨ĮǪN̨ ̨ ̨W̨įt̨h̨ ̨t̨h̨ę ̨ęv̨ęr̨ ̨įn̨c̨r̨ęąs̨įn̨g̨ ̨ąv̨ąįl̨ąb̨įl̨įt̨y̨ ̨ǫf̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨(̨B̨N̨C̨s̨)̨ ̨s̨h̨ǫw̨ ̨g̨r̨ęąt̨ ̨p̨ǫt̨ęn̨t̨įąl̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨c̨ąn̨ ̨b̨ę ̨l̨ęąr̨n̨ęd̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę,̨ ̨į.̨ę.̨ ̨w̨įt̨h̨ǫųt̨ ̨h̨ąv̨įn̨g̨ ̨t̨ǫ ̨h̨ǫl̨d̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨įn̨ ̨m̨ąįn̨ ̨m̨ęm̨ǫr̨y̨.̨ ̨T̨h̨įs̨ ̨c̨ąn̨ ̨b̨ę ̨d̨ǫn̨ę ̨įn̨ ̨ą ̨d̨įs̨c̨r̨įm̨įn̨ąt̨įv̨ę ̨f̨ąs̨h̨įǫn̨,̨ ̨f̨ǫr̨ ̨ęx̨ąm̨p̨l̨ę,̨ ̨T̨ĄN̨ ̨<̨c̨įt̨.̨>̨,̨ ̨k̨D̨B̨ ̨<̨c̨įt̨.̨>̨ ̨ąn̨d̨ ̨S̨ęl̨ęc̨t̨įv̨ę ̨k̨D̨B̨ ̨(̨S̨k̨D̨B̨)̨ ̨<̨c̨įt̨.̨>̨ ̨ąs̨ ̨w̨ęl̨l̨ ̨ąs̨ ̨g̨ęn̨ęr̨ąt̨įv̨ęl̨y̨,̨ ̨ųs̨įn̨g̨ ̨f̨įx̨ęd̨-̨s̨t̨r̨ųc̨t̨ųr̨ę ̨m̨ǫd̨ęl̨s̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨<̨c̨įt̨.̨>̨ ̨ąn̨d̨ ̨ąv̨ęr̨ąg̨ę ̨n̨-̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨ęs̨t̨įm̨ąt̨ǫr̨s̨ ̨–̨ ̨Ąn̨D̨Ę ̨<̨c̨įt̨.̨>̨.̨ ̨Įn̨ ̨c̨ǫn̨t̨r̨ąs̨t̨,̨ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨s̨ ̨(̨R̨F̨s̨)̨ ̨<̨c̨įt̨.̨>̨,̨ ̨ąr̨ę ̨n̨ǫt̨ ̨ęąs̨įl̨y̨ ̨l̨ęąr̨n̨ęd̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨r̨ęq̨ųįr̨ę ̨ęįt̨h̨ęr̨ ̨r̨ęp̨ęąt̨ęd̨ ̨s̨ǫr̨t̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨ǫr̨ ̨s̨ąm̨p̨l̨įn̨g̨.̨ ̨S̨t̨ąn̨d̨ąr̨d̨ ̨įm̨p̨l̨ęm̨ęn̨t̨ąt̨įǫn̨s̨ ̨s̨įd̨ę-̨s̨t̨ęp̨ ̨t̨h̨ę ̨p̨r̨ǫb̨l̨ęm̨ ̨ęįt̨h̨ęr̨ ̨b̨y̨ ̨ęn̨s̨ųr̨įn̨g̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨t̨r̨ąįn̨įn̨g̨ ̨s̨ęt̨s̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨t̨r̨ęę ̨ǫf̨ ̨t̨h̨ę ̨f̨ǫr̨ęs̨t̨ ̨įs̨ ̨s̨m̨ąl̨l̨ ̨ęn̨ǫųg̨h̨ ̨t̨ǫ ̨b̨ę ̨įn̨-̨c̨ǫr̨ę ̨<̨c̨įt̨.̨>̨,̨ ̨ǫr̨ ̨b̨y̨ ̨r̨ęl̨y̨įn̨g̨ ̨ǫn̨ ̨ǫn̨-̨d̨įs̨k̨ ̨ǫp̨ęr̨ąt̨įǫn̨s̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨C̨ǫn̨s̨t̨r̨ąįn̨t̨s̨ ̨ǫn̨ ̨t̨h̨ę ̨n̨ęt̨w̨ǫr̨k̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ǫf̨ ̨B̨N̨C̨s̨ ̨ąr̨ę ̨ųs̨ųąl̨l̨y̨ ̨c̨ǫn̨s̨įd̨ęr̨ęd̨ ̨t̨ǫ ̨b̨ę ̨t̨h̨ę ̨m̨ąįn̨ ̨c̨ǫn̨t̨r̨ǫl̨ ̨ǫn̨ ̨t̨h̨ęįr̨ ̨b̨įąs̨-̨v̨ąr̨įąn̨c̨ę ̨t̨r̨ąd̨ę-̨ǫf̨f̨.̨ ̨Įf̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨ ̨f̨ǫr̨ ̨n̨ǫd̨ęs̨ ̨įs̨ ̨r̨ęs̨t̨r̨įc̨t̨ęd̨ ̨t̨ǫ ̨ą ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨l̨ǫw̨ ̨n̨ųm̨b̨ęr̨,̨ ̨t̨h̨ęn̨ ̨b̨įąs̨ ̨w̨įl̨l̨ ̨g̨ęn̨ęr̨ąl̨l̨y̨ ̨b̨ę ̨h̨įg̨h̨ ̨ąn̨d̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨ǫn̨ ̨t̨h̨ęįr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨l̨ǫw̨ ̨(̨w̨ę ̨w̨įl̨l̨ ̨ąc̨t̨ųąl̨l̨y̨ ̨s̨h̨ǫw̨ ̨įn̨ ̨t̨h̨ę ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨c̨ąn̨ ̨b̨ę ̨h̨įg̨h̨ ̨ęv̨ęn̨ ̨f̨ǫr̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨ ̨w̨įt̨h̨ ̨l̨ǫw̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨)̨.̨ ̨F̨ǫr̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨l̨ǫw̨ęr̨ ̨b̨įąs̨ ̨ǫr̨ ̨h̨įg̨h̨ęr̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨įs̨ ̨p̨r̨ęf̨ęr̨ąb̨l̨ę ̨b̨ęc̨ąųs̨ę ̨įt̨ ̨ąl̨l̨ǫw̨s̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨s̨ ̨t̨ǫ ̨m̨ǫr̨ę ̨p̨r̨ęc̨įs̨ęl̨y̨ ̨c̨ąp̨t̨ųr̨ę ̨f̨įn̨ę ̨d̨ęt̨ąįl̨ ̨įn̨ ̨t̨h̨ę ̨d̨ąt̨ą,̨ ̨t̨r̨ąn̨s̨l̨ąt̨įn̨g̨ ̨įn̨t̨ǫ ̨h̨įg̨h̨ęr̨ ̨ąc̨c̨ųr̨ąc̨y̨ ̨(̨ęx̨ęm̨p̨l̨įf̨įęd̨ ̨b̨y̨ ̨t̨h̨ę ̨s̨ųc̨c̨ęs̨s̨ ̨ǫf̨ ̨d̨ęęp̨ ̨n̨ęt̨w̨ǫr̨k̨s̨)̨.̨ ̨T̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨įn̨c̨r̨ęąs̨ęs̨ ̨ęx̨p̨ǫn̨ęn̨t̨įąl̨l̨y̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨ ̨ąl̨l̨ǫw̨ęd̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę;̨ ̨t̨h̨ųs̨,̨ ̨f̨ǫr̨ ̨l̨ąr̨g̨ęr̨ ̨m̨ǫd̨ęl̨s̨,̨ ̨ąc̨c̨ųr̨ąt̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨b̨ęc̨ǫm̨ęs̨ ̨c̨r̨įt̨įc̨ąl̨.̨ ̨ ̨W̨ę ̨n̨ǫw̨ ̨t̨ųr̨n̨ ̨t̨ǫ ̨t̨h̨ę ̨ąįm̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨c̨ųr̨r̨ęn̨t̨ ̨p̨ąp̨ęr̨.̨ ̨Ǫn̨ę ̨ǫf̨ ̨t̨h̨ę ̨m̨ąįn̨ ̨įs̨s̨ųęs̨ ̨w̨įt̨h̨ ̨l̨ǫw̨-̨b̨įąs̨ ̨l̨ęąr̨n̨ęr̨s̨ ̨įs̨ ̨t̨h̨ęįr̨ ̨v̨ąr̨įąn̨c̨ę;̨ ̨įt̨ ̨įs̨ ̨l̨ǫg̨įc̨ąl̨ ̨t̨h̨ąt̨ ̨w̨h̨ęn̨ ̨įn̨c̨r̨ęąs̨įn̨g̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨f̨r̨ęę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨,̨ ̨ęv̨ęn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨l̨ąr̨g̨ęs̨t̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨d̨ąt̨ąs̨ęt̨,̨ ̨t̨h̨ęr̨ę ̨w̨įl̨l̨ ̨b̨ę ̨ą ̨p̨ǫįn̨t̨ ̨ąt̨ ̨w̨h̨įc̨h̨ ̨s̨ǫm̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨w̨įl̨l̨ ̨n̨ǫt̨ ̨h̨ąv̨ę ̨s̨ųf̨f̨įc̨įęn̨t̨ ̨ęx̨ąm̨p̨l̨ęs̨ ̨t̨ǫ ̨b̨ę ̨l̨ęąr̨n̨ęd̨ ̨w̨įt̨h̨ ̨p̨r̨ęc̨įs̨įǫn̨.̨ ̨V̨ąr̨įąn̨c̨ę ̨įs̨ ̨t̨h̨ųs̨ ̨n̨ǫt̨ ̨j̨ųs̨t̨ ̨ą ̨p̨r̨ǫb̨l̨ęm̨ ̨f̨ǫr̨ ̨s̨m̨ąl̨l̨ ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨b̨ųt̨ ̨c̨ąn̨ ̨r̨ęąp̨p̨ęąr̨ ̨w̨h̨ęn̨ ̨d̨ęs̨įg̨n̨įn̨g̨ ̨ęf̨f̨ęc̨t̨įv̨ę ̨l̨ęąr̨n̨ęr̨s̨ ̨f̨ǫr̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨r̨ęq̨ųįr̨ę ̨l̨ǫw̨ ̨b̨įąs̨.̨ ̨W̨h̨ęn̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨ęx̨ąm̨p̨l̨ęs̨ ̨p̨ęr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨d̨ęc̨r̨ęąs̨ęs̨,̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨įn̨c̨r̨ęąs̨ęs̨ ̨b̨ęc̨ąųs̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ąįl̨s̨ ̨t̨ǫ ̨d̨ęr̨įv̨ę ̨ąc̨c̨ųr̨ąt̨ę ̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨T̨h̨įs̨,̨ ̨ǫf̨ ̨c̨ǫųr̨s̨ę,̨ ̨įs̨ ̨w̨h̨y̨ ̨m̨ąx̨įm̨ųm̨-̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨(̨M̨L̨Ęs̨)̨ ̨ąr̨ę ̨n̨ǫt̨ ̨ǫf̨t̨ęn̨ ̨ųs̨ęd̨ ̨w̨įt̨h̨ ̨l̨ǫw̨-̨b̨įąs̨ ̨l̨ęąr̨n̨ęr̨s̨ ̨ųn̨l̨ęs̨s̨ ̨ęn̨s̨ęm̨b̨l̨ęs̨ ̨ąr̨ę ̨ąl̨s̨ǫ ̨įn̨v̨ǫl̨v̨ęd̨.̨ ̨ ̨R̨ęm̨ąr̨k̨ąb̨l̨y̨,̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨įn̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨f̨ǫr̨ ̨n̨ęt̨w̨ǫr̨k̨s̨ ̨ąs̨ ̨s̨įm̨p̨l̨ę ̨ąs̨ ̨T̨ĄN̨ ̨(̨w̨h̨ęr̨ę ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨h̨ąs̨ ̨t̨w̨ǫ ̨p̨ąr̨ęn̨t̨s̨ ̨ąt̨ ̨m̨ǫs̨t̨)̨,̨ ̨w̨h̨įc̨h̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨ųn̨d̨ęr̨p̨ęr̨f̨ǫr̨m̨ ̨R̨F̨s̨ ̨w̨h̨ęn̨ ̨ųs̨įn̨g̨ ̨L̨ąp̨l̨ąc̨ę ̨s̨m̨ǫǫt̨h̨įn̨g̨,̨ ̨c̨ąn̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨R̨F̨s̨ ̨ǫn̨c̨ę ̨m̨ǫr̨ę ̨c̨ąr̨ęf̨ųl̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨įs̨ ̨p̨ęr̨f̨ǫr̨m̨ęd̨.̨ ̨ ̨T̨h̨įs̨ ̨įs̨ ̨p̨ąr̨t̨įc̨ųl̨ąr̨l̨y̨ ̨s̨ųr̨p̨r̨įs̨įn̨g̨ ̨b̨ęc̨ąųs̨ę ̨ǫn̨ę ̨w̨ǫųl̨d̨n̨'̨t̨ ̨ęx̨p̨ęc̨t̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨t̨ǫ ̨b̨ę ̨h̨įg̨h̨ ̨f̨ǫr̨ ̨m̨ǫd̨ęl̨s̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨T̨ĄN̨.̨ ̨ ̨T̨h̨įs̨ ̨įs̨ ̨d̨ųę ̨t̨ǫ ̨t̨h̨ę ̨f̨ąc̨t̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨įs̨ ̨n̨ǫt̨ ̨ęv̨ęn̨ ̨ąm̨ǫn̨g̨ ̨ąl̨l̨ ̨c̨ǫm̨b̨įn̨ąt̨įǫn̨s̨ ̨ǫf̨ ̨f̨ęąt̨ųr̨ę ̨v̨ąl̨ųęs̨ ̨ąn̨d̨ ̨c̨ąn̨ ̨įn̨d̨ęęd̨ ̨b̨ę ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨h̨įg̨h̨ ̨f̨ǫr̨ ̨s̨ǫm̨ę ̨ǫf̨ ̨t̨h̨ęm̨.̨ ̨ ̨W̨ę ̨w̨įl̨l̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąųt̨ǫm̨ąt̨įc̨ąl̨l̨y̨ ̨ąd̨ąp̨t̨ ̨t̨ǫ ̨c̨ąs̨ęs̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ ̨ǫr̨ ̨l̨ǫw̨ ̨v̨ąr̨įąn̨c̨ę ̨b̨y̨ ̨c̨ąr̨ęf̨ųl̨ ̨ųs̨ę ̨ǫf̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨(̨H̨D̨P̨)̨.̨ ̨ ̨D̨r̨ąw̨įn̨g̨ ̨t̨h̨ę ̨l̨įn̨k̨ ̨b̨ęt̨w̨ęęn̨ ̨B̨N̨C̨s̨ ̨ąn̨d̨ ̨H̨D̨P̨:̨ ̨S̨ąy̨ ̨y̨ǫų ̨w̨ąn̨t̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ę ̨c̨ąn̨c̨ęr̨ ̨r̨ąt̨ę ̨įn̨ ̨ą ̨p̨ǫp̨ųl̨ąt̨įǫn̨ ̨ąn̨d̨ ̨y̨ǫų ̨ąr̨ę ̨ǫn̨l̨y̨ ̨g̨įv̨ęn̨ ̨1̨0̨ ̨s̨ąm̨p̨l̨ęs̨;̨ ̨y̨ǫų ̨w̨įl̨l̨ ̨g̨ęt̨ ̨ą ̨v̨ęr̨y̨ ̨c̨r̨ųd̨ę ̨ęs̨t̨įm̨ąt̨ę.̨ ̨ ̨Įn̨ ̨ęf̨f̨ęc̨t̨,̨ ̨t̨h̨įs̨ ̨h̨ąp̨p̨ęn̨s̨ ̨1̨0̨0̨'̨s̨ ̨ǫf̨ ̨t̨įm̨ęs̨ ̨ǫv̨ęr̨ ̨ąt̨ ̨ęąc̨h̨ ̨l̨ęąf̨ ̨ǫf̨ ̨ą ̨d̨ęc̨įs̨įǫn̨ ̨t̨r̨ęę ̨ǫr̨ ̨c̨l̨įq̨ųę ̨ǫf̨ ̨ą ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨w̨h̨ęn̨ ̨d̨ąt̨ą ̨įs̨ ̨n̨ǫt̨ ̨ąb̨ųn̨d̨ąn̨t̨ ̨ąt̨ ̨t̨h̨ę ̨n̨ǫd̨ę.̨ ̨F̨ǫr̨ ̨n̨-̨g̨r̨ąm̨ ̨m̨ǫd̨ęl̨s̨,̨ ̨w̨h̨ęr̨ę ̨ǫn̨ę ̨w̨įs̨h̨ęs̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨ęx̨t̨r̨ęm̨ęl̨y̨ ̨l̨ǫw̨-̨b̨įąs̨ ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨ąn̨d̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨v̨ęr̨y̨ ̨f̨ęw̨ ̨ęx̨ąm̨p̨l̨ęs̨ ̨p̨ęr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ąr̨ę ̨ąv̨ąįl̨ąb̨l̨ę,̨ ̨M̨L̨Ęs̨ ̨h̨ąv̨ę ̨l̨ǫn̨g̨ ̨s̨įn̨c̨ę ̨b̨ęęn̨ ̨ąb̨ąn̨d̨ǫn̨ęd̨ ̨įn̨ ̨f̨ąv̨ǫųr̨ ̨ǫf̨ ̨s̨ǫp̨h̨įs̨t̨įc̨ąt̨ęd̨ ̨s̨m̨ǫǫt̨h̨įn̨g̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨m̨ǫd̨įf̨įęd̨ ̨K̨n̨ęs̨ęr̨-̨N̨ęy̨ ̨<̨c̨įt̨.̨>̨.̨ ̨T̨h̨ęs̨ę,̨ ̨h̨ǫw̨ęv̨ęr̨,̨ ̨h̨ąv̨ę ̨c̨ǫm̨p̨l̨ęx̨ ̨b̨ąc̨k̨-̨ǫf̨f̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨h̨ąt̨ ̨n̨ęęd̨ ̨t̨ǫ ̨b̨ę ̨s̨ęt̨.̨ ̨F̨ǫr̨ ̨ǫųr̨ ̨m̨ǫr̨ę ̨g̨ęn̨ęr̨ąl̨ ̨ąn̨d̨ ̨h̨ęt̨ęr̨ǫg̨ęn̨ęǫųs̨ ̨c̨ǫn̨t̨ęx̨t̨ ̨ǫf̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨t̨ąb̨l̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨,̨ ̨t̨h̨ęr̨ę ̨ęx̨įs̨t̨ ̨n̨ǫ ̨t̨ęc̨h̨n̨įq̨ųęs̨ ̨t̨ǫ ̨s̨ęt̨ ̨t̨h̨ęs̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨.̨ ̨H̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨P̨įt̨m̨ąn̨-̨Y̨ǫr̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨(̨H̨P̨Y̨P̨)̨ ̨įs̨ ̨t̨h̨ę ̨B̨ąy̨ęs̨įąn̨ ̨v̨ęr̨s̨įǫn̨ ̨ǫf̨ ̨K̨n̨ęs̨ęr̨-̨N̨ęy̨ ̨s̨m̨ǫǫt̨h̨įn̨g̨;̨ ̨įt̨ ̨w̨ąs̨ ̨įn̨t̨r̨ǫd̨ųc̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨ ̨ąn̨d̨ ̨ųs̨ęs̨ ̨ęm̨p̨įr̨įc̨ąl̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨s̨.̨ ̨ ̨T̨h̨įs̨ ̨h̨ąs̨ ̨b̨ęęn̨ ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨ęd̨ ̨t̨ǫ ̨b̨ę ̨v̨ęr̨y̨ ̨ęf̨f̨ęc̨t̨įv̨ę ̨<̨c̨įt̨.̨>̨.̨ ̨H̨P̨Y̨P̨ ̨įs̨ ̨w̨ęl̨l̨-̨s̨ųįt̨ęd̨ ̨f̨ǫr̨ ̨Z̨įp̨f̨įąn̨ ̨c̨ǫn̨t̨ęx̨t̨s̨:̨ ̨ ̨w̨h̨ęr̨ę ̨d̨įs̨c̨r̨ęt̨ę ̨v̨ąr̨įąb̨l̨ęs̨ ̨h̨ąv̨ę ̨h̨ųn̨d̨r̨ęd̨s̨ ̨ǫr̨ ̨m̨ǫr̨ę ̨ǫųt̨c̨ǫm̨ęs̨ ̨w̨įt̨h̨ ̨v̨ęr̨y̨ ̨b̨įąs̨ęd̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨.̨ ̨S̨įn̨c̨ę ̨w̨ę ̨h̨ąv̨ę ̨d̨įs̨c̨r̨ęt̨ę ̨v̨ąr̨įąb̨l̨ęs̨ ̨w̨įt̨h̨ ̨m̨ǫs̨t̨l̨y̨ ̨f̨ęw̨ęr̨ ̨ǫųt̨c̨ǫm̨ęs̨ ̨w̨ę ̨d̨ǫ ̨n̨ǫt̨ ̨ųs̨ę ̨t̨h̨ę ̨H̨P̨Y̨P̨,̨ ̨ąn̨d̨ ̨p̨r̨ęf̨ęr̨ ̨t̨h̨ę ̨l̨ǫw̨ęr̨-̨v̨ąr̨įąn̨c̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨(̨H̨D̨P̨)̨ ̨<̨c̨įt̨.̨>̨–̨ ̨įt̨ ̨įs̨ ̨ęq̨ųįv̨ąl̨ęn̨t̨ ̨t̨ǫ ̨H̨P̨Y̨P̨ ̨w̨įt̨h̨ ̨d̨įs̨c̨ǫųn̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨įx̨ęd̨ ̨t̨ǫ ̨0̨.̨ ̨ ̨ ̨ ̨Įn̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨,̨ ̨w̨ę ̨p̨r̨ǫp̨ǫs̨ę ̨t̨ǫ ̨ąd̨ąp̨t̨ ̨t̨h̨ę ̨m̨ęt̨h̨ǫd̨ ̨ǫf̨ ̨<̨c̨įt̨.̨>̨ ̨f̨ǫr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨n̨-̨g̨r̨ąm̨ ̨m̨ǫd̨ęl̨s̨ ̨ąn̨d̨ ̨ąp̨p̨l̨y̨ ̨įt̨ ̨t̨ǫ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨.̨ ̨R̨ąt̨h̨ęr̨ ̨t̨h̨ąn̨ ̨t̨h̨ę ̨H̨P̨Y̨P̨ ̨ųs̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨ ̨w̨ę ̨ųs̨ę ̨t̨h̨ę ̨m̨ǫr̨ę ̨c̨ǫm̨p̨ųt̨ąt̨įǫn̨ąl̨l̨y̨ ̨ęf̨f̨įc̨įęn̨t̨ ̨H̨D̨P̨.̨ ̨Įn̨ ̨t̨h̨įs̨ ̨c̨ǫn̨t̨ęx̨t̨,̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨ ̨įs̨ ̨s̨įm̨p̨l̨ęr̨ ̨b̨ęc̨ąųs̨ę ̨ą ̨H̨D̨P̨ ̨w̨įt̨h̨ ̨ą ̨f̨įn̨įt̨ę ̨d̨įs̨c̨r̨ęt̨ę ̨b̨ąs̨ę ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨įs̨ ̨b̨y̨ ̨d̨ęf̨įn̨įt̨įǫn̨ ̨ęq̨ųįv̨ąl̨ęn̨t̨ ̨t̨ǫ ̨ą ̨D̨įr̨įc̨h̨l̨ęt̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨,̨ ̨t̨h̨ąt̨ ̨įs̨ ̨H̨D̨P̨s̨ ̨b̨ęc̨ǫm̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨įn̨ ̨ǫųr̨ ̨c̨ǫn̨t̨ęx̨t̨.̨ ̨W̨h̨įl̨ę ̨c̨ǫn̨c̨ęp̨t̨ųąl̨l̨y̨ ̨s̨įm̨p̨l̨ęr̨,̨ ̨w̨ę ̨s̨t̨įl̨l̨ ̨ųs̨ę ̨H̨D̨P̨ ̨s̨t̨y̨l̨ę ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨,̨ ̨ąl̨b̨įęt̨ ̨m̨ǫr̨ę ̨r̨ęc̨ęn̨t̨ ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨,̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨ąr̨ę ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨ęf̨f̨įc̨įęn̨t̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨t̨ǫ ̨t̨h̨ę ̨ǫl̨d̨ęr̨ ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨s̨t̨y̨l̨ę ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨H̨ąv̨įn̨g̨ ̨s̨h̨ǫw̨n̨ ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ąp̨p̨r̨ǫąc̨h̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨s̨ ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨B̨N̨C̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨,̨ ̨w̨ę ̨ųs̨ę ̨R̨F̨ ̨ąs̨ ̨ąn̨ ̨ęx̨ęm̨p̨l̨ąr̨ ̨ǫf̨ ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨m̨ąc̨h̨įn̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨b̨ęc̨ąųs̨ę ̨įt̨ ̨įs̨ ̨ą ̨w̨įd̨ęl̨y̨ ̨ųs̨ęd̨ ̨l̨ęąr̨n̨įn̨g̨ ̨m̨ęt̨h̨ǫd̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨t̨y̨p̨ęs̨ ̨ǫf̨ ̨t̨ąb̨ųl̨ąr̨ ̨d̨ąt̨ą ̨t̨ǫ ̨w̨h̨įc̨h̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨ąr̨ę ̨s̨ųįt̨ęd̨ ̨w̨h̨įc̨h̨ ̨c̨ąn̨ ̨b̨ę ̨ųs̨ęd̨ ̨ǫųt̨ ̨ǫf̨ ̨t̨h̨ę ̨b̨ǫx̨ ̨w̨įt̨h̨ǫųt̨ ̨n̨ęęd̨ ̨f̨ǫr̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨.̨ ̨W̨ę ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ǫr̨ ̨ąl̨l̨ǫw̨s̨ ̨B̨N̨C̨s̨ ̨t̨ǫ ̨c̨ǫm̨p̨ęt̨ę ̨ąg̨ąįn̨s̨t̨ ̨R̨F̨s̨ ̨ǫn̨ ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨ąt̨ąs̨ęt̨s̨.̨ ̨F̨ųr̨t̨h̨ęr̨m̨ǫr̨ę,̨ ̨b̨ęc̨ąųs̨ę ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨įs̨ ̨c̨ǫm̨p̨l̨ęt̨ęl̨y̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę,̨ ̨w̨ę ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨ę ̨t̨h̨ąt̨ ̨w̨ę ̨c̨ąn̨ ̨ǫb̨t̨ąįn̨ ̨r̨ęs̨ųl̨t̨s̨ ̨ǫn̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨ǫn̨ ̨s̨t̨ąn̨d̨ąr̨d̨ ̨c̨ǫm̨p̨ųt̨ęr̨s̨ ̨w̨įt̨h̨ ̨w̨h̨įc̨h̨ ̨R̨F̨ ̨c̨ąn̨n̨ǫt̨ ̨ęv̨ęn̨ ̨b̨ę ̨t̨r̨ąįn̨ęd̨ ̨ųs̨įn̨g̨ ̨s̨t̨ąn̨d̨ąr̨d̨ ̨p̨ąc̨k̨ąg̨ęs̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨W̨ęk̨ą.̨ ̨Ǫųr̨ ̨m̨ǫd̨ęl̨s̨ ̨c̨ąn̨ ̨ąl̨s̨ǫ ̨c̨l̨ąs̨s̨įf̨y̨ ̨ǫr̨d̨ęr̨s̨ ̨ǫf̨ ̨m̨ąg̨n̨įt̨ųd̨ę ̨f̨ąs̨t̨ęr̨ ̨t̨h̨ąn̨ ̨R̨F̨.̨ ̨ ̨T̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨įs̨ ̨ǫr̨g̨ąn̨įz̨ęd̨ ̨ąs̨ ̨f̨ǫl̨l̨ǫw̨s̨.̨ ̨Įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨w̨ę ̨r̨ęv̨įęw̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨(̨B̨N̨C̨s̨)̨.̨ ̨ ̨Įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨w̨ę ̨m̨ǫt̨įv̨ąt̨ę ̨ǫųr̨ ̨ųs̨ę ̨ǫf̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨P̨r̨ǫc̨ęs̨s̨ęs̨ ̨(̨H̨D̨P̨s̨)̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨'̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨.̨ ̨ ̨W̨ę ̨p̨r̨ęs̨ęn̨t̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨ąn̨d̨ ̨r̨ęl̨ąt̨ęd̨ ̨w̨ǫr̨k̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨W̨ę ̨h̨ąv̨ę ̨c̨ǫn̨d̨ųc̨t̨ęd̨ ̨ęx̨t̨ęn̨s̨įv̨ę ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨,̨ ̨r̨ęp̨ǫr̨t̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨§̨ ̨S̨T̨ĄN̨D̨ĄR̨D̨ ̨B̨ĄY̨ĘS̨ĮĄN̨ ̨N̨ĘT̨W̨ǪR̨K̨ ̨C̨L̨ĄS̨S̨ĮF̨ĮĘR̨S̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨N̨ǫt̨ąt̨įǫn̨s̨ ̨L̨ęt̨ ̨=̨ ̨{̨^̨(̨1̨)̨,̨⋯̨,̨^̨(̨N̨)̨}̨ ̨b̨ę ̨ą ̨d̨ąt̨ąs̨ęt̨ ̨w̨įt̨h̨ ̨N̨ ̨ǫb̨j̨ęc̨t̨s̨.̨ ̨Ęąc̨h̨ ̨d̨ąt̨ųm̨ ̨=̨ ̨⟨̨ ̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨⟩̨ ̨įs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨ǫv̨ęr̨ ̨r̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨X̨_̨1̨,̨⋯̨,̨X̨_̨n̨.̨ ̨T̨h̨ę ̨f̨ǫl̨l̨ǫw̨įn̨g̨ ̨f̨r̨ąm̨ęw̨ǫr̨k̨ ̨c̨ąn̨ ̨b̨ę ̨f̨ǫųn̨d̨ ̨įn̨ ̨t̨ęx̨t̨s̨ ̨ǫn̨ ̨l̨ęąr̨n̨įn̨g̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨s̨,̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨<̨c̨įt̨.̨>̨.̨ ̨Ą ̨=̨ ̨⟨̨,̨Θ̨⟩̨,̨ ̨įs̨ ̨c̨h̨ąr̨ąc̨t̨ęr̨įz̨ęd̨ ̨b̨y̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ ̨(̨ą ̨d̨įr̨ęc̨t̨ęd̨ ̨ąc̨y̨c̨l̨įc̨ ̨g̨r̨ąp̨h̨,̨ ̨w̨h̨ęr̨ę ̨ęąc̨h̨ ̨v̨ęr̨t̨ęx̨ ̨į ̨įs̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨t̨ǫ ̨ą ̨r̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨X̨_̨į)̨,̨ ̨ąn̨d̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨Θ̨,̨ ̨t̨h̨ąt̨ ̨q̨ųąn̨t̨įf̨įęs̨ ̨t̨h̨ę ̨d̨ęp̨ęn̨d̨ęn̨c̨įęs̨ ̨w̨įt̨h̨įn̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę.̨ ̨T̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ǫb̨j̨ęc̨t̨ ̨Θ̨,̨ ̨c̨ǫn̨t̨ąįn̨s̨ ̨ą ̨s̨ęt̨ ̨ǫf̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨v̨ęr̨t̨ęx̨ ̨įn̨ ̨:̨ ̨θ̨_̨x̨_̨į|̨Π̨_̨į(̨)̨,̨ ̨w̨h̨ęr̨ę ̨Π̨_̨į(̨.̨)̨ ̨įs̨ ̨ą ̨f̨ųn̨c̨t̨įǫn̨ ̨w̨h̨įc̨h̨ ̨g̨įv̨ęn̨ ̨t̨h̨ę ̨d̨ąt̨ųm̨ ̨=̨ ̨⟨̨ ̨x̨_̨1̨,̨…̨,̨x̨_̨n̨ ̨⟩̨ ̨ąs̨ ̨įt̨s̨ ̨įn̨p̨ųt̨,̨ ̨r̨ęt̨ųr̨n̨s̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨t̨h̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨w̨h̨įc̨h̨ ̨ąr̨ę ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨s̨ ̨ǫf̨ ̨n̨ǫd̨ę ̨į ̨įn̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨.̨ ̨ ̨N̨ǫt̨ę,̨ ̨ęąc̨h̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨įs̨ ̨ą ̨r̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨X̨_̨į ̨ąn̨d̨ ̨x̨_̨į ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨v̨ąl̨ųę ̨ǫf̨ ̨t̨h̨ąt̨ ̨r̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę.̨ ̨ ̨F̨ǫr̨ ̨n̨ǫt̨ąt̨įǫn̨ąl̨ ̨s̨įm̨p̨l̨įc̨įt̨y̨ ̨w̨ę ̨w̨r̨įt̨ę ̨θ̨_̨x̨_̨į|̨Π̨_̨į(̨)̨ ̨įn̨s̨t̨ęąd̨ ̨ǫf̨ ̨θ̨_̨X̨_̨į ̨=̨ ̨x̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨.̨ ̨W̨ę ̨ąl̨s̨ǫ ̨ųs̨ę ̨θ̨_̨X̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨ ̨t̨ǫ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨h̨ę ̨f̨ųl̨l̨ ̨v̨ęc̨t̨ǫr̨ ̨ǫf̨ ̨v̨ąl̨ųęs̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨x̨_̨į.̨ ̨Ą ̨ ̨c̨ǫm̨p̨ųt̨ęs̨ ̨t̨h̨ę ̨j̨ǫįn̨t̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨ąs̨ ̨ ̨ ̨ ̨ ̨ ̨_̨(̨)̨ ̨=̨ ̨∏̨_̨į=̨1̨^̨n̨θ̨_̨x̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨.̨ ̨T̨h̨ę ̨g̨ǫąl̨ ̨ǫf̨ ̨d̨ęv̨ęl̨ǫp̨įn̨g̨ ̨ą ̨ ̨c̨l̨ąs̨s̨įf̨įęr̨ ̨įs̨ ̨t̨ǫ ̨p̨r̨ęd̨įc̨t̨ ̨t̨h̨ę ̨v̨ąl̨ųę ̨ǫf̨ ̨ąn̨ ̨ąd̨d̨įt̨įǫn̨ąl̨ ̨v̨ąr̨įąb̨l̨ę ̨X̨_̨0̨=̨Y̨:̨ ̨X̨_̨0̨ ̨įs̨ ̨t̨h̨ę ̨r̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨ąn̨d̨ ̨w̨ę ̨ąl̨s̨ǫ ̨d̨ęn̨ǫt̨ę ̨įt̨ ̨b̨y̨ ̨Y̨ ̨ąn̨d̨ ̨įt̨s̨ ̨v̨ąl̨ųęs̨ ̨b̨y̨ ̨y̨∈̨𝒴̨.̨ ̨T̨h̨ę ̨d̨ąt̨ą ̨t̨h̨ęn̨ ̨t̨ąk̨ęs̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ ̨=̨ ̨{̨ ̨(̨y̨^̨(̨1̨)̨,̨^̨(̨1̨)̨)̨,̨…̨,̨ ̨(̨y̨^̨(̨N̨)̨,̨^̨(̨N̨)̨)̨ ̨}̨,̨ ̨t̨h̨ę ̨n̨ęt̨w̨ǫr̨k̨ ̨t̨ąk̨ęs̨ ̨ąn̨ ̨ąd̨d̨įt̨įǫn̨ąl̨ ̨n̨ǫd̨ę ̨ąn̨d̨ ̨w̨ę ̨c̨ąn̨ ̨w̨r̨įt̨ę:̨ ̨ ̨ ̨ ̨ ̨_̨(̨y̨|̨)̨ ̨=̨ ̨ ̨_̨(̨y̨,̨)̨/̨_̨(̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨=̨ ̨θ̨_̨y̨ ̨|̨Π̨_̨0̨(̨)̨∏̨_̨į=̨1̨^̨n̨θ̨_̨x̨_̨į ̨|̨ ̨y̨,̨ ̨Π̨_̨į(̨)̨/̨∑̨_̨y̨'̨∈̨𝒴̨θ̨_̨y̨'̨ ̨|̨ ̨Π̨_̨0̨(̨)̨∏̨_̨į=̨1̨^̨n̨θ̨_̨x̨_̨į ̨|̨ ̨y̨'̨,̨ ̨Π̨_̨į(̨)̨.̨ ̨F̨ǫr̨ ̨s̨įm̨p̨l̨įc̨įt̨y̨,̨ ̨įn̨ ̨t̨h̨ę ̨f̨ǫl̨l̨ǫw̨įn̨g̨,̨ ̨w̨ę ̨ųs̨ę ̨θ̨_̨y̨ ̨t̨ǫ ̨d̨ęn̨ǫt̨ę ̨θ̨_̨y̨|̨Π̨_̨0̨(̨)̨.̨ ̨M̨ǫs̨t̨ ̨n̨ǫt̨ąt̨įǫn̨s̨ ̨ąr̨ę ̨s̨ųm̨m̨ąr̨įs̨ęd̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨S̨t̨r̨ųc̨t̨ųr̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨ ̨ ̨M̨ǫs̨t̨ ̨ąp̨p̨r̨ǫąc̨h̨ęs̨ ̨t̨ǫ ̨l̨ęąr̨n̨įn̨g̨ ̨B̨N̨C̨s̨ ̨l̨ęąr̨n̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨f̨įr̨s̨t̨ ̨ąn̨d̨ ̨t̨h̨ęn̨ ̨l̨ęąr̨n̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąs̨ ̨ą ̨s̨ęp̨ąr̨ąt̨ę ̨s̨t̨ęp̨.̨ ̨N̨ųm̨ęr̨ǫųs̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨h̨ąv̨ę ̨b̨ęęn̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨f̨ǫr̨ ̨l̨ęąr̨n̨įn̨g̨ ̨B̨N̨C̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę.̨ ̨T̨h̨ę ̨k̨ęy̨ ̨d̨įf̨f̨ęr̨ęn̨c̨ę ̨t̨h̨ąt̨ ̨d̨įs̨t̨įn̨g̨ųįs̨h̨ęs̨ ̨B̨N̨C̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨ ̨f̨r̨ǫm̨ ̨n̨ǫr̨m̨ąl̨ ̨B̨N̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨įs̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨p̨r̨ęc̨įs̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨ǫs̨t̨ęr̨įǫr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨_̨(̨y̨|̨)̨ ̨ ̨m̨ąt̨t̨ęr̨s̨ ̨r̨ąt̨h̨ęr̨ ̨t̨h̨ąn̨ ̨t̨h̨ę ̨p̨r̨ęc̨įs̨įǫn̨ ̨ǫf̨ ̨_̨(̨y̨,̨)̨.̨ ̨Ąs̨ ̨ą ̨r̨ęs̨ųl̨t̨,̨ ̨įt̨ ̨įs̨ ̨ųs̨ųąl̨l̨y̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨t̨ǫ ̨ęn̨s̨ųr̨ę ̨t̨h̨ąt̨ ̨ąl̨l̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨'̨ ̨M̨ąr̨k̨ǫv̨ ̨b̨l̨ąn̨k̨ęt̨ ̨ąr̨ę ̨c̨ǫn̨n̨ęc̨t̨ęd̨ ̨d̨įr̨ęc̨t̨l̨y̨ ̨t̨ǫ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨ǫr̨ ̨įt̨s̨ ̨c̨h̨įl̨d̨r̨ęn̨.̨ ̨Ąs̨ ̨ą ̨c̨ǫn̨s̨ęq̨ųęn̨c̨ę,̨ ̨įt̨ ̨įs̨ ̨c̨ǫm̨m̨ǫn̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨ ̨t̨ǫ ̨c̨ǫn̨n̨ęc̨t̨ ̨ąl̨l̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨t̨ǫ ̨t̨h̨ę ̨c̨l̨ąs̨s̨.̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ǫb̨s̨ęr̨v̨ęd̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨0̨p̨t̨,̨c̨įr̨c̨l̨ę,̨d̨r̨ąw̨=̨b̨l̨ųę!̨5̨0̨,̨f̨įl̨l̨=̨b̨l̨ųę!̨2̨0̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ųn̨ǫb̨s̨ęr̨v̨ęd̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨0̨p̨t̨,̨c̨įr̨c̨l̨ę,̨d̨r̨ąw̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨h̨y̨p̨ęr̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨p̨t̨,̨c̨įr̨c̨l̨ę,̨f̨įl̨l̨=̨b̨l̨ąc̨k̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨p̨ǫs̨t̨/̨.̨s̨t̨y̨l̨ę=̨-̨>̨,̨>̨=̨s̨t̨ęąl̨t̨h̨'̨,̨s̨ęm̨įt̨h̨įc̨k̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨1̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨1̨,̨0̨)̨ ̨X̨_̨2̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨2̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨2̨.̨5̨,̨0̨)̨ ̨X̨_̨4̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨3̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨4̨,̨0̨)̨ ̨X̨_̨1̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨4̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨5̨.̨5̨,̨0̨)̨ ̨X̨_̨3̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨y̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨3̨.̨2̨5̨,̨1̨.̨5̨)̨ ̨Y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨3̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨4̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨-̨>̨,̨ǫp̨ąc̨įt̨y̨=̨0̨.̨0̨]̨ ̨ ̨(̨1̨,̨-̨0̨.̨5̨5̨)̨–̨(̨5̨.̨5̨,̨-̨0̨.̨5̨5̨)̨ ̨n̨ǫd̨ę[̨b̨ęl̨ǫw̨,̨m̨įd̨w̨ąy̨]̨ ̨D̨ęc̨r̨ęąs̨įn̨g̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨Y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ǫb̨s̨ęr̨v̨ęd̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨2̨p̨t̨,̨c̨įr̨c̨l̨ę,̨d̨r̨ąw̨=̨b̨l̨ųę!̨5̨0̨,̨f̨įl̨l̨=̨b̨l̨ųę!̨2̨0̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ųn̨ǫb̨s̨ęr̨v̨ęd̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨2̨p̨t̨,̨c̨įr̨c̨l̨ę,̨d̨r̨ąw̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨h̨y̨p̨ęr̨/̨.̨s̨t̨y̨l̨ę=̨m̨įn̨įm̨ųm̨ ̨s̨įz̨ę=̨1̨p̨t̨,̨c̨įr̨c̨l̨ę,̨f̨įl̨l̨=̨b̨l̨ąc̨k̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨p̨ǫs̨t̨/̨.̨s̨t̨y̨l̨ę=̨-̨>̨,̨>̨=̨s̨t̨ęąl̨t̨h̨'̨,̨s̨ęm̨įt̨h̨įc̨k̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨1̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨1̨,̨0̨)̨ ̨X̨_̨2̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨2̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨2̨.̨5̨,̨0̨)̨ ̨X̨_̨4̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨3̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨4̨,̨0̨)̨ ̨X̨_̨1̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨4̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨5̨.̨5̨,̨0̨)̨ ̨X̨_̨3̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨y̨)̨ ̨[̨ǫb̨s̨ęr̨v̨ęd̨]̨ ̨ąt̨ ̨(̨3̨.̨2̨5̨,̨1̨.̨5̨)̨ ̨Y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨3̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨y̨)̨ ̨–̨ ̨(̨x̨4̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨1̨)̨ ̨ęd̨g̨ę ̨(̨x̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨1̨)̨ ̨ęd̨g̨ę ̨[̨b̨ęn̨d̨ ̨l̨ęf̨t̨]̨ ̨(̨x̨3̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨3̨)̨ ̨ęd̨g̨ę ̨(̨x̨4̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨-̨>̨]̨ ̨ ̨(̨1̨,̨-̨0̨.̨5̨5̨)̨–̨(̨5̨.̨5̨,̨-̨0̨.̨5̨5̨)̨ ̨n̨ǫd̨ę[̨b̨ęl̨ǫw̨,̨m̨įd̨w̨ąy̨]̨ ̨D̨ęc̨r̨ęąs̨įn̨g̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨Y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨Ęx̨ąm̨p̨l̨ę ̨B̨N̨C̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨:̨ ̨(̨ą)̨ ̨N̨ąį̈v̨ę ̨B̨ąy̨ęs̨,̨ ̨(̨b̨)̨ ̨k̨D̨B̨-̨1̨ ̨ ̨N̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨(̨N̨B̨ ̨-̨ ̨s̨ęę ̨ę.̨g̨.̨ ̨<̨c̨įt̨.̨>̨)̨ ̨įs̨ ̨ą ̨p̨ǫp̨ųl̨ąr̨ ̨B̨N̨C̨ ̨t̨h̨ąt̨ ̨m̨ąk̨ęs̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨ǫf̨ ̨ąl̨l̨ ̨ǫt̨h̨ęr̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨ąn̨d̨ ̨įn̨c̨l̨ųd̨ęs̨ ̨n̨ǫ ̨ǫt̨h̨ęr̨ ̨ęd̨g̨ęs̨.̨ ̨T̨h̨ę ̨r̨ęs̨ųl̨t̨įn̨g̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨įs̨ ̨įl̨l̨ųs̨t̨r̨ąt̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨ą)̨ ̨ąn̨d̨ ̨ąs̨s̨ųm̨ęs̨ ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨įn̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨b̨ęt̨w̨ęęn̨ ̨ąl̨l̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨c̨ǫn̨d̨įt̨įǫn̨ęd̨ ̨ǫn̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨.̨ ̨Ąs̨ ̨ą ̨c̨ǫn̨s̨ęq̨ųęn̨c̨ę,̨ ̨ ̨ ̨_̨(̨y̨|̨)̨ ̨ ̨∝̨ ̨θ̨_̨y̨∏̨_̨į=̨1̨^̨n̨θ̨_̨x̨_̨į ̨|̨ ̨y̨ ̨.̨ ̨ ̨T̨r̨ęę-̨ąųg̨m̨ęn̨t̨ęd̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨(̨T̨ĄN̨)̨ ̨<̨c̨įt̨.̨>̨ ̨ąd̨d̨s̨ ̨ą ̨f̨ųr̨t̨h̨ęr̨ ̨p̨ąr̨ęn̨t̨ ̨t̨ǫ ̨ęąc̨h̨ ̨n̨ǫn̨-̨c̨l̨ąs̨s̨ ̨ąt̨t̨r̨įb̨ųt̨ę,̨ ̨s̨ęęk̨įn̨g̨ ̨t̨ǫ ̨ąd̨d̨r̨ęs̨s̨ ̨t̨h̨ę ̨g̨r̨ęąt̨ęs̨t̨ ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨įn̨t̨ęr̨d̨ęp̨ęn̨d̨ęn̨c̨įęs̨.̨ ̨Įt̨ ̨ųs̨ęs̨ ̨t̨h̨ę ̨C̨h̨ǫw̨-̨L̨įų ̨<̨c̨įt̨.̨>̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨ ̨t̨ǫ ̨f̨įn̨d̨ ̨t̨h̨ę ̨m̨ąx̨įm̨ųm̨-̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨t̨r̨ęę ̨ǫf̨ ̨d̨ęp̨ęn̨d̨ęn̨c̨įęs̨ ̨ąm̨ǫn̨g̨ ̨t̨h̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨įn̨ ̨p̨ǫl̨y̨n̨ǫm̨įąl̨ ̨t̨įm̨ę.̨ ̨ ̨K̨-̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨B̨ąy̨ęs̨ ̨(̨k̨D̨B̨)̨ ̨<̨c̨įt̨.̨>̨ ̨ąl̨l̨ǫw̨s̨ ̨ęąc̨h̨ ̨n̨ǫn̨-̨c̨l̨ąs̨s̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨t̨ǫ ̨h̨ąv̨ę ̨ųp̨ ̨t̨ǫ ̨k̨ ̨p̨ąr̨ęn̨t̨s̨,̨ ̨w̨įt̨h̨ ̨k̨ ̨b̨ęįn̨g̨ ̨ą ̨ųs̨ęr̨-̨s̨ęt̨ ̨v̨ąl̨ųę.̨ ̨Įt̨ ̨f̨įr̨s̨t̨ ̨s̨ǫr̨t̨s̨ ̨t̨h̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨ǫn̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨.̨ ̨Ęąc̨h̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨x̨_̨į ̨įs̨ ̨ąs̨s̨įg̨n̨ęd̨ ̨t̨h̨ę ̨k̨ ̨p̨ąr̨ęn̨t̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨t̨h̨ąt̨ ̨m̨ąx̨įm̨įz̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨(̨C̨M̨Į)̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨,̨ ̨C̨M̨Į(̨y̨,̨x̨_̨į|̨Π̨_̨į(̨)̨)̨,̨ ̨ǫųt̨ ̨ǫf̨ ̨t̨h̨ǫs̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ęr̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨.̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨b̨)̨ ̨s̨h̨ǫw̨s̨ ̨k̨D̨B̨-̨1̨ ̨(̨f̨ǫr̨ ̨k̨=̨1̨)̨.̨ ̨ ̨S̨ęl̨ęc̨t̨įv̨ę ̨k̨D̨B̨ ̨(̨S̨k̨D̨B̨)̨ ̨<̨c̨įt̨.̨>̨ ̨s̨ęl̨ęc̨t̨s̨ ̨v̨ąl̨ųęs̨ ̨n̨^̨*̨≤̨ ̨n̨ ̨ąn̨d̨ ̨k̨^̨*̨≤̨ ̨k̨ ̨s̨ųc̨h̨ ̨t̨h̨ąt̨ ̨ą ̨k̨D̨B̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨n̨^̨*̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ęs̨t̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨ąn̨d̨ ̨ųs̨įn̨g̨ ̨k̨^̨*̨ ̨įn̨ ̨p̨l̨ąc̨ę ̨ǫf̨ ̨k̨ ̨m̨ąx̨įm̨įz̨ęs̨ ̨s̨ǫm̨ę ̨ųs̨ęr̨ ̨s̨ęl̨ęc̨t̨ęd̨ ̨m̨ęąs̨ųr̨ę ̨ǫf̨ ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę ̨(̨įn̨ ̨t̨h̨ę ̨c̨ųr̨r̨ęn̨t̨ ̨w̨ǫr̨k̨,̨ ̨R̨M̨S̨Ę)̨ ̨ąs̨s̨ęs̨s̨ęd̨ ̨ųs̨įn̨g̨ ̨įn̨c̨r̨ęm̨ęn̨t̨ąl̨ ̨c̨r̨ǫs̨s̨ ̨v̨ąl̨įd̨ąt̨įǫn̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨t̨r̨ąįn̨įn̨g̨ ̨d̨ąt̨ą.̨ ̨ ̨Ǫt̨h̨ęr̨ ̨d̨įs̨c̨r̨įm̨įn̨ąt̨įv̨ę ̨s̨c̨ǫr̨įn̨g̨ ̨s̨c̨h̨ęm̨ęs̨ ̨h̨ąv̨ę ̨b̨ęęn̨ ̨s̨t̨ųd̨įęd̨,̨ ̨s̨ęę ̨f̨ǫr̨ ̨ęx̨ąm̨p̨l̨ę ̨t̨h̨ę ̨w̨ǫr̨k̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨.̨ ̨Ą ̨r̨ęc̨ęn̨t̨ ̨r̨ęv̨įęw̨ ̨ǫf̨ ̨B̨N̨C̨s̨ ̨w̨ąs̨ ̨w̨r̨įt̨t̨ęn̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨M̨ąx̨įm̨ųm̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ ̨G̨įv̨ęn̨ ̨d̨ąt̨ą ̨p̨ǫįn̨t̨s̨ ̨=̨ ̨{̨ ̨(̨y̨^̨(̨1̨)̨,̨^̨(̨1̨)̨)̨,̨…̨,̨ ̨(̨y̨^̨(̨N̨)̨,̨^̨(̨N̨)̨)̨ ̨}̨,̨ ̨t̨h̨ę ̨l̨ǫg̨-̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨ǫf̨ ̨ ̨įs̨:̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨∑̨_̨j̨=̨1̨^̨N̨l̨ǫg̨_̨(̨y̨^̨(̨j̨)̨,̨^̨(̨j̨)̨)̨ ̨ ̨ ̨ ̨ ̨=̨ ̨∑̨_̨j̨=̨1̨^̨N̨(̨ ̨l̨ǫg̨θ̨_̨y̨^̨(̨j̨)̨ ̨|̨ ̨Π̨_̨0̨(̨^̨(̨j̨)̨)̨ ̨+̨ ̨∑̨_̨į=̨1̨^̨n̨l̨ǫg̨θ̨_̨X̨_̨į^̨(̨j̨)̨ ̨|̨ ̨y̨^̨(̨j̨)̨,̨Π̨_̨į(̨^̨(̨j̨)̨)̨)̨,̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨w̨įt̨h̨ ̨∑̨_̨y̨ ̨∈̨𝒴̨θ̨_̨y̨|̨Π̨_̨0̨(̨)̨ ̨=̨ ̨1̨,̨ ̨ ̨ ̨ ̨ ̨ąn̨d̨ ̨∑̨_̨X̨_̨į ̨∈̨_̨įθ̨_̨X̨_̨į|̨y̨,̨Π̨_̨į(̨)̨ ̨=̨ ̨1̨.̨ ̨ ̨M̨ąx̨įm̨įz̨įn̨g̨ ̨t̨h̨ę ̨l̨ǫg̨-̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨t̨ǫ ̨ǫp̨t̨įm̨įz̨ę ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨(̨Θ̨)̨ ̨y̨įęl̨d̨s̨ ̨t̨h̨ę ̨w̨ęl̨l̨-̨k̨n̨ǫw̨n̨ ̨M̨L̨Ęs̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨s̨.̨ ̨ ̨M̨ǫs̨t̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨l̨y̨,̨ ̨M̨L̨Ęs̨ ̨f̨ąc̨t̨ǫr̨įz̨ę ̨įn̨t̨ǫ ̨įn̨d̨ęp̨ęn̨d̨ęn̨t̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę,̨ ̨ąs̨ ̨d̨ǫ ̨m̨ǫs̨t̨ ̨s̨t̨ąn̨d̨ąr̨d̨ ̨m̨ąx̨įm̨ųm̨ ̨ąp̨ǫs̨t̨ęr̨įǫr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨<̨c̨įt̨.̨>̨.̨ ̨<̨c̨įt̨.̨>̨ ̨W̨įt̨h̨įn̨ ̨t̨h̨ę ̨c̨ǫn̨s̨t̨r̨ąįn̨t̨s̨ ̨įn̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨įs̨ ̨m̨ąx̨įm̨įz̨ęd̨ ̨w̨h̨ęn̨ ̨θ̨_̨x̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨ ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨s̨ ̨t̨ǫ ̨ęm̨p̨įr̨įc̨ąl̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫf̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨d̨ąt̨ą,̨ ̨t̨h̨ąt̨ ̨įs̨,̨ ̨θ̨_̨y̨ ̨|̨ ̨Π̨_̨0̨(̨)̨ ̨=̨ ̨_̨ ̨(̨y̨ ̨|̨ ̨Π̨_̨0̨(̨)̨)̨ ̨ąn̨d̨ ̨θ̨_̨X̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨ ̨=̨ ̨_̨ ̨(̨X̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨)̨.̨ ̨ ̨T̨h̨ųs̨ ̨ǫųr̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨d̨ęc̨ǫm̨p̨ǫs̨ę ̨t̨h̨ę ̨p̨r̨ǫb̨l̨ęm̨ ̨įn̨t̨ǫ ̨s̨ęp̨ąr̨ąt̨ę ̨s̨ųb̨-̨p̨r̨ǫb̨l̨ęm̨s̨,̨ ̨ǫn̨ę ̨f̨ǫr̨ ̨ęąc̨h̨ ̨θ̨_̨X̨_̨į ̨|̨y̨,̨ ̨Π̨_̨į(̨)̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Ęf̨f̨įc̨įęn̨c̨y̨ ̨ǫf̨ ̨B̨N̨C̨ ̨l̨ęąr̨n̨įn̨g̨ ̨ ̨Ǫn̨ę ̨ǫf̨t̨ęn̨ ̨ųn̨d̨ęr̨-̨ąp̨p̨r̨ęc̨įąt̨ęd̨ ̨ąs̨p̨ęc̨t̨ ̨ǫf̨ ̨m̨ąn̨y̨ ̨B̨N̨C̨ ̨l̨ęąr̨n̨įn̨g̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨įs̨ ̨t̨h̨ęįr̨ ̨c̨ǫm̨p̨ųt̨ąt̨įǫn̨ąl̨ ̨ęf̨f̨įc̨įęn̨c̨y̨.̨ ̨M̨ąn̨y̨ ̨B̨N̨C̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨c̨ąn̨ ̨b̨ę ̨l̨ęąr̨n̨ęd̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę,̨ ̨ąv̨ǫįd̨įn̨g̨ ̨t̨h̨ę ̨ǫv̨ęr̨h̨ęąd̨s̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨r̨ęt̨ąįn̨įn̨g̨ ̨t̨h̨ę ̨t̨r̨ąįn̨įn̨g̨ ̨d̨ąt̨ą ̨įn̨ ̨m̨ęm̨ǫr̨y̨.̨ ̨ ̨ ̨N̨B̨ ̨r̨ęq̨ųįr̨ęs̨ ̨ǫn̨l̨y̨ ̨ą ̨s̨įn̨g̨l̨ę ̨p̨ąs̨s̨ ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨t̨ǫ ̨l̨ęąr̨n̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨,̨ ̨c̨ǫųn̨t̨įn̨g̨ ̨t̨h̨ę ̨j̨ǫįn̨t̨ ̨f̨r̨ęq̨ųęn̨c̨y̨ ̨ǫf̨ ̨ęąc̨h̨ ̨p̨ąįr̨ ̨ǫf̨ ̨ą ̨c̨l̨ąs̨s̨ ̨ąn̨d̨ ̨ąn̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨v̨ąl̨ųę.̨ ̨ ̨T̨ĄN̨ ̨ąn̨d̨ ̨k̨D̨B̨ ̨r̨ęq̨ųįr̨ę ̨t̨w̨ǫ ̨p̨ąs̨s̨ęs̨ ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨d̨ąt̨ą.̨ ̨T̨h̨ę ̨f̨įr̨s̨t̨ ̨c̨ǫl̨l̨ęc̨t̨s̨ ̨t̨h̨ę ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨r̨ęq̨ųįr̨ęd̨ ̨t̨ǫ ̨l̨ęąr̨n̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę,̨ ̨ąn̨d̨ ̨t̨h̨ę ̨s̨ęc̨ǫn̨d̨ ̨t̨h̨ę ̨j̨ǫįn̨t̨ ̨f̨r̨ęq̨ųęn̨c̨y̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨r̨ęq̨ųįr̨ęd̨ ̨t̨ǫ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ę ̨t̨h̨ąt̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę.̨ ̨S̨k̨D̨B̨ ̨r̨ęq̨ųįr̨ęs̨ ̨t̨h̨r̨ęę ̨p̨ąs̨s̨ęs̨ ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨d̨ąt̨ą.̨ ̨T̨h̨ę ̨f̨įr̨s̨t̨ ̨t̨w̨ǫ ̨c̨ǫl̨l̨ęc̨t̨ ̨t̨h̨ę ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨r̨ęq̨ųįr̨ęd̨ ̨t̨ǫ ̨l̨ęąr̨n̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ąn̨d̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨,̨ ̨ąs̨ ̨p̨ęr̨ ̨s̨t̨ąn̨d̨ąr̨d̨ ̨k̨D̨B̨.̨ ̨T̨h̨ę ̨t̨h̨įr̨d̨ ̨p̨ęr̨f̨ǫr̨m̨s̨ ̨ąn̨ ̨įn̨c̨r̨ęm̨ęn̨t̨ąl̨ ̨c̨r̨ǫs̨s̨ ̨v̨ąl̨įd̨ąt̨įǫn̨ ̨t̨ǫ ̨s̨ęl̨ęc̨t̨ ̨ą ̨s̨ųb̨s̨ęt̨ ̨ǫf̨ ̨t̨h̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨ąn̨d̨ ̨t̨h̨ę ̨k̨^̨*̨≤̨ ̨k̨ ̨t̨ǫ ̨b̨ę ̨ųs̨ęd̨ ̨įn̨ ̨p̨l̨ąc̨ę ̨ǫf̨ ̨k̨.̨ ̨ ̨ ̨ ̨§̨ ̨W̨H̨Y̨ ̨ĄN̨D̨ ̨H̨ǪW̨ ̨ĄR̨Ę ̨W̨Ę ̨ŲS̨ĮN̨G̨ ̨H̨D̨P̨S̨?̨ ̨ ̨T̨h̨ę ̨k̨ęy̨ ̨c̨ǫn̨t̨r̨įb̨ųt̨įǫn̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨įs̨ ̨t̨ǫ ̨ųs̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ęs̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨θ̨_̨X̨_̨į|̨Π̨_̨į(̨)̨,̨ ̨w̨h̨įc̨h̨ ̨y̨įęl̨d̨s̨ ̨b̨ąc̨k̨-̨ǫf̨f̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨h̨ąt̨ ̨n̨ąt̨ųr̨ąl̨l̨y̨ ̨s̨m̨ǫǫt̨h̨ ̨t̨h̨ę ̨ęm̨p̨įr̨įc̨ąl̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąt̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨.̨ ̨ ̨T̨h̨ę ̨įn̨t̨ųįt̨įǫn̨ ̨f̨ǫr̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨įs̨ ̨t̨h̨ąt̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨ ̨s̨h̨ǫųl̨d̨ ̨s̨h̨ąr̨ę ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ęįr̨ ̨n̨ęąr̨ ̨n̨ęįg̨h̨b̨ǫųr̨s̨.̨ ̨ ̨S̨ųp̨p̨ǫs̨ę ̨y̨ǫų ̨w̨įs̨h̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨ą ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨t̨ąb̨l̨ę ̨(̨C̨P̨T̨)̨ ̨f̨ǫr̨ ̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨3̨)̨ ̨f̨r̨ǫm̨ ̨d̨ąt̨ą ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨f̨ęąt̨ųr̨ęs̨ ̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨3̨ ̨t̨ąk̨ę ̨ǫn̨ ̨v̨ąl̨ųęs̨ ̨{̨1̨,̨2̨,̨3̨,̨4̨}̨.̨ ̨T̨h̨įs̨ ̨C̨P̨T̨ ̨c̨ąn̨ ̨b̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨ąs̨ ̨ą ̨t̨r̨ęę:̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨n̨ǫd̨ę ̨b̨r̨ąn̨c̨h̨ęs̨ ̨ǫn̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨x̨_̨1̨ ̨ąn̨d̨ ̨h̨ąs̨ ̨4̨ ̨b̨r̨ąn̨c̨h̨ęs̨,̨ ̨t̨h̨ę ̨2̨^̨n̨d̨ ̨ąn̨d̨ ̨3̨^̨r̨d̨ ̨l̨ęv̨ęl̨ ̨n̨ǫd̨ęs̨ ̨t̨ęs̨t̨ ̨x̨_̨2̨ ̨ąn̨d̨ ̨x̨_̨3̨ ̨ąn̨d̨ ̨h̨ąv̨ę ̨4̨ ̨b̨r̨ąn̨c̨h̨ęs̨.̨ ̨T̨h̨ę ̨4̨^̨t̨h̨ ̨l̨ęv̨ęl̨ ̨c̨ǫn̨s̨įs̨t̨s̨ ̨ǫf̨ ̨l̨ęąv̨ęs̨ ̨ąn̨d̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨h̨ąs̨ ̨ą ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨ ̨f̨ǫr̨ ̨y̨ ̨t̨h̨ąt̨ ̨w̨ę ̨w̨įs̨h̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę.̨ ̨T̨h̨ę ̨s̨h̨ąr̨įn̨g̨ ̨įn̨t̨ųįt̨įǫn̨ ̨s̨ąy̨s̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨l̨ęąf̨ ̨n̨ǫd̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨įn̨g̨ ̨(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨1̨)̨ ̨s̨h̨ǫųl̨d̨ ̨h̨ąv̨ę ̨s̨įm̨įl̨ąr̨ ̨v̨ąl̨ųęs̨ ̨t̨ǫ ̨t̨h̨ę ̨l̨ęąf̨ ̨f̨ǫr̨ ̨(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨2̨)̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨h̨ąv̨ę ̨ą ̨c̨ǫm̨m̨ǫn̨ ̨p̨ąr̨ęn̨t̨,̨ ̨b̨ųt̨ ̨s̨h̨ǫųl̨d̨ ̨n̨ǫt̨ ̨b̨ę ̨s̨ǫ ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ ̨(̨y̨|̨x̨_̨1̨=̨3̨,̨x̨_̨2̨=̨1̨,̨x̨_̨3̨=̨2̨)̨,̨ ̨w̨h̨įc̨h̨ ̨ǫn̨l̨y̨ ̨s̨h̨ąr̨ęs̨ ̨ą ̨g̨r̨ęąt̨ ̨g̨r̨ąn̨d̨p̨ąr̨ęn̨t̨.̨ ̨ ̨W̨ę ̨ąc̨h̨įęv̨ę ̨t̨h̨įs̨ ̨s̨h̨ąr̨įn̨g̨ ̨b̨y̨ ̨ųs̨įn̨g̨ ̨ą ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨p̨r̨įǫr̨.̨ ̨S̨ǫ ̨w̨ę ̨h̨ąv̨ę ̨v̨ęc̨t̨ǫr̨s̨ ̨(̨Y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨ų)̨ ̨(̨f̨ǫr̨ ̨ų=̨1̨,̨2̨,̨3̨,̨4̨)̨ ̨t̨h̨ąt̨ ̨ąr̨ę ̨g̨ęn̨ęr̨ąt̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨p̨r̨įǫr̨ ̨w̨įt̨h̨ ̨ą ̨c̨ǫm̨m̨ǫn̨ ̨m̨ęąn̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨,̨ ̨s̨ąy̨ ̨q̨(̨Y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨)̨.̨ ̨N̨ǫw̨ ̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨3̨)̨ ̨c̨ąn̨ ̨ǫf̨t̨ęn̨ ̨b̨ę ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ ̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨ ̨w̨h̨įc̨h̨ ̨įn̨ ̨t̨ųr̨n̨ ̨c̨ąn̨ ̨ǫf̨t̨ęn̨ ̨b̨ę ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ ̨(̨y̨|̨x̨_̨1̨)̨ ̨ąn̨d̨ ̨įn̨ ̨t̨ųr̨n̨ ̨t̨ǫ ̨(̨y̨)̨.̨ ̨H̨ǫw̨ęv̨ęr̨,̨ ̨s̨t̨r̨įc̨t̨l̨y̨ ̨s̨p̨ęąk̨įn̨g̨,̨ ̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨,̨ ̨(̨y̨|̨x̨_̨1̨)̨ ̨ąn̨d̨ ̨(̨y̨)̨ ̨ąr̨ę ̨ąg̨g̨r̨ęg̨ąt̨ę ̨v̨ąl̨ųęs̨ ̨h̨ęr̨ę ̨d̨ęr̨įv̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨ųn̨d̨ęr̨l̨y̨įn̨g̨ ̨m̨ǫd̨ęl̨ ̨w̨h̨įc̨h̨ ̨s̨p̨ęc̨įf̨įęs̨ ̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨2̨)̨.̨ ̨S̨ǫ,̨ ̨t̨ǫ ̨m̨ǫd̨ęl̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨s̨įm̨įl̨ąr̨įt̨y̨ ̨w̨įt̨h̨ ̨ą ̨H̨D̨P̨,̨ ̨įn̨s̨t̨ęąd̨ ̨ǫf̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨d̨ęr̨įv̨ęd̨ ̨(̨y̨|̨x̨_̨1̨,̨ ̨x̨_̨2̨)̨,̨ ̨(̨y̨|̨x̨_̨1̨)̨ ̨ąn̨d̨ ̨(̨y̨)̨ ̨įn̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨p̨r̨įǫr̨,̨ ̨w̨ę ̨įn̨t̨r̨ǫd̨ųc̨ę ̨s̨ǫm̨ę ̨l̨ąt̨ęn̨t̨ ̨(̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨)̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨,̨ ̨s̨ąy̨ ̨q̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨,̨ ̨q̨(̨y̨|̨x̨_̨1̨)̨ ̨ąn̨d̨ ̨q̨(̨y̨)̨.̨ ̨ ̨T̨h̨įs̨ ̨įn̨d̨ęęd̨ ̨įs̨ ̨t̨h̨ę ̨įn̨n̨ǫv̨ąt̨įǫn̨ ̨ǫf̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨ ̨Įn̨ ̨ǫųr̨ ̨c̨ąs̨ę ̨w̨ę ̨ųs̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨v̨ąr̨įąb̨l̨ęs̨ ̨ąr̨ę ̨ąl̨l̨ ̨d̨įs̨c̨r̨ęt̨ę ̨ąn̨d̨ ̨f̨įn̨įt̨ę,̨ ̨b̨ųt̨ ̨t̨h̨ę ̨ąl̨g̨ǫr̨įt̨h̨m̨ ̨r̨ęl̨įęs̨ ̨ǫn̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨f̨ǫr̨ ̨ą ̨H̨D̨P̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Įn̨t̨ųįt̨įǫn̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨f̨ǫr̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨ ̨Įm̨ąg̨įn̨ę ̨ą ̨s̨įm̨p̨l̨ę ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨s̨ųc̨h̨ ̨ąs̨ ̨įl̨l̨ųs̨t̨r̨ąt̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨ą)̨:̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨įs̨ ̨t̨h̨ę ̨s̨ǫl̨ę ̨p̨ąr̨ęn̨t̨ ̨ǫf̨ ̨ęv̨ęr̨y̨ ̨n̨ǫd̨ę ̨įn̨ ̨.̨ ̨Įn̨ ̨t̨h̨įs̨ ̨c̨ąs̨ę,̨ ̨w̨ę ̨ųs̨ę ̨ą ̨(̨n̨ǫn̨-̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨)̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨ąs̨ ̨s̨ųg̨g̨ęs̨t̨ęd̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨<̨c̨įt̨.̨>̨,̨ ̨f̨ǫr̨ ̨į=̨1̨,̨⋯̨,̨n̨ ̨ąn̨d̨ ̨ąl̨l̨ ̨y̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨θ̨_̨X̨_̨į|̨y̨∼̨ϕ̨_̨X̨_̨į,̨α̨_̨į ̨,̨ ̨ ̨w̨h̨ęr̨ę ̨α̨_̨į ̨įs̨ ̨ą ̨(̨D̨įr̨įc̨h̨l̨ęt̨)̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨n̨ǫd̨ę ̨į ̨(̨w̨ę ̨w̨įl̨l̨ ̨l̨ąt̨ęr̨ ̨d̨ęv̨ęl̨ǫp̨ ̨h̨ǫw̨ ̨w̨ę ̨t̨įę ̨t̨h̨ęs̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨įn̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨s̨ ̨įn̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨c̨ąs̨ę)̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ę ̨n̨ǫn̨-̨s̨t̨ąn̨d̨ąr̨d̨ ̨n̨ǫt̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨D̨įr̨įc̨h̨l̨ęt̨:̨ ̨f̨ǫr̨ ̨c̨ǫn̨v̨ęn̨įęn̨c̨ę ̨w̨ę ̨s̨ęp̨ąr̨ąt̨ę ̨t̨h̨ę ̨v̨ęc̨t̨ǫr̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ϕ̨_̨X̨_̨į ̨ąn̨d̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨α̨_̨į,̨ ̨m̨ąk̨įn̨g̨ ̨įt̨ ̨ą ̨2̨-̨ąr̨g̨ųm̨ęn̨t̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨.̨[̨S̨ǫm̨ę ̨p̨ąp̨ęr̨s̨ ̨w̨ǫųl̨d̨ ̨ ̨ ̨ųs̨ę ̨t̨h̨ę ̨n̨ǫt̨ąt̨įǫn̨ ̨α̨_̨įϕ̨_̨X̨_̨į ̨ǫr̨ ̨s̨ęp̨ąr̨ąt̨ę ̨t̨h̨ę ̨v̨ęc̨t̨ǫr̨ ̨ ̨ ̨(̨α̨_̨įϕ̨_̨X̨_̨į)̨ ̨įn̨t̨ǫ ̨įt̨s̨ ̨|̨X̨_̨į|̨ ̨ąr̨g̨ųm̨ęn̨t̨s̨.̨]̨ ̨ ̨ ̨W̨ę ̨c̨ąn̨ ̨t̨h̨įn̨k̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨m̨ǫd̨ęl̨ ̨įn̨ ̨t̨w̨ǫ ̨w̨ąy̨s̨:̨ ̨w̨ę ̨ąd̨d̨ ̨ą ̨b̨įąs̨ ̨t̨ǫ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨t̨h̨ąt̨ ̨ęn̨c̨ǫųr̨ąg̨ęs̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫf̨ ̨ęąc̨h̨ ̨θ̨_̨X̨_̨į|̨y̨ ̨t̨ǫ ̨h̨ąv̨ę ̨ą ̨c̨ǫm̨m̨ǫn̨ ̨m̨ęąn̨ ̨ϕ̨_̨X̨_̨į ̨f̨ǫr̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨y̨.̨ ̨Ąl̨t̨ęr̨n̨ąt̨įv̨ęl̨y̨,̨ ̨w̨ę ̨ęx̨p̨ęc̨t̨ ̨θ̨_̨X̨_̨į|̨y̨ ̨f̨ǫr̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųęs̨ ̨y̨ ̨t̨ǫ ̨b̨ę ̨s̨įm̨įl̨ąr̨.̨ ̨Įf̨ ̨t̨h̨ęy̨ ̨ąr̨ę ̨s̨įm̨įl̨ąr̨,̨ ̨įt̨ ̨įs̨ ̨n̨ąt̨ųr̨ąl̨ ̨t̨ǫ ̨t̨h̨įn̨k̨ ̨t̨h̨ąt̨ ̨t̨h̨ęy̨ ̨h̨ąv̨ę ̨ą ̨c̨ǫm̨m̨ǫn̨ ̨m̨ęąn̨,̨ ̨įn̨ ̨t̨h̨įs̨ ̨c̨ąs̨ę ̨ϕ̨_̨X̨_̨į.̨ ̨N̨ǫt̨ę,̨ ̨h̨ǫw̨ęv̨ęr̨,̨ ̨t̨h̨ąt̨ ̨ϕ̨_̨X̨_̨į ̨įs̨ ̨ą ̨ ̨p̨r̨įǫr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨,̨ ̨įn̨t̨r̨ǫd̨ųc̨ęd̨ ̨ąb̨ǫv̨ę ̨ąs̨ ̨q̨(̨·̨)̨,̨ ̨ąn̨d̨ ̨d̨ǫęs̨ ̨n̨ǫt̨ ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨ ̨t̨ǫ ̨t̨h̨ę ̨m̨ęąn̨ ̨ęs̨t̨įm̨ąt̨ęd̨ ̨b̨y̨ ̨m̨ąr̨g̨įn̨ąl̨įs̨įn̨g̨ ̨w̨įt̨h̨ ̨∑̨_̨y̨ ̨p̨̨̂(̨y̨)̨θ̨_̨X̨_̨į|̨y̨ ̨r̨ęąd̨įl̨y̨ ̨ęs̨t̨įm̨ąt̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨d̨ąt̨ą.̨ ̨T̨h̨ę ̨ϕ̨_̨X̨_̨į ̨įs̨ ̨ą ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ę ̨ąn̨d̨ ̨ą ̨B̨ąy̨ęs̨įąn̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨s̨ąm̨p̨l̨ęr̨ ̨įs̨ ̨r̨ęq̨ųįr̨ęd̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨įt̨.̨ ̨ ̨T̨h̨ę ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨ ̨α̨_̨į,̨ ̨c̨ąl̨l̨ęd̨ ̨ą ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨,̨ ̨c̨ǫn̨t̨r̨ǫl̨s̨ ̨h̨ǫw̨ ̨s̨įm̨įl̨ąr̨ ̨t̨h̨ę ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨θ̨_̨X̨_̨į|̨y̨ ̨ąn̨d̨ ̨ϕ̨_̨X̨_̨į ̨s̨h̨ǫųl̨d̨ ̨b̨ę:̨ ̨įf̨ ̨α̨_̨į ̨įs̨ ̨l̨ąr̨g̨ę,̨ ̨t̨h̨ęn̨ ̨ęąc̨h̨ ̨θ̨_̨X̨_̨į|̨y̨ ̨v̨įr̨t̨ųąl̨l̨y̨ ̨r̨ęp̨r̨ǫd̨ųc̨ęs̨ ̨ϕ̨_̨X̨_̨į;̨ ̨c̨ǫn̨v̨ęr̨s̨ęl̨y̨,̨ ̨θ̨_̨X̨_̨į|̨y̨ ̨c̨ąn̨ ̨v̨ąr̨y̨ ̨m̨ǫr̨ę ̨f̨r̨ęęl̨y̨ ̨ąs̨ ̨α̨_̨į ̨t̨ęn̨d̨s̨ ̨t̨ǫ ̨0̨.̨ ̨Ęs̨t̨įm̨ąt̨įǫn̨ ̨ąl̨s̨ǫ ̨įn̨v̨ǫl̨v̨ęs̨ ̨ęs̨t̨įm̨ąt̨įn̨g̨ ̨t̨h̨ę ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨s̨,̨ ̨ąs̨ ̨d̨įs̨c̨ųs̨s̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Įn̨t̨ųįt̨įǫn̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨f̨ǫr̨ ̨ ̨ ̨Ąs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨k̨D̨B̨-̨1̨ ̨r̨ęl̨ąx̨ęs̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨'̨ ̨ąs̨s̨ųm̨p̨t̨įǫn̨ ̨ąb̨ǫųt̨ ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨įn̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨(̨g̨įv̨ęn̨ ̨y̨)̨ ̨b̨ęt̨w̨ęęn̨ ̨t̨h̨ę ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨b̨y̨ ̨ąl̨l̨ǫw̨įn̨g̨ ̨ǫn̨ę ̨ęx̨t̨r̨ą-̨p̨ąr̨ęn̨t̨ ̨p̨ęr̨ ̨n̨ǫd̨ę ̨ ̨ąs̨ ̨p̨r̨ęs̨ęn̨t̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨.̨ ̨T̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨s̨t̨ąr̨t̨s̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨N̨B̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę.̨ ̨T̨h̨ęn̨ ̨įt̨ ̨ǫr̨d̨ęr̨s̨ ̨t̨h̨ę ̨n̨ǫd̨ęs̨ ̨b̨y̨ ̨h̨įg̨h̨ęs̨t̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨t̨ǫ ̨b̨ę ̨r̨ąn̨k̨ęd̨ ̨f̨įr̨s̨t̨,̨ ̨⟨̨ ̨x̨_̨2̨,̨x̨_̨4̨,̨x̨_̨1̨,̨x̨_̨3̨⟩̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨.̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨įt̨ ̨c̨ǫn̨s̨įd̨ęr̨s̨ ̨ąl̨l̨ ̨c̨ąn̨d̨įd̨ąt̨ę ̨p̨ąr̨ęn̨t̨s̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ęr̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨t̨h̨ąn̨ ̨įt̨s̨ęl̨f̨ ̨(̨b̨ęf̨ǫr̨ę ̨įn̨ ̨t̨h̨ę ̨ǫr̨d̨ęr̨)̨,̨ ̨ąn̨d̨ ̨c̨h̨ǫǫs̨ęs̨ ̨t̨h̨ę ̨ǫn̨ę ̨t̨h̨ąt̨ ̨ǫf̨f̨ęr̨s̨ ̨t̨h̨ę ̨h̨įg̨h̨ęs̨t̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨b̨ęt̨w̨ęęn̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨ąn̨d̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨n̨ǫd̨ę ̨w̨h̨ęn̨ ̨c̨ǫn̨d̨įt̨įǫn̨ęd̨ ̨ǫn̨ ̨įt̨.̨ ̨ ̨W̨ę ̨k̨ęęp̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨įd̨ęą ̨f̨ǫr̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨θ̨_̨X̨_̨į|̨Π̨_̨į(̨)̨ ̨ąs̨ ̨įn̨ ̨t̨h̨ę ̨N̨B̨ ̨c̨ąs̨ę,̨ ̨(̨<̨r̨ęf̨>̨)̨,̨ ̨ęx̨c̨ęp̨t̨ ̨t̨h̨ąt̨ ̨n̨ǫw̨ ̨X̨_̨į ̨h̨ąs̨ ̨2̨ ̨p̨ąr̨ęn̨t̨s̨:̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨ąn̨d̨ ̨ąn̨ǫt̨h̨ęr̨ ̨c̨ǫv̨ąr̨įąt̨ę.̨ ̨T̨h̨įs̨ ̨t̨r̨ąn̨s̨l̨ąt̨ęs̨ ̨įn̨t̨ǫ ̨t̨h̨ę ̨f̨ǫl̨l̨ǫw̨įn̨g̨,̨ ̨f̨ǫr̨ ̨į=̨1̨,̨⋯̨,̨n̨ ̨ąn̨d̨ ̨ąl̨l̨ ̨y̨,̨Π̨(̨į)̨ ̨ ̨ ̨ ̨ ̨θ̨_̨X̨_̨į|̨y̨,̨Π̨(̨į)̨∼̨ϕ̨_̨X̨_̨į|̨y̨,̨α̨_̨į|̨y̨ ̨,̨ ̨ ̨w̨h̨ęr̨ę ̨Π̨(̨į)̨ ̨ǫn̨l̨y̨ ̨c̨ǫm̨p̨r̨įs̨ęs̨ ̨ą ̨s̨įn̨g̨l̨ę ̨n̨ǫd̨ę ̨f̨ǫr̨ ̨ąl̨l̨ ̨į>̨1̨ ̨(̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨n̨ǫd̨ę ̨h̨ąs̨ ̨ǫn̨l̨y̨ ̨y̨ ̨ąs̨ ̨ą ̨p̨ąr̨ęn̨t̨)̨.̨ ̨N̨ǫw̨ ̨w̨ę ̨c̨ǫųl̨d̨ ̨h̨ąv̨ę ̨ųs̨ęd̨ ̨ϕ̨_̨X̨_̨į ̨ąs̨ ̨t̨h̨ę ̨l̨ąt̨ęn̨t̨ ̨p̨ąr̨ęn̨t̨,̨ ̨s̨ǫ ̨įt̨ ̨įs̨ ̨įn̨d̨ęp̨ęn̨d̨ęn̨t̨ ̨ǫf̨ ̨y̨,̨ ̨b̨ųt̨ ̨t̨h̨įs̨ ̨w̨ǫųl̨d̨ ̨m̨ęąn̨ ̨ąl̨l̨ ̨l̨ęąv̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨h̨ąv̨ę ̨s̨įm̨įl̨ąr̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨s̨.̨ ̨T̨h̨įs̨ ̨įs̨ ̨ą ̨s̨t̨r̨ǫn̨g̨ęr̨ ̨s̨t̨ąt̨ęm̨ęn̨t̨ ̨t̨h̨ąn̨ ̨w̨ę ̨n̨ęęd̨;̨ ̨ ̨r̨ąt̨h̨ęr̨ ̨w̨ę ̨p̨r̨ęf̨ęr̨ ̨ąd̨j̨ąc̨ęn̨t̨ ̨n̨ǫd̨ęs̨ ̨ǫn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨t̨ǫ ̨b̨ę ̨s̨įm̨įl̨ąr̨,̨ ̨n̨ǫt̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨.̨ ̨W̨įt̨h̨ ̨ą ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨m̨ǫd̨ęl̨ ̨w̨ę ̨ąd̨d̨ ̨ąn̨ǫt̨h̨ęr̨ ̨l̨ęv̨ęl̨ ̨ǫf̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨,̨ ̨m̨ąk̨įn̨g̨ ̨t̨h̨ę ̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨ǫn̨ ̨y̨ ̨ąn̨d̨ ̨r̨ęq̨ųįr̨ę ̨ą ̨f̨ųr̨t̨h̨ęr̨ ̨p̨ąr̨ęn̨t̨ ̨ąb̨ǫv̨ę ̨f̨ǫr̨ ̨į=̨1̨,̨⋯̨,̨n̨ ̨ąn̨d̨ ̨ąl̨l̨ ̨y̨ ̨ ̨ ̨ ̨ ̨ϕ̨_̨X̨_̨į|̨y̨∼̨ϕ̨_̨X̨_̨į,̨α̨_̨į|̨1̨ ̨.̨ ̨ ̨T̨h̨įs̨ ̨m̨ęąn̨s̨ ̨t̨h̨ąt̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨b̨r̨ąn̨c̨h̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨c̨ąn̨ ̨h̨ąv̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨m̨ęąn̨s̨,̨ ̨ąn̨d̨ ̨t̨h̨ųs̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨ ̨įs̨ ̨m̨ǫr̨ę ̨f̨l̨ęx̨įb̨l̨ę ̨(̨ąn̨d̨ ̨h̨ąs̨ ̨h̨ęn̨c̨ę ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨l̨ǫw̨ ̨b̨įąs̨)̨.̨ ̨ ̨Ǫųr̨ ̨B̨ąy̨ęs̨įąn̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨h̨ąn̨d̨l̨ęs̨ ̨t̨h̨ęs̨ę ̨ąd̨d̨įt̨įǫn̨ąl̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąn̨d̨ ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąn̨d̨ ̨l̨įm̨įt̨s̨ ̨t̨h̨ę ̨ęf̨f̨ęc̨t̨ ̨ǫf̨ ̨v̨ąr̨įąn̨c̨ę ̨ǫn̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨.̨ ̨ ̨ ̨T̨h̨ę ̨m̨ǫd̨ęl̨ ̨n̨ąt̨ųr̨ąl̨l̨y̨ ̨d̨ęf̨įn̨ęs̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨g̨įv̨ęn̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨,̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ųl̨ą ̨ąb̨ǫv̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨b̨y̨ ̨t̨h̨ę ̨g̨r̨ąp̨h̨įc̨ąl̨ ̨m̨ǫd̨ęl̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨ą)̨.̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨p̨l̨ąt̨ę/̨.̨s̨t̨y̨l̨ę=̨,̨s̨c̨ąl̨ę=̨.̨8̨ ̨ ̨ ̨ ̨ ̨ ̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨į)̨ ̨ąt̨ ̨(̨0̨,̨0̨)̨ ̨ϕ̨_̨X̨_̨į;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨įg̨y̨)̨ ̨ąt̨ ̨(̨0̨,̨-̨2̨)̨ ̨ϕ̨_̨X̨_̨į|̨y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨įg̨y̨p̨1̨)̨ ̨ąt̨ ̨(̨0̨,̨-̨4̨.̨3̨)̨ ̨θ̨_̨X̨_̨į|̨y̨,̨Π̨(̨į)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ą0̨)̨ ̨ąt̨ ̨(̨1̨.̨5̨,̨0̨)̨ ̨α̨_̨į|̨0̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ąį)̨ ̨ąt̨ ̨(̨2̨.̨5̨,̨0̨)̨ ̨α̨_̨į|̨1̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ąįg̨y̨)̨ ̨ąt̨ ̨(̨2̨,̨-̨4̨.̨3̨)̨ ̨α̨_̨į|̨y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨į)̨ ̨–̨ ̨(̨x̨įg̨y̨)̨;̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨)̨ ̨–̨ ̨(̨x̨įg̨y̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ąį)̨ ̨–̨ ̨(̨x̨įg̨y̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ą0̨)̨ ̨–̨ ̨(̨x̨į)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ąįg̨y̨)̨ ̨–̨ ̨(̨x̨įg̨y̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨ ̨s̨h̨ąp̨ę=̨r̨ęc̨t̨ąn̨g̨l̨ę,̨ ̨r̨ǫųn̨d̨ęd̨ ̨c̨ǫr̨n̨ęr̨s̨=̨0̨.̨5̨ęx̨,̨ ̨t̨h̨įc̨k̨,̨ąl̨įg̨n̨=̨r̨įg̨h̨t̨,̨ ̨įn̨n̨ęr̨ ̨s̨ęp̨=̨5̨p̨t̨,̨ ̨įn̨n̨ęr̨ ̨y̨s̨ęp̨=̨1̨3̨p̨t̨,̨l̨ąb̨ęl̨=̨[̨x̨s̨h̨įf̨t̨=̨-̨3̨0̨p̨t̨,̨y̨s̨h̨įf̨t̨=̨1̨4̨p̨t̨]̨s̨ǫųt̨h̨ ̨ęąs̨t̨:̨|̨Π̨(̨į)̨|̨,̨ ̨f̨įt̨=̨(̨x̨įg̨y̨p̨1̨)̨]̨(̨p̨l̨ąt̨ę1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨ ̨s̨h̨ąp̨ę=̨r̨ęc̨t̨ąn̨g̨l̨ę,̨ ̨r̨ǫųn̨d̨ęd̨ ̨c̨ǫr̨n̨ęr̨s̨=̨0̨.̨5̨ęx̨,̨ ̨t̨h̨įc̨k̨,̨ąl̨įg̨n̨=̨r̨įg̨h̨t̨,̨ ̨įn̨n̨ęr̨ ̨s̨ęp̨=̨8̨p̨t̨,̨ ̨įn̨n̨ęr̨ ̨y̨s̨ęp̨=̨1̨6̨p̨t̨,̨l̨ąb̨ęl̨=̨[̨x̨s̨h̨įf̨t̨=̨-̨2̨0̨p̨t̨,̨y̨s̨h̨įf̨t̨=̨-̨1̨7̨p̨t̨]̨n̨ǫr̨t̨h̨ ̨ęąs̨t̨:̨|̨Y̨|̨,̨ ̨f̨įt̨=̨(̨x̨įg̨y̨p̨1̨)̨ ̨(̨x̨įg̨y̨)̨ ̨(̨ąl̨p̨h̨ąįg̨y̨)̨]̨(̨p̨l̨ąt̨ę2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨p̨l̨ąt̨ę/̨.̨s̨t̨y̨l̨ę=̨,̨s̨c̨ąl̨ę=̨.̨8̨ ̨ ̨ ̨ ̨ ̨ ̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨į)̨ ̨ąt̨ ̨(̨0̨,̨0̨)̨ ̨ϕ̨_̨X̨_̨į;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ą0̨)̨ ̨ąt̨ ̨(̨2̨,̨0̨)̨ ̨α̨_̨į|̨0̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨įg̨y̨)̨ ̨ąt̨ ̨(̨0̨,̨-̨2̨)̨ ̨ϕ̨_̨X̨_̨į|̨y̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨x̨įg̨y̨p̨1̨)̨ ̨ąt̨ ̨(̨0̨,̨-̨4̨.̨3̨)̨ ̨θ̨_̨X̨_̨į|̨y̨,̨Π̨(̨į)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ą1̨)̨ ̨ąt̨ ̨(̨2̨,̨-̨2̨)̨ ̨α̨_̨į|̨1̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨c̨įr̨c̨l̨ę]̨ ̨(̨ąl̨p̨h̨ą2̨)̨ ̨ąt̨ ̨(̨2̨,̨-̨4̨.̨3̨)̨ ̨α̨_̨į|̨2̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨į)̨ ̨–̨ ̨(̨x̨įg̨y̨)̨;̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨)̨ ̨–̨ ̨(̨x̨įg̨y̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ą1̨)̨ ̨–̨ ̨(̨x̨įg̨y̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ą0̨)̨ ̨–̨ ̨(̨x̨į)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨ąl̨p̨h̨ą2̨)̨ ̨–̨ ̨(̨x̨įg̨y̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨ ̨s̨h̨ąp̨ę=̨r̨ęc̨t̨ąn̨g̨l̨ę,̨ ̨r̨ǫųn̨d̨ęd̨ ̨c̨ǫr̨n̨ęr̨s̨=̨0̨.̨5̨ęx̨,̨ ̨t̨h̨įc̨k̨,̨ąl̨įg̨n̨=̨r̨įg̨h̨t̨,̨ ̨įn̨n̨ęr̨ ̨s̨ęp̨=̨5̨p̨t̨,̨ ̨įn̨n̨ęr̨ ̨y̨s̨ęp̨=̨1̨3̨p̨t̨,̨l̨ąb̨ęl̨=̨[̨x̨s̨h̨įf̨t̨=̨-̨3̨0̨p̨t̨,̨y̨s̨h̨įf̨t̨=̨1̨4̨p̨t̨]̨s̨ǫųt̨h̨ ̨ęąs̨t̨:̨|̨Π̨(̨į)̨|̨,̨ ̨f̨įt̨=̨(̨x̨įg̨y̨p̨1̨)̨]̨(̨p̨l̨ąt̨ę1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨ ̨s̨h̨ąp̨ę=̨r̨ęc̨t̨ąn̨g̨l̨ę,̨ ̨r̨ǫųn̨d̨ęd̨ ̨c̨ǫr̨n̨ęr̨s̨=̨0̨.̨5̨ęx̨,̨ ̨t̨h̨įc̨k̨,̨ąl̨įg̨n̨=̨r̨įg̨h̨t̨,̨ ̨įn̨n̨ęr̨ ̨s̨ęp̨=̨8̨p̨t̨,̨ ̨įn̨n̨ęr̨ ̨y̨s̨ęp̨=̨1̨6̨p̨t̨,̨l̨ąb̨ęl̨=̨[̨x̨s̨h̨įf̨t̨=̨-̨2̨0̨p̨t̨,̨y̨s̨h̨įf̨t̨=̨-̨1̨7̨p̨t̨]̨n̨ǫr̨t̨h̨ ̨ęąs̨t̨:̨|̨Y̨|̨,̨ ̨f̨įt̨=̨(̨x̨įg̨y̨p̨1̨)̨ ̨(̨x̨įg̨y̨)̨]̨(̨p̨l̨ąt̨ę2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨Ǫųr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨m̨ǫd̨ęl̨ ̨f̨ǫr̨ ̨ǫn̨ę ̨X̨_̨į ̨ąn̨d̨ ̨k̨D̨B̨-̨1̨.̨ ̨(̨ą)̨ ̨T̨y̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨ąt̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨.̨ ̨(̨b̨)̨ ̨T̨y̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨ąt̨ ̨t̨h̨ę ̨l̨ęv̨ęl̨.̨ ̨D̨ęt̨ąįl̨s̨ ̨ǫn̨ ̨t̨y̨įn̨g̨ ̨ąr̨ę ̨g̨įv̨ęn̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Įn̨t̨ųįt̨įǫn̨ ̨–̨ ̨g̨ęn̨ęr̨ąl̨ ̨f̨r̨ąm̨ęw̨ǫr̨k̨ ̨ ̨T̨h̨ę ̨įn̨t̨ųįt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨f̨r̨ąm̨ęw̨ǫr̨k̨ ̨f̨ǫr̨ ̨k̨D̨B̨-̨1̨ ̨n̨ąt̨ųr̨ąl̨l̨y̨ ̨ęx̨t̨ęn̨d̨s̨ ̨t̨ǫ ̨B̨N̨s̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ęr̨ ̨n̨ųm̨b̨ęr̨s̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨.̨ ̨ ̨W̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ǫf̨ ̨ęąc̨h̨ ̨f̨ąc̨t̨ǫr̨ ̨“̨c̨h̨įl̨d̨ ̨g̨įv̨ęn̨ ̨p̨ąr̨ęn̨t̨s̨”̨ ̨t̨ǫ ̨h̨ąv̨ę ̨ą ̨h̨įęr̨ąr̨c̨h̨y̨ ̨w̨įt̨h̨ ̨ąs̨ ̨m̨ąn̨y̨ ̨l̨ęv̨ęl̨s̨ ̨ąs̨ ̨t̨h̨ę ̨n̨ǫd̨ę ̨h̨ąs̨ ̨p̨ąr̨ęn̨t̨s̨.̨ ̨ ̨Ąt̨ ̨ęąc̨h̨ ̨l̨ęv̨ęl̨,̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨y̨ ̨b̨r̨ąn̨c̨h̨ęs̨ ̨ǫn̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųęs̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨n̨ęw̨l̨y̨ ̨įn̨t̨r̨ǫd̨ųc̨ęd̨ ̨p̨ąr̨ęn̨t̨ ̨t̨ąk̨ęs̨:̨ ̨ǫn̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨y̨ ̨ąt̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨l̨ęv̨ęl̨,̨ ̨ǫn̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨p̨ąr̨ęn̨t̨ ̨ąt̨ ̨t̨h̨ę ̨s̨ęc̨ǫn̨d̨ ̨l̨ęv̨ęl̨,̨ ̨ęt̨c̨.̨ ̨ ̨Ǫn̨c̨ę ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨įs̨ ̨s̨ęt̨,̨ ̨ąl̨l̨ ̨w̨ę ̨n̨ęęd̨ ̨įs̨ ̨t̨ǫ ̨h̨ąv̨ę ̨ąn̨ ̨ǫr̨d̨ęr̨ ̨b̨ęt̨w̨ęęn̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨s̨.̨ ̨ ̨F̨ǫr̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨,̨ ̨t̨h̨ęr̨ę ̨įs̨ ̨ǫn̨l̨y̨ ̨ǫn̨ę ̨p̨ąr̨ęn̨t̨ ̨–̨y̨.̨ ̨ ̨F̨ǫr̨ ̨t̨r̨ęę-̨ąųg̨m̨ęn̨t̨ęd̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨(̨T̨ĄN̨)̨,̨ ̨ąs̨ ̨n̨ǫd̨ęs̨ ̨c̨ąn̨n̨ǫt̨ ̨h̨ąv̨ę ̨m̨ǫr̨ę ̨t̨h̨ąn̨ ̨ą ̨s̨įn̨g̨l̨ę ̨p̨ąr̨ęn̨t̨ ̨ąp̨ąr̨t̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨,̨ ̨w̨ę ̨p̨l̨ąc̨ę ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨f̨įr̨s̨t̨ ̨ąn̨d̨ ̨įt̨s̨ ̨ǫt̨h̨ęr̨ ̨p̨ąr̨ęn̨t̨ ̨s̨ęc̨ǫn̨d̨.̨ ̨ ̨F̨ǫr̨ ̨ąl̨l̨ ̨ǫt̨h̨ęr̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨,̨ ̨w̨ę ̨p̨l̨ąc̨ę ̨y̨ ̨ąs̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨p̨ąr̨ęn̨t̨ ̨ąn̨d̨ ̨t̨h̨ęn̨ ̨ǫr̨d̨ęr̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨s̨ ̨Π̨_̨į ̨b̨y̨ ̨h̨įg̨h̨ęs̨t̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨b̨ęt̨w̨ęęn̨ ̨t̨h̨ęm̨ ̨ąn̨d̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨c̨ǫn̨d̨įt̨įǫn̨ęd̨ ̨ǫn̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨.̨ ̨ ̨T̨h̨įs̨ ̨f̨ǫl̨l̨ǫw̨s̨ ̨b̨ǫt̨h̨ ̨t̨h̨ę ̨N̨L̨P̨ ̨f̨r̨ąm̨ęw̨ǫr̨k̨ ̨f̨ǫr̨ ̨n̨-̨g̨r̨ąm̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ąn̨d̨ ̨k̨D̨B̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨l̨ęąr̨n̨įn̨g̨:̨ ̨ ̨p̨ǫs̨įt̨įǫn̨ ̨f̨įr̨s̨t̨ ̨įn̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨y̨ ̨t̨h̨ę ̨n̨ǫd̨ęs̨ ̨t̨h̨ąt̨ ̨ąr̨ę ̨m̨ǫs̨t̨ ̨l̨įk̨ęl̨y̨ ̨t̨ǫ ̨h̨ąv̨ę ̨ąn̨ ̨įn̨f̨l̨ųęn̨c̨ę ̨ǫn̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ę.̨ ̨ ̨P̨ǫs̨įt̨įǫn̨įn̨g̨ ̨t̨h̨ę ̨c̨l̨ąs̨s̨ ̨f̨įr̨s̨t̨ ̨ąl̨l̨ǫw̨s̨ ̨ųs̨ ̨t̨ǫ ̨p̨ųl̨l̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨ǫ ̨b̨ę ̨m̨ǫs̨t̨ ̨ąc̨c̨ųr̨ąt̨ę ̨įn̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨s̨p̨ąc̨ę ̨t̨h̨ąt̨ ̨įs̨ ̨n̨ęąr̨ ̨(̨y̨|̨)̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ǫųr̨ ̨f̨įn̨ąl̨ ̨t̨ąr̨g̨ęt̨ ̨f̨ǫr̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨,̨ ̨ąs̨ ̨w̨ę ̨ąr̨ę ̨n̨ǫt̨ ̨r̨ęąl̨l̨y̨ ̨įn̨t̨ęr̨ęs̨t̨ęd̨ ̨įn̨ ̨ǫb̨t̨ąįn̨įn̨g̨ ̨ąc̨c̨ųr̨ąt̨ę ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫf̨ ̨(̨X̨_̨į|̨y̨,̨Π̨(̨į)̨)̨ ̨įn̨ ̨p̨ąr̨t̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨s̨p̨ąc̨ę ̨t̨h̨ąt̨ ̨ąr̨ę ̨ųn̨r̨ęl̨ąt̨ęd̨ ̨t̨ǫ ̨y̨.̨ ̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨t̨h̨ę ̨l̨ąt̨ęn̨t̨/̨p̨r̨įǫr̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨s̨ ̨ ̨ϕ̨_̨X̨_̨į|̨y̨,̨Π̨_̨į(̨)̨ ̨d̨ǫ ̨n̨ǫt̨ ̨m̨ǫd̨ęl̨ ̨ǫb̨s̨ęr̨v̨ęd̨ ̨d̨ąt̨ą,̨ ̨ąs̨ ̨t̨h̨ę ̨ ̨θ̨_̨X̨_̨į|̨y̨,̨Π̨_̨į(̨)̨ ̨d̨ǫ.̨ ̨W̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨h̨ęm̨ ̨w̨įt̨h̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨s̨y̨m̨b̨ǫl̨s̨ ̨(̨ϕ̨ ̨v̨ęr̨s̨ųs̨ ̨θ̨)̨ ̨t̨ǫ ̨h̨įg̨h̨l̨įg̨h̨t̨ ̨t̨h̨įs̨ ̨f̨ųn̨d̨ąm̨ęn̨t̨ąl̨ ̨d̨įf̨f̨ęr̨ęn̨c̨ę.̨ ̨ ̨ ̨ ̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨n̨ǫt̨ę ̨t̨h̨ąt̨ ̨įn̨ ̨t̨h̨ę ̨f̨įn̨įt̨ę ̨d̨įs̨c̨r̨ęt̨ę ̨c̨ǫn̨t̨ęx̨t̨,̨ ̨D̨P̨s̨ ̨ąr̨ę ̨ęq̨ųįv̨ąl̨ęn̨t̨ ̨t̨ǫ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨<̨c̨įt̨.̨>̨,̨ ̨s̨ǫ ̨w̨ę ̨p̨r̨ęs̨ęn̨t̨ ̨ǫųr̨ ̨m̨ǫd̨ęl̨s̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨D̨įr̨įc̨h̨l̨ęt̨s̨,̨ ̨b̨ųt̨ ̨t̨h̨ę ̨įn̨f̨ęr̨ęn̨c̨ę ̨įs̨ ̨d̨ǫn̨ę ̨ęf̨f̨įc̨įęn̨t̨l̨y̨ ̨ųs̨įn̨g̨ ̨ą ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨ ̨H̨D̨P̨s̨ ̨<̨c̨įt̨.̨>̨.̨ ̨T̨h̨ęs̨ę ̨r̨ęc̨ęn̨t̨ ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨B̨ąy̨ęs̨įąn̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨ąr̨ę ̨c̨ǫn̨s̨įd̨ęr̨ąb̨l̨y̨ ̨m̨ǫr̨ę ̨ęf̨f̨įc̨įęn̨t̨ ̨ąn̨d̨ ̨ąc̨c̨ųr̨ąt̨ę ̨ąn̨d̨ ̨s̨ǫ ̨d̨ǫ ̨n̨ǫt̨ ̨s̨ųf̨f̨ęr̨ ̨t̨h̨ę ̨w̨ęl̨l̨-̨k̨n̨ǫw̨n̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨įc̨ ̨įs̨s̨ųęs̨ ̨ǫf̨ ̨ǫr̨įg̨įn̨ąl̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨<̨c̨įt̨.̨>̨.̨ ̨N̨ǫt̨ę ̨h̨ǫw̨ęv̨ęr̨ ̨t̨h̨ąt̨,̨ ̨ųn̨l̨įk̨ę ̨s̨ǫm̨ę ̨ąp̨p̨l̨įc̨ąt̨įǫn̨s̨ ̨ǫf̨ ̨H̨D̨P̨s̨,̨ ̨t̨h̨ęr̨ę ̨ąr̨ę ̨n̨ǫ ̨`̨ąt̨ǫm̨s̨'̨ ̨g̨ęn̨ęr̨ąt̨ęd̨ ̨ąt̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨ǫf̨ ̨t̨h̨ę ̨H̨D̨P̨ ̨h̨įęr̨ąr̨c̨h̨y̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨r̨ǫǫt̨ ̨įs̨ ̨j̨ųs̨t̨ ̨ą ̨D̨įr̨įc̨h̨l̨ęt̨,̨ ̨w̨h̨įc̨h̨ ̨ęf̨f̨ęc̨t̨įv̨ęl̨y̨ ̨h̨ąs̨ ̨t̨h̨ę ̨f̨įn̨įt̨ę ̨d̨įs̨c̨r̨ęt̨ę ̨s̨ęt̨ ̨ǫf̨ ̨ąt̨ǫm̨s̨ ̨ąl̨r̨ęąd̨y̨ ̨p̨r̨ęs̨ęn̨t̨.̨ ̨T̨h̨ę ̨H̨D̨P̨ ̨f̨ǫr̨m̨ąl̨įs̨m̨ ̨įs̨ ̨ųs̨ęd̨ ̨t̨ǫ ̨p̨r̨ǫv̨įd̨ę ̨ąn̨ ̨ęf̨f̨įc̨įęn̨t̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨ ̨ąs̨ ̨ą ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨v̨ęr̨s̨įǫn̨ ̨ǫf̨ ̨ą ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨ęr̨.̨ ̨ ̨ ̨ ̨§̨ ̨ǪŲR̨ ̨F̨R̨ĄM̨ĘW̨ǪR̨K̨:̨ ̨H̨D̨P̨S̨ ̨F̨ǪR̨ ̨B̨N̨C̨S̨ ̨ ̨T̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨ ̨r̨ęv̨įęw̨s̨ ̨ǫųr̨ ̨m̨ǫd̨ęl̨ ̨ąn̨d̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ąp̨p̨r̨ǫąc̨h̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨M̨ǫd̨ęl̨ ̨C̨ǫn̨s̨įd̨ęr̨ ̨t̨h̨ę ̨c̨ąs̨ę ̨ǫf̨ ̨ęs̨t̨įm̨ąt̨įn̨g̨ ̨(̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨)̨ ̨w̨h̨ęr̨ę ̨X̨_̨c̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨v̨ąr̨įąb̨l̨ę ̨ǫf̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨ąr̨ę ̨t̨r̨y̨įn̨g̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨,̨ ̨ąn̨d̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ąr̨ę ̨r̨ęs̨p̨ęc̨t̨įv̨ęl̨y̨ ̨ųs̨ęd̨ ̨t̨ǫ ̨d̨ęn̨ǫt̨ę ̨t̨h̨ę ̨v̨ąr̨įąb̨l̨ę ̨v̨ąl̨ųęs̨ ̨Y̨=̨y̨,̨X̨_̨1̨=̨v̨_̨1̨,̨⋯̨,̨X̨_̨n̨=̨v̨_̨n̨ ̨.̨ ̨T̨h̨ę ̨v̨ąr̨įąb̨l̨ęs̨ ̨X̨_̨1̨,̨⋯̨,̨X̨_̨n̨ ̨f̨ǫr̨ ̨n̨≥̨ ̨0̨ ̨ąr̨ę ̨ǫr̨d̨ęr̨ęd̨ ̨b̨y̨ ̨m̨ųt̨ųąl̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨X̨_̨c̨ ̨ąs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨p̨r̨ęv̨įǫųs̨l̨y̨.̨ ̨L̨ąt̨ęr̨,̨ ̨w̨ę ̨w̨įl̨l̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨X̨_̨c̨ ̨w̨įl̨l̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨v̨ąr̨įąb̨l̨ę ̨įn̨ ̨t̨h̨ę ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨ǫf̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨w̨ąn̨t̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨g̨įv̨ęn̨ ̨įt̨s̨ ̨p̨ąr̨ęn̨t̨s̨ ̨v̨ąl̨ųęs̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨.̨ ̨ ̨W̨ę ̨c̨ąn̨ ̨p̨r̨ęs̨ęn̨t̨ ̨t̨h̨įs̨ ̨ąs̨ ̨ą ̨d̨ęc̨įs̨įǫn̨ ̨t̨r̨ęę ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨r̨ǫǫt̨ ̨n̨ǫd̨ę ̨b̨ąn̨c̨h̨ęs̨ ̨ǫn̨ ̨y̨ ̨(̨ǫn̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨Y̨)̨,̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨ąt̨ ̨t̨h̨ę ̨1̨^̨s̨t̨ ̨l̨ęv̨ęl̨ ̨b̨r̨ąn̨c̨h̨ ̨ǫn̨ ̨x̨_̨1̨ ̨(̨ǫn̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨t̨ąk̨ęn̨ ̨b̨y̨ ̨X̨_̨1̨)̨,̨ ̨ąt̨ ̨t̨h̨ę ̨2̨^̨n̨d̨ ̨l̨ęv̨ęl̨ ̨t̨ęs̨t̨ ̨x̨_̨2̨ ̨ąn̨d̨ ̨s̨ǫ ̨f̨ǫr̨t̨h̨.̨ ̨Ą ̨n̨ǫd̨ę ̨ąt̨ ̨t̨h̨ę ̨l̨ęąf̨ ̨(̨t̨h̨ę ̨n̨+̨1̨-̨t̨h̨ ̨l̨ęv̨ęl̨)̨ ̨h̨ąs̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨v̨ęc̨t̨ǫr̨ ̨θ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨f̨ǫr̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨g̨įv̨ęn̨ ̨b̨y̨ ̨įt̨s̨ ̨b̨r̨ąn̨c̨h̨ ̨ǫn̨ ̨t̨h̨ę ̨t̨r̨ęę.̨ ̨Ą ̨n̨ǫd̨ę ̨ąt̨ ̨t̨h̨ę ̨į-̨t̨h̨ ̨l̨ęv̨ęl̨ ̨(̨f̨ǫr̨ ̨į=̨1̨,̨…̨,̨n̨)̨ ̨h̨ąs̨ ̨ą ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į–̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ą ̨l̨ąt̨ęn̨t̨ ̨p̨r̨įǫr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨–̨ ̨w̨h̨ęr̨ę ̨ąg̨ąįn̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ąr̨ę ̨g̨įv̨ęn̨ ̨b̨y̨ ̨įt̨s̨ ̨b̨r̨ąn̨c̨h̨ ̨ǫn̨ ̨t̨h̨ę ̨t̨r̨ęę.̨ ̨T̨h̨ę ̨f̨ųl̨l̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨m̨ǫd̨ęl̨ ̨įs̨ ̨g̨įv̨ęn̨ ̨b̨y̨ ̨ ̨ ̨ ̨ ̨ ̨θ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨∼̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨-̨1̨,̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨∼̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨,̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨į=̨1̨,̨…̨,̨n̨-̨1̨ ̨ ̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨ ̨ ̨ ̨∼̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨,̨α̨_̨y̨ ̨ ̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨ ̨ ̨ ̨∼̨ ̨ ̨ ̨1̨/̨|̨X̨_̨c̨|̨1̨̨⃗,̨α̨_̨0̨ ̨.̨ ̨N̨ǫt̨ę ̨ęąc̨h̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨h̨ąs̨ ̨ą ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ąs̨ ̨ą ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨,̨ ̨ąn̨d̨ ̨d̨ęn̨ǫt̨ę ̨t̨h̨ę ̨f̨ųl̨l̨ ̨s̨ęt̨ ̨ǫf̨ ̨t̨h̨ęs̨ę ̨b̨y̨ ̨α̨_̨*̨.̨ ̨T̨h̨ęs̨ę ̨ąr̨ę ̨k̨n̨ǫw̨n̨ ̨t̨ǫ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨c̨h̨ąn̨g̨ę ̨t̨h̨ę ̨c̨h̨ąr̨ąc̨t̨ęr̨įs̨t̨įc̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨,̨ ̨s̨ǫ ̨t̨h̨ęy̨ ̨m̨ųs̨t̨ ̨b̨ę ̨ęs̨t̨įm̨ąt̨ęd̨ ̨ąs̨ ̨w̨ęl̨l̨.̨ ̨W̨ę ̨d̨įs̨c̨ųs̨s̨ ̨b̨ęl̨ǫw̨,̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨h̨ǫw̨ ̨w̨ę ̨c̨ąn̨ ̨t̨įę ̨t̨h̨ęs̨ę ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨α̨_̨*̨ ̨s̨ǫ ̨t̨h̨ąt̨ ̨t̨h̨ęy̨ ̨ąr̨ę ̨n̨ǫt̨ ̨ąl̨l̨ ̨d̨įs̨t̨įn̨c̨t̨.̨ ̨Ęx̨p̨ęr̨įęn̨c̨ę ̨h̨ąs̨ ̨s̨h̨ǫw̨n̨ ̨ųs̨ ̨t̨h̨ąt̨ ̨t̨h̨ęr̨ę ̨s̨h̨ǫųl̨d̨ ̨n̨ǫt̨ ̨b̨ę ̨j̨ųs̨t̨ ̨ǫn̨ę ̨v̨ąl̨ųę ̨įn̨ ̨t̨h̨ę ̨ęn̨t̨įr̨ę ̨t̨r̨ęę,̨ ̨n̨ǫr̨ ̨s̨h̨ǫųl̨d̨ ̨t̨h̨ęr̨ę ̨b̨ę ̨ą ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨v̨ąl̨ųę ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨P̨ǫs̨t̨ęr̨įǫr̨ ̨įn̨f̨ęr̨ęn̨c̨ę ̨T̨ǫ ̨c̨ǫn̨s̨įd̨ęr̨ ̨h̨ǫw̨ ̨p̨ǫs̨t̨ęr̨įǫr̨ ̨įn̨f̨ęr̨ęn̨c̨ę ̨įs̨ ̨d̨ǫn̨ę ̨w̨įt̨h̨ ̨t̨h̨įs̨ ̨m̨ǫd̨ęl̨,̨ ̨f̨įr̨s̨t̨ ̨c̨ǫn̨s̨įd̨ęr̨ ̨t̨h̨ę ̨s̨įm̨p̨l̨ęs̨t̨ ̨c̨ąs̨ę ̨ǫf̨ ̨ą ̨s̨įn̨g̨l̨ę ̨n̨ǫd̨ę ̨w̨įt̨h̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨ ̨w̨h̨ęr̨ę ̨ą ̨d̨ąt̨ą ̨v̨ęc̨t̨ǫr̨ ̨n̨_̨X̨_̨c̨|̨y̨ ̨įs̨ ̨s̨ąm̨p̨l̨ęd̨ ̨w̨įt̨h̨ ̨t̨ǫt̨ąl̨ ̨c̨ǫųn̨t̨ ̨n̨_̨·̨|̨y̨=̨∑̨_̨x̨_̨c̨n̨_̨x̨_̨c̨|̨y̨:̨ ̨ ̨ ̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨ ̨ ̨ ̨∼̨ ̨ ̨ ̨ϕ̨_̨X̨_̨c̨,̨α̨_̨y̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨n̨_̨X̨_̨c̨|̨y̨ ̨ ̨ ̨∼̨ ̨ ̨ ̨(̨ϕ̨_̨X̨_̨c̨|̨y̨,̨n̨_̨·̨|̨y̨)̨ ̨ ̨.̨ ̨F̨ǫr̨ ̨ęx̨ąm̨p̨l̨ę,̨ ̨įn̨ ̨D̨ąt̨ąs̨ęt̨ ̨1̨ ̨g̨įv̨ęn̨ ̨l̨ąt̨ęr̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨,̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨n̨_̨x̨_̨1̨|̨y̨ ̨ąr̨ę ̨ąs̨ ̨f̨ǫl̨l̨ǫw̨s̨,̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨v̨ąl̨ųę ̨ǫf̨ ̨X̨_̨1̨ ̨ąn̨d̨ ̨Y̨:̨ ̨ ̨ ̨[̨ ̨n̨_̨0̨|̨0̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨ ̨2̨;̨ ̨n̨_̨1̨|̨0̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨ ̨0̨;̨ ̨n̨_̨0̨|̨1̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨2̨0̨;̨ ̨n̨_̨1̨|̨1̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨ ̨5̨ ̨]̨ ̨ ̨ ̨T̨h̨ęs̨ę ̨ąr̨ę ̨c̨ǫn̨t̨ąįn̨ęd̨ ̨įn̨t̨ǫ ̨t̨w̨ǫ ̨v̨ęc̨t̨ǫr̨s̨ ̨f̨ǫr̨ ̨Y̨=̨0̨ ̨ąn̨d̨ ̨Y̨=̨1̨:̨ ̨ ̨ ̨[̨ ̨n̨_̨X̨_̨1̨|̨0̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨[̨2̨,̨0̨]̨;̨ ̨n̨_̨X̨_̨1̨|̨1̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨[̨2̨0̨,̨5̨]̨ ̨]̨ ̨ ̨ ̨T̨h̨ę ̨t̨ǫt̨ąl̨ ̨c̨ǫųn̨t̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨v̨ęc̨t̨ǫr̨ ̨ąr̨ę ̨t̨h̨ųs̨ ̨r̨ęs̨p̨ęc̨t̨įv̨ęl̨y̨ ̨n̨_̨.̨|̨0̨=̨2̨ ̨ąn̨d̨ ̨n̨_̨.̨|̨1̨=̨2̨5̨.̨ ̨T̨h̨ę ̨m̨ąr̨g̨įn̨ąl̨įs̨ęd̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨f̨ǫr̨ ̨t̨h̨įs̨,̨ ̨w̨h̨įc̨h̨ ̨m̨ąr̨g̨įn̨ąl̨įs̨ęs̨ ̨ǫųt̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨ ̨t̨ąk̨ęs̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ ̨<̨c̨įt̨.̨>̨ ̨ ̨ ̨ ̨ ̨(̨n̨_̨X̨_̨c̨|̨y̨|̨ϕ̨_̨X̨_̨c̨,̨α̨_̨y̨,̨n̨_̨·̨|̨y̨)̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨ ̨ ̨n̨_̨·̨|̨y̨ ̨n̨_̨X̨_̨c̨|̨y̨Γ̨(̨α̨_̨y̨)̨/̨∏̨_̨x̨_̨c̨Γ̨(̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨)̨∏̨_̨x̨_̨c̨Γ̨(̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨+̨ ̨n̨_̨x̨_̨c̨|̨y̨)̨/̨Γ̨(̨α̨_̨y̨+̨n̨_̨·̨|̨y̨)̨ ̨.̨ ̨w̨h̨ęr̨ę ̨x̨_̨c̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨v̨ąl̨ųęs̨ ̨t̨ąk̨ęn̨ ̨b̨y̨ ̨X̨_̨c̨.̨ ̨Ǫųr̨ ̨g̨ǫąl̨ ̨įn̨ ̨t̨h̨įs̨ ̨įs̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ę ̨ϕ̨_̨X̨_̨c̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨.̨ ̨Ąs̨ ̨įt̨ ̨s̨t̨ąn̨d̨s̨,̨ ̨t̨h̨įs̨ ̨įs̨ ̨g̨ǫįn̨g̨ ̨t̨ǫ ̨b̨ę ̨v̨ęr̨y̨ ̨c̨ǫs̨t̨l̨y̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨ąp̨p̨ęąr̨ ̨įn̨ ̨ą ̨c̨ǫm̨p̨l̨ęx̨ ̨f̨ǫr̨m̨ ̨įn̨s̨įd̨ę ̨g̨ąm̨m̨ą ̨f̨ųn̨c̨t̨įǫn̨s̨,̨ ̨∏̨_̨x̨_̨c̨Γ̨(̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨+̨ ̨n̨_̨x̨_̨c̨|̨y̨)̨/̨Γ̨(̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨)̨.̨ ̨N̨ęw̨ ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨f̨ǫr̨ ̨H̨D̨P̨s̨ ̨d̨ęąl̨ ̨w̨įt̨h̨ ̨t̨h̨įs̨ ̨p̨r̨ǫb̨l̨ęm̨ ̨b̨y̨ ̨m̨ǫd̨įf̨y̨įn̨g̨ ̨įt̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨įn̨t̨r̨ǫd̨ųc̨t̨įǫn̨ ̨ǫf̨ ̨n̨ęw̨ ̨(̨l̨ąt̨ęn̨t̨)̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨t̨h̨ąt̨ ̨m̨ąk̨ę ̨t̨h̨ę ̨g̨ąm̨m̨ą ̨f̨ųn̨c̨t̨įǫn̨s̨ ̨d̨įs̨ąp̨p̨ęąr̨.̨ ̨ ̨W̨h̨įl̨ę ̨ǫn̨ę ̨c̨ąn̨ ̨f̨ǫr̨m̨ąl̨įs̨ę ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨ųs̨įn̨g̨ ̨H̨D̨P̨s̨,̨ ̨įn̨ ̨t̨h̨įs̨ ̨c̨ąs̨ę ̨ą ̨d̨įr̨ęc̨t̨ ̨ąųg̨m̨ęn̨t̨ąt̨įǫn̨ ̨c̨ąn̨ ̨b̨ę ̨d̨ǫn̨ę ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨įd̨ęn̨t̨įt̨y̨ ̨(̨f̨ǫr̨ ̨n̨∈̨ ̨N̨^̨+̨)̨ ̨ ̨ ̨ ̨ ̨ ̨Γ̨(̨α̨+̨n̨)̨/̨Γ̨(̨α̨)̨ ̨ ̨ ̨ ̨ ̨=̨ ̨∑̨_̨t̨=̨1̨^̨n̨ ̨α̨^̨t̨ ̨S̨^̨n̨_̨t̨ ̨ ̨w̨h̨ęr̨ę ̨S̨^̨n̨_̨t̨ ̨įs̨ ̨ąn̨ ̨ųn̨s̨įg̨n̨ęd̨ ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨k̨įn̨d̨.̨ ̨T̨h̨ę ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨ ̨įs̨ ̨ą ̨c̨ǫm̨b̨įn̨ąt̨ǫr̨įc̨ ̨q̨ųąn̨t̨įt̨y̨ ̨t̨h̨ąt̨ ̨įs̨ ̨ęąs̨įl̨y̨ ̨t̨ąb̨ųl̨ąt̨ęd̨ ̨<̨c̨įt̨.̨>̨ ̨ąn̨d̨ ̨s̨įm̨p̨l̨ę ̨ąs̨y̨m̨p̨t̨ǫt̨įc̨ ̨f̨ǫr̨m̨ųl̨ą ̨ęx̨įs̨t̨ ̨<̨c̨įt̨.̨>̨.̨T̨h̨įs̨ ̨įs̨ ̨s̨ǫm̨ęt̨įm̨ęs̨ ̨c̨ǫn̨v̨ęr̨t̨ęd̨ ̨įn̨t̨ǫ ̨t̨h̨ę ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨(̨C̨R̨D̨)̨ ̨įn̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ ̨ ̨ ̨ ̨ ̨ ̨(̨t̨|̨C̨R̨D̨,̨n̨,̨α̨)̨ ̨=̨ ̨Γ̨(̨α̨)̨/̨Γ̨(̨α̨+̨n̨)̨α̨^̨t̨ ̨S̨^̨n̨_̨t̨ ̨ ̨ąn̨d̨ ̨n̨ǫt̨ę ̨t̨h̨ę ̨n̨ǫr̨m̨ąl̨įs̨ąt̨įǫn̨ ̨ǫf̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨įs̨ ̨s̨h̨ǫw̨n̨ ̨b̨y̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨w̨h̨ęr̨ę ̨t̨∈̨{̨1̨,̨.̨.̨.̨,̨n̨}̨ ̨f̨ǫr̨ ̨n̨>̨0̨.̨T̨ǫ ̨s̨įm̨p̨l̨įf̨y̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨m̨ųl̨t̨įp̨l̨y̨ ̨t̨h̨ę ̨L̨H̨S̨ ̨b̨y̨ ̨∏̨_̨x̨_̨c̨(̨t̨_̨x̨_̨c̨|̨y̨|̨C̨R̨D̨,̨n̨_̨x̨_̨c̨|̨y̨,̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨)̨ ̨ąn̨d̨ ̨t̨h̨ę ̨R̨H̨S̨ ̨b̨y̨ ̨t̨h̨ę ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨įn̨g̨ ̨R̨H̨S̨s̨ ̨f̨r̨ǫm̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨T̨h̨įs̨ ̨įs̨ ̨c̨ąl̨l̨ęd̨ ̨ąn̨ ̨ąųg̨m̨ęn̨t̨ąt̨įǫn̨ ̨b̨ęc̨ąųs̨ę ̨w̨ę ̨ąr̨ę ̨ ̨ąr̨ę ̨įn̨t̨r̨ǫd̨ųc̨įn̨g̨ ̨n̨ęw̨ ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨t̨_̨x̨_̨c̨|̨y̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨x̨_̨c̨,̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨įn̨ ̨ǫųr̨ ̨n̨ǫt̨ąt̨įǫn̨ ̨ąs̨ ̨t̨_̨X̨_̨c̨|̨y̨.̨ ̨T̨h̨ę ̨t̨ęr̨m̨s̨ ̨įn̨ ̨Γ̨(̨ϕ̨_̨x̨_̨c̨|̨y̨α̨_̨y̨)̨ ̨ęt̨c̨.̨,̨ ̨c̨ąn̨c̨ęl̨ ̨ǫųt̨ ̨y̨įęl̨d̨įn̨g̨ ̨ ̨ ̨ ̨ ̨(̨n̨_̨X̨_̨c̨|̨y̨,̨t̨_̨X̨_̨c̨|̨y̨|̨ϕ̨_̨X̨_̨c̨,̨α̨_̨y̨,̨n̨_̨·̨|̨y̨)̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨n̨_̨·̨|̨y̨ ̨n̨_̨X̨_̨c̨|̨y̨Γ̨(̨α̨_̨y̨)̨/̨Γ̨(̨α̨_̨y̨+̨n̨_̨·̨|̨y̨)̨∏̨_̨x̨_̨c̨ ̨(̨α̨_̨y̨ϕ̨_̨x̨_̨c̨)̨^̨t̨_̨x̨_̨c̨|̨y̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨_̨t̨_̨x̨_̨c̨|̨y̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨n̨_̨·̨|̨y̨ ̨n̨_̨X̨_̨c̨|̨y̨α̨_̨y̨^̨t̨_̨·̨|̨y̨/̨α̨_̨y̨^̨(̨n̨_̨·̨|̨y̨)̨∏̨_̨x̨_̨c̨ϕ̨_̨x̨_̨c̨^̨t̨_̨x̨_̨c̨|̨y̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨_̨t̨_̨x̨_̨c̨|̨y̨ ̨ ̨w̨h̨ęr̨ę ̨ ̨α̨^̨(̨n̨)̨=̨α̨(̨α̨+̨1̨)̨⋯̨(̨α̨+̨n̨-̨1̨)̨ ̨įs̨ ̨ą ̨r̨įs̨įn̨g̨ ̨f̨ąc̨t̨ǫr̨įąl̨.̨ ̨ ̨N̨ǫt̨įc̨ę ̨w̨h̨ąt̨ ̨h̨ąs̨ ̨b̨ęęn̨ ̨d̨ǫn̨ę ̨h̨ęr̨ę ̨f̨ǫr̨ ̨t̨h̨ę ̨c̨ųr̨r̨ęn̨t̨ ̨n̨ǫd̨ęs̨ ̨X̨_̨c̨:̨ ̨ ̨ ̨ ̨*̨ ̨ ̨ ̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨ ̨ϕ̨_̨X̨_̨c̨ ̨n̨ǫw̨ ̨ąp̨p̨ęąr̨ ̨įn̨ ̨ą ̨s̨įm̨p̨l̨ę ̨ ̨ ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨∏̨_̨x̨_̨c̨ϕ̨_̨x̨_̨c̨^̨t̨_̨x̨_̨c̨|̨y̨,̨ ̨ ̨ ̨ ̨*̨ ̨ ̨ ̨ ̨t̨h̨ęįr̨ ̨p̨r̨įǫr̨ ̨c̨ǫm̨p̨l̨ęx̨ ̨f̨ǫr̨m̨ ̨įn̨s̨įd̨ę ̨g̨ąm̨m̨ą ̨f̨ųn̨c̨t̨įǫn̨s̨ ̨h̨ąs̨ ̨b̨ęęn̨ ̨ęl̨įm̨įn̨ąt̨ęd̨,̨ ̨ ̨ ̨ ̨*̨ ̨ ̨ ̨ ̨b̨ųt̨ ̨ąt̨ ̨t̨h̨ę ̨ęx̨p̨ęn̨s̨ę ̨ǫf̨ ̨įn̨t̨r̨ǫd̨ųc̨įn̨g̨ ̨n̨ęw̨ ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨t̨_̨X̨_̨c̨|̨y̨.̨ ̨ ̨T̨h̨įs̨ ̨ǫp̨ęr̨ąt̨įǫn̨ ̨f̨ǫr̨m̨s̨ ̨t̨h̨ę ̨b̨ąs̨įs̨ ̨f̨ǫr̨ ̨s̨įm̨p̨l̨įf̨y̨įn̨g̨ ̨ą ̨f̨ųl̨l̨ ̨t̨r̨ęę ̨ǫf̨ ̨s̨ųc̨h̨ ̨n̨ǫd̨ęs̨ ̨r̨ęc̨ųr̨s̨įv̨ęl̨y̨,̨ ̨p̨r̨ęs̨ęn̨t̨ęd̨ ̨įn̨ ̨t̨h̨ę ̨n̨ęx̨t̨ ̨s̨ęc̨t̨įǫn̨.̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨w̨ąs̨ ̨ǫr̨įg̨įn̨ąl̨l̨y̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨ąn̨d̨ ̨ųs̨ęd̨ ̨įn̨ ̨t̨h̨ę ̨c̨ǫn̨t̨ęx̨t̨ ̨ǫf̨ ̨t̨h̨ę ̨H̨D̨P̨,̨ ̨b̨ųt̨ ̨t̨h̨ę ̨ąb̨ǫv̨ę ̨ąl̨t̨ęr̨n̨ąt̨įv̨ę ̨d̨ęr̨įv̨ąt̨įǫn̨ ̨įs̨ ̨ąd̨ęq̨ųąt̨ę ̨f̨ǫr̨ ̨ǫųr̨ ̨p̨ųr̨p̨ǫs̨ęs̨.̨Ǫn̨ę ̨c̨ąn̨ ̨t̨h̨įn̨k̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨B̨ąy̨ęs̨įąn̨ ̨įn̨f̨ęr̨ęn̨c̨ę ̨ǫn̨ ̨ą ̨D̨ĄG̨ ̨w̨h̨ęr̨ę ̨ęv̨įd̨ęn̨c̨ę ̨f̨ųn̨c̨t̨įǫn̨s̨ ̨ąr̨ę ̨p̨ąs̨s̨ęd̨ ̨b̨ęt̨w̨ęęn̨ ̨n̨ǫd̨ęs̨.̨ ̨Įn̨s̨t̨ęąd̨ ̨ǫf̨ ̨p̨ąs̨s̨įn̨g̨ ̨t̨h̨ę ̨ęv̨įd̨ęn̨c̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨b̨y̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨f̨r̨ǫm̨ ̨n̨ǫd̨ęs̨ ̨X̨_̨c̨ ̨t̨ǫ ̨p̨ąr̨ęn̨t̨ ̨y̨,̨ ̨w̨ę ̨p̨ąs̨s̨ ̨t̨h̨ę ̨ęv̨įd̨ęn̨c̨ę ̨∏̨_̨x̨_̨c̨ϕ̨_̨x̨_̨c̨^̨t̨_̨x̨_̨c̨|̨y̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨j̨ųs̨t̨ ̨ą ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨s̨ǫ ̨įt̨ ̨c̨ąn̨ ̨b̨ę ̨c̨ǫm̨b̨įn̨ęd̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨p̨r̨įǫr̨ ̨įn̨ ̨t̨h̨ę ̨ųs̨ųąl̨ ̨m̨ąn̨n̨ęr̨.̨ ̨S̨ǫ ̨f̨ǫr̨ ̨ęv̨ęr̨y̨ ̨c̨ǫųn̨t̨ ̨n̨_̨x̨_̨c̨|̨y̨>̨0̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę,̨ ̨ǫn̨ę ̨įs̨ ̨įn̨t̨r̨ǫd̨ųc̨įn̨g̨ ̨ą ̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨t̨_̨x̨_̨c̨|̨y̨ ̨ąs̨ ̨ą ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ę,̨ ̨w̨h̨ęr̨ę ̨1̨≤̨ ̨t̨_̨x̨_̨c̨|̨y̨≤̨ ̨n̨_̨x̨_̨c̨|̨y̨.̨ ̨ ̨ ̨ ̨ ̨H̨ǫw̨ ̨d̨ǫęs̨ ̨t̨h̨įs̨ ̨r̨ęl̨ąt̨ę ̨t̨ǫ ̨ą ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨(̨C̨R̨P̨)̨?̨ ̨S̨ųp̨p̨ǫs̨ę ̨w̨ę ̨h̨ąv̨ę ̨ą ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨w̨įt̨h̨ ̨b̨ąs̨ę ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨ϕ̨_̨X̨_̨c̨ ̨ąn̨d̨ ̨w̨ę ̨s̨ąm̨p̨l̨ę ̨n̨_̨·̨|̨y̨ ̨d̨ąt̨ą ̨g̨ęn̨ęr̨ąt̨įn̨g̨ ̨ą ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨,̨ ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨n̨_̨·̨|̨y̨ ̨s̨ąm̨p̨l̨ę ̨p̨ǫįn̨t̨s̨ ̨ąr̨ę ̨d̨įs̨t̨r̨įb̨ųt̨ęd̨ ̨ǫv̨ęr̨ ̨ą ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨t̨ąb̨l̨ęs̨.̨ ̨T̨h̨ęn̨ ̨t̨h̨ę ̨t̨_̨x̨_̨c̨|̨y̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨ąb̨ǫv̨ę ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨s̨ ̨t̨ǫ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨t̨ąb̨l̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨f̨ǫr̨ ̨d̨ąt̨ą ̨x̨_̨c̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨b̨y̨ ̨d̨ęf̨įn̨įt̨įǫn̨ ̨b̨ęt̨w̨ęęn̨ ̨1̨ ̨ąn̨d̨ ̨n̨_̨x̨_̨c̨|̨y̨ ̨w̨h̨ęn̨ ̨n̨_̨x̨_̨c̨|̨y̨>̨0̨<̨c̨įt̨.̨>̨.̨ ̨Įn̨d̨ęęd̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ǫf̨ ̨t̨h̨ę ̨C̨R̨D̨ ̨ąb̨ǫv̨ę ̨įs̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ųl̨ą ̨f̨ǫr̨ ̨ą ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨C̨R̨P̨ ̨<̨c̨įt̨.̨>̨,̨ ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨s̨ ̨ǫf̨ ̨d̨ąt̨ą ̨ąt̨ ̨ęąc̨h̨ ̨t̨ąb̨l̨ę ̨ąr̨ę ̨m̨ąr̨g̨įn̨ąl̨įs̨ęd̨ ̨ǫųt̨,̨ ̨ǫn̨l̨y̨ ̨k̨ęęp̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫųn̨t̨ ̨ǫf̨ ̨t̨ąb̨l̨ęs̨.̨ ̨T̨h̨įs̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨ą ̨h̨ųg̨ę ̨ąd̨v̨ąn̨t̨ąg̨ę ̨c̨ǫm̨p̨ųt̨ąt̨įǫn̨ąl̨l̨y̨ ̨b̨ęc̨ąųs̨ę ̨ǫn̨ę ̨ǫn̨l̨y̨ ̨n̨ęęd̨s̨ ̨t̨ǫ ̨s̨t̨ǫr̨ę ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨t̨ąb̨l̨ęs̨ ̨ąt̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę,̨ ̨n̨ǫt̨ ̨t̨h̨ę ̨f̨ųl̨l̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨ ̨ǫf̨ ̨c̨ųs̨t̨ǫm̨ęr̨s̨ ̨ąt̨ ̨t̨ąb̨l̨ęs̨.̨ ̨ ̨T̨h̨įs̨ ̨ęl̨įm̨įn̨ąt̨ęs̨ ̨t̨h̨ę ̨n̨ęęd̨ ̨f̨ǫr̨ ̨d̨y̨n̨ąm̨įc̨ ̨m̨ęm̨ǫr̨y̨ ̨t̨h̨ąt̨ ̨b̨ųr̨d̨ęn̨s̨ ̨ą ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨C̨R̨P̨.̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨C̨ǫn̨t̨ęx̨t̨ ̨t̨r̨ęę ̨–̨ ̨d̨ąt̨ą ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ ̨T̨h̨ę ̨įn̨t̨ųįt̨įǫn̨ ̨ǫf̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨įs̨ ̨t̨h̨ąt̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨θ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ǫr̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨p̨ąs̨s̨ęs̨ ̨ųp̨ ̨s̨ǫm̨ę ̨f̨r̨ąc̨t̨įǫn̨ ̨ǫf̨ ̨įt̨s̨ ̨ǫw̨n̨ ̨d̨ąt̨ą ̨ąs̨ ̨ą ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨t̨ǫ ̨įt̨s̨ ̨p̨ąr̨ęn̨t̨.̨ ̨S̨ǫ ̨t̨h̨ę ̨n̨ǫd̨ęs̨ ̨w̨įl̨l̨ ̨h̨ąv̨ę ̨ą ̨v̨ęc̨t̨ǫr̨ ̨ǫf̨ ̨s̨ųf̨f̨įc̨įęn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨n̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨r̨ęc̨ǫr̨d̨ęd̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę.̨ ̨T̨h̨ęs̨ę ̨h̨ąv̨ę ̨ą ̨v̨įr̨t̨ųąl̨ ̨C̨R̨P̨ ̨w̨įt̨h̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨ǫn̨l̨y̨ ̨r̨ęc̨ǫr̨d̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨t̨ąb̨l̨ęs̨ ̨t̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į,̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨r̨ęf̨ęr̨ ̨t̨ǫ ̨ąs̨ ̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨.̨ ̨T̨h̨ę ̨c̨ǫųn̨t̨s̨ ̨t̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨f̨r̨ąc̨t̨įǫn̨ ̨ǫf̨ ̨n̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨t̨h̨ąt̨ ̨įs̨ ̨p̨ąs̨s̨ęd̨ ̨(̨įn̨ ̨ą ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨)̨ ̨ųp̨ ̨t̨ǫ ̨įt̨s̨ ̨p̨ąr̨ęn̨t̨ ̨n̨ǫd̨ę,̨ ̨ąs̨ ̨d̨įc̨t̨ąt̨ęd̨ ̨b̨y̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨Ąn̨ ̨ęx̨ąm̨p̨l̨ę ̨ǫf̨ ̨c̨ǫn̨t̨ęx̨t̨ ̨t̨r̨ęę ̨f̨ǫr̨ ̨k̨D̨B̨1̨ ̨įs̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨,̨ ̨w̨h̨įc̨h̨ ̨s̨įm̨p̨l̨y̨ ̨ųn̨f̨ǫl̨d̨s̨ ̨t̨h̨ę ̨p̨l̨ąt̨ę ̨n̨ǫt̨ąt̨įǫn̨s̨ ̨ųs̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨ ̨ąn̨d̨ ̨ąd̨d̨s̨ ̨t̨h̨ę ̨t̨ ̨ąn̨d̨ ̨n̨ ̨v̨ąr̨įąb̨l̨ęs̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨[̨ ̨ ̨ ̨ ̨ ̨p̨l̨ąt̨ę/̨.̨s̨t̨y̨l̨ę=̨,̨s̨c̨ąl̨ę=̨.̨7̨,̨r̨ǫųn̨d̨ęd̨ ̨c̨ǫr̨n̨ęr̨s̨ ̨ ̨ ̨ ̨ ̨]̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨į)̨ ̨ąt̨ ̨(̨0̨,̨0̨)̨ ̨[̨ ̨ϕ̨_̨X̨_̨į;̨ ̨n̨_̨X̨_̨į;̨ ̨t̨_̨X̨_̨į ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨1̨)̨ ̨ąt̨ ̨(̨-̨4̨,̨-̨3̨)̨ ̨[̨ ̨ϕ̨_̨X̨_̨į|̨y̨_̨1̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨1̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨1̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨2̨)̨ ̨ąt̨ ̨(̨4̨,̨-̨3̨)̨ ̨[̨ ̨ϕ̨_̨X̨_̨į|̨y̨_̨2̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨2̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨2̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨1̨p̨1̨)̨ ̨ąt̨ ̨(̨-̨6̨,̨-̨6̨.̨3̨)̨ ̨[̨ ̨θ̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨1̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨1̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨1̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨1̨p̨2̨)̨ ̨ąt̨ ̨(̨-̨2̨,̨-̨6̨.̨3̨)̨ ̨[̨ ̨θ̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨2̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨2̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨1̨,̨Π̨(̨į)̨_̨2̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨2̨p̨1̨)̨ ̨ąt̨ ̨(̨2̨,̨-̨6̨.̨3̨)̨ ̨[̨ ̨θ̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨1̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨1̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨1̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨[̨d̨r̨ąw̨,̨r̨ęc̨t̨ąn̨g̨l̨ę]̨ ̨(̨x̨įg̨y̨2̨p̨2̨)̨ ̨ąt̨ ̨(̨6̨,̨-̨6̨.̨3̨)̨ ̨[̨ ̨θ̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨2̨;̨ ̨n̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨2̨;̨ ̨t̨_̨X̨_̨į|̨y̨_̨2̨,̨Π̨(̨į)̨_̨2̨ ̨]̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨į)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨l̨ęf̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨į1̨)̨ ̨Y̨=̨y̨_̨1̨ ̨(̨x̨įg̨y̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨į)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨r̨įg̨h̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨į2̨)̨ ̨Y̨=̨y̨_̨2̨ ̨(̨x̨įg̨y̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨įg̨y̨1̨)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨l̨ęf̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨įg̨y̨1̨1̨)̨ ̨Π̨(̨į)̨_̨1̨ ̨(̨x̨įg̨y̨1̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨įg̨y̨1̨)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨r̨įg̨h̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨įg̨y̨1̨2̨)̨ ̨Π̨(̨į)̨_̨2̨ ̨(̨x̨įg̨y̨1̨p̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨įg̨y̨2̨)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨l̨ęf̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨įg̨y̨2̨1̨)̨ ̨Π̨(̨į)̨_̨1̨ ̨(̨x̨įg̨y̨2̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨ ̨(̨x̨įg̨y̨2̨)̨ ̨–̨ ̨n̨ǫd̨ę[̨m̨įd̨w̨ąy̨,̨r̨įg̨h̨t̨]̨ ̨(̨ąr̨r̨ǫw̨x̨įg̨y̨2̨2̨)̨ ̨Π̨(̨į)̨_̨2̨ ̨(̨x̨įg̨y̨2̨p̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨į)̨ ̨–̨ ̨(̨x̨įg̨y̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨į)̨ ̨–̨ ̨(̨x̨įg̨y̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨1̨)̨ ̨–̨ ̨(̨x̨įg̨y̨1̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨1̨)̨ ̨–̨ ̨(̨x̨įg̨y̨1̨p̨2̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨2̨)̨ ̨–̨ ̨(̨x̨įg̨y̨2̨p̨1̨)̨;̨ ̨ ̨ ̨ ̨ ̨[̨t̨h̨įc̨k̨,̨-̨>̨]̨ ̨(̨x̨įg̨y̨2̨)̨ ̨–̨ ̨(̨x̨įg̨y̨2̨p̨2̨)̨;̨ ̨ ̨ ̨C̨ǫn̨t̨ęx̨t̨ ̨t̨r̨ęę ̨f̨ǫr̨ ̨ǫųr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨m̨ǫd̨ęl̨ ̨f̨ǫr̨ ̨k̨D̨B̨1̨ ̨ąn̨d̨ ̨ǫn̨ę ̨X̨_̨į.̨ ̨ ̨Ąs̨ ̨w̨įt̨h̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨C̨R̨P̨s̨,̨ ̨t̨h̨ęs̨ę ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąr̨ę ̨r̨ęl̨ąt̨ęd̨ ̨f̨ǫr̨ ̨į≥̨ ̨0̨:̨ ̨ ̨ ̨ ̨ ̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨=̨ ̨∑̨_̨x̨_̨į ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į,̨ ̨ ̨ąn̨d̨ ̨m̨ǫr̨ęǫv̨ęr̨ ̨t̨h̨ę ̨b̨ąs̨ę ̨c̨ąs̨ę ̨n̨_̨x̨_̨c̨ ̨=̨ ̨∑̨_̨y̨ ̨t̨_̨x̨_̨c̨|̨y̨.̨ ̨T̨h̨ę ̨c̨ǫųn̨t̨s̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨h̨ęr̨ę ̨ǫn̨l̨y̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨r̨ęąl̨ ̨d̨ąt̨ą ̨c̨ǫųn̨t̨s̨ ̨ąt̨ ̨t̨h̨ę ̨l̨ęąf̨ ̨n̨ǫd̨ęs̨.̨ ̨Ąt̨ ̨įn̨t̨ęr̨n̨ąl̨ ̨n̨ǫd̨ęs̨,̨ ̨t̨h̨ę ̨n̨_̨*̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨ǫt̨ąl̨s̨ ̨ǫf̨ ̨p̨s̨ųęd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨n̨ǫd̨ęs̨,̨ ̨ąs̨ ̨p̨ąs̨s̨ęd̨ ̨ųp̨ ̨b̨y̨ ̨t̨h̨ę ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨ęv̨įd̨ęn̨c̨ę ̨m̨ęs̨s̨ąg̨ęs̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨r̨ęn̨.̨ ̨ ̨T̨h̨ę ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨w̨įt̨h̨ ̨t̨h̨įs̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨ ̨c̨ąn̨ ̨b̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨w̨įt̨h̨ ̨θ̨ ̨ąn̨d̨ ̨ąl̨l̨ ̨ ̨b̨ųt̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨ϕ̨ ̨m̨ąr̨g̨įn̨ąl̨įs̨ęd̨ ̨ǫųt̨:̨ ̨ ̨ ̨ ̨ ̨ ̨(̨,̨n̨,̨t̨|̨ϕ̨_̨X̨_̨c̨,̨α̨)̨=̨(̨∏̨_̨x̨_̨c̨ϕ̨_̨x̨_̨c̨^̨n̨_̨x̨_̨c̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨∏̨_̨į=̨0̨^̨n̨ ̨(̨∏̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨įα̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į^̨t̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į/̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į^̨(̨n̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į)̨∏̨_̨x̨_̨c̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į_̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į)̨ ̨,̨ ̨ ̨ąn̨d̨ ̨t̨h̨ę ̨`̨d̨ǫt̨'̨ ̨n̨ǫt̨ąt̨įǫn̨ ̨įs̨ ̨ųs̨ęd̨ ̨t̨ǫ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨ǫt̨ąl̨s̨,̨ ̨s̨ǫ ̨n̨_̨·̨|̨y̨=̨∑̨_̨x̨_̨c̨ ̨n̨_̨x̨_̨c̨|̨y̨.̨ ̨T̨h̨ę ̨m̨ųl̨t̨įn̨ǫm̨įąl̨ ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨ǫn̨ ̨ϕ̨_̨X̨_̨c̨ ̨ ̨c̨ąn̨ ̨ąl̨s̨ǫ ̨b̨ę ̨m̨ąr̨g̨įn̨ąl̨įs̨ęd̨ ̨ǫųt̨ ̨w̨įt̨h̨ ̨ą ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨įǫr̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ę ̨f̨ǫr̨m̨ųl̨ą ̨c̨ąn̨ ̨b̨ę ̨s̨ęęn̨ ̨t̨ǫ ̨b̨ę ̨d̨ęr̨įv̨ęd̨ ̨b̨y̨ ̨r̨ęc̨ųr̨s̨įv̨ę ̨ąp̨p̨l̨įc̨ąt̨įǫn̨ ̨(̨b̨ǫt̨t̨ǫm̨ ̨ųp̨)̨ ̨ǫf̨ ̨t̨h̨ę ̨f̨ǫr̨m̨ųl̨ą ̨įn̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨Ǫn̨c̨ę ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨h̨ąv̨ę ̨b̨ęęn̨ ̨ęs̨t̨įm̨ąt̨ęd̨ ̨(̨d̨ęs̨c̨r̨įb̨ęd̨ ̨įn̨ ̨t̨h̨ę ̨n̨ęx̨t̨ ̨s̨ųb̨-̨s̨ęc̨t̨įǫn̨)̨,̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨θ̨ ̨c̨ąn̨ ̨b̨ę ̨ęs̨t̨įm̨ąt̨ęd̨ ̨r̨ęc̨ųr̨s̨įv̨ęl̨y̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨s̨t̨ąn̨d̨ąr̨d̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨C̨R̨P̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨m̨ųl̨ą:̨ ̨ ̨ ̨ ̨ ̨ ̨ϕ̨̨̂_̨x̨_̨c̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨n̨_̨x̨_̨c̨ ̨+̨ ̨1̨/̨|̨X̨_̨c̨|̨α̨_̨0̨/̨n̨_̨·̨ ̨+̨ ̨α̨_̨0̨ ̨ ̨ ̨ ̨ ̨ϕ̨̨̂_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨+̨ ̨ϕ̨̨̂_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į/̨n̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨+̨ ̨ ̨ ̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨ ̨θ̨̨̂_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨=̨ ̨ ̨ ̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨+̨ ̨ϕ̨̨̂_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨-̨1̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨/̨n̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨+̨ ̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ ̨N̨ǫt̨ę,̨ ̨įn̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨t̨h̨ę ̨c̨ǫųn̨t̨s̨ ̨n̨_̨*̨ ̨ąr̨ę ̨d̨ęr̨įv̨ęd̨ ̨q̨ųąn̨t̨įt̨įęs̨ ̨(̨s̨ųm̨m̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ęįr̨ ̨c̨h̨įl̨d̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨)̨ ̨ąn̨d̨ ̨ąl̨l̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨*̨ ̨ąr̨ę ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨t̨h̨ąt̨ ̨ąr̨ę ̨s̨ąm̨p̨l̨ęd̨ ̨ųs̨įn̨g̨ ̨ą ̨G̨įb̨b̨s̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨.̨ ̨M̨ǫr̨ęǫv̨ęr̨,̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨θ̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ąn̨d̨ ̨ϕ̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ąr̨ę ̨ęs̨t̨įm̨ąt̨ęd̨ ̨r̨ęc̨ųr̨s̨įv̨ęl̨y̨ ̨f̨r̨ǫm̨ ̨ϕ̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨ąn̨d̨ ̨t̨h̨ę ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨įn̨g̨ ̨c̨ǫųn̨t̨s̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨s̨t̨ąn̨d̨ąr̨d̨ ̨C̨R̨P̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨Ęq̨ųąt̨įǫn̨s̨ ̨<̨r̨ęf̨>̨-̨<̨r̨ęf̨>̨.̨ ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨*̨ ̨ąn̨d̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨α̨_̨*̨ ̨įs̨ ̨d̨ǫn̨ę ̨ąn̨d̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨θ̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨įs̨ ̨m̨ąd̨ę ̨p̨ęr̨įǫd̨įc̨ąl̨l̨y̨ ̨t̨ǫ ̨ǫb̨t̨ąįn̨ ̨ąn̨ ̨M̨C̨M̨C̨ ̨ęs̨t̨įm̨ąt̨ę ̨f̨ǫr̨ ̨įt̨.̨ ̨T̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨ ̨t̨h̨ęn̨ ̨d̨įs̨c̨ųs̨s̨ęs̨ ̨h̨ǫw̨ ̨t̨h̨ę ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ęs̨ę ̨ąr̨ę ̨d̨ǫn̨ę.̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨.̨§̨ ̨S̨ąm̨p̨l̨įn̨g̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨*̨ ̨ ̨W̨ę ̨ųs̨ę ̨ą ̨d̨įr̨ęc̨t̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨f̨ǫr̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨t̨h̨ę ̨t̨_̨*̨,̨ ̨s̨w̨ęęp̨įn̨g̨ ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ęąc̨h̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨įn̨d̨įv̨įd̨ųąl̨l̨y̨ ̨ųs̨įn̨g̨ ̨ą ̨f̨ǫr̨m̨ųl̨ą ̨d̨ęr̨įv̨ęd̨ ̨f̨r̨ǫm̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨:̨ ̨ ̨ ̨ ̨ ̨ ̨(̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į|̨,̨n̨_̨*̨,̨t̨_̨*̨^̨-̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į,̨ϕ̨_̨X̨,̨α̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨∝̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į^̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į/̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨^̨(̨n̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨_̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į_̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨,̨ ̨ ̨w̨h̨ęr̨ę ̨t̨_̨*̨^̨-̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨_̨*̨ ̨-̨ ̨{̨ ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į}̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ęx̨įs̨t̨s̨ ̨įm̨p̨l̨įc̨įt̨l̨y̨ ̨įn̨ ̨t̨h̨ę ̨t̨w̨ǫ ̨s̨ųm̨s̨ ̨n̨_̨·̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨ąn̨d̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨ ̨d̨ųę ̨t̨ǫ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨T̨h̨įs̨ ̨s̨w̨ęęp̨ ̨įs̨ ̨m̨ąd̨ę ̨ęf̨f̨įc̨įęn̨t̨ ̨b̨ęc̨ąųs̨ę ̨c̨ǫm̨p̨ųt̨įn̨g̨ ̨t̨h̨ę ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨s̨ ̨įs̨ ̨ą ̨t̨ąb̨l̨ę ̨l̨ǫǫk̨ųp̨,̨ ̨ąn̨d̨ ̨t̨h̨ę ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨s̨ ̨ąr̨ę ̨s̨h̨ąr̨ęd̨ ̨ąm̨ǫn̨g̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨t̨r̨ęęs̨,̨ ̨s̨ǫ ̨t̨h̨ęy̨ ̨ąr̨ę ̨ǫn̨l̨y̨ ̨c̨ąl̨c̨ųl̨ąt̨ęd̨ ̨ǫn̨c̨ę ̨f̨ǫr̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨ǫf̨ ̨t̨h̨ę ̨B̨N̨C̨.̨ ̨ ̨ ̨ ̨T̨h̨ę ̨b̨ąs̨ę ̨c̨ąs̨ę,̨ ̨į=̨0̨ ̨įs̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨r̨ǫǫt̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨v̨ęc̨t̨ǫr̨ ̨ϕ̨_̨X̨_̨c̨ ̨įs̨ ̨m̨ąr̨g̨įn̨ąl̨įs̨ęd̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨D̨įr̨įc̨h̨l̨ęt̨ ̨įn̨t̨ęg̨r̨ąl̨:̨ ̨ ̨ ̨ ̨ ̨ ̨(̨t̨_̨x̨_̨c̨|̨y̨|̨,̨n̨,̨t̨^̨-̨x̨_̨c̨|̨y̨,̨α̨)̨ ̨∝̨Γ̨(̨n̨_̨x̨_̨c̨|̨y̨+̨α̨_̨0̨/̨|̨X̨_̨c̨|̨)̨/̨Γ̨(̨n̨_̨·̨|̨y̨+̨α̨_̨0̨)̨α̨_̨y̨^̨t̨_̨x̨_̨c̨|̨y̨ ̨S̨^̨n̨_̨x̨_̨c̨|̨y̨_̨t̨_̨x̨_̨c̨|̨y̨ ̨.̨ ̨ ̨T̨h̨ęs̨ę ̨t̨w̨ǫ ̨s̨ąm̨p̨l̨įn̨g̨ ̨f̨ǫr̨m̨ųl̨ą,̨ ̨ąs̨ ̨t̨h̨ęy̨ ̨s̨t̨ąn̨d̨,̨ ̨ąr̨ę ̨ąl̨s̨ǫ ̨įn̨ęf̨f̨įc̨įęn̨t̨ ̨b̨ęc̨ąųs̨ę ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨r̨ąn̨g̨ęs̨ ̨ǫv̨ęr̨ ̨1̨,̨⋯̨,̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨w̨h̨ęn̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į>̨0̨.̨ ̨ ̨F̨r̨ǫm̨ ̨D̨P̨ ̨t̨h̨ęǫr̨y̨,̨ ̨w̨ę ̨k̨n̨ǫw̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨h̨ąv̨ę ̨ą ̨s̨t̨ąn̨d̨ąr̨d̨ ̨d̨ęv̨įąt̨įǫn̨ ̨g̨įv̨ęn̨ ̨b̨y̨ ̨Ǫ(̨l̨ǫg̨^̨1̨/̨2̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į)̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨v̨ęr̨y̨ ̨s̨m̨ąl̨l̨,̨ ̨t̨h̨ųs̨ ̨įn̨ ̨p̨r̨ąc̨t̨įc̨ę ̨t̨h̨ę ̨f̨ųl̨l̨ ̨r̨ąn̨g̨ę ̨įs̨ ̨ąl̨m̨ǫs̨t̨ ̨c̨ęr̨t̨ąįn̨l̨y̨ ̨n̨ęv̨ęr̨ ̨ųs̨ęd̨.̨ ̨M̨ǫr̨ęǫv̨ęr̨,̨ ̨n̨ǫt̨ę ̨t̨h̨ę ̨m̨ęąn̨ ̨ǫf̨ ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨c̨h̨ąn̨g̨ęs̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨,̨ ̨s̨ǫ ̨įn̨ ̨ęf̨f̨ęc̨t̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨įs̨ ̨c̨ǫųp̨l̨ęd̨ ̨ąn̨d̨ ̨l̨ąr̨g̨ę ̨m̨ǫv̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨“̨s̨ęąr̨c̨h̨”̨ ̨m̨ąy̨ ̨n̨ǫt̨ ̨b̨ę ̨ęf̨f̨ęc̨t̨įv̨ę.̨ ̨Ąs̨ ̨ą ̨s̨ąf̨ę ̨ąn̨d̨ ̨ęf̨f̨įc̨įęn̨t̨ ̨ǫp̨t̨įǫn̨,̨ ̨w̨ę ̨ǫn̨l̨y̨ ̨s̨ąm̨p̨l̨ę ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨w̨įt̨h̨įn̨ ̨ą ̨w̨įn̨d̨ǫw̨ ̨ǫf̨ ̨±̨ ̨1̨0̨ ̨ǫf̨ ̨t̨h̨ęįr̨ ̨c̨ųr̨r̨ęn̨t̨ ̨v̨ąl̨ųę.̨ ̨W̨ę ̨h̨ąv̨ę ̨t̨ęs̨t̨ęd̨ ̨t̨h̨įs̨ ̨ęm̨p̨įr̨įc̨ąl̨l̨y̨,̨ ̨ąn̨d̨ ̨d̨ųę ̨t̨ǫ ̨t̨h̨ę ̨s̨t̨ąn̨d̨ąr̨d̨ ̨d̨ęv̨įąt̨įǫn̨s̨,̨ ̨įt̨ ̨įs̨ ̨s̨ąf̨ęr̨ ̨ąs̨ ̨t̨h̨ę ̨M̨ǫn̨t̨ę ̨C̨ąr̨l̨ǫ ̨s̨ąm̨p̨l̨įn̨g̨ ̨c̨ǫn̨v̨ęr̨g̨ęs̨ ̨ąn̨d̨ ̨s̨m̨ąl̨l̨ęr̨ ̨m̨ǫv̨ęs̨ ̨ąr̨ę ̨t̨y̨p̨įc̨ąl̨.̨ ̨ ̨M̨ǫr̨ęǫv̨ęr̨,̨ ̨t̨ǫ ̨įn̨įt̨įąl̨įs̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨įn̨ ̨t̨h̨ę ̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨ęr̨,̨ ̨w̨ę ̨ųs̨ę ̨t̨h̨ę ̨ęx̨p̨ęc̨t̨ęd̨ ̨v̨ąl̨ųę ̨ǫf̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨f̨ǫr̨ ̨ą ̨H̨D̨P̨ ̨g̨įv̨ęn̨ ̨t̨h̨ę ̨c̨ųr̨r̨ęn̨t̨ ̨c̨ǫųn̨t̨ ̨ąn̨d̨ ̨t̨h̨ę ̨r̨ęl̨ęv̨ąn̨t̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨s̨:̨ ̨ ̨ ̨ ̨ ̨ ̨t̨ ̨←̨{̨[̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨n̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨įf̨ ̨n̨⩽̨ ̨1̨;̨ ̨m̨ąx̨(̨1̨,̨⌊̨α̨(̨ ̨ψ̨_̨0̨(̨α̨+̨n̨)̨-̨ψ̨_̨0̨(̨α̨)̨)̨⌋̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨įf̨ ̨n̨ ̨>̨ ̨1̨;̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨]̨.̨ ̨ ̨T̨h̨įs̨ ̨r̨ęq̨ųįr̨ęs̨ ̨s̨w̨ęęp̨įn̨g̨ ̨ųp̨ ̨t̨h̨ę ̨t̨r̨ęę ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨ąt̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨;̨ ̨ψ̨_̨0̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨d̨įg̨ąm̨m̨ą ̨f̨ųn̨c̨t̨įǫn̨:̨ ̨ψ̨_̨0̨(̨x̨)̨=̨Γ̨ ̨'̨(̨x̨)̨/̨Γ̨ ̨(̨x̨)̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨.̨§̨ ̨S̨ąm̨p̨l̨įn̨g̨ ̨ąn̨d̨ ̨t̨y̨įn̨g̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨s̨ ̨α̨_̨*̨ ̨N̨ǫ ̨p̨r̨ǫp̨ęr̨ ̨m̨ęn̨t̨įǫn̨ ̨h̨ąs̨ ̨b̨ęęn̨ ̨m̨ąd̨ę ̨y̨ęt̨ ̨ǫf̨ ̨h̨ǫw̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨α̨_̨*̨ ̨ąr̨ę ̨s̨ąm̨p̨l̨ęd̨.̨ ̨T̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨įn̨f̨l̨ųęn̨c̨ę ̨h̨ǫw̨ ̨s̨įm̨įl̨ąr̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨w̨įl̨l̨ ̨b̨ę ̨t̨ǫ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨.̨ ̨W̨ę ̨k̨n̨ǫw̨ ̨t̨h̨įs̨ ̨b̨ęc̨ąųs̨ę ̨D̨įr̨įc̨h̨l̨ęt̨ ̨t̨h̨ęǫr̨y̨ ̨t̨ęl̨l̨s̨ ̨ųs̨,̨ ̨l̨ǫǫk̨įn̨g̨ ̨ąt̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨ ̨ ̨ ̨ ̨ ̨(̨θ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨)̨ ̨ ̨ ̨ ̨ ̨≈̨1̨/̨α̨_̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨-̨1̨(̨1̨-̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨-̨1̨)̨ ̨ ̨S̨įn̨c̨ę ̨w̨ę ̨c̨ąn̨n̨ǫt̨ ̨b̨ę ̨s̨ųr̨ę ̨ǫf̨ ̨h̨ǫw̨ ̨l̨ąr̨g̨ę ̨t̨h̨įs̨ ̨w̨įl̨l̨ ̨b̨ę,̨ ̨w̨ę ̨ąl̨s̨ǫ ̨s̨ąm̨p̨l̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨.̨ ̨Ęx̨p̨ęr̨įęn̨c̨ę ̨w̨įt̨h̨ ̨ǫt̨h̨ęr̨ ̨m̨ǫd̨ęl̨s̨ ̨ųs̨įn̨g̨ ̨H̨D̨P̨s̨ ̨ąl̨ęr̨t̨s̨ ̨ųs̨ ̨t̨h̨ąt̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨ ̨įm̨p̨r̨ǫv̨ęm̨ęn̨t̨s̨ ̨s̨h̨ǫųl̨d̨ ̨b̨ę ̨p̨ǫs̨s̨įb̨l̨ę ̨b̨y̨ ̨j̨ųd̨įc̨įǫųs̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨<̨c̨įt̨.̨>̨.̨N̨ǫt̨ę ̨w̨ę ̨ęx̨p̨ęc̨t̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę ̨t̨ǫ ̨g̨ęt̨ ̨s̨m̨ąl̨l̨ęr̨ ̨ąs̨ ̨w̨ę ̨g̨ǫ ̨d̨ǫw̨n̨ ̨t̨h̨ę ̨t̨r̨ęę,̨ ̨s̨ǫ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨s̨h̨ǫųl̨d̨ ̨b̨ę ̨l̨ąr̨g̨ęr̨ ̨f̨ųr̨t̨h̨ęr̨ ̨d̨ǫw̨n̨ ̨t̨h̨ę ̨t̨r̨ęę.̨T̨y̨įn̨g̨:̨S̨įn̨c̨ę ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨α̨_̨*̨ ̨įs̨ ̨ęq̨ųąl̨ ̨t̨ǫ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨n̨ǫd̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę,̨ ̨t̨h̨ęr̨ę ̨ąr̨ę ̨p̨ǫs̨s̨įb̨l̨y̨ ̨t̨ǫǫ ̨m̨ąn̨y̨ ̨t̨ǫ ̨s̨ąm̨p̨l̨ę.̨ ̨S̨ǫ ̨r̨ąt̨h̨ęr̨ ̨t̨h̨ąn̨ ̨ųs̨įn̨g̨ ̨ą ̨s̨ęp̨ąr̨ąt̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨α̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨f̨ǫr̨ ̨ęv̨ęr̨y̨ ̨n̨ǫd̨ę,̨ ̨w̨ę ̨t̨įę ̨s̨ǫm̨ę,̨ ̨w̨h̨įc̨h̨ ̨m̨ęąn̨s̨ ̨t̨h̨ąt̨ ̨w̨ę ̨m̨ąk̨ę ̨t̨h̨ęįr̨ ̨v̨ąl̨ųęs̨ ̨ęq̨ųąl̨ ̨f̨ǫr̨ ̨s̨ǫm̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨n̨ǫd̨ęs̨.̨F̨įg̨ųr̨ęs̨ ̨<̨r̨ęf̨>̨(̨ą)̨ ̨ąn̨d̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨b̨)̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨ ̨t̨w̨ǫ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨ǫf̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨.̨ ̨T̨h̨ę ̨f̨įr̨s̨t̨ ̨ǫn̨ę ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨s̨ ̨t̨ǫ ̨t̨y̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨s̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨t̨h̨ąt̨ ̨s̨h̨ąr̨ę ̨ą ̨p̨ąr̨ęn̨t̨ ̨n̨ǫd̨ę:̨ ̨t̨h̨ęr̨ę ̨w̨įl̨l̨ ̨t̨h̨ųs̨ ̨b̨ę ̨ą ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨b̨ųt̨ ̨t̨h̨ę ̨l̨ǫw̨ęs̨t̨ ̨ǫn̨ę.̨ ̨T̨h̨ę ̨s̨ęc̨ǫn̨d̨ ̨ǫn̨ę ̨h̨ąs̨ ̨ǫn̨l̨y̨ ̨ǫn̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨l̨ęv̨ęl̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę.̨ ̨T̨y̨įn̨g̨ ̨įs̨ ̨ǫn̨l̨y̨ ̨d̨ǫn̨ę ̨w̨įt̨h̨įn̨ ̨ǫn̨ę ̨c̨ǫn̨t̨ęx̨t̨-̨t̨r̨ęę,̨ ̨į.̨ę.̨ ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąr̨ę ̨įn̨f̨ęr̨r̨ęd̨ ̨c̨ǫm̨p̨l̨ęt̨ęl̨y̨ ̨įn̨d̨ęp̨ęn̨d̨ęn̨t̨l̨y̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨θ̨_̨X̨_̨į ̨|̨ ̨Π̨_̨į(̨)̨.̨ ̨Ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨ǫn̨ ̨t̨h̨ę ̨t̨y̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ęs̨ę ̨h̨y̨p̨ęr̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąr̨ę ̨p̨r̨ęs̨ęn̨t̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨b̨ęl̨ǫw̨ ̨įt̨ęr̨ąt̨ęs̨ ̨ǫv̨ęr̨ ̨ąl̨l̨ ̨t̨h̨ę ̨t̨įęd̨ ̨n̨ǫd̨ęs̨ ̨(̨s̨ęę ̨j̨)̨;̨ ̨s̨ǫ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨ǫn̨l̨y̨ ̨ąf̨f̨ęc̨t̨ ̨t̨h̨ę ̨n̨ǫd̨ęs̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨r̨ųn̨s̨ ̨ǫv̨ęr̨.̨ ̨S̨ąm̨p̨l̨įn̨g̨:̨W̨ę ̨ųs̨ę ̨ąn̨ ̨ąųg̨m̨ęn̨t̨ąt̨įǫn̨ ̨d̨ęt̨ąįl̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨4̨.̨3̨ ̨ǫf̨ ̨<̨c̨įt̨.̨>̨.̨ ̨T̨h̨įs̨ ̨įn̨t̨r̨ǫd̨ųc̨ęs̨ ̨ą ̨n̨ęw̨ ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ę ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę,̨ ̨ąn̨d̨ ̨t̨h̨ęn̨ ̨ą ̨g̨ąm̨m̨ą ̨s̨ąm̨p̨l̨ę ̨c̨ąn̨ ̨b̨ę ̨t̨ąk̨ęn̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨t̨įęd̨ ̨v̨ąr̨įąb̨l̨ę ̨ąf̨t̨ęr̨ ̨s̨ųm̨m̨įn̨g̨ ̨t̨h̨ę ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąc̨r̨ǫs̨s̨ ̨t̨h̨ę ̨t̨įęd̨ ̨n̨ǫd̨ęs̨.̨ ̨T̨h̨ę ̨g̨ęn̨ęr̨ąl̨ ̨f̨ǫr̨m̨ ̨ǫf̨ ̨t̨h̨ę ̨l̨įk̨ęl̨įh̨ǫǫd̨ ̨f̨ǫr̨ ̨ą ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨,̨ ̨α̨,̨ ̨f̨r̨ǫm̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨įs̨ ̨∏̨_̨j̨α̨^̨t̨_̨j̨/̨α̨^̨(̨n̨_̨j̨)̨ ̨w̨h̨ęr̨ę ̨j̨ ̨r̨ųn̨s̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨t̨įęd̨ ̨n̨ǫd̨ęs̨ ̨ąn̨d̨ ̨ ̨(̨n̨_̨j̨,̨t̨_̨j̨)̨ ̨ąr̨ę ̨t̨h̨ę ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨įn̨g̨ ̨c̨ǫųn̨t̨s̨ ̨ąt̨ ̨t̨h̨ę ̨n̨ǫd̨ęs̨.̨ ̨T̨ǫ ̨s̨ąm̨p̨l̨ę ̨α̨ ̨w̨ę ̨n̨ęęd̨ ̨t̨ǫ ̨ąųg̨m̨ęn̨t̨ ̨t̨h̨ę ̨d̨ęn̨ǫm̨įn̨ąt̨ǫr̨ ̨t̨ęr̨m̨s̨ ̨α̨^̨(̨n̨_̨j̨)̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨h̨ąv̨ę ̨n̨ǫ ̨m̨ąt̨c̨h̨ ̨t̨ǫ ̨ą ̨k̨n̨ǫw̨n̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨.̨ ̨T̨h̨įs̨ ̨įs̨ ̨d̨ǫn̨ę ̨b̨y̨ ̨ąd̨d̨įn̨g̨ ̨ą ̨n̨ęw̨ ̨t̨ęr̨m̨ ̨ǫn̨ ̨b̨ǫt̨h̨ ̨s̨įd̨ęs̨ ̨(̨q̨|̨α̨,̨n̨)̨ ̨w̨h̨įc̨h̨ ̨įn̨t̨r̨ǫd̨ųc̨ęs̨ ̨q̨_̨j̨|̨α̨∼̨(̨α̨,̨n̨_̨j̨)̨,̨ ̨ ̨ ̨t̨h̨ęn̨ ̨t̨h̨ę ̨j̨ǫįn̨t̨ ̨p̨ǫs̨t̨ęr̨įǫr̨ ̨įs̨ ̨d̨ęr̨įv̨ęd̨ ̨ąs̨ ̨f̨ǫl̨l̨ǫw̨s̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨α̨|̨,̨n̨,̨t̨)̨(̨q̨|̨α̨,̨n̨)̨ ̨ ̨ ̨∝̨ ̨ ̨ ̨(̨α̨)̨ ̨(̨∏̨_̨j̨α̨^̨t̨_̨j̨/̨α̨^̨(̨n̨_̨j̨)̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨q̨|̨α̨,̨n̨)̨ ̨ ̨ ̨ ̨ ̨(̨α̨,̨q̨|̨,̨n̨,̨t̨)̨ ̨ ̨ ̨∝̨ ̨ ̨ ̨(̨α̨)̨ ̨∏̨_̨j̨α̨^̨t̨_̨j̨/̨α̨^̨(̨n̨_̨j̨)̨∏̨_̨j̨ ̨q̨_̨j̨^̨α̨-̨1̨(̨1̨-̨q̨_̨j̨)̨^̨n̨_̨j̨Γ̨(̨α̨+̨n̨_̨j̨)̨/̨Γ̨(̨α̨)̨Γ̨(̨n̨_̨j̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨∝̨ ̨ ̨ ̨(̨α̨)̨ ̨ ̨∏̨_̨j̨α̨^̨t̨_̨j̨q̨_̨j̨^̨α̨-̨1̨(̨1̨-̨q̨_̨j̨)̨^̨n̨_̨j̨ ̨.̨ ̨ ̨ ̨ ̨L̨ǫǫk̨įn̨g̨ ̨c̨l̨ǫs̨ęl̨y̨ ̨ąt̨ ̨t̨h̨įs̨,̨ ̨ǫn̨ę ̨c̨ąn̨ ̨s̨ęę ̨α̨ ̨įn̨ ̨t̨h̨ę ̨ ̨ ̨ąųg̨m̨ęn̨t̨ęd̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨ ̨h̨ąs̨ ̨ą ̨g̨ąm̨m̨ą ̨l̨įk̨ęl̨įh̨ǫǫd̨.̨ ̨ ̨ ̨T̨h̨ųs̨,̨ ̨ųs̨įn̨g̨ ̨ą ̨g̨ąm̨m̨ą ̨p̨r̨įǫr̨ ̨ ̨ ̨α̨ ̨∼̨(̨ν̨_̨0̨,̨μ̨_̨0̨)̨ ̨m̨ąk̨ęs̨ ̨ęv̨ęr̨y̨t̨h̨įn̨g̨ ̨w̨ǫr̨k̨ ̨s̨įm̨p̨l̨y̨.̨ ̨ ̨ ̨T̨h̨ę ̨d̨ęr̨įv̨ęd̨ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨ ̨f̨ǫr̨ ̨α̨ ̨įs̨ ̨ąs̨ ̨f̨ǫl̨l̨ǫw̨įn̨g̨:̨ ̨ ̨ ̨ ̨*̨ ̨ ̨ ̨ ̨s̨ąm̨p̨l̨ę ̨q̨_̨j̨∼̨(̨α̨,̨n̨_̨j̨)̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨j̨,̨ ̨t̨h̨ęn̨ ̨ ̨ ̨ ̨*̨ ̨ ̨ ̨ ̨s̨ąm̨p̨l̨ę ̨α̨∼̨(̨ν̨_̨0̨+̨∑̨_̨j̨ ̨t̨_̨j̨,̨ ̨μ̨_̨0̨+̨∑̨_̨j̨ ̨l̨ǫg̨ ̨1̨/̨q̨_̨j̨)̨.̨ ̨ ̨N̨ǫt̨ę ̨f̨ǫr̨ ̨ǫųr̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨w̨ę ̨ųs̨ę ̨ąn̨ ̨ęm̨p̨įr̨įc̨ąl̨ ̨B̨ąy̨ęs̨įąn̨ ̨ąp̨p̨r̨ǫąc̨h̨,̨ ̨s̨ǫ ̨ν̨_̨0̨=̨μ̨_̨0̨=̨0̨,̨ ̨ąn̨d̨ ̨l̨ęąv̨ę ̨t̨h̨ę ̨įs̨s̨ųę ̨ǫf̨ ̨s̨ęl̨ęc̨t̨įn̨g̨ ̨ąn̨ ̨ąp̨p̨r̨ǫp̨r̨įąt̨ę ̨p̨r̨įǫr̨ ̨ąs̨ ̨f̨ųr̨t̨h̨ęr̨ ̨r̨ęs̨ęąr̨c̨h̨.̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Ąl̨g̨ǫr̨įt̨h̨m̨įc̨ ̨d̨ęs̨c̨r̨įp̨t̨įǫn̨ ̨W̨ę ̨p̨r̨ęs̨ęn̨t̨ ̨h̨ęr̨ę ̨ą ̨h̨įg̨h̨-̨l̨ęv̨ęl̨ ̨d̨ęs̨c̨r̨įp̨t̨įǫn̨ ̨ǫf̨ ̨ǫųr̨ ̨s̨ąm̨p̨l̨ęr̨ ̨ąn̨d̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨H̨D̨P̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨įn̨ ̨Ąl̨g̨ǫr̨įt̨h̨m̨s̨ ̨<̨r̨ęf̨>̨ ̨t̨ǫ ̨<̨r̨ęf̨>̨.̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨įs̨ ̨t̨h̨ę ̨m̨ąįn̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨:̨ ̨įt̨ ̨t̨ąk̨ęs̨ ̨ąs̨ ̨ąn̨ ̨įn̨p̨ųt̨ ̨ą ̨d̨ąt̨ąs̨ęt̨ ̨ąn̨d̨ ̨r̨ęt̨ųr̨n̨s̨ ̨ą ̨c̨ǫn̨t̨ęx̨t̨ ̨t̨r̨ęę ̨c̨ǫn̨t̨ąįn̨įn̨g̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ę.̨ ̨Įt̨ ̨s̨t̨ąr̨t̨s̨ ̨b̨y̨ ̨c̨r̨ęąt̨įn̨g̨ ̨t̨h̨ę ̨t̨r̨ęę ̨b̨ąs̨ęd̨ ̨ǫn̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨,̨ ̨į.̨ę.̨ ̨c̨r̨ęąt̨įn̨g̨ ̨t̨h̨ę ̨b̨r̨ąn̨c̨h̨ęs̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨c̨ąs̨ęs̨ ̨p̨r̨ęs̨ęn̨t̨ ̨įn̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨,̨ ̨ąs̨ ̨w̨ęl̨l̨ ̨ąs̨ ̨s̨t̨ǫr̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫųn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąt̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨.̨ ̨T̨h̨ę ̨t̨r̨ęę ̨įs̨ ̨ą ̨t̨y̨p̨įc̨ąl̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨w̨įt̨h̨ ̨ą ̨r̨ǫǫt̨ ̨n̨ǫd̨ę;̨ ̨n̨ǫd̨ęs̨ ̨c̨ǫn̨t̨ąįn̨ ̨t̨h̨ę ̨c̨ǫųn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨t̨_̨⋆̨ ̨ąn̨d̨ ̨n̨_̨⋆̨,̨ ̨ą ̨l̨įn̨k̨ ̨t̨ǫ ̨įt̨s̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨α̨ ̨ąn̨d̨ ̨ą ̨l̨įn̨k̨ ̨t̨ǫ ̨ą ̨t̨ąb̨l̨ę ̨ǫf̨ ̨c̨h̨įl̨d̨r̨ęn̨ ̨(̨ǫn̨ę ̨c̨h̨įl̨d̨ ̨p̨ęr̨ ̨v̨ąl̨ųę ̨ǫf̨ ̨t̨h̨ę ̨b̨r̨ąn̨c̨h̨įn̨g̨ ̨v̨ąr̨įąb̨l̨ę ̨ąt̨ ̨t̨h̨ąt̨ ̨n̨ǫd̨ę)̨.̨ ̨Įt̨ ̨t̨h̨ęn̨ ̨c̨ąl̨l̨s̨ ̨t̨h̨ę ̨įn̨įt̨įąl̨įs̨ąt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨⋆̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę,̨ ̨ąn̨d̨ ̨c̨r̨ęąt̨ęs̨ ̨ąn̨ ̨ąr̨r̨ąy̨ ̨ǫf̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨h̨ąt̨ ̨ąr̨ę ̨t̨įęd̨ ̨ąt̨ ̨ęąc̨h̨ ̨l̨ęv̨ęl̨.̨ ̨Įt̨ ̨t̨h̨ęn̨ ̨p̨r̨ǫc̨ęęd̨s̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨p̨r̨ǫc̨ęs̨s̨.̨ ̨F̨ǫr̨ ̨ęąc̨h̨ ̨įt̨ęr̨ąt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨p̨r̨ǫc̨ęs̨s̨,̨ ̨w̨ę ̨f̨įr̨s̨t̨ ̨s̨ąm̨p̨l̨ę ̨t̨h̨ę ̨t̨_̨⋆̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨ ̨ųp̨ ̨t̨ǫ ̨t̨h̨ę ̨r̨ǫǫt̨,̨ ̨t̨h̨ęn̨ ̨w̨ę ̨s̨ąm̨p̨l̨ę ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨(̨ǫn̨ę ̨p̨ęr̨ ̨l̨ęv̨ęl̨ ̨ęx̨c̨ęp̨t̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨n̨ǫd̨ę,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨n̨ǫt̨ ̨s̨ąm̨p̨l̨ęd̨)̨.̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨ąf̨t̨ęr̨ ̨t̨h̨ę ̨b̨ųr̨n̨-̨įn̨ ̨p̨ęr̨įǫd̨ ̨h̨ąs̨ ̨p̨ąs̨s̨ęd̨,̨ ̨w̨ę ̨r̨ęc̨ǫr̨d̨ ̨ąn̨d̨ ̨ąv̨ęr̨ąg̨ę ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ąt̨ ̨t̨h̨ę ̨c̨ųr̨r̨ęn̨t̨ ̨įt̨ęr̨ąt̨įǫn̨.̨ ̨W̨h̨ęn̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨įs̨ ̨t̨ęr̨m̨įn̨ąt̨ęd̨,̨ ̨t̨h̨ęs̨ę ̨ąv̨ęr̨ąg̨ęd̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨(̨s̨t̨ǫr̨ęd̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę)̨ ̨c̨ǫn̨s̨t̨įt̨ųt̨ę ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨;̨ ̨t̨h̨ęy̨ ̨c̨ąn̨ ̨b̨ę ̨ąc̨c̨ęs̨s̨ęd̨ ̨b̨y̨ ̨q̨ųęr̨y̨įn̨g̨ ̨t̨h̨ę ̨t̨r̨ęę.̨ ̨F̨ǫr̨ ̨b̨r̨ęv̨įt̨y̨,̨ ̨w̨ę ̨d̨ǫ ̨n̨ǫt̨ ̨d̨ęs̨c̨r̨įb̨ę ̨t̨h̨ę ̨f̨ǫl̨l̨ǫw̨įn̨g̨ ̨s̨įm̨p̨l̨ę ̨f̨ųn̨c̨t̨įǫn̨s̨:̨ ̨ ̨ ̨ ̨ ̨ ̨*̨ ̨:̨ ̨r̨ęt̨ųr̨n̨įn̨g̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨ąt̨ ̨ą ̨g̨įv̨ęn̨ ̨d̨ęp̨t̨h̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ ̨ ̨ ̨ ̨ ̨*̨ ̨:̨ ̨c̨r̨ęąt̨įn̨g̨ ̨t̨h̨ę ̨b̨r̨ąn̨c̨h̨ęs̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę ̨d̨ǫw̨n̨ ̨t̨ǫ ̨t̨h̨ę ̨l̨ęąv̨ęs̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨d̨ąt̨ą ̨ęx̨įs̨t̨s̨ ̨ ̨ ̨ ̨ ̨ ̨*̨ ̨:̨ ̨c̨r̨ęąt̨įn̨g̨ ̨ąn̨ ̨ąr̨r̨ąy̨ ̨ǫf̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨ǫb̨j̨ęc̨t̨s̨ ̨ǫf̨ ̨g̨įv̨ęn̨ ̨s̨įz̨ę ̨ ̨ ̨ ̨ ̨ ̨*̨ ̨:̨ ̨ąv̨ęr̨ąg̨įn̨g̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ ̨ ̨:̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨n̨Įt̨ęr̨s̨:̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨t̨ǫ ̨r̨ųn̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨n̨B̨ųr̨n̨Įn̨:̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨b̨ųr̨n̨-̨įn̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨b̨ęf̨ǫr̨ę ̨s̨t̨ąr̨t̨įn̨g̨ ̨t̨ǫ ̨ąv̨ęr̨ąg̨ę ̨ǫųt̨ ̨t̨h̨ę ̨θ̨s̨Ęs̨t̨įm̨ąt̨ęP̨r̨ǫb̨H̨D̨B̨(̨d̨ąt̨ą,̨ ̨n̨Įt̨ęr̨s̨,̨ ̨n̨B̨ųr̨n̨Įn̨)̨ ̨t̨r̨ęę ̨←̨ ̨įn̨įt̨T̨r̨ęęW̨įt̨h̨D̨ąt̨ąs̨ęt̨(̨)̨ ̨*̨c̨r̨ęąt̨ę ̨t̨r̨ęę ̨w̨įt̨h̨ ̨ąv̨ąįl̨.̨ ̨d̨ąt̨ą ̨įn̨įt̨P̨ąr̨ąm̨ęt̨ęr̨s̨R̨ęc̨ųr̨s̨įv̨ęl̨y̨(̨t̨r̨ęę.̨r̨ǫǫt̨)̨*̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨t̨ąb̨l̨ę ̨ǫf̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨s̨,̨ ̨ǫn̨ę ̨p̨ęr̨ ̨l̨ęv̨ęl̨ ̨(̨ ̨t̨y̨įn̨g̨)̨ ̨c̨T̨ąb̨ ̨←̨ ̨c̨r̨ęąt̨ęC̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨Ąr̨r̨ąy̨(̨t̨r̨ęę.̨d̨ęp̨t̨h̨)̨ ̨d̨ęp̨t̨h̨1̨t̨r̨ęę.̨d̨ęp̨t̨h̨n̨ǫd̨ę∈̨t̨r̨ęę.̨g̨ęt̨N̨ǫd̨ęs̨Ąt̨D̨ęp̨t̨h̨(̨d̨ęp̨t̨h̨)̨ ̨ ̨ ̨ ̨ ̨n̨ǫd̨ę.̨α̨c̨T̨ąb̨[̨d̨ęp̨t̨h̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨(̨*̨[̨f̨]̨G̨įb̨b̨s̨ ̨s̨ąm̨p̨l̨ęr̨)̨įt̨ęr̨←̨ ̨1̨n̨Įt̨ęr̨s̨s̨ąm̨p̨l̨įn̨g̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨n̨ǫd̨ęs̨ ̨b̨ǫt̨t̨ǫm̨-̨ųp̨d̨ęp̨t̨h̨←̨t̨r̨ęę.̨d̨ęp̨t̨h̨1̨ ̨ ̨ ̨ ̨ ̨n̨ǫd̨ę∈̨t̨r̨ęę.̨g̨ęt̨N̨ǫd̨ęs̨Ąt̨D̨ęp̨t̨h̨(̨d̨ęp̨t̨h̨)̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨s̨ąm̨p̨l̨ęN̨ǫd̨ę(̨n̨ǫd̨ę,̨1̨0̨,̨c̨T̨ąb̨[̨d̨ęp̨t̨h̨]̨)̨ ̨*̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨(̨*̨[̨f̨]̨s̨ąm̨p̨l̨įn̨g̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨s̨)̨l̨ęv̨ęl̨←̨ ̨2̨t̨r̨ęę.̨d̨ęp̨t̨h̨ ̨ ̨ ̨ ̨ ̨s̨ąm̨p̨l̨ęC̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨(̨α̨,̨t̨r̨ęę.̨g̨ęt̨N̨ǫd̨ęs̨Ąt̨D̨ęp̨t̨h̨(̨l̨ęv̨ęl̨)̨)̨*̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨ ̨ ̨įt̨ęr̨>̨n̨B̨ųr̨n̨Įn̨ ̨ ̨ ̨ ̨ ̨r̨ęc̨ǫr̨d̨P̨r̨ǫb̨ąb̨įl̨įt̨y̨R̨ęc̨ųr̨s̨įv̨ęl̨y̨(̨t̨r̨ęę.̨r̨ǫǫt̨)̨ ̨ ̨ ̨ ̨t̨r̨ęę ̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨d̨ęs̨c̨r̨įb̨ęs̨ ̨t̨h̨ę ̨įn̨įt̨įąl̨įs̨ąt̨įǫn̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę'̨s̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨p̨ęr̨f̨ǫr̨m̨ęd̨ ̨b̨ǫt̨t̨ǫm̨-̨ųp̨.̨ ̨S̨t̨ąr̨t̨įn̨g̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨,̨ ̨w̨ę ̨p̨r̨ǫp̨ąg̨ąt̨ę ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨t̨_̨⋆̨,̨ ̨w̨h̨įc̨h̨ ̨c̨ǫn̨s̨t̨įt̨ųt̨ęs̨ ̨t̨h̨ę ̨n̨_̨⋆̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨ ̨n̨ǫd̨ęs̨ ̨(̨l̨įn̨ęs̨ ̨1̨–̨9̨)̨.̨ ̨Įn̨įt̨įąl̨įs̨ąt̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨⋆̨ ̨įs̨ ̨d̨ǫn̨ę ̨f̨ǫl̨l̨ǫw̨įn̨g̨ ̨Ęq̨ųąt̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨n̨ǫd̨ę:̨ ̨n̨ǫd̨ę ̨ǫf̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨w̨ąn̨t̨ ̨t̨ǫ ̨įn̨įt̨įąl̨įs̨ę ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨įn̨įt̨P̨ąr̨ąm̨ęt̨ęr̨s̨R̨ęc̨ųr̨s̨įv̨ęl̨y̨(̨n̨ǫd̨ę)̨(̨*̨[̨f̨]̨įn̨įt̨.̨ ̨c̨h̨įl̨d̨r̨ęn̨ ̨ąn̨d̨ ̨c̨ǫl̨l̨ęc̨t̨ ̨s̨t̨ąt̨s̨)̨n̨ǫd̨ę ̨įs̨ ̨n̨ǫt̨ ̨ą ̨l̨ęąf̨c̨h̨įl̨d̨∈̨n̨ǫd̨ę.̨c̨h̨įl̨d̨r̨ęn̨ ̨ ̨ ̨ ̨ ̨įn̨įt̨P̨ąr̨ąm̨ęt̨ęr̨s̨R̨ęc̨ųr̨s̨įv̨ęl̨y̨(̨c̨h̨įl̨d̨)̨ ̨k̨ ̨ ̨1̨|̨X̨_̨c̨|̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨n̨ǫd̨ę.̨n̨[̨k̨]̨n̨ǫd̨ę.̨n̨[̨k̨]̨ ̨+̨c̨h̨įl̨d̨.̨t̨[̨k̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨n̨ǫd̨ę.̨n̨n̨ǫd̨ę.̨n̨ ̨+̨n̨ǫd̨ę.̨n̨[̨k̨]̨*̨m̨ąr̨g̨įn̨ąl̨n̨ǫd̨ę ̨įs̨ ̨r̨ǫǫt̨∀̨ ̨k̨,̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨m̨įn̨(̨1̨,̨n̨ǫd̨ę.̨n̨[̨k̨]̨)̨k̨ ̨ ̨1̨|̨X̨_̨c̨|̨n̨ǫd̨ę.̨n̨[̨k̨]̨⩽̨ ̨1̨n̨ǫd̨ę.̨t̨[̨k̨]̨n̨ǫd̨ę.̨n̨[̨k̨]̨n̨ǫd̨ę.̨t̨[̨k̨]̨m̨ąx̨(̨1̨,̨⌊̨n̨ǫd̨ę.̨α̨(̨ ̨ψ̨_̨0̨(̨n̨ǫd̨ę.̨α̨+̨n̨ǫd̨ę.̨n̨)̨-̨ψ̨_̨0̨(̨n̨ǫd̨ę.̨α̨)̨)̨⌋̨)̨n̨ǫd̨ę.̨t̨ ̨∑̨_̨k̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨*̨m̨ąr̨g̨įn̨ąl̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨d̨ęs̨c̨r̨įb̨ęs̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨t̨_̨⋆̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨ą ̨n̨ǫd̨ę,̨ ̨į.̨ę.̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨t̨h̨ąt̨ ̨s̨h̨ǫųl̨d̨ ̨b̨ę ̨p̨r̨ǫp̨ąg̨ąt̨ęd̨ ̨ųp̨ ̨t̨ǫ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨n̨ǫd̨ę.̨ ̨S̨ąm̨p̨l̨įn̨g̨ ̨h̨ąp̨p̨ęn̨s̨ ̨įf̨ ̨ąn̨d̨ ̨ǫn̨l̨y̨ ̨įf̨ ̨t̨h̨ę ̨n̨ǫd̨ę ̨įs̨ ̨n̨ǫt̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨n̨ǫd̨ę,̨ ̨ąn̨d̨ ̨t̨h̨ę ̨n̨_̨⋆̨ ̨c̨ǫųn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąr̨ę ̨s̨t̨r̨įc̨t̨l̨y̨ ̨g̨r̨ęąt̨ęr̨ ̨t̨h̨ąn̨ ̨1̨.̨[̨Įf̨ ̨n̨_̨⋆̨=̨0̨,̨ ̨t̨h̨ęn̨ ̨n̨ǫ ̨d̨ąt̨ą ̨h̨ąs̨ ̨b̨ęęn̨ ̨p̨r̨ǫp̨ąg̨ąt̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨c̨h̨įl̨d̨r̨ęn̨,̨ ̨ąn̨d̨ ̨h̨ęn̨c̨ę ̨n̨ǫ ̨d̨ąt̨ą ̨c̨ąn̨ ̨b̨ę ̨p̨r̨ǫp̨ąg̨ąt̨ęd̨ ̨ųp̨ ̨t̨ǫ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨.̨ ̨Įf̨ ̨n̨_̨⋆̨=̨1̨,̨ ̨t̨h̨ęn̨ ̨t̨h̨ąt̨ ̨d̨ąt̨ąp̨ǫįn̨t̨ ̨h̨ąs̨ ̨t̨ǫ ̨b̨ę ̨p̨r̨ǫp̨ąg̨ąt̨ęd̨ ̨t̨ǫ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨ąn̨d̨ ̨h̨ęn̨c̨ę ̨n̨ęęd̨s̨ ̨n̨ǫ ̨s̨ąm̨p̨l̨įn̨g̨.̨]̨ ̨T̨h̨ę ̨p̨s̨ęųd̨ǫ ̨c̨ǫųn̨t̨ ̨įs̨ ̨t̨h̨ęn̨ ̨s̨ąm̨p̨l̨ęd̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨w̨įn̨d̨ǫw̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨;̨ ̨v̨ąl̨ųęs̨ ̨ęįt̨h̨ęr̨ ̨ǫųt̨s̨įd̨ę ̨t̨h̨įs̨ ̨w̨įn̨d̨ǫw̨,̨ ̨ǫr̨ ̨įm̨p̨ǫs̨s̨įb̨l̨ę ̨g̨įv̨ęn̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨ąt̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨ ̨g̨ęt̨ ̨ąs̨s̨įg̨n̨ęd̨ ̨ą ̨0̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ǫf̨ ̨b̨ęįn̨g̨ ̨s̨ąm̨p̨l̨ęd̨ ̨(̨s̨ęę ̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨)̨.̨ ̨V̨ąl̨įd̨ ̨v̨ąl̨ųęs̨ ̨w̨įt̨h̨įn̨ ̨t̨h̨ę ̨w̨įn̨d̨ǫw̨ ̨ąr̨ę ̨s̨ąm̨p̨l̨ęd̨ ̨f̨ǫl̨l̨ǫw̨įn̨g̨ ̨t̨h̨ę ̨Ęq̨ųąt̨įǫn̨s̨ ̨p̨r̨ęs̨ęn̨t̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨n̨ǫd̨ę:̨ ̨n̨ǫd̨ę ̨ǫf̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨w̨ąn̨t̨ ̨t̨ǫ ̨s̨ąm̨p̨l̨ę ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨w̨:̨ ̨w̨įn̨d̨ǫw̨ ̨f̨ǫr̨ ̨s̨ąm̨p̨l̨įn̨g̨α̨:̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨t̨ǫ ̨ąs̨s̨įg̨n̨ ̨t̨ǫ ̨n̨ǫd̨ęs̨ąm̨p̨l̨ęN̨ǫd̨ę(̨n̨ǫd̨ę,̨w̨,̨α̨)̨n̨ǫd̨ę ̨įs̨ ̨r̨ǫǫt̨∀̨ ̨k̨,̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨m̨įn̨(̨1̨,̨n̨ǫd̨ę.̨n̨[̨k̨]̨)̨*̨n̨ǫ ̨s̨ąm̨p̨l̨įn̨g̨ ̨ ̨ ̨n̨ǫd̨ę.̨α̨α̨*̨ąs̨s̨įg̨n̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨t̨ǫ ̨n̨ǫd̨ęk̨ ̨ ̨1̨ ̨⋯̨|̨X̨_̨c̨|̨n̨ǫd̨ę.̨n̨[̨k̨]̨⩽̨ ̨1̨n̨ǫd̨ę.̨t̨[̨k̨]̨n̨ǫd̨ę.̨n̨[̨k̨]̨*̨v̨ąl̨ųę ̨f̨įx̨ęd̨m̨įn̨T̨k̨m̨ąx̨(̨1̨,̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨-̨w̨ ̨ ̨)̨ ̨m̨ąx̨T̨k̨m̨įn̨(̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨+̨w̨,̨n̨ǫd̨ę.̨n̨[̨k̨]̨ ̨)̨ ̨C̨ǫn̨s̨t̨r̨ųc̨t̨įn̨g̨ ̨ą ̨v̨ęc̨t̨ǫr̨ ̨t̨ǫ ̨s̨ąm̨p̨l̨ę ̨n̨ǫd̨ę.̨t̨[̨k̨]̨ ̨f̨r̨ǫm̨v̨̨⃗0̨̨⃗*̨l̨ęn̨g̨t̨h̨ ̨įs̨ ̨(̨n̨ǫd̨ę.̨n̨[̨k̨]̨+̨1̨)̨t̨m̨įn̨T̨k̨⋯̨m̨ąx̨T̨k̨v̨̨⃗_̨t̨c̨h̨ąn̨g̨ęT̨k̨Ąn̨d̨G̨ęt̨P̨r̨ǫb̨ąb̨įl̨įt̨y̨(̨n̨ǫd̨ę,̨k̨,̨t̨)̨*̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨∀̨ ̨t̨,̨v̨̨⃗_̨t̨v̨̨⃗_̨t̨/̨∑̨_̨t̨ ̨v̨̨⃗_̨t̨*̨N̨ǫr̨m̨ąl̨įz̨ę ̨v̨ęc̨t̨ǫr̨t̨ ̨∼̨(̨ ̨v̨̨⃗)̨ ̨c̨h̨ąn̨g̨ęT̨k̨Ąn̨d̨G̨ęt̨P̨r̨ǫb̨ąb̨įl̨įt̨y̨(̨n̨ǫd̨ę,̨k̨,̨t̨)̨*̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨b̨ǫt̨h̨ ̨c̨h̨ąn̨g̨ęs̨ ̨t̨h̨ę ̨v̨ąl̨ųę ̨ǫf̨ ̨ą ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨t̨_̨⋆̨ ̨ąt̨ ̨ą ̨n̨ǫd̨ę ̨ąn̨d̨ ̨r̨ęt̨ųr̨n̨s̨ ̨įt̨s̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨.̨ ̨Ąs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨ąb̨ǫv̨ę,̨ ̨įt̨ ̨s̨t̨ąr̨t̨s̨ ̨b̨y̨ ̨c̨h̨ęc̨k̨įn̨g̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨n̨ęw̨ ̨v̨ąl̨ųę ̨f̨ǫr̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨įs̨ ̨v̨ąl̨įd̨ ̨(̨ęl̨s̨ę ̨d̨ǫęs̨ ̨n̨ǫt̨ ̨d̨ǫ ̨t̨h̨ę ̨c̨h̨ąn̨g̨ę ̨ąn̨d̨ ̨r̨ęt̨ųr̨n̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨0̨)̨.̨ ̨Įt̨ ̨t̨h̨ęn̨ ̨ųp̨d̨ąt̨ęs̨ ̨t̨h̨ę ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨ ̨f̨ǫr̨ ̨t̨h̨ąt̨ ̨n̨ǫd̨ę,̨ ̨ąn̨d̨ ̨t̨h̨ę ̨c̨ǫųn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨ ̨n̨_̨⋆̨ ̨ąt̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨.̨ ̨Įt̨ ̨f̨įn̨ąl̨l̨y̨ ̨r̨ęt̨ųr̨n̨s̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ąs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨n̨ǫd̨ę:̨ ̨n̨ǫd̨ę ̨ǫf̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨w̨ąn̨t̨ ̨t̨ǫ ̨s̨ąm̨p̨l̨ę ̨t̨h̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨k̨:̨ ̨įn̨d̨ęx̨ ̨ǫf̨ ̨t̨h̨ę ̨v̨ąl̨ųę ̨w̨ę ̨w̨ąn̨t̨ ̨t̨ǫ ̨c̨h̨ąn̨g̨ę ̨įn̨ ̨t̨n̨ęw̨V̨ąl̨ųę:̨ ̨v̨ąl̨ųę ̨t̨ǫ ̨r̨ęp̨l̨ąc̨ę ̨t̨_̨k̨ ̨b̨y̨,̨ ̨įf̨ ̨p̨ǫs̨s̨įb̨l̨ęc̨h̨ąn̨g̨ęT̨k̨Ąn̨d̨G̨ęt̨P̨r̨ǫb̨ąb̨įl̨įt̨y̨(̨n̨ǫd̨ę,̨k̨,̨n̨ęw̨V̨ąl̨ųę)̨įn̨c̨n̨ęw̨V̨ąl̨ųę ̨-̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨ ̨(̨*̨[̨f̨]̨c̨h̨ęc̨k̨ ̨įf̨ ̨v̨ąl̨įd̨ ̨f̨ǫr̨ ̨p̨ąr̨ęn̨t̨)̨įn̨c̨<̨ ̨0̨n̨ǫd̨ę ̨įs̨ ̨n̨ǫt̨ ̨r̨ǫǫt̨ ̨ąn̨d̨(̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨[̨k̨]̨+̨įn̨c̨)̨<̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨t̨[̨k̨]̨ ̨0̨ ̨n̨ǫd̨ę.̨t̨[̨k̨]̨n̨ǫd̨ę.̨t̨[̨k̨]̨ ̨+̨ ̨įn̨c̨ ̨n̨ǫd̨ę.̨t̨n̨ǫd̨ę.̨t̨ ̨+̨ ̨įn̨c̨*̨m̨ąr̨g̨įn̨ąl̨(̨*̨[̨f̨]̨ųp̨d̨ąt̨ę ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąt̨ ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨)̨n̨ǫd̨ę ̨įs̨ ̨n̨ǫt̨ ̨r̨ǫǫt̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨[̨k̨]̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨[̨k̨]̨ ̨+̨ ̨įn̨c̨ ̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨ ̨+̨ ̨įn̨c̨*̨m̨ąr̨g̨įn̨ąl̨n̨ǫd̨ę.̨α̨^̨n̨ǫd̨ę.̨t̨[̨k̨]̨·̨ ̨S̨^̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨[̨k̨]̨_̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨t̨[̨k̨]̨·̨ ̨S̨^̨n̨ǫd̨ę.̨n̨[̨k̨]̨_̨n̨ǫd̨ę.̨t̨[̨k̨]̨/̨r̨įs̨įn̨g̨_̨f̨ąc̨t̨ǫr̨įąl̨(̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨α̨,̨n̨ǫd̨ę.̨p̨ąr̨ęn̨t̨.̨n̨[̨k̨]̨)̨ ̨ ̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨Ąl̨g̨ǫr̨įt̨h̨m̨ ̨<̨r̨ęf̨>̨ ̨d̨ęs̨c̨r̨įb̨ęs̨ ̨ą ̨s̨įm̨p̨l̨ę ̨s̨ąm̨p̨l̨įn̨g̨ ̨ǫf̨ ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę,̨ ̨ąs̨s̨ųm̨įn̨g̨ ̨t̨h̨ąt̨ ̨t̨y̨įn̨g̨ ̨įs̨ ̨d̨ǫn̨ę ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨ ̨s̨t̨r̨ąt̨ęg̨y̨.̨ ̨Ąs̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨t̨y̨įn̨g̨ ̨r̨ęq̨ųįr̨ęs̨ ̨t̨ǫ ̨įt̨ęr̨ąt̨ę ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨t̨_̨⋆̨ ̨ąn̨d̨ ̨n̨_̨⋆̨ ̨ǫf̨ ̨t̨h̨ę ̨`̨t̨įęd̨'̨ ̨n̨ǫd̨ęs̨.̨ ̨F̨ǫr̨ ̨ąl̨l̨ ̨t̨h̨ę ̨`̨t̨įęd̨'̨ ̨n̨ǫd̨ęs̨,̨ ̨įt̨ ̨t̨h̨ųs̨ ̨p̨ęr̨f̨ǫr̨m̨s̨ ̨ą ̨c̨h̨ąn̨g̨ę ̨ǫf̨ ̨v̨ąr̨įąb̨l̨ę ̨t̨ǫ ̨q̨ ̨ąn̨d̨ ̨t̨h̨ęn̨ ̨s̨ąm̨p̨l̨ęs̨ ̨t̨h̨ę ̨n̨ęw̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨.̨ ̨Ǫt̨h̨ęr̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨ąr̨ę ̨g̨įv̨ęn̨ ̨įn̨ ̨t̨h̨ę ̨s̨ǫųr̨c̨ę-̨c̨ǫd̨ę ̨f̨ųn̨c̨t̨įǫn̨ ̨(̨s̨ęę ̨b̨ęg̨įn̨n̨įn̨g̨ ̨ǫf̨ ̨S̨ęc̨t̨įǫn̨ ̨6̨.̨1̨ ̨f̨ǫr̨ ̨l̨įn̨k̨ ̨t̨ǫ ̨s̨ǫųr̨c̨ę ̨c̨ǫd̨ę)̨.̨ ̨α̨:̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨t̨ǫ ̨s̨ąm̨p̨l̨ęn̨ǫd̨ęs̨:̨ ̨n̨ǫd̨ęs̨ ̨s̨h̨ąr̨įn̨g̨ ̨t̨h̨įs̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨(̨t̨y̨įn̨g̨)̨s̨ąm̨p̨l̨ęC̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨(̨α̨,̨ ̨n̨ǫd̨ęs̨)̨ ̨r̨ąt̨ę ̨0̨ ̨n̨ǫd̨ę∈̨n̨ǫd̨ęs̨q̨∼̨(̨α̨,̨n̨ǫd̨ę.̨n̨)̨*̨c̨h̨ąn̨g̨ę ̨ǫf̨ ̨v̨ąr̨įąb̨l̨ę,̨ ̨s̨ąm̨p̨l̨ę ̨q̨ ̨ ̨ ̨r̨ąt̨ę ̨ ̨r̨ąt̨ę ̨-̨l̨ǫg̨(̨q̨)̨ ̨ ̨ ̨α̨∼̨(̨∑̨_̨n̨∈̨ ̨n̨ǫd̨ęs̨n̨.̨t̨,̨ ̨r̨ąt̨ę)̨*̨s̨ąm̨p̨l̨ę ̨α̨(̨*̨[̨f̨]̨ąs̨s̨įg̨n̨ ̨n̨ęw̨ ̨α̨ ̨t̨ǫ ̨n̨ǫd̨ęs̨)̨n̨ǫd̨ę∈̨n̨ǫd̨ęs̨n̨ǫd̨ę.̨α̨α̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨W̨ǫr̨k̨ęd̨ ̨ęx̨ąm̨p̨l̨ę ̨ ̨ ̨ ̨W̨ę ̨h̨ąv̨ę ̨n̨ǫw̨ ̨f̨ųl̨l̨y̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨ǫųr̨ ̨H̨D̨P̨-̨b̨ąs̨ęd̨ ̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨ ̨ ̨Įn̨ ̨t̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨,̨ ̨w̨ę ̨d̨r̨ąw̨ ̨ąl̨l̨ ̨t̨h̨ę ̨t̨h̨ęǫr̨y̨ ̨t̨ǫg̨ęt̨h̨ęr̨ ̨ąn̨d̨ ̨s̨h̨ǫw̨ ̨h̨ǫw̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨ąp̨p̨l̨įęs̨ ̨t̨ǫ ̨t̨w̨ǫ ̨s̨įm̨p̨l̨ę ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨h̨įg̨h̨l̨įg̨h̨t̨ęd̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨ ̨B̨ǫt̨h̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨h̨ąv̨ę ̨t̨w̨ǫ ̨b̨įn̨ąr̨y̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨X̨_̨1̨ ̨ąn̨d̨ ̨Y̨,̨ ̨ąn̨d̨ ̨ą ̨s̨įm̨p̨l̨ę ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę,̨ ̨w̨ę ̨f̨ǫc̨ųs̨ ̨ǫn̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨(̨X̨_̨1̨|̨Y̨)̨.̨ ̨ ̨ ̨ ̨Ąl̨t̨h̨ǫųg̨h̨ ̨t̨h̨įs̨ ̨s̨įm̨p̨l̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨d̨ǫęs̨ ̨n̨ǫt̨ ̨g̨įv̨ę ̨f̨ųl̨l̨ ̨j̨ųs̨t̨įc̨ę ̨t̨ǫ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨d̨ęęp̨ęr̨ ̨h̨įęr̨ąr̨c̨h̨įęs̨,̨ ̨w̨ę ̨f̨ęęl̨ ̨t̨h̨ąt̨ ̨s̨ųc̨h̨ ̨ąn̨ ̨ęx̨ąm̨p̨l̨ę ̨h̨ęl̨p̨s̨ ̨ųn̨d̨ęr̨s̨t̨ąn̨d̨įn̨g̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨c̨ǫm̨p̨ǫn̨ęn̨t̨s̨ ̨ǫf̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨D̨ąt̨ąs̨ęt̨ ̨ ̨ ̨V̨ąl̨ųę ̨ ̨ ̨F̨r̨ęq̨ųęn̨c̨y̨ ̨ ̨ ̨3̨c̨p̨̨̂(̨X̨_̨1̨|̨Y̨)̨ ̨4̨-̨6̨ ̨ ̨ ̨f̨ǫr̨ ̨Y̨ ̨ ̨ ̨n̨_̨X̨_̨1̨|̨y̨ ̨ ̨ ̨M̨L̨Ę ̨ ̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨(̨m̨=̨1̨)̨ ̨ ̨ ̨ ̨H̨D̨P̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨1̨ ̨ ̨ ̨0̨ ̨ ̨ ̨[̨2̨,̨0̨]̨ ̨ ̨ ̨[̨1̨.̨0̨0̨,̨0̨.̨0̨0̨]̨ ̨ ̨ ̨[̨0̨.̨8̨3̨,̨0̨.̨1̨7̨]̨ ̨ ̨ ̨[̨0̨.̨8̨9̨,̨0̨.̨1̨1̨]̨ ̨ ̨ ̨ ̨1̨ ̨ ̨ ̨[̨2̨0̨,̨5̨]̨ ̨ ̨ ̨[̨0̨.̨8̨0̨,̨0̨.̨2̨0̨]̨ ̨ ̨ ̨[̨0̨.̨7̨9̨,̨0̨.̨2̨1̨]̨ ̨ ̨ ̨[̨0̨.̨7̨9̨,̨0̨.̨2̨0̨]̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨2̨ ̨ ̨ ̨0̨ ̨ ̨ ̨[̨2̨,̨0̨]̨ ̨ ̨ ̨[̨1̨.̨0̨0̨,̨0̨.̨0̨0̨]̨ ̨ ̨ ̨[̨0̨.̨8̨3̨,̨0̨.̨1̨7̨]̨ ̨ ̨ ̨[̨0̨.̨8̨6̨,̨0̨.̨1̨4̨]̨ ̨ ̨ ̨ ̨1̨ ̨ ̨ ̨[̨4̨,̨9̨]̨ ̨ ̨ ̨[̨0̨.̨3̨1̨,̨0̨.̨6̨9̨]̨ ̨ ̨ ̨[̨0̨.̨3̨2̨,̨0̨.̨6̨8̨]̨ ̨ ̨ ̨[̨0̨.̨3̨4̨,̨0̨.̨6̨6̨]̨Ęx̨ąm̨p̨l̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨w̨įt̨h̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨ęs̨t̨įm̨ąt̨ęs̨Ǫųr̨ ̨ąįm̨ ̨įs̨ ̨t̨ǫ ̨h̨įg̨h̨l̨įg̨h̨t̨ ̨h̨ǫw̨ ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨įs̨ ̨s̨h̨ąr̨ęd̨ ̨b̨ęt̨w̨ęęn̨ ̨(̨X̨_̨1̨|̨Y̨=̨0̨)̨ ̨ąn̨d̨ ̨(̨X̨_̨1̨|̨Y̨=̨1̨)̨ ̨t̨h̨r̨ǫųg̨h̨ ̨t̨h̨ę ̨m̨ąr̨g̨įn̨ąl̨ ̨(̨m̨ęąn̨)̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨(̨X̨_̨1̨)̨.̨ ̨L̨ęt̨ ̨ųs̨ ̨d̨ęs̨c̨r̨įb̨ę ̨t̨h̨ę ̨t̨w̨ǫ ̨d̨ąt̨ąs̨ęt̨s̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨:̨ ̨D̨ąt̨ąs̨ęt̨ ̨#̨1̨ ̨h̨ąs̨ ̨(̨X̨_̨1̨|̨Y̨=̨0̨)̨≈̨(̨X̨_̨1̨|̨Y̨=̨1̨)̨–̨ ̨b̨ųt̨ ̨w̨įt̨h̨ ̨ǫn̨l̨y̨ ̨l̨įt̨t̨l̨ę ̨d̨ąt̨ą ̨ąv̨ąįl̨ąb̨l̨ę ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨(̨X̨_̨1̨|̨Y̨=̨0̨)̨ ̨–̨ ̨w̨h̨įl̨ę ̨D̨ąt̨ąs̨ęt̨ ̨#̨2̨ ̨h̨ąs̨ ̨(̨X̨_̨1̨|̨Y̨=̨0̨)̨ ̨≉̨(̨X̨_̨1̨|̨Y̨=̨1̨)̨.̨L̨ęt̨ ̨ųs̨ ̨s̨t̨ąr̨t̨ ̨b̨y̨ ̨t̨h̨ę ̨ąn̨ąl̨y̨s̨įs̨ ̨ǫf̨ ̨t̨h̨ę ̨c̨ąs̨ęs̨ ̨w̨įt̨h̨ ̨Y̨=̨0̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨t̨w̨ǫ ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨c̨ąs̨ęs̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨ąv̨ąįl̨ąb̨l̨ę ̨įs̨ ̨įd̨ęn̨t̨įc̨ąl̨.̨ ̨T̨h̨ę ̨f̨įr̨s̨t̨ ̨t̨h̨įn̨g̨ ̨t̨ǫ ̨ǫb̨s̨ęr̨v̨ę ̨įs̨ ̨t̨h̨ąt̨,̨ ̨ąs̨ ̨t̨h̨ę ̨f̨r̨ęq̨ųęn̨c̨y̨ ̨įs̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨f̨ǫr̨ ̨b̨ǫt̨h̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨c̨ąs̨ęs̨ ̨w̨įt̨h̨ ̨Y̨=̨0̨,̨ ̨s̨ǫ ̨ąr̨ę ̨t̨h̨ę ̨M̨L̨Ęs̨ ̨ąn̨d̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨[̨M̨ǫr̨ę ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨ąb̨ǫųt̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨įs̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨]̨,̨ ̨r̨ęs̨p̨ęc̨t̨įv̨ęl̨y̨.̨ ̨M̨L̨Ęs̨ ̨ąn̨d̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąr̨ę ̨ąg̨n̨ǫs̨t̨įc̨ ̨ǫf̨ ̨t̨h̨ę ̨m̨ąr̨g̨įn̨ąl̨;̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫn̨l̨y̨ ̨p̨ųl̨l̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨ǫw̨ąr̨d̨ ̨ą ̨ųn̨įf̨ǫr̨m̨ ̨p̨r̨įǫr̨.̨ ̨S̨ęc̨ǫn̨d̨,̨ ̨w̨ę ̨c̨ąn̨ ̨ǫb̨s̨ęr̨v̨ę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨D̨ąt̨ąs̨ęt̨ ̨#̨1̨ ̨ąr̨ę ̨c̨l̨ǫs̨ęr̨ ̨t̨ǫ ̨t̨h̨ę ̨M̨L̨Ęs̨ ̨t̨h̨ąn̨ ̨t̨ǫ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨T̨h̨įs̨ ̨įs̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨d̨ąt̨ą ̨ąv̨ąįl̨ąb̨l̨ę ̨f̨ǫr̨ ̨Y̨=̨1̨ ̨`̨c̨ǫr̨r̨ǫb̨ǫr̨ąt̨ęs̨'̨ ̨t̨h̨ę ̨f̨ąc̨t̨ ̨t̨h̨ąt̨ ̨(̨X̨_̨1̨=̨0̨|̨Y̨)̨ ̨įs̨ ̨m̨ųc̨h̨ ̨g̨r̨ęąt̨ęr̨ ̨t̨h̨ąn̨ ̨(̨X̨_̨1̨=̨1̨|̨Y̨)̨.̨ ̨F̨ǫr̨ ̨D̨ąt̨ąs̨ęt̨ ̨#̨2̨ ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨t̨w̨ǫ ̨c̨ąs̨ęs̨ ̨f̨ǫr̨ ̨Y̨ ̨d̨įf̨f̨ęr̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ę ̨f̨ǫr̨ ̨Y̨=̨0̨ ̨įs̨ ̨c̨l̨ǫs̨ęr̨ ̨t̨ǫ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ąn̨ ̨įt̨ ̨w̨ąs̨ ̨f̨ǫr̨ ̨D̨ąt̨ąs̨ęt̨ ̨#̨1̨ ̨ąl̨t̨h̨ǫųg̨h̨ ̨t̨h̨ę ̨f̨r̨ęq̨ųęn̨c̨įęs̨ ̨ąr̨ę ̨t̨h̨ę ̨s̨ąm̨ę;̨ ̨t̨h̨įs̨ ̨įs̨ ̨b̨ęc̨ąųs̨ę ̨n̨ǫw̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨ąv̨ąįl̨ąb̨l̨ę ̨f̨ǫr̨ ̨Y̨=̨1̨ ̨d̨ǫęs̨ ̨n̨ǫt̨ ̨s̨ųp̨p̨ǫr̨t̨ ̨t̨h̨ę ̨h̨y̨p̨ǫt̨h̨ęs̨įs̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨m̨ąr̨g̨įn̨ąl̨ ̨(̨X̨_̨1̨)̨ ̨įs̨ ̨h̨ęl̨p̨f̨ųl̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨(̨X̨_̨1̨|̨Y̨=̨0̨)̨ ̨w̨h̨įl̨ę ̨h̨ąv̨įn̨g̨ ̨l̨įt̨t̨l̨ę ̨d̨ąt̨ą ̨ąv̨ąįl̨ąb̨l̨ę.̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ę ̨f̨ǫr̨ ̨(̨X̨_̨1̨|̨Y̨=̨1̨)̨ ̨įn̨ ̨D̨ąt̨ąs̨ęt̨ ̨#̨2̨ ̨g̨ǫęs̨ ̨ęv̨ęn̨ ̨f̨ųr̨t̨h̨ęr̨ ̨t̨h̨ąn̨ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨ąn̨d̨ ̨p̨ųl̨l̨s̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ę ̨ęv̨ęn̨ ̨c̨l̨ǫs̨ęr̨ ̨t̨ǫ ̨ą ̨ųn̨įf̨ǫr̨m̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨.̨ ̨T̨h̨įs̨ ̨įs̨ ̨ąg̨ąįn̨ ̨h̨ęr̨ę ̨b̨ęc̨ąųs̨ę ̨ǫf̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨f̨ǫr̨ ̨Y̨=̨0̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨§̨ ̨R̨ĘL̨ĄT̨ĘD̨ ̨W̨ǪR̨K̨ ̨ ̨Ęx̨t̨ęn̨s̨įv̨ę ̨d̨įs̨c̨ųs̨s̨įǫn̨s̨ ̨ǫf̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨f̨ǫr̨ ̨D̨P̨ ̨ąn̨d̨ ̨P̨Y̨P̨ ̨h̨įęr̨ąr̨c̨h̨įęs̨ ̨ąr̨ę ̨p̨r̨ęs̨ęn̨t̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨.̨ ̨S̨t̨ąn̨d̨ąr̨d̨ ̨C̨h̨įn̨ęs̨ę ̨r̨ęs̨t̨ąųr̨ąn̨t̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨(̨C̨R̨P̨)̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨<̨c̨įt̨.̨>̨ ̨ųs̨ę ̨d̨y̨n̨ąm̨įc̨ ̨m̨ęm̨ǫr̨y̨ ̨s̨ǫ ̨ąr̨ę ̨c̨ǫm̨p̨ųt̨ąt̨įǫn̨ąl̨l̨y̨ ̨d̨ęm̨ąn̨d̨įn̨g̨,̨ ̨ąn̨d̨ ̨n̨ǫt̨ ̨b̨ęįn̨g̨ ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨ąl̨s̨ǫ ̨m̨ąk̨ęs̨ ̨t̨h̨ęm̨ ̨c̨ǫn̨s̨įd̨ęr̨ąb̨l̨y̨ ̨s̨l̨ǫw̨ęr̨.̨ ̨<̨c̨įt̨.̨>̨ ̨d̨ęąl̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨ąs̨ę ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨c̨ǫųn̨t̨s̨ ̨ąt̨ ̨t̨h̨ę ̨l̨ęąv̨ęs̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ąr̨ę ̨l̨ąt̨ęn̨t̨,̨ ̨ąn̨d̨ ̨t̨h̨ųs̨ ̨ąr̨ę ̨n̨ǫt̨ ̨ąp̨p̨l̨įc̨ąb̨l̨ę ̨t̨ǫ ̨ǫųr̨ ̨c̨ǫn̨t̨ęx̨t̨.̨ ̨T̨h̨ę ̨d̨įr̨ęc̨t̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨ǫf̨ ̨<̨c̨įt̨.̨>̨,̨ ̨w̨h̨įc̨h̨ ̨ąr̨ę ̨ąl̨s̨ǫ ̨c̨ǫl̨l̨ąp̨s̨ęd̨ ̨C̨R̨P̨ ̨s̨ąm̨p̨l̨ęr̨s̨,̨ ̨ąr̨ę ̨m̨ǫr̨ę ̨ęf̨f̨įc̨įęn̨t̨ ̨t̨h̨ąn̨ ̨C̨R̨P̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨ąn̨d̨ ̨t̨h̨ǫs̨ę ̨ǫf̨ ̨<̨c̨įt̨.̨>̨ ̨įn̨ ̨t̨h̨ę ̨c̨ųr̨r̨ęn̨t̨ ̨c̨ǫn̨t̨ęx̨t̨.̨ ̨<̨c̨įt̨.̨>̨ ̨d̨ęąl̨t̨ ̨w̨įt̨h̨ ̨ą ̨P̨Y̨P̨ ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨d̨įs̨c̨ǫųn̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨c̨h̨ąn̨g̨ę ̨f̨r̨ęq̨ųęn̨t̨l̨y̨ ̨s̨ǫ ̨d̨įr̨ęc̨t̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨w̨ęr̨ę ̨įn̨ęf̨f̨įc̨įęn̨t̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨c̨ąc̨h̨ę ̨ǫf̨ ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨s̨ ̨n̨ęęd̨ęd̨ ̨c̨ǫn̨s̨t̨ąn̨t̨ ̨r̨ęc̨ǫm̨p̨ųt̨ąt̨įǫn̨.̨ ̨Ǫn̨-̨t̨h̨ę-̨f̨l̨y̨ ̨s̨ąm̨p̨l̨ęr̨s̨ ̨h̨ąv̨ę ̨ąl̨s̨ǫ ̨b̨ęęn̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨ ̨f̨ǫr̨ ̨P̨Y̨P̨ ̨h̨įęr̨ąr̨c̨h̨įęs̨,̨ ̨m̨ąk̨įn̨g̨ ̨įt̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨t̨ǫ ̨ųs̨ę ̨P̨Y̨P̨ ̨f̨ǫr̨ ̨d̨ęęp̨ ̨t̨r̨ęęs̨ ̨ąn̨d̨ ̨l̨ąr̨g̨ę ̨d̨ąt̨ąs̨ęt̨ ̨s̨įz̨ęs̨.̨ ̨ ̨T̨h̨įs̨ ̨h̨ǫw̨ęv̨ęr̨ ̨d̨ǫęs̨ ̨n̨ǫt̨ ̨c̨h̨ąn̨g̨ę ̨t̨h̨ę ̨įs̨s̨ųę ̨ǫf̨ ̨c̨ǫn̨s̨t̨ąn̨t̨ ̨r̨ęc̨ǫm̨p̨ųt̨ąt̨įǫn̨ ̨ǫf̨ ̨S̨t̨įr̨l̨įn̨g̨ ̨n̨ųm̨b̨ęr̨s̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨w̨h̨y̨ ̨įn̨įt̨įąl̨įs̨ąt̨įǫn̨s̨ ̨b̨ąs̨ęd̨ ̨ǫn̨ ̨m̨ǫd̨įf̨įęd̨ ̨K̨n̨ęs̨ęr̨-̨N̨ęy̨ ̨h̨ąv̨ę ̨b̨ęęn̨ ̨d̨ęv̨ęl̨ǫp̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨ ̨T̨h̨ę ̨ųs̨ę ̨ǫf̨ ̨D̨P̨ ̨ąn̨d̨ ̨P̨Y̨P̨ ̨h̨įęr̨ąr̨c̨h̨įęs̨ ̨f̨ǫr̨ ̨r̨ęg̨r̨ęs̨s̨įǫn̨ ̨ąn̨d̨ ̨c̨l̨ųs̨t̨ęr̨įn̨g̨ ̨–̨ ̨ąs̨ ̨ǫp̨p̨ǫs̨ęd̨ ̨t̨ǫ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨įn̨ ̨ǫųr̨ ̨c̨ąs̨ę ̨–̨ ̨h̨ąs̨ ̨b̨ęęn̨ ̨s̨t̨ųd̨įęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨,̨ ̨r̨ęs̨p̨ęc̨t̨įv̨ęl̨y̨.̨ ̨ ̨ ̨R̨ęl̨ąt̨ęd̨ ̨w̨ǫr̨k̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨ ̨w̨ąs̨ ̨d̨įs̨c̨ųs̨s̨ęd̨ ̨įn̨ ̨<̨r̨ęf̨>̨.̨ ̨T̨h̨ęr̨ę ̨ąr̨ę ̨ǫt̨h̨ęr̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨f̨ǫr̨ ̨įm̨p̨r̨ǫv̨įn̨g̨ ̨B̨N̨C̨s̨.̨ ̨Ą ̨s̨įm̨p̨l̨ę ̨b̨ąc̨k̨-̨ǫf̨f̨ ̨s̨t̨r̨ąt̨ęg̨y̨,̨ ̨b̨ąc̨k̨įn̨g̨ ̨ǫf̨f̨ ̨t̨ǫ ̨t̨h̨ę ̨r̨ǫǫt̨,̨ ̨įs̨ ̨p̨r̨ǫp̨ǫs̨ęd̨ ̨b̨y̨ ̨<̨c̨įt̨.̨>̨.̨ ̨M̨ǫr̨ęǫv̨ęr̨,̨ ̨f̨ǫr̨ ̨s̨ǫm̨ę ̨s̨įm̨p̨l̨ę ̨c̨l̨ąs̨s̨ęs̨ ̨ǫf̨ ̨n̨ęt̨w̨ǫr̨k̨s̨,̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨T̨ĄN̨,̨ ̨ą ̨d̨įs̨c̨įm̨įn̨ąt̨įv̨ę ̨g̨ęn̨ęr̨ąl̨įs̨ąt̨įǫn̨ ̨ǫf̨ ̨l̨ǫg̨įs̨t̨įc̨ ̨r̨ęg̨r̨ęs̨s̨įǫn̨ ̨c̨ąn̨ ̨b̨ę ̨ųs̨ęd̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨ǫp̨t̨įm̨įs̨ąt̨įǫn̨ ̨s̨ųr̨f̨ąc̨ę ̨įs̨ ̨c̨ǫn̨v̨ęx̨ ̨<̨c̨įt̨.̨>̨.̨ ̨N̨ęįt̨h̨ęr̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨ ̨ąr̨ę ̨ąp̨p̨l̨įc̨ąb̨l̨ę ̨t̨ǫ ̨t̨h̨ę ̨m̨ǫr̨ę ̨c̨ǫm̨p̨l̨ęx̨ ̨B̨N̨C̨s̨ ̨w̨ę ̨c̨ǫn̨s̨įd̨ęr̨.̨ ̨ ̨B̨ąy̨ęs̨įąn̨ ̨m̨ǫd̨ęl̨ ̨ąv̨ęr̨ąg̨įn̨g̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨ąr̨ę ̨c̨ǫm̨m̨ǫn̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨l̨ęąr̨n̨įn̨g̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨Ąv̨ęr̨ąg̨ę ̨n̨-̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨ęs̨t̨įm̨ąt̨ǫr̨s̨ ̨–̨ ̨Ąn̨D̨Ę ̨<̨c̨įt̨.̨>̨,̨ ̨ąn̨ǫt̨h̨ęr̨ ̨ęn̨s̨ęm̨b̨l̨ę ̨m̨ęt̨h̨ǫd̨,̨ ̨įs̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ę ̨f̨ǫr̨ ̨s̨m̨ąl̨l̨ęr̨ ̨d̨ąt̨ą ̨s̨ęt̨s̨ ̨b̨ųt̨ ̨c̨ąn̨n̨ǫt̨ ̨c̨ǫm̨p̨ęt̨ę ̨ąg̨ąįn̨s̨t̨ ̨S̨k̨D̨B̨ ̨f̨ǫr̨ ̨l̨ąr̨g̨ęr̨ ̨d̨ąt̨ą ̨s̨ęt̨s̨ ̨<̨c̨įt̨.̨>̨.̨ ̨ ̨Ęįt̨h̨ęr̨ ̨w̨ąy̨,̨ ̨t̨h̨ęs̨ę ̨įn̨v̨ąr̨įąb̨l̨y̨ ̨ųs̨ę ̨t̨h̨ę ̨s̨ąm̨ę ̨L̨ąp̨l̨ąc̨įąn̨ ̨p̨r̨įǫr̨ ̨ąs̨ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨r̨ęp̨ǫr̨t̨ęd̨ ̨h̨ęr̨ę ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨§̨ ̨ĘX̨P̨ĘR̨ĮM̨ĘN̨T̨S̨ ̨ ̨T̨h̨ę ̨ąįm̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨ ̨įs̨ ̨t̨ǫ ̨ąs̨s̨ęs̨s̨ ̨ǫųr̨ ̨H̨D̨P̨-̨b̨ąs̨ęd̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨(̨B̨N̨C̨s̨)̨.̨ ̨ ̨Įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨w̨ę ̨g̨įv̨ę ̨t̨h̨ę ̨g̨ęn̨ęr̨ąl̨ ̨s̨ęt̨t̨įn̨g̨s̨ ̨t̨h̨ąt̨ ̨ąr̨ę ̨n̨ęc̨ęs̨s̨ąr̨y̨ ̨t̨ǫ ̨ųn̨d̨ęr̨s̨t̨ąn̨d̨ ̨ąn̨d̨ ̨r̨ęp̨r̨ǫd̨ųc̨ę ̨ǫųr̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨.̨ ̨T̨h̨ęn̨,̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨w̨ę ̨s̨t̨ąr̨t̨ ̨b̨y̨ ̨s̨t̨ųd̨y̨įn̨g̨ ̨h̨ǫw̨ ̨t̨ǫ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ę ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨:̨ ̨į.̨ę.̨ ̨b̨y̨ ̨s̨t̨ųd̨y̨įn̨g̨ ̨t̨h̨ę ̨įn̨f̨l̨ųęn̨c̨ę ̨ǫf̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨ąn̨d̨ ̨t̨h̨ę ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨ųs̨ęd̨.̨ ̨Įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨,̨ ̨w̨ę ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨ę ̨t̨h̨ę ̨s̨ųp̨ęr̨įǫr̨įt̨y̨ ̨ǫf̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨s̨t̨ąt̨ę ̨ǫf̨ ̨t̨h̨ę ̨ąr̨t̨ ̨ąc̨r̨ǫs̨s̨ ̨8̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨B̨N̨C̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨.̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨h̨ąv̨įn̨g̨ ̨ǫb̨t̨ąįn̨ęd̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨ ̨įm̨p̨r̨ǫv̨ęm̨ęn̨t̨s̨ ̨ǫv̨ęr̨ ̨t̨h̨ę ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨,̨ ̨w̨ę ̨t̨h̨ęn̨ ̨t̨ųr̨n̨ ̨t̨ǫ ̨c̨ǫm̨p̨ąr̨įn̨g̨ ̨t̨h̨ę ̨b̨ęs̨t̨-̨p̨ęr̨f̨ǫr̨m̨įn̨g̨ ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨ ̨(̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨ ̨w̨įt̨h̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨)̨ ̨w̨įt̨h̨ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨ ̨(̨R̨F̨)̨ ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨.̨ ̨W̨ę ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ę ̨ąl̨l̨ǫw̨s̨ ̨ęv̨ęn̨ ̨m̨ǫd̨ęl̨s̨ ̨ąs̨ ̨s̨įm̨p̨l̨ę ̨ąs̨ ̨T̨ĄN̨ ̨t̨ǫ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨R̨F̨ ̨(̨w̨įt̨h̨ ̨s̨t̨ąt̨įs̨t̨įc̨ąl̨ ̨s̨įg̨n̨įf̨įc̨ąn̨c̨ę)̨,̨ ̨w̨h̨įl̨ę ̨s̨t̨ąn̨d̨ąr̨d̨ ̨ąp̨p̨r̨ǫąc̨h̨ęs̨ ̨t̨ǫ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ąr̨ę ̨b̨ęąt̨ęn̨ ̨b̨y̨ ̨R̨F̨.̨ ̨W̨ę ̨c̨ǫn̨c̨l̨ųd̨ę ̨t̨h̨ę ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨w̨įt̨h̨ ̨ą ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨įǫn̨ ̨ǫf̨ ̨ǫųr̨ ̨s̨y̨s̨t̨ęm̨'̨s̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨c̨ąp̨ąb̨įl̨įt̨y̨ ̨ąn̨d̨ ̨s̨h̨ǫw̨ ̨r̨ęs̨ųl̨t̨s̨ ̨ǫb̨t̨ąįn̨ęd̨ ̨ǫn̨ ̨t̨h̨ę ̨S̨p̨l̨įc̨ę ̨d̨ąt̨ąs̨ęt̨ ̨w̨įt̨h̨ ̨5̨0̨ ̨m̨įl̨l̨įǫn̨ ̨t̨r̨ąįn̨įn̨g̨ ̨ęx̨ąm̨p̨l̨ęs̨,̨ ̨ą ̨q̨ųąn̨t̨įt̨y̨ ̨t̨h̨ąt̨ ̨R̨F̨ ̨c̨ąn̨n̨ǫt̨ ̨h̨ąn̨d̨l̨ę ̨ǫn̨ ̨m̨ǫs̨t̨ ̨m̨ąc̨h̨įn̨ęs̨.̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Ęx̨p̨ęr̨įm̨ęn̨t̨ąl̨ ̨d̨ęs̨įg̨n̨ ̨ąn̨d̨ ̨s̨ęt̨t̨įn̨g̨ ̨D̨ęs̨įg̨n̨:̨ ̨Ąl̨l̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨ąr̨ę ̨c̨ąr̨r̨įęd̨ ̨ǫųt̨ ̨ǫn̨ ̨ą ̨t̨ǫt̨ąl̨ ̨ǫf̨ ̨6̨8̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨ŲC̨Į ̨ąr̨c̨h̨įv̨ę ̨<̨c̨įt̨.̨>̨;̨ ̨3̨8̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨w̨įt̨h̨ ̨l̨ęs̨s̨ ̨t̨h̨ąn̨ ̨1̨0̨0̨0̨ ̨įn̨s̨t̨ąn̨c̨ęs̨,̨ ̨2̨3̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨w̨įt̨h̨ ̨įn̨s̨t̨ąn̨c̨ęs̨ ̨b̨ęt̨w̨ęęn̨ ̨1̨0̨0̨0̨ ̨ąn̨d̨ ̨1̨0̨0̨0̨0̨,̨ ̨ąn̨d̨ ̨7̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨w̨įt̨h̨ ̨m̨ǫr̨ę ̨t̨h̨ąn̨ ̨1̨0̨0̨0̨0̨ ̨įn̨s̨t̨ąn̨c̨ęs̨.̨ ̨ ̨T̨h̨ę ̨l̨įs̨t̨ ̨ąn̨d̨ ̨d̨ęs̨c̨r̨įp̨t̨įǫn̨ ̨ǫf̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨įs̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨ ̨ąt̨ ̨t̨h̨ę ̨ęn̨d̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨.̨ ̨ ̨F̨ǫr̨ ̨ąl̨l̨ ̨m̨ęt̨h̨ǫd̨s̨,̨ ̨n̨ųm̨ęr̨įc̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨ąr̨ę ̨d̨įs̨c̨r̨ęt̨įz̨ęd̨ ̨b̨y̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨m̨įn̨įm̨ųm̨ ̨d̨ęs̨c̨r̨įp̨t̨įǫn̨ ̨l̨ęn̨g̨t̨h̨ ̨(̨M̨D̨L̨)̨ ̨d̨įs̨c̨r̨ęt̨įz̨ąt̨įǫn̨ ̨m̨ęt̨h̨ǫd̨ ̨<̨c̨įt̨.̨>̨.̨ ̨Ą ̨m̨įs̨s̨įn̨g̨ ̨v̨ąl̨ųę ̨įs̨ ̨t̨r̨ęąt̨ęd̨ ̨ąs̨ ̨ą ̨s̨ęp̨ąr̨ąt̨ę ̨ąt̨t̨r̨įb̨ųt̨ę ̨v̨ąl̨ųę ̨ąn̨d̨ ̨t̨ąk̨ęn̨ ̨įn̨t̨ǫ ̨ąc̨c̨ǫųn̨t̨ ̨ęx̨ąc̨t̨l̨y̨ ̨l̨įk̨ę ̨ǫt̨h̨ęr̨ ̨v̨ąl̨ųęs̨.̨ ̨Ęąc̨h̨ ̨ąl̨g̨ǫr̨įt̨h̨m̨ ̨įs̨ ̨t̨ęs̨t̨ęd̨ ̨ǫn̨ ̨ęąc̨h̨ ̨d̨ąt̨ąs̨ęt̨ ̨ųs̨įn̨g̨ ̨2̨-̨f̨ǫl̨d̨ ̨c̨r̨ǫs̨s̨ ̨v̨ąl̨įd̨ąt̨įǫn̨ ̨r̨ęp̨ęąt̨ęd̨ ̨5̨ ̨t̨įm̨ęs̨.̨ ̨W̨ę ̨ąs̨s̨ęs̨s̨ ̨t̨h̨ę ̨r̨ęs̨ųl̨t̨s̨ ̨b̨y̨ ̨r̨ęp̨ǫr̨t̨įn̨g̨ ̨0̨-̨1̨ ̨L̨ǫs̨s̨ ̨ąn̨d̨ ̨R̨M̨S̨Ę,̨ ̨ąn̨d̨ ̨r̨ęp̨ǫr̨t̨ ̨W̨įn̨-̨D̨r̨ąw̨-̨L̨ǫs̨s̨ ̨(̨W̨-̨D̨-̨L̨)̨ ̨r̨ęs̨ųl̨t̨s̨ ̨w̨h̨ęn̨ ̨c̨ǫm̨p̨ąr̨įn̨g̨ ̨t̨h̨ę ̨0̨-̨1̨ ̨L̨ǫs̨s̨ ̨ąn̨d̨ ̨R̨M̨S̨Ę ̨ǫf̨ ̨t̨w̨ǫ ̨m̨ǫd̨ęl̨s̨.̨ ̨ ̨Ą ̨t̨w̨ǫ-̨t̨ąįl̨ ̨b̨įn̨ǫm̨įąl̨ ̨s̨įg̨n̨ ̨t̨ęs̨t̨ ̨įs̨ ̨ųs̨ęd̨ ̨t̨ǫ ̨d̨ęt̨ęr̨m̨įn̨ę ̨t̨h̨ę ̨s̨įg̨n̨įf̨įc̨ąn̨c̨ę ̨ǫf̨ ̨t̨h̨ę ̨r̨ęs̨ųl̨t̨s̨,̨ ̨ųs̨įn̨g̨ ̨p̨ ̨≤̨ ̨0̨.̨0̨5̨.̨ ̨ ̨N̨ǫt̨ę ̨t̨h̨ę ̨R̨M̨S̨Ę ̨įs̨ ̨r̨ęl̨ąt̨ęd̨ ̨t̨ǫ ̨t̨h̨ę ̨B̨r̨įęr̨ ̨s̨c̨ǫr̨ę,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ą ̨p̨r̨ǫp̨ęr̨ ̨s̨c̨ǫr̨įn̨g̨ ̨r̨ųl̨ę ̨f̨ǫr̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ąn̨d̨ ̨t̨h̨ųs̨ ̨g̨ęn̨ęr̨ąl̨l̨y̨ ̨p̨r̨ęf̨ęr̨ąb̨l̨ę ̨t̨ǫ ̨ęr̨r̨ǫr̨,̨ ̨ęs̨p̨ęc̨įąl̨l̨y̨ ̨įn̨ ̨t̨h̨ę ̨c̨ǫn̨t̨ęx̨t̨ ̨ǫf̨ ̨ųn̨ęq̨ųąl̨l̨y̨ ̨ǫc̨c̨ųr̨r̨įn̨g̨ ̨c̨l̨ąs̨s̨ęs̨ ̨ǫr̨ ̨ųn̨ęq̨ųąl̨ ̨c̨ǫs̨t̨s̨.̨ ̨Įt̨ ̨m̨ęąs̨ųr̨ęs̨ ̨h̨ǫw̨ ̨w̨ęl̨l̨ ̨c̨ąl̨įb̨r̨ąt̨ęd̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąr̨ę.̨ ̨W̨ę ̨ųs̨ę ̨įt̨ ̨b̨ęc̨ąųs̨ę ̨w̨ę ̨s̨ųs̨p̨ęc̨t̨ęd̨ ̨t̨h̨ąt̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨s̨ ̨c̨ǫųl̨d̨ ̨įm̨p̨r̨ǫv̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨b̨ųt̨ ̨n̨ǫt̨ ̨n̨ęc̨ęs̨s̨ąr̨įl̨y̨ ̨ęr̨r̨ǫr̨s̨.̨ ̨ ̨S̨ǫf̨t̨w̨ąr̨ę:̨ ̨T̨ǫ ̨ęn̨s̨ųr̨ę ̨r̨ęp̨r̨ǫd̨ųc̨įb̨įl̨įt̨y̨ ̨ǫf̨ ̨ǫųr̨ ̨w̨ǫr̨k̨ ̨ąn̨d̨ ̨ąl̨l̨ǫw̨ ̨ǫt̨h̨ęr̨ ̨r̨ęs̨ęąr̨c̨h̨ęr̨s̨ ̨t̨ǫ ̨ęąs̨įl̨y̨ ̨b̨ųįl̨d̨ ̨ǫn̨ ̨ǫųr̨ ̨r̨ęs̨ęąr̨c̨h̨,̨ ̨w̨ę ̨h̨ąv̨ę ̨m̨ąd̨ę ̨ǫųr̨ ̨s̨ǫųr̨c̨ę ̨c̨ǫd̨ę ̨f̨ǫr̨ ̨H̨D̨P̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ąv̨ąįl̨ąb̨l̨ę ̨ǫn̨ ̨h̨t̨t̨p̨s̨:̨/̨/̨g̨įt̨h̨ųb̨.̨c̨ǫm̨/̨f̨p̨ęt̨įt̨j̨ęąn̨/̨H̨įęr̨ąr̨c̨h̨įc̨ąl̨D̨įr̨įc̨h̨l̨ęt̨P̨r̨ǫc̨ęs̨s̨Ęs̨t̨įm̨ąt̨įǫn̨G̨įt̨h̨ųb̨.̨ ̨ ̨ ̨C̨ǫm̨p̨ąr̨ęd̨ ̨m̨ęt̨h̨ǫd̨s̨:̨ ̨ ̨W̨ę ̨ąs̨s̨ęs̨s̨ ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨8̨ ̨B̨N̨C̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨ ̨w̨įt̨h̨ ̨g̨r̨ǫw̨įn̨g̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨.̨ ̨Ǫųr̨ ̨B̨N̨C̨ ̨s̨t̨r̨ųc̨t̨ųr̨ęs̨ ̨ąr̨ę:̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨(̨N̨B̨)̨,̨ ̨t̨r̨ęę-̨ąųg̨m̨ęn̨t̨ęd̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨(̨T̨ĄN̨)̨ ̨<̨c̨įt̨.̨>̨,̨ ̨k̨-̨d̨ęp̨ęn̨d̨ęn̨c̨ę ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨(̨k̨D̨B̨)̨ ̨<̨c̨įt̨.̨>̨ ̨w̨įt̨h̨ ̨k̨=̨1̨ ̨t̨ǫ ̨5̨ ̨ąn̨d̨ ̨s̨ęl̨ęc̨t̨įv̨ę ̨k̨D̨B̨ ̨(̨S̨k̨D̨B̨)̨ ̨<̨c̨įt̨.̨>̨ ̨w̨įt̨h̨ ̨m̨ąx̨įm̨ųm̨ ̨k̨ ̨s̨ęt̨ ̨t̨ǫ ̨5̨ ̨ąl̨s̨ǫ.̨[̨W̨ę ̨d̨ǫ ̨n̨ǫt̨ ̨c̨ǫn̨s̨įd̨ęr̨ ̨h̨įg̨h̨ęr̨ ̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨k̨,̨ ̨b̨ęc̨ąųs̨ę ̨(̨1̨)̨ ̨f̨ǫr̨ ̨k̨D̨B̨ ̨w̨ę ̨w̨įl̨l̨ ̨s̨ęę ̨įn̨ ̨S̨ęc̨t̨įǫn̨ ̨<̨r̨ęf̨>̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨s̨ųp̨ęr̨įǫr̨įt̨y̨ ̨ǫf̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨įs̨ ̨s̨t̨ąt̨įs̨t̨įc̨ąl̨l̨y̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨ ̨f̨ųr̨t̨h̨ęr̨ ̨įn̨c̨r̨ęąs̨ęs̨ ̨w̨įt̨h̨ ̨k̨;̨ ̨(̨2̨)̨ ̨f̨ǫr̨ ̨S̨k̨D̨B̨,̨ ̨9̨5̨%̨ ̨ǫf̨ ̨t̨h̨ę ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨s̨ęę ̨įt̨ ̨c̨h̨ǫǫs̨ę ̨ą ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨w̨įt̨h̨ ̨k̨<̨5̨,̨ ̨d̨įf̨f̨ęr̨ęn̨c̨ęs̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ęr̨ ̨k̨ ̨w̨ǫųl̨d̨ ̨t̨h̨ųs̨ ̨b̨ę ̨m̨įn̨įm̨ąl̨.̨]̨ ̨W̨h̨ęn̨ ̨c̨ǫm̨p̨ąr̨įn̨g̨ ̨t̨ǫ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨ ̨(̨R̨F̨)̨,̨ ̨w̨ę ̨ųs̨ę ̨t̨h̨ę ̨W̨ęk̨ą ̨d̨ęf̨ąųl̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ąt̨įǫn̨,̨ ̨į.̨ę.̨ ̨s̨ęl̨ęc̨t̨įn̨g̨ ̨l̨ǫg̨_̨2̨(̨n̨)̨+̨1̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨įn̨ ̨ęąc̨h̨ ̨t̨r̨ęę,̨[̨S̨ęl̨ęc̨t̨įn̨g̨ ̨√̨(̨n̨)̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨p̨r̨ǫd̨ųc̨ęs̨ ̨s̨įm̨įl̨ąr̨ ̨r̨ęs̨ųl̨t̨s̨ ̨ąn̨d̨ ̨c̨ǫn̨c̨l̨ųs̨įǫn̨,̨ ̨s̨ǫ ̨t̨h̨ę ̨r̨ęs̨ųl̨t̨s̨ ̨ąr̨ę ̨l̨ęf̨t̨ ̨ǫųt̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨f̨ǫr̨ ̨c̨ǫn̨c̨įs̨įǫn̨.̨]̨ ̨n̨ǫ ̨m̨įn̨įm̨ųm̨ ̨l̨ęąf̨ ̨s̨įz̨ę ̨ąn̨d̨ ̨ųs̨įn̨g̨ ̨1̨0̨0̨ ̨d̨ęc̨įs̨įǫn̨ ̨t̨r̨ęęs̨ ̨įn̨ ̨t̨h̨įs̨ ̨w̨ǫr̨k̨.̨ ̨ ̨F̨ǫr̨ ̨B̨N̨C̨s̨,̨ ̨w̨ę ̨c̨ǫm̨p̨ąr̨ę ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨ǫ ̨s̨ǫ-̨c̨ąl̨l̨ęd̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨[̨Ąl̨s̨ǫ ̨k̨n̨ǫw̨n̨ ̨ąs̨ ̨S̨c̨h̨ųr̨m̨ąn̨n̨-̨G̨r̨ąs̨s̨b̨ęr̨g̨ęr̨'̨s̨ ̨L̨ąw̨ ̨w̨h̨ęn̨ ̨m̨=̨1̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ą ̨p̨ąr̨t̨įc̨ųl̨ąr̨ ̨c̨ąs̨ę ̨ǫf̨ ̨L̨įd̨s̨t̨ǫn̨ę'̨s̨ ̨l̨ąw̨ ̨<̨c̨įt̨.̨>̨ ̨w̨įt̨h̨ ̨λ̨=̨1̨/̨|̨X̨_̨į|̨,̨ ̨ąl̨s̨ǫ ̨b̨ąs̨ęd̨ ̨ǫn̨ ̨ą ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨įǫr̨.̨]̨<̨c̨įt̨.̨>̨ ̨ąs̨ ̨f̨ǫl̨l̨ǫw̨s̨:̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨p̨̨̂(̨x̨_̨į|̨Π̨(̨į)̨)̨ ̨=̨ ̨c̨ǫųn̨t̨s̨(̨x̨_̨į,̨Π̨(̨į)̨)̨+̨m̨/̨|̨X̨_̨į|̨/̨c̨ǫųn̨t̨s̨(̨Π̨(̨į)̨)̨+̨m̨ ̨ ̨ ̨w̨h̨ęr̨ę ̨Π̨(̨į)̨ ̨ąr̨ę ̨t̨h̨ę ̨p̨ąr̨ęn̨t̨-̨v̨ąl̨ųęs̨ ̨ǫf̨ ̨X̨_̨į.̨ ̨T̨h̨ę ̨v̨ąl̨ųę ̨ǫf̨ ̨m̨ ̨įs̨ ̨s̨ęt̨ ̨b̨y̨ ̨c̨r̨ǫs̨s̨-̨v̨ąl̨įd̨ąt̨įǫn̨ ̨ǫn̨ ̨ą ̨h̨ǫl̨d̨ǫųt̨ ̨s̨ęt̨ ̨ǫf̨ ̨s̨įz̨ę ̨m̨įn̨(̨N̨/̨1̨0̨,̨5̨0̨0̨0̨)̨ ̨ąm̨ǫn̨g̨ ̨w̨įt̨h̨ ̨m̨∈̨{̨0̨,̨0̨.̨0̨5̨,̨0̨.̨2̨,̨1̨,̨5̨,̨2̨0̨}̨.̨ ̨ ̨ ̨C̨ǫųn̨t̨ ̨s̨t̨ąt̨įs̨t̨įc̨s̨ ̨ąr̨ę ̨s̨t̨ǫr̨ęd̨ ̨įn̨ ̨ą ̨p̨r̨ęf̨įx̨ ̨t̨r̨ęę;̨ ̨f̨ǫr̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨įf̨ ̨z̨ęr̨ǫ ̨c̨ǫųn̨t̨s̨ ̨ąr̨ę ̨f̨ǫųn̨d̨,̨ ̨w̨ę ̨b̨ąc̨k̨ ̨ǫf̨f̨ ̨ąs̨ ̨m̨ąn̨y̨ ̨l̨ęv̨ęl̨s̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęę ̨ąs̨ ̨n̨ęc̨ęs̨s̨ąr̨y̨ ̨t̨ǫ ̨f̨įn̨d̨ ̨ąt̨ ̨l̨ęąs̨t̨ ̨ǫn̨ę ̨c̨ǫųn̨t̨.̨ ̨F̨ǫr̨ ̨įn̨s̨t̨ąn̨c̨ę,̨ ̨įf̨ ̨c̨ǫųn̨t̨s̨(̨x̨_̨4̨,̨x̨_̨0̨,̨x̨_̨3̨)̨ ̨įs̨ ̨ęq̨ųąl̨ ̨t̨ǫ ̨z̨ęr̨ǫ,̨ ̨t̨h̨ęn̨ ̨p̨̨̂(̨x̨_̨4̨|̨x̨_̨0̨)̨ ̨įs̨ ̨c̨ǫn̨s̨įd̨ęr̨ęd̨ ̨įn̨s̨t̨ęąd̨ ̨ǫf̨ ̨p̨̨̂(̨x̨_̨4̨|̨x̨_̨0̨,̨x̨_̨3̨)̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨n̨ǫt̨ ̨ųs̨įn̨g̨ ̨t̨h̨įs̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨d̨ęg̨r̨ąd̨ęs̨ ̨t̨h̨ę ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę ̨ǫf̨ ̨B̨N̨C̨s̨ ̨w̨h̨ęn̨ ̨ųs̨įn̨g̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨(̨f̨ǫr̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨t̨h̨ę ̨įn̨t̨ęr̨m̨ęd̨įąt̨ę ̨n̨ǫd̨ęs̨ ̨ϕ̨ ̨ąr̨ę ̨c̨ǫn̨s̨įd̨ęr̨ęd̨ ̨l̨ąt̨ęn̨t̨ ̨ąn̨d̨ ̨t̨h̨ųs̨ ̨įn̨f̨ęr̨r̨ęd̨ ̨d̨įr̨ęc̨t̨l̨y̨ ̨d̨ųr̨įn̨g̨ ̨s̨ąm̨p̨l̨įn̨g̨)̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨T̨y̨įn̨g̨ ̨ąn̨d̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨ ̨B̨ęf̨ǫr̨ę ̨p̨r̨ǫc̨ęęd̨įn̨g̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨c̨ǫm̨p̨ąr̨įs̨ǫn̨ ̨ǫf̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨t̨ǫ ̨t̨h̨ę ̨s̨t̨ąt̨ę ̨ǫf̨ ̨t̨h̨ę ̨ąr̨t̨,̨ ̨įt̨ ̨įs̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨t̨ǫ ̨s̨t̨ųd̨y̨ ̨t̨w̨ǫ ̨ęl̨ęm̨ęn̨t̨s̨:̨ ̨(̨1̨)̨ ̨f̨ǫr̨ ̨h̨ǫw̨ ̨m̨ąn̨y̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨t̨ǫ ̨r̨ųn̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨ąn̨d̨ ̨(̨2̨)̨ ̨h̨ǫw̨ ̨t̨ǫ ̨t̨įę ̨t̨h̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨.̨ ̨T̨h̨ęs̨ę ̨t̨w̨ǫ ̨ęl̨ęm̨ęn̨t̨s̨ ̨ąr̨ę ̨d̨įr̨ęc̨t̨l̨y̨ ̨r̨ęl̨ąt̨ęd̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ę ̨l̨ęs̨s̨ ̨t̨y̨įn̨g̨,̨ ̨t̨h̨ę ̨m̨ǫr̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨ǫ ̨įn̨f̨ęr̨,̨ ̨w̨h̨įc̨h̨ ̨m̨ęąn̨s̨ ̨t̨h̨ąt̨ ̨w̨ę ̨ęx̨p̨ęc̨t̨ ̨t̨ǫ ̨h̨ąv̨ę ̨t̨ǫ ̨r̨ųn̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨ ̨m̨ǫr̨ę ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨ ̨ ̨ ̨W̨ę ̨c̨ǫn̨s̨įd̨ęr̨ ̨t̨h̨r̨ęę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨:̨ ̨ ̨ ̨ ̨*̨ ̨ ̨S̨ąm̨ę ̨P̨ąr̨ęn̨t̨ ̨(̨S̨P̨)̨:̨ ̨c̨h̨įl̨d̨r̨ęn̨ ̨ǫf̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨s̨h̨ąr̨ę ̨t̨h̨ę ̨s̨ąm̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨–̨ ̨įl̨l̨ųs̨t̨r̨ąt̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨ą)̨.̨ ̨ ̨ ̨ ̨ ̨*̨ ̨ ̨L̨ęv̨ęl̨ ̨(̨L̨)̨:̨ ̨w̨ę ̨ųs̨ę ̨ǫn̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨l̨ęv̨ęl̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨r̨ęę ̨–̨ ̨įl̨l̨ųs̨t̨r̨ąt̨ęd̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨b̨)̨.̨ ̨ ̨ ̨ ̨ ̨*̨ ̨ ̨S̨įn̨g̨l̨ę ̨(̨S̨)̨:̨ ̨ąl̨l̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨įęd̨ ̨t̨ǫg̨ęt̨h̨ęr̨.̨ ̨N̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨:̨ ̨Ąs̨y̨m̨p̨t̨ǫt̨įc̨ąl̨l̨y̨,̨ ̨t̨h̨ę ̨ąc̨c̨ųr̨ąc̨y̨ ̨ǫf̨ ̨t̨h̨ę ̨ęs̨t̨įm̨ąt̨ęs̨ ̨įm̨p̨r̨ǫv̨ęs̨ ̨ąs̨ ̨w̨ę ̨įn̨c̨r̨ęąs̨ę ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨T̨h̨ę ̨q̨ųęs̨t̨įǫn̨ ̨įs̨ ̨h̨ǫw̨ ̨q̨ųįc̨k̨l̨y̨ ̨t̨h̨ęy̨ ̨ąs̨y̨m̨p̨t̨ǫt̨ę.̨ ̨W̨ę ̨t̨h̨ųs̨ ̨s̨t̨ųd̨įęd̨ ̨t̨h̨ę ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę ̨ǫf̨ ̨ǫųr̨ ̨t̨w̨ǫ ̨f̨l̨ąg̨s̨h̨įp̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨–̨ ̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨ ̨–̨ ̨ǫn̨ ̨ąl̨l̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨ąs̨ ̨w̨ę ̨įn̨c̨r̨ęąs̨ę ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨f̨r̨ǫm̨ ̨5̨0̨0̨ ̨t̨ǫ ̨5̨0̨,̨0̨0̨0̨.̨ ̨F̨ǫr̨ ̨ęąc̨h̨ ̨c̨ǫm̨b̨įn̨ąt̨įǫn̨ ̨ǫf̨ ̨c̨l̨ąs̨s̨įf̨įęr̨×̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨,̨ ̨w̨ę ̨ąs̨s̨ęs̨s̨ ̨t̨h̨ę ̨w̨įn̨-̨l̨ǫs̨s̨ ̨p̨r̨ǫf̨įl̨ę ̨f̨ǫr̨ ̨x̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨v̨ęr̨s̨ųs̨ ̨5̨0̨,̨0̨0̨0̨.̨ ̨T̨h̨ę ̨r̨ęs̨ųl̨t̨įn̨g̨ ̨w̨įn̨-̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨ ̨s̨h̨ǫw̨s̨ ̨t̨h̨ąt̨ ̨ąc̨r̨ǫs̨s̨ ̨ąl̨l̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨ąn̨d̨ ̨m̨ǫd̨ęl̨s̨,̨ ̨r̨ųn̨n̨įn̨g̨ ̨ǫųr̨ ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨įs̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨b̨ęt̨t̨ęr̨ ̨t̨h̨ąn̨ ̨w̨įt̨h̨ ̨f̨ęw̨ęr̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨Ęv̨ęn̨ ̨f̨ǫr̨ ̨m̨ǫd̨ęl̨s̨ ̨ąs̨ ̨s̨įm̨p̨l̨ę ̨ąs̨ ̨T̨ĄN̨ ̨w̨įt̨h̨ ̨ą ̨S̨įn̨g̨l̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨,̨ ̨r̨ųn̨n̨įn̨g̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨ ̨5̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨w̨įn̨s̨ ̨1̨3̨ ̨t̨įm̨ęs̨ ̨ąn̨d̨ ̨l̨ǫs̨ęs̨ ̨4̨2̨ ̨t̨įm̨ęs̨ ̨ąs̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨t̨ǫ ̨r̨ųn̨n̨įn̨g̨ ̨įt̨ ̨f̨ǫr̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨Ųn̨l̨ęs̨s̨ ̨s̨p̨ęc̨įf̨įęd̨ ̨ǫt̨h̨ęr̨w̨įs̨ę,̨ ̨w̨ę ̨t̨h̨ųs̨ ̨r̨ųn̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨f̨ǫr̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨W̨ę ̨s̨ųr̨m̨įs̨ę ̨t̨h̨ąt̨ ̨ęv̨ęn̨ ̨m̨ǫr̨ę ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨c̨ǫųl̨d̨ ̨f̨ųr̨t̨h̨ęr̨ ̨įm̨p̨r̨ǫv̨ę ̨ąc̨c̨ųr̨ąc̨y̨ ̨b̨ųt̨ ̨l̨ęąv̨ę ̨t̨h̨įs̨ ̨f̨ǫr̨ ̨f̨ųt̨ųr̨ę ̨r̨ęs̨ęąr̨c̨h̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨W̨įn̨/̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨ ̨ǫn̨ ̨R̨M̨S̨Ę ̨f̨ǫr̨ ̨ęąc̨h̨ ̨c̨ǫm̨b̨įn̨ąt̨įǫn̨ ̨ǫf̨ ̨(̨f̨l̨ąg̨s̨h̨įp̨ ̨c̨l̨ąs̨s̨įf̨įęr̨)̨ ̨×̨ ̨(̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨)̨.̨ ̨C̨ǫm̨p̨ąr̨įs̨ǫn̨ ̨įs̨ ̨f̨ǫr̨ ̨r̨ųn̨n̨įn̨g̨ ̨ęąc̨h̨ ̨c̨ǫm̨b̨įn̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨x̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨v̨s̨ ̨5̨0̨,̨0̨0̨0̨ ̨ąn̨d̨ ̨įn̨c̨l̨ųd̨ę ̨S̨įn̨g̨l̨ę,̨ ̨L̨ęv̨ęl̨ ̨ąn̨d̨ ̨S̨ąm̨ęP̨ąr̨ęn̨t̨.̨T̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨:̨ ̨ ̨H̨ąv̨įn̨g̨ ̨s̨ęęn̨ ̨t̨h̨ąt̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨s̨ęęm̨s̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨r̨ęg̨ąr̨d̨l̨ęs̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨,̨ ̨w̨ę ̨h̨ęr̨ę ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨t̨y̨įn̨g̨ ̨p̨ęr̨ ̨L̨ęv̨ęl̨ ̨s̨ęęm̨s̨ ̨t̨ǫ ̨b̨ę ̨t̨h̨ę ̨b̨ęs̨t̨ ̨d̨ęf̨ąųl̨t̨ ̨s̨t̨r̨ąt̨ęg̨y̨.̨ ̨Įt̨ ̨įs̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨t̨ǫ ̨n̨ǫt̨ę ̨t̨h̨ąt̨ ̨w̨ę ̨d̨ǫ ̨n̨ǫt̨ ̨įn̨t̨ęn̨d̨ ̨t̨ǫ ̨g̨įv̨ę ̨ą ̨d̨ęf̨įn̨įt̨įv̨ę ̨ąn̨s̨w̨ęr̨ ̨v̨ąl̨įd̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨d̨ǫm̨ąįn̨s̨ ̨h̨ęr̨ę,̨ ̨b̨ųt̨ ̨ąr̨ę ̨s̨įm̨p̨l̨y̨ ̨g̨įv̨įn̨g̨ ̨ą ̨r̨ęąs̨ǫn̨ąb̨l̨ę ̨`̨d̨ęf̨ąųl̨t̨'̨ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ąt̨įǫn̨.̨ ̨T̨h̨ę ̨L̨ęv̨ęl̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨w̨ąs̨ ̨įl̨l̨ųs̨t̨r̨ąt̨ęd̨ ̨f̨ǫr̨ ̨k̨D̨B̨-̨1̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨(̨b̨)̨.̨ ̨ ̨T̨ǫ ̨įl̨l̨ųs̨t̨r̨ąt̨ę ̨t̨h̨įs̨ ̨w̨ę ̨c̨ǫm̨p̨ąr̨ę ̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ęd̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨p̨ąr̨ęn̨t̨ ̨(̨S̨P̨)̨ ̨ąn̨d̨ ̨s̨įn̨g̨l̨ę ̨(̨S̨)̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨v̨ęr̨s̨ųs̨ ̨ųs̨įn̨g̨ ̨t̨h̨ę ̨l̨ęv̨ęl̨ ̨(̨L̨)̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨ąc̨r̨ǫs̨s̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨n̨ųm̨b̨ęr̨s̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨ ̨g̨įv̨ęs̨ ̨t̨h̨ę ̨w̨įn̨-̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨.̨ ̨W̨ę ̨s̨ęę ̨t̨h̨ąt̨ ̨L̨ ̨p̨r̨ǫv̨įd̨ęs̨ ̨ą ̨ųn̨įf̨ǫr̨m̨l̨y̨ ̨g̨ǫǫd̨ ̨s̨ǫl̨ųt̨įǫn̨ ̨p̨r̨ǫv̨įd̨įn̨g̨ ̨b̨ǫt̨h̨ ̨t̨h̨ę ̨b̨ęs̨t̨ ̨r̨ęs̨ųl̨t̨s̨ ̨w̨įt̨h̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨b̨ųt̨ ̨ąl̨s̨ǫ ̨p̨r̨ǫv̨įd̨įn̨g̨ ̨s̨ǫl̨įd̨ ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ęs̨ ̨ąs̨ ̨ęąr̨l̨y̨ ̨ąs̨ ̨5̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨.̨ ̨Įt̨ ̨įs̨ ̨w̨ǫr̨t̨h̨ ̨n̨ǫt̨įn̨g̨ ̨t̨h̨ąt̨ ̨f̨ǫr̨ ̨T̨ĄN̨,̨ ̨t̨h̨ę ̨L̨ ̨ąn̨d̨ ̨S̨ ̨s̨t̨r̨ąt̨ęg̨įęs̨ ̨ąr̨ę ̨v̨ęr̨y̨ ̨s̨įm̨įl̨ąr̨,̨ ̨ǫn̨l̨y̨ ̨d̨įf̨f̨ęr̨įn̨g̨ ̨b̨y̨ ̨ǫn̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨.̨ ̨T̨h̨ę ̨S̨P̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨s̨ęęm̨s̨ ̨t̨ǫ ̨c̨l̨ęąr̨l̨y̨ ̨ųn̨d̨ęr̨p̨ęr̨f̨ǫr̨m̨ ̨L̨,̨ ̨ąl̨l̨ ̨t̨h̨ę ̨m̨ǫr̨ę ̨w̨h̨ęn̨ ̨t̨h̨ę ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨ǫf̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨ ̨įn̨c̨r̨ęąs̨ęs̨,̨ ̨w̨h̨įc̨h̨ ̨m̨ąk̨ęs̨ ̨s̨ęn̨s̨ę ̨g̨įv̨ęn̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨įn̨c̨r̨ęąs̨ęs̨ ̨ęx̨p̨ǫn̨ęn̨t̨įąl̨l̨y̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨d̨ęp̨t̨h̨ ̨ǫf̨ ̨t̨h̨ę ̨p̨r̨ęf̨įx̨ ̨t̨r̨ęę,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨m̨ǫs̨t̨l̨y̨ ̨c̨ǫn̨t̨r̨ǫl̨l̨ęd̨ ̨b̨y̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨į.̨ ̨Įt̨ ̨įs̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨t̨h̨ąt̨ ̨f̨ǫr̨ ̨l̨ąr̨g̨ę ̨ąm̨ǫųn̨t̨s̨ ̨ǫf̨ ̨d̨ąt̨ą,̨ ̨t̨h̨ę ̨S̨P̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨w̨ǫųl̨d̨ ̨ǫf̨f̨ęr̨ ̨ą ̨b̨ęt̨t̨ęr̨ ̨b̨įąs̨/̨v̨ąr̨įąn̨c̨ę ̨t̨r̨ąd̨ęǫf̨f̨ ̨b̨ųt̨ ̨s̨ųc̨h̨ ̨ą ̨s̨t̨ųd̨y̨ ̨f̨ąl̨l̨s̨ ̨ǫųt̨ ̨ǫf̨ ̨t̨h̨ę ̨s̨c̨ǫp̨ę ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨.̨ ̨W̨ę ̨t̨h̨ųs̨ ̨ųs̨ę ̨L̨ ̨ąs̨ ̨ą ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨r̨ęm̨ąįn̨d̨ęr̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨W̨įn̨/̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨ ̨ǫf̨ ̨ęąc̨h̨ ̨c̨ǫm̨b̨įn̨ąt̨įǫn̨ ̨ǫf̨ ̨(̨f̨l̨ąg̨s̨h̨įp̨ ̨c̨l̨ąs̨s̨įf̨įęr̨)̨ ̨×̨ ̨(̨S̨ ̨ǫr̨ ̨S̨P̨ ̨t̨y̨įn̨g̨ ̨s̨t̨r̨ąt̨ęg̨y̨)̨ ̨v̨ęr̨s̨ųs̨ ̨t̨y̨įn̨g̨ ̨ąt̨ ̨l̨ęv̨ęl̨ ̨(̨L̨)̨.̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨H̨D̨P̨ ̨v̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ ̨S̨ǫ ̨f̨ąr̨,̨ ̨w̨ę ̨h̨ąv̨ę ̨ǫn̨l̨y̨ ̨ąs̨s̨ęs̨s̨ęd̨ ̨t̨h̨ę ̨r̨ęl̨ąt̨įv̨ę ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę ̨ǫf̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨w̨įt̨h̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨įz̨ąt̨įǫn̨s̨.̨ ̨H̨ąv̨įn̨g̨ ̨s̨ęt̨t̨l̨ęd̨ ̨ǫn̨ ̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨ąn̨d̨ ̨p̨ęr̨ ̨L̨ęv̨ęl̨ ̨t̨y̨įn̨g̨,̨ ̨w̨ę ̨n̨ǫw̨ ̨t̨ųr̨n̨ ̨t̨ǫ ̨t̨h̨ę ̨f̨ųl̨l̨ ̨c̨ǫm̨p̨ąr̨įs̨ǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨įn̨ ̨s̨m̨ǫǫt̨h̨įn̨g̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨:̨ ̨ųs̨įn̨g̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨v̨ąl̨ųę ̨ǫf̨ ̨m̨ ̨c̨r̨ǫs̨s̨-̨v̨ąl̨įd̨ąt̨ęd̨ ̨ǫn̨ ̨ą ̨h̨ǫl̨d̨ǫųt̨ ̨s̨ęt̨.̨ ̨W̨ę ̨ąl̨s̨ǫ ̨r̨ęm̨įn̨d̨ ̨t̨h̨ę ̨r̨ęąd̨ęr̨ ̨t̨h̨ąt̨,̨ ̨t̨ǫ ̨p̨r̨ǫv̨įd̨ę ̨t̨h̨ę ̨b̨ęs̨t̨ ̨c̨ǫm̨p̨ęt̨įt̨ǫr̨,̨ ̨w̨ę ̨ąl̨s̨ǫ ̨ąd̨d̨ęd̨ ̨t̨h̨ę ̨b̨ąc̨k̨-̨ǫf̨f̨ ̨s̨t̨r̨ąt̨ęg̨y̨ ̨d̨ęs̨c̨r̨įb̨ęd̨ ̨ąb̨ǫv̨ę,̨ ̨w̨įt̨h̨ǫųt̨ ̨w̨h̨įc̨h̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨c̨ąn̨n̨ǫt̨ ̨c̨ǫm̨p̨ęt̨ę ̨ąt̨ ̨ąl̨l̨.̨ ̨ ̨W̨ę ̨r̨ęp̨ǫr̨t̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨ ̨t̨h̨ę ̨w̨įn̨-̨d̨r̨ąw̨-̨l̨ǫs̨s̨ ̨ǫf̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨v̨ęr̨s̨ųs̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąc̨r̨ǫs̨s̨ ̨8̨ ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨B̨N̨C̨s̨ ̨f̨r̨ǫm̨ ̨n̨ąį̈v̨ę ̨B̨ąy̨ęs̨ ̨ąn̨d̨ ̨T̨ĄN̨ ̨t̨ǫ ̨k̨D̨B̨ ̨w̨įt̨h̨ ̨1̨⩽̨ ̨k̨⩽̨ ̨5̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨.̨ ̨ ̨W̨įn̨/̨D̨r̨ąw̨/̨L̨ǫs̨s̨ ̨f̨ǫr̨ ̨8̨ ̨B̨N̨C̨s̨ ̨f̨ǫr̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ę ̨v̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę.̨ ̨S̨t̨ąt̨.̨ ̨s̨įg̨.̨ ̨(̨p̨<̨0̨.̨0̨5̨)̨ ̨r̨ęs̨ųl̨t̨s̨ ̨ąr̨ę ̨d̨ęp̨įc̨t̨ęd̨ ̨įn̨ ̨b̨ǫl̨d̨f̨ąc̨ę.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨C̨l̨ąs̨s̨įf̨įęr̨ ̨ ̨ ̨2̨l̨W̨įn̨–̨d̨r̨ąw̨–̨l̨ǫs̨s̨ ̨f̨ǫr̨ ̨H̨D̨P̨ ̨v̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨2̨-̨3̨ ̨ ̨ ̨ ̨0̨/̨1̨-̨l̨ǫs̨s̨ ̨ ̨ ̨ ̨ ̨R̨M̨S̨Ę ̨ ̨ ̨N̨ąįv̨ę ̨B̨ąy̨ęs̨ ̨ ̨ ̨4̨1̨–̨4̨–̨2̨3̨ ̨ ̨ ̨4̨0̨–̨0̨–̨2̨8̨ ̨T̨ĄN̨ ̨ ̨ ̨4̨5̨–̨4̨–̨1̨9̨ ̨ ̨ ̨5̨2̨–̨1̨–̨1̨5̨ ̨k̨D̨B̨-̨1̨ ̨ ̨ ̨4̨5̨–̨4̨–̨1̨9̨ ̨ ̨ ̨5̨0̨–̨1̨–̨1̨7̨ ̨k̨D̨B̨-̨2̨ ̨ ̨ ̨5̨4̨–̨2̨–̨1̨2̨ ̨ ̨ ̨5̨4̨–̨0̨–̨1̨4̨ ̨k̨D̨B̨-̨3̨ ̨ ̨ ̨5̨2̨–̨4̨–̨1̨2̨ ̨ ̨ ̨5̨3̨–̨2̨–̨1̨3̨ ̨k̨D̨B̨-̨4̨ ̨ ̨ ̨5̨6̨–̨4̨–̨0̨8̨ ̨ ̨ ̨5̨6̨–̨0̨–̨1̨2̨ ̨k̨D̨B̨-̨5̨ ̨ ̨ ̨6̨0̨–̨4̨–̨0̨4̨ ̨ ̨ ̨6̨0̨–̨2̨–̨0̨6̨ ̨S̨k̨D̨B̨ ̨ ̨ ̨4̨5̨–̨4̨–̨1̨9̨ ̨ ̨ ̨5̨4̨–̨0̨–̨1̨4̨ ̨ ̨Įt̨ ̨įs̨ ̨c̨l̨ęąr̨ ̨f̨r̨ǫm̨ ̨t̨h̨įs̨ ̨t̨ąb̨l̨ę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąr̨ę ̨f̨ąr̨ ̨s̨ųp̨ęr̨įǫr̨ ̨t̨ǫ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨Įt̨ ̨įs̨ ̨ęv̨ęn̨ ̨q̨ųįt̨ę ̨s̨ųr̨p̨r̨įs̨įn̨g̨ ̨t̨ǫ ̨s̨ęę ̨ǫųr̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨w̨įt̨h̨ ̨m̨ǫd̨ęl̨s̨ ̨ąs̨ ̨s̨įm̨p̨l̨ę ̨ąs̨ ̨N̨ąį̈v̨ę ̨B̨ąy̨ęs̨,̨ ̨w̨h̨ęr̨ę ̨ǫųr̨ ̨h̨įęr̨ąr̨c̨h̨y̨ ̨ǫn̨l̨y̨ ̨h̨ąs̨ ̨ǫn̨ę ̨s̨įn̨g̨l̨ę ̨l̨ęv̨ęl̨.̨ ̨M̨ǫr̨ęǫv̨ęr̨,̨ ̨ąs̨ ̨t̨h̨ę ̨m̨ǫd̨ęl̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨įn̨c̨r̨ęąs̨ęs̨ ̨(̨t̨h̨ę ̨m̨ąx̨įm̨ųm̨ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨p̨ąr̨ęn̨t̨s̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę)̨,̨ ̨t̨h̨įs̨ ̨d̨įf̨f̨ęr̨ęn̨c̨ę ̨įn̨c̨r̨ęąs̨ęs̨.̨ ̨T̨h̨ę ̨s̨c̨ąt̨t̨ęr̨-̨p̨l̨ǫt̨ ̨f̨ǫr̨ ̨k̨D̨B̨-̨5̨ ̨H̨D̨P̨ ̨v̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨įs̨ ̨g̨įv̨ęn̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨ ̨ąn̨d̨ ̨s̨h̨ǫw̨s̨ ̨ąg̨ąįn̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨t̨r̨ęn̨d̨ ̨w̨įt̨h̨ ̨H̨D̨P̨ ̨s̨įg̨n̨įf̨įc̨ąn̨t̨l̨y̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨įn̨g̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę.̨ ̨Ąs̨ ̨ųs̨ųąl̨ ̨w̨h̨ęn̨ ̨d̨ęąl̨įn̨g̨ ̨w̨įt̨h̨ ̨ą ̨b̨r̨ǫąd̨ ̨r̨ąn̨g̨ę ̨ǫf̨ ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨t̨h̨ęr̨ę ̨ąr̨ę ̨ą ̨f̨ęw̨ ̨p̨ǫįn̨t̨s̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨H̨D̨P̨ ̨l̨ǫs̨ęs̨.̨ ̨Įn̨t̨ęr̨ęs̨t̨įn̨g̨l̨y̨,̨ ̨t̨h̨ę ̨m̨ǫs̨t̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨l̨ǫs̨s̨ ̨įs̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨ ̨d̨ąt̨ąs̨ęt̨,̨ ̨w̨h̨įc̨h̨ ̨c̨ǫn̨t̨ąįn̨ ̨ǫn̨l̨y̨ ̨5̨4̨0̨ ̨s̨ąm̨p̨l̨ęs̨,̨ ̨ąn̨d̨ ̨t̨h̨ųs̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨w̨ǫųl̨d̨ ̨h̨ąv̨ę ̨ęx̨p̨ęc̨t̨ęd̨ ̨t̨h̨ąt̨ ̨s̨m̨ǫǫt̨h̨įn̨g̨ ̨w̨ǫųl̨d̨ ̨b̨ę ̨įm̨p̨ǫr̨t̨ąn̨t̨;̨ ̨d̨ęt̨ąįl̨ęd̨ ̨įn̨s̨p̨ęc̨t̨įǫn̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨d̨ąt̨ąs̨ęt̨ ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨5̨4̨0̨ ̨c̨ąs̨ęs̨ ̨s̨ęęm̨ ̨t̨ǫ ̨b̨ę ̨r̨ęl̨ąt̨įv̨ęl̨y̨ ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ ̨ęąc̨h̨ ̨ǫt̨h̨ęr̨ ̨(̨įn̨ ̨w̨h̨įc̨h̨ ̨c̨ąs̨ę ̨t̨h̨ę ̨c̨r̨ǫs̨s̨-̨v̨ąl̨įd̨ąt̨įǫn̨ ̨ųs̨ęd̨ ̨f̨ǫr̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨h̨ęl̨p̨ ̨d̨įs̨c̨ǫv̨ęr̨ ̨t̨h̨įs̨)̨.̨ ̨ ̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨[̨]̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨(̨ą)̨ ̨S̨c̨ąt̨t̨ęr̨ ̨p̨l̨ǫt̨ ̨ǫn̨ ̨R̨M̨S̨Ę ̨f̨ǫr̨ ̨k̨D̨B̨-̨5̨ ̨f̨ǫr̨ ̨H̨D̨P̨ ̨v̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę.̨ ̨(̨b̨)̨ ̨W̨įn̨/̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨ ̨ǫf̨ ̨k̨D̨B̨-̨5̨ ̨v̨s̨ ̨k̨D̨B̨-̨x̨ ̨f̨ǫr̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨v̨s̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ǫn̨ęs̨.̨ ̨ ̨ ̨Įt̨ ̨įs̨ ̨ąl̨s̨ǫ ̨įn̨t̨ęr̨ęs̨t̨įn̨g̨ ̨t̨ǫ ̨s̨t̨ųd̨y̨ ̨t̨h̨ę ̨c̨ąp̨ąc̨įt̨y̨ ̨ǫf̨ ̨H̨D̨P̨ ̨t̨ǫ ̨p̨r̨ęv̨ęn̨t̨ ̨ǫv̨ęr̨f̨įt̨t̨įn̨g̨ ̨ąs̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨t̨ǫ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ę ̨(̨w̨įt̨h̨ ̨m̨ ̨c̨r̨ǫs̨s̨-̨v̨ąl̨įd̨ąt̨ęd̨)̨.̨ ̨ ̨Įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨,̨ ̨w̨ę ̨r̨ęp̨ǫr̨t̨ ̨f̨ǫr̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨h̨ę ̨w̨įn̨-̨l̨ǫs̨s̨ ̨p̨l̨ǫt̨ ̨f̨ǫr̨ ̨k̨D̨B̨-̨5̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨t̨ǫ ̨k̨D̨B̨s̨ ̨w̨įt̨h̨ ̨įn̨c̨r̨ęąs̨įn̨g̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨f̨r̨ǫm̨ ̨0̨ ̨(̨k̨D̨B̨-̨0̨ ̨įs̨ ̨N̨B̨)̨ ̨t̨ǫ ̨4̨.̨ ̨G̨įv̨ęn̨ ̨t̨h̨ąt̨ ̨k̨D̨B̨-̨5̨ ̨h̨ąs̨ ̨g̨ęn̨ęr̨ąl̨l̨y̨ ̨l̨ǫw̨ęr̨ ̨b̨įąs̨ ̨t̨h̨ąn̨ ̨k̨D̨B̨ ̨∀̨ ̨k̨⩽̨ ̨4̨,̨ ̨w̨ę ̨c̨ąn̨ ̨t̨y̨p̨įc̨ąl̨l̨y̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨įt̨s̨ ̨l̨ǫs̨s̨ęs̨ ̨t̨ǫ ̨ǫv̨ęr̨f̨įt̨t̨įn̨g̨.̨ ̨S̨t̨ąr̨t̨įn̨g̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨b̨ǫt̨t̨ǫm̨ ̨l̨įn̨ę,̨ ̨w̨h̨įc̨h̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨s̨ ̨t̨h̨ę ̨b̨ęh̨ąv̨įǫųr̨ ̨ǫf̨ ̨ųs̨įn̨g̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨k̨D̨B̨-̨5̨ ̨g̨ęn̨ęr̨ąl̨l̨y̨ ̨l̨ǫs̨ęs̨ ̨t̨ǫ ̨l̨ǫw̨ęr̨ ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨k̨D̨B̨s̨.̨ ̨T̨h̨ę ̨m̨ąx̨įm̨ųm̨ ̨d̨įf̨f̨ęr̨ęn̨c̨ę ̨įs̨ ̨w̨įt̨h̨ ̨k̨D̨B̨-̨3̨ ̨w̨h̨įc̨h̨ ̨s̨ęęm̨s̨ ̨t̨ǫ ̨g̨l̨ǫb̨ąl̨l̨y̨ ̨h̨ąv̨ę ̨ą ̨n̨įc̨ę ̨b̨įąs̨/̨v̨ąr̨įąn̨c̨ę ̨t̨r̨ąd̨ęǫf̨f̨ ̨ǫn̨ ̨t̨h̨įs̨ ̨c̨ǫl̨l̨ęc̨t̨įǫn̨ ̨ǫf̨ ̨d̨ąt̨ąs̨ęt̨s̨.̨ ̨ ̨C̨ǫn̨v̨ęr̨s̨ęl̨y̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨(̨t̨ǫp̨-̨c̨ųr̨v̨ę ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨)̨ ̨ ̨ąl̨l̨ǫw̨s̨ ̨ųs̨ ̨t̨ǫ ̨n̨įc̨ęl̨y̨ ̨c̨ǫn̨t̨r̨ǫl̨ ̨f̨ǫr̨ ̨ǫv̨ęr̨f̨įt̨t̨įn̨g̨.̨ ̨ ̨W̨h̨ąt̨ ̨h̨ąp̨p̨ęn̨s̨ ̨įs̨ ̨t̨h̨ąt̨ ̨w̨ę ̨m̨ąk̨ę ̨t̨h̨ę ̨m̨ǫs̨t̨ ̨ǫf̨ ̨t̨h̨ę ̨l̨ǫw̨-̨b̨įąs̨ęd̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ǫf̨f̨ęr̨ęd̨ ̨b̨y̨ ̨k̨D̨B̨,̨ ̨w̨h̨įl̨ę ̨n̨ǫt̨ ̨b̨ęįn̨g̨ ̨ǫv̨ęr̨l̨y̨ ̨p̨r̨ǫn̨ę ̨t̨ǫ ̨ǫv̨ęr̨f̨įt̨t̨įn̨g̨.̨ ̨Įn̨ ̨s̨ǫm̨ę ̨s̨ęn̨s̨ę,̨ ̨ǫųr̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨m̨ąk̨ęs̨ ̨įt̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨t̨ǫ ̨p̨ųl̨l̨ ̨t̨h̨ę ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨ǫw̨ąr̨d̨s̨ ̨h̨įg̨h̨ęr̨-̨l̨ęv̨ęl̨ ̨n̨ǫd̨ęs̨ ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨h̨ąv̨ę ̨m̨ǫr̨ę ̨d̨ąt̨ą,̨ ̨ąn̨d̨ ̨t̨h̨įs̨ ̨ąųt̨ǫm̨ąt̨įc̨ąl̨l̨y̨ ̨d̨ęp̨ęn̨d̨įn̨g̨ ̨ǫn̨ ̨t̨h̨ę ̨d̨ąt̨ąs̨ęt̨.̨ ̨ ̨Įt̨ ̨s̨ęęm̨s̨ ̨t̨h̨ąt̨ ̨įt̨ ̨m̨ąk̨ęs̨ ̨įt̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨t̨ǫ ̨b̨ę ̨l̨ęs̨s̨ ̨s̨t̨r̨įc̨t̨ ̨ąb̨ǫųt̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ąn̨d̨ ̨t̨ǫ ̨b̨ę ̨p̨ǫw̨ęr̨f̨ųl̨ ̨ąt̨ ̨c̨ǫn̨t̨r̨ǫl̨l̨įn̨g̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨v̨ąr̨įąn̨c̨ę.̨ ̨Įn̨ ̨f̨ąc̨t̨,̨ ̨c̨ǫn̨t̨r̨ǫl̨l̨įn̨g̨ ̨f̨ǫr̨ ̨ǫv̨ęr̨f̨įt̨t̨įn̨g̨ ̨įs̨ ̨w̨h̨ąt̨ ̨s̨ęl̨ęc̨t̨įv̨ę ̨k̨D̨B̨ ̨(̨S̨k̨D̨B̨)̨ ̨t̨r̨įęs̨ ̨t̨ǫ ̨ąc̨h̨įęv̨ę;̨ ̨įn̨ ̨ǫųr̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨,̨ ̨k̨D̨B̨5̨-̨H̨D̨P̨ ̨h̨ąs̨ ̨ą ̨s̨l̨įg̨h̨t̨ ̨ęd̨g̨ę ̨ǫv̨ęr̨ ̨S̨k̨D̨B̨5̨-̨H̨D̨P̨ ̨w̨įt̨h̨ ̨ą ̨w̨įn̨-̨d̨r̨ąw̨-̨l̨ǫs̨s̨ ̨ǫf̨ ̨3̨3̨–̨5̨–̨3̨0̨ ̨ǫn̨ ̨R̨M̨S̨Ę.̨ ̨N̨ęv̨ęr̨t̨h̨ęl̨ęs̨s̨,̨ ̨įt̨ ̨r̨ęm̨ąįn̨s̨ ̨t̨h̨ąt̨ ̨H̨D̨P̨ ̨l̨ąr̨g̨ęl̨y̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨w̨įt̨h̨ ̨ą ̨w̨įn̨-̨l̨ǫs̨s̨ ̨–̨ ̨f̨ǫr̨ ̨S̨k̨D̨B̨ ̨–̨ ̨ǫf̨ ̨6̨0̨ ̨t̨ǫ ̨8̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨F̨įn̨ąl̨l̨y̨,̨ ̨w̨ę ̨p̨r̨ęs̨ęn̨t̨ ̨s̨ǫm̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨c̨ųr̨v̨ęs̨ ̨f̨ǫr̨ ̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨ ̨ǫn̨ ̨ą ̨s̨ǫm̨ę ̨l̨ąr̨g̨ęr̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨įn̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨.̨ ̨Ęąc̨h̨ ̨p̨ǫįn̨t̨ ̨c̨ǫr̨r̨ęs̨p̨ǫn̨d̨s̨ ̨t̨h̨ę ̨m̨ęąn̨ ̨R̨M̨S̨Ę ̨f̨ǫr̨ ̨q̨ųąn̨t̨įt̨y̨ ̨ǫf̨ ̨d̨ąt̨ą ̨x̨ ̨ǫv̨ęr̨ ̨1̨0̨ ̨r̨ųn̨s̨.̨ ̨G̨l̨ǫb̨ąl̨l̨y̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨s̨ęęm̨ ̨t̨ǫ ̨`̨l̨ęąr̨n̨'̨ ̨f̨ąs̨t̨ęr̨,̨ ̨į.̨ę.̨ ̨ǫv̨ęr̨f̨įt̨ ̨l̨ęs̨s̨.̨ ̨F̨ǫr̨ ̨t̨h̨ę ̨ ̨d̨ąt̨ąs̨ęt̨,̨ ̨S̨k̨D̨B̨-̨H̨D̨P̨ ̨d̨ǫm̨įn̨ąt̨ęs̨ ̨ąl̨l̨ ̨t̨h̨ę ̨w̨ąy̨ ̨t̨h̨r̨ǫųg̨h̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨c̨ę ̨įn̨ ̨R̨M̨S̨Ę ̨g̨ęt̨t̨įn̨g̨ ̨s̨m̨ąl̨l̨ęr̨ ̨ąs̨ ̨t̨h̨ę ̨q̨ųąn̨t̨įt̨y̨ ̨ǫf̨ ̨d̨ąt̨ą ̨įn̨c̨r̨ęąs̨ęs̨.̨ ̨F̨ǫr̨ ̨,̨ ̨w̨ę ̨c̨ąn̨ ̨ǫb̨s̨ęr̨v̨ę ̨t̨h̨ę ̨s̨ąm̨ę ̨b̨ęh̨ąv̨įǫųr̨ ̨f̨ǫr̨ ̨S̨k̨D̨B̨.̨ ̨Įn̨t̨ęr̨ęs̨t̨įn̨g̨l̨y̨,̨ ̨f̨ǫr̨ ̨T̨ĄN̨ ̨ǫn̨ ̨t̨h̨įs̨ ̨d̨ąt̨ąs̨ęt̨,̨ ̨ąl̨t̨h̨ǫųg̨h̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨d̨ǫ ̨l̨ęąr̨n̨ ̨f̨ąs̨t̨ęr̨,̨ ̨t̨h̨ęy̨ ̨ąr̨ę ̨ǫv̨ęr̨t̨ąk̨ęn̨ ̨b̨y̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąf̨t̨ęr̨ ̨1̨0̨,̨0̨0̨0̨ ̨d̨ąt̨ąp̨ǫįn̨t̨s̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨L̨ęąr̨n̨įn̨g̨ ̨c̨ųr̨v̨ęs̨ ̨ǫn̨ ̨R̨M̨S̨Ę ̨f̨ǫr̨ ̨H̨D̨P̨ ̨ąn̨d̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ę.̨ ̨ ̨T̨h̨ę ̨x̨-̨ąx̨įs̨ ̨įs̨ ̨d̨ąt̨ąs̨ęt̨ ̨s̨įz̨ę,̨ ̨t̨h̨ę ̨y̨-̨ąx̨įs̨ ̨įs̨ ̨R̨M̨S̨Ę.̨ ̨ ̨ ̨§̨.̨§̨ ̨B̨N̨C̨s̨ ̨w̨įt̨h̨ ̨H̨D̨P̨ ̨v̨s̨ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨ ̨ ̨H̨ąv̨įn̨g̨ ̨s̨h̨ǫw̨n̨ ̨t̨h̨ąt̨ ̨ǫųr̨ ̨ąp̨p̨r̨ǫąc̨h̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨s̨ ̨t̨h̨ę ̨s̨t̨ąt̨ę ̨ǫf̨ ̨t̨h̨ę ̨ąr̨t̨ ̨f̨ǫr̨ ̨B̨N̨C̨s̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨,̨ ̨w̨ę ̨c̨ǫm̨p̨ąr̨ę ̨B̨N̨C̨s̨ ̨ųs̨įn̨g̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąg̨ąįn̨s̨t̨ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨ ̨(̨R̨F̨)̨.̨ ̨ ̨T̨h̨ę ̨ąįm̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨ ̨įs̨ ̨n̨ǫt̨ ̨t̨ǫ ̨s̨ųg̨g̨ęs̨t̨ ̨t̨h̨ąt̨ ̨B̨N̨C̨s̨ ̨s̨h̨ǫųl̨d̨ ̨r̨ęp̨l̨ąc̨ę ̨R̨F̨,̨ ̨b̨ųt̨ ̨r̨ąt̨h̨ęr̨ ̨t̨h̨ąt̨ ̨B̨N̨C̨s̨ ̨c̨ąn̨ ̨p̨ęr̨f̨ǫr̨m̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ęl̨y̨.̨ ̨ ̨ ̨B̨ęf̨ǫr̨ę ̨p̨r̨ǫc̨ęęd̨įn̨g̨,̨ ̨įt̨ ̨įs̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨t̨ǫ ̨r̨ęc̨ąl̨l̨ ̨t̨h̨ąt̨ ̨R̨F̨ ̨įs̨ ̨r̨ųn̨ ̨ǫn̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨d̨ąt̨ąs̨ęt̨s̨ ̨ąs̨ ̨ǫųr̨ ̨B̨N̨C̨s̨ ̨w̨įt̨h̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨į.̨ę.̨,̨ ̨w̨įt̨h̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨d̨įs̨c̨r̨ęt̨įz̨ęd̨ ̨w̨h̨ęn̨ ̨n̨ęc̨ęs̨s̨ąr̨y̨.̨ ̨ ̨ ̨ ̨W̨ę ̨r̨ęp̨ǫr̨t̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨ ̨ąn̨d̨ ̨F̨įg̨ųr̨ę ̨<̨r̨ęf̨>̨ ̨t̨h̨ę ̨r̨ęs̨ųl̨t̨s̨ ̨ǫf̨ ̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨.̨ ̨F̨r̨ǫm̨ ̨t̨h̨įs̨ ̨t̨ąb̨l̨ę ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨R̨F̨ ̨įs̨ ̨g̨ęn̨ęr̨ąl̨l̨y̨ ̨m̨ǫr̨ę ̨ąc̨c̨ųr̨ąt̨ę ̨t̨h̨ąn̨ ̨t̨h̨ę ̨B̨N̨C̨s̨ ̨w̨įt̨h̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨C̨ǫn̨v̨ęr̨s̨ęl̨y̨,̨ ̨w̨ę ̨c̨ąn̨ ̨s̨ęę ̨t̨h̨ąt̨ ̨B̨N̨C̨s̨ ̨w̨įt̨h̨ ̨H̨D̨P̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨R̨F̨ ̨m̨ǫr̨ę ̨ǫf̨t̨ęn̨,̨ ̨ęv̨ęn̨ ̨w̨įt̨h̨ ̨ą ̨m̨ǫd̨ęl̨ ̨ąs̨ ̨s̨įm̨p̨l̨ę ̨ąs̨ ̨T̨ĄN̨.̨ ̨ ̨T̨h̨įs̨ ̨r̨ęs̨ųl̨t̨ ̨įs̨ ̨įm̨p̨ǫr̨t̨ąn̨t̨ ̨b̨ęc̨ąųs̨ę ̨ǫųr̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨ ̨ąr̨ę ̨ąl̨l̨ ̨c̨ǫm̨p̨l̨ęt̨ęl̨y̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨ąn̨d̨ ̨d̨ǫ ̨n̨ǫt̨ ̨n̨ęęd̨ ̨t̨ǫ ̨r̨ęt̨ąįn̨ ̨t̨h̨ę ̨d̨ąt̨ą ̨įn̨ ̨m̨ąįn̨ ̨m̨ęm̨ǫr̨y̨,̨ ̨ąs̨ ̨d̨ǫ ̨m̨ǫs̨t̨ ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨l̨ęąr̨n̨ęr̨s̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨c̨ǫm̨p̨ąr̨įn̨g̨ ̨0̨-̨1̨ ̨l̨ǫs̨s̨ ̨įs̨ ̨p̨r̨ǫb̨ąb̨l̨y̨ ̨f̨ąįr̨ęr̨ ̨t̨ǫ ̨R̨F̨,̨ ̨b̨ęc̨ąųs̨ę ̨R̨F̨ ̨įs̨ ̨n̨ǫt̨ ̨ą ̨p̨r̨ǫb̨ąb̨įl̨įs̨t̨įc̨ ̨m̨ǫd̨ęl̨ ̨(̨ęv̨ęn̨ ̨įf̨ ̨p̨l̨ąįn̨ ̨R̨F̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąs̨ ̨w̨ę ̨d̨ǫ ̨h̨ąv̨ę ̨b̨ęęn̨ ̨r̨ęp̨ǫr̨t̨ęd̨ ̨t̨ǫ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ ̨ǫt̨h̨ęr̨ ̨R̨F̨ ̨v̨ąr̨įąt̨įǫn̨s̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨R̨M̨S̨Ę ̨<̨c̨įt̨.̨>̨)̨.̨ ̨ ̨ ̨ ̨ ̨ ̨<̨ ̨g̨ ̨r̨ ̨ą ̨p̨ ̨h̨ ̨į ̨c̨ ̨s̨ ̨>̨ ̨0̨-̨1̨ ̨l̨ǫs̨s̨ ̨s̨c̨ąt̨t̨ęr̨ ̨p̨l̨ǫt̨ ̨ǫf̨ ̨S̨k̨D̨B̨ ̨w̨įt̨h̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨ę ̨v̨s̨ ̨R̨ąn̨d̨ǫm̨ ̨F̨ǫr̨ęs̨t̨ ̨ ̨Ǫb̨v̨įǫųs̨l̨y̨,̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨l̨ąr̨g̨ęr̨ ̨d̨ąt̨ąs̨ęt̨s̨,̨ ̨R̨F̨ ̨c̨ąt̨c̨h̨ęs̨ ̨ųp̨ ̨t̨ǫ ̨T̨ĄN̨-̨H̨D̨P̨ ̨(̨w̨h̨įc̨h̨ ̨h̨ąs̨ ̨ą ̨h̨įg̨h̨-̨b̨įąs̨ ̨s̨t̨r̨ųc̨t̨ųr̨ę)̨ ̨b̨ųt̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨1̨0̨ ̨l̨ąr̨g̨ęs̨t̨ ̨d̨ąt̨ąs̨ęt̨s̨ ̨w̨ę ̨c̨ǫn̨s̨įd̨ęr̨ęd̨,̨ ̨T̨ĄN̨-̨H̨D̨P̨ ̨s̨t̨įl̨l̨ ̨w̨įn̨s̨ ̨6̨ ̨t̨įm̨ęs̨ ̨(̨1̨ ̨d̨r̨ąw̨)̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨-̨H̨D̨P̨ ̨įs̨ ̨ęx̨t̨r̨ęm̨ęl̨y̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ę ̨w̨įt̨h̨ ̨ą ̨w̨įn̨-̨d̨r̨ąw̨-̨l̨ǫs̨s̨ ̨ǫf̨ ̨7̨–̨0̨–̨3̨.̨ ̨ ̨W̨įn̨/̨D̨r̨ąw̨/̨L̨ǫs̨s̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨ąn̨d̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨ąs̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨w̨įt̨h̨ ̨R̨ąn̨d̨ǫm̨ ̨F̨ǫr̨ęs̨t̨.̨ ̨W̨ę ̨ųs̨ę ̨ǫųr̨ ̨2̨ ̨f̨l̨ąg̨s̨h̨įp̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨T̨ĄN̨ ̨ąn̨d̨ ̨S̨k̨D̨B̨.̨ ̨S̨t̨ąt̨.̨ ̨s̨įg̨.̨ ̨r̨ęs̨ųl̨t̨s̨ ̨(̨p̨<̨0̨.̨0̨5̨)̨ ̨ąr̨ę ̨d̨ęp̨įc̨t̨ęd̨ ̨įn̨ ̨b̨ǫl̨d̨f̨ąc̨ę.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨C̨ǫm̨p̨ąr̨ęd̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ ̨ ̨2̨l̨W̨įn̨–̨d̨r̨ąw̨–̨l̨ǫs̨s̨ ̨2̨-̨3̨ ̨ ̨ ̨ ̨0̨/̨1̨-̨l̨ǫs̨s̨ ̨ ̨ ̨ ̨ ̨R̨M̨S̨Ę ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨T̨ĄN̨-̨m̨ ̨v̨s̨ ̨R̨F̨ ̨ ̨ ̨ ̨2̨6̨–̨3̨–̨3̨9̨ ̨ ̨ ̨2̨5̨–̨0̨–̨4̨3̨ ̨ ̨ ̨ ̨S̨k̨D̨B̨-̨m̨ ̨v̨s̨ ̨R̨F̨ ̨ ̨ ̨ ̨2̨7̨–̨3̨–̨3̨8̨ ̨ ̨ ̨2̨9̨–̨1̨–̨3̨8̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨T̨ĄN̨-̨H̨D̨P̨ ̨v̨s̨ ̨R̨F̨ ̨ ̨ ̨ ̨4̨2̨–̨3̨–̨2̨3̨ ̨ ̨ ̨4̨2̨–̨0̨–̨2̨6̨ ̨ ̨ ̨ ̨S̨k̨D̨B̨-̨H̨D̨P̨ ̨v̨s̨ ̨R̨F̨ ̨ ̨ ̨ ̨ ̨3̨5̨–̨3̨–̨3̨0̨ ̨ ̨ ̨ ̨4̨4̨–̨0̨–̨2̨4̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨Ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨c̨ąp̨ąc̨įt̨y̨ ̨ ̨Ǫųr̨ ̨l̨ąs̨t̨ ̨s̨ęt̨ ̨ǫf̨ ̨ęx̨p̨ęr̨įm̨ęn̨t̨s̨ ̨ąįm̨s̨ ̨ąt̨ ̨s̨h̨ǫw̨c̨ąs̨įn̨g̨ ̨t̨h̨ę ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨c̨ąp̨ąc̨įt̨y̨ ̨ǫf̨ ̨ǫųr̨ ̨s̨y̨s̨t̨ęm̨.̨ ̨W̨ę ̨r̨ųn̨ ̨S̨k̨D̨B̨ ̨ǫn̨ ̨t̨h̨ę ̨S̨p̨l̨įc̨ę ̨d̨ąt̨ąs̨ęt̨ ̨<̨c̨įt̨.̨>̨–̨ ̨w̨h̨įc̨h̨ ̨c̨ǫn̨t̨ąįn̨s̨ ̨5̨0̨ ̨m̨įl̨l̨įǫn̨ ̨t̨r̨ąįn̨įn̨g̨ ̨ęx̨ąm̨p̨l̨ęs̨ ̨ąn̨d̨ ̨įs̨ ̨p̨r̨ǫv̨įd̨ęd̨ ̨w̨įt̨h̨ ̨ą ̨t̨ęs̨t̨ ̨d̨ąt̨ąs̨ęt̨ ̨w̨įt̨h̨ ̨5̨M̨ ̨s̨ąm̨p̨l̨ęs̨ ̨–̨ ̨ąn̨d̨ ̨c̨ǫm̨p̨ąr̨ę ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨t̨ǫ ̨t̨h̨ę ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨t̨h̨įs̨ ̨d̨ąt̨ąs̨ęt̨ ̨įs̨ ̨įm̨b̨ąl̨ąn̨c̨ęd̨ ̨w̨įt̨h̨ ̨ǫn̨l̨y̨ ̨1̨%̨ ̨ǫf̨ ̨ęx̨ąm̨p̨l̨ęs̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨p̨ǫs̨įt̨įv̨ę ̨c̨l̨ąs̨s̨.̨ ̨Ǫn̨ ̨t̨h̨įs̨ ̨d̨ąt̨ąs̨ęt̨,̨ ̨R̨F̨ ̨c̨ǫųl̨d̨ ̨n̨ǫt̨ ̨r̨ųn̨ ̨ųs̨įn̨g̨ ̨W̨ęk̨ą ̨d̨ęf̨ąųl̨t̨s̨,̨ ̨r̨ęq̨ųįr̨įn̨g̨ ̨m̨ǫr̨ę ̨t̨h̨ąn̨ ̨ǫųr̨ ̨l̨įm̨įt̨ ̨ǫf̨ ̨1̨3̨8̨G̨B̨ ̨ǫf̨ ̨R̨ĄM̨.̨ ̨W̨ę ̨t̨h̨ųs̨ ̨ųs̨ęd̨ ̨įn̨s̨t̨ęąd̨ ̨X̨G̨B̨ǫǫs̨t̨ ̨<̨c̨įt̨.̨>̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨t̨h̨ę ̨s̨t̨ąt̨ę ̨ǫf̨ ̨t̨h̨ę ̨ąr̨t̨ ̨f̨ǫr̨ ̨s̨c̨ąl̨ąb̨l̨ę ̨m̨įx̨t̨ųr̨ę ̨ǫf̨ ̨t̨r̨ęęs̨ ̨(̨h̨ęr̨ę ̨b̨ǫǫs̨t̨įn̨g̨)̨ ̨ąn̨d̨ ̨ųs̨ęd̨ ̨w̨įd̨ęl̨y̨ ̨b̨y̨ ̨d̨ąt̨ą ̨s̨c̨įęn̨t̨įs̨t̨s̨ ̨t̨ǫ ̨ąc̨h̨įęv̨ę ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨r̨ęs̨ųl̨t̨s̨ ̨ǫn̨ ̨m̨ąn̨y̨ ̨m̨ąc̨h̨įn̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨c̨h̨ąl̨l̨ęn̨g̨ęs̨ ̨(̨X̨G̨B̨ǫǫs̨t̨ ̨w̨ąs̨ ̨ųs̨ęd̨ ̨įn̨ ̨1̨7̨ ̨ǫųt̨ ̨ǫf̨ ̨2̨9̨ ̨w̨įn̨n̨įn̨g̨ ̨s̨ǫl̨ųt̨įǫn̨s̨ ̨įn̨ ̨t̨h̨ę ̨m̨ąc̨h̨įn̨ę ̨l̨ęąr̨n̨įn̨g̨ ̨c̨ǫm̨p̨ęt̨įt̨įǫn̨ ̨s̨įt̨ę ̨K̨ąg̨g̨l̨ę ̨įn̨ ̨2̨0̨1̨5̨ ̨<̨c̨įt̨.̨>̨)̨.̨ ̨W̨ę ̨ųs̨ę ̨X̨G̨B̨ǫǫs̨t̨'̨s̨ ̨d̨ęf̨ąųl̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨s̨ ̨ąs̨ ̨p̨ęr̨ ̨v̨ęr̨s̨įǫn̨ ̨0̨.̨6̨ ̨–̨ ̨w̨ę ̨ųs̨ę ̨m̨ąx̨įm̨ųm̨ ̨d̨ęp̨t̨h̨ ̨ǫf̨ ̨6̨ ̨ąn̨d̨ ̨5̨0̨ ̨r̨ǫųn̨d̨s̨ ̨ǫf̨ ̨b̨ǫǫs̨t̨įn̨g̨.̨ ̨S̨įm̨įl̨ąr̨l̨y̨ ̨t̨ǫ ̨t̨h̨ę ̨p̨r̨ęv̨įǫųs̨,̨ ̨t̨h̨ę ̨ąįm̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨s̨ęc̨t̨įǫn̨ ̨įs̨ ̨n̨ǫt̨ ̨t̨ǫ ̨s̨ųg̨g̨ęs̨t̨ ̨t̨h̨ąt̨ ̨B̨N̨C̨s̨ ̨s̨h̨ǫųl̨d̨ ̨r̨ęp̨l̨ąc̨ę ̨X̨G̨B̨ǫǫs̨t̨,̨ ̨b̨ųt̨ ̨r̨ąt̨h̨ęr̨ ̨t̨ǫ ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨B̨N̨C̨s̨ ̨ąr̨ę ̨ąn̨ ̨įn̨t̨ęr̨ęs̨t̨įn̨g̨ ̨s̨ęt̨ ̨ǫf̨ ̨m̨ǫd̨ęl̨s̨ ̨t̨h̨ąt̨ ̨c̨ąn̨ ̨p̨ęr̨f̨ǫr̨m̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę ̨ąn̨d̨ ̨p̨ęr̨f̨ǫr̨m̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ęl̨y̨ ̨w̨h̨ęn̨ ̨ųs̨įn̨g̨ ̨ǫųr̨ ̨H̨D̨P̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨ ̨ ̨T̨h̨ę ̨r̨ęs̨ųl̨t̨s̨ ̨ąr̨ę ̨r̨ęp̨ǫr̨t̨ęd̨ ̨įn̨ ̨T̨ąb̨l̨ę ̨<̨r̨ęf̨>̨.̨ ̨T̨h̨ęy̨ ̨s̨h̨ǫw̨ ̨t̨h̨ąt̨ ̨H̨D̨P̨ ̨d̨r̨ąm̨ąt̨įc̨ąl̨l̨y̨ ̨įm̨p̨r̨ǫv̨ęs̨ ̨b̨ǫt̨h̨ ̨0̨-̨1̨ ̨l̨ǫs̨s̨ ̨ąn̨d̨ ̨R̨M̨S̨Ę ̨ąs̨ ̨c̨ǫm̨p̨ąr̨ęd̨ ̨t̨ǫ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨.̨ ̨ ̨N̨ǫt̨ę ̨t̨h̨ąt̨ ̨m̨-̨ęs̨t̨įm̨ąt̨ęs̨ ̨w̨ǫųl̨d̨ ̨ęv̨ęn̨ ̨b̨ę ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨ęd̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨ęr̨r̨ǫr̨-̨r̨ąt̨ę ̨b̨y̨ ̨s̨įm̨p̨l̨y̨ ̨p̨r̨ęd̨įc̨t̨įn̨g̨ ̨t̨h̨ę ̨m̨ąj̨ǫr̨įt̨y̨ ̨c̨l̨ąs̨s̨.̨ ̨C̨ǫm̨p̨ąr̨įs̨ǫn̨ ̨w̨įt̨h̨ ̨X̨G̨B̨ǫǫs̨t̨ ̨įs̨ ̨įn̨t̨ęr̨ęs̨t̨įn̨g̨,̨ ̨įt̨ ̨s̨h̨ǫw̨s̨ ̨t̨h̨ąt̨ ̨S̨k̨D̨B̨5̨ ̨w̨įt̨h̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨c̨ǫm̨ęs̨ ̨v̨ęr̨y̨ ̨c̨l̨ǫs̨ę ̨t̨ǫ ̨X̨G̨B̨ǫǫs̨t̨ ̨įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨0̨-̨1̨ ̨l̨ǫs̨s̨.̨ ̨Įn̨ ̨t̨ęr̨m̨s̨ ̨ǫf̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨c̨ąl̨įb̨r̨ąt̨įǫn̨ ̨ǫųr̨ ̨H̨D̨P̨ ̨ęs̨t̨įm̨ąt̨ęs̨ ̨ęv̨ęn̨ ̨p̨ųs̨h̨ ̨B̨N̨C̨s̨ ̨b̨ęy̨ǫn̨d̨ ̨X̨G̨B̨ǫǫs̨t̨'̨s̨ ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę,̨ ̨ąs̨ ̨ęv̨įd̨ęn̨c̨ęd̨ ̨b̨y̨ ̨t̨h̨ę ̨R̨M̨S̨Ę.̨ ̨ ̨R̨ęs̨ųl̨t̨s̨ ̨ǫn̨ ̨t̨h̨ę ̨S̨p̨l̨įc̨ę ̨d̨ąt̨ąs̨ęt̨ ̨ǫn̨ ̨w̨h̨įc̨h̨ ̨R̨F̨ ̨c̨ąn̨n̨ǫt̨ ̨r̨ųn̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨C̨l̨ąs̨s̨įf̨įęr̨ ̨ ̨ ̨ ̨0̨/̨1̨-̨l̨ǫs̨s̨ ̨ ̨ ̨ ̨R̨M̨S̨Ę ̨ ̨S̨k̨D̨B̨5̨-̨m̨ ̨ ̨ ̨ ̨1̨.̨4̨9̨9̨%̨ ̨ ̨ ̨ ̨0̨.̨1̨0̨9̨3̨ ̨ ̨S̨k̨D̨B̨5̨-̨H̨D̨P̨ ̨ ̨ ̨ ̨ ̨0̨.̨3̨1̨8̨%̨ ̨ ̨ ̨ ̨0̨.̨0̨5̨4̨4̨ ̨X̨G̨B̨ǫǫs̨t̨ ̨ ̨ ̨0̨.̨3̨1̨4̨%̨ ̨ ̨ ̨0̨.̨0̨5̨9̨4̨ ̨ ̨ ̨ ̨§̨.̨§̨ ̨R̨ųn̨n̨įn̨g̨ ̨t̨įm̨ę ̨ ̨Ąl̨t̨h̨ǫųg̨h̨ ̨r̨ųn̨n̨įn̨g̨ ̨t̨įm̨ę ̨įs̨ ̨n̨ǫt̨ ̨d̨įr̨ęc̨t̨l̨y̨ ̨ą ̨f̨ǫc̨ųs̨ ̨ǫf̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨,̨ ̨w̨ę ̨g̨įv̨ę ̨b̨ęl̨ǫw̨ ̨s̨ǫm̨ę ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨ǫb̨s̨ęr̨v̨ąt̨įǫn̨s̨:̨ ̨ ̨ ̨ ̨*̨ ̨ ̨T̨r̨ąįn̨įn̨g̨ ̨t̨įm̨ę ̨c̨ǫm̨p̨l̨ęx̨įt̨y̨ ̨įn̨c̨r̨ęąs̨ęs̨ ̨l̨įn̨ęąr̨l̨y̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨ ̨r̨ųn̨s̨ ̨f̨ǫr̨,̨ ̨l̨įn̨ęąr̨l̨y̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨c̨ǫv̨ąr̨įąt̨ęs̨ ̨ąn̨d̨ ̨l̨įn̨ęąr̨l̨y̨ ̨w̨įt̨h̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨n̨ǫd̨ęs̨ ̨įn̨ ̨t̨h̨ę ̨t̨r̨ęęs̨ ̨(̨w̨h̨įc̨h̨ ̨įn̨c̨r̨ęąs̨ęs̨ ̨ęx̨p̨ǫn̨ęn̨t̨įąl̨l̨y̨ ̨w̨įt̨h̨ ̨d̨ęp̨t̨h̨)̨.̨ ̨ ̨ ̨ ̨*̨ ̨ ̨T̨r̨ąįn̨įn̨g̨ ̨t̨įm̨ę ̨įs̨ ̨r̨ęąs̨ǫn̨ąb̨l̨ę.̨ ̨Ąs̨ ̨ąn̨ ̨ęx̨ąm̨p̨l̨ę,̨ ̨t̨r̨ąįn̨įn̨g̨ ̨ǫf̨ ̨ ̨S̨k̨D̨B̨_̨5̨-̨H̨D̨P̨ ̨(̨w̨įt̨h̨ ̨m̨ąx̨K̨=̨5̨)̨ ̨ǫn̨ ̨S̨p̨l̨įc̨ę ̨w̨įt̨h̨ ̨5̨0̨ ̨m̨įl̨l̨įǫn̨ ̨s̨ąm̨p̨l̨ęs̨ ̨t̨ǫǫk̨ ̨ųn̨d̨ęr̨ ̨4̨ ̨h̨ǫųr̨s̨,̨ ̨ąm̨ǫn̨g̨ ̨w̨h̨įc̨h̨ ̨1̨.̨5̨ ̨h̨ǫųr̨s̨ ̨ąr̨ę ̨s̨p̨ęn̨t̨ ̨t̨ǫ ̨l̨ęąr̨n̨ ̨t̨h̨ę ̨s̨t̨r̨ųc̨t̨ųr̨ę ̨ǫf̨ ̨t̨h̨ę ̨B̨N̨.̨ ̨ ̨S̨k̨D̨B̨_̨5̨ ̨įm̨p̨l̨įęd̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨1̨4̨0̨ ̨įn̨d̨ęp̨ęn̨d̨ęn̨t̨ ̨h̨įęr̨ąr̨c̨h̨įęs̨ ̨h̨ąv̨ę ̨ą ̨d̨ęp̨t̨h̨ ̨ǫf̨ ̨6̨ ̨ąn̨d̨ ̨w̨ę ̨r̨ųn̨ ̨5̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨ǫf̨ ̨t̨h̨ę ̨s̨ąm̨p̨l̨ęr̨.̨ ̨T̨h̨įs̨ ̨ąl̨s̨ǫ ̨įm̨p̨l̨įęs̨ ̨t̨h̨ąt̨ ̨ ̨S̨k̨D̨B̨_̨5̨-̨m̨ ̨t̨ąk̨ęs̨ ̨ą ̨b̨įt̨ ̨m̨ǫr̨ę ̨t̨h̨ąn̨ ̨1̨.̨5̨ ̨h̨ǫųr̨s̨ ̨t̨ǫ ̨b̨ę ̨t̨r̨ąįn̨ęd̨.̨ ̨X̨G̨B̨ǫǫs̨t̨ ̨–̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ą ̨h̨įg̨h̨l̨y̨ ̨ǫp̨t̨įm̨įs̨ęd̨ ̨p̨ąc̨k̨ąg̨ę ̨–̨ ̨ǫn̨ ̨S̨p̨l̨įc̨ę ̨r̨ęq̨ųįr̨ęd̨ ̨j̨ųs̨t̨ ̨ųn̨d̨ęr̨ ̨ǫn̨ę ̨h̨ǫųr̨ ̨ǫf̨ ̨c̨ǫm̨p̨ųt̨ąt̨įǫn̨.̨ ̨ ̨ ̨ ̨ ̨*̨ ̨ ̨F̨ǫr̨ ̨t̨h̨ę ̨Ąd̨ųl̨t̨ ̨d̨ąt̨ąs̨ęt̨ ̨t̨r̨ąįn̨įn̨g̨ ̨ ̨S̨k̨D̨B̨_̨5̨ ̨w̨įt̨h̨ ̨2̨5̨k̨ ̨s̨ąm̨p̨l̨ęs̨ ̨ąn̨d̨ ̨5̨0̨,̨0̨0̨0̨ ̨įt̨ęr̨ąt̨įǫn̨s̨ ̨w̨įt̨h̨ ̨l̨ęv̨ęl̨ ̨t̨y̨įn̨g̨ ̨t̨ǫǫk̨ ̨8̨6̨ ̨s̨ęc̨ǫn̨d̨s̨,̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨Ąb̨ąl̨ǫn̨ę ̨d̨ąt̨ąs̨ęt̨ ̨t̨r̨ąįn̨įn̨g̨ ̨w̨įt̨h̨ ̨2̨k̨ ̨s̨ąm̨p̨l̨ęs̨ ̨t̨ǫǫk̨ ̨6̨ ̨s̨ęc̨ǫn̨d̨s̨ ̨–̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨t̨įm̨ę ̨t̨ąk̨ęs̨ ̨l̨ęs̨s̨ ̨t̨h̨ąn̨ ̨1̨s̨ ̨t̨ǫ ̨c̨l̨ąs̨s̨įf̨y̨ ̨2̨5̨k̨ ̨s̨ąm̨p̨l̨ęs̨,̨ ̨w̨h̨įc̨h̨ ̨įs̨ ̨ǫn̨ę ̨ǫf̨ ̨t̨h̨ę ̨s̨t̨r̨ęn̨g̨t̨h̨ ̨ǫf̨ ̨B̨N̨C̨s̨:̨ ̨ǫn̨c̨ę ̨l̨ęąr̨n̨ęd̨,̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨įs̨ ̨ą ̨s̨įm̨p̨l̨ę ̨l̨ǫǫk̨-̨ųp̨ ̨f̨ǫr̨ ̨ęąc̨h̨ ̨f̨ąc̨t̨ǫr̨.̨ ̨T̨h̨įs̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨t̨įm̨ę ̨įs̨ ̨ąc̨t̨ųąl̨l̨y̨ ̨ųn̨d̨ęr̨ ̨1̨s̨ ̨f̨ǫr̨ ̨ąl̨l̨ ̨m̨ǫd̨ęl̨s̨ ̨c̨ǫn̨s̨įd̨ęr̨ęd̨ ̨įn̨ ̨t̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨Ąd̨ųl̨t̨ ̨d̨ąt̨ąs̨ęt̨.̨ ̨ ̨ ̨ ̨ ̨ ̨§̨ ̨C̨ǪN̨C̨L̨ŲS̨ĮǪN̨S̨ ̨ ̨T̨h̨įs̨ ̨p̨ąp̨ęr̨ ̨p̨r̨ęs̨ęn̨t̨s̨ ̨ąc̨c̨ųr̨ąt̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨f̨ǫr̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨ ̨c̨l̨ąs̨s̨įf̨įęr̨s̨ ̨ųs̨įn̨g̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨D̨įr̨įc̨h̨l̨ęt̨ ̨p̨r̨ǫc̨ęs̨s̨ ̨ęs̨t̨įm̨ąt̨ęs̨,̨ ̨c̨ǫm̨b̨įn̨įn̨g̨ ̨t̨h̨ęs̨ę ̨w̨ęl̨l̨-̨r̨ęs̨ęąr̨c̨h̨ęd̨ ̨ąr̨ęąs̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨f̨įr̨s̨t̨ ̨t̨įm̨ę.̨ ̨W̨ę ̨h̨ąv̨ę ̨d̨ęm̨ǫn̨s̨t̨r̨ąt̨ęd̨ ̨t̨h̨ąt̨ ̨H̨D̨P̨s̨ ̨ąr̨ę ̨n̨ǫt̨ ̨ǫn̨l̨y̨ ̨c̨ąp̨ąb̨l̨ę ̨ǫf̨ ̨ǫųt̨p̨ęr̨f̨ǫr̨m̨įn̨g̨ ̨s̨t̨ąt̨ę-̨ǫf̨-̨t̨h̨ę-̨ąr̨t̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨t̨ęc̨h̨n̨įq̨ųęs̨,̨ ̨b̨ųt̨ ̨d̨ǫ ̨s̨ǫ ̨w̨h̨įl̨ę ̨f̨ųn̨c̨t̨įǫn̨įn̨g̨ ̨c̨ǫm̨p̨l̨ęt̨ęl̨y̨ ̨ǫųt̨-̨ǫf̨-̨c̨ǫr̨ę.̨ ̨W̨ę ̨h̨ąv̨ę ̨ąl̨s̨ǫ ̨s̨h̨ǫw̨ęd̨ ̨t̨h̨ąt̨,̨ ̨f̨ǫr̨ ̨c̨ąt̨ęg̨ǫr̨įc̨ąl̨ ̨d̨ąt̨ą,̨ ̨t̨h̨įs̨ ̨m̨ąk̨ęs̨ ̨įt̨ ̨p̨ǫs̨s̨įb̨l̨ę ̨t̨ǫ ̨m̨ąk̨ę ̨B̨N̨C̨s̨ ̨h̨įg̨h̨l̨y̨ ̨c̨ǫm̨p̨ęt̨įt̨įv̨ę ̨w̨įt̨h̨ ̨r̨ąn̨d̨ǫm̨ ̨f̨ǫr̨ęs̨t̨.̨ ̨W̨ę ̨n̨ǫt̨ę ̨t̨h̨ąt̨ ̨w̨h̨įl̨ę ̨B̨N̨C̨s̨ ̨ąr̨ę ̨n̨ǫt̨ ̨c̨ųr̨r̨ęn̨t̨l̨y̨ ̨s̨t̨ąt̨ę ̨ǫf̨ ̨t̨h̨ę ̨ąr̨t̨ ̨f̨ǫr̨ ̨c̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨,̨ ̨t̨h̨ęy̨ ̨ąr̨ę ̨s̨t̨įl̨l̨ ̨p̨ǫp̨ųl̨ąr̨ ̨įn̨ ̨ąp̨p̨l̨įc̨ąt̨įǫn̨s̨.̨ ̨W̨įt̨h̨ ̨t̨h̨įs̨ ̨įm̨p̨r̨ǫv̨ęm̨ęn̨t̨ ̨įn̨ ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę,̨ ̨ąn̨d̨ ̨ųs̨ąb̨l̨ę ̨įm̨p̨l̨ęm̨ęn̨t̨ąt̨įǫn̨s̨ ̨ ̨įn̨ ̨p̨ąc̨k̨ąg̨ęs̨ ̨s̨ųc̨h̨ ̨ąs̨ ̨R̨,̨ ̨B̨N̨C̨s̨ ̨w̨įl̨l̨ ̨b̨ę ̨f̨ąr̨ ̨m̨ǫr̨ę ̨ųs̨ęf̨ųl̨ ̨įn̨ ̨r̨ęąl̨-̨w̨ǫr̨l̨d̨ ̨ąp̨p̨l̨įc̨ąt̨įǫn̨s̨ ̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨ąr̨ę ̨r̨ęąd̨įl̨y̨ ̨įm̨p̨l̨ęm̨ęn̨t̨ęd̨ ̨ǫn̨ ̨h̨įg̨h̨ ̨p̨ęr̨f̨ǫr̨m̨ąn̨c̨ę ̨d̨ęs̨k̨t̨ǫp̨s̨,̨ ̨ąn̨d̨ ̨d̨ǫ ̨n̨ǫt̨ ̨r̨ęq̨ųįr̨ę ̨ą ̨c̨l̨ųs̨t̨ęr̨.̨ ̨ ̨T̨h̨įs̨ ̨w̨ǫr̨k̨ ̨n̨ąt̨ųr̨ąl̨l̨y̨ ̨ǫp̨ęn̨s̨ ̨ųp̨ ̨ą ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨ǫp̨p̨ǫr̨t̨ųn̨įt̨įęs̨ ̨f̨ǫr̨ ̨f̨ųt̨ųr̨ę ̨r̨ęs̨ęąr̨c̨h̨.̨ ̨F̨įr̨s̨t̨,̨ ̨w̨ę ̨w̨ǫųl̨d̨ ̨l̨įk̨ę ̨t̨ǫ ̨p̨ęr̨f̨ęc̨t̨ ̨ǫųr̨ ̨s̨ąm̨p̨l̨ęr̨ ̨b̨y̨ ̨ąs̨s̨ęs̨s̨įn̨g̨ ̨t̨h̨ę ̨įn̨f̨l̨ųęn̨c̨ę ̨ǫf̨ ̨t̨h̨ę ̨d̨įf̨f̨ęr̨ęn̨t̨ ̨r̨ųn̨t̨įm̨ę ̨c̨ǫn̨f̨įg̨ųr̨ąt̨įǫn̨s̨ ̨ǫf̨ ̨ǫųr̨ ̨s̨y̨s̨t̨ęm̨ ̨įn̨c̨l̨ųd̨įn̨g̨:̨ ̨h̨ǫw̨ ̨ǫf̨t̨ęn̨ ̨s̨h̨ǫųl̨d̨ ̨w̨ę ̨s̨ąm̨p̨l̨ę ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨,̨ ̨w̨įd̨ęn̨įn̨g̨ ̨t̨h̨ę ̨w̨įn̨d̨ǫw̨ ̨ǫf̨ ̨p̨s̨ęųd̨ǫ-̨c̨ǫųn̨t̨s̨ ̨ąt̨ ̨t̨h̨ę ̨s̨t̨ąr̨t̨ ̨ǫf̨ ̨t̨h̨ę ̨s̨y̨s̨t̨ęm̨ ̨ąn̨d̨ ̨b̨ųr̨n̨-̨įn̨.̨ ̨S̨ęc̨ǫn̨d̨,̨ ̨w̨ę ̨w̨ǫųl̨d̨ ̨l̨įk̨ę ̨t̨ǫ ̨ęx̨t̨ęn̨d̨ ̨t̨h̨įs̨ ̨w̨ǫr̨k̨ ̨t̨ǫ ̨P̨įt̨m̨ąn̨-̨Y̨ǫr̨ ̨p̨r̨ǫc̨ęs̨s̨ęs̨,̨ ̨w̨h̨įc̨h̨ ̨ǫf̨f̨ęr̨ ̨ąn̨ ̨ęx̨c̨įt̨įn̨g̨ ̨ąv̨ęn̨ųę ̨f̨ǫr̨ ̨r̨ęs̨ęąr̨c̨h̨,̨ ̨įn̨ ̨p̨ąr̨t̨įc̨ųl̨ąr̨ ̨f̨ǫr̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨w̨įt̨h̨ ̨h̨įg̨h̨ ̨c̨ąr̨d̨įn̨ąl̨įt̨y̨.̨ ̨ ̨T̨h̨įr̨d̨,̨ ̨w̨ę ̨w̨ǫųl̨d̨ ̨l̨įk̨ę ̨t̨ǫ ̨ęx̨t̨ęn̨d̨ ̨t̨h̨įs̨ ̨f̨r̨ąm̨ęw̨ǫr̨k̨ ̨t̨ǫ ̨t̨h̨ę ̨g̨ęn̨ęr̨ąl̨ ̨c̨l̨ąs̨s̨ ̨ǫf̨ ̨B̨ąy̨ęs̨įąn̨ ̨n̨ęt̨w̨ǫr̨k̨s̨.̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨N̨ǫt̨ąt̨įǫn̨ ̨ ̨ ̨ ̨ ̨D̨ęs̨c̨r̨įp̨t̨įǫn̨ ̨ ̨n̨ ̨ ̨ ̨ ̨N̨ųm̨b̨ęr̨ ̨ǫf̨ ̨ąt̨t̨r̨įb̨ųt̨ęs̨ ̨–̨ ̨ąl̨s̨ǫ ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨v̨ąr̨įąb̨l̨ęs̨ ̨ųs̨ęd̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨N̨ ̨ ̨ ̨ ̨N̨ųm̨b̨ęr̨ ̨ǫf̨ ̨d̨ąt̨ą ̨p̨ǫįn̨t̨s̨ ̨įn̨ ̨ ̨Y̨ ̨ ̨ ̨ ̨R̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨c̨l̨ąs̨s̨ ̨l̨ąb̨ęl̨ ̨–̨ ̨ąl̨s̨ǫ ̨X̨_̨0̨ ̨y̨ ̨ ̨ ̨ ̨v̨ąl̨ųę ̨t̨ąk̨ęn̨ ̨b̨y̨ ̨Y̨ ̨|̨Y̨|̨ ̨ ̨ ̨ ̨N̨ųm̨b̨ęr̨ ̨ǫf̨ ̨c̨l̨ąs̨s̨ęs̨ ̨X̨_̨į ̨ ̨ ̨ ̨R̨ąn̨d̨ǫm̨ ̨v̨ąr̨įąb̨l̨ę ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨ąt̨t̨r̨įb̨ųt̨ę ̨į ̨x̨_̨į ̨ ̨ ̨ ̨v̨ąl̨ųę ̨t̨ąk̨ęn̨ ̨b̨y̨ ̨X̨_̨į ̨X̨_̨c̨ ̨ ̨ ̨ ̨c̨h̨įl̨d̨ ̨v̨ąr̨įąb̨l̨ę ̨f̨ǫr̨ ̨w̨h̨įc̨h̨ ̨w̨ę ̨ąr̨ę ̨ęs̨t̨įm̨ąt̨įn̨g̨ ̨t̨h̨ę ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨ ̨θ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨v̨ęc̨t̨ǫr̨ ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨l̨ęąf̨ ̨n̨ǫd̨ę ̨(̨ąt̨ ̨l̨ęv̨ęl̨ ̨n̨+̨1̨)̨ ̨f̨ǫr̨ ̨v̨ąl̨ųęs̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ϕ̨_̨X̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨l̨ąt̨ęn̨t̨ ̨p̨r̨įǫr̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨n̨ǫd̨ę ̨ąt̨ ̨l̨ęv̨ęl̨ ̨į ̨ąs̨s̨ǫc̨įąt̨ęd̨ ̨w̨įt̨h̨ ̨b̨r̨ąn̨c̨h̨įn̨g̨ ̨v̨ąl̨ųęs̨ ̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨α̨ ̨ ̨ ̨ ̨c̨ǫn̨c̨ęn̨t̨r̨ąt̨įǫn̨ ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨f̨ǫr̨ ̨t̨h̨ę ̨D̨įr̨įc̨h̨l̨ęt̨ ̨d̨įs̨t̨r̨įb̨ųt̨įǫn̨s̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨ ̨ ̨ ̨l̨ęąf̨-̨n̨ǫd̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨įn̨g̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨d̨ąt̨ą ̨p̨ǫįn̨t̨s̨ ̨w̨įt̨h̨ ̨v̨ąl̨ųęs̨ ̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨n̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨įn̨t̨ęr̨m̨ęd̨įąt̨ę-̨n̨ǫd̨ę ̨p̨ąr̨ąm̨ęt̨ęr̨ ̨r̨ęp̨r̨ęs̨ęn̨t̨įn̨g̨ ̨t̨h̨ę ̨n̨ųm̨b̨ęr̨ ̨ǫf̨ ̨d̨ąt̨ą ̨p̨ǫįn̨t̨s̨ ̨r̨ęc̨ęįv̨ęd̨ ̨f̨r̨ǫm̨ ̨įt̨s̨ ̨c̨h̨įl̨d̨r̨ęn̨ ̨n̨ǫd̨ęs̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į-̨1̨=̨∑̨_̨x̨_̨į ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨l̨ąt̨ęn̨t̨ ̨v̨ąr̨įąb̨l̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨įn̨g̨ ̨t̨h̨ę ̨f̨r̨ąc̨t̨įǫn̨ ̨ǫf̨ ̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨t̨h̨ąt̨ ̨įs̨ ̨p̨ąs̨s̨ęd̨ ̨ųp̨ ̨t̨ǫ ̨įt̨s̨ ̨p̨ąr̨ęn̨t̨ ̨n̨_̨.̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨m̨ąr̨g̨įn̨ąl̨ ̨c̨ǫųn̨t̨ ̨n̨_̨.̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į=̨∑̨_̨x̨_̨c̨n̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨t̨_̨.̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨ ̨ ̨ ̨m̨ąr̨g̨įn̨ąl̨ ̨c̨ǫųn̨t̨ ̨t̨_̨.̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į=̨∑̨_̨x̨_̨c̨t̨_̨x̨_̨c̨|̨y̨,̨x̨_̨1̨,̨⋯̨,̨x̨_̨į ̨L̨įs̨t̨ ̨ǫf̨ ̨s̨y̨m̨b̨ǫl̨s̨ ̨ųs̨ęd̨.̨D̨ąt̨ąs̨ęt̨s̨ ̨=̨1̨.̨0̨p̨t̨ ̨ ̨D̨ǫm̨ąįn̨ ̨ ̨ ̨ ̨ ̨C̨ąs̨ę ̨ ̨ ̨ ̨ ̨Ąt̨t̨ ̨ ̨ ̨ ̨ ̨C̨l̨ąs̨s̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨ ̨D̨ǫm̨ąįn̨ ̨ ̨ ̨ ̨ ̨C̨ąs̨ę ̨ ̨ ̨ ̨ ̨Ąt̨t̨ ̨ ̨ ̨ ̨ ̨C̨l̨ąs̨s̨ ̨ ̨ ̨ ̨C̨ǫn̨n̨ęc̨t̨-̨4̨Ǫp̨ęn̨įn̨g̨ ̨ ̨ ̨6̨7̨5̨5̨7̨ ̨ ̨ ̨4̨3̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨P̨įm̨ąĮn̨d̨įąn̨s̨D̨įąb̨ęt̨ęs̨ ̨ ̨ ̨7̨6̨8̨ ̨ ̨ ̨9̨ ̨ ̨ ̨2̨ ̨ ̨ ̨S̨t̨ąt̨l̨ǫg̨(̨S̨h̨ųt̨t̨l̨ę)̨ ̨ ̨ ̨5̨8̨0̨0̨0̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨7̨ ̨ ̨ ̨ ̨ ̨ ̨B̨r̨ęąs̨t̨C̨ąn̨c̨ęr̨(̨W̨įs̨c̨ǫn̨s̨įn̨)̨ ̨ ̨ ̨6̨9̨9̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨Ąd̨ųl̨t̨ ̨ ̨ ̨4̨8̨8̨4̨2̨ ̨ ̨ ̨1̨5̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨C̨r̨ęd̨įt̨S̨c̨r̨ęęn̨įn̨g̨ ̨ ̨ ̨6̨9̨0̨ ̨ ̨ ̨1̨6̨ ̨ ̨ ̨2̨ ̨ ̨ ̨L̨ęt̨t̨ęr̨R̨ęc̨ǫg̨n̨įt̨įǫn̨ ̨ ̨ ̨2̨0̨0̨0̨0̨ ̨ ̨ ̨1̨7̨ ̨ ̨ ̨2̨6̨ ̨ ̨ ̨ ̨ ̨ ̨B̨ąl̨ąn̨c̨ęS̨c̨ąl̨ę ̨ ̨ ̨6̨2̨5̨ ̨ ̨ ̨5̨ ̨ ̨ ̨3̨ ̨ ̨ ̨M̨ĄG̨ĮC̨G̨ąm̨m̨ąT̨ęl̨ęs̨c̨ǫp̨ę ̨ ̨ ̨1̨9̨0̨2̨0̨ ̨ ̨ ̨1̨1̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨S̨y̨n̨c̨ǫn̨ ̨ ̨ ̨6̨0̨0̨ ̨ ̨ ̨6̨1̨ ̨ ̨ ̨6̨ ̨ ̨ ̨N̨ųr̨s̨ęr̨y̨ ̨ ̨ ̨1̨2̨9̨6̨0̨ ̨ ̨ ̨9̨ ̨ ̨ ̨5̨ ̨ ̨ ̨ ̨ ̨ ̨C̨h̨ęs̨s̨ ̨ ̨ ̨5̨5̨1̨ ̨ ̨ ̨4̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨S̨įg̨n̨ ̨ ̨ ̨1̨2̨5̨4̨6̨ ̨ ̨ ̨9̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨C̨y̨l̨įn̨d̨ęr̨ ̨ ̨ ̨5̨4̨0̨ ̨ ̨ ̨4̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨P̨ęn̨D̨įg̨įt̨s̨ ̨ ̨ ̨1̨0̨9̨9̨2̨ ̨ ̨ ̨1̨7̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨ ̨ ̨ ̨M̨ųs̨k̨1̨ ̨ ̨ ̨4̨7̨6̨ ̨ ̨ ̨1̨6̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨T̨h̨y̨r̨ǫįd̨ ̨ ̨ ̨9̨1̨6̨9̨ ̨ ̨ ̨3̨0̨ ̨ ̨ ̨2̨0̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ǫųs̨ęV̨ǫt̨ęs̨8̨4̨ ̨ ̨ ̨4̨3̨5̨ ̨ ̨ ̨1̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨M̨ųs̨h̨r̨ǫǫm̨s̨ ̨ ̨ ̨8̨1̨2̨4̨ ̨ ̨ ̨2̨3̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ǫr̨s̨ęC̨ǫl̨įc̨ ̨ ̨ ̨3̨6̨8̨ ̨ ̨ ̨2̨2̨ ̨ ̨ ̨2̨ ̨ ̨ ̨M̨ųs̨k̨2̨ ̨ ̨ ̨6̨5̨9̨8̨ ̨ ̨ ̨1̨6̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨D̨ęr̨m̨ąt̨ǫl̨ǫg̨y̨ ̨ ̨ ̨3̨6̨6̨ ̨ ̨ ̨3̨5̨ ̨ ̨ ̨6̨ ̨ ̨ ̨S̨ąt̨ęl̨l̨įt̨ę ̨ ̨ ̨6̨4̨3̨5̨ ̨ ̨ ̨3̨7̨ ̨ ̨ ̨6̨ ̨ ̨ ̨ ̨ ̨ ̨Įǫn̨ǫs̨p̨h̨ęr̨ę ̨ ̨ ̨3̨5̨1̨ ̨ ̨ ̨3̨5̨ ̨ ̨ ̨2̨ ̨ ̨ ̨Ǫp̨t̨įc̨ąl̨D̨įg̨įt̨s̨ ̨ ̨ ̨5̨6̨2̨0̨ ̨ ̨ ̨4̨9̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨ ̨ ̨ ̨L̨įv̨ęr̨D̨įs̨ǫr̨d̨ęr̨s̨(̨B̨ųp̨ą)̨ ̨ ̨ ̨3̨4̨5̨ ̨ ̨ ̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨P̨ąg̨ęB̨l̨ǫc̨k̨s̨C̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨ ̨ ̨5̨4̨7̨3̨ ̨ ̨ ̨1̨1̨ ̨ ̨ ̨5̨ ̨ ̨ ̨ ̨ ̨ ̨P̨r̨įm̨ąr̨y̨T̨ųm̨ǫr̨ ̨ ̨ ̨3̨3̨9̨ ̨ ̨ ̨1̨8̨ ̨ ̨ ̨2̨2̨ ̨ ̨ ̨W̨ąl̨l̨-̨f̨ǫl̨l̨ǫw̨įn̨g̨ ̨ ̨ ̨5̨4̨5̨6̨ ̨ ̨ ̨2̨5̨ ̨ ̨ ̨4̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ąb̨ęr̨m̨ąn̨'̨s̨S̨ųr̨v̨įv̨ąl̨ ̨ ̨ ̨3̨0̨6̨ ̨ ̨ ̨4̨ ̨ ̨ ̨2̨ ̨ ̨ ̨N̨ęt̨t̨ąl̨k̨(̨P̨h̨ǫn̨ęm̨ę)̨ ̨ ̨ ̨5̨4̨3̨8̨ ̨ ̨ ̨8̨ ̨ ̨ ̨5̨2̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ęąr̨t̨D̨įs̨ęąs̨ę(̨C̨l̨ęv̨ęl̨ąn̨d̨)̨ ̨ ̨ ̨3̨0̨3̨ ̨ ̨ ̨1̨4̨ ̨ ̨ ̨2̨ ̨ ̨ ̨W̨ąv̨ęf̨ǫr̨m̨-̨5̨0̨0̨0̨ ̨ ̨ ̨5̨0̨0̨0̨ ̨ ̨ ̨4̨1̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ųn̨g̨ąr̨įąn̨ ̨ ̨ ̨2̨9̨4̨ ̨ ̨ ̨1̨4̨ ̨ ̨ ̨2̨ ̨ ̨ ̨S̨p̨ąm̨b̨ąs̨ę ̨ ̨ ̨4̨6̨0̨1̨ ̨ ̨ ̨5̨8̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨Ąųd̨įǫl̨ǫg̨y̨ ̨ ̨ ̨2̨2̨6̨ ̨ ̨ ̨7̨0̨ ̨ ̨ ̨2̨4̨ ̨ ̨ ̨Ąb̨ąl̨ǫn̨ę ̨ ̨ ̨4̨1̨7̨7̨ ̨ ̨ ̨9̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨N̨ęw̨-̨T̨h̨y̨r̨ǫįd̨ ̨ ̨ ̨2̨1̨5̨ ̨ ̨ ̨6̨ ̨ ̨ ̨3̨ ̨ ̨ ̨H̨y̨p̨ǫt̨h̨y̨r̨ǫįd̨(̨G̨ąr̨ąv̨ąn̨)̨ ̨ ̨ ̨3̨7̨7̨2̨ ̨ ̨ ̨3̨0̨ ̨ ̨ ̨4̨ ̨ ̨ ̨ ̨ ̨ ̨G̨l̨ąs̨s̨Įd̨ęn̨t̨įf̨įc̨ąt̨įǫn̨ ̨ ̨ ̨2̨1̨4̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨3̨ ̨ ̨ ̨S̨įc̨k̨-̨ęųt̨h̨y̨r̨ǫįd̨ ̨ ̨ ̨3̨7̨7̨2̨ ̨ ̨ ̨3̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨S̨ǫn̨ąr̨C̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨ ̨ ̨2̨0̨8̨ ̨ ̨ ̨6̨1̨ ̨ ̨ ̨2̨ ̨ ̨ ̨K̨įn̨g̨-̨r̨ǫǫk̨-̨v̨s̨-̨k̨įn̨g̨-̨p̨ąw̨n̨ ̨ ̨ ̨3̨1̨9̨6̨ ̨ ̨ ̨3̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨Ąųt̨ǫĮm̨p̨ǫr̨t̨s̨ ̨ ̨ ̨2̨0̨5̨ ̨ ̨ ̨2̨6̨ ̨ ̨ ̨7̨ ̨ ̨ ̨S̨p̨l̨įc̨ę-̨j̨ųn̨c̨t̨įǫn̨G̨ęn̨ęS̨ęq̨ųęn̨c̨ęs̨ ̨ ̨ ̨3̨1̨9̨0̨ ̨ ̨ ̨6̨2̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨W̨įn̨ęR̨ęc̨ǫg̨n̨įt̨įǫn̨ ̨ ̨ ̨1̨7̨8̨ ̨ ̨ ̨1̨4̨ ̨ ̨ ̨3̨ ̨ ̨ ̨S̨ęg̨m̨ęn̨t̨ ̨ ̨ ̨2̨3̨1̨0̨ ̨ ̨ ̨2̨0̨ ̨ ̨ ̨7̨ ̨ ̨ ̨ ̨ ̨ ̨H̨ęp̨ąt̨įt̨įs̨ ̨ ̨ ̨1̨5̨5̨ ̨ ̨ ̨2̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨C̨ąr̨Ęv̨ąl̨ųąt̨įǫn̨ ̨ ̨ ̨1̨7̨2̨8̨ ̨ ̨ ̨8̨ ̨ ̨ ̨4̨ ̨ ̨ ̨ ̨ ̨ ̨T̨ęąc̨h̨įn̨g̨Ąs̨s̨įs̨t̨ąn̨t̨Ęv̨ąl̨ųąt̨įǫn̨ ̨ ̨ ̨1̨5̨1̨ ̨ ̨ ̨6̨ ̨ ̨ ̨3̨ ̨ ̨ ̨V̨ǫl̨c̨ąn̨ǫęs̨ ̨ ̨ ̨1̨5̨2̨0̨ ̨ ̨ ̨4̨ ̨ ̨ ̨4̨ ̨ ̨ ̨ ̨ ̨ ̨Įr̨įs̨C̨l̨ąs̨s̨įf̨įc̨ąt̨įǫn̨ ̨ ̨ ̨1̨5̨0̨ ̨ ̨ ̨5̨ ̨ ̨ ̨3̨ ̨ ̨ ̨Y̨ęąs̨t̨ ̨ ̨ ̨1̨4̨8̨4̨ ̨ ̨ ̨9̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨ ̨ ̨ ̨L̨y̨m̨p̨h̨ǫg̨r̨ąp̨h̨y̨ ̨ ̨ ̨1̨4̨8̨ ̨ ̨ ̨1̨9̨ ̨ ̨ ̨4̨ ̨ ̨ ̨C̨ǫn̨t̨r̨ąc̨ęp̨t̨įv̨ęM̨ęt̨h̨ǫd̨C̨h̨ǫįc̨ę ̨ ̨ ̨1̨4̨7̨3̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨ ̨ ̨Ęc̨h̨ǫc̨ąr̨d̨įǫg̨r̨ąm̨ ̨ ̨ ̨1̨3̨1̨ ̨ ̨ ̨7̨ ̨ ̨ ̨2̨ ̨ ̨ ̨G̨ęr̨m̨ąn̨ ̨ ̨ ̨1̨0̨0̨0̨ ̨ ̨ ̨2̨1̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨P̨r̨ǫm̨ǫt̨ęr̨G̨ęn̨ęS̨ęq̨ųęn̨c̨ęs̨ ̨ ̨ ̨1̨0̨6̨ ̨ ̨ ̨5̨8̨ ̨ ̨ ̨2̨ ̨ ̨ ̨L̨ĘD̨ ̨ ̨ ̨1̨0̨0̨0̨ ̨ ̨ ̨8̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨ ̨ ̨ ̨Z̨ǫǫ ̨ ̨ ̨1̨0̨1̨ ̨ ̨ ̨1̨7̨ ̨ ̨ ̨7̨ ̨ ̨ ̨V̨ǫw̨ęl̨ ̨ ̨ ̨9̨9̨0̨ ̨ ̨ ̨1̨4̨ ̨ ̨ ̨1̨1̨ ̨ ̨ ̨ ̨ ̨ ̨P̨ǫs̨t̨ǫp̨ęr̨ąt̨įv̨ęP̨ąt̨įęn̨t̨ ̨ ̨ ̨9̨0̨ ̨ ̨ ̨9̨ ̨ ̨ ̨3̨ ̨ ̨T̨įc̨-̨T̨ąc̨-̨T̨ǫęĘn̨d̨g̨ąm̨ę ̨ ̨ ̨9̨5̨8̨ ̨ ̨ ̨1̨0̨ ̨ ̨ ̨2̨ ̨ ̨ ̨ ̨ ̨ ̨L̨ąb̨ǫr̨N̨ęg̨ǫt̨įąt̨įǫn̨s̨ ̨ ̨ ̨5̨7̨ ̨ ̨ ̨1̨7̨ ̨ ̨ ̨2̨ ̨ ̨Ąn̨n̨ęąl̨įn̨g̨ ̨ ̨ ̨8̨9̨8̨ ̨ ̨ ̨3̨9̨ ̨ ̨ ̨6̨ ̨ ̨ ̨ ̨ ̨ ̨L̨ųn̨g̨C̨ąn̨c̨ęr̨ ̨ ̨ ̨3̨2̨ ̨ ̨ ̨5̨7̨ ̨ ̨ ̨3̨ ̨ ̨ ̨V̨ęh̨įc̨l̨ę ̨ ̨ ̨8̨4̨6̨ ̨ ̨ ̨1̨9̨ ̨ ̨ ̨4̨ ̨ ̨ ̨ ̨ ̨ ̨C̨ǫn̨t̨ąc̨t̨-̨l̨ęn̨s̨ęs̨ ̨ ̨ ̨2̨4̨ ̨ ̨ ̨5̨ ̨ ̨ ̨3̨ ̨ ̨ ̨ ̨§̨ ̨C̨ǪM̨P̨L̨ĮĄN̨C̨Ę ̨W̨ĮT̨H̨ ̨ĘT̨H̨ĮC̨ĄL̨ ̨S̨T̨ĄN̨D̨ĄR̨D̨S̨ ̨ ̨T̨h̨įs̨ ̨w̨ǫr̨k̨ ̨w̨ąs̨ ̨s̨ųp̨p̨ǫr̨t̨ęd̨ ̨b̨y̨ ̨t̨h̨ę ̨Ąųs̨t̨r̨ąl̨įąn̨ ̨R̨ęs̨ęąr̨c̨h̨ ̨C̨ǫųn̨c̨įl̨ ̨ųn̨d̨ęr̨ ̨ąw̨ąr̨d̨s̨ ̨D̨Ę1̨7̨0̨1̨0̨0̨0̨3̨7̨ ̨ąn̨d̨ ̨D̨P̨1̨4̨0̨1̨0̨0̨0̨8̨7̨.̨ ̨ ̨T̨h̨ę ̨ąųt̨h̨ǫr̨s̨ ̨w̨ǫųl̨d̨ ̨l̨įk̨ę ̨t̨ǫ ̨t̨h̨ąn̨k̨ ̨J̨ǫąn̨ ̨C̨ąp̨d̨ęv̨įl̨ą ̨P̨ųj̨ǫl̨ ̨ąn̨d̨ ̨ąn̨ǫn̨y̨m̨ǫųs̨ ̨r̨ęv̨įęw̨ęr̨s̨ ̨f̨ǫr̨ ̨h̨ęl̨p̨įn̨g̨ ̨ųs̨ ̨s̨t̨r̨ęn̨g̨t̨h̨ęn̨ ̨t̨h̨ę ̨ǫr̨įg̨įn̨ąl̨ ̨m̨ąn̨ųs̨c̨r̨įp̨t̨.̨ ̨ ̨ ̨ ̨ ̨s̨p̨b̨ąs̨įc̨ ̨ ̨§̨ ̨S̨T̨ŲF̨F̨ ̨T̨ĄK̨ĘN̨ ̨ǪŲT̨ ̨ǪF̨ ̨ĮN̨T̨R̨ǪD̨ŲC̨T̨ĮǪN̨ ̨ ̨T̨h̨ę ̨įn̨t̨ųįt̨įǫn̨ ̨f̨ǫr̨ ̨ǫųr̨ ̨m̨ęt̨h̨ǫd̨ ̨įs̨ ̨t̨h̨ąt̨ ̨ęs̨t̨įm̨ąt̨įǫn̨ ̨ǫf̨ ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨įęs̨ ̨s̨h̨ǫųl̨d̨ ̨s̨h̨ąr̨ę ̨įn̨f̨ǫr̨m̨ąt̨įǫn̨ ̨w̨įt̨h̨ ̨t̨h̨ęįr̨ ̨n̨ęąr̨ ̨n̨ęįg̨h̨b̨ǫųr̨s̨ ̨įn̨ ̨t̨h̨ę ̨h̨įęr̨ąr̨c̨h̨y̨.̨ ̨S̨ųp̨p̨ǫs̨ę ̨y̨ǫų ̨w̨įs̨h̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę ̨ą ̨c̨ǫn̨d̨įt̨įǫn̨ąl̨ ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨t̨ąb̨l̨ę ̨(̨C̨P̨T̨)̨ ̨f̨ǫr̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨3̨)̨f̨r̨ǫm̨ ̨d̨ąt̨ą ̨w̨h̨ęr̨ę ̨t̨h̨ę ̨f̨ęąt̨ųr̨ęs̨x̨_̨1̨,̨x̨_̨2̨,̨x̨_̨3̨t̨ąk̨ę ̨ǫn̨ ̨v̨ąl̨ųęs̨1̨,̨2̨,̨3̨,̨4̨.̨ ̨T̨h̨įs̨ ̨C̨P̨T̨ ̨c̨ąn̨ ̨b̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨ęd̨ ̨ąs̨ ̨ą ̨t̨r̨ęę:̨ ̨t̨h̨ę ̨r̨ǫǫt̨ ̨n̨ǫd̨ę ̨t̨ęs̨t̨s̨x̨_̨1̨ąn̨d̨ ̨h̨ąs̨ ̨4̨ ̨b̨r̨ąn̨c̨h̨ęs̨,̨ ̨t̨h̨ę ̨2̨n̨d̨ ̨ąn̨d̨ ̨3̨r̨d̨ ̨l̨ęv̨ęl̨ ̨n̨ǫd̨ęs̨ ̨t̨ęs̨t̨x̨_̨2̨ąn̨d̨x̨_̨3̨ąn̨d̨ ̨h̨ąv̨ę ̨4̨ ̨b̨r̨ąn̨c̨h̨ęs̨.̨ ̨T̨h̨ę ̨4̨t̨h̨ ̨l̨ęv̨ęl̨ ̨c̨ǫn̨s̨įs̨t̨s̨ ̨ǫf̨ ̨l̨ęąv̨ęs̨ ̨ąn̨d̨ ̨ęąc̨h̨ ̨n̨ǫd̨ę ̨h̨ąs̨ ̨ą ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨ ̨f̨ǫr̨y̨t̨h̨ąt̨ ̨w̨ę ̨w̨įs̨h̨ ̨t̨ǫ ̨ęs̨t̨įm̨ąt̨ę.̨ ̨T̨h̨ę ̨s̨h̨ąr̨įn̨g̨ ̨įn̨t̨ųįt̨įǫn̨ ̨s̨ąy̨s̨ ̨t̨h̨ąt̨ ̨t̨h̨ę ̨l̨ęąf̨ ̨n̨ǫd̨ę ̨r̨ęp̨r̨ęs̨ęn̨t̨įn̨g̨(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨1̨)̨s̨h̨ǫųl̨d̨ ̨h̨ąv̨ę ̨s̨įm̨įl̨ąr̨ ̨v̨ąl̨ųęs̨ ̨t̨ǫ ̨t̨h̨ę ̨l̨ęąf̨ ̨f̨ǫr̨(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨2̨)̨b̨ęc̨ąųs̨ę ̨t̨h̨ęy̨ ̨h̨ąv̨ę ̨ą ̨c̨ǫm̨m̨ǫn̨ ̨p̨ąr̨ęn̨t̨,̨ ̨b̨ųt̨ ̨s̨h̨ǫųl̨d̨ ̨n̨ǫt̨ ̨b̨ę ̨s̨ǫ ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ(̨y̨|̨x̨_̨1̨=̨3̨,̨x̨_̨2̨=̨1̨,̨x̨_̨3̨=̨2̨)̨,̨ ̨w̨h̨įc̨h̨ ̨ǫn̨l̨y̨ ̨s̨h̨ąr̨ęs̨ ̨ą ̨g̨r̨ęąt̨-̨g̨r̨ęąt̨ ̨g̨r̨ąn̨d̨p̨ąr̨ęn̨t̨.̨ ̨W̨ę ̨ąc̨h̨įęv̨ę ̨t̨h̨įs̨ ̨s̨h̨ąr̨įn̨g̨ ̨b̨y̨ ̨ųs̨įn̨g̨ ̨ą ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨p̨r̨įǫr̨.̨ ̨S̨ǫ ̨w̨ę ̨h̨ąv̨ę(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨,̨x̨_̨3̨=̨X̨_̨3̨)̨(̨f̨ǫr̨X̨_̨4̨=̨1̨,̨2̨,̨3̨,̨4̨)̨ ̨ąr̨ę ̨g̨ęn̨ęr̨ąt̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨s̨ąm̨ę ̨p̨r̨įǫr̨ ̨w̨h̨įc̨h̨ ̨h̨ąs̨ ̨ą ̨p̨r̨ǫb̨ąb̨įl̨įt̨y̨ ̨v̨ęc̨t̨ǫr̨q̨(̨y̨|̨x̨_̨1̨=̨1̨,̨x̨_̨2̨=̨2̨)̨ąs̨ ̨ą ̨m̨ęąn̨.̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨c̨ąn̨ ̨ǫf̨t̨ęn̨ ̨b̨ę ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ(̨y̨|̨x̨_̨1̨)̨w̨h̨įc̨h̨ ̨įn̨ ̨t̨ųr̨n̨ ̨c̨ąn̨ ̨ǫf̨t̨ęn̨ ̨b̨ę ̨s̨įm̨įl̨ąr̨ ̨t̨ǫ(̨y̨)̨.̨ ̨H̨ǫw̨ęv̨ęr̨,̨ ̨s̨t̨r̨įc̨t̨l̨y̨ ̨s̨p̨ęąk̨įn̨g̨,̨(̨y̨|̨x̨_̨1̨)̨ąn̨d̨(̨y̨)̨ąr̨ę ̨ąg̨g̨r̨ęg̨ąt̨ę ̨v̨ąl̨ųęs̨ ̨h̨ęr̨ę ̨d̨ęr̨įv̨ęd̨ ̨f̨r̨ǫm̨ ̨t̨h̨ę ̨ųn̨d̨ęr̨l̨y̨įn̨g̨ ̨m̨ǫd̨ęl̨ ̨w̨h̨įc̨h̨ ̨s̨p̨ęc̨įf̨įęs̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨.̨ ̨ ̨S̨ǫ,̨ ̨t̨ǫ ̨m̨ǫd̨ęl̨ ̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨ ̨s̨įm̨įl̨ąr̨įt̨y̨ ̨w̨įt̨h̨ ̨ą ̨H̨D̨P̨,̨(̨y̨|̨x̨_̨1̨)̨ąn̨d̨(̨y̨)̨ąr̨ę ̨m̨ǫd̨ęl̨l̨ęd̨ ̨ąs̨ ̨l̨ąt̨ęn̨t̨ ̨(̨h̨įęr̨ąr̨c̨h̨įc̨ąl̨)̨ ̨p̨r̨įǫr̨s̨ ̨f̨ǫr̨(̨y̨|̨x̨_̨1̨,̨x̨_̨2̨)̨.̨ | http://arxiv.org/abs/1708.07581v3 | {
"authors": [
"Francois Petitjean",
"Wray Buntine",
"Geoffrey I. Webb",
"Nayyar Zaidi"
],
"categories": [
"cs.LG",
"stat.ML",
"68Q87"
],
"primary_category": "cs.LG",
"published": "20170825002049",
"title": "Accurate parameter estimation for Bayesian Network Classifiers using Hierarchical Dirichlet Processes"
} |
APS/[email protected] Indian Institute of Science, Bangalore 560012, IndiaIn this paper, we seek to finda modified theory of gravity that accounts for the back-reaction of QED on curved spacetime. It is already known that vacuum fluctuations induce interactions between gravity and photons. An effective action for electromagnetism, which encodes the details of such quantum process, is utilized to get a set of modified Maxwell's equation and a new Lagrangian for the dynamics of gravity. Imposition of Cauchy predictability on these modified Maxwell's equations lead us to a new effective metric, from which the Lagrangian can be calculated. The new Lagrangian for gravity turns out to be a function of not just the higher order derivatives of the metric tensor but also the polarization of the photon. This immediately results in phenomenon such as gravitational birefringence and existence of black holes with polarization dependent event horizons.Valid PACS appear here Back-reaction of quantum processes and modified gravitational dynamics Vyshnav Mohan December 30, 2023 ======================================================================§ INTRODUCTION Quantum electrodynamics is one of the most successful theories of theoretical physics and has made many significant predictions about the way our universe behaves. One of the consequences of QED is that it provides corrections to local photon propagation on a curved spacetime. This effect, produced by vacuum polarization, on the velocity of light and the light cone structure has already been analyzed in various background spacetimes by Drummond and Hathrell<cit.>. Working in one-loop approximation, they found out an effective action for the theory. We can physically motivate this modification as the vacuum polarization are processes in which the photon exists as a virtual electron and positron pair and these species behave differently from photons in a general curved spacetime. The same line of thought can be seen in the works of Klaus Scharnhorst and others like Sergei Odintsov. <cit.>However, if we want these modified matter equations to be Cauchy predictable, the dynamics of the underlying gravitational field becomes constrained as shown in Ref. <cit.>. This is because suitable initial data surface are required to uniquely evolve the initial data. It is imperative that a classical theory must be predictable and thus, the constraints on gravity arise naturally. It turns out that the geometric metric tensor ceases to determine the dynamics of the gravitational field and some quantum corrections are required to fix this issue. The details of these corrections are worked out in the next section. Thus, the predictability condition helps us to assess the influence of QED vacuum on the gravitational field.This process of finding a modified theory of gravity is quite analogous to that of finding the electric field inside a sphere of linear dielectric material in classical electrodynamics. Assume that there is a uniform static electric field and the sphere is embedded in it. The field polarizes the sphere and produces bound charges. These charges in turn produce a field of its own and impart more polarization to the sphere. This process continues and each iteration further improves the accuracy of the field calculated inside the object and one can see in Ref. <cit.> that it converges to the conventional value obtained by other methods. Analogously, we start off by putting quantum fields on a static background spacetime and look at the modification of the matter equations and use this information to find the corrections to the background gravitational field. The details of these calculations are given in the rest of the paper.§THE EFFECTIVE METRIC The Drummond and Hathrell effective action Γ_DH can be calculated by accounting for one-loop corrections in QED <cit.>i.e.Γ_DH = Γ_M +ln det S(x, x^') Here, Γ_M is the free Maxwell action and S(x, x^') is the electron propagator on the curved spacetime. Suppressing higher order derivatives of the curvature tensors relative to O(λ/L), where λ is the photon wavelength and L is the typical curvature scale, we can find out the effective action as in Ref.<cit.>. After varying the action, we obtain the following modified Maxwell's equation <cit.>:D_μ F^μν -1m^2 [2σ R_μνD^μ F^λν + 4 ζ g^νκ R_μκλρD^μ F^λρ]Here, σ =13α/360π , ζ=-α/360π where α is the fine structure constant and m is the mass of the electron. All the physical quantities are in natural units and F^μν is the electromagnetic field tensor as usual. In order to focus on the characteristics of photon propagation, we use geometric optics approximation,where the electromagnetic field tensor is given by a product ofslowly varying amplitude and rapidly varying phase as <cit.>F_μν =f_μν e^i θand the wave vector is given by k_μ = ∂_μθ. Together with Bianchi identities D_[μ F_λν]=0, we get the following constraintf_μν =k_μa_ν -k_νa_μ Where a^ν is a vector in the direction of the polarization of the photon and can be assumed to obey the relation k_μa^μ= 0. Thus,Eq. (<ref>) gives the modified light cone condition as in <cit.> given byg_ab k^a k^b +2 σm^2 R_ab k^a k^b-8 ζm^2 R_acbdk^a k^ba^c a^d =0 .Here, the polarization 4-vector is assumed to be space like normalized with respect to g_ab and the latin indices run from 0 to 3.LetG_ab := g_ab +2 σm^2 R_ab-8 ζm^2R_acbda^c a^dThus, the modified light cone condition can be written as G_abk^a k^b = 0This is strikingly similar to the light cone condition in general relativity, given by:g_abk^a k^b = 0 The predictability of the matter equations translates to the condition that the principle polynomial of the matter field should be a bi-hyperbolic and energy distinguishing homogeneous polynomial [See Ref.<cit.> for detailed discussions]. From the light cone condition or from the coefficients of the highest-order derivative in Eq. (<ref>), we can read off the dual polynomial for the system asP^# (x,v) = G_ab v^a v^b.Certain properties of G_ab can easily be seen from the definition of the tensor. Due to the symmetries of Ricci and Riemann-Christoffel tensor, we find that G_ab is a billinear, symmetric, non- degenerate map on that tangent space T_xM of the manifold M at every point x. Thus, G_ab by itself can be treated as an effective metric of the manifold, provided an inverse metric exist at all points on the spacetime. It turns out that it is even possible to define G _ab as a metric tensor of the manifold and find appropriate g_ab which solves for G _ab and satisfies its the non-degeneracy condition. Thus, we calculate the principal polynomial to beP (x,k) = G^ab k_a k_b. where G^ab is the inverse of the metric G_ab. Another compelling reason to consider G_ab as an effective metric is the following: The wave vector k_μ is defined as a one-form belonging to T_x^∗(M)at every point of spacetime while the momentum p^ν of the photon belongs to the tangent space T_x(M). In this modified theory, the relationship between them is not trivial asp^ν = g^μν k_μ. Instead the wave vector is mapped to the momenta up to a factor by the Gauss map (See Ref.<cit.>) given by [k] ↦[∂ P∂ k_a(x,k)] where [X] represents the projective equivalence class of all vectors collinear with X. Thus, we see that the correct relationship between p^ν andk_μ is:p^a =∂ ( G^ab k_a k_b)∂ k_a = G^ab k_b This shows that G^ab takes the role of g^ab in raising and lowering the indices of vectors and co-vectors. Thus, G^ab can be treated as an effective metric to the manifold. For the modified Maxwell's equations to have a well posed initial value problem, the principal polynomial should be hyperbolic. Since the underlying geometry is determined by an effective metric G^ab, the hyperbolicity of dual polynomial reduces to the condition that there exist some vector field h such that P^#(h)>0 and for any vector field q, the following equation has only two real rootsP^# (x,h + λ q) =0 . Expanding the equation in terms of G_ab results in a quadratic equation of λ as in Ref.<cit.>. Thus, the condition boils down to the discriminant of the quadratic equation being positive i.e.G_ab h^a q^b -G_ab h^a h^b G_cd q^c q^d >0 .Let us choose a vector basis {ϵ_α} such that ϵ_0 := h and G_abϵ^a_0ϵ^b _α = 0As G_abϵ^a_0ϵ^b _0 > 0, the discriminant becomesq^α q^βG_abϵ^a_αϵ^b _β< 0. As g_ab is lorentzian, we find that if g_ab h^a h^b >0 , then q^α q^βg_abϵ^a_αϵ^b _β< 0⇒2 σm^2R_αβ -8 ζm^2 R_α c β d a^c a^d < 02 σm^2 R_00-8 ζm^2 R_0 c 0d a^c a^d > 0 Here α,β runs from 1 to 3 and R_αβ,R_α c β d are the components of the tensors w.r.t basis{ϵ_α}Multiplying inequality Eq. (<ref>) byg^αβ and multiplying inequality Eq. (<ref>) byg^00summing up the indices, we get,2 σm^2g^αβR_αβ -8 ζm^2 g^αβ R_α c β d a^c a^d > 0 2 σm^2 g^00 R_00-8 ζm^2 g^00 R_0 c 0d a^c a^d > 0 These inequalities are satisfied if 2 σm^2 R > 8 ζm^2 R_ c d a^c a^d Conversely, if inequality (3) is satisfied along with the condition that the metricg_ab is lorentzian, the matter equations will havea well posed initial value problem.§THE LAGRANGIANObtaining the Lagrangian from the principal polynomial is quite tedious [3]. However, as the geometry of the spacetime is described by a metric G_ab ,we can use the hindsight from Einstein-Hilbert action that the gravitational part of the new Lagrangian must contain a term proportional to the Ricci scalar of the metric G_ab.∴ℒ= ∫[ 12κ( ℛ - 2 Λ) + ℒ_M] √(-g) d^4 x ,whereκ = 8 π Gℛ is the Ricci tensor obtained from the metric G_ab, Λ is the cosmological constant and ℒ_M is the Lagrangian of any matter fields appearing in the theory.ℛ contains the Ricci scalar of the metricg_ab and higher order derivatives of Ricci and Riemann–Christoffel tensor of themetricg_ab .§ MODIFIED EINSTEIN'S EQUATION The equation of motion of the system can be obtained by varying the Lagrangian w.r.t the metric g_ab. As the ℛ term is messy, it is difficult to approach the problem directly. So, we vary the Lagrangian withG_ab and then change the variation from G_ab to g_ab. Focusing on the gravitational part of the Lagrangian and varying w.r.t to G_ab, we get ∫[ 12κ( δ Rδ G_ab +R√(-G)δ√(-G)δ G_ab) ] δ G_ab√(-G) d^4xHowever, from the definition of G_ab. we haveδ G_ab =δ g_ab +2 σm^2δ R_ab-8 ζm^2δ R_acbda^c a^d We analyze each term separately. We know that from Ref. <cit.>δ R^h_cld =∇_l (δΓ^h_dc) - ∇_d (δΓ^h_lc) andδΓ^α_βμ= 12 g^αλ (∇_βδ g_λμ+ ∇_μδ g_βλ- ∇_λδ g_βμ) ∴δ R^h_cld =∇_l (12 g^he (∇_dδ g_ec+ ∇_cδ g_de- ∇_eδ g_dc) )- ∇_d (12 g^he (∇_lδ g_ec+ ∇_cδ g_le- ∇_eδ g_lc) ) . =12 g^he (∇_l ∇_dδ g_ec+∇_l∇_cδ g_de- ∇_l ∇_eδ g_dc - ∇_d ∇_lδ g_ec+ ∇_d ∇_cδ g_le- ∇_d∇_eδ g_lc) . Andδ R_acbd = δ (g_kh R^h_cld) =δ g_kh R^h_cld+ g_khδR^h_cld,From pallatani identities [8], we haveδ R_kl = 12 g^ab (∇_a ∇_lδ g_kb +∇_a ∇_kδ g_lb-∇_k ∇_lδ g_ab - ∇_a ∇_bδ g_kl) . Therefore the variation can be written asδℒ = ∫12κ( 𝒢^klδ g_kl +𝒢^kl2 σm^2δ R_kl -𝒢^kl8 ζm^2δ R_kclda^c a^d) √(-G) d^4x . where 𝒢^kl is the Einstein tensor associated with metric G_ab. Consider the second term of the integral. ∫12κ𝒢^kl2 σm^2δ R_kl√(-G) d^4x= ∫12κ𝒢^kl2 σm^2 (12 g^ab (∇_a ∇_lδ g_kb+∇_a ∇_kδ g_lb -∇_k ∇_lδ g_ab - ∇_a ∇_bδ g_kl) ) √(-G) d^4x .Now, we use integration by parts twice to shift the covariant derivative from the variation of the metric and modify the dummy indices to get :∫12κ2 σm^2∇_b ∇_a( g^al𝒢^bk -12 g^kl𝒢^ab-12 g^ab𝒢^kl) δ g_kl√(-G) d^4x .Now consider the third term ∫12κ𝒢^kl8 ζm^2δ R_kclda^c a^d√(-G) d^4x= ∫12κ𝒢^kl8 ζm^2(δ g_kh R^h_cld+ g_khδR^h_cld )a^c a^d√(-G) d^4xLet us focus on the second term of the above expression i.e.∫12κ𝒢^kl8 ζm^2g_khδR^h_cld a^c a^d√(-G) d^4x =∫12κ𝒢^kl8 ζm^2 g_kh (12 g^he (∇_l ∇_dδ g_ec+∇_l ∇_cδ g_de -∇_l∇_eδ g_dc -∇_d ∇_lδ g_ec+ ∇_d ∇_cδ g_le -∇_d ∇_eδ g_lc )) a^c a^d√(-G) d^4x .Using the identity δ g_ec_;ld-δ g_ec_;dl=R^a_eldδ g_ac + R^a_cldδ g_ae for first and fourth terms and using integration by parts twice on other terms and then changing the dummy indices, we get∫12κ𝒢^kl8 ζm^2g_khδR^h_cld a^c a^d√(-G) d^4x =∫12κ(𝒢^ac4 ζm^2R^k_acd a^l a^d + 𝒢^la4 ζm^2R^k_cad a^c a^d - ∇_d ∇_c( 𝒢^dc4 ζm^2a^l a^k)+∇_c ∇_d( 𝒢^ld4 ζm^2a^c a^k) )- ∇_c ∇_d( 𝒢^ck4 ζm^2a^l a^d)+∇_c ∇_d( 𝒢^kl4 ζm^2a^c a^d) )δ g_kl√(-G) d^4x) .∴ The total variation isδℒ = ∫12κ( 𝒢^kl + 2 σm^2∇_b ∇_a( g^al𝒢^bk-12 g^kl𝒢^ab-12 g^ab𝒢^kl)- 𝒢^ac4 ζm^2R^k_acd a^l a^d- 𝒢^la4 ζm^2R^k_cad a^c a^d- 𝒢^kh8 ζm^2R^l_chd a^c a^d- ∇_d ∇_c( 𝒢^dc4 ζm^2a^l a^k) +∇_c ∇_d( 𝒢^ld4 ζm^2a^c a^k ) + ∇_c ∇_d( 𝒢^ck4 ζm^2a^l a^d) -∇_c ∇_d( 𝒢^kl4 ζm^2a^c a^d) ) δ g_kl√(-G) d^4x .!= 0 ( By principle of least action.) Thus, the integrand must vanish identically at all points. Therefore, the modified Einstein's equation is𝒢^kl + 2 σm^2∇_b ∇_a( g^al𝒢^bk - 12 g^kl𝒢^ab-12 g^ab𝒢^kl)- 𝒢^ac4 ζm^2R^k_acd a^l a^d - 𝒢^la4ζm^2R^k_cad a^c a^d - 𝒢^kh8 ζm^2R^l_chd a^c a^d- ∇_d ∇_c( 𝒢^dc4 ζm^2a^l a^k) +∇_c ∇_d( 𝒢^ld4 ζm^2a^c a^k ) + ∇_c ∇_d( 𝒢^ck4 ζm^2a^l a^d) -∇_c ∇_d( 𝒢^kl4 ζm^2a^c a^d)= κ T^klWhere the T^ kl is the stress-energy tensor.§ CORRECTIONS TO SCHWARZSCHILD METRIC Solving directly for a homogeneous, static and spherically symmetric spacetime turns out to be a very difficult problem.Therefore, we linearize the metric and obtain the solutions to the modified equations. Thus, the metric is assumed to take the form g = B(r) dt⊗dt+A(r) dr⊗dr +r^2( dθ⊗dθ +sin^2θ dϕ⊗dϕ )where B(r) = -(1 +k (r)) and A(r) = 1 +j (r). Herek (r) and j (r) are assumed to be infinitesimal in magnitude and we neglect their higher powers . We assume this applies to the higher order derivatives of the functions as well. In this way, we neglect the product or products of the function with their higher derivatives.This implies that the inverses are approximately given byB ^-1 (r) = -1 + k (r) andA ^-1 (r) = 1 -j (r) However, upon close inspection of the modified Einstein's equations, we find that these differential equations are of the order of six in terms ofk(r) and j(r).But we can bring the order of the equations down by using the fact that the we keep only terms that are linear in k(r) and j(r)and their derivatives. The calculation of the components of Riemann-Christoeffel tensor shows that all the non-vanishing terms are of infinitesimal magnitude. Hence, we can neglect the terms like𝒢^kh R^l_chd a^c a^d as 𝒢^kh is also an infinitesimal quantity.Thus, the final equations comprise of terms containing 𝒢^khand a^d and their covariant derivatives.This motivates to us to start from G_ab rather than g_ab. So , we define G = ℬ(r) dt⊗dt+ 𝒜(r) dr⊗dr +ℛ(r)^2( dθ⊗dθ +sin^2θ dϕ⊗dϕ ) The component functions ℬ(r), 𝒜(r) and ℛ(r) can be calculated from the definition of G_ab in terms of k (r) and j (r), but instead we define it as the following ℬ(r) = 1 + v (r), 𝒜 (r) = 1 +w (r) ℛ (r) =r^2 +h (r) and solve for v (r), w (r) and h (r).In the case of Schwarzschild solution for Einstein's field equations, we have the vanishing of all the components of the Einstein tensors as T^ kl is zero. Using this information and by inspecting our modified equations, we can make an educated guess for a solution given by 𝒢^rr =𝒢^θθ = 0 It follows from the symmetry of the metric that 𝒢^ϕϕ =r^2 sin^2θ𝒢^θθ = 0Thus the modified equations reduce to a single equation given by 𝒢^tt+ σm^2∇_t ∇_t(g^tt𝒢^tt )-2 σm^2∇_b ∇_a( 12 g^ab𝒢^tt) - ∇_t ∇_t( 𝒢^tt4 ζm^2(a^t)^2) +∇_c ∇_t( 𝒢^tt4 ζm^2a^c a^t )+ ∇_t ∇_d( 𝒢^tt4 ζm^2a^t a^d) -∇_c ∇_d( 𝒢^tt4 ζm^2a^c a^d)= 0Thus, it is clear that the solution to the modified Einstein's equation depends upon the polarization of the photon.In fact, this turns out to be the most important feature of the new gravity theory. To obtain a nice analytic solution and for the sake of simplicity, we look at a case where a^r =(i m^2)/4 ζ anda^θ = a^ϕ = 0. Thus, the equation becomes 𝒢^tt +σm^2∇_t ∇_t(g^tt𝒢^tt )-2 σm^2∇_b ∇_a( 12 g^ab𝒢^tt) + ∇_r ∇_r( 𝒢^ttm^24 ζ)= 0 Calculating the covariant derivative w.r.t g_ab, and neglecting product of infinitesimal terms, we obtain𝒢^tt +[m^24 ζ- σm^2]d^2 𝒢^ttd r^2-2 σm^2∇_θ∇_θ( 𝒢^ttr^2 ) = 0This can be further simplified to get :𝒢^tt +[m^24 ζ- σm^2]d^2 𝒢^ttd r^2-2 σm^2[1rd𝒢^ttd r - 2r^3d 𝒢^ttd r +6 G^tt r^4]= 0 However, these equations are too complicated to have a well-behaved analytic solution. But, the equation can be reduced to a simpler form if we make some sensible approximations. Plugging in the values of the ζ , σand m, we find that2σm^2∼10^-32[m^24 ζ- σm^2]∼ 10^-16 Thus, for very large r, the contribution of the third term in square bracket of Eq. (<ref>) is negligible and we get the following equation :𝒢^tt +[m^24 ζ- σm^2]d^2 𝒢^ttd r^2 = 0 This is an ordinary differential equation in one variable which can be solved easily. A physical solution is 𝒢^tt =e^- k r ,where k= 1/√([-m^24 ζ+ σm^2]) Herek is a constant and as ζ is negative, k is real and positive. It is imperative that we check the consistency of the solution obtained before proceeding any further. We can easily see that only at very large r,𝒢^tt turns out to be infinitesimal in magnitude and this is necessary for our effective metric G_ab tosatisfy theEq. (<ref>)This condition renders the contribution of the third term of Eq. (<ref>) negligible and we end up gettingEq. (<ref>) Starting from the effective metricG_ab, we can calculate the components of the Einstein tensors given by𝒢^tt = -w'r +h”r^2 -h'2r^3 -3 h 2 r^4- wr^2 𝒢^rr = -v'r -h'2r^3 -3 h 2 r^4+ wr^2𝒢^θθ =-v”2 r^2 -(v'-w')2r^3 +3 h 2 r^6+h'2r^5 𝒢^ϕϕ =r^2 sin^2θ𝒢^θθ Since the last three equations vanish, we can freely set h(r)=0 as w(r) = r v'(r) solves both the second and third equation. Thus, first equation implies-w'r - wr^2 =e^- k r This can be easily solved to get the solution w(r) = fr + e^-k r(k^2 r^2+2 k r+2)k^3 r The constant of integration can be fixed by taking the classical limit of r →∞ and identifying the leading order term of the metric component as that of the Schwarzschild metric for a very large r ,viz. (1 + 2 G Mr). Thus, we getw(r) = 2 G Mr + e^-k r(k^2 r^2+2 k r+2)k^3 r From w(r) = r v'(r) , we get v(r) = - 2 G Mr-e^-k r (kr+2)k^3r Thus, we observe that the photon “sees" a modified Schwarzschild metric, with a very small quantum correction. However, there is a peculiar property to this solution which is absent in the Schwarzschild solution of general relativity. Let us solve for the radius at which the radial component of velocity of the photon vanishes. This radial distance corresponds to the event horizon of the black hole. Since the photon travels along null geodesics in geometric optics approximation, we have G_ab u^a u^b = ℬ(r) ṫ^2+ 𝒜(r) ṙ^2 =0Here, u^a is the tangent vector along the trajectory of the photon, parametrized by some τ and the dot is taken w.r.t τ . The expression translates todrdt = √(- ℬ(r)/𝒜(r)) = √((1+v(r)) (1-w(r))) The LHS is the radial component of the co-ordinate velocity of the photon. The dependence of the velocity with r can analyzed using Eq. (<ref>). Plugging in the values of the functions, we can easily see that the radial component vanishes not at the Schwarzschild radius, but at a slightly closer value of r. This implies that the event horizon depends upon the direction of polarization of the light.Since the metric G_ab is lorentzian, the predictability of the matter equations is guaranteed and the principle polynomial becomes bi-hyperbolic. Thus, the solution doesn't violate the initial assumptions of the theory.Now let us consider a photon which is not polarized in any particular direction. Thus, the components a^k will be varying in both negate and positive directions. As the frequency of this variation is greater than the reaction rate of the metric to the fluctuations of the electromagnetic field, we take a statistical average and find that ⟨a^r⟩=⟨ a^t⟩ =⟨a^θ⟩ = ⟨a^ϕ⟩ = 0. Thus, if we set𝒢^rr =𝒢^θθ = 0, the new equations reduce to the Einstein's equations. Hence, we get back the Schwarzschild solution. This is an interesting phenomenon as some photons “see" the event horizon at a particular distance while other photons with different polarization see the horizon at another radial co-ordinate. § DISCUSSIONWe have studied how the interaction of photons and the metric modifies the Maxwell's equation in QED and how this leads to new gravitational dynamics. The alteration of local photon propagation has lead to the modification of the Maxwell's equation. For these new matter equations to have a well-posed Cauchy problem, it is necessary for the underlying gravitational field to have a modified dynamics. These constraints lead to a new modified theory of gravity. The most surprising feature of the theory is that the modified Einstein's equation is dependent on the polarization of the photon. This is analogous to the dependence of wavelength of light in the rainbow gravity theory[4]. However, it should be noted that when we were formulating the theory, we started off with just electromagnetic field. We did not include other forms of matter to determine the Lagrangian for gravity. Light cones of the photons will cease to dictate the trajectories of massive fields upon the inclusion of other such matter fields. This is because massive fields couple to the metric through their fields or the components of the fields in some representation space of their symmetry group and may behave non-trivially.It should also be noted that only kinematics of photons of very long wavelengths are described by the theory as the modified equations are derived using geometric optics ansatz. Thus, it is necessary to account for higher energyphotons in developing a better approximation to this modified theory .Another consequence of the modifications is that for a homogeneous and isotropic universe, the dependence of polarization of the metric translates to the fact that if we trace out the past of each photon, we end up getting different singularities in spacetime. This has a direct implication that a particle sees the universe older or younger than another particle with a different polarization. Thinking along the lines of Ref.<cit.>, we find that this could solve the Horizon problem or even takes us further away from a solution. The event horizon of a black hole becomes dependent on the polarization of the photon. The exact location of horizon of the black hole is different for different polarization of light. Thus, we end up getting an effective event horizon rather than an absolute event horizon. This dependence of polarization might give us some clues in solving the information loss paradox <cit.> and improve our understanding of black holes.§ ACKNOWLEDGMENTS I am very grateful to Aninda Sinha and Chethan Krishnan of Center for High Energy Physics, Indian Institute of Science for their useful and insightful comments on the paper. * | http://arxiv.org/abs/1708.07997v2 | {
"authors": [
"Vyshnav Mohan"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170826170721",
"title": "Back-reaction of quantum processes and modified gravitational dynamics"
} |
Department of Science and Technology (ITN), Linköping University, Campus Norrköping, 60174 Norrköping, SwedenSchool of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, UKSchool of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, UKLULI, Ecole Polytechnique, CNRS, CEA, UPMC, 91128 Palaiseau, FranceSchool of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, UKUniversité de Lyon, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, FranceUniversité de Lyon, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, FranceETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario de Ciudad Real, 13071 Ciudad Real, SpainSchool of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, UK corresponding authorThe expansion of a radial blast shell into an ambient plasma is modeled with a particle-in-cell (PIC) simulation. The unmagnetized plasma consists of electrons and protons. The formation and evolution of an electrostatic shock is observed, which is trailed by ion-acoustic solitary waves that grow on the beam of the blast shell ions in the post-shock plasma. In spite of the initially radially symmetric outflow, the solitary waves become twisted and entangled and, hence, they break the radial symmetry of the flow. The waves and their interaction with the shocked ambient ions slows down the blast shell protons and brings the post-shock plasma closer to an equilibrium. Expansion of a radial plasma blast shell into an ambient plasma M. Borghesi December 30, 2023 ===============================================================The ablation of a solid target by an intense laser pulse yields a dense and hot blast shell <cit.>. Collisions between the plasma particles do not frequently occur on the time-scales of interest and the plasma remains far from a thermal equilibrium. The hot and light electrons expand faster than the ions and the charge separation results in an electric field that accelerates the ions of the blast shell. Depending on the duration and intensity of the laser pulse, they can reach speeds of the order of 10^5-10^7 m/s via this process, often referred to as target normal sheath acceleration (TNSA) <cit.>. Radiation from the target ionizes any residual gas that was present in the experimental vessel prior to the laser shot. This ambient plasma will resist the expansion of the blast shell. During the blast shell's free expansion phase its thermal pressure exceeds by far that of the ambient plasma. The blast shell expands in the form of a rarefaction wave <cit.>, which piles up the ambient plasma ahead of it. A forward shock forms between the piled-up ambient plasma and the pristine ambient plasma. This shock is mediated by collective electrostatic forces if no background magnetic field is present <cit.>.The forward shock increases the thermal pressure of the ambient plasma by heating and compressing it. An expansion of the radially symmetric blast shell furthermore implies a density profile that decreases rapidly with increasing radius r. The pressure of the shock-compressed ambient plasma will become large enough at some r to slow down the blast shell. Laboratory experiments show that in collisionless plasma ion-acoustic solitons (IAS's) <cit.> and electrostatic shocks <cit.> emerge in the region where the blast shell interacts most effectively with the ambient plasma. Here we perform a PIC simulation to test if we can observe these structures. We model for this purpose the expansion of a circular blast shell in a two-dimensional simulation box with the PIC code EPOCH <cit.>. The radial symmetry, the absence of any strong background magnetic field and the usage of reflecting boundary conditions for particles and fields implies that we only need to resolve one quadrant of the expanding blast shell. We model the quadrant defined by 0≤ x ≤ L and 0 ≤ y ≤ L, where the side length L = 1.2 mm of the simulation box is resolved by 1500 grid cells along each direction. A uniformly distributed plasma, which consists of electrons and protons with the number density n_0 = 3× 10^16cm^-3, fills the simulation box at the time t=0. The electron density n_0 is comparable to that in the ambient plasma in laser-plasma experiments. The electrons (protons) are represented by 100 (200) computational particles (CPs) per cell and the electron (proton) temperature is set to T_0 = 1 keV (T_0/10). The plasma frequency of the ambient medium is ω_p ≡(n_0e^2/m_eϵ_0)^1/2≈ 10^13s^-1 (e, m_e, ϵ_0: elementary charge, electron mass and vacuum permittivity). The blast shell is modeled by superimposing a second plasma on top of the ambient one in the interval 0 ≤ r ≤ L/4, where r^2=x^2+y^2. The densities of the electrons and protons of this second plasma are 24n_0 and the electrons (protons) of this plasma are represented by 400 (800) CPs per cell. The proton temperature of the blast shell plasma matches that of the ambient plasma while the electron temperature is 4.5T_0. We adjust the numerical weights of the CP's that represent the electrons and protons such that the plasma is initially charge-neutral. No net current is present at t=0 and we set 𝐄(x,y) and 𝐁(x,y) to zero. The simulation resolves 530 ps by 3× 10^5 time steps. The simulation provides us with the spatio-temporal distributions of the proton density n(x,y,t) (normalized to n_0) and the normalized energy density E_E(x,y,t)=e^2ϵ_0(E_x^2(x,y,t)+E_y^2(x,y,t))/(2m_e^2c^2ω_p^2) of the in-plane electric field. We re-sample both distributions in radial coordinates (r,α: radius and azimuth angle relative to y=0), which gives n(r,α,t) and E_E (r,α,t). Thermal diffusion results in a net flow of electrons from a dense into a dilute plasma and the blast shell plasma goes on a positive potential relative to the ambient one. The ambipolar electric field, which sustains the potential difference, accelerates the ambient electrons that enter the blast shell and decelerates the blast shell electrons that escape into the ambient plasma. Two-stream instabilities, which would otherwise develop in the dense plasma <cit.>, are suppressed by the larger initial temperature of the blast shell electrons. The ambipolar electric field will accelerate protons towards increasing r. Figure <ref> visualizes this expansion with the help of n(r,t) and E_E(r,t), which are the azimuthal averages of n(r,α,t) and E_E(r,α,t). The contour n(r,t)=20 in Fig. <ref>(a) moves in time to lower r and reaches r=0 at t = 530 ps. Rarefaction waves propagate into the dense plasma and accelerate protons towards the dilute plasma. The density distribution at t=530 ps decreases approximately exponentially with increasing r<400μ m. A new density bump separates itself from the rarefaction wave at r≈ 350μ m and t≈ 100 ps (lower white circle), which is confined by two boundaries across which the density changes to its maximum ≈ 3. The right boundary propagates from r=300μ m at t=0 to r = 750μ m at t= 530 ps. Its speed decreases in time and its average is v_fs≈ 8.5 × 10^5 m/s. Given that the ion acoustic speed c_s=(k_B(5T_0/3 + 3T_0/10)/m_p) (k_B,m_p: Boltzmann constant and proton mass) of the unperturbed ambient medium is c_s ≈ 4.3 × 10^5 m/s, this front is an electrostatic shock with the Mach number v_fs/c_s ≈ 2. A density pulse forms at t≈ 200 ps in Fig. <ref>(a) (upper white circle), which detaches itself from the main bump and reaches r≈ 530 μ m at t=530 ps (black circle). A peak of E_E(r,t) forms in Fig. <ref>(b) at the blast shell boundary r=300 μ m immediately after the simulation started. It is the ambipolar electric field that is driven by the initial density jump. Its magnitude exceeds the displayed color range by the factor 10. This initial pulse spreads out and elevated values of E_E(r,t) are present in the interval, which is delimited by the line r=300μ m and the line that starts at the same position and goes to r=150 μ m at t=530 ps. This electric field patch outlines the density gradient of the rarefaction wave. Statistical fluctuations of the particle number in a volume element in PIC simulation or in real plasma yield fluctuations in the charge- and current density and, hence, electromagnetic fluctuations <cit.>. The field energy density increases with the particle's thermal energy density. The latter is large in the blast shell plasma, causing elevated level of E_E(r,t) at low r in Fig. <ref>(b).The large electric field energy in the density bump at large r is related to the thermalization processes in collisionless plasma. A sharp propagating electric pulse is observed that travels from 300μ m at t≈ 0 to 400μ m at t=100 ps, after which it starts to become more diffuse. The speed of this pulse is 2.3c_s for 0 < t < 100 ps and it is thus the forward shock.Figure <ref> shows the spatial distributions of n(r,α,t) and E_E(r,α,t).A sharp electric field pulse is present at r≈ 400 μ m in Fig. <ref>(a), which coincides with the density jump between the expanding blast shell and the dilute ambient medium in Fig. <ref>(b). This pulse is the electrostatic shock. The shock has propagated to r ≈ 490μ m at t=200 ps in Fig. <ref>(c). It has lost its sharpness and waves are observed upstream of it. Such a fragmentation is typical for shocks that reflect a significant part of the inflowing upstream ions, which triggers ion acoustic instabilities <cit.>. The onset of such instabilities explains why the shock has become diffuse in Fig. <ref>(b). Another structure has emerged at r≈ 410μ m in Fig. <ref>(c), which is close to the left boundary of the density bump in Fig. <ref>(a). It reveals two stripes with a large electric field energy density that are separated by a minimum, which coincides with a density spike in Fig. <ref>(d). We refer with ion solitary wave (ISW) to such a structure. The ISW and the forward shock at r≈ 490μ m enclose a turbulent region with an elevated plasma density. Figures <ref>(e,f) show E_E(r,α) and n(r,α) at t=530 ps. We observe two strong ISWs at r≈ 540μ m and at r≈ 570μ m and entangled ones at larger r. The average density increases with increasing r in the interval 550 μ m ≤ r ≤ 650μ m, in which we find the entangled ISWs. The trailing ISW had the practically constant value r=410μ m at t=200 ps, while it shows strong variations of r with α at 540μ m and t=530 ps. The wavelength of the oscillation at r≈ 540 μ m and α≈ 70 degrees is about 20 degrees, which corresponds for this value of r to an arc length of ≈ 190μ m. The amplitude and wave length of this oscillation are close to the values observed at a thin shell of dense ions in a laboratory plasma with similar conditions <cit.>.The nonlinear plasma structures can be identified unambiguously with the help of the phase space density distribution of the protons. Figure <ref> showsthe phase space density distribution f_p(r,α,(|v|/v_th)^2) of the protons at the time 200 ps, where v_th=(k_BT_0/10m_p)^1/2 is the proton thermal speed ≈ 10^5 m/s.The mean velocity of the rarefaction wave increases linearly with r in the interval 300 μ m ≤ r ≤ 360 μ m and its velocity increase slows down between r=360μ m and r = 400μ m. The reduced acceleration is caused by the presence of shocked ambient protons at this location. The density contribution of these protons decreases the proton density gradient in this interval and, hence, the ambipolar electric field that accelerates the protons of the rarefaction wave. The mean velocity of the blast shell protons oscillates in the interval 400 μ m ≤ r ≤ 450μ m with an amplitude that exceeds v_th significantly. These velocity oscillations correspond to the previously observed ISWs. They are immersed in hot protons, which originate from the shock-heated ambient protons. They form a dilute cloud with a large thermal spread in the interval 400μ m ≤ r ≤ 500 μ m. A forward shock is located at r≈ 500μ m, which moves to increasing r. The ambient protons that cross this shock are heated to the downstream temperature. A fraction of the protons is reflected by the shock potential. These protons feed the dilute low-energy part of the energetic proton beam in the interval r>500 μ m. Reflected protons are a characteristic of electrostatic shocks. Blast wave protons that crossed the downstream region and were accelerated to larger energies by the shock potential form the denser high-energy part of the fast proton beam ahead of the shock. The shock thus also acts as a double layer <cit.>. The interaction between the energetic proton beam and the ambient protons in the interval r>500 μ m causes an ion-ion instability <cit.>, which replaces the narrow uni-polar electric field pulse at r=400 μ m in Fig. <ref>(a) by the broad turbulent layer that starts to form in Fig. <ref>(c) at r≈ 500μ m.Figure <ref> shows f_p(r,α,(|v|/v_th)) for α = 45^∘ and t=530 ps. The blast shell protons enter with the speed 8 × 10^5 m/s at r=450μ m and are slowed down by the ISW at r ≈ 530 μ m, which is the density band to the left in Fig. <ref>(f). The amplitude of its velocity modulation is 4v_th, which is close to c_s, and its width is about 5 electron Debye lengths λ_D ≡(ϵ_0 k_B T_0/n_0e^2)^1/2=1.35 μ m of the ambient plasma. The large amplitude of the ISW is close to the limit, at which it changes into a shock <cit.>. The ISW has moved from r≈ 410μ m in Fig. <ref>(d) to ≈ 530μ m in Fig. <ref>(f) and its speed 3.6× 10^5 m/s is approximately constant (See Fig. <ref>). The ISW propagates at the speed ≈ c_s towards lower values of r in the rest frame of the proton beam that moves with the speed ∼ 8 v_th in Fig. <ref>. The local ion acoustic speed exceeds c_s because the electron temperature, averaged over the interval 525 μ m < r < 540 μ m in which the ISW is located, is 30% larger than T_0 (not shown). Even the strongest ISW is thus not an IAS, which would require it to propagate faster than the local c_s <cit.>. The blast shell protons traverse the ISW and encounter a second one at r ≈ 560μ m. More ISW's are observed to the right, which form the entangled ISW's in Fig. <ref>(f). The size of the ISW's and the density of the blast shell protons decreases with each ISW crossing and the hot proton background gets denser. The hot low-energetic protons have a density minimum at the location of each ISW; the electric potential of each ISW repels protons. In conclusion we have modeled the expansion of a radial blast shell into a uniform plasma. A shock formed, which moved at more than twice the ion acoustic speed and compressed, heated and accelerated the ambient protons. ISW's formed in the post-shock plasma, which consisted of a dense beam of blast shell protons and the shock-heated ambient protons. The ISW's grew in a turbulent plasma and, hence, they were non-planar to start with. An instability amplified their initial oscillations. The electric field distributions of these entangled ISW's and their interaction with the shocked ambient protons slowed down and compressed the blast shell protons and helped confining the shock-heated ambient protons.M. E. D. acknowledges financial support by a visiting fellowship of CRAL. M. B. and G. S. acknowledge financial support by the EPSRC grants: EP/P010059/1 and EP/K022415/1.The simulations were performed on resources provided by the Grand Equipement National de Calcul Intensif (GENCI) through grant x2016046960. 100 Maksimchuk00 A. Maksimchuk, S. Gu, K. Flippo, D. Umstadter, and V. Y. Bychenkov, Phys. Rev. Lett. 84, 4108 (2000). Wilks01 S. C. Wilks, A. B. Langdon, T. E. Cowan, M. Roth, M. Singh, S. Hatchett, M. H. Key, D. Pennington, A. MacKinnon, and R. A. Snavely, Phys. Plasmas 8 542 (2001). Romagnani05 L. Romagnani, J. Fuchs, M. Borghesi, P. Antici, P. Audebert, F. Ceccherini, T. Cowan, T. Grismayer, S. Kar, A. Macchi, P. Mora, G. Pretzler, A. Schiavi, T. Toncian, and O. Willi, Phys. Rev. Lett. 96 195001 (2005). Auer75 J. E. Crow, P. L. Auer, and J. E. Allen, J. Plasma Phys. 14 65 (1975). Mora09 P. Mora, and T. Grismayer, Phys. Rev. Lett. 102 145001 (2009). Bardotti70 G. Bardotti, and S. E. Segre, Plasma Phys. 12 247 (1970). Forslund70 D. W. Forslund, and C. R. Shonk, Phys. Rev. Lett. 25 1699 (1970). Forslund71 D. W. Forslund, and J. P. Freidberg, Phys. Rev. Lett. 27, 1189 (1971). Eliasson06 B. Eliasson, and P. K. Shukla, Phys. Rep. 422 225 (2006). Romagnani08 L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Loewenbrueck, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O, Willi, Phys. Rev. Lett. 101 025004 (2008). Honzawa73 T. Honzawa, Plasma Phys. Controll. Fusion 15 467 (1973). Ahmed13 H. Ahmed, M. E. Dieckmann, L. Romagnani, D. Doria, G. Sarri, M. Cerchez, E. Ianni, I. Kourakis, A. L. Giesecke, M. Notley, R. Prasad, K. Quinn, O. Willi, and M. Borghesi, Phys. Rev. Lett. 110 205001 (2013). Arber15 T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, Plasma Phys. Controll. Fusion 57 113001 (2015). Dieckmann10 M. E. Dieckmann, G. Sarri, L. Romagnani, I. Kourakis, and M. Borghesi, Plasma Phys. Controll. Fusion 52 025001 (2010). Dieckmann04 M. E. Dieckmann, A. Ynnerman, S. C. Chapman, G. Rowlands, and N. Andersson, Phys. Scr. 69 456 (2004). Karimabadi91 H. Karimabadi, N. Omidi, and K. B. Quest, Geophys. Res. Lett. 18 1813 (1991). Kato10 T. N. Kato, and H. Takabe, Phys. Plasmas 17 032114 (2010). Hamad17 H. Ahmed, D. Doria, M. E. Dieckmann, G. Sarri, L. Romagnani, A. Bret, M. Cerchez, A. L. Giesecke, E. Ianni, S. Kar, M. Notley, R. Prasad, K. Quinn, O. Willi, and M. Borghesi, Astrophys. J. 834 L21 (2017). Hershkowitz81 N. Hershkowitz, J. Geophys. Res. 86 3307 (1981). Forslund70b D. W. Forslund, and C. R. Shonk, Phys. Rev. Lett. 25 281 (1970).Malkov15 M. A. Malkov, R. Z. Sagdeev, G. I. Dudnikova, T. V. Liseykina, P. H. Diamond, K. Papadopoulos, C. S. Liu, and J. J. Su, Phys. Plasmas 23 043105 (2016). Tran79 M. Q. Tran, Phys. Scripta 20 317 (1979). Dieckmann15 M. E. Dieckmann, H. Ahmed, D. Doria, G. Sarri, R. Walder, D. Folini, A. Bret, A. Ynnerman, and M. Borghesi, Phys. Rev. E 92 031101 (2015). | http://arxiv.org/abs/1708.07991v1 | {
"authors": [
"M E Dieckmann",
"D Doria",
"H Ahmed",
"L Romagnani",
"G Sarri",
"D Folini",
"R Walder",
"A Bret",
"M Borghesi"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170826161317",
"title": "Expansion of a radial plasma blast shell into an ambient plasma"
} |
Design of Configurable Sequential Circuits in Quantum-dot Cellular AutomataM. Goswami et. al.M. Goswami, M. R. Chowdhury and B. Sen Department of Computer Science and Engineering,National Institute of Technology, Durgapur,West Bengal - 713209, India [email protected] Design of Configurable Sequential Circuits in Quantum-dot Cellular Automata Mrinal Goswami Mayukh Roy Chowdhury Bibhash Sen Received: date / Accepted: date =========================================================================== Quantum-dot cellular automata (QCA) is a likely candidate for future low power nano-scale electronic devices. Sequential circuits in QCA attract more attention due to its numerous application in digital industry. On the other hand, configurable devices provide low device cost and efficient utilization of device area. Since the fundamental building block of any sequential logic circuit is flip flop, hence constructing configurable, multi-purpose QCA flip-flops are one of the prime importance of current research. This work proposes a design of configurable flip-flop (CFF) which is the first of its kind in QCA domain. The proposed flip-flop can be configured to D, T and JK flip-flop by configuring its control inputs. In addition, to make more efficient configurable flip-flop, a clock pulse generator (CPG) is designed which can trigger all types of edges (falling, rising and dual) of a clock. The same CFF design is used to realize an edge configurable (dual/rising/falling) flip-flop with the help of CPG. The biggest advantage of using edge configurable (dual/rising/falling) flip-flop is that it can be used in 9 different ways using the same single circuit. All the proposed designs are verified using QCADesigner simulator. § INTRODUCTIONCMOS based VLSI designs play a very important role in digital design. The operation of computer chips has shown outstanding growth in the last few decade. But the exponential increase of computing power over the last decades has relied on shrinking the transistor. The recent survey of International Technology Roadmap for Semiconductors (ITRS-2015) shows that the transistor could face the challenge of feature size limitation by 2021 <cit.>. Quantum-dot Cellular Automata (QCA) is an emerging nanotechnology with extremely small feature size and ultra low power consumption which is suggested as an alternative technology of current CMOS <cit.>. Flip-flop plays a significant role in any sequential circuits. Sequential circuits in QCA gain maximum attention due to its realization difficulties in terms of architectural complexity and synchronous mechanism <cit.>. Till date, various studies have been performed on flip-flop design. However, all the previous designs realized different flip-flops with disparate designs and thus they differ in terms of area and latency. No regular, uniform or symmetric design paradigm is addressed so far for a simple implementation of flip-flops in QCA. The disparity of designs (SR, D, T and JK) leads to various problems in a single circuit having multiple flip-flops. The inter-connection of the different flip-flops (SR, D, T and JK) is another big issue. So, to tackle these issues, a uniform design methodology is to be explored for flip-flops having same area and latency.In the recent past, conventional QCA based flip-flops were studied extensively <cit.> but, most of the previous designs are not robust and highly error prone due to defective wire-crossing employed in their designs <cit.>. Generally,two well known wire-crossing techniques have been used in QCA technology: multi-layer and coplanar wire-crossing (using rotated QCA cells). Multi-layer wire-crossing has several fabrication limitations <cit.> <cit.> due to which it is not considered for our proposed designs. On the other hand, coplanar wire-crossing in QCA can be realized with 45^0 (normal QCA cell) cells and 90^0 (rotated QCA cell) cells. However, production of 90^0 cell needs a high level of precision in placing which increases the overall cost and the implementation complexity <cit.> <cit.>. On the other hand, flexibility and performance are the two big issues of any digital logic circuit <cit.>. The intermediate trade-off between flexibility and performance is the utmost necessity for current digital logic circuit which can be attained by reconfigurability <cit.>. It is observed that reconfigurability in the nanoscale era can lead to the design of various low power and energy efficient system which is the need of the hour <cit.>. All these above factors motivate us to design configurable, robust flip-flop structures for QCA which can address the irregular behaviour of designs as well as reliability issue of wire-crossing in QCA. The main contribution of this research is as follows: * A configurable level triggered QCA flip-flop (CFF) is designed which can be configured to JK, D and T flip-flop. This is the very first attempt to design a configurable flip-flop in QCA.* To remove the wire-crossing difficulties, clock-zone based coplanar crossover technique is employed, which is the most robust wire-crossing technique in QCA technology <cit.><cit.>.* Considering CFF as a basic element, an edge configurable (dual/rising/falling) flip-flop is designed using a clock pulse generator (CPG).* Based on the edge configurable (dual/rising/falling) flip-flop, an n-bit edge configurable (dual/rising/falling) counter/shift register is also proposed. The rest of the paper is organized as follows: section <ref> introduces fundamentals of quantum-dot cellular automata (QCA). Existing works based on reconfigurable logic are discussed in section <ref>. The proposed design of level triggered configurable flip-flop (CFF) is introduced in section <ref>. The design of edge configurable flip-flop is introduced in section <ref>. The higher order QCA configurable circuits are proposed in section <ref> employing edge configurable flip-flop as the basic element. Finally, section <ref>, concludes the paper. § BASICS OF QCA A QCA cell comprises of four quantum dots (Figure <ref>(a)) which can carry two free electrons. These free electrons can move between the four quantum dots. As shown in Figure <ref>(b), the two polarization states of a QCA cell can be represented as P=-1 (logic 0) and P=+1 (logic 1). The basic structures of QCA technology are majority voter (Figure <ref>(c)) and inverter (Figure <ref>(d)). The QCA majority voter (MV) can be expressed as, MV(A, B, C) = AB + BC + CA A majority voter can serve as a 2-input AND gate if one of the inputs is fixed at P=-1. Alternatively, if any one input of the majority voter is fixed at P=+1 then the modified majority voter can serve as a 2-input OR gate. There are two different types of inverter available in QCA technology as shown in Figure <ref>(d). The possible two orientations (“+") and (“×") (90^ 0 and 45^ 0 respectively) of QCA cell are shown in Figure <ref>(e). There are two fundamental wire-crossing techniques (co-planar and multi-layer) available in QCA. As shown in Figure <ref>(e), the co-planar wire-crossing can be implemented with the mix combination of two different orientations (90^0 and 45^0) of QCA cells. The multi-layer wire-crossing can be implemented with the help of two or more different layers. As discussed earlier, co-planar and multi-layer wire-crossing techniques are facing serious issues <cit.> <cit.> due to which we consider clock-zone based wire-crossing technique which can be implemented using non-adjacent clock zones (phase difference is equal to 180^0)on the same plane <cit.><cit.> as shown in Figure <ref>. This eliminates the problem of interference between the QCA cells. The QCA clock controls the flow of information within the circuit. The circuit information is carried from one end to another with the help of clock. The clock zones of QCA are distinct and 90^0 phase shifted <cit.> <cit.>. At the time of computation, the previous clock zone must hold its output, which can be achieved by splitting the clock into four phases: switch, hold, release and relax <cit.> which is shown in Figure <ref>(f). Four types of implementation technology exists in QCA: (a) metal-island <cit.>; (b) semiconductor <cit.>; (c) molecular <cit.> <cit.>; and (d) magnetic <cit.> <cit.>. The metal-island is the first implementation technology created to demonstrate the behaviour of QCA which requires cryogenic temperature to operate <cit.>. The semiconductor technology, now-a- days becomes operable in room temperature <cit.>. The semiconductor QCA technology is adopted as the implementation technology for our proposed QCA designs. The molecular technology, not yet implemented, is a single molecule implementation technology. This technology is highly promising due to its highly symmetric QCA cell structure, very high switching speeds, extremely high device density, operation at room temperature and even the possibility of mass-production by means of self-assembly. Magnetic QCA (MQCA) is based on the interaction between the nanoparticles of the magnet. The MQCA depends on the quantum mechanical nature of magnetic interactions, which can be operated at room temperature <cit.> <cit.>. § RELATED WORK Limited attempts have been made to realize reconfigurable designs in QCA <cit.>. An And-OR-Inverter (AOI) logic is proposed in <cit.> where by fixing one or more input various logic functions can be implemented such as OR-AND, NAND-OR, NOR-NAND etc. which is shown in Figure <ref>(a). In <cit.>, a complex 7-input QCA configurable gate (out of 7 inputs, 3 inputs are used as control inputs) is proposed by cascading three 3-input majority voter as shown in Figure <ref>(b). The sum of product, product of sum, four-input AND, four-input OR logic etc. can be constructed by fixing control inputs to “0" or “1". In <cit.>, under the presence of different QCAdefects, a symmetric configurable fault tolerant reconfigurable gate (RFTG) has been proposed as shown in Figure <ref>(c). A novel configurable QCA design is proposed in <cit.>. The design (Figure <ref>(d)) is capable of realizing various logic functions such as 2-input OR, 3-input AND, 2-input OR, 3-input OR etc. A reconfigurable majority gate (Figure <ref>(e)) is presented in <cit.>. It is apparent that most of the previous work is limited to implement small logic functions only. Configurable memory structures in QCA are yet to be explored. § PROPOSED LEVEL TRIGGERED CONFIGURABLE FLIP-FLOPThe main advantage of realizing configurable hardware is low device cost and efficient utilization of device area.Till date, all the existing configurable designs in QCA strictly follow design rules mentioned below as in <cit.>: * Fixing input cells (control inputs) to logic “0" or “1" to produce different functions.* Displacing some of the input cells to change the distance between driver cells of the device to generate different functionality.The first method is adopted to realize the proposed configurable designs as mentioned in <cit.>. To realize a configurable flip-flop, we first choose a JK flip-flop. The characteristic equation of JK flip-flop is Q(t+1) = JQ + KQ. To produce D and T flip-flop, either a complemented value of input J or an un-complemented value of input J need to pass to both the inputs of JK flip-flop. The output function of XNOR is F = A.B + A.B. If we choose B as a control input then we can use XNOR function as a simple QCA wire or an inverter. If B is zero then it will act as an inverter otherwise it will act as a simple QCA wire. So, the XNOR function can be used to produce a complemented value as well as an un-complemented value of its inputs controlling one of the inputs to zero or one. Moreover, a multiplexer is the best possible choice to select any one input between two input values.The QCA representation of the proposed configurable flip-flop (CFF) is shown in Figure <ref>. The proposed CFF has 5 inputs (A, B, C1, C2 and CLK) and two outputs (Q and Q) where C1, C2 are control inputs. The primary output of the proposed CFF is as follows:Q_t+1 = { A.Q_t +(B.C1 + A.C2. C1 + A. C1. C2)Q_t} CLK +CLK. Q_t Q_t+1 = { A.Q_t + {A. B(C1+C2) + A.B(C1+C2) +B.C1 + C1(A.C2 + A.C2)} Q_t} CLK + CLK.Q_t §.§ Case 1:If C1 = C2 = 0 and CLK=1 then the equation <ref> will be Q_t+1 = { A.Q_t + { A.B + A } Q_t}Q_t+1 = A.Q_t + A.Q_t Q_t+1 = A §.§ Case 2:If C1 = 0, C2 = 1 and CLK = 1 then the equation <ref> will be Q_t+1 = A.Q_t + {A.B + A} Q_tQ_t+1 = A.Q_t + A.Q_t §.§ Case 3:If C1 = 1, C2 = X (Don't Care) and CLK = 1 then the equation <ref> will be Q_t+1 = A.Q_t + {A.B + A.B + B} Q_tQ_t+1 = A.Q_t + B.Q_tThe Table <ref> shows the different functionality of the proposed CFF based on control inputs C1 and C2. The CFF will obey the rules of T FF if C1C2 = 01 and if C1C2 = 00 then CFF will behave as D FF. The CFF will behave as JK FF if C1C2 = 1X where X means don't care. The truth table of the proposed CFF is shown in Table <ref>. §.§.§ Module 1 (XNOR)The function produced by this module is: F1= A.C2 + A.C2. The output of module 1 is passed to module 2 as shown in Figure <ref>. Depending on the value of input C2, module 1 generates a complemented or un-complemented value of input A to module 2. If C2=0 then a complemented value of the input A is passed to module 2 otherwise uncomplemented value of input A is passed to module 2.§.§.§ Module 2 (2:1 MUX)The output function of this module is : F2=B.C1+F1.C1. If C1=1, input B will be selected and module 3 will behave according to JK flip-flop. On the other hand, if C1=0, the module 3 will behave either D flip-flop or T flip-flop depending on the output F1. If F1= A then the module 3 will behave as a D flip-flop otherwise it will behave as a T flip-flop. §.§.§ Module 3 (JK FF)This module follows the instructions of module 1 and 2. Module 3 behaves as T flip-flop if module 1produces F1 = A and at the same time module 2 produces F2 = F1 (Figure <ref>(a)). Alternatively, if module 1 produces F1= A and at the same time module 2 produces F2 = F1 then module 3 behaves as a D flip-flop. Finally, if module 2 produce F2 = B (module 1: F1= don't care ) then module 3 behaves like a JK flip-flop. The advantage of configurable memory units lies in its application domain. These can be used to design circuits with architectural similarity eliminating the need to design separate hardware performing a specific function. For example, counters and shift registers can be implemented in a single design with the option to configure the flip-flop according to the need. The biggest advantage of the proposed CFF is that single module can serve as a D, T and JK FF. Table <ref> shows the performance of CFF with the existing D, T and JK FF. The proposed CFF consists of 2.75 delay covering an area of 0.20 μ m^2 with 159 QCA cells. Although, the proposed CFF exceeds in all the respect compared to existing designs, but these small overheads can be accepted due to the configurable superiority of CFF over the existing designs. All the previous designs <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> <cit.> are not configurable in nature and can produce only one function. On the other hand, the CFF can produce three different functions using the same circuit. Moreover, the CFF allows us to present an all in one flip-flop, which can also be perceived as a standard universal design. A universal, scalable, efficient (USE) realistic clock distribution scheme for QCA is proposed in <cit.>. The CFF layout under the USE clocking scheme (CFF-USE) is shown in Figure <ref>. The proposed CFF-USE consists of 541 QCA cells covering an area of 1.81 μ m^2. §.§ Memory design using proposed level triggered flip-flopIn this section, a 1-bit RAM is designed to test the proposed CFF functionality. All the pervious memory (RAM) designs in QCA can be categorized into two groups: line based and loop based. In the loop-based RAM design, four clock zones are utilized to store the previous value but in the line-based RAM design, a QCA wire is used to store the previous value of the output <cit.>. The proposed RAM design also utilizes the property of loop based design which is shown in Figure <ref>(b). The incoming input value is circulated inside the memory loop if the Enable input is set to 1 and at the same time Write/Read input is set to 0. If the Enable input is set to 1 and Write/Read input is 1 then the current stored value inside the memory loop is fed to the output. In both the cases i.e. read or write, the control input value of C2 is set to zero. The simulation result of the proposed RAM is shown in Figure <ref>. The inputs Enable and Write/Read are configured in such a way that they can provide the values needed for C1 and B input (Figure <ref>(a)). The proposed 1-bit RAM has 209 QCA cells covering an area of 0.31 μ m^2.§ DESIGN OFEDGE CONFIGURABLE FLIP-FLOPThe level triggered JK flip-flop is susceptible to noise due to which it leads to race-round condition <cit.>. In order to avoid such unstable phenomenon, edge triggered (falling, rising and dual) flip-flops are extensively studied. There are two well-known schemes in QCA to implement edge triggered flip-flop, the clock pulse generator scheme and the MUX/latch scheme <cit.>. In QCA, the clock pulse generator scheme is explored more than the MUX/latch scheme. Figure <ref> shows the proposed pulse generator which can produce clock pulses for the falling edge, rising edge as well as dual edges with the help of the two control inputs C3 and C4. To detect a falling edge it takes the help of the previous clock pulse (CLK_old) and compares it with the current clock pulse (CLK). The CLK_old is produced by using four consecutive clock zones (one clock cycle delay) as shown in Figure <ref>(b). The majority voter representation of the falling edge operation is as follows:Out1= MV(MV(CLK, C3, -1), CLK_old, -1) The value of (CLK. CLK_old) will give a resultant boolean value of 1 which triggers a pulse in the output if the control inputs are set to C3=1 and C4=0 (Table <ref>). At the same time, the proposed clock pulse generator can generate rising edge (CLK (CLK_old)) which results to 1) if C3=0 and C4=1 (Table <ref>) and hence producing a trigger in the final output. The majority voter representation of the rising edge operation is as follows:Out2= MV(MV(CLK, C4, -1), CLK_old, -1) The output of Out1 and Out2 (Figure <ref>(b)) is passed through an OR gate to produce the final output. If both the control inputs (C3=1 and C4=1) are set to 1 then the proposed clock pulse generator can produce pulses for both the edges (rising as well as falling) and hence works as a dual edge triggered clock pulse generator (Table <ref>). The majority voter representation of the dual edge operation is as follows: Output=MV(Out1, Out2, 1) The job of the control inputs (C3 and C4) is to activate the pulse generator to get the necessary output pulse. The function of the control inputs are as follows: 1. If C3=0 and C4=1 then the pulse generator acts as a rising edge triggered pulse generator.2. If C3=1 and C4=0 then the pulse generator acts as a falling edge triggered pulse generator.3. If C3=1 and C4=1 then the pulse generator acts as a dual edge pulse generator.The simulation waveform of the proposed clock pulse generator is shown in Figure <ref> which establishes the correctness of the proposed clock pulse generator. The proposed edge configurable flip-flop (ECFF) (Figure <ref>) can be implemented with the help of a clock pulse generator (Figure <ref>) using CFF as the basic element. Efforts have been made to increase the functionality of ECFF by trying to incorporate a clock pulse generator whose effectiveness is determined by implementing all three types of pulses generated in a single design. The additional design is incorporated within the proposed structure to devise a configurable edge triggered flip-flop. The manifold advantage being that 9 different types of flip-flop designs usually done separately can be incorporated within one. This means the proposed ECFF can be reconfigured to falling edge D/T/JK FF, rising edge D/T/JK FF and dual edge D/T/JK FF using the same circuit. The simulation results of falling/rising/dual edge triggered JK FF is shown in Figure <ref>. The performance of the proposed ECFF is shown in Table <ref>. The existing dual edge triggered flip-flops are compared with the proposed ECFF which can be configured to falling edge D/T/JK FF, rising edge D/T/JK FF and dual edge D/T/JK FF using the same circuit. This means 9 different functions can be produced using the same ECFF whereas the existing flip-flops can produce only one function.§ REALIZATION OF EDGE CONFIGURABLE N-BIT COUNTER/SHIFT REGISTERIn this section, the edge configurable flip-flop is used as a basic element to realize the proposed counter/ shift register. In order to synchronize the clock of the proposed configurable counter/shift register one additional delay control circuit (DCC) (Figure <ref>) is introduced. The additional delay circuitry is used to delay the clock (fixing C3=0) by one complete cycle so that the clock can be synchronized. The function of the delay circuit is shown in Table <ref>.The proposed configurable 2-bit counter/ shift register is shown in Figure <ref>. It can be configured to a shift register or a counter as necessary. The proposed configurable 2-bit counter/ shift register has 7 control inputs (C1, C1, C2, C2, C3, C4, C5), two primary inputs (A, B) and two primary outputs (Q_0 and Q_1). The basic unit of the proposed counter/ shift register is the edge configurable flip-flop (CFF). A counter can be constructed using T or JK FF whereas shift register can be constructed using D FF. In the proposed 2-bit counter/ shift register, C1 and C2 inputs are used to adjust the CFF module to the required flip-flop whereas C5 input defines the behaviour of the circuit i.e. if C5=0 then the circuit behaves as a shift register otherwise it behaves as a counter. The two ECFF (configured as a D FF; C1=C1=0, C2=C2=0 and C5=0) are cascaded to design the proposed 2-bit shift register. The 3-bit design of the proposed configurable counter/ shift register is shown in Figure <ref>. To synchronize the input and output of the 3-bit configurable counter/ shift register circuit, additional three delay control circuits need to be added to the ECFF module. The top delay control circuit is used to synchronize the output of the first module with rest modules whereas the bottom two delay control circuits are used to synchronize clock signal (CLK) of the proposed circuit. In the case of a 3-bit proposed edge configurable counter/ shift register, if C5=C6=0 then the circuit behaves as a shift register whereas if C5=C6=1 then the circuit will behave as a counter. Similarly, by cascading n-CFF modules, we can construct n-bit edge configurable counter/ shift register as shown in Figure <ref> which can be configured to (dual/rising/falling) edge shift register as well as (dual/rising/falling) edge counter. The QCA representation of the proposed n-bit edge configurable counter/ shift register is shown in Figure <ref>. The performance of the proposed edge configurable counter/shift register is shown in Table <ref>. If we want to construct a shift register then there is no extra space requirement for the proposed configurable counter/ shift register due to its configurable nature hence it is more cost effective than the conventional designs. The main benefit of the configurable n-bit counter/shift register is that the same circuit can be configured to six different forms, (Dual/Falling/Rising) counter as well as (Dual/Falling/Rising) shift register. § SIMULATION SETUPQCADesigner (version 2.0.3) <cit.> has been used to verify the functional correctness of all the proposed QCA designs using both coherence vector and bistable approximation simulation engines using all the default parameters. § CONCLUSIONThis paper attempts for the very first time, to design a level triggered configurable flip-flop (CFF) which can be used as a D, T and JK flip-flop as per the requirement. A 1-bit RAM is designed considering CFF as a basic element. Moreover, an edge configurable flip-flop (ECFF) is designed with the help of a clock pulse generator (CPG). The CPG can produce three different (rising/falling and dual) types of pulses using the same circuit. The edge configurable flip-flop (ECFF) can be used as 9 different flip-flops controlling only single module. An efficient edge configurable n-bit counter/shift register is proposed which can be configured as an n-bit counter or n-bit shift register as per the requirement.§ ACKNOWLEDGEMENTThis research is supported by the Department of Electronics and Information Technology, Ministry of Communications and IT, Government of India under the Visvesvaraya PhD Scheme administered by Media Lab Asia. spmpsci | http://arxiv.org/abs/1708.07616v1 | {
"authors": [
"Mrinal Goswami",
"Mayukh Roy Chowdhury",
"Bibhash Sen"
],
"categories": [
"cs.ET"
],
"primary_category": "cs.ET",
"published": "20170825052521",
"title": "Design of Configurable Sequential Circuits in Quantum-dot Cellular Automata"
} |
𝐱 𝐛̱ 𝐟 Ç 𝒮 𝒰 ℳ ℋ̋e b 1/2 𝐭 𝐧 𝐮̆ 𝐯̌ 𝐈 𝐕𝐊 ℝ𝐀 𝐆 𝐅 S ℤ Pτ̅thmTheoremlemLemmaelsarticle-numcadaddress]Dengyang Zhaocadaddress]Ming Limycorrespondingauthorcadaddress]Yusheng Liu[mycorrespondingauthor]Corresponding author: [email protected][cadaddress]State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou, China The paper presents a topology optimization approach that designs an optimal structure, called a self-supporting structure, which is ready to be fabricated via additive manufacturing without the usage of additional support structures. Such supports in general have to be created during the fabricating process so that the primary object can be manufactured layer by layer without collapse, which is very time-consuming and waste of material.The proposed approach resolves this problem by formulating the self-supporting requirements as a novel explicit quadratic continuous constraint in the topology optimization problem, or specifically, requiring the number of unsupported elements (in terms of the sum of squares of their densities) to be zero. Benefiting form such novel formulations, computing sensitivity of the self-supporting constraint with respect to the design density is straightforward, which otherwise would require lots of research efforts in general topology optimization studies. The derived sensitivity for each element is only linearly dependent on its sole density, which, different from previous layer-based sensitivities, consequently allows for a parallel implementation and possible higher convergence rate. In addition, a discrete convolution operator is also designed to detect the unsupported elements as involved in each step of optimization iteration, and improves the detection process 100 times as compared with simply enumerating these elements. The approach works for cases of general overhang angle, or general domain, and produces an optimized structures, and their associated optimal compliance, very close to that of the reference structure obtained without considering the self-supporting constraint, as demonstrated by extensive 2D and 3D benchmark examples.self-supportingtopology optimization explicit quadratic constraintsadditive manufacturingdiscrete convolution § INTRODUCTIONTopology optimization aims to generate an optimal material distribution within a design domain under certain geometric or physical constraints. Since its introduction in late the 1980s <cit.>, this problem has attracted wide industrial and academic interests due to its large potentiality in engineering applications and its intrinsic mathematical challenges. Topology optimization has developed in many different forms, such as: homogenization <cit.>, density (SIMP) <cit.>, evolutionary approaches (BESO) <cit.>, level set <cit.>, or more recently IGA (iso-geometric analysis) <cit.>, to name a few. See <cit.> for a recent and comprehensive review on this topic.The complex geometric designs produced by topology optimization show the approach's superiority in balancing the geometric distribution and the target physical performance. Such designs are however very difficult to be manufactured directly via traditional subtractive or formative manufacturing techniques <cit.>. On the other hand, rapidly developing additive manufacturing technologies have the promise to overcome the barrier between the potentiality that the topology optimization approaches can provide and the limitations that traditional manufacturing technologies can fabricate. In reality, additive manufacturing is a natural counterpart to topology optimization in that they have very versatile capability to quickly generate and realize new components not existing before <cit.>. Despite the enhanced geometric freedom associated with additive manufacturing, specific design rules must still be satisfied in order to ensure manufacturability. The fabrication overhang angle is an example of a rule which is of paramount importance to ensure that the part will not collapse when fabricating the designed structure layer by layer. A structure satisfying such an overhang angle constraint is called self-supporting. For example, Thomas <cit.> identified 45 degree as the typical maximum overhang angle with a large number of experiments. For a non self-supporting structure, its geometry has to be modified or additional support structures need to be generated. Modifying the geometry will ultimately reduce the structure's physical performance, while additional support raises the issue of automatic and minimum volume support design <cit.>, and further post-processing activities to remove the unwanted supports. In the case that the support is made of the same material as the main component, such as the selective laser melting (SLM) process using metals, it is extremely difficult to remove out the support structure. Particularly, when the generated supports are embedded within a closed volume of the model, it is impossible to remove them.The best strategy to resolve the issue of topology optimization for additive manufacturing is perhaps to design a completely self-supporting structure, via topology optimization, that can be fabricated directly without the usage of support materials. Brackett et al first suggested including the overhang angle constraints into the topology optimization process <cit.>, but does not produce a complete self-supporting structure. The first self-supporting structure built from topology optimization is due to the pioneering work of Gaynor and Guest in 2014 <cit.> (and a very recent journal version <cit.>), which is achieved via introducing a wedge-shaped filter during the topology optimization process. Also very recently, an excellent work was also conducted by Langelaar in 2D <cit.> and in 3D <cit.> via introducing a novel self-supporting filter into the topology optimization process, which is achieved via building smooth approximation to the minimum and maximum functions. Impressive 2D and 3D examples were also shown in these studies <cit.>.In this paper, an alternative novel self-supporting topology optimization approach is proposed to generate a structure of optimal physical performance that does not need any additional support materials. It is achieved via carefully formulating the self-supporting constraint as an explicit quadratic function with respect to the design density, specifically, requiring the number of unsupported elements (in terms of the sum of their densities) to be zero. Benefiting from the novel quadratic formulation, the self-supporting sensitivity for each element is straightforward to compute, and is only linearly dependent on density of the element itself; notice that designing a proper filter and computing the associated sensitivity usually requires lots of research efforts in general topology optimization framework <cit.>. The derived sensitivity does not involve density information of any other elements, and thus allows for a parallel implementation, which is particularly important for 3D problem of high DOFs. Previous approaches <cit.> have a nonlinear layer-based sensitivity expressions, and may thus inhibit parallel implementations, as also explained by the authors <cit.>. In addition, a discrete convolution operator is also designed to detect the unsupported elements as involved in each step of optimization iteration, and improves the detection process 100 times as compared with simply enumerating these elements. Comparisons between the proposed approach and previous studies <cit.> are also summarized in Table 1.The remainder of the paper is organized as follows. The novel formulation of self-supporting topology optimization, together with its overall numerical procedure, is presented in Section <ref>. Several numerical techniques behind the proposed approach are detailed in Section <ref>. Extensive 2D and 3D examples are demonstrated in Section <ref>. pFinally the paper is concluded in Section <ref>.§ PROBLEM STATEMENT AND APPROACH OVERVIEWIn the section, the self-supporting constraint is formulated as a quadratic continuous function in terms of the element density, and is integrated within the classical SIMP framework <cit.> for self-supporting topology optimization. Following on from this, the proposed numerical approach to resolve the problem is outlined. §.§ Supported and unsupported elementsThe supported elements generally stand for the structural elements that can be fabricated via an additive manufacturing technology without collapse with respect to the fabrication process. They are defined here using the concepts of a maximum printable supporting angle, or overhang angle, which is first assumed to be 45 degree following previous study <cit.>. Extensions of the approach to general overhang angles are also explained later. We also assume that the printing direction is following the positive-y axis direction in both 2D and 3D for ease of explanation.First consider a 2D discrete structured mesh modelconsisting of square elements e(n,m), that is,={e(n,m)| 1≤ n≤ N, 1≤ m≤ M },where n,m are the indices increasing along the x and y axes respectively. Without confusion, we also use e to represent a square element without explicitly mentioning its indices n,m. In addition, a density matrixof size N× M is also associated to , where an entry value ρ(n,m)=1 or 0 respectively represents a solid or void element e(n,m) of .As illustrated in Fig. <ref>(a), given a solid element e(n,m) in(in orange), it is supported, or called a supported element, if one of the three blue elements below it is solid. We formulate the self-supporting condition in a continuous form as follows: an element e(n,m)∈ is supported if∑_n-1≤ r≤ n+1ρ(r,m-1)> 0. Correspondingly, the supporting set_S of modelis the set of all supported elements within , that is,_S={e(n,m)∈ | m=1∑_n-1≤ r≤ n+1ρ(r,m-1)> 0}. Similarly, given a 3D structured mesh modelconsisting of cubic elements,={e(n,m,l)| 1≤ n≤ N, 1≤ m≤ M, 1≤ l≤ L},where n,m,l are the indices increasing along the x,y,z axes respectively, the supporting set ofis similarly defined (see also Fig. <ref>(b)):_S= {e(n,m,l)∈ | m=1 ∑_n-1≤ r≤ n+1,l-1≤ s≤ l+1ρ(r,m-1,s)> 0}. Note also here that only five elements are included here as other element do not form an appropriate overhang angle with the orange element.Correspondingly, the set of unsupported elements of a modelis_U=∖_S.§.§ Formulation of self-supporting topology optimizationThe self-supporting topology optimization problem aims to find the optimal material distribution within a design domain under certain boundary conditions. As widely studied before, the problem of minimum compliance or equivalently maximum stiffness is examined here. Following the classical SIMP framework <cit.>, the problem of self-supporting topology optimization is formulated here as an optimization problem with an additional explicit self-supporting constraint. The constraint is carefully reformulated using a simple quadratic function with respect to the density, specifically, requiring the number of unsupported elements (in terms of the sum of square of their densities) is zero. Details are explained below.The problem of self-supporting topology optimization is stated as: find the optimal density distribution ,min_∈^N× Mc (,̆),s.t.{[ ( )=̆( ),;V()/V_0 ≤ f,;U() = ∑_e ∈_Uρ_e^2 ≤ϵ,; 0< ρ_e≤ 1,e=1,…, s, ].whereis the vector of design variables (element densities) to be computed, $̆ is the vector of global displacements andis the global stiffness matrix. The objective functionc(,̆)is the structure's compliance, defined asc(,̆)=^̆T.̆()is the nodal force vector,V()andV_0are the material volume and design domain volume,fis the prescribed volume fraction,_Uis the index set of unsupported elements as defined in (<ref>), andϵ>0is a small parameter closed to0. A penalty parameterp, usually set asp=3, is applied here for the 0,1 convergence of, or specifically,_e=ρ_e^p_e^0,where_e^0is the element stiffness matrix associated with an elementein the modelandρ_ethe associated element density.The only difference of the above conventions in (<ref>) with previous SIMP-based formulations is that it has an additional constraintU() = ∑_e ∈ρ_e^2 ≤ϵto meet the self-supporting requirement. This condition is based on the observation that when the sum of the element densities of unsupported elements tends to0, all the elements are self-supported; the square is used here so that its derivative is not constant. The simple quadratic expression allows for a straightforward sensitivity derivation of the self-supporting constraint, and ultimately results in a linear expression.§.§ Approach overview The self-supporting topology optimization problem (<ref>) is ready-to-solve using the MMA approach noticing that the sensitivity is straightforward to compute (as can be further seen in Section <ref>). On the other hand, in order to further improve the approach's convergence and computational efficiency, the overall optimization process is carefully designed, as plotted in Fig. <ref> and detailed below.Firstly, whether an element is self-supporting is dependent on the printing direction. Different printing directions produce different optimization structures. Thus, if the printing direction is chosen arbitrarily, it may produce a structure totally different from the benchmark support-needed structure obtained without considering the self-supporting constraint, with a possible worse physical performance. In some very special case, the optimization approach may not converge. Thus, an appropriate printing direction is first set via generating a coarse structure via topology optimization without considering the self-supporting constraints. It chooses the coordinate axis direction with the least number of unsupported elements as the print direction.The criteria to generate this initial coarse structure, or to stop the initial optimization process stops, is based on the measure of non-discretenessM_nd, proposed in <cit.>. It stops whenM_nd<M_nd^0,whereM_nd=∑_e=1^N4ρ_e(1-ρ_e)/n× 100%,andnis the number of elements of the domain,ρ_eis the density an elemente. The valve parameterM_nd^0is set0.36, which corresponds to a structure of average density of0.1or0.9.The generated coarse structure is then set as an initial structure for self-supporting topology optimization, together with the chosen printing direction. During the optimization process, in order to prevent the self-supporting constraint hindering the formation of load-carrying structures, the constraint is imposed in a soft way using a similar strategy as previously performed in work <cit.>. Specifically, the value of the tolerance parameterϵin (<ref>) is decreased continuously from a relatively large value to the predefined toleranceϵ_0during the course of optimization.During the process of self-supporting topology optimization, a black-white filter is additionally applied in the last few steps to produce a totally 0-1 density distribution so that the discrete convolution finds exactly the self-supporting elements without the influence of gray elements. This is achieved using the Heaviside projection filter as originally designed by Guest et al <cit.>. Otherwise, gray elements may not be strong enough to support a black element above it, and special care has to be considered <cit.>.Lastly, in order to have a completely self-supporting structure (without any unsupported elements), the self-supporting constraint is added in a strict way in the last few iteration steps (when the number of unsupported elements is no longer decreasing) via removing non-self supporting elements (whose number is at most 5 is all tests given in this paper). This strategy has an ignorable influence on the final structure's compliance, noticing the nature of optimization approach, i.e. the approach may fluctuate between self-supporting constraints and target optimization and the volume faction constraint. Consequently, all these remaining elements are not essential in determining the structure's physical compliance. This is very different from removing unsupported elements from the support-needed structure in a post-processing step. § NUMERICAL ASPECTS§.§ Sensitivity analysisThe sensitivity of the self-support constraintU()is straightforward to compute from (<ref>), and given as:∂ U/∂ρ_e=2ρ_e ife ∈_U 0 ife ∉_U,where_Uis the set of unsupported elements. Note here that the sensitivity for each elementeis only dependent on the densityρ_eof the element itself, without information of density of any other elements, and can be easily implemented in parallel. Previous approaches <cit.> have shown skilled techniques in designing self-supporting filters, and the self-supporting sensitivity of each element was expressed in terms of the densities information of the layers below it. The approach has its great freedom in computing self-supporting structure, but also requires a further improvement to overcome it inhabitation of parallel processing, as also explained by the authors <cit.>.Derivations of the sensitivities of the objective function(̧)or of the volume constraintV(), involved in (<ref>), are totally the same as those done in previous studies <cit.>. Integrating these sensitivities with thickness control can be achieved using the Heaviside filter <cit.>, as will be demonstrated in Section <ref>. Details are not further explained here. §.§ Discrete convolution for efficient unsupported element detectionComputing the sensitivity (<ref>) of the self-supporting constraints requires detecting the set_Uof all the unsupported elements. They can be easily detected via enumerating all the discrete elementseofnot satisfying the property in (<ref>) or (<ref>). Such process is however very time-consuming. In order to further accelerate this process, a novel convolution operator is designed for such detections, which acceleration the process of detecting the unsupported element with a speedup of 100 times as compared with the approach of directly enumerating them element-wise. pGiven a discrete structureof sizeN× Min 2D, we can see from (<ref>) that an elemente(n,m)∈is supported if the summation of the densities of its supporting elements is larger than zero, or specifically,∑_n-1≤ r≤ n+1ρ(r,m-1)> 0. | http://arxiv.org/abs/1708.07364v1 | {
"authors": [
"Dengyang Zhao",
"Ming Li",
"Yusheng Liu"
],
"categories": [
"cs.CE"
],
"primary_category": "cs.CE",
"published": "20170824115515",
"title": "Self-supporting Topology Optimization for Additive Manufacturing"
} |
James' Submodule Theorem and the Steinberg Module James' Submodule Theoremand the Steinberg Module[This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-2018.htmlhttps://www.emis.de/journals/SIGMA/symmetric-groups-2018.html] Meinolf GECK M. Geck IAZ - Lehrstuhl für Algebra, Universität Stuttgart,Pfaffenwaldring 57, D–70569 Stuttgart, Germany mailto:[email protected]@mathematik.uni-stuttgart.de <http://www.mathematik.uni-stuttgart.de/ geckmf/>Received August 29, 2017, in final form November 28, 2017; Published online December 05, 2017James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.groups with a BN-pair; Steinberg representation; modular representations20C33; 20C20 § INTRODUCTION Let G be a finite group of Lie type and _k be the Steinberg representation of G, defined over an arbitrary field k; see <cit.>. We shall be concerned here with the case where _k is reducible. There is only very little general knowledge about the structure of _k in this case; see, e.g., <cit.> and the references there. Using the theory of Gelfand–Graev representations of G, Hiss <cit.> showed that _k always has a certain distinguished composition factor with multiplicity 1. It appears to be extremely difficult to determine further properties of this composition factor, e.g., its dimension. The purpose of this note is to show that this composition factor can be defined in a somewhat more intrinsic way through a version of James' submodule theorem <cit.>; see Remark <ref>.§ GROUPS WITH A SPLIT BN-PAIRLet G be a finite group and B,N⊆ G be subgroups which satisfy the axioms for an “algebraic group with a split BN-pair” in <cit.>. We just recall explicitly those properties of G which will be important for us in the sequel. Firstly, there is a prime number p such that we have a semidirect product decomposition B=U⋊ H where H=B∩ N is an abelian group of order prime to p and U is a normal p-subgroup of B. The group H is normal in N and W=N/H is a finite Coxeter group with a canonically defined generating set S; let l W→_0 be the corresponding length function. For each w∈ W, let n_w∈ N be such that Hn_w=w. Let w_0∈ W be the unique element of maximal length; we have w_0^2=1. Let n_0∈ N be a representative of w_0 and V:=n_0^-1Un_0; then U∩ V=H. For w∈ W, let U_w:=U∩ n_w^-1Vn_w. (Note that V, U_w do not depend on the choices of n_0, n_w.) Then we have the following sharp form of the Bruhat decomposition:G=∐_w∈ W Bn_wU_w,,that is, every g∈ Bn_wB can be uniquely written as g=bn_wu where b∈ B and u∈ U_w. Note that, since w_0^2=1, we have U_w_0=U and Bn_0B=Bn_0U=Un_0B, in both cases with uniqueness of expressions.Let k be any field and kG be the group algebra of G. All our kG-modules will be finite-dimensional and left modules.Let :=∑_b∈ B b∈ kG. Then kG is a left kG-module which is canonically isomorphic to the induced module _B^G(k_B); here, k_B denotes the trivial kB-module. Now kG carries a natural symmetric bilinear form ⟨ , ⟩ kG× kG→ k such that, for any g,g'∈ G, we have⟨ g,g'⟩=1,,0,.This form is non-singular and G-invariant. For any subset X⊆ kG, we denoteX^⊥:={a∈ kG | ⟨ a,x⟩=0 }.If X is a kG-submodule of kG, then so is X^⊥. Let σ U→ k^× be a group homomorphism and set_σ =∑_u∈ Uσ(u)u ∈ kG.Then Γ_σ:= kG_σ is a left kG-module which is isomorphic to the induced module _U^G(k_σ), where k_σ denotes the 1-dimensional kU-module corresponding to σ. Note that _σ^2=|U|_σ. Hence, if |U|1_k≠ 0, then Γ_σ is a projective kG-module. We say that σ is non-degenerate if the restriction of σ to U_s is non-trivial for every s∈ S. Note that this can only occur if |U|1_k≠ 0. In the case where G=_n(q), the following result is contained in James <cit.>; see also Szechtman <cit.>.Assume that |U|1_k≠ 0 and σ is non-degenerate. Then the subspace _σ kG is 1-dimensional and spanned by _σ n_0. Furthermore, _σ n_w=0 for all w∈ W such that w≠ w_0.By the Bruhat decomposition, we can write any g∈ G in the form g=un_wb where u∈ U, w∈ W and b∈ B. Now note that b= for all b∈ B and _σ u=σ(u)^-1_σ for all u∈ U. Thus, _σ kG is spanned by {_σ n_w|w∈ W}. Now let w∈ W be such that w≠ w_0. We shall show that _σ n_w =0. For this purpose, we use the factorisation U=U_wU_w_0w where U_w∩ U_w_0w={1}; see <cit.>. Since w≠ w_0, there exists some s∈ S such that l(ws)>l(w) and so U_s⊆ U_w_0w; see <cit.>. (Note that U_s is denoted by X_i in [loc. cit.].) Since n_w^-1∈ n_w^-1H, we obtain_σ n_w^-1=_σ n_w^-1=(∑_u_1∈ U_wσ(u_1)u_1)n_w^-1(∑_u_2∈ U_w_0wσ(u_2) n_wu_2n_w^-1).By the definition of U_w_0w, we have n_wU_w_0wn_w^-1⊆ U and so n_wu_2n_w^-1= for every fixed u_2∈ U_w_0w. Hence, we obtain∑_u_2∈ U_w_0wσ(u_2) n_wu_2n_w^-1= (∑_u_2∈ U_w_0wσ(u_2)).Finally, since U_s⊆ U_w_0w and the restriction of σ to U_s is non-trivial, the above sum evaluates to 0. Thus, _σ n_w^-1=0 for all w≠ w_0. Since w_0=w_0^-1, this also implies that _σ n_w=0 for all w≠ w_0, as required.Hence, _σ kG is spanned by _σ n_0. Finally, by the sharp form of the Bruhat decomposition, every element of Bn_0B has a unique expression of the form un_0b where u∈ U and b∈ B. In particular, _σ n_0 ≠ 0 and so _σ kG=1. Assume that |U|1_k≠ 0 and σ is non-degenerate. Then the map φΓ_σ→ kG, γ↦γ n_0, is a non-zero kG-module homomorphism and every homomorphism Γ_σ→ kG is a scalar multiple of φ.The fact that φ, as defined above, is a kG-module homomorphism is clear; it is non-zero since φ(_σ)=_σ n_0≠ 0 by Lemma <ref>. Since _σ is a non-zero scalar multiple of an idempotent (see Remark <ref>), we have _kG (Γ_σ, kG) ≅_σ kG and this is 1-dimensional by Lemma <ref>. § THE SUBMODULE THEOREM We keep the notation of the previous section and assume throughout that |U|1_k≠ 0. For any group homomorphism σ U → k^×, we denote by σ^*U→ k^× the group homomorphism given by σ^*(u)= σ(u)^-1 for all u∈ U. Note that, if σ is non-degenerate, then so is σ^*. We can now state the following version of James's “submodule theorem” <cit.>, <cit.>.Let σ U→ k^× be non-degenerate and consider the submodule _σ:=kG_σ n_0 ⊆ kG. Then the following hold. =0pt(i) If M⊆ kG is any submodule, then either _σ⊆ M or M⊆_σ^*^⊥.(ii) We have _σ⊈_σ^*^⊥ and _σ∩_σ^*^⊥⫋_σ is the unique maximal submodule of _σ.(iii) The kG-module D_σ:=_σ/(_σ∩_σ^*^⊥) is absolutely irreducible and isomorphic to the contragredient dual of D_σ^*.(iv) D_σ occurs with multiplicity 1 as a composition factor of kG and of _σ. Having established Lemma <ref> and Corollary <ref>, this readily follows from the general results in <cit.>; the only difference is that James also assumes that _σ= _σ^*. As our notation and setting are somewhat different from those in <cit.>, we recall the most important steps of the argument.(i) Since M⊆ kG, it follows from Lemma <ref> that, for any m∈ M, there exists some c_m∈ k such that _σ m=c_m _σ n_0. If there exists some m∈ M with c_m≠ 0, then _σ n_0= c_m^-1_σ m∈ M and, hence, we have _σ⊆ M in this case. Now assume that c_m=0 for all m∈ M; that is, we have _σ M={0}. Let m∈ M and g∈ G. Using the definition of _σ, _σ^* and the G-invariance of ⟨ , ⟩, we obtain⟨ m,g_σ^*n_0⟩=⟨ g^-1m,_σ^* n_0⟩=⟨_σ(g^-1m),n_0⟩=0,where the last equality holds since _σ M={0}. Thus, M ⊆_σ^*^⊥ in this case.(ii) For u,u'∈ U, we have un_0B=u'n_0B if and only if u=u', by the sharp form of the Bruhat decomposition. Thus, we obtain⟨_σ n_0,_σ^*n_0⟩= ∑_u,u'∈ Uσ(u)σ^*(u')⟨ un_0,u'n_0⟩ =|U|1_k≠ 0,which means that _σ⊈_σ^*^⊥ and so _σ∩_σ^*^⊥⫋_σ. Now let M⫋_σ be any maximal submodule. Then (i) immediately implies that M=_σ∩_σ^*^⊥.(iii) By (ii), we already know that D_σ is irreducible. The remaining assertions then follow exactly as in the proof of <cit.>.(iv) By construction, D_σ occurs at least once in kG and in _σ. Let φΓ_σ→ kG be as in Corollary <ref>. Since Γ_σ is projective and φ(Γ_σ)=_σ, some indecomposable direct summand of Γ_σ is a projective cover of D_σ and so the desired multiplicity of D_σ is at most _kG(Γ_σ, kG)=1, where the last equality holds by Corollary <ref>. We now relate the modules _σ, D_σ in Proposition <ref> to the Steinberg module _k of G, as defined in <cit.>. Recall that_k=kG⊆ kG,:= ∑_w∈ W (-1)^l(w)n_w.We have _k=|U|; a basis of _k is given by {u |u∈ U}.We have _σ=kG_σ⊆_k. Consequently, D_σ is a composition factor (with multiplicity 1) of _k.By Lemma <ref>, we have the identity_σ=∑_w∈ W (-1)^l(w)_σ n_w=(-1)^l(w_0)_σ n_0and so _σ=kG_σ n_0=kG_σ⊆ kG=_k. The statement about D_σ then follows from Proposition <ref>(iv). Gow <cit.> gives an explicit formula for the restriction of the bilinear form ⟨ , ⟩ kG× kG→ k (see Remark <ref>) to _k. For this purpose, he first works overand then rescales to obtain a non-zero form over k. One can also proceed directly, as follows. We have⟨ u_1,u_2⟩=c_W(u_1^-1u_2)1_k ,where c_W(u):=|{w∈ W |n_w^-1un_w∈ U}| for u∈ U. Indeed, since ⟨ , ⟩ is G-invariant, it is enough to show that ⟨,u⟩=c_W(u)1_k for u∈ U. Now, we have⟨,u⟩=∑_w,w'∈ W (-1)^l(w)+l(w')⟨ n_w, un_w'⟩=∑_w∈ W⟨ n_w, un_w⟩,where the second equality holds since n_wB=un_w'B if and only if w=w'. Furthermore, n_wB=un_wB if and only if n_w^-1un_w∈ B. Since n_w^-1un_w is a p-element, the latter condition is equivalent to n_w^-1un_w∈ U. This yields the desired formula. Since |U|1_k≠ 0, the module Γ_σ in Remark <ref> is projective. Also note that _σ=φ(Γ_σ) where φ is defined in Corollary <ref>. Using also Proposition <ref>, we conclude that _kG(Γ_σ,_k)=_kG(Γ_σ,kG )=1. So there is a unique indecomposable direct summand P_ of Γ_σ such that_kG(P_,_k)≠ 0.Being projective indecomposable, P_ has a unique simple quotient whose Brauer character is denoted by σ_G by Hiss <cit.>. By Proposition <ref>(iv) (and its proof), we now see that σ_G is the Brauer character of D_σ. It is known that the socle of _k is simple; see <cit.>. We claim that this simple socle is contained in _σ∩_σ^*^⊥, unless _k is irreducible. Indeed, assume that _σ∩_σ^*^⊥={0}. Then D_σ⊆_k⊆ kG and, hence, D_σ belongs to the Harish-Chandra series defined by the pair (∅, k_H) (notation of Hiss <cit.>). By Remark <ref> and the argument in <cit.>, it then follows that [G:B]1_k≠ 0. So _k is irreducible by <cit.>. § EXAMPLESWe keep the setting of the previous section. We also assume now that G is a true finite group of Lie type, as in <cit.>. Thus, using the notation in [loc. cit.], we have G=^F where is a connected reductive algebraic groupover _p and F→ is an endomorphism such that some power of F is a Frobenius map. Then the ingredients of the BN-pair in G will also be derived from : we have B=^F whereis an F-stable Borel subgroup ofand H=^F whereis an F-stable maximal torus contained in ; furthermore, N=N_ ()^F and U=^F whereis the unipotent radical of . In this setting, one can single out a certain class of non-degenerate group homomorphisms σ U → k^×, as follows.The commutator subgroup [,] is an F-stable closed connected normal subgroup of . We define the subgroup U^*:=[,]^F ⊆ U; then [U,U] ⊆ U^*. Furthermore, we shall fix a group homomorphism σ U→ k^× which is a regular character, that is, we have U^*⊆(σ) and the restriction of σ to U_s is non-trivial for all s∈ S. (Such characters always exist; see <cit.> and <cit.>.) Then the corresponding module Γ_σ= kG_σ is called a Gelfand–Graev module for G. Let h ∈ H and σ^h U → k^× be defined by σ^h(u):=σ(h^-1uh) for u ∈ U. Then σ^h also is a regular character and_σ^h=∑_u ∈ Uσ^h(u)u=h_σ h^-1.Hence, right multiplication by h^-1 defines an isomorphism between the corresponding Gelfand–Graev modules Γ_σ and Γ_σ^h.Let Z(G) be the center of G. Then Z(G) ⊆ H and Z(G)=Z()^F; see <cit.>. Assume now that Z() is connected. Then there are precisely |H/Z(G)| regular characters and they are all conjugate under the action of H; see <cit.>. For any h ∈ H we have_σ^hn_0=h_σ h^-1 n_0=h_σ n_0h^-1= h_σ n_0and so _σ=_σ^h. It follows that _σ= _σ^*=_σ' for all regular characters σ, σ' of U, and we can denote this submodule simply by _0. By Proposition <ref>(iii), the simple module D_0:=_0/(∩_0^⊥) is now self-dual. Furthermore, we have_0 ≥ |H/Z(G)|.Indeed, we have _σ n_0∈_0 for all regular characters σ. Since pairwise distinct group homomorphisms U→ k^× are linearly independent, the elements _σ n_0 (where σ runs over all regular characters of U) form a set of |H/Z(G)| linearly independent elements in _0. Let G=_n(q) where q is a prime power. Then our module _0 is S_λ in James' notation <cit.>, where λ is the partition of n with all parts equal to 1. We claim that _0=_k in this case.Indeed, by <cit.>, _0 is independent of the field k, as long as (k)≠ p. Since _ is irreducible, we conclude that _0=_k and the claim follows. Consequently, by Proposition <ref>(ii), D_0 is the unique simple quotient of _k. (The facts that _0=_k and that this module has a unique simple quotient are also shown by Szechtman <cit.>.) However, D_0 may certainly vary as the field k varies; see the tables in <cit.>. See <cit.> for further examples where _0=_k. On the other hand, Gow <cit.> gives examples (where G=Sp_4(q)) where _k does not have a unique simple quotient, and so _0 ⫋_k. Here is a further example.Let G=Ree(q^2) be the Ree group of type ^2G_2, where q is an odd power of √(3). Then G has a BN-pair of rank 1 and so [G:B]=_k+1. Let k be a field of characteristic 2. Then kG and _k have socle series as follows:kGk_Gφ_2 φ_4 φ_3 φ_5,φ_2 k_G _k φ_2 φ_4 φ_3 φ_5.φ_2 k_GHere, φ_i (i=1,2,3,4,5) are simple kG-modules and φ_4 is the contragredient dual of φ_5; see Landrock–Michler <cit.>. By Proposition <ref>, we have D_0≅φ_3 and _0 is the uniserial submodule with composition factors k_G, φ_2, φ_3. [1]Referencesref 99 =0ptCa2 Carter R.W., Finite groups of Lie type. Conjugacy classes and complexcharacters, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester,1993.DiMi Digne F., Michel J., Representations of finite groups of Lie type,https://doi.org/10.1017/CBO9781139172417London Mathematical Society Student Texts, Vol. 21, CambridgeUniversity Press, Cambridge, 1991.myst Geck M., On the modular composition factors of the Steinberg representation,https://doi.org/10.1016/j.jalgebra.2015.11.005J. Algebra 475 (2017), 370–391.gow Gow R., The Steinberg lattice of a finite Chevalley group and its modularreduction, https://doi.org/10.1112/S0024610703004101J. London Math. Soc. 67 (2003), 593–608.hiss2 Hiss G., Harish-Chandra series of Brauer characters in a finite group witha split BN-pair, https://doi.org/10.1112/jlms/s2-48.2.219J. London Math. Soc. 48 (1993),219–228.Jam1 James G.D., Representations of general linear groups, https://doi.org/10.1017/CBO9780511661921LondonMathematical Society Lecture Note Series, Vol. 94, Cambridge UniversityPress, Cambridge, 1984.LaMi Landrock P., Michler G.O., Principal 2-blocks of the simple groups of Reetype, https://doi.org/10.2307/1999877Trans. Amer. Math. Soc. 260 (1980), 83–111.St57 Steinberg R., Prime power representations of finite linear groups. II,https://doi.org/10.4153/CJM-1957-041-1Canad. J. Math. 9 (1957), 347–351.Sz1 Szechtman F., Steinberg lattice of the general linear group and its modularreduction, https://doi.org/10.1515/JGT.2010.067J. Group Theory 14 (2011), 603–635,https://arxiv.org/abs/0812.2232arXiv:0812.2232.tin1 Tinberg N.B., The Steinberg component of a finite group with a split(B,N)-pair, https://doi.org/10.1016/0021-8693(86)90242-5J. Algebra 104 (1986), 126–134. | http://arxiv.org/abs/1708.07782v2 | {
"authors": [
"Meinolf Geck"
],
"categories": [
"math.RT",
"20C33"
],
"primary_category": "math.RT",
"published": "20170825154036",
"title": "James' Submodule Theorem and the Steinberg Module"
} |
A Stochastic Analysis of Bike Sharing Systems Shuang Tao School of Operations Research and Information Engineering Cornell University 293 Rhodes Hall, Ithaca, NY [email protected] Jamol Pender School of Operations Research and Information Engineering Cornell University 228 Rhodes Hall, Ithaca, NY [email protected] December 30, 2023 ==============================================================================================================================================================================================================================================================================================================================As more people move back into densely populated cities, bike sharing is emerging as an important mode of urban mobility.In a typical bike sharing system, riders arrive at a station and take a bike if it is available. After retrieving a bike, they ride it for a while, then return it to a station near their final destinations.Since space is limited in cities, each station has a finite capacity of docks, which cannot hold more bikes than its capacity.In this paper, we study bike sharing systems with stations having a finite capacity.By an appropriate scaling of our stochastic model, we prove a mean field limit and a central limit theorem for an empirical process of the number of stations with k bikes.The mean field limit and the central limit theorem provides insight on the mean, variance, and sample path dynamics of large scale bike sharing systems.We also leverage our results to estimate confidence intervals for various performance measures such as the proportion of empty stations, the proportion of full stations, and the number of bikes in circulation.These performance measures have the potential to inform the operations and design of future bike sharing systems. § INTRODUCTION Bike sharingis an emerging mode of eco friendly transportation that have launched in over 400 cities around the world (<cit.>).In the United States, we are witnessing a transition where more people are deciding to live in large and densely populated cities.As more people transition from the sprawling suburbs into densely populated cities, bike sharing programs will continue to grow in popularity since they provide easy transportation for citizens of these large cities.As more people use these bike sharing systems(BSS), less people will drive motor vehicles on the road.This reduction of vehicles on the road due to BSS has the potential to also reduce the growing traffic congestion in these growing cities.BSS also promote healthy living as biking is a great form of exercise. They are environmentally friendly and they have the potential to reduce carbon emissions if operated correctly and efficiently, see for example <cit.> and their references within for more information on BSS.For a typical system, riders simply arrive at a station and select a bike if there is one available for them to take.If there is no bike available for the rider, the rider will leave the system.Otherwise, the rider will take a bike and ride it for a while before returning it to another station near their final destination, if there is available space. If no space is available, the rider must find a nearby station to return the bike.If there were an infinite supply of docks to store the bikes, then our model would be reduced to a network of infinite server queues, which is more tractable to analyze.However, since the number of bike docks have finite capacity, the model become less tractable especially for systems with a large number of stations.Much of the complexity inherent in BSS lies in the scarcity of resources to move all riders around each city at all times of the day. Riders can encounter the scarcity of resources in two fundamental ways.First, a rider can encounter insufficient resources by finding an empty station with no bikes when a rider needs one.Secondly, a rider can find a station full with bikes when attempting to return a bike.Thus, from a managerial point of view, having stations with no bikes or too many bikes are both problematic for riders.There are several reasons why bike stations either have no or too many bikes.One main reason why stations might have too many bikes or too few bikes is that the system is highly inhomogeneous.Not only is the arrival rate a non-constant function of the time of day (see Figure <ref>), but also riders do not evenly distribute themselves amongst the available stations.For example, many stations that are located in residential areas have fewer bikes available for riding as many riders take bikes to more commercial areas.Another example that illustrates the inhomogeneous dynamics is that riders tend to take bikes from up-hill stations to go to down-hill stations, however, very few riders take bikes from down-hill stations to go up-hill.Thus, as more bikes flow from residential to commercial areas during rush hours, or up-hill to down-hill stations, this causes the system to be more imbalanced over time. Since BSS are quite complex, researchers have been inspired to study these systems in great depth.The subsequent analysis of BSS has generated many insights on these systems, especially for rebalancing the fleet of bikes.Although there is a large community that studies these systems, few analytical models have been proposed, especially stochastic analytical models.This is primarily because the stochastic models for BSS are often very complex and are rather intractable to analyze without making strong assumptions.Nevertheless, the analysis of such stochastic and mathematical models could provide insights on the behavior of these BSS and how to manage them effectively. In fact, a deeper analysis ofstochastic models for such systems could help researchers understand the impact of different incentive algorithms for taking or returning bikes, which can lead to significant improvements in the overall system performance. Most BSS can be viewed as closed queueing networks. One of the first papers to model the bike sharing system as a closed queueing network is by<cit.>. However, in the bike sharing context, the customers are replaced by bikes. The number of bikes, also known as the fleet size, is fixed and remains constant.In this model, the bikes can go to two types of stations.The first type of station is a single-server queue where the service times are the user inter-arrival times to this station. The second type of station is an infinite-server queue where the service times are the trip times on a route from station i to station j. The main drawback of the model by <cit.> is that it is based on infinite capacity queues.This means that the model does not take into account the finite capacity of the stations and the related strategies of the users to return their bikes.To overcome this major drawback, the model proposed in this paper allow the finite capacity at stations.Thus, we are able to model the real system where customers are blocked from returning bikes to the stations that is nearest to their destination.In our model, we model the bike sharing system as state dependent M/M/1/K_i queueing networks.When joining a saturated single-server queue, the user reattempts in another queue, after a time with the same distribution as the trip time, until he returns his bike.Although this model seems to model the behavior of the network, it is not practical since it scales with the number of stations.Thus, we follow an approach developed by <cit.> to study the bike sharing network's empirical process instead.The empirical process still allows us to derive important performance measures of the original system, however, it scales with the maximum station capacity and not the number of stations, which is more practical for large networks like CitiBike.§.§ Main Contributions of Paper The contributions of this work can be summarized as follows: * We construct a stochastic bike sharing queueing model that incorporates the finite capacity of stations. Since our model is difficult to analyze for a large number of stations, we propose to analyze an empirical process that describes the proportion of stations that have a certain number of bikes. * We prove a mean field limit and a central limit theorem for our stochastic bike sharing empirical process, showing that the mean field limit and the variance of the empirical process can be described by a system of 1/2(K+4)(K+1) differential equations where K is the maximum station capacity.* Using the mean field limit of the empirical process, we are able to approximate the mean proportion of empty and full stations.Furthermore, with the central limit theorem of the empirical process, we are able to construct confidence intervals around the mean field limit for the same performance measures.* We compare the mean field limit and the central limit theorem to a simulation and show that the differential equations approximate the mean and variance of the empirical process extremely well. §.§ Organization of Paper The remainder of this paper is organized as follows. Section <ref> provides a brief history of bike sharing programs and a literature review on the research streams concerning BSS. Section <ref> introduces our bike sharing model and notation of the paper.In Section <ref>, we derive amd prove the mean field limit of the empirical measure process of the distribution of stations with different bikes and utilization rate. In Section <ref>, we derive the diffusion limit and prove a functional CLT for our model. We show that the diffusion process is a centered Gaussian OU process and we also obtain a closed form expression of the diffusion limit process. In Section <ref>, we extend our analysis in Section <ref> to a broader case with non-uniform routing probabilities and capacities, and derive the mean field limit and diffusion limit in this extended case. In Section <ref>, we discuss the simulation results of our model, with both stationary and non-stationary arrival processes. We also give a comparison between real CitiBike empirical measure and simulated ones using our model to show how well our model is in capturing reality.Finally, in Section <ref>, we give concluding remarks and provide some future directions of research that we intend to pursue later.§ HISTORY AND LITERATURE REVIEW The literature that focuses on the analysis and operations of BSS is increasing rapidly. Early research that studied the history of BSS includes <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.These papers provide a history of bike sharing and how it has evolved over time.The beginning of bike sharing can be traced back to the first generation of white bikes (or free bikes) in Amsterdam, The Netherlands as early as 1965. However, this first generation of BSS failed due to a large amount of bike theft.The launch of Bycyklen in Copenhagen in 1995 marked the second generation of BSS.Bycyklen was the first BSS to implement docking stations and coin-deposit systems to unlock bikes.These coin-deposit systems helped with the bike theft problem and thus made the BSS more reliable. However, even with these improvements, the Bycyklen could not eliminate bike thefts mainly due to the fact that customer still remained anonymous and there were no time limits on how long a customer could use a bike.The failures of the second generation of BSS inspired the present-day or the third generation of BSS by combining docks with information technology. These new systems incorporate information technology for bicycle reservations, pickups, and drop-offs.This new technology has enabled many BSS to keep better track of bicycles and the users that use them, thus eliminating virtually all bike theft.One example of a third generation BSS is the Paris’ bike sharing program Velib.Velib was launched in Paris in July 2007 and has emerged as the most prominent example of a successful bike sharing program in the modern world. As a result of the success of Velib,many cities like New York City (Citi Bike), Chicago (Divvy), and even Ithaca (Big Red Bikes) have implemented large-scale BSS and bike sharing has become a widely used form of transportation in these cities.For the interested reader,<cit.> provides a comprehensive survey of the vehicle/bike sharing literature.Rebalancing is currently the biggest stream of research concerning BSS.In rebalancing operations, there are two methods of rebalancing: (1) deploying a truck fleet or (2) providing user incentives, where deploying a truck fleet is often referred to as bike repositioning. Both methods involve static and dynamic cases.Static repositioning usually is moving bikes during the night when traffic flow is low, while dynamic repositioning is moving bikes during the day based on current state of the system. Most research on this area focuses on the static case, partly because it is easier to model and also because the impact of repositioning is more important during the night (<cit.>). <cit.> is one of the first papers to study static repositioning of BSS, using mixed integer linear programming by maximizing customer demand satisfaction. <cit.> consider a similar problem, where a single truck repositions bikes to bring the inventory of each station to a predetermined value.However, their objective is to minimize the routing cost as opposed to maximizing customer satisfaction. In the case of dynamic repositioning, <cit.> and <cit.> consider the case when the trucks respond in real time to the current state of the system.However, <cit.> and <cit.> consider the situation where the time-dependent demand is known a priori and the rebalancing operations are computed in an off-line fashion. Yet, none of these papers really explore stochastic dynamics and they for the most part mainly exploit optimization techniques to tackle the problem.Unlike the rebalancing literature, our paper falls into the performance analysis literature with an emphasis in supply analysis. We focus on analyzing the most salient performance measures such as the mean, variance, covariance and sample path dynamics of the bike distributions in a large-scale BSS.There is not much literature that explores the fluctuations of BSS around the mean field limit.In this paper, we prove a mean field limit and a functional central limit theorem under some smoothness conditions and show that the diffusion limit is characterized by a multi-dimensional Ornstein-Uhlenbeck (OU) process.The functional central limit theorem not only gives us information about the sample path fluctuations of the queue length process, but it also allows us to construct approximate confidence intervals for various performance measures such as the proportion of empty stations, the proportion of full stations, and the mean number of bikes in circulation. Unlike the previous literature, our paper also considers non-stationary arrivals to stations, which is much more realistic given the user patterns we observe in the historical data from Citi Bike. In Figure <ref>, we plot the empirical mean of the number of trips (5 minute intervals) during the week for the time period Jan 1st-Dec 31st, 2015. We observe from Figure <ref> that the arrival rate is non-stationary and clearly reflects the morning rush and evening rush during the peak times. Thus, analyzing the non-stationary dynamics is crucial for understanding the impact of system inhomogeneity since it can provide useful guidelines for rebalancing operations. § BIKE-SHARING QUEUEING MODELIn this section, we construct a Markovian bike sharing queueing model where customers can pick-up and drop-off bikes at each station if there is available capacity.Figure <ref> provides an illustration of a typical Citi Bike station in New York City (NYC).As one can see in Figure <ref>, the bikes are attached to docks and the number of docks is finite with roughly 40 docks.Figure <ref> shows a map of Citi Bike stations, the nation's largest bike sharing program, with over 10,000 bikes and 600 stations across Manhattan, Brooklyn, Queens and Jersey City. Citi Bike was designed for quick, affordable and convenient trips, and has become an essential part of the transportation infrastructure in NYC.Motivated by the Citi Bike bike sharing system, we consider a bikes sharing system with N stations and a fleet of M bikes in total. We assume that the arrival of customers to the stations are independent Poisson processes with rate λ_i for station i.When a customer arrives at a station, if there is no bikes available, they will then leave the system and are immediately blocked and lost. Otherwise, the customer will take a bike and ride to station j with probability P_j.We assume that the travel time of the rider is exponentially distributed with mean 1/μ, for every transition from one station to another.Since we are concerned with finite capacity stations, we assume that station i has a bike capacity of K_i, which is assumed to be finite for all stations. Thus, when a customer arrives at station j, if there are less than K_j bikes in this station, he returns his bike and leaves the system. If there are exactly K_j bikes (i.e. the station is full), the customer randomly chooses another station k with probability P_k and goes to that station to drop the bike off.As before, it takes a time that is exponentially distributed with mean 1/μ. Finally, the customer rides like this again until he can return his bike to a station that is not full. Below, we provide Table <ref> for the reader's convenience so that they understand the notation that we will use throughout the paper. To avoid cumbersome notation, throughout Sections <ref>, <ref> and <ref>, we assume without loss of generality that the service rate μ is equal to 1 and that the routing probability from each station is uniform, i.e. P_i=1/N. We also assume that capacities across all stations are equal, i.e. K_i=K, fori=1,...,N. With our notation in hand, we are ready to develop our stochastic model for our bike sharing network.At first glance, these assumptions seem restrictive, however, we explain in Section <ref>how our model can be extended seamlessly to more complex settings with non-uniform routing probabilities and capacities.We should mention, however, the extension to inter-station transition probabilities i.e probabilities that depends on the departing station as well as the returning station are non-Markovian models.In Section <ref> we discuss more in detail how one can extend the state space, by tracking the number of in-transit bikes coming from each station, to make the queueing model Markovian, however this extended model makes the problem even more high-dimensional and adds difficulty to the analysis. We define 𝐗(t)=(X_1(t),⋯, X_N(t)), where X_i(t) is the number of bikes at station i at time t. Then X_i(t) is a continuous time Markov chain(CTMC), in particular a state dependent M/M/1/K_i queue. In this model, the rate of dropping off bikes at station i is equal to μ P_i(M-∑_k=1^NX_k(t))1{X_i(t)<K_i}, and the rate of retrieving bikes at station i is equal to λ_i1{X_i(t)>0}.Using these rates, we can construct the functional forward equations for our stochastic bike sharing model.This construction is given below in the following proposition: For any integrablefunction f: ℤ_+^N→ℝ, the CTMC 𝐗(t) satisfy the following functional forward equation, 𝔼[f(𝐗(t)) | 𝐗(0) = 𝐱]≡d/dt𝔼[f(𝐗(t)) | 𝐗(0) = 𝐱]= ∑_i=1^N𝔼[(f(𝐗(t)-1_i)-f(𝐗(t))λ_i1_{X_i(t)>0}]+∑_i=1^N𝔼[(f(𝐗(t)+1_i)-f(𝐗(t))μ P_i(M-∑_k=1^NX_k(t))1_{X_i(t)<K_i}] .The proof is found in the Appendix. The time derivatives of the mean, variance, and covariance of 𝐗(t) are given by𝔼[X_i(t)]= 𝔼[ μ P_i(M-∑_k=1^NX_k(t))1{X_i(t)<K_i}]- λ_iℙ[ X_i(t) > 0 ] , Var[X_i(t)] = 𝔼[X_i^2(t)]-2𝔼[X_i(t)]𝔼[X_i(t)]=2 Cov[X_i(t),μ P_i(M-∑_k=1^NX_k(t))1{X_i(t)<K_i}]-2 Cov[X_i(t),λ_i1{ X_i(t) > 0 }] + 𝔼[ μ P_i(M-∑_k=1^NX_k(t))1{X_i(t)<K_i}] +λ_iℙ[ X_i(t) > 0 ],Cov[X_i(t), X_j(t)] = Cov[μ P_j(M-∑_k=1^NX_k(t))1_{X_j(t)<K_j}-λ_j1_{X_j(t)>0},X_i(t) ]+Cov[μ P_i(M-∑_k=1^NX_k(t))1_{X_i(t)<K_i}-λ_i1_{X_i(t)>0},X_j(t) ],for i,j=1,⋯,N and i≠ j. §.§ Intractability of Individual Stations ModelAlthough the functional forward equations given in Corollary <ref> describe the exact dynamics of the mean, variance and covariance of the bike sharing system, the system of differential equations are not closed.This non-closure property of the functional forward equations in this model arises from the fact that the bike sharing system has finite capacity.More importantly, it also implies that we need to know a priori the full distribution of the whole stochastic process 𝐗(t) in order to calculate the mean or variance or any moment for that matter. Work by <cit.> could yield useful and accurate closure approximations for making the system closed.Moreover, withe exception of <cit.>, thereare no error bounds on the accuracy of various closure approximations.Thus, it is not clear how well the closure approximations would perform over a variety of parameter settings.Finally if we even wanted to solve these equations and knew the entire distribution of 𝐗(t) a priori, there are still O(N^2) differential equations(around 180,900 equations in the CitiBike case) that would need to be numerically integrated.This is very computationally expensive and thus, we must take a different approach to analyze our bike sharing system.Moreover, if we want to analyze the limiting behavior of {X_i}_i=1^N asCTMCs, as we let N go to ∞, the mean field limit would become infinite dimensional, which is quite complicated.However, if we instead analyze an empirical measure process for 𝐗(t), we can use the finite capacity nature of the bike sharing system to our advantage and have a finite dimensional CTMC for the empircal measure process.§.§ An Empirical Measure ModelFollowing the model of <cit.>, we construct an empirical measure process that counts the proportions of stations with n bikes and utilization r.This empirical measure process is given below by the following equation: Y_t^N(r,n)=1/N∑_i=1^N1{r_i^N=r,X_i^N(t)=n}. We further define that Y_t^N(n)=∑_rY_t^N(r,n). By observing the empirical measure process, we notice that Y_t^N=(Y_t^N(0),⋯,Y_t^N(K)) ∈ [0,1]^K+1.Thus, for our empirical measure process, we only need to solve O(K^2) differential equations for understanding the mean and variance dynamics of the bike sharing system, where K<<N. More importantly, the empirical measure will also allow us to obtain salient performance measures such as Y_t^N(0) (the proportion of stations with no bikes), Y_t^N(K) (the proportion of stations that are full of bikes), M - ∑^K_j=0 j · Y_t^N(j)N (the number of bikes in circulation), among others.Conditioning on Y_t^N(r,n)=y(r,n) and given our assumptions that μ=1 and P_i=1/N, the transition rates of y are specified as follows:When a customer arrives to a station with n bikes and relative utilization r to retrieve a bike, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n-1 bikes goes up by 1/N, and the transition rate Q^N is Q^N(y,y+1/N(1_(r,n-1)-1_(r,n)) )= y(r,n)λ_r N 1_n>0=y(r,n)μ P_i/RN 1_n>0=y(r,n)1/NR N1_n>0= y(r,n)/rR^N_max1_n>0.When a customer returns a bike to a station with n bikes and relative utilization r, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n+1 bikes goes up by 1/N, and the transition rate Q^N is Q^N(y,y+1/N(1_(r,n+1)-1_(r,n)) )= y(r,n) ·μ·(M-∑_n'∑_r'n'y(r',n')N)1_n<K= y(r,n)N(M/N-∑_n'∑_r'n'y(r',n'))1_n<K.Similarly, we have the functional forward equations for Y_t^N(r,n).For any integrable function f: [0,1]^K+1→ℝ, Y_t^N(r)=(Y_t^N(r,0),⋯,Y_t^N(r,K)) satisfies the following functional forward equation, 𝔼(f(Y_t^N(r)) | Y_0^N(r) = y_0(r)]= ∑_n=0^K𝔼[(f(Y_t^N(r)+1/N(1_r,n-1-1_r,n))-f(Y_t^N(r)))Y^N_t(r,n)/rR_max1_n>0]+ ∑_n=0^K𝔼[(f(Y_t^N(r)+1/N(1_r,n+1-1_r,n))-f(Y_t^N(r)))Y^N_t(r,n)N(M/N..- ..∑_n'∑_r'n'Y_t^N(r',n'))1_n<K] The proof is similar to the proof of Proposition <ref>.The time derivative of the mean of Y_t^N(r,n) is given by𝔼[Y_t^N(r,n)] = 𝔼[ 1/rNR^N_max(Y^N_t(r,n+1)1_n<K-Y^N_t(r,n)1_n>0) ] + 𝔼[ (M/N-∑_n'∑_r'n'Y_t^N(r',n'))(Y^N_t(r,n-1)1_n>0-Y^N_t(r,n)1_n<K) ]. for n=0,⋯,K. Denote Σ_i,j^N(r,t)=Cov[Y_t^N(r,i),Y_t^N(r,j)]. When i=j,the time derivative of the variance term Var[Y_t^N(r,i)] is given byVar[Y_t^N(r,i)] = 𝔼[Y_t^N(r,i)^2]-2𝔼[Y_t^N(r,i)]𝔼[Y_t^N(r,i)]= 2/rNR^N_max(Σ_i,i+1^N(r,t)1_i<K-Σ_i,i^N(r,t)1_i>0)+2M/N(Σ^N_i,i-1(r,t)1_i>0 -Σ^N_i,i(r,t)1_i<K) - 2 ∑_n'∑_r'n'(Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,i-1)]1_i>0. .-Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,i) ]1_i<K) + 1/rN^2R^N_max(𝔼[Y^N_t(r,i+1) ]1_i<K+𝔼[Y^N_t(r,i)]1_i>0) +M/N^2(𝔼[Y^N_t(r,i-1)]1_i>0+𝔼[Y^N_t(r,i)]1_i<K)-1/N∑_n'∑_r'n'(𝔼[Y_t^N(r',n')Y^N_t(r,i-1)]1_i>0+𝔼[Y_t^N(r',n')Y^N_t(r,i) ]1_i<K).When |i-j|>1, the time derivative of the covariance term Cov[Y_t^N(r,i), Y_t^N(r,j)] is given by Cov[Y_t^N(r,i), Y_t^N(r,j)]= 𝔼[Y_t^N(r,i)Y_t^N(r,j)]-𝔼[Y_t^N(r,i)]𝔼[Y_t^N(r,j)] -𝔼[Y_t^N(r,j)]𝔼[Y_t^N(r,i)]= 1/rNR^N_max[Σ_i+1,j^N(r,t)1_i<K+Σ_i,j+1^N(r,t)1_j<K-Σ_i,j^N(r,t)(1_j>0+1_i>0)] +M/N[Σ^N_i-1,j(r,t)1_i>0+Σ^N_i,j-1(r,t)1_j>0 -Σ^N_i,j(r,t)(1_i<K+1_j<K)] -∑_n'∑_r'n'(Cov[Y_t^N(r,j),Y_t^N(r',n')Y^N_t(r,i-1)]1_i>0-Cov[Y_t^N(r,j),Y_t^N(r',n')Y^N_t(r,i) ]1_i<K) -∑_n'∑_r'n'(Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,j-1)]1_j>0-Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,j) ]1_j<K).and when j=i+1, the time derivative of the covariance term Cov[Y_t^N(r,i), Y_t^N(r,i+1)] is given byCov[Y_t^N(r,i), Y_t^N(r,i+1)]= 𝔼[Y_t^N(r,i)Y_t^N(r,i+1)]-𝔼[Y_t^N(r,i)]𝔼[Y_t^N(r,i+1)] -𝔼[Y_t^N(r,i+1)]𝔼[Y_t^N(r,i)]= 1/rNR^N_max[Σ_i+1,i+1^N(r,t)+Σ_i,i+2^N(r,t)1_i<K-1-Σ_i,i+1^N(r,t)(1+1_i>0)] +M/N[Σ^N_i-1,i+1(r,t)1_i>0+Σ^N_i,i(r,t) -Σ^N_i,i+1(r,t)(1+1_i<K-1)] -∑_n'∑_r'n'(Cov[Y_t^N(r,i+1),Y_t^N(r',n')Y^N_t(r,i-1)]1_i>0. .-Cov[Y_t^N(r,i+1),Y_t^N(r',n')Y^N_t(r,i) ]) -∑_n'∑_r'n'(Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,i)]-Cov[Y_t^N(r,i),Y_t^N(r',n')Y^N_t(r,i+1) ]1_i<K-1) -𝔼[Y_t^N(r,i+1)]/rN^2R_max-M/N^2𝔼[Y_t^N(r,i)]+1/N∑_n'∑_r'n'𝔼[Y_t^N(r',n')Y_t^N(r,i)] . The time derivatives of 𝔼[Y_t^N(r,i)] and Var[Y_t^N(r,i)] come directly from applying Proposition <ref> with f(Y_t(r))=Y_t(r,i), Y_t^2(r,i) respectively. For the covariance term, we let f(Y_t^N(r))=f(Y^N_t(r,i),Y^N_t(r,j)) to prove the result. Although we have equations for the moments of the empirical process and individual stations, it is still difficult to analyze them and gain insights from them directly.One reason is that even though we have reduced the dimensionality of the analysis significantly, we have not removed the non-closure property of the differential equations.Thus, we need to develop a new approach that will allow us to get around this complication.The method that we choose to use is asymptotic analysis and will be described in more details in the sequel.However, before we get to the asymptotic analysis we state some technicalities about weak convergence.§.§ Preliminaries of Weak Convergence Following <cit.>, we assume that all random variables in this paper are defined on a common probability space (Ω, ℱ, ℙ).Moreover, for all positive integers k, we let 𝒟([0 , ∞), ℝ^k) be the space of right continuous functions with left limits (RCLL) in ℝ^k that have a time domain in [0, ∞).As is usual, we endow the space 𝒟([0 , ∞), ℝ^k) with the usual Skorokhod J_1 topology, and let M^k be defined as the Borel σ-algebra associated with the J_1 topology. We also assume that all stochastic processes are measurable functions from our common probability space (Ω, ℱ, ℙ) into (𝒟([0 , ∞), ℝ^k), M^k). Thus, if {ζ}^∞_n=1 is a sequence of stochastic processes, then the notation ζ^n →ζ implies that the probability measures that are induced by the ζ^n's on the space (𝒟([0 , ∞), ℝ^k), M^k) converge weakly to the probability measure on the space (𝒟([0 , ∞), ℝ^k), M^k) induced by ζ.For any x ∈ (𝒟([0 , ∞), ℝ^k), M^k)and any T > 0, we define ||x||_T ≡sup_0 ≤ t ≤ T max_i = 1,2,...,k |x_i(t)|and note that ζ^n converges almost surely to a continuous limit process ζ in the J_1 topology if and only if ||ζ^n - ζ||_T → 0a.s.for every T > 0.§ MEAN FIELD LIMITIn this section, we prove the mean field limit for our bike sharing model.A mean field limit describes the large station dynamics of the bike sharing network over time.Deriving the mean field limit allows us to obtain new insights on average system dynamics, when the demand for bikes and the number of stations are very large. Thus, we avoid the need to study an N-dimensional CTMC and compute its steady state distribution in this high dimensional setting. First, we state the important assumptions that will be used through out the paper, to ensure the existence of a mean field limit of our model.There exists a probability measure I(r,k)on ]0,1] ×ℕ with finite support and Λ > 0 such that, as N tends to infinity, we have i) 1/N∑_i=1^N1 _(r_i^N,K_i^N)⇒ I(r,k),ii)NR_max^N→Λ^-1,iii)M/N→γ.Now we state the main theorem in this section that proves the convergence of empirical process to its mean field limit.Let |· | denote the Euclidean norm in ℝ^K+1. Under Assumption <ref>, suppose that Y_0^N y_0,then we have for any ϵ>0 and t_0>0,lim_N→∞P(sup_t≤ t_0|Y_t^N-y_t|>ϵ)=0.Here y_t=(y_t(0),⋯,y_t(K)), where y_t(k)=∫_0^1dy_t(r,k) for k=0,⋯,K. And y_t is the unique solution to the following differential equation starting at y_0y_t=b(y_t) where b:[0,1]^K+1→ℝ^K+1 is a vector field satisfiesb(y_t) = ∬_]0,1]× [0,...,K][ Λ/r(1_(r,n-1)-1_(r,n))1_n>0+(γ-∑_n∫_0^1 ndy_t(r,n) )(1_(r,n+1)-1_(r,n))1_n<K] dy_t(r,n),or componentwiseb(y_t)(0)=-∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) )dy_t(r,0)_return a bike to a no-bike station+∫_0^1Λ/rdy_t(r,1)_retrieve a bike from a 1-bike station, b(y_t)(k)= ∫_0^1Λ/rdy_t(r,k+1)_retrieve a bike from a k+1-bike station-∫_0^1(Λ/r+γ-∑_n∫_0^1 ndy_t(r,n) )dy_t(r,k)_retrieve and return a bike to a k-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) )dy_t(r,k-1)_return a bike to a k-1-bike station, for k=1,...,K-1, and b(y_t)(K)=-∫_0^1Λ/r dy_t(r,K)_retrieve a bike from a K-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) )dy_t(r,K-1)_return a bike to a K-1-bike station. A similar theorem is given in the paper of <cit.>, however, a proof is not given in their work.Thus, to make our paper self contained, we provide a full proof of the mean field limit for the convenience of the reader as it is essential for our future results.Our proof exploits Doob's inequality for martingales and Gronwall's lemma. Moreover, we use Proposition <ref>, Proposition <ref>, and Proposition <ref> in the proof, and they are stated after the proof of Theorem <ref>. Since Y_t^N is a semi-martingale, we have the following decomposition of Y_t^N ,Y_t^N=Y_0^N_initial condition+M_t^N_martingale+∫_0^tβ(Y_s^N)_drift termdswhere Y_0^N is the initial condition and M_t^N is a family of martingales.Moreover, ∫_0^tβ(Y_s^N)ds is the integral of the drift term where the drift term is given by β: [0,1]^K+1→ℝ^K+1 orβ(y) =∑_x≠ y(x-y)Q(y,x)= ∑_n,r[1/rNR_max(1_(r,n-1)-1_(r,n))1_n>0+(M/N-∑_n'∑_r'n'y(r',n'))(1_(r,n+1)-1_(r,n))1_n<K]y(r,n).We want to compare the empirical measure Y_t^N with the mean field limit y_t defined byy_t=y_0+∫_0^tb(y_s)ds. The remaining of the proof of this theorem can be found in the Appendix. For any stopping time T such that 𝔼(T)<∞, we have𝔼(sup_t≤ T|M_t^N|^2)≤ 4𝔼∫_0^Tα(Y_t^N)dt. The proof is found in the Appendix. The drift function b(y) given in Equation (<ref>) is a Lipschitz function with respect to the Euclidean norm in ℝ^K+1. The proof is found in the Appendix. Under Assumption <ref>, we have for any ϵ>0 and s≥ 0,lim_N→∞ℙ(|β(Y_s^N)-b(Y_s^N)|>ϵ)= 0. The proof is found in the Appendix.We have proved mean field limit for our bike sharing model.Our analysis yields that as the number of stations goes towards infinity, we can solve a set of ordinary differential equations to obtain important performance measure information. The performance measures that we can approximate are the mean proportion of empty or saturated stations, and the average number of bikes in circulation.Moreover, we can analyze how factors such as fleet size and capacity change the value of the performance measures.However, just knowing the mean field limit is not enough.One reason is that we would like to know more about the stochastic variability of the system, i.e. the fluctuations around the mean field limit.The mean field limit cannot explain the stochastic fluctuations of the BSS and therefore, we need to analyze the BSS in a different way. Thus, in the subsequent section we develop a functional central limit theorem for our bike sharing model, and explain why it is important for understanding stochastic fluctuations of bike sharing networks. § DIFFUSION LIMITIn this section, we derive the diffusion limit of our stochastic empirical process bike sharing model.Diffusion limits are critical for obtaining a deep understanding of the sample path behavior of stochastic processes.One reason is that diffusion limits describe the fluctuations around the mean field limit and can help understand the variance or the asymptotic distribution of the stochastic process being analyzed.We define our diffusion scaled bike sharing model by subtracting the mean field limit from the empirical measure process and rescaling it by √(N).Thus, we obtain the following expression for the diffusion scaled bike sharing empirical processD_t^N=√(N)(Y_t^N-y_t) . Unlike many other ride-sharing systems such as Lyft or Uber, bike sharing programs cannot use pricing as a mechanism for redistributing bikes to satisfy demand in real-time.For this reason, it is essential to understand the dynamics and behavior of D_t^N.D_t^N can be useful for describing the probability that the proportion of stations with i bikes exceeds a threshold i.e. ℙ(Y^N_t > x) for some x ∈ [0,1]^K+1.It also describes this probability in a situation where there is no control or rebalancing of bikes in the system.This knowledge of the uncontrolled system is especially important for newly-started bike sharing systems who are still in the process of gathering information about the system demand. The diffusion limit helps managers of BSS to understand the system dynamics and stability, which in turn helps them make short term and long term managerial decisions. It is also helpful in the case when the operators of the bike sharing system have no money for rebalancing the system to meetreal-time demand. Using the semi-martingale decomposition of Y^N_t given in Equation (<ref>), we can write a similar decomposition for D_t^N as follows:D_t^N =√(N)(Y_0^N-y_0)+√(N)M_t^N+∫_0^t√(N)[β(Y_s^N)-b(y_s)]ds=D_0^N+√(N)M_t^N+∫_0^t√(N)[β(Y_s^N)-b(Y_s^N)]ds+∫_0^t√(N)[b(Y_s^N)-b(y_s)]ds. Define D_t=D_0+∫_0^tb'(y_s)D_s ds+M_twhere b'(y)=(∂ b(y)(i)/∂ y(j))_ij∈ℝ^(K+1)× (K+1) and M_t=(M_t(0),⋯,M_t(K))∈ℝ^K+1 is a real continuous centered Gaussian martingale, with Doob-Meyer brackets given byM(k) _t = ∫_0^t(b_+(y_s)(k)+b_-(y_s)(k))ds,M(k),M(k+1) _t = -∫_0^t[∫_0^1Λ/rdy_s(r,k+1).+ .∫_0^1(γ-∑_n∫_0^1ndy_s(r,n)) dy_s(r,k)]ds fork<K,M(k),M(j) _t = 0for|k-j|>1.Here b_+(y)=max(b(y),0) and b_-(y)=-min(b(y),0) denote the positive and the negative parts of function b(y) respectively.Now we state the functional central limit theorem for the empirical measure process as follows,Consider D_t^N in 𝔻(ℝ_+,ℝ^K+1) with the Skorokhod J_1 topology, and suppose that1) lim sup_N→∞√(N)(min_iλ_i^N-Λ)<∞,2) lim sup_N→∞√(N)(M/N-γ)< ∞.Then if D_0^N converges in distribution to D_0, then D_t^N converges to the unique OU process solving D_t=D_0+∫_0^tb'(y_s)D_s ds+M_t in distribution.To prove Theorem <ref>, we take the following 4 steps,1).√(N)M_t^N is a family of martingales independent of D_0^N with Doob-Meyer brackets given by√(N)M^N(k)_t = ∫_0^t(β_+(Y_s^N)(k)+β_-(Y_s^N)(k))ds, √(N)M^N(k),√(N)M^N(k+1)_t = -∫_0^t∑_r[1/rNR_maxY_s^N(r,k+1). .+(M/N-∑_n'∑_r'n'Y_s^N(r',n'))Y_s^N(r,k)] ds fork<K, √(N)M^N(k),√(N)M^N(j)_t = 0for|k-j|>1.2).For any T≥ 0,lim sup_N→∞𝔼(|D_0^N|^2)<∞⇒lim sup_N→∞𝔼(sup_0≤ t≤ T|D_t^N|^2)<∞. 3). If (D_0^N)_N=1^∞ is tight then (D^N)_N=1^∞ is tight and its limit points are continuous.4). If D_0^N converges to D_0 in distribution, then D_t^N converges to the unique OU process solving D_t=D_0+∫_0^tb'(y_s)D_s ds+M_t in distribution. √(N)M_t^N is a family of martingales independent of D_0^N with Doob-Meyer brackets given by√(N)M^N(k)_t = ∫_0^t(β_+(Y_s^N)(k)+β_-(Y_s^N)(k))ds, √(N)M^N(k),√(N)M^N(k+1)_t = -∫_0^t∑_r[1/rNR_maxY_s^N(r,k+1). .+(M/N-∑_n'∑_r'n'Y_s^N(r',n'))Y_s^N(r,k)] ds fork<K, √(N)M^N(k),√(N)M^N(j)_t = 0for|k-j|>1.The proof is found in the Appendix. For any s≥ 0,lim sup_N→∞√(N)|β(Y_s^N)-b(Y_s^N)|<∞. The proof is found in the Appendix.For any T≥ 0, iflim sup_N→∞𝔼(|D_0^N|^2 ) < ∞ ,then we havelim sup_N→∞𝔼(sup_0≤ t≤ T|D_t^N|^2 ) < ∞ .The proof is found in the Appendix. If (D_0^N)_N=1^∞ is tight then (D^N)_N=1^∞ is tight and its limit points are continuous. The proof is found in the Appendix.b(y) is continuously differentiable with the derivatives∂ b(y(r,i))/∂ y(r,j) as follows,∂ b(y(r,0))/∂ y(r,j) = j· y(r,0)+Λ/r1_{j=1}-(γ-∑_n=0^Kn(∑_r'y(r',n)))1_{j=0}, ∂ b(y(r,k))/∂ y(r,j) = j·( y(k)-y(k-1))+Λ/r(1_{j=k+1}-1_{j=k}) +(γ-∑_n=0^Kn(∑_r'y(r',n)))(1_{j=k-1}-1_{j=k})for0<k<K, ∂ b(y(r,K))/∂ y(r,j) = -j· y(K-1)-Λ/r1_{j=K}+(γ-∑_n=0^Kn(∑_r'y(r',n)))1_{j=K-1}.The above equations can be obtained by directly taking derivatives to Equation (<ref>). We can see that the derivatives of b(y) are linear in y. Thus we can conclude that b(y) is continuously differentiable with respect to y.By Theorem 4.1 in Chapter 7 of <cit.>,it suffices to prove that the following condition holdssup_t≤ T|∫_0^t{√(N)[b(Y_s^N)-b(y_s)]-b'(y_s)D_s^N}ds| 0.By Proposition <ref>, we know that b(y_t) is continuously differentiable with respect to y_t. By the mean value theorem, for every 0≤ s≤ t there exists a vector Z_s^N in between Y_s^N and y_s such thatb(Y_s^N)-b(y_s)=b'(Z_s^N)(Y_s^N-y_s).Therefore∫_0^t{√(N)[b(Y_s^N)-b(y_s)]-b'(y_s)D_s^N}ds=∫_0^t[b'(Z_s^N)-b'(y_s)]D_s^Nds.We know thatlim_N→∞sup_t≤ T|b'(Z_s^N)-b'(y_s)|=0in probabilityby the mean field limit convergence and the uniform continuity of b'. By applying Chebyshev inequality we have that D_s^N is bounded in probability, then by Lemma 5.6 in <cit.>,sup_t≤ T|∫_0^t{√(N)[b(Y_s^N)-b(y_s)]-b'(y_s)D_s^N}ds| 0. The SDE (<ref>) has a unique solution D_t=e^∫_0^tb'(y_s)dsD_0+∫_0^te^∫_s^tb'(y_u)dudM_s. Define 𝒜(t)=b'(y_t), ℬ(t)=(d/dt M(i),M(j)_t)_ij, then the expectation E(D_t) is𝔼[D_t]=e^∫_0^t𝒜(s)ds𝔼[D_0], and the covariance matrix Σ (t)=Cov[D_t,D_t] isΣ (t)=e^∫_0^t𝒜(s)dsΣ(0)e^∫_0^t𝒜^⊤(s)ds+∫_0^te^∫_s^t𝒜(u)duℬ(s)e^∫_s^t𝒜^⊤(u)duds.Moreover, differentiation with respect to t yieldsd𝔼[D_t]/dt=𝒜(t)𝔼[D_t],dΣ(t)/dt=Σ(t)𝒜(t)^⊤+𝒜(t)Σ(t)+ℬ(t). The proof is found in the Appendix. § EXTENSIONS In Section <ref>, we assumed WLOG that the routing probabilities and capacities are uniform throughout all stations. If one views our statement as assumptions, then it seems like our model is limiting, however, we emphasize that these are not assumptions and that our analysis extends to the broader case of non-uniform routing probabilities and capacities.Thus, in the non-uniform setting, we are still able to prove the same fluid and diffusion limits and more importantly reduce the dimensionality of the bike sharing network. §.§ Extensions to non-uniform routing probabilitiesand capacitiesThe first possible extension of our current model is to incorporate non-uniform routing probabilities, and take into account the origin-destination pairs. That is, after a customer picks up a bike from station i, the probability that he will drop off at station j is equal to P_ij. However, to preserve the Markovian property of the queueing process, we also need to track the number of bikes in use which originated from station i, denoted as U^N_t(i), in addition to the empirical measure process Y^N_t. As the scale of the system N goes to infinity, the dimensionality of the new queueing process (Y^N_t, U^N_t) will also go to infinity, therefore we are back to the same infinite-dimensionality problem with X^N(t), which is illustrated more in details in Section <ref>, and lose the benefit of finite-dimensionality we get by studying the empirical measure Y^N_t.However, we can still extend our model to non-uniform routing probabilities and capacities without considering the origin-destination pairs. In particular, we assume the probability that a bike being dropped off at station i is equals to P_i, and the capacity at station i is equal to K_i. We define relative routing probability at station i as p_i=P_i/max_iP_i. We consider the empirical measure process Y^N_t(r,n,p,k) that counts the proportion of stations that have n bikes, relative utilization r, routing probability p, and capacity k,Y^N_t(r,n,p,k)=1/N∑_i^N1{X_i^N(t)=n, r_i^N=r, p_i^N=p,K_i^N=k}. Conditioning on Y_t^N(r,n,p,k)=y(r,n,p,k), the transition rates of y are specified as follows:When a customer arrives to a station with n bikes,relative utilization r, relative routing probability p, and capacity k, to retrieve a bike, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n-1 bikes goes up by 1/N, and the transition rate Q^N is Q^N(y,y+1/N(1_(r,n-1,p,k)-1_(r,n,p,k)) )= y(r,n,p,k)λ_r N 1_n>0=y(r,n,p,k)μ P/RN 1_n>0=y(r,n,p,k)pP^N_max/rR^N_maxN 1_n>0.When a customer returns a bike to a station with n bikes ,relative utilization r, relative routing probability p, and capacity k, to retrieve a bike,, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n+1 bikes goes up by 1/N, and the transition rate Q^N is Q^N(y,y+1/N(1_(r,n+1,p,k)-1_(r,n,p,k)) )= y(r,n,p,k) N·μ· P(M-∑_n'∑_r',p',k'n'y(r',n',p',k')N)1_n<k= y(r,n,p,k)N^2pP^N_max(M/N-∑_n'∑_r',p',k'n'y(r',n',p',k'))1_n<k. We have the following functional forward equations for Y_t^N(r,n,p,k).For any integrable function f: [0,1]^k+1→ℝ, andY_t^N(r,p,k)=(Y_t^N(r,0,p,k),⋯,Y_t^N(r,k,p,k)) satisfies the following functional forward equation, 𝔼(f(Y_t^N(r,p,k)) | Y_0^N(r,p,k) = y_0(r,p,k)]= ∑_n=0^k𝔼[(f(Y_t^N(r,p,k)+1/N(1_r,n-1,p,k-1_r,n,p,k))-f(Y_t^N(r,p,k)))Y^N_t(r,n,p,k)NpP^N_max/rR^N_max1_n>0]+ ∑_n=0^k𝔼[(f(Y_t^N(r,p,k)+1/N(1_r,n+1,p,k-1_r,n,p,k))-f(Y_t^N(r,p,k)))Y^N_t(r,n,p,k)N^2pP^N_max(M/N..- ..∑_n'∑_r',p',k'n'Y_t^N(r',n',p',k'))1_n<k]§.§.§ Mean Field LimitWe first state the following assumptions that we use throughout this section, There exists a probability measure I(r,p,k)on [0,1]× [0,1] ×ℕ and Λ > 0 such that, as N tends to infinity, we have i) 1/N∑_i=1^N1 _(r_i^N,p_i^N,K_i^N)⇒ I(r,p,k),ii)P^N_max/R_max^N→Λ, NP^N_max→𝒫, M/N→γ,iii)The set 𝒦=⋃_N=1^∞{K_i^N,i=1,...,N} is finite.Now we state the main theorem in this section that proves the convergence of empirical process to its mean field limit.Let |· | denote the Euclidean norm in ℝ^K_max+1. Under Assumption <ref>, suppose that Y_0^N y_0,then we have for any ϵ>0 and t_0>0,lim_N→∞ℙ(sup_t≤ t_0|Y_t^N-y_t|>ϵ)=0.Here y_t=(y_t(0),⋯,y_t(K_max)), where y_t(j)=∑_k∈𝒦∫_0^1∫ _0^1dy_t(r,j,p,k) for j=0,⋯,K_max. And y_t is the unique solution to the following differential equation starting at y_0y_t=b(y_t) where b:[0,1]^K+1→ℝ^K_max+1 is a vector field satisfiesb(y_t) = ∑_k∈𝒦∑_n=0^k∬_]0,1]× [0,1][ pΛ/r(1_(r,n-1,p,k)-1_(r,n,p,k))1_n>0+. .p𝒫(γ-∑_n∑_k≥ n∬_]0,1]×[0,1] ndy_t(r,n,p,k) )(1_(r,n+1,p,k)-1_(r,n,p,k))1_n<k] dy_t(r,n,p,k),or componentwise b(y_t)(0) = -∑_k∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_k∈𝒦∬_]0,1]× [0,1] ndy_t(r,n,p,k) )dy_t(r,0,p,k)_return a bike to a no-bike station +∑_k∈𝒦∬_]0,1]× [0,1]pΛ/rdy_t(r,1,p,k)_retrieve a bike from a 1-bike station, b(y_t)(j)= ∑_k∈𝒦∬_]0,1]× [0,1]pΛ/rdy_t(r,j+1,p,k)_retrieve a bike from a j+1-bike station -∑_k∈𝒦∬_]0,1]× [0,1](pΛ/r+p𝒫(γ-∑_n∑_k∈𝒦∬_]0,1]× [0,1] ndy_t(r,n,p,k) ))dy_t(r,j,p,k)_retrieve and return a bike to a j-bike station+∑_k∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_k∈𝒦∬_]0,1]× [0,1] ndy_t(r,n,p,k) )dy_t(r,j-1,p,k)_return a bike to a j-1-bike station, for j=1,...,K_max-1, and b(y_t)(K_max) = -∑_k∈𝒦∬_]0,1]× [0,1]pΛ/r dy_t(r,K_max,p,k)_retrieve a bike from a K_max-bike station +∑_k∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_k∈𝒦∬_]0,1]× [0,1] ndy_t(r,n,p,k) )dy_t(r,K_max-1,p,k)_return a bike to a K_max-1-bike station.The only things we need to prove in this extension case are that b(y) is Lipschitz and that β(Y_t^N) b(Y_t^N) for any t≥ 0. The rest of the proof stays the same as in Section <ref>.The drift function b(y) given in Equation (<ref>) is a Lipschitz function with respect to the Euclidean norm in ℝ^K_max+1. The proof is found in the Appendix. Under Assumption <ref>, we have for any ϵ>0 and s≥ 0,lim_N→∞ℙ(|β(Y_s^N)-b(Y_s^N)|>ϵ)= 0. The proof is found in the Appendix. §.§.§ Diffusion LimitNow we state the functional central limit theorem for the empirical measure process in the extension case as follows,Consider D_t^N in 𝔻(ℝ_+,ℝ^K_max+1) with the Skorokhod J_1 topology, and suppose that1) lim sup_N→∞√(N)(P_max^N/R_max^N-Λ)<∞,2) lim sup_N→∞√(N)(M/N-γ)< ∞,3) lim sup_N→∞√(N)(NP_max^N-𝒫)< ∞.Then if D_0^N converges in distribution to D_0, then D_t^N converges to the unique OU process solving D_t=D_0+∫_0^tb'(y_s)D_s ds+M_t in distribution. Here b'(y)=(∂ b(y)(i)/∂ y(j))_ij∈ℝ^(K_max+1)× (K_max+1) and M_t=(M_t(0),⋯,M_t(K_max))∈ℝ^K_max+1 is a real continuous centered Gaussian martingale, with Doob-Meyer brackets given byM(k) _t = ∫_0^t(b_+(y_s)(k)+b_-(y_s)(k))ds,M(k),M(k+1) _t = -∫_0^t[∑_K∈𝒦∬_]0,1]× [0,1]pΛ/rdy_s(r,k+1,p,K).+ .∑_K∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_K∈𝒦∬_]0,1]× [0,1]ndy_s(r,n,p,K)) dy_s(r,k,p,K)]dsfork<K_max,M(k),M(j) _t = 0for|k-j|>1.Here b_+(y)=max(b(y),0) and b_-(y)=-min(b(y),0) denote the positive and the negative parts of function b(y) respectively. The only things we need to prove in this extension case are listed as propositions below. The rest of the proof stays the same as in Section <ref>.√(N)M_t^N is a family of martingales independent of D_0^N with Doob-Meyer brackets given byM(k) _t = ∫_0^t(b_+(y_s)(k)+b_-(y_s)(k))ds,M(k),M(k+1) _t = -∫_0^t[∑_K∈𝒦∬_]0,1]× [0,1]pΛ/rdy_s(r,k+1,p,K).+ .∑_K∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_K∈𝒦∬_]0,1]× [0,1]ndy_s(r,n,p,K)) dy_s(r,k,p,K)]dsfork<K_max,M(k),M(j) _t = 0for|k-j|>1. For any s≥ 0, lim sup_N→∞√(N)|β(Y_s^N-b(Y_s^N)|< ∞ The proof is found in the Appendix. For k<K_maxlim_N→∞ℙ(sup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k),M(k+1)_t|>ϵ)=0 The proof is found in the Appendix.b(y) is continously differentiable with the derivatives∂ b(y(r,i,p,K))/∂ y(r,j,p,K) as follows,∂ b(y(r,0,p,K))/∂ y(r,j,p,K) = p𝒫j· y(r,0,p,K)+pΛ/r1_{j=1}-p𝒫(γ-∑_n=0^K_maxn(∑_r',p',K'y(r',n,p',K')))1_{j=0}, ∂ b(y(r,k,p,K))/∂ y(r,j,p,K) = p𝒫 j·( y(r,k,p,K)-y(r,k-1,p,K))+pΛ/r(1_{j=k+1}-1_{j=k}) +p𝒫(γ-∑_n=0^K_maxn(∑_r',p',K'y(r',n,p',K')))(1_{j=k-1}-1_{j=k})for0<k<K_max, ∂ b(y(r,K,p,K))/∂ y(r,j,p,K) = -p𝒫 j· y(K_max-1)-pΛ/r1_{j=K_max} +p𝒫(γ-∑_n=0^K_maxn(∑_r',p',K'y(r',n,p',K')))1_{j=K_max-1}.The above equations can be obtained by directly taking derivatives to Equation (<ref>). We can see that the derivatives of b(y) are linear in y. Thus we can conclude that b(y) is continuously differentiable with respect to y.§.§ Extensions to adding bike repositioningWe can also extend our model to incorporate repositioning of bikes, which is adopted by most bike-sharing companies to rebalance the network over time. To avoid cumbersome notations, we restrict ourselves to the uniform routing probabilities and capacities scenario for the model discussed in this section.We may assume that "rebalancers" will arrive to the system according to a independent Poisson process with rate λ_RY^N_t(K)N, where λ_R>0 is a constant and Y^N_t(K) is the proportion of full stations at time t. They pick a full station uniformly at random among all full stations, and remove a certain number C of the bikes from that station. In terms of repositioning these bikes, we assume that C number of bikes are being put back to stations with bikes ranging from 0 to K-C. The probability of repositioning to a station with j bikes is equal to Y^N_t(j)g_j/∑_i=0^K-CY^N_t(i)g_i, where {g_i}_i are positive constants and g_0>>g_1>>g_2⋯>>g_K-C>0. This is a smoothed version of choosing the station with minimum number of bikes when repositioniong, and it preserves the Lipschitz property of the drift function, which is essential for deriving the fluid limit and the diffusion limit. We assume additionally that M<N(K-C) to avoid the case where the denominator ∑_i=0^K-CY^N_t(i)g_i can be 0. This can be easily shown through a proof by contradiction. For simplicity of the model, we assume repositioning time are exponentially distributed with rate μ_R. Finally, we use R^N_t to denote the number of bikes currently in repositioning by the rebalancers at time t.Conditioning on (Y_t^N(r,n),R^N_t)=(y(r,n),R), , we have the following transition rate for the four types of events:When a customer arrives to a station with n bikes,relative utilization r, to retrieve a bikes, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n-1 bikes goes up by 1/N, and the transition rate Q^N is Q^N((y,R),(y+1/N(1_(r,n-1)-1_(r,n)),R) )= y(r,n)/rR_max^N1_n>0.When a customer returns a bike to a station with n bikes and relative utilization r, the proportion of stations having n bikes goes down by 1/N, the proportion of stations having n+1 bikes goes up by 1/N, and the transition rate Q^N is Q^N((y,R),(y+1/N(1_(r,n+1)-1_(r,n)),R) )= y(r,n) ·μ·(M-∑_n'∑_r'n'y(r',n')N-R)1_n<K.When a rebalancer arrives to a station with K bikes,relative utilization r, to retrieve C bikes, the proportion of stations having K bikes goes down by 1/N, the proportion of stations having K-C bikes goes up by 1/N, and the transition rate Q^N is Q^N((y,R),(y+1/N(1_(r,K-C)-1_(r,K)),R+C) )= y(r,K)λ_R N. When a rebalancer arrives to a station with j bikes and relative utilization r, where 0≤ j≤ K-C, to put back C bikes, the proportion of stations having j bikes goes down by 1/N, the proportion of stations having j+C bikes goes up by 1/N, and the transition rate Q^N is Q^N((y,R),(y+1/N(1_(r,j+C)-1_(r,j)),R-C) )=μ_R y(r,j)g_j/∑_i=0^K-Cy(r,i)g_i·R/C.We have the following functional forward equations for (Y_t^N(r,n),R).For any integrable function f: [0,1]^K+1× [0,∞)→ℝ,Y_t^N(r)=(Y_t^N(r,0),⋯,Y_t^N(r,K)) satisfies the following functional forward equation, 𝔼(f(Y_t^N(r),R) | Y_0^N(r) = y_0(r), R_0^N=R_0]= ∑_n=0^K𝔼[(f(Y_t^N(r)+1/N(1_r,n-1-1_r,n),R_t^N)-f(Y_t^N(r),R_t^N))Y^N_t(r,n)/rR^N_max1_n>0]+ ∑_n=0^K𝔼[(f(Y_t^N(r)+1/N(1_r,n+1-1_r,n),R_t^N)-f(Y_t^N(r)),R_t^N)Y^N_t(r,n)N(M/N..- ..∑_n'∑_r'n'Y_t^N(r',n')-R^N_t/N)1_n<K]+ 𝔼[(f(Y_t^N(r)+1/N(1_r,K-C-1_r,K),R^N_t+C)-f(Y_t^N(r),R^N_t))Y^N_t(r,K)λ_R N]+ ∑_n=0^K-C𝔼[(f(Y_t^N(r)+1/N(1_r,n+C-1_r,n),R^N_t-C)-f(Y_t^N(r),R^N_t))μ_R Y^N_t(r,n)g_n/∑_i=0^K-CY^N_t(r,i)g_i·R^N_t/C].§.§.§ Mean Field LimitNow we show the mean field limit result for the empirical process with rebalancing.Let |·| denote the Euclidean norm in ℝ^K+2. Under Assumption <ref>, suppose that (Y_0^N,R_0^N/N) (y_0,r̅_0) ,then we have for any ϵ>0 and t_0>0,lim_N→∞P(sup_t≤ t_0|(Y_t^N,R^N_t/N)-(y_t,r̅_t)|>ϵ)=0.Here y_t=(y_t(0),⋯,y_t(K)), where y_t(k)=∫_0^1dy_t(r,k) for k=0,⋯,K. And (y_t,r̅_t) is the unique solution to the following differential equation starting at (y_0,r̅_0)y_t = b_1(y_t,r̅_t), r̅_t = b_2(y_t,r̅_t),where b_1:[0,1]^K+1→ℝ^K+1 and b_2:[0,∞)→ℝ are vector fields satisfyb_1(y_t,r̅_t)= ∬_]0,1]× [0,...,K][ Λ/r(1_(r,n-1)-1_(r,n))1_n>0+(γ-∑_n∫_0^1 ndy_t(r,n)-r̅_t )(1_(r,n+1)-1_(r,n))1_n<K] dy_t(r,n)+ ∫_]0,1]λ_R 1_(r,K-C-1_(r,K))dy_t(r,K)+∬_]0,1]× [0,...,K-C]μ_R g_n/∑_i=0^K-Cy_t(r,i)g_i·r̅_t/C (1_(r,n+C)-1_(r,n)) dy_t(r,n),or componentwiseb_1(y_t,r̅_t)(0) = -∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,0)_return a bike to a no-bike station+∫_0^1Λ/rdy_t(r,1)_retrieve a bike from a 1-bike station - ∫_0^1μ_R g_0/∑_i=0^K-Cy_t(r,i)g_i·r̅_t/Cdy_t(r,0)_rebalancer put back C bikes to a 0-bike station, b_1(y_t,r̅_t)(k)= ∫_0^1Λ/rdy_t(r,k+1)_retrieve a bike from a k+1-bike station-∫_0^1(Λ/r+γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,k)_retrieve and return a bike to a k-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,k-1)_return a bike to a k-1-bike station-∫_0^1μ_R g_k/∑_i=0^K-Cy_t(r,i)g_i·r̅_t/Cdy_t(r,k)_rebalancer put back C bikes to a k-bike station,for k=1,...,K-C-1, and b_1(y_t,r̅_t)(k)= ∫_0^1Λ/rdy_t(r,k+1)_retrieve a bike from a k+1-bike station-∫_0^1(Λ/r+γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,k)_retrieve and return a bike to a k-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,k-1)_return a bike to a k-1-bike station+∫_0^1μ_R g_k-C/∑_i=0^K-Cy_t(r,i)g_i·r̅_t/Cdy_t(r,k-C)_rebalancer put back C bikes to a k-C-bike station +∫_0^1 Cλ_R dy_t(r,K)1_k=K-C_rebalancers retrieve C bikes from a K-bike station,for k=K-C,...,K-1, and b_1(y_t,r̅_t)(K) = -∫_0^1Λ/r dy_t(r,K)_retrieve a bike from a K-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) -r̅_t )dy_t(r,K-1)_return a bike to a K-1-bike station -∫_0^1 Cλ_R dy_t(r,K)1_k=K-C_rebalancers retrieve C bikes from a K-bike station,andb_2(y_t,r̅_t)=∫_0^1 Cλ_R dy_t(r,K)_rebalancers retrieve C bikes from a K-bike station-∑_n=0^K-C∫_0^1μ_R g_n/∑_i=0^K-Cy_t(r,i)g_ir̅_t dy_t(r,n)_rebalancers put back C bikes to a n-bike station.This result can be shown by similar techniques used in Section <ref>. § NUMERICAL EXAMPLES AND SIMULATIONIn this section,we confirm our theoretical results of our bike sharing model with a stochastic simulation. We perform simulations with both stationaryand non-stationary arrival rates, which are discussed substantially in the following subsections.Figures <ref> and <ref> provide the user patterns of BSS from the historical data of Citi Bike. In Figure <ref>, we can see that the arrivalrate of users to CitiBike stations varies not only on the time of the day, but also on the day of the week. So in the subsequent numerical examples, we consider both stationary and non-stationary arrival rates to provide insights for the behavior of such systems under different demand and usage conditions.However, unlike the arrival rate, in Figure <ref>, we observe that the mean travel duration of Citi Bike users does not vary significantly as a function of the time of day or the day of the week.Therefore, in the subsequent numerical examples we assume a constant mean travel time 1/μ. §.§ Simulation Experiments with a Stationary Arrival Rate In this section, we provide the results of the simulation studies of our stochastic bike sharing model with a stationary arrival process.Some of the common model parameters that we use in all of our simulation experiments are given as follows:N=100, λ=1, μ=1. The number of sample paths we use is 50. Other parameters are specified in the illustration of each figure below.Figure <ref>: Figure <ref> provides the simulation result of the different components of Y_t^N and their mean field limits y_t when K=10. Each station has 5 bikes at the begining of the simulation. The solid lines of Figure <ref> represent the different components of Y_t^N, and the dashed lines of the same color represent the corresponding mean field limit y_t. It is surprising to notice that even though we only have 100 stations in the simulation, the mean field limit provides a high quality approximation to the empirical measure on each component. Furthermore, since arrival rate is constant, the empirical measure converges to its equilibrium point as time tends towards infinity. Figures <ref> - <ref>: Figures <ref> - <ref> show the simulation results of Y_t^N with a 95% confidence interval, vs. the mean field limit y_t with twostandard deviations of the unscaled diffusion limit, when K=3. The initial distribution of bikes is given as Y_0^N=(0,0.5,0.5,0). In Figure <ref>, the black curve shows Y_t^N(0), the simulated dynamics (average of 50 sample paths) of the proportion of stations with no bikes. The red dashed line shows y_t(0), the mean field limit of the proportion of stations with no bikes over time, which is obtained by solving the system of ODEs derived in Theorem <ref>. The green curves show 2σ(Y_t^N(0)) or two standard deviations of Y_t^N(0) from the sample mean. The blue dashed lines show 2σ(D_t)/N or two standard deviations of the unscaled diffusion limit from the mean field limit y_t(0). Here σ(D_t) is obtained from solving the system of ODEs in Theorem <ref> Equation (<ref>).We observe that the mean field limit and the diffusion limit provide high quality approximations for our stochastic bike sharing model.The high quality of the approximations serve to provide additional evidence for using the mean field and diffusion limits we derived earlier.Similarly, Figures <ref>, <ref>, and <ref> show the dynamics of the proportion of stations with 1, 2, and 3 bikes respectively, like Figure <ref> does for the proportion of stations with no bikes. Figures <ref> :Figure <ref> shows the simulation results of the variance of D_t^N vs. the numerical solution for the system of ODEs regarding the covariance matrix of D_t (see Equation (<ref>)) when K=3. Again, the approximation of the variance of diffusion limit to the variance of the actual diffusion process is pretty good. Figure <ref>:Figure <ref> shows the number of bikes in circulation over time, which is denoted as C_t^N≜ M-∑_j=0^Kj· Y_t^N(j)N. We use N(γ-∑_j=0^Kj· y_t(j)) to approximate 𝔼[C_t^N], the expectation of the number of bikes in circulation. We use Var[M-∑_j=0^Kj· Y_t^N(j)N ] = Var[∑_j=0^Kj· Y_t^N(j)N ]=N^2[∑_j=0^Kj^2 ·Var[Y^N_t(j)]+∑_i=0^K∑_j=i+1^K2ij·Cov[Y^N_t(i),Y^N_t(j)]]=N^2[∑_j=0^Kj^2 ·Var[y_t(j)+1/√(N)D_t^N(j)]. .+∑_i=0^K∑_j=i+1^K2ij·Cov[y_t(i)+1/√(N)D_t^N(i),y_t(j)+1/√(N)D_t^N(j)]]≈N[∑_j=0^Kj^2 ·Var[D_t(j)]+∑_i=0^K∑_j=i+1^K2ij·Cov[D_t(i),D_t(j)]]to approximate Var[C^N_t], the variance of the number of bikes in circulation. The black curve shows the simulated result of C_t^N, the average number of bikes in circulation.The red dashed line shows the mean field limit of the number of bikes in circulation over time, which is obtained by solving the ODEs derived in Theorem <ref>. The green curves show 2σ(C_t^N) or two standard deviations of the simulated (50 sample paths) number of bikes in circulation from its sample mean. The blue dashed lines show our approximation of2σ(C_t^N)using the diffusion limit (see Equation (<ref>)), which can be computed from solving theODEs derived in Theorem <ref>Equation (<ref>). We also see that the mean field limit and the diffusion limit are good approximations for the actual dynamics of the average number of bikes in circulation. In this specific example, the number of bikes in circulation in equilibrium is roughly 60. Figure <ref>:Figure <ref> shows that the effect of increasing the arrival rate on the average number of bikes across stations. From Figure <ref>, we see that the average number of bikes decreases as arrival rate increases, and the intuition is that there will be fewer bikes at stations when more people arrive at the stations to retrieve bikes. From an operator's perspective, this plot illustrates that operators of BSS should be more aware of empty stations when usage of the bike sharing platform increases, and vice versa. Figure <ref>: Figure <ref> shows the bike distribution in equilibrium for different values of γ when K=20 . We can see that in equilibrium, the shape of the bike distribution depends on the value of γ. The distribution is right-skewed when γ is small, and left-skewed when γ is large. In particular, the distribution is roughly symmetric when γ is equal to 11.§.§ Simulation Experiments with a Non-stationary Arrival RateIn this section, we provide the results of the simulation studies of our stochastic bike sharing model with a non-stationary arrival process.We perform our simulation with the following parameters: N=100, λ(t)=1+0.5sin(t/2), μ=1. The number of sample paths we use in our simulation is 50. Other parameters are specified in the illustration of each figure below. For the time-varying arrival rate, we use a periodic function λ(t)=1+0.5sin(t/2) to mimic the patterns of morning rush and the afternoon-evening rush that we observe from the Citi Bike data (See Figure <ref>).Figure <ref>: In Figure <ref>, we simulate the different components of Y_t^N and their subsequent mean field limits y_t when K=10. Each station has 5 bikes at the beginning of the simulation. The solid lines represent the simulation results of different components of Y_t^N, and the dashed lines of the same color represent the corresponding mean field limit y_t. We also add the purple dashed line to represent the time-varying arrival rate. We observe that the mean field limit provides an accurate approximation to the empirical measure on each component.We also notice that the empirical measure is indeed affected by the non-stationary arrival rate and is also non-stationary.Figures <ref> - <ref>:Figures <ref> - <ref> show the simulation results of Y^N_t with a 95% confidence interval, vs. the mean field limit y(t) withtwostandard deviations of the unscaled diffusion limit, when K=3. The initial distribution of bikes is given as Y_0^N=(0,0.5,0.5,0). In Figure <ref>, the black curve shows Y_t^N(0), the simulated proportion of stations with no bikes over time. To produce the simulation, we take an average of 50 independent sample paths. The red dashed line shows y_t(0), the mean field limit of the proportion of stations with no bikes over time, which is obtained from solving the system of ODEs in Theorem <ref>. The green curves show 2σ(Y_t^N(0)) or two standard deviations of Y_t^N(0), from the 50 sample paths in simulation. Lastly, the blue dashed lines show 2σ(D_t)/N or two standard deviations of the unscaled diffusion limit, which is obtained from solving the system of ODEs from Theorem <ref>Equation (<ref>). We also add a purple dashed line to represent the time-varying arrival rate, with black dashed vertical lines to indicate the peaks and valleys of the arrival rate and the corresponding mean field limit.Again, we observe that the mean field limit and the diffusion limit provide an accurate approximation to the simulation results.This accuracy serves to validate the correctness of the mean field and diffusion limits we proved earlier. Moreover, with a small time lag, the simulation results of Y_t^N(0) are positively associated with movements of the arrival rate. Figures <ref>, <ref>, and <ref> show the dynamics of the proportion of stations with 1, 2, and 3 bikes respectively. Unlike the case of zero bikes, we observe that the dynamics are negatively associated with movements of the arrival rate.To observe these dynamics in one graph, Figure <ref> combines Figures <ref> - <ref> in one graph, but only keeps the mean field limit curves. This allows the reader to visualize the dynamics of the mean field empirical measure under a time-varying arrival rate. Figure <ref>: Figure <ref> shows the simulation results of the variance of D_t^N vs. the numerical solution for the system of ODEs regarding the covariance matrix of D_t (see Equation (<ref>)) when K=3. The approximation of the variance of diffusion limit to the variance of the actual diffusion process is quite accurate. We also observe that as arrival rate increases, the variance of the proportion of stations with no bikes increases, while the proportion of stations with 1, 2, or 3 bikes decreases, and vice versa. Figure <ref>:Figure <ref> shows the the average number of bikes in circulation over time, which is denoted as C_t^N≜ M-∑_j=0^Kj· Y_t^N(j)N. We use N(γ-∑_j=0^Kj· y_t(j)) to approximate 𝔼[C_t^N], the expectation of the number of bikes in circulation. We use Var[M-∑_j=0^Kj· Y_t^N(j)N ] = Var[∑_j=0^Kj· Y_t^N(j)N ]=N^2[∑_j=0^Kj^2 ·Var[Y^N_t(j)]+∑_i=0^K∑_j=i+1^K2ij·Cov[Y^N_t(i),Y^N_t(j)]]=N^2[∑_j=0^Kj^2 ·Var[y_t(j)+1/√(N)D_t^N(j)]. .+∑_i=0^K∑_j=i+1^K2ij·Cov[y_t(i)+1/√(N)D_t^N(i),y_t(j)+1/√(N)D_t^N(j)]]≈N[∑_j=0^Kj^2 ·Var[D_t(j)]+∑_i=0^K∑_j=i+1^K2ij·Cov[D_t(i),D_t(j)]]to approximate Var[C^N_t], the variance of the number of bikes in circulation. The black curve shows the simulated result of the average number of bikes in circulation.The red dashed line is the mean field limit of the number of bikes in circulation over time, from solving the system of ODEs obtained by Theorem <ref>. The green curves are 2σ(C_t^N), or two standard deviations from the mean of the number of bikes in circulation we obtain by simulating 50 independent sample paths of the stochastic bike sharing model. Lastly, the blue dashed lines showour approximation of2σ(C_t^N)using the diffusion limit (see Equation (<ref>)), which can be computed from solving the system of ODEs from Theorem <ref> Equation (<ref>). We also add a purple dashed line to represent the time-varing arrival rate, with black dashed vertical lines to indicate the peaks and valleys of the arrival rate and the corresponding mean field limit. Again, we can see that the mean field limit and the diffusion limit provide a high quality approximation for the actual dynamics of the average number of bikes in circulation.Moreover, we observe two important things.First, the average number of bikes in circulation is also time varying and fluctuates between 40 bikes and 80 bikes.Second, the average number of bikes in circulation lags slightly behind the arrival rate, which is a common phenomenon in non-stationary queues.§.§.§ Additional Commentary on the Lag Effect In the case where the arrival process is non-stationary, we observe a lag effect on the change of empirical measure in response to the change of the arrival rate. Here we explain what the dynamics of the lag effect are in this non-stationary case.We observe in Figure <ref> and Figure <ref> that after time 5, when the effect of the transient behavior is reduced,the proportion of stations with no bikes increases as arrival rate increases. However, the proportion of stations with 3 bikes goes down as arrival rate increases. The intuition is that when more people starting picking up bikes at stations, we end up with more empty stations and less full stations.Similarly, as the arrival rate decreases, the proportion of stations with no bikes decreases while the proportion of stations with 3 bikes increases. The intuition is that when less people starting picking up bikes at stations, we end up with less empty stations and more full stations.We also observe a lag between the peak(valley) of arrival and the peak(valley) of the proportion of stations with no bikes (Figure <ref>). There is also a lag between the peak(valley) of arrival and the valley(peak) of the proportion of stations with 3 bikes (Figure <ref>). This is because it takes time for the empirical process to respond to the change in the arrival rate.In Figure <ref>, we plot the size of the lag for different values of the service rate μ.We see that the lag effect has the most impact on the proportion of stations with one bike.It also has an effect on the proportion of stations with no bikes, 2 bikes and 3 bikes, however, the time lag is much smaller than that of one bike.We also find that the lag effect does not affect maximums and minimums the same way.We observe in all of the plots, except for proportion of stations with one bike, that the lag effect is more pronounced for minimums than maximums.This is especially true in the case of the proportion of stations having two bikes. §.§ Simulation Experiments with Heterogeneous Arrival Rates and Capacities In this section, we provide the results of the simulation studies of our stochastic bike sharing model with heterogeneous arrival rates and capacities. We also extend the scale of the system in our example to make it closer to reality. Some of the common model parameters that we use in all of our simulation experiments are given as follows:N=800, μ=1, and for each station, K=20,30,40, or 50 with equal probabilities. The number of sample paths we use is 30. Other parameters are specified in the illustration of each figure below.Figure <ref> :Figure <ref> shows the distribution of station capacities of Citi Bike (December 2017). We observe that most station capacities range from20 to 50. Thus we set the parameter K=20,30,40, or 50 with equal probabilities.Figure <ref> : Figure <ref> shows the average empirical distribution of CitiBike in a day (September 25th, 2017), with one standard deviation. This is exactly the empirical measure y we try to analyze in our bike sharing model. Studying the empirical measure and the its fluctuation over time is of great importance to understanding the system dynamics, providing prediction and guidance for rebalancing and strategic design. Figure <ref> : In Figure <ref>, we simulate the real Citi Bike network (New York City area) and show the empirical measure results from the simulation. The network setup is exactly the same as CitiBike over the month of September, 2017, where we have N=695 number of stations with capacities being the same as Citi Bike. We use Fourier regression to provide a fit for arrival functions for each station, and use them as heterogeneous time-varying arrival rates in our bike sharing model. We used a constant travel time rate μ=3.75, which comes from taking average of the trip duration data. We run the simulation for a 24-hour period and plot the average empirical measure from the simulation.Figure <ref> : Figure <ref> shows the difference in average empirical measure between simulation and real CitiBike data. We can see that the simulated empirical measure have at most 0.02 difference from the real CitiBike data. We can also observe that stations with 10-28 bikes have a positive difference while stations with less than 10 bikes have a negative difference. This is because rebalancing happens in the real CitiBike network, where bikes will be moved from stations with more bikes to stations with less bikes during peak hours to make the system more balanced, something we are not able to capture in our model. This causes the result from our model to have a larger number ofstations with more bikes, and a smaller number of stations with less stations, compared to reality. Besides, we also didn't consider the effect of information on CitiBike customers in our model, such as smartphone app that tells people the number of bikes and docks at stations in real time. In reality, a lot of people rely on the app to find stations for picking up and dropping off, which drives people to stations that have more bikes or docks. This would cause the same effect in making simulation results slightly different from reality. Overall, we can conclude from the simulation results that the bike sharing model proposed in our paper is doing well in capturing the real-life situation at CitiBike, which shows that our mean field limit and diffusion limit results will be of great value to the understanding the bike sharing systems in real life. Figure <ref> :This figure shows the empirical measure from simulation and its mean field limit in the heterogeneous arrivals and capacities case where arrivals are stationary. The arrival rates are set up to be λ=0.25,0.5,0.75, or 1 with equal probabilities. In this figure we showed the the empirical measure and its corresponding mean field limit with 95% confidence interval at a given time point. The blue bars show the empirical measure from simulation, which is an average of 30 sample paths. The red bars show the corresponding mean field limit we get from solving the ODEs in (<ref>). The error bars show the 95% confidence interval from the simulation results. We can see that the mean field limit is fitting well with empirical measure, which shows that our model is able to capture heterogeneous arrivals and capacities very well, and it can be easily adapted to more complex models, such as those with non-uniform routing probabilities, as shown in Section <ref>. Figure <ref> : This figure shows the evolution of different components of the empirical measure over time in the heteogeneous arrivals and capacities setting with the same parameters setup as in Figure <ref>.The solid lines represents proportion of stations with k bikes where k=5, 10, 15, 20, 25. The dashed lines represents their corresponding mean field limits. Again the mean field limits are fitting well with the empirical measure. Figure <ref> :This figure shows the empirical measure from simulation and its mean field limit in the heterogeneous arrivals and capacities case where arrivals are non-stationary. The arrival rates are set up to be λ(t)=4λ_0(1+sin(2t)) where λ_0=0.25,0.5,0.75,1 with equal probabilities. In this figure we showed the the empirical measure and its corresponding mean field limit with 95% confidence interval at a given time point. The blue bars show the empirical measure from simulation, which is an average of 30 sample paths. The red bars show the corresponding mean field limit we get from solving the ODEs in (<ref>). The error bars show the 95% confidence interval from the simulation results. We can see that the mean field limit is fitting well with empirical measure, which shows that our model is able to capture heterogeneous arrivals that are also non-stationary very well, and it can be easily adapted to more complex models, such as those with non-uniform routing probabilities, as shown in Section <ref>.Figure <ref> : This figure shows the evolution of different components of the empirical measure over time in the heterogeneous arrivals and capacities setting with the same parameters setup as in Figure <ref>.The solid lines represents proportion of stations with k bikes where k=0, 10, 20, 30, 40, 50. The dashed lines represents their corresponding mean field limits. We see that the empirical measure is also non-stationary and that the mean field limits are fitting well with the empirical measure. Computational cost: Another major benefit of our model compared to just using simulation to study the bike sharing system is that our model is extremely computationally inexpensive, given that it only involves numerically solving ODEs. The computational time for running the simulation example in Figure <ref>, with 10 sample paths for 8 unit times is 13.5 hours, while getting the mean field limit for the same example only takes less than 30 seconds. § APPLIED AND PRACTICAL VALUE OF OUR WORKThe results presented in this paper have great value to the operations of bike sharing systems. First of all, not only is the empirical measure itself an important performance measure to the bike sharing systems, but it also allows us to obtain salient performance measures such as y_t(0) (the proportion of stations with no bikes), y_t(K) (the proportion of stations that are full), M - ∑^K_j=0 j · y_t(j)N (the number of bikes in circulation), among others.The empirical measure approach is a significant reduction in computational complexity when compared to the full stochastic model.Moreover, it is the first time that a diffusion limit of the empirical measure of a inhomogeneous bike sharing systems is derived, which gives great insights to the fluctuation of the systems performance over time from a queueing and risk management perspective. It is a much more computationally efficient way to study the system behaviors in strategic planning stage, where given the design of the system and parameters such as arrival rate, travel time distribution and fleet size, you can easily get the proportion of problematic stations (empty and full stations) over time, which is a key measure that we want to minimize in a bike sharing system. More importantly, the diffusion limit of the empirical measure provides a refinement to the mean field limit, which gives us a better understanding to system fluctuations over time. By using the diffusion limit, we are able to build confidence intervals for the proportion of problematic stations. This helps us design a BSS with low blocking experiences, not just in expectation, but with high probability.This is especially important for managers of BSSs who want to control the dynamics of bike stations and reduce the volatility of station fluctuations.Another benefit of our work is that deriving the mean field limit and diffusion limit provides a way to formulate optimization problems associated with BSS. For example, the mean field and diffusion limit for the proportion of problematic stations can be used as objective functions that we try to minimize given system parameters. The expectation and variance of the number of bikes in circulation can computed through the mean field limit and diffusion limit of the empirical process, and therefore can be used to determine optimal fleet size.The current literature only uses mean field limits and our diffusion limits can be used to determine optimal fleet size under a more complex stochastic setting instead of a deterministic setting.Our analysis also benefits the rebalancing of bike sharing systems. It can be used to provide short-term prediction of the empirical measure of the system, which helps the operators of BSS identify when key measures such as proportion of problematic stations, or number of bikes in circulation, will go beyond a threshold and act beforehand. Different from traditional data analysis methods that predict patterns of BSS solely using history data, our method analyzes the system behaviors from a more fundamental way, one that does not heavily rely on data.Most importantly, as peak hours only last a few hours, decisions for rebalancing need to be made fast. In this case our method is much more computationally efficient and effective, than just doing simulations, which tends to be very slow and intractable for large scale systems like CitiBike.Another major benefit of using the empirical measure approach is that if CitiBike chooses to add a station, then in the empirical measure approach the dimensionality will only increase if the number of docks at the new station is larger than all of the rest.However, in the individual station model, it will automatically increase by one.Even though systems like CitiBike are large, they continue to add stations and increase the complexity of simulating the system.The empirical measure approach that we advocate in this work does not get worse when the management chooses to add stations. From a broader perspective, the framework of mean field and diffusion limits we established in this paper provide aneffective and efficient way to analyze different problems associated with BSS, such as, designing reasonable architecture of a BSS, finding a better path scheduling, improving inventory management, redistributing the bikes among stations or clusters, price optimization, application of intelligent information technologies and so forth.Our work serves as the initial step to exploring these important problems facing BSS. § CONCLUSION In this paper, we construct a bike sharing queueing model that incorporates the finite capacity of stations. Since our model is intractable to analyze directly, especially for a large number of stations, we propose to analyze the limiting dynamics of an empirical process that describes the proportion of stations that have a certain number of bikes.We prove a mean field limit and a functional central limit theorem for our stochastic bike sharing empirical process, showing that the mean field limit and the variance of the empirical process can be described by a system of 1/2(K+4)(K+1) differential equations where K is the maximum station capacity.We compare the mean field limit and the functional central limit theorem with simulation and show that the differential equations approximate the mean and variance of the empirical process extremely well. There are many directions for future work.The first direction would be to generalize the arrival and service distribution to follow general distributions.As Figure <ref> shows, the trip durations are not exponential and are closer to a lognormal distribution.An extension to general distributions would aid in showing how the non-exponential distributions affect the dynamics of the empirical process.Recent work by <cit.> provides a Poisson process representation of phase type distributions and Markovian arrival processes.This work might be useful in deriving new limit theorems for the queueing process with non-renewal arrival and service processes.In the non-stationary context, it is not only important to understand the dynamics of the mean field limit, but also it is important to know various properties of the mean field limit. For example, it would be informative to know the size of the amplitude and the frequency of the mean field limit when the arrival rate is periodic. One way to analyze the amplitude and the frequency is to exploit methods from non-linear dynamics like Lindstedt's method and the two-variable expansion method in <cit.>.Lastly, it is also interesting to consider a spatial model of arrivals to the bike sharing network.In this case, we would consider customers arriving to the system via a spatial Poisson process and customers would choose among the nearest stations to retrieve a bike.This spatial process can model the real choices that riders make and would model the real spatial dynamics of bike sharing networks.We intend to pursue these extensions in future work. § ACKNOWLEDGMENTSJamol Pender gratefully acknowledges the support of National Science Foundation (NSF) for Jamol Pender's Career Award CMMI # 1751975. § APPENDIXThe time derivative of the expectation 𝔼[f(𝐗(t))] can be derived by the following discretization method. Taking the expectation on f(𝐗(t+Δ)) conditioned on 𝐗(0)=𝐱 for some small Δ>0, we have 𝔼[f(𝐗(t+Δ)|𝐗(0)=𝐱] = ∑_i=1^Nf(𝐗(t))[1-λ_iΔ1_{X_i(t)>0}-μ P_i(M-∑_k=1^NX_k(t))Δ1_{X_i(t)<K_i}] +∑_i=1^N[f(𝐗(t)-1_i)λ_iΔ1_{X_i(t)>0}+ f(𝐗(t)+1_i)μ P_i(M-∑_k=1^NX_k(t))Δ1_{X_i(t)<K_i}] +o(Δ). Then𝔼[f(𝐗(t+Δ)|𝐗(0)=𝐱]- f(𝐗(t))= ∑_i=1^N[f(𝐗(t)-1_i)-f(𝐗(t)]λ_iΔ1_{X_i(t)>0}+∑_i=1^N[f(𝐗(t)+1_j)-f(𝐗(t)]μ P_i(M-∑_k=1^NX_k(t))Δ1_{X_i(t)<K_i} +o(Δ). By dividing by Δ and taking the expectation on both sides of Equation(<ref>) we get 𝔼[f(𝐗(t+Δ))]-𝔼[f(𝐗(t))]/Δ= ∑_i=1^N𝔼[(f(𝐗(t)-1_i)-f(𝐗(t))λ_i1_{X_i(t)>0}]+∑_i=1^N𝔼[(f(𝐗(t)+1_i)-f(𝐗(t))μ P_i(M-∑_k=1^NX_k(t))1_{X_i(t)<K_i}] +o(Δ)/Δ . Taking Δ→ 0 yields d/dt𝔼[f(𝐗(t))]= ∑_i=1^N𝔼[(f(𝐗(t)-1_i)-f(𝐗(t))λ_i1_{X_i(t)>0}]+∑_i=1^N𝔼[(f(𝐗(t)+1_i)-f(𝐗(t))μ P_i(M-∑_k=1^NX_k(t))1_{X_i(t)<K_i}]. Let f(𝐗(t))=X_i(t) for i=1,⋯,N, we have the following functional forward equations to each component of 𝐗(t), 𝔼[f(X_i(t)) | X_i(0) = x_i]≡d/dt𝔼[f(X_i(t)) | X_i(0) = x_i]= 𝔼[ ( f(X_i(t) +1) - f(X_i(t)) ) ·(μ P_i(M-∑_k=1^NX_k(t))1{X_i(t)<K_i}) ] +𝔼[ ( f(X_i(t) -1) - f(X_i(t)) ) ·( λ_i1{ X_i(t) > 0 }) ] . Let |·| denote the Euclidean norm in ℝ^K+1, then|Y_t^N-y_t| = |Y_0^N+M_t^N+∫_0^tβ (Y_s^N)ds-y_0-∫_0^tb(y_s)ds| = |Y_0^N-y_0+M_s^N+∫_0^t(β (Y_s^N)-b(Y_s^N))ds+∫_0^t(b(Y_s^N)-b(y_s))ds|.Now define the random function f^N(t)=sup_s≤ t|Y_s^N-y_s|, we havef^N(t)≤ |Y_0^N-y_0|+sup_s≤ t|M_s^N|+∫_0^t|β(Y_s^N)-b(Y_s^N)|ds+∫_0^t|b(Y_s^N)-b(y_s)|ds.By Proposition <ref>, b(y) is Lipschitz with respect to Euclidean norm. Let L be the Lipschitz constant of b(y), thenf^N(t) ≤|Y_0^N-y_0|+sup_s≤ t|M_s^N|+∫_0^t|β(Y_s^N)-b(Y_s^N)|ds+∫_0^t|b(Y_s^N)-b(y_s)|ds ≤|Y_0^N-y_0|+sup_s≤ t|M_s^N|+∫_0^t|β(Y_s^N)-b(Y_s^N)|ds+L∫_0^t|Y_s^N-y_s|ds ≤|Y_0^N-y_0|+sup_s≤ t|M_s^N|+∫_0^t|β(Y_s^N)-b(Y_s^N)|ds+L∫_0^tf^N(s)ds. By Gronwall's lemma (See <cit.>), f^N(t) ≤(|Y_0^N-y_0|+sup_s≤ t|M_s^N|+∫_0^t|β(Y_s^N)-b(Y_s^N)|ds)e^Lt. Nowto bound f^N(t) term by term, we define the function α: [0,1]^K+1→ℝ^K+1 asα(y) = ∑_x≠ y|x-y|^2Q(y,x)= 1/N∑_n∑_r[1/rNR_max(1_(r,n-1)+1_(r,n))1_n>0. + . (M/N-∑_n'∑_r'n'y(r',n'))(1_(r,n+1)+1_(r,n))1_n<K] · y(r,n)and considerthe following four sets Ω_0= {|Y_0^N-y_0|≤δ},Ω_1= {∫_0^t_0|β(Y_s^N)-b(Y_s^N)|ds≤δ},Ω_2= {∫_0^t_0α(Y_t^N)dt ≤ A(N)t_0 },Ω_3= {sup_t≤ t_0|M_t^N|≤δ} ,where δ=ϵ e^-Lt_0/3. Here the set Ω_1 is to bound the initial condition, the set Ω_2 is to bound the drift term β and the limit of drift term b, andthe sets Ω_2,Ω_3 are to bound the martingale M_t^N.Therefore on the event Ω_0∩Ω_1∩Ω_3,f^N(t_0)≤ 3δ e^Lt_0=ϵ. Since lim_N→∞M/N=γ and lim_N→∞NR_max=1/Λ, we can chooseN large enough such thatM/N≤ 2γ, NR_max≥1/2Λ.And by the proof of Proposition <ref>, there exists C>0 such that lim_N→∞r_i^N≥Λ/C. See the proof of Proposition <ref> in the Appendix for details.Thusα(y)≤ 1/N∑_n∑_r(2Λ/r· 2+2γ· 2 )· y(r,n)≤ 1/N( 4Λ/Λ/C+4γ)∑_n∑_ry(r,n)≤ 4/N(C+γ).Consider the stopping timeT=t_0∧inf{t≥ 0:∫ _0^tα(Y_s^N)ds>A(N)t_0}.By Proposition <ref>,𝔼(sup_t≤ T|M_t^N|^2)≤ 4𝔼∫_0^Tα(Y_t^N)dt≤ 4A(N)t_0.On Ω_2, we have T =t_0, soΩ_2 ∩Ω_3^c⊂{sup_t≤ T|M^N_t|>δ}. By Chebyshev's inequality, ℙ(Ω_2 ∩Ω_3^c)≤ℙ(sup_t≤ T|M^N_t|>δ)≤𝔼(sup_t≤ T|M_t^N|^2)/δ^2≤ 4A(N)t_0/δ^2.Thus, by Equation (<ref>), we have the following result,ℙ(sup_t≤ t_0|Y_t^N-y_t|>ϵ) ≤ℙ(Ω_0^c∪Ω_1^c∪Ω_3^c)≤ℙ(Ω_2 ∩Ω_3^c)+ℙ(Ω_0^c∪Ω_1^c∪Ω_2^c)≤ 4A(N)t_0/δ^2+ℙ(Ω_0^c∪Ω_1^c∪Ω_2^c)=36A(N)t_0 e^2Lt_0/ϵ^2+ℙ(Ω_0^c∪Ω_1^c∪Ω_2^c). Let A(N)=4(C+γ)/N, then Ω_2^c=∅. And since Y_0^N y_0, lim_N→∞ℙ(Ω_2^c)=0. Therefore we havelim_N→∞ℙ(sup_t≤ t_0|Y_t^N-y_t|>ϵ)=lim_N→∞ℙ(Ω_1^c) .By Proposition <ref>, lim_N→∞ℙ(Ω_1^c)=0. Thus, we proved the final resultlim_N→∞ℙ(sup_t≤ t_0|Y_t^N-y_t|>ϵ)=0.<ref>[Bounding Martingale] For any stopping time T such that 𝔼(T)<∞, we have𝔼(sup_t≤ T|M_t^N|^2)≤ 4𝔼∫_0^Tα(Y_t^N)dt. Let μ̃ be the jump measure of Y_t^N, and ν be its compensator, defined on (0,∞)× [0,1]by μ̃=∑_t:Y_t^N≠ Y_t-^Nδ(t,Y_t^N), ν(dt,B)=Q(Y_t-^N,B)dt ∀ B∈ℬ([0,1]).Let Ỹ_m,t^N be the jump chain of Y_t^N, J_m be the jump time, then we have for any t∈ [0,∞), J_n≤ t<J_n+1 for some n≥ 0. The martingale M_t^N can be written asM_t^N = Y^N_t-Y_0^N-∫_0^tβ(Y_s^N)ds = ∑_m=0^n-1(Ỹ^N_m+1-Ỹ^N_m)-∫_0^t∫_0^1(y-Y_s-^N)Q(Y_s-^N,dy)ds= ∫_0^t∫_0^1(y-Y_s-^N)μ̃(ds,dy)-∫_0^t∫_0^1(y-Y_s-^N)ν(ds,dy)= ∫_0^t∫_0^1(y-Y_s-^N)(μ̃-ν)(ds,dy).Note the following identity(M_t^N)^2 = 2∫_0^t∫_0^1M_s-(y-Y_s-^N)(μ̃-ν)(ds,dy)+∫_0^t∫_0^1(y-Y_s-^N)^2μ̃(ds,dy).This can be established by verifying that the jumps of the left and right hand sides agree, and that their derivatives agree between jump times. Then we can write(M_t^N)^2=N_t^N+∫_0^tα(Y_t^N)dswhereN_t^N=∫_0^t∫_0^1H(s,y)(μ̃-ν)(ds,dy),andH(s,y)=2M_s-(y-Y_s-^N)+(y-Y_s-^N)^2.Consider the previsible processH_2(t,y)=H(t,y)1_{t≤ T∧ T_n}where T_n=inf{t≥ 0: β(Y_t^N)>n}∧ n.ThenN_T∧ T_n=∫_0^∞∫_0^1H_2(t,y)(μ̃-ν)(dt,dy),and𝔼∫_0^∞∫_0^1|H_2(s,y)|ν(ds,dy) = 𝔼∫_0^T∧ T_n∫_0^1|2M_s-(y-Y_s-^N)+(y-Y_s-^N)^2|ν(ds,dy) =𝔼∫_0^T∧ T_n(2|M_t^N|β(Y_t^N)+α(Y_t^N))dt ≤ 𝔼∫_0^T∧ T_n2(2+n^2)ndt+𝔼∫_0^Tα(Y_t^N)dt ≤ 2n^4+4n^2+4(C+γ)/N𝔼(T)<∞.By Theorem 8.4 in <cit.>, we have that N^T∧ T_n is a martingale. Replace t by T∧ T_n in Equation (<ref>) and take expectation to obtain𝔼(|M_T∧ T_n|^2)=𝔼(N_T∧ T_n)+𝔼∫_0^T∧ T_nα(Y_t^N)dt.Since N^T∧ T_n is a martingale, 𝔼(N_T∧ T_n)=𝔼(N_0)=0.Apply Doob's L^2-inequality to the martingale M^T∧ T_n to obtain𝔼(sup_t≤ T∧ T_n|M_t|^2) ≤4𝔼(|M_T∧ T_n|^2)=4𝔼∫_0^T∧ T_nα(Y_t^N)dt. <ref>[Asymptotic Drift is Lipschitz] The drift function b(y) given in Equation (<ref>) is a Lipschitz function with respect to the Euclidean norm in ℝ^K+1. Assume that max_i(λ_i^N)≤ C <∞ for all N,thenr_i^N=R_i^N/R_max=1/Nλ_i^N/max_i(1/Nλ_i^N)=min_iλ_i^N/λ_i^N.By Assumption <ref>,NR_max=1/min_iλ_i^N→1/Λ>0.Thereforelim_N→∞r_i^N≥lim_N→∞Λ/max_iλ_i^N≥Λ/C.Thus the integral that defines function b should start from Λ/C instead of 0, i.e.b(y)=∬_[Λ/C,1]× [0,...,K][ Λ/r(1_(r,n-1)-1_(r,n))1_n>0+(γ-∑_n∫ ndy(r,n) )(1_(r,n+1)-1_(r,n))1_n<K] dy(r,n).Now consider y,ỹ∈ [0,1]^K+1,|b(y)-b(ỹ)|≤ 2(Λ/Λ/C+γ)|y-ỹ|=2(C+γ)|y-ỹ|≜ L|y-ỹ|where |·| denotes the Euclidean norm.<ref>[Asymptotic Drift is Lipschitz] The drift function b(y) given in Equation (<ref>) is a Lipschitz function with respect to the Euclidean norm in ℝ^K_max+1. Assume that max_i(λ_i^N)≤ C <∞ for all N,thenp_i^N/r_i^N=P_i^N/P_max^N/R_i^N/R^N_max=λ_i^NR^N_max/P_max^NBy Assumption <ref>,R^N_max/P^N_max→1/Λ>0.Thereforelim sup_N→∞p_i^N/r_i^N≤ C/Λ.Thus the function to be integrated in b is bounded,b(y) = ∑_k∈𝒦∬_[0,1]× [0,1][ pΛ/r(1_(r,n-1,p,k)-1_(r,n,p,k))1_n>0. +.p𝒫(γ-∑_n∑_k∈𝒦∬_[0,1]× [0,1] ndy(r,n,p,k) )(1_(r,n+1,p,k)-1_(r,n,p,k))1_n<k] dy(r,n,p,k). Now consider y,ỹ∈ [0,1]^K_max+1,|b(y)-b(ỹ)|≤ 2(C/Λ·Λ+𝒫γ)|y-ỹ|=2(C+𝒫γ)|y-ỹ|≜ L|y-ỹ|where |·| denotes the Euclidean norm. <ref>[Drift is Asymptotically Close to a Lipschitz Drift] Under Assumption <ref>, we have for any ϵ>0 and s≥ 0,lim_N→∞ℙ(|β(Y_s^N)-b(Y_s^N)|>ϵ)= 0.|β(Y_s)-b(Y_s)|^2 ≤ 2^2∑_n|∑_r1/rNR_maxY_s(r,n)-∫_0^1Λ/rdY_s(r,n)|^2+2^2∑_n|∑_rM/NY_s(r,n)-∫_0^1γ dY_s(r,n)|^2+2^2∑_n|∑_r(∑_n'∑_r'n'Y_s(r',n'))Y_s(r,n)-∫_0^1(∑_n'∫_0^1n'dY_s(r',n'))dY_s(r,n)|^2.It suffices to show each term goes to zero as N→∞.Since Y_s^N is a discrete random variable, we have for each n,∫_0^1f(r)dY_s(r,n)=∑_rf(r)Y_s(r,n) holds for any function f.Then|∑_r1/rNR_maxY_s(r,n)-∫_0^1Λ/rdY_s(r,n)| ≤ |∑_r(1/rNR_max-Λ/r) Y_s(r,n)|+|∑_rΛ/rY_s(r,n)-∫_0^1Λ/rdY_s(r,n)|=|1/NR_max-Λ|∑_r1/rY_s(r,n) ≤ |1/NR_max-Λ| C/Λ∑_rY_s(r,n) ≤ |1/NR_max-Λ|C/Λ→ 0.Similarly, |∑_rM/NY_s(r,n)-∫_0^1γ dY_s(r,n)| ≤ |∑_r(M/N-s) Y_s(r,n)|+|∑_rγ Y_s(r,n)-∫_0^1γ dY_s(r,n)| ≤ |M/N-γ|→ 0. The last term is zero since Y_s^N is discrete and ∫_0^1f(r)dY_s(r,n)=∑_rf(r)Y_s(r,n)for any function f. <ref>[Drift is Asymptotically Close to a Lipschitz Drift] Under Assumption <ref>, we have for any ϵ>0 and s≥ 0,lim_N→∞ℙ(|β(Y_s^N)-b(Y_s^N)|>ϵ)= 0.|β(Y_s)-b(Y_s)|^2≤2^2∑_n|∑_r,p,kpP^N_max/rR^N_maxY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]pΛ/rdY_s(r,n,p,k)|^2 +2^2∑_n|∑_r,p,kpNP^N_maxM/NY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]p𝒫γ dY_s(r,n,p,k)|^2 +2^2∑_n|∑_r,p,k(∑_n'∑_r',p',k'n'Y_s(r',n',p',k'))Y_s(r,n,p,k). -.∑_k∈𝒦∬_[0,1]× [0,1](∑_n'∑_k∈𝒦∬_[0,1]× [0,1]n'dY_s(r',n',p',k'))dY_s(r,n,p,k)|^2.It suffices to show each term goes to zero as N→∞.Since Y_s^N is a discrete random variable, we have for each n,∑_k∈𝒦∬_[0,1]× [0,1]f(r,p,k)dY_s(r,n,p,k)=∑_k∈𝒦∬_[0,1]× [0,1]f(r,p,k)Y_s(r,n,p,k) holds for any function f.Then|∑_r,p,kpP^N_max/rR^N_maxY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]Λ/rdY_s(r,n,p,k)| ≤ |∑_r,p,k(pP^N_max/rR^N_max-pΛ/r) Y_s(r,n,p,k)|+|∑_r,p,kpΛ/rY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]pΛ/rdY_s(r,n,p,k)|=|P^N_max/R^N_max-Λ|∑_r,p,kp/rY_s(r,n,p,k) ≤ |P^N_max/R^N_max-Λ| C/Λ∑_r,p,kY_s(r,n,p,k) ≤ |P^N_max/R^N_max-Λ|C/Λ→ 0.Similarly,|∑_r,p,kpNP^N_maxM/NY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]p𝒫γ dY_s(r,n,p,k)| ≤ |∑_r,p,k(NP^N_maxM/N-𝒫γ) pY_s(r,n,p,k)|+|∑_r,p,kp𝒫γ Y_s(r,n)-∑_k∈𝒦∬_[0,1]× [0,1]p𝒫γ dY_s(r,n,p,k)| ≤ |NP^N_maxM/N-𝒫γ|→ 0. The last term is zero since Y_s^N is discrete and ∑_k∈𝒦∬_[0,1]× [0,1]f(r,p,k)dY_s(r,n,p,k)=∑_k∈𝒦∬_[0,1]× [0,1]f(r,p,k)Y_s(r,n,p,k)for any function f.By Dynkin's formula,√(N)M^N(k)_t= ∫_0^tN∑_x≠ Y_s^N|x(k)-Y_s^N(k)|^2 Q(Y_s^N,x)ds= N∫_0^tα(Y_s^N)(k)ds= ∫_0^t∑_r[1/rNR_max(Y_s^N(r,k+1)1_k<K+Y_s^N(r,k)1_k>0).+.(M/N-∑_n'∑_r'n'Y_s^N(r',n'))(Y_s^N(r,k)1_k<K+Y_s^N(r,k-1)1_k>0)]ds= ∫_0^t(β_+(Y_s^N)(k)+β_-(Y_s^N)(k))ds. To compute√(N)M^N(k),√(N)M^N(k+1)_t for k<K, sinceM^N(k)+M^N(k+1)_t= ∫_0^t∑_x≠ Y_s^N|x(k)+x(k+1)-Y_s^N(k)-Y_s^N(k+1)|^2 Q(Y_s^N,x)ds= 1/N∫_0^t∑_r[1/rNR_max(Y_s^N(r,k+2)1_k<K-1+Y_s^N(r,k)1_k>0).+.(M/N-∑_n'∑_r'n'Y_s^N(r',n'))(Y_s^N(r,k+1)1_k<K-1+Y_s^N(r,k-1)1_k>0)]ds.We have that√(N)M^N(k),√(N)M^N(k+1)_t= N/2[ M^N(k)+M^N(k+1)_t- M^N(k)_t- M^N(k+1)_t]= 1/2∫_0^t∑_r[1/rNR_max(Y_s^N(r,k+2)1_k<K-1+Y_s^N(r,k)1_k>0).+.(M/N-∑_n'∑_r'n'Y_s^N(r',n'))(Y_s^N(r,k+1)1_k<K-1+Y_s^N(r,k-1)1_k>0)]ds-1/2∫_0^t(β_+(Y_s^N)(k)+β_+(Y_s^N)(k+1)+β_-(Y_s^N)(k)+β_-(Y_s^N)(k+1))ds= -∫_0^t∑_r[1/rNR_maxY_s^N(r,k+1)+(M/N-∑_n'∑_r'n'Y_s^N(r',n'))Y_s^N(r,k)] ds.When |k-j|>1, M^N(k) and M^N(j) are independent, thus √(N)M^N(k),√(N)M^N(j)_t=0.<ref> For any s≥ 0,lim sup_N→∞√(N)|β(Y_s^N)-b(Y_s^N)|<∞.|β(Y_s)-b(Y_s)| ≤ 2∑_n|∑_r1/rNR_maxY_s(r,n)-∫_0^1Λ/rdY_s(r,n)|+2∑_n|∑_rM/NY_s(r,n)-∫_0^1γ dY_s(r,n)|+2∑_n|∑_r(∑_n'∑_r'n'Y_s(r',n'))Y_s(r,n)-∫_0^1(∑_n'∫_0^1n'dY_s(r',n'))dY_s(r,n)|.By Equation (<ref>) and (<ref>),|β(Y_s)-b(Y_s)|≤ 2(K+1)(|1/NR_max-Λ|C/Λ+|M/N-γ|). By the assumptions in Theorem <ref>, we have lim sup_N→∞√(N)(min_iλ_i^N-Λ)<∞,lim sup_N→∞√(N)(M/N-γ)< ∞.Thus lim sup_N→∞√(N)|β(Y_s^N)-b(Y_s^N)| ≤ lim sup_N→∞2(K+1)√(N)(|1/NR_max-Λ|C/Λ+|M/N-γ|) < ∞.By Proposition <ref>, √(N)|β(Y_s^N)-b(Y_s^N)|=O(1), then|D_t^N| ≤ |D_0^N|+√(N)|M_t^N|+O(1)t+∫_0^t√(N) |b(Y_s^N)-b(y_s)|ds≤|D_0^N|+√(N)|M_t^N|+O(1)t+∫_0^t√(N)L|Y_s^N-y_s|ds=|D_0^N|+√(N)|M_t^N|+O(1)t+∫_0^tL|D_s^N|ds.By Gronwall's Lemma,sup_0≤ t≤ T|D_t^N|≤ e^LT(|D_0^N|+O(1)T+sup_0≤ t≤ T|√(N)M_t^N|).Thenlim sup_N→∞𝔼(sup_0≤ t ≤ T|D_t^N|^2)≤ e^2LT[lim sup_N→∞𝔼(|D_0^N|)+O(1)T+lim sup_N→∞𝔼(sup_0≤ t≤ T√(N)|M_t^N|)]^2.By Jensen's inequality and Proposition <ref>, we have that [𝔼(sup_0≤ t≤ T√(N)|M_t^N|)]^2≤ N𝔼(sup_0≤ t≤ T|M_t^N|^2)≤ 4NA(N)T,and that A(N)=O(1/N). Thereforelim sup_N→∞𝔼(sup_0≤ t≤ T√(N)|M_t^N| )<∞.Together with our assumption lim sup_N→∞𝔼(|D_0^N|^2)<∞, we havelim sup_N→∞𝔼(sup_0≤ t ≤ T|D_t^N|^2 )<∞. To prove the tightness of (D^N)_N=1^∞ and the continuity of the limit points, we only need to show that the following two tightness conditions hold for each T>0 and ϵ>0, (i) lim_K→∞lim sup_N→∞ℙ(sup_0≤ t≤ T|D_t^N|>K )=0, (ii) lim_δ→ 0lim sup_N→∞ℙ(w(D^N,δ,T)≥ϵ)=0 where for x∈𝔻^d,w(x,δ,T)=sup{sup_u,v∈[t,t+δ]|x(u)-x(v)|:0≤ t≤ t+δ≤ T}.By Lemma <ref>, there exists C_0>0 such thatlim_K→∞lim sup_N→∞ℙ(sup_0≤ t≤ T|D_t^N|>K )≤ lim_K→∞lim sup_N→∞𝔼(sup_0≤ t≤ T|D_t^N|^2 )/K^2≤ lim_K→∞C_0/K^2= 0,which proves condition (i).For condition (ii), we have that D^N_u - D^N_v = √(N)· ( M^N_u - M^N_v)_first term + ∫^u_v√(N)( β(Y^N_z) - b(Y^N_z)) dz_second term+ ∫^u_v√(N)( b(Y^N_z) - b(y_z)) dz _third termfor any 0<t≤ u<v≤ t+δ≤ T.Now it suffices to show that each of the three terms of D^N_u - D^N_v satisfies condition (ii).In what follows, we will show that each of the three terms satisfies condition (ii) to complete the proof of tightness.For the first term √(N)· ( M^N_u - M^N_v), we would like to show that the limiting sample path of √(N)M_t^N is a continuous Brownian motion,by using the martingale central limit theorem. Similar to the proof of Proposition <ref>, we can show that sup_t≤ T|β_+(Y_t^N)-b_+(Y_t^N)|0, sup_t≤ T|β_-(Y_t^N)-b_-(Y_t^N)|0.And by the proof of Proposition <ref>, b_+(y), b_-(y) are also Lipschitz with constant L, then by the fact that the composition of Lipschitz functions are also Lipschitz,max{sup_t≤ T|b_+(Y_t^N)-b_+(y_t)|,sup_t≤ T|b_-(Y_t^N)-b_-(y_t)|}≤ Lsup_t≤ T|Y_t^N-y_t|.By Theorem <ref>, sup_t≤ T|Y_t^N-y_t| 0.Thus combining Equations (<ref>), (<ref>) and (<ref>), we have lim_N→∞ℙ(sup_t≤ T|√(N)M^N(k)_t-M(k)_t|>ϵ)= lim_N→∞ℙ(sup_t≤ T|∫_0^t(β_+(Y_s^N)(k)+β_-(Y_s^N)(k)- b_+(y_s)(k)-b_-(y_s)(k))ds|>ϵ)≤ lim_N→∞ℙ(sup_t≤ TT|β_+(Y_t^N)(k)- b_+(Y^N_t)(k)|>ϵ/4)+lim_N→∞ℙ(sup_t≤ TT|b_+(Y_t^N)(k)- b_+(y_t)(k)|>ϵ/4) + lim_N→∞ℙ(sup_t≤ TT|β_-(Y_t^N)(k)- b_-(Y^N_t)(k)|>ϵ/4)+lim_N→∞ℙ(sup_t≤ TT|b_-(Y_t^N)(k)- b_-(y_t)(k)|>ϵ/4)≤ lim_N→∞ℙ(sup_t≤ TT|β_+(Y_t^N)(k)- b_+(Y^N_t)(k)|>ϵ/4)+2lim_N→∞ℙ(sup_t≤ TLT|Y_t^N- y_t|>ϵ/4 )+ lim_N→∞ℙ(sup_t≤ TT|β_-(Y_t^N)(k)- b_-(Y^N_t)(k)|>ϵ/4)=0,for any ϵ>0 and 0≤ k≤ K. This result implies thatsup_t≤ T|√(N)M^N(k)_t-M(k)_t| 0. For the adjacent terms, we havelim_N→∞ℙ(sup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k),M(k+1)_t|>ϵ)= lim_N→∞ℙ(sup_t≤ T|∫_0^t[∑_r(1/rNR_maxY_s^N(r,k+1)+(M/N-∑_n'∑_r'n'Y_s^N(r',n'))Y_s^N(r,k)). ..- ...(∫_0^1Λ/rdy_s(r,k+1)+∫_0^1(γ-∑_n∫_0^1ndy_s(r,n)) dy_s(r,k))]ds|>ϵ)≤ lim_N→∞ℙ(sup_t≤ TT|∑_r1/rNR_maxY_t^N(r,k+1)-∫_0^1Λ/rdY_t(r,k+1) |>ϵ/3)+ lim_N→∞ℙ(sup_t≤ TT|(M/N-∑_n',r'n'Y_t^N(r',n'))Y_t^N(r,k)- ∫_0^1(γ-∑_n∫_0^1ndY_t(r,n)) dY_t(r,k)|>ϵ/3)+ lim_N→∞ℙ(sup_t≤ T2LT|Y_t^N- y_t|>ϵ/3 )≤ lim_N→∞ℙ(sup_t≤ TT|C/Λ NR_max-Λ/Λ /C|∑_rY_s(r,k+1) >ϵ/3)+ lim_N→∞ℙ(sup_t≤ TT|M/N-γ| ∑_rY_s(r,k)>ϵ/3)≤ lim_N→∞ℙ(sup_t≤ TCT/Λ|1/ NR_max-Λ| >ϵ/3)+lim_N→∞ℙ(sup_t≤ TT|M/N-γ| >ϵ/3)= 0,which impliessup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k), M(k+1)_t| 0.Since for all |i-j|>1, √(N)M^N(i),√(N)M^N(j)_t = M(i), M(j)_t=0.We can conclude that sup_t≤ T|√(N)M^N(i),√(N)M^N(j)_t-M(i), M(j)_t| 0.for all 0≤ i,j≤ K.We also know that the jump size of Y^N_t is 1/N, thereforelim_N→∞𝔼[sup_0<t≤ T|√(N)M^N_t-√(N)M^N_t-| ]=lim_N→∞𝔼[sup_0<t≤ T|√(N)Y^N_t-√(N)Y^N_t-| ]=0.By Theorem 1.4 in Chapter 7 of <cit.>,√(N)M^N_t converges to the continuous Brownian motion M_t in distribution in 𝔻(ℝ_+,ℝ^K+1). By Prohorov's theorem, (√(N)M^N)_N=1^∞ is tight. This automatically implies the tightness condition ii). Thuslim_δ→ 0lim sup_N→∞ℙ(sup_u,v∈ [t,t+δ]| √(N)(M_u^N-M_v^N)|>ϵ)≤ lim_δ→ 0lim sup_N→∞ℙ(sup_u,v∈ [t,t+δ]| √(N)M_u^N-M_u|>ϵ/3)+lim_δ→ 0lim sup_N→∞ℙ(sup_u,v∈ [t,t+δ]|√(N)M_v^N-M_v|>ϵ/3) +lim_δ→ 0lim sup_N→∞ℙ(sup_u,v∈ [t,t+δ]| M_u-M_v|>ϵ/3)≤ lim_δ→ 0ℙ(sup_u,v∈ [t,t+δ]| M_u-M_v|>ϵ/3)=0For the second term ∫^u_v√(N)( β(Y^N_z) - b(Y^N_z)) dz, we have by Proposition <ref> that the quantity √(N)( β(Y^N_z) - b(Y^N_z)) is bounded for any value of z∈ [0,T].Therefore, there exists some constant C_1 that does not depend on N such thatsup_z∈ [0,T]√(N)| β(Y^N_z) - b(Y^N_z)|≤ C_1.Thenlim_δ→ 0lim_N→∞ℙ(sup_u,v∈ [0,T],|u-v|≤δ∫^u_v√(N)| β(Y^N_z) - b(Y^N_z)| dz > ϵ)≤ lim_δ→ 0lim_N→∞ℙ(δsup_z∈ [0,T]√(N)| β(Y^N_z) - b(Y^N_z)|> ϵ)≤ lim_δ→ 0ℙ(δ C_1> ϵ)=0. Thus, we have proved the oscillation bound for the second term.Finally for the third term we have that ∫^u_v√(N)| b(Y^N_z) - b(y_z)| dz≤ ∫^u_v√(N) L| Y^N_z - y_z| dz= ∫^u_v L ·|D^N_z| dz ≤Lδsup_t∈ [0,T]|D^N_t|.By Lemma <ref>,lim_δ→ 0lim_N→∞ℙ(sup_u,v∈ [0,T],|u-v|≤δ∫^u_v√(N)| b(Y^N_z) - b(y_z)| dz>ϵ)≤ lim_δ→ 0lim_N→∞ℙ(Lδsup_t∈ [0,T]|D^N_t|>ϵ)≤ lim_δ→ 0lim_N→∞𝔼(sup_t∈ [0,T]|D^N_t|^2)/(ϵ/L δ)^2≤ lim_δ→ 0C_0(Lδ)^2/ϵ^2=0, which implies that the oscillation bound holds for the third term. To prove the existence and uniqueness of the SDE (<ref>), we show the following two conditions hold: There exists a constant H>0 such that (1)Lipschitz condition: for any D,D̃∈ℝ^K+1, any t∈ (t_0,T), |b'(y_t)D-b'(y_t)D̃|≤ H|D-D̃|. (2)Linear growth condition: for any D∈ℝ^K+1, any t∈ (t_0,T),|b_+(y_t)+b_-(y_t)|≤ H(1+|D|),|b'(y_t)D|≤ H(1+|D|).By Proposition <ref>, b(y) is Lipschitz, and by our assumption in Theorem <ref>, b(y) is also continuously differentiable, thus b'(y) is bounded. Then conditions (1)and (2) follow, which prove that there exist a unique solution to the SDE (<ref>).Take expectation on both sides of Equation (<ref>), since 𝔼[ ∫_0^te^∫_s^tb'(y_u)dudM_s]=0, we have𝔼[D_t]=e^∫_0^t𝒜(s)ds𝔼[D_0].ThereforeD_t-𝔼[D_t] = e^∫_0^t𝒜(s)ds(D_0-𝔼[D_0])+∫_0^te^∫_s^t𝒜(u)dudM_s, Σ(t) = E[(D_t-𝔼[D_t])(D_t-𝔼[D_t])^⊤]= e^∫_0^t𝒜(s)dsE[(D_0-𝔼[D_0])(D_0-𝔼[D_0])^⊤] (e^∫_0^t𝒜(s)ds)^⊤+ (∫_0^te^∫_s^t𝒜(u)dudM_s)(∫_0^te^∫_s^t𝒜(u)dudM_s)^⊤= e^∫_0^t𝒜(s)dsΣ(0)e^∫_0^t𝒜^⊤(s)ds+∫_0^te^∫_s^t𝒜(u)duℬ(s)e^∫_s^t𝒜^⊤(u)duds.<ref> For any s≥ 0,lim sup_N→∞√(N)|β(Y_s^N)-b(Y_s^N)|<∞.|β(Y_s)-b(Y_s)| ≤ 2∑_n|∑_r,p,kpP^N_max/rR^N_maxY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]pΛ/rdY_s(r,n,p,k)|+2∑_n|∑_r,p,kpNP^N_maxM/NY_s(r,n,p,k)-∑_k∈𝒦∬_[0,1]× [0,1]p𝒫γ dY_s(r,n,p,k)|+2∑_n|∑_r,p,k(∑_n'∑_r',p',k'n'Y_s(r',n',p',k'))Y_s(r,n,p,k).-.∑_k∈𝒦∬_[0,1]× [0,1](∑_n'∑_k∈𝒦∬_[0,1]× [0,1]n'dY_s(r',n',p',k'))dY_s(r,n,p,k)|.By Equation (<ref>) and (<ref>),|β(Y_s)-b(Y_s)|≤ 2(K_max+1)(|P_max/R_max-Λ|C/Λ+|NP_maxM/N-𝒫γ|). By the assumptions in Theorem <ref>, we have lim sup_N→∞√(N)(P_max^N/R_max^N-Λ)<∞, lim sup_N→∞√(N)(M/N-γ)< ∞, lim sup_N→∞√(N)(NP_max^N-𝒫)< ∞.Thus lim sup_N→∞√(N)|β(Y_s^N)-b(Y_s^N)| ≤ lim sup_N→∞2(K_max+1)√(N)(|P_max/R_max-Λ|C/Λ+|NP_maxM/N-𝒫γ|) < ∞.<ref> For k<K_max,lim_N→∞ℙ(sup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k),M(k+1)_t|>ϵ)=0. For the adjacent terms, we havelim_N→∞ℙ(sup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k),M(k+1)_t|>ϵ)= lim_N→∞ℙ(sup_t≤ T|∫_0^t[∑_r,p,K∈𝒦(pP_max/rR_maxY_s^N(r,k+1,p,K)+(pNP_maxM/N-∑_n'∑_r',p',K'n'Y_s^N(r',n',p',K')). ... .. . . Y_s^N(r,k,p,K))-(∑_K∈𝒦∬_[0,1]× [0,1]pΛ/rdY_s(r,k+1,p,K). ...+.. . . ∑_K∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_K∈𝒦∬_]0,1]× [0,1]ndy_s(r,n,p,K)) dy_s(r,k,p,K))]ds|>ϵ)≤ lim_N→∞ℙ(sup_t≤ TT|∑_r,p,KpP_max/rR_maxY_t^N(r,k+1,P,K)-∑_K∈𝒦∬_[0,1]× [0,1]pΛ/rdY_s(r,k+1,p,K) |>ϵ/3)+ lim_N→∞ℙ(sup_t≤ TT|∑_r,p,K∈𝒦(pNP_maxM/N-∑_n'∑_r',p',K'n'Y_s^N(r',n',p',K'))Y_t^N(r,k,p,K).. .. - ∑_K∈𝒦∬_]0,1]× [0,1]p𝒫(γ-∑_n∑_K∈𝒦∬_]0,1]× [0,1]ndy_s(r,n,p,K)) dy_s(r,k,p,K)|>ϵ/3)+ lim_N→∞ℙ(sup_t≤ T2LT|Y_t^N- y_t|>ϵ/3 )≤ lim_N→∞ℙ(sup_t≤ TT|CP_max/Λ R_max-Λ/Λ /C|∑_r,p,KY_s(r,k+1,p,K) >ϵ/3)+ lim_N→∞ℙ(sup_t≤ TT| NP_maxM/N-𝒫γ| ∑_r,p,KpY_s(r,k,p,K)>ϵ/3)≤ lim_N→∞ℙ(sup_t≤ TCT/Λ|P_max/ R_max-Λ| >ϵ/3)+lim_N→∞ℙ(sup_t≤ TT|NP_maxM/N-𝒫γ| >ϵ/3)= 0,which impliessup_t≤ T|√(N)M^N(k),√(N)M^N(k+1)_t-M(k), M(k+1)_t| 0.<ref> b(y) is continously differentiable with the derivatives∂/∂ y(j)b(y)(i) as follows,∂/∂ y(j)b(y)(0) = j· y(0)+1/y(j)∫_0^1Λ/rdy(r,1)1_{j=1}-(γ-∑_n=0^Kny(n))1_{j=0}, ∂/∂ y(j)b(y)(k) = j·( y(k)-y(k-1))+1/y(j)(∫_0^1Λ/rdy(r,k+1)1_{j=k+1}-∫_0^1Λ/rdy(r,k)1_{j=k}) +(γ-∑_n=0^Kny(n))(1_{j=k-1}-1_{j=k})for0<k<K, ∂/∂ y(j)b(y)(K) = -j· y(K-1)-1/y(j)∫_0^1Λ/rdy(r,K)1_{j=K}+(γ-∑_n=0^Kny(n))1_{j=K-1}. Since y_t(k)=∫_0^1dy_t(r,k) for k=0,⋯, K, we haveb(y_t)(k)= ∫_0^1Λ/rdy_t(r,k+1)1_k<K_retrieve a bike from a k+1-bike station-∫_0^1(Λ/r1_k>0+(γ-∑_n∫_0^1 ndy_t(r,n) )1_k<K)dy_t(r,k)_retrieve and return a bike to a k-bike station+∫_0^1(γ-∑_n∫_0^1 ndy_t(r,n) )dy_t(r,k-1)1_k>0_return a bike to a k-1-bike station=y_t(k+1)1_k<K∫_0^1Λ/rdy_t(r,k+1)/y_t(k+1)-y_t(k)1_k>0∫_0^1Λ/rdy_t(r,k)/y_t(k)+(γ-∑_n ny_t(n) )(y_t(k-1)1_k>0-y_t(k)1_k<K).By condition i) in Assumption <ref>, 1/N∑_i=1^N1 _(r_i^N,K_i^N)⇒ I(r,k).As mentioned in the begining of Section <ref>,we assume without loss of generality that K_i^N=K. This is valid because when {K_i^N}_iare different, we can still carry out the proof of the mean field and diffusion limits by simply replacing r with (r,K). Thus, in this case the limiting measure I(r,k)=I(r,K) is one dimensional.By our definition, r_i^N, the relative utilization at station i,only denpends on the station number i. It does not depend on the number of bikes at that station. This means that the two components of the empirical measure y(r,n), the relative utilization r and the number of bikes n,are independent. Then∫_0^1Λ/rdy_t(r,k)/y_t(k)=∫_0^1Λ/rdI(r,K),which no longer depends on y_t(k). Therefore we have the derivatives of b(y_t)(k) as follows,∂/∂ y_t(j)b(y_t)(k) = j·( y_t(k)1_k<K-y_t(k-1)1_k>0)+1/y_t(j)(∫_0^1Λ/rdy_t(r,k+1)1_{j=k+1}. .-∫_0^1Λ/rdy_t(r,k)1_{j=k>0})+(γ-∑_n=0^Kny_t(n))(1_{j=k-1}-1_{j=k<K}) ,for 0≤ k,j ≤ K. We can also write the derivatives of b(y) seperately as follows,∂/∂ y(j)b(y)(0) = j· y(0)+1/y(j)∫_0^1Λ/rdy(r,1)1_{j=1}-(γ-∑_n=0^Kny(n))1_{j=0}, ∂/∂ y(j)b(y)(k) = j·( y(k)-y(k-1))+1/y(j)(∫_0^1Λ/rdy(r,k+1)1_{j=k+1}-∫_0^1Λ/rdy(r,k)1_{j=k}) +(γ-∑_n=0^Kny(n))(1_{j=k-1}-1_{j=k})for0<k<K, ∂/∂ y(j)b(y)(K) = -j· y(K-1)-1/y(j)∫_0^1Λ/rdy(r,K)1_{j=K}+(γ-∑_n=0^Kny(n))1_{j=K-1}.We can see that the derivatives of b(y) are linear in y=(y(0),⋯,y(K)). Thus we can conclude that b(y) is continuously differentiable with respect to y. plainnat | http://arxiv.org/abs/1708.08052v2 | {
"authors": [
"Shuang Tao",
"Jamol Pender"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170827051804",
"title": "A Stochastic Analysis of Bike Sharing Systems"
} |
Topological Edge Solitons in Polaritonic Lattice Antoni Szczurek[Also at Faculty of Mathematics and Natural Sciences, University of Rzeszow, ul. Pigonia 1, 35-310 Rzeszów, Poland.] December 30, 2023 ======================================================================================================================================= D. R. Gulevich^1, D. Yudin^1, D. V. Skryabin^1,2, I. V. Iorsh^1,and I. A. Shelykh^1,3^1ITMO University, St. Petersburg 197101, Russia^2Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom^3Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, IcelandWe study discrete nonlinear edge excitations of polaritonic kagome lattice. We show that when nontrivial topological phase of polaritons is realized, the kagome lattice permits propagation of bright solitons formed from topological edge states. Introduction Mapping condensed matter electronic systems to photonic structures is powerful tool for simulation of complex condensed matter phenomena since it allows to use well-controllable photonic systems to study their far less controllable solid state counterparts <cit.>. It is therefore not surprising that topological ideas have been widely explored in the domain of photonics since pioneer work of Raghu and Haldane <cit.> who brought forward the ideas of photonic chiral edge states and Berry curvature for the photonic bands. Propagation of topological edge modes was then studied in real photonic crystals <cit.> and concepts of the photonic analogues of the Hall states <cit.> and topological insulators <cit.> were introduced. This allowed extensive theoretical and experimental study of the effects of non-trivial topology in electromagnetic systems in the broad spectral range from microwave to optical frequencies <cit.>.Despite this success,optical circuits miss one important property inherent to solid state: strong inter-particle interactions. Exciton-polarions <cit.>, quasi-particles originating from strong coupling of quantum well-excitons and cavity photons in semiconductor microcavities, can be exploited to fill this gap. Due to their mixed light-matter nature, they can be both effectively controlled by well developed methods of optical confinement (similar to those used in photonic lattices), and demonstrate extremely strong nonlinear response (due to exciton-exciton interactions). Exciton-polaritons thus offer a unique laboratory for study of non-trivial topological phases in presence of interactions. The emergence of the non-trivial topology and the existence of the topologically protected edge states in polaritonic lattices of different geometries has been shown theoretically in a number of works <cit.>. Recently, the focus of attention in the study of the effects of non-trivial topology has shifted towards systems with nonlinearity <cit.>, where exciton-polaritons, due to their unique properties, play a special role. Among these works are studies of self-localized states <cit.>, self-induced topological transitions <cit.>, topological Bogoliubov excitations <cit.>, suppression of topological phases <cit.>, vortices in lattices <cit.>, spin-Meissner states in ring resonators <cit.>, solitons in lattices <cit.>, dimer chains <cit.>, dark solitons in kagome lattice <cit.>. In the present work we present our results for propagation of solitons formed from topological edge states of polaritonic kagome lattice. Results The tight-binding Hamiltonian for polaritons confined inside an array of coupled microcavity pillars forming kagome lattice reads <cit.>:Ĥ= Ω∑_i,σσ â_i,σ^†â_i,σ - ∑_⟨ ij⟩,σ( J â_i,σ^†â_j,σ+ δ J e^2iφ_ijσ̅â_i,σ^†â_j,σ̅ + h.c.) + ∑_i,σ(α_1/2 â_i,σ^†â_i,σâ_i,σ^†â_i,σ+ α_2 â_i,σ^†â_i,σâ_i,σ̅^†â_i,σ̅). Here, the operators â_i,σ^† (â_i,σ) create (annihilate) exciton-polariton of circular polarization σ=± at site i of the lattice, the summation ⟨ ij⟩ goes over nearest neighbors (NN), the angles φ_ij specify the directions of vectors connecting the neighboring sites. The first term in (<ref>) describes Zeeman energy splitting (2Ω) of the circular polarized components induced by the external magnetic field, the second term describes the NN hopping with conservation and inversion of circular polarization (the latter comes from TE-TM splitting), and the last term describes the on-site polariton-polariton interactions with effective constants α_1 and α_2. In what follows, we will use normalized units where J is a unit of the energy and the interpillar distance is unit of the length.First, consider the linear regime when interactions in (<ref>) can be neglected. In Ref. <cit.> it was shown that the presence of the finite TE-TM splitting δ J and magnetic field Ω opens a gap which separates the Bloch bands into two bundles with nontrivial topology, see Fig. <ref>. According to the bulk-boundary correspondence there must exist topological edge states propagating along the interface of a finite (or, semi-finite) lattice. The peculiar shape of the dispersion in Fig. <ref> makes favourable the existence of nonlinear excitations of the edge states in the form of solitons. Numerically calculated dynamics of bright soliton formed by topological edge statein presence of the interactions is shown in Fig. <ref>. The initial condition for the numerical simulations was choosen as bell-shape profile with fixed quasimomentum in the vicinity of the maximum of the dispersion in Fig.1. For the specifically chosen shape of the initial excitation and nonlinearity strength, the excitation keeps its form and starts propagating along the boundary. The speed and the direction of the propagation depend on the chosen value of quasimomentum of the topological edge dispersion. Conclusion We demonstrated propagation of bright solitons formed from topological edge states. Such solitons are formed from wavepackets with quasimomentum in the vicinity of the local maximum of the edge state dispersion.AcknowledgementsD.R.G., D.Y. and I.V.I. acknowledge support from the grant 3.8884.2017 of the Ministry of Education and Science of the Russian Federation. D.Y. acknowledges support from RFBR project 16-32-60040. I.A.S. acknowledge support from the Icelandic Research Fund, Grant No. 163082-051, mega-grant № 14.Y26.31.0015 of the Ministry of Education and Science of the Russian Federation and Horizon2020 RISE project CoExAN, Grant No. 644076.99Georgescu2014 Georgescu, I., Ashhab, S. & Nori, F, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).Raghu-2008 Raghu, S. & Haldane, F. D. M., Analogs of quantum-hall-effect edge states in photonic crystals, Phys. Rev. A 78, 033834 (2008).Wang-2009 Wang, Z., Chong, Y., Joannopoulos, F. D. & Soljačić M., Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461, 772–775 (2009).Haldane2008a Haldane, F. & Raghu, S., Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100, 013904 (2008). Wang2008 Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Phys. Rev. Lett. 100, 013905 (2008).Hafezi-2011 Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M., Robust optical delay lines with topological protection, Nat. Phys. 7, 907–912 (2011).Fang-2012 Fang, K., Yu, Z. & Fan, S., Realizing effective magnetic field for photons by controlling the phase of dynamic modulation, Nat. Photon. 6, 782 (2012).Rechtsman-2013 Rechtsman, M. C. et al., Photonic floquet topological insulators, Nature 496, 196 (2013).Khanikaev2013 Khanikaev, A. B. et al., Photonic topological insulators, Nat. Mater. 12, 233–239 (2013).Lu-2014 Lu, L., Joannopoulos, J. D. & Soljačić, M., Topological photonics, Nat. Photon. 8, 821 (2014).Carusotto2013 Carusotto, I. & Ciuti, C. Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).Karzig-PRX-2015 Karzig, T., Bardyn, C.-E., Lindner, N. H. & Refael, G., Topological polaritons. Phys. Rev. X 5, 031001 (2015).Bardyn-PRB-2015 Bardyn, C.-E., Karzig, T., Refael, G. & Liew, T. C. H., Topological polaritons and excitons in garden-variety systems. Phys. Rev. B 91, 161413 (2015).Nalitov-Z Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G., Polariton Z topological insulator, Phys. Rev. Lett. 114, 116401 (2015).Yi-PRB-2016 Yi, K. & Karzig, T., Topological polaritons from photonic dirac cones coupled to excitons in a magnetic field. Phys. Rev. B 93, 104303 (2016).Bardyn-PRB-2016 Bardyn, C.-E., Karzig, T., Refael, G. & Liew, T. C. H., Chiral bogoliubov excitations in nonlinear bosonic systems. Phys. Rev. B 93, 020502 (2016).Janot-PRB-2016 Janot, A., Rosenow, B. & Refael, G., Topological polaritons in a quantum spin hall cavity. Phys. Rev. B 93, 161111 (2016).Gulevich-kagome Gulevich, D. R., Yudin, D., Iorsh, I. V. & Shelykh, I. A., Kagome lattice from an exciton-polariton perspective. Phys. Rev. B 94, 115437 (2016).Lumer Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M., Self-localized states in photonic topological insulators. Phys. Rev. Lett. 111, 243905 (2013).Ostrovskaya Ostrovskaya, E. A., Abdullaev, J., Fraser, M. D., Desyatnikov, A. S. & Kivshar, Y. S., Self-localization of polariton condensates in periodic potentials. Phys. Rev. Lett. 110, 170407 (2013).Hadad Hadad, Y., Khanikaev, A. B. & Alú, A., Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Phys. Rev. B 93, 155112 (2016).Bleu-PRB-2016 Bleu, O., Solnyshkov, D. D., and Malpuech, G., Interacting quantum fluid in a polariton Chern insulator, Phys. Rev. B 93, 085438 (2016).Kart Kartashov, Y. V. & Skryabin, D. V., Two-dimensional lattice solitons in polariton condensates with spin-orbit coupling. Opt. Lett. 41, 5043–5046 (2016).Gulevich-Meissner Gulevich, D. R., Skryabin, D. V., Alodjants, A. P. & Shelykh, I. A., Topological spin Meissner effect in spinor exciton- polariton condensate: Constant amplitude solutions, half-vortices, and symmetry breaking. Phys. Rev. B 94, 115407 (2016).Ablowitz Ablowitz, M. J., Curtis, C. W. & Ma, Y.-P., Linear and nonlinear traveling edge waves in optical honeycomb lattices. Phys. Rev. A 90, 023813 (2014).Leykam Leykam, D. & Chong, Y. D., Edge solitons in nonlinear-photonic topological insulators. Phys. Rev. Lett. 117, 143901 (2016).Kartashov-Optica Kartashov, Y. V. & Skryabin, D. V., Modulational instability and solitary waves in polariton topological insulators. Optica 3 1228 (2016).CerdaMendez2016 Cerda-Méndez, E. A. et al., Exciton-polariton gap solitons in two-dimensional lattices, Phys. Rev. Lett. 111, 146401 (2013).Soln-PRL-2017 Solnyshkov, D. D., Bleu, O., Teklu, B., & Malpuech, G., Chirality of topological gap solitons in bosonic dimer chains, Phys. Rev. Lett. 118, 023901 (2017).Gorlach-PRA-2017 Gorlach, M. A. and Poddubny, A. N. Interaction-induced two-photon edge states in an extended Hubbard model realized in a cavity array, Phys. Rev. A 95 033831 (2017).DGulevich-SciRep Gulevich, D. R., Yudin, D., Skryabin, D. V., Iorsh, I. V. & Shelykh, I. A., Exploring nonlinear topological states of matter with exciton-polaritons: Edge solitons in kagome lattice, Sci. Rep. 7, 1780 (2017). | http://arxiv.org/abs/1708.07736v1 | {
"authors": [
"D. R. Gulevich",
"D. Yudin",
"D. V. Skryabin",
"I. V. Iorsh",
"I. A. Shelykh"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170825134854",
"title": "Topological Edge Solitons in Polaritonic Lattice"
} |
LGR LGRLGR [1]#1 LGR[1]126#1 | http://arxiv.org/abs/1708.08331v4 | {
"authors": [
"Ali Övgün",
"Izzet Sakalli",
"Joel Saavedra"
],
"categories": [
"physics.gen-ph"
],
"primary_category": "physics.gen-ph",
"published": "20170824235859",
"title": "Quasinormal Modes of a Schwarzschild Black Hole Immersed in an Electromagnetic Universe"
} |
Reinforcement Mechanism Design for e-commerce Yiwei Zhang December 30, 2023 ============================================= The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distributionover locality-sensitive hash functions that partition space. For a collection of n points, after preprocessing, the query time is dominated by O(n^ρlog n) evaluations of hash functions fromand O(n^ρ) hash table lookups and distance computations where ρ∈ (0,1) is determined by the locality-sensitivity properties of .It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to O(log^2 n), leaving the query time to be dominated by O(n^ρ) distance computations and O(n^ρlog n) additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework, making it a viable replacement in practice.Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to O(n^ρ). § INTRODUCTION The approximate near neighbor problem is the problem of preprocessing a collection P of n points in a space (X, ) into a data structure such that, for parameters r_1 < r_2 and given a query point q ∈ X, if there exists a point x ∈ P with (q, x) ≤ r_1, then the data structure is guaranteed to return a point x' ∈ P such that (q, x') < r_2.Indyk and Motwani <cit.> introduced a general framework for constructing solutions to the approximate near neighbor problem using a technique known as locality-sensitive hashing (LSH).The framework takes a distribution over hash functionswith the property that near points are more likely to collide under a random h ∼.During preprocessing a number of locality-sensitive hash functions are sampled fromand used to hash the points of P into buckets.The query algorithm evaluates the same hash functions on the query point and looks into the associated buckets to find an approximate near neighbor.The locality-sensitive hashing framework of Indyk and Motwani has had a large impact in both theory and practice (see surveys <cit.> and <cit.> for an introduction), and many of the best known solutions to the approximate near neighbor problem in high-dimensional spaces, such as Euclidean space <cit.>, the unit sphere under inner product similarity <cit.>, and sets under Jaccard similarity <cit.> come in the form of families of locality-sensitive hash functions that can be plugged into the Indyk-Motwani LSH framework.Let (X, ) be a distance space and letbe a distribution over functions hX → R. We say thatis (r_1, r_2, p_1, p_2)-sensitive if for x, y ∈ X and h ∼ we have that: * If (x, y) ≤ r_1 then [h(x) = h(y)] ≥ p_1.* If (x, y) ≥ r_2 then [h(x) = h(y)] ≤ p_2.The Indyk-Motwani framework takes a (r_1, r_2, p_1, p_2)-sensitive familyand constructs a data structure that solves the approximate near neighbor problem for parameters r_1 < r_2 with some positive constant probability of success.We will refer to this randomized approximate version of the near neighbor problem as the (r_1, r_2)-near neighbor problem, where we require queries to succeed with probability at least 1/2 (see Definition <ref>). To simplify the exposition we will assume throughout the introduction, unless otherwise stated, that 0 < p_1 < p_2 < 1 are constant, that a hash function h ∈ can be stored in n / log n words of space, and for ρ = log(1/p_1) / log(1/p_2) ∈ (0,1) that a point x ∈ X can be stored in O(n^ρ) words of space. The assumption of a constant gap between p_1 and p_2 allows us to avoid performing distance computations by instead using the 1-bit sketching scheme of Li and König <cit.> together with the familyto approximate distances (see Section <ref> for details). In the remaining part of the paper we will state our results without any such assumptions to ensure, for example, that our results hold in the important case where p_1, p_2 may depend on n or the dimensionality of the space <cit.>. Letbe (r_1, r_2, p_1, p_2)-sensitive and let ρ = log(1/p_1)/log(1/p_2), then there exists a solution to the (r_1, r_2)-near neighbor problem using O(n^1+ρ) words of space and with query time dominated by O(n^ρlog n) evaluations of functions from . The query time of the Indyk-Motwani framework is dominated by the number of evaluations of locality-sensitive hash functions.To make matters worse, almost all of the best known and most widely used locality-sensitive families have an evalution time that is at least linear in the dimensionality of the underlying space <cit.>. Significant effort has been devoted to the problem of reducing the evaluation complexity of locality-sensitive hash families <cit.>, while the question of how many independent locality-sensitive hash functions are actually needed to solve the (r_1, r_2)-near neighbor problem has received relatively little attention <cit.>.This paper aims to bring attention to, strengthen, generalize, and simplify results that reduce the number of locality-sensitive hash functions used to solve the (r_1, r_2)-near neighbor problem. In particular, we will extract a general framework from a technique introduced by Dahlgaard et al. <cit.> in the context of set similarity search under Jaccard similarity, showing that the number of locality-sensitive hash functions can be reduced to O(log^2 n) in general.We further show how to reduce the word-RAM complexity of the general framework from O(n^ρlog n) to O(n^ρ) by combining techniques from Dahlgaard et al. and Andoni and Indyk <cit.>. Reducing the number of locality-sensitive hash functions allows us to spend time O(n^ρ / log^2 n) per hash function evaluation without increasing the overall complexity of the query algorithm — something which is particularly useful in Euclidean space where the best known LSH upper bounds offer a tradeoff between the ρ-value that can be achieved and the evaluation complexity of the locality-sensitive hash function <cit.>. §.§ Related work Indyk-Motwani.The Indyk-Motwani framework uses L = O(n^ρ) independent partitions of space, each formed by overlaying k = O(log n) random partitions induced by k random hash functions from a locality-sensitive family . The parameter k is chosen such that a random partition has the property that a pair of points x,y ∈ X with (x, y) ≤ r_1 has probability n^-ρ of ending up in the same part of the partition, while a pair of points with (x, y) ≥ r_2 has probability n^-1 of colliding.By randomly sampling L = O(n^ρ) such partitions we are able to guarantee that a pair of near points will collide with constant probability in at least one of them.Applying these L partitions to our collection of data points P and storing the result of each partition of P in a hash table we obtain a data structure that solves the (r_1, r_2)-near neighbor problem as outlined in Theorem <ref> above.Section <ref> and <ref> contains a more complete description of LSH-based frameworks and the Indyk-Motwani framework.Andoni-Indyk.As previously mentioned, many locality-sensitive hash functions happen to have a super-constant evaluation time.This motivated Andoni and Indyk to introduce a replacement to the Indyk-Motwani framework in a paper on substring near neighbor search <cit.>. The key idea is to re-use hash functions from a small collection of size m ≪ L by forming all combinations of mt hash functions.This technique is also known as tensoring and has seen some use in the work on alternative solutions to the approximate near neighbor problem, in particular the work on locality-sensitive filtering <cit.>. By applying the tensoring technique the Andoni-Indyk framework reduces the number of hash functions to O(exp(√(ρlog n loglog n))) = n^o(1) as stated in Theorem <ref>. Letbe (r_1, r_2, p_1, p_2)-sensitive and let ρ = log(1/p_1)/log(1/p_2), then there exists a solution to the (r_1, r_2)-near neighbor problem using O(n^1+ρ) words of space and with query time dominated by O(exp(√(ρlog n loglog n))) evaluations of functions fromand O(n^ρ) other word-RAM operations.The paper by Andoni and Indyk did not state this result explicitly as a theorem in the same form as the Indyk-Motwani framework; the analysis made some implicit restrictive assumptions on p_1, p_2 and ignored integer constraints.Perhaps for these reasons the result does not appear to have received much attention, although it has seen some limited use in practice <cit.>. In Section <ref> we present a slightly different version of the Andoni-Indyk framework together with an analysis that satisfies integer constraints, providing a more accurate assessment of the performance of the framework in the general, unrestricted case. Dahlgaard-Knudsen-Throup.The paper by Dahlgaard et al. <cit.> introduced a different technique for constructing the L hash functions/partitions from a smaller collection of m hash functions from .Instead of forming all combinations of subsets of size t as the Andoni-Indyk framework they instead sample k hash functions from the collection to form each of the L partitions. The paper focused on a particular application to set similarity search under Jaccard similarity, and stated the result in terms of a solution to this problem.In Section <ref> we provide a simplified and tighter analysis to yield a general framework: Letbe (r_1, r_2, p_1, p_2)-sensitive and let ρ = log(1/p_1)/log(1/p_2), then there exists a solution to the (r_1, r_2)-near neighbor problem using O(n^1+ρ) words of space and with query time dominated by O(log^2 n) evaluations of functions fromand O(n^ρlog n) other word-RAM operations. The analysis of <cit.> indicates that the Dahlgaard-Knudsen-Thorup framework, when compared to the Indyk-Motwani framework, would use at least 50 times as many partitions (and a corresponding increase in the number of hash table lookups and distance computations) to solve the (r_1, r_2)-near neighbor problem with success probability at least 1/2.Using elementary tools, the analysis in this paper shows that we only have to use twice as many partitions as the Indyk-Motwani framework to obtain the same guarantee of success.Number of hash functions in practice.To provide some idea of what the number of hash functions H used by the different frameworks would be in practice, Figure <ref> shows the value of log_2 H that is obtained by actual implementations of the Indyk-Motwani (IM), Andoni-Indyk (AI), and Dahlgaard-Knudsen-Thorup (DKT) frameworks according to the analysis in Section <ref> for p_1 = 1/2 and every value of 0 < p_2 < 1/2 for a solution to the (r_1, r_2)-near neighbor problem on a collection of n = 2^30 points.Figure <ref> reveals that the number of hash functions used by the Indyk-Motwani framework exceeds 2^30, the size of the collection of points P, as p_2 approaches p_1.In addition, locality-sensitive hash functions used in practice such as Charikar's SimHash <cit.> and p-stable LSH <cit.> have evaluation time O(d) for points in ℝ^d.These two factors might help explain why a linear scan over sketches of the entire collection of points is a popular approach to solve the approximate near neighbor problem in practice <cit.>. The Andoni-Indyk framework reduces the number of hash functions by several orders of magnitude, and the Dahlgaard-Knudsen-Thorup framework presents another improvement of several orders of magnitude.Since the word-RAM complexity of the DKT framework matches the the number of hash functions used by the IM framework, the gap between the solid line (DKT) and the dotted line (IM) gives some indication of the time we can spend on evaluating a single hash function in the DKT framework without suffering a noticeable increase in the query time.§.§ Contribution Improved word-RAM complexity.In addition to our work on the Andoni-Indyk and Dahlgaard-Knudsen-Thorup frameworks as mentioned above, we show how the word-RAM complexity of the DKT framework can be reduced by a logarithmic factor.The solution is a simple combination of the DKT sampling technique and the AI tensoring technique:First we use the DKT sampling technique twice to construct two collections of √(L) partitions. Then we use the AI tensoring technique to form L = √(L)×√(L) pairs of partitions from the two collections.Below we state our main Theorem <ref> in its general form where we make no implicit assumptions about(p_1 and p_2 are not assumed to be constant and can depend on for example n) or about the complexity of storing a point or a hash function, or computing the distance between pairs of points in the space (X, ). Letbe (r_1, r_2, p_1, p_2)-sensitive and let ρ = log(1/p_1) / log(1/p_2), then there exists a solution to the (r_1, r_2)-near neighbor with the following properties: * The query complexity is dominated by O(log_1/p_2^2(n)/p_1) evaluations of functions from , O(n^ρ) distance computations, and O(n^ρ / p_1) other word-RAM operations. * The solution uses O(n^1 + ρ /p_1) words of space in addition to the space required to store the data and O(log_1/p_2^2(n)/p_1) functions from . Under the same simplifying assumptions used in the statements of Theorem <ref>, <ref>, and <ref>, our main Theorem <ref> can be stated as Theorem <ref> with the word-RAM complexity reduced by a logarithmic factor to O(n^ρ).This improvement in the word-RAM complexity comes at the cost of a (rather small) constant factor increase in the number of hash functions, lookups, and distance computations compared to the DKT framework. By varying the size m of the collection of hash functions fromand performing independent repetitions we can obtain a tradeoff between the number of hash functions and the number of lookups. In Section <ref> we remark on some possible improvements in the case where p_2 is large.Distance sketching using LSH.Finally, we combine Theorem <ref> with the 1-bit sketching scheme of Li and König <cit.> where we use the locality-sensitive hash family to create sketches that allow us to leverage word-level parallelism and avoid direct distance computations.This sketching technique is well known and has been used before in combination with LSH-based approximate similarity search <cit.>, but we believe there is some value in the simplicity of the analysis and in a clear statement of the combination of the two results as given in Theorem <ref>, for example in the important case where 0 < p_2 < p_1 < 1 are constant. Letbe (r_1, r_2, p_1, p_2)-sensitive and let ρ = log(1/p_1) / log(1/p_2), then there exists a solution to the (r_1, r_2)-near neighbor with the following properties: * The complexity of the query operation is dominated by O(log^2(n)/(p_1 - p_2)^2) evaluations of hash functions fromand O(n^ρ/(p_1 - p_2)^2) other word-RAM operations. * The solution uses O(n^1 + ρ/p_1 + n/(p_1 - p_2)^2) words of space in addition to the space required to store the data and O(log^2(n)/(p_1 - p_2)^2) hash functions from .§ PRELIMINARIES Problem and dynamization.We begin by defining the version of the approximate near neighbor problem that the frameworks presented in this paper will be solving:Let P ⊆ X be a collection of |P| = n points in a distance space (X, ). A solution to the (r_1, r_2)-near neighbor problem is a data structure that supports the following query operation: Given a query point q ∈ X, if there exists a point x ∈ P with (q, x) ≤ r_1,then, with probability at least 1/2, return a point x' ∈ P such that (q, x') < r_2.We aim for solutions with a failure probability that is upper bounded by 1/2. The standard trick of using η independent repetitions of the data structure allows us to reduce the probability of failure to 1/2^η. For the sake of simplicity we restrict our attention to static solutions, meaning that we do not concern ourselves with the complexity of updates to the underlying set P, although it is simple to modify the static solutions presented in this paper to dynamic solutions where the update complexity essentially matches the query complexity <cit.> LSH powering.The Indyk-Motwani framework and the Andoni-Indyk framework will make use of the following standard powering technique described in the introduction as “overlaying partitions”. Let k ≥ 1 be an integer and letdenote a locality-sensitive family of hash functions as in Definition <ref>.We will use the notation ^k to denote the distribution over functions h'X → R^k whereh'(x) = (h_1(x), …, h_k(x))and h_1, …, h_k are sampled independently at random from . It is easy to see that ^k is (r_1, r_2, p_1^k, p_2^k)-sensitive. To deal with some special cases we define ^0 to be the family consisting of a single constant function.Model of computation.We will work in the standard word-RAM model of computation <cit.> with a word length of Θ(log n) bits where n denotes the size of the collection P to be searched in the (r_1, r_2)-near neighbor problem. During the preprocessing stage of our solutions we will assume access to a source of randomness that allows us to sample independently from a familyand to seed pairwise independent hash functions <cit.>.The latter can easily be accomplished by augmenting the model with an instruction that generates a uniformly random word in constant time and using that to seed the tables of a Zobrist hash function <cit.>.§ FRAMEWORKS Overview.We will describe frameworks that take as input a (r_1, r_2, p_1, p_2)-sensitive familyand a collection P of n points and constructs a data structure that solves the (r_1, r_2)-near neighbor problem. The frameworks described in this paper all use the same high-level technique of constructing L hash functions g_1,…,g_L that are used to partition space such that a pair of points x, y with (x, y) ≤ r_1 will end up in the same part of one of the L partitions with probability at least 1/2.That is, for x, y with (x, y) ≤ r_1 we have that [∃ l ∈ [L]g_l(x) = g_l(y)] ≥ 1/2 where [L] is used to denote the set {1,2,…,L}. At the same time we ensure that the expected number of collisions between pairs of points x,y with (x, y) ≥ r_2 is at most one in each partition.Preprocessing and queries.During the preprocessing phase, for each of the L hash functions g_1, …,g_L we compute the partition of the collection of points P induced by g_l and store it in a hash table in the form of key-value pairs (z, { x ∈ P | g_l(x) = z }). To reduce space usage we store only a single copy of the collection P and store references to P in our L hash tables. To guarantee lookups in constant time we can use the perfect hashing scheme by Fredman et al. <cit.> to construct our hash tables. We will assume that hash values z = g_l(x) fit into O(1) words.If this is not the case we can use universal hashing <cit.> to operate on fingerprints of the hash values.We perform a query for a point q as follows: for l = 1, …, L we compute g_l(q),retrieve the set of points { x ∈ P | g_l(x) = g_l(q) }, and compute the distance between q and each point in the set. If we encounter a point x' with (q, x') < r_2 then we return x' and terminate. If after querying the L sets no such point is encountered we return a special symbol ∅ and terminate.We will proceed by describing and analyzing the solutions to the (r_1, r_2)-near neighbor problem for different approaches to sampling, storing, and computing the L hash functions g_1, …, g_L, resulting in the different frameworks as mentioned in the introduction. §.§ Indyk-Motwani To solve the (r_1, r_2)-near neighbor problem using the Indyk-Motwani framework we sample L hash functions g_1, …,g_L independently at random from the family ^k where we set k = ⌈log(n) / log(1/p_2) ⌉ and L = ⌈ (ln 2)/p_1^k ⌉. Correctness of the data structure follows from the observation that the probability that a pair of points x, y with (x,y) ≤ r_1 does not collide under a randomly sampled g_l ∼^k is at most 1 - p_1^k.We can therefore upper bound the probability that a near pair of points does not collide under any of the hash functions by (1-p_1^k)^L ≤exp(-p_1^k L) ≤ 1/2 using a standard bound stated as Lemma <ref> in Appendix <ref>.In the worst case, the query operation computes L hash functions from ^k corresponding to Lk hash functions from .For a query point q the expected number of points x' ∈ P with (q, x') ≥ r_2 that collide with q under a randomly sampled g_l ∼^k is at most np_2^k ≤ np_2^log(n) / log (1/p_2) = 1. It follows from linearity of expectation that the total expected number of distance computations during a query is at most L. The result is summarized in Theorem <ref> from which the simplified Theorem <ref> follows.Given a (r_1, r_2, p_1, p_2)-sensitive familywe can construct a data structure that solves the (r_1, r_2)-near neighbor problem such that for k = ⌈log(n) / log(1/p_2) ⌉ and L = ⌈ (ln 2)/p_1^k ⌉ the data structure has the following properties: * The query operation uses at most Lk evaluations of hash functions from , expected L distance computations, and O(Lk) other word-RAM operations.* The data structure uses O(nL) words of space in addition to the space required to store the data and Lk hash functions from . Theorem <ref> gives a bound on the expected number of distance computations while the simplified version stated in Theorem <ref> uses Markov's inequality and independent repetitions to remove the expectation from the bound by treating an excessive number of distance computations as a failure. §.§ Andoni-Indyk In 2006 Andoni and Indyk, as part of a paper on the substring near neighbor problem, introduced an improvement to the Indyk-Motwani framework that reduces the number of locality-sensitive hash functions <cit.>. Their improvement comes from the use of a technique that we will refer to as tensoring: setting the hash functions g_1, …, g_L to be all t-tuples from a collection of m functions sampled from ^k/t where m ≪ L. The analysis in <cit.> shows that by setting m = n^ρ/t and repeating the entire scheme t! times, the total number of hash functions can be reduced to O(exp(√(ρlog n loglog n))) when setting t = √(ρlog n/loglog n). This analysis ignores integer constraints on t, k, and m, and implicitly place restrictions on p_1 and p_2 in relation to n (e.g. 0 < p_2 < p_1 < 1 are constant). We will introduce a slightly different scheme that takes into account integer constraints and analyze it without restrictions on the properties of .Assume that we are given a (r_1, r_2, p_1, p_2)-sensitive family . Let η, t, k_1, k_2, m_1, m_2 be non-negative integer parameters. Each of the L hash functions g_1, …, g_L will be formed by concatenating one hash function from each of t collections of m_1 hash functions from ^k_1 and concatenating a last hash function from a collection of m_2 hash functions from ^k_2. We take all m_1^t m_2 hash functions of the above form and repeat η times for a total of L = η m_1^t m_2 hash functions constructed from a total of H = η(m_1 k_1 t + m_2 k_2) hash functions from . In Appendix <ref> we set parameters, leaving t variable, and provide an analysis of this scheme, showing that L matches the Indyk-Motwani framework bound of O(1/p_1^k) up to a constant where k = ⌈log(n) / log(1/p_2) ⌉ as in Theorem <ref>.Setting t.It remains to show how to set t to obtain a good bound on the number of hash functions H. Note that in practice we can simply set t = _t H by trying t = 1,…,k.If we ignore integer constraints and place certain restrictions ofas in the original tensoring scheme by Andoni and Indyk we want to set t to minimize the expression t^t n^ρ/t.This minimum is obtained when setting t such that t^2 log t = ρlog n. We therefore cannot do much better than setting t = √(ρlog(n) / loglog n) which gives the bound H = O(exp(√(ρlog(n) loglog n))) as shown in <cit.>.To allow for easy comparison with the Indyk-Motwani framework without placing restrictions onwe set t = ⌈√(k)⌉, resulting in Theorem <ref>. Given a (r_1, r_2, p_1, p_2)-sensitive familywe can construct a data structure that solves the (r_1, r_2)-near neighbor problem such that for k = ⌈log(n) / log(1/p_2) ⌉, H = k(√(k)/p_1)^√(k), and L = ⌈ 1/p_1^k ⌉ the data structure has the following properties: * The query operation uses O(H) evaluations of functions from , O(L) distance computations, and O(L + H) other word-RAM operations. * The data structure uses O(nL) words of space in addition to the space required to store the data and O(H) hash functions from . Thus, compared to the Indyk-Motwani framework we have gone from using O(k(1/p_1)^k) locality-sensitive hash functions to O(k(√(k)/p_1)^√(k)) locality-sensitive hash functions.Figure <ref> shows the actual number of hash functions of the revised version of the Andoni-Indyk scheme as analyzed in Appendix <ref> when t is set to minimize H. §.§ Dahlgaard-Knudsen-Thorup In a recent paper Dahlgaard et al. <cit.> introduce a different technique for reducing the number of locality-sensitive hash functions. The idea is to construct each hash value g_l(x) by sampling and concatenating k hash values from a collection of km pre-computed hash functions from . Dahlgaard et al. applied this technique to provide a fast solution the the approximate near neighbor problem for sets under Jaccard similarity. In this paper we use the same technique to derive a general framework solution that works with every family of locality-sensitive hash functions, reducing the number of locality-sensitive hash functions compard to the Indyk-Motwani and Andoni-Indyk frameworks.Let [n] denote the set of integers {1,2,…,n}. For i ∈ [k] and j ∈ [m] let h_i,j∼ denote a hash function in our collection. To sample from the collection we use k pairwise independent hash functions <cit.> of the form f_i[L] → [m] and setg_l(x) = (h_1,f_1(l)(x), …, h_k,f_k(l)(x)).To show correctness of this scheme we will use make use of an elementary one-sided version of Chebyshev's inequality stating that for a random variable Z with mean μ > 0 and variance σ^2 < ∞ we have that [Z ≤ 0] ≤σ^2 / (μ^2 + σ^2).For completeness we have included the proof of this inequality in Lemma <ref> in Appendix <ref>. We will apply this inequality to lower bound the probability that there are no collisions between close pairs of points. For two points x and y let Z_l = {g_l(x) = g_l(y)} so that Z = ∑_l = 1^L Z_l denotes the sum of collisions under the L hash functions.To apply the inequality we need to derive an expression for the expectation and the variance of the random variable Z. Let p = _h ∼[h(x) = h(y)] then by linearity of expectation we have that μ = [Z] = L p^k. To bound σ^2 = [Z^2] - μ^2 we proceed by bounding [Z^2] where we note that Z_l = Π_i = 1^k Y_l, i for Y_l, i = 1{h_i,f_i(l)(x) = h_i,f_i(l)(x)} and make use of the independence between Y_l,i and Y_l', i' for i ≠ i'.[Z^2]= ∑_l, l' ∈ [L] l ≠ l'[Z_l Z_l'] + ∑_l = 1^L [Z_l] = (L^2 - L) [Z_l Z_l'] + μ≤ L^2 [ Π_i = 1^k Y_l, i Y_l', i] + μ= L^2 ( [Y_l, i Y_l', i] )^k + μ.We have that [Y_l, i Y_l', i] = [f_i(l) = f_i(l')] p + [f_i(l) ≠ f_i(l')] p^2 = (1/m)p + (1-1/m)p^2 which follows from the pairwise independence of f_i.Let ε > 0 and set m = ⌈1-p_1/p_1k/ln(1+ε)⌉ then for p ≥ p_1 we have that ( [Y_l, i Y_l', i] )^k ≤ (1 + ε)p^2k. This allows us to bound the variance of Z by σ^2 ≤εμ^2 + μ resulting in the following lower bound on the probability of collision between similar points.For ε > 0 let m ≥⌈1-p_1/p_1k/ln(1+ε)⌉, then for every pair of points x, y with (x, y) ≤ r_1 we have that [∃ l ∈ [L]g_l(x) = g_l(y)] ≥1 + εμ/1 + (1 + ε)μ.By setting ε = 1/4 and L = ⌈ (2 ln(2))/p_1^k ⌉ we obtain an upper bound on the failure probability of 1/2. Setting the size of each of the k collections of pre-computed hash values to m = ⌈ 5k/p_1 ⌉ is sufficient to yield the following solution to the (r_1, r_2)-near neighbor problem where provide exact bounds on the number of lookups L and hash functions H: Given a (r_1, r_2, p_1, p_2)-sensitive familywe can construct a data structure that solves the (r_1, r_2)-near neighbor problem such that for k = ⌈log(n) / log(1/p_2) ⌉, H = k ⌈ 5k / p_1 ⌉, and L = ⌈ (2 ln(2))/p_1^k ⌉ the data structure has the following properties: * The query operation uses at most H evaluations of hash functions from , expected L distance computations, and O(Lk) other word-RAM operations.* The data structure uses O(nL) words of space in addition to the space required to store the data and H hash functions from .Compared to the Indyk-Motwani framework we have reduced the number of locality-sensitive hash functions H from O(k (1/p_1)^k) to O(k^2 / p_1) at the cost of using twice as many lookups.To reduce the number of lookups further we can decrease ε and perform several independent repetitions. This comes at the cost of an increase in the number of hash functions H.§ REDUCING THE WORD-RAM COMPLEXITYOne drawback of the DKT framework is that each hash value g_l(x) still takes O(k) word-RAM operations to compute, even after the underlying locality-sensitive hash functions are known.This results in a bound on the total number of additional word-RAM operations of O(Lk). We show how to combine the DKT universal hashing technique with the AI tensoring technique to ensure that the running time is dominated by O(L) distance computations and O(H) hash function evaluations.The idea is to use the DKT scheme to construct two collections of respectively L_1 and L_2 hash functions, and then to use the AI tensoring approach to form g_1, …, g_L as the L = L_1 × L_2 combinations of functions from the two collections. The number of lookups can be reduced by applying tensoring several times in independent repetitions, but for the sake of simplicity we use a single repetition. For the usual setting of k = ⌈log(n) / log(1/p_2) ⌉ let k_1 =⌈ k/2 ⌉ and k_2 = ⌊ k/2 ⌋. Set L_1 = ⌈ 6 (1/p_1)^k_1⌉ and L_2 = ⌈ 6 (1/p_1)^k_2⌉. According to Lemma <ref> if we set ε = 1/6 the success probability of each collection is at least 3/4 and by a union bound the probability that either collection fails to contain a colliding hash function is at most 1/2. This concludes the proof of our main Theorem <ref>. §.§ Sketching The theorems of the previous section made no assumptions on the word-RAM complexity of distance computations and instead stated the number of distance computations as part of the query complexity.We can use a (r_1, r_2, p_1, p_2)-sensitive familyto create sketches that allows us to efficiently approximate the distance between pairs of points, provided that the gap between p_1 and p_2 is sufficiently large.In this section we will re-state the results of Theorem <ref> when applying the familyto create sketches using the 1-bit sketching scheme of Li and König <cit.>. Let b be a positive integer denoting the length of the sketches in bits. The advantage of this scheme is that we can use word level parallelism to evaluate a sketch of b bits in time O(b/log n) in our word-RAM model with word length Θ(log n). For i = 1, …, b let h_iX → R denote a randomly sampled locality-sensitive hash function fromand let f_iR →{0,1} denote a randomly sampled universal hash function. We let s(x) ∈{0,1}^b denote the sketch of a point x ∈ X where we set the ith bit of the sketch s(x)_i = f_i(h(x)). For two points x, y ∈ X the probability that they agree on the ith bit is 1 if the points collide under h_i and 1/2 otherwise. [s(x)_i = s(y)_i] = [h_i(x) = h_i(y)] + (1 - [h_i(x) = h_i(y)])/2 = (1 + [h_i(x) = h_i(y)])/2.We will apply these sketches during our query procedure instead of direct distance computations when searching through the points in the L buckets, comparing them to our query point q. Let λ∈ (0,1) be a parameter that will determine whether we report a point or not. For sketches of length b we will return a point x if s(q) - s(x)_1 > λ b. An application of Hoeffiding's inequality gives us the following properties of the sketch: Letbe a (r_1, r_2, p_1, p_2)-sensitive family and let λ = (1+p_2)/2 + (p_1-p_2)/4, then for sketches of length b ≥ 1 and for every pair points x, y ∈ X: * If (x ,y) ≤ r_1 then [s(x) - s(y)_1 ≤λ b] ≤ e^b(p_1 - p_2)^2 / 8. * If (x ,y) ≥ r_2 then [s(x) - s(y)_1 > λ b] ≤ e^b(p_1 - p_2)^2 / 8. If we replace the exact distance computations with sketches we want to avoid two events:Failing to report a point with (q,x) ≤ r_1 and reporting a point x with (q, x) ≥ r_2. By setting b = O(ln(n) / (p_1 - p_2)^2) and applying a union bound over the n events that the sketch fails for a point in our collection P we obtain Theorem <ref>. § THE NUMBER OF HASH FUNCTIONS IN CORNER CASES When the collision probabilities of the (r_1, r_2, p_1, p_2)-sensitive familyare close to one we get the behavior displayed in Figure <ref> where we have set p_1 = 0.9.Here it may be possible to reduce the number of hash functions by applying the DKT framework to the family ^τ for some positive integer τ. That is, instead of applying the DKT technique directly towe first apply the powering trick to produce the family ^τ. The number of locality-sensitive hash functions fromused by the DKT framework is given by H = O( (log(n) / log(1/p_2))^2 / p_1). If we instead use the family ^τ the expression becomes H = O( τ(log(n) / log(1/p_2^τ))^2 / p_1^τ) = O((log(n) / log(1/p_2))^2 / τ p_1^τ).Ignoring integer constraints, the value of τ that maximizes τ p_1^τ, thereby minimizing H, is given by τ = 1 / ln(1/p_1). Discretizing, the resulting number of hash functions when setting τ = ⌈ 1 / ln(1/p_1)⌉ is given by H = O(ρ (log n)^2 / (p_1 log(1/p_2))). For constant ρ and large p_2 this reduces the number of hash functions by a factor 1/log(1/p_2).The behavior for small values of p_1 is displayed in Figure <ref> where we have set p_1 = 0.1. § CONCLUSION AND OPEN PROBLEMS We have shown that there exists a simple and general framework for solving the (r_1, r_2)-near neighbor problem using only few locality-sensitive hash functions and with a reduced word-RAM complexity matching the number of lookups. The analysis in this paper indicates that the performance of the Dahlgaard-Knudsen-Thorup framework is highly competitive compared to the Indyk-Motwani framework in practice, especially when locality-sensitive hash functions are expensive to evaluate, as is often the case.An obvious open problem is the question of whether the number of locality-sensitive hash functions can be reduced even below O(k^2 / p_1). Another possible direction for future research would be to obtain similar framework results in the context of solutions to the (r_1, r_2)-near neighbor problem that allow for space-time tradeoffs <cit.>.Acknowledgements. I want to thank Rasmus Pagh commenting on an earlier version of this manuscript and for making me aware of the application of the tensoring technique in <cit.> that led me to the Andoni-Indyk framework <cit.>.§ INEQUALITIES We make use of the following standard inequalities for the exponential function.See <cit.> for more details. Let n, t ∈ℝ such that n ≥ 1 and |t| ≤ n then e^-t(1-t^2 /n) ≤ (1 - t/n)^n ≤ e^-t. For t ≥ 0 we have that e^-t≤ 1 - t + t^2 / 2. We make use of a one-sided version of Chebyshev's inequality to show correctness of the Dahlgaard-Knudsen-Thorup LSH framework. Let Z be a random variable with [Z] = μ > 0 and [Z] = σ^2 < ∞ then [Z ≤ 0] ≤σ^2/(μ^2 + σ^2). For every s ∈ℝ we have that [Z ≤ 0] = [-(Z - μ) + s ≥μ + s] ≤[(-(Z - μ) + s)^2 ≥ (μ + s)^2].Next we apply Markov's inequality [(-(Z - μ) + s)^2 ≥ (μ + s)^2] ≤[(-(Z - μ) + s)^2]/ (μ + s)^2 = (σ^2 + s^2)/ (μ + s)^2Set s = σ^2 / μ and use that σ^2 = sμ to simplify (σ^2 + s^2)/ (μ + s)^2 = (sμ + s^2)/ (μ + s)^2 = σ^2 / (μ^2 + σ^2).To analyze the 1-bit sketching scheme by Li and König we make use of Hoeffding's inequality: Let X_1, X_2, …, X_n be independent random variables satisfying 0 ≤ X_i ≤ 1 for i ∈ [n]. Define X̅ = (X_1 + X_2 + … + X_n)/n and μ = [X̅], then: - For 0 < ε < 1 - μ we have that [X̅ - μ≥ε] ≤ e^-2nε^2. - For 0 < ε < μ we have that [X̅ - μ≤ - ε] ≤ e^-2nε^2. § ANALYSIS OF THE ANDONI-INDYK FRAMEWORK Let φ denote the probability that a pair of points x, y with (x,y) ≤ r_1 collide in a single repetition of the scheme. A collision occurs if and only if there there exists at least one hash function in each of the underlying t + 1 collections where the points collide. It follows thatφ = (1 - (1-p_1^k_1)^m_1)^t (1 - (1-p_1^k_2)^m_2).To guarantee a collision with probability at least 1/2 it suffices to set η = ⌈ln(2) / φ⌉.We will proceed by analyzing this scheme where we let t ≥ 1 be variable and set parameters as followers:k= ⌈log(n) / log(1/p_2) ⌉ k_1= ⌊ k/t ⌋ k_2= k - tk_1 m_1= ⌈ 1/ t p_1^k_1⌉ m_2= ⌈ 1/ p_1^k_2⌉ η = ⌈ln(2) / φ⌉.To upper bound L we begin by lower bounding φ. The second part of φ can be lower bounded using Lemma <ref> to yield (1 - (1-p_1^k_2)^m_2) ≥ 1 - 1/e. To lower bound (1 - (1-p_1^k_1)^m_1)^t we first note that in the case where p_1^k_1 > 1/t we have m_1 = 1 and the expression can be lower bounded by p_1^k_1 t = (p_1^k_1 m_1)^t ≥(p_1^k_1 m_1)^t / 2e. The same lower bound holds in the case there t = 1. In the case where p_1^k_1≤ 1/t and t ≥ 2 we make use of Lemma <ref> and <ref> to derive the lower bound. 1 - (1-p_1^k_1)^m_1 ≥ 1 - e^-p_1^k_1 m_1≥ 1 - (1 - p_1^k_1 m_1 + (p_1^k_1 m_1)^2/2) ≥ p_1^k_1 m_1 (1 - p_1^k_1(1/t p_1^k_1 + 1)/2) ≥ p_1^k_1 m_1 (1 - 1/t).Using the bound (p_1^k_1 m_1 (1 - 1/t))^t ≥ (p_1^k_1 m_1)^t / 2e we have that φ≥ (p_1^k_1 m_1)^t / 4e ≥ (1/t)^t / 4e.We can then bound the number of lookups and the expected number of distance computationsL = η m_1^t m_2 ≤ (4e / (p_1^k_1 m_1)^t + 1) m_1^t (1/p_1^k_2 + 1) ≤ 16e (1/p_1^k).Note that this matches the upper bound of the Indyk-Motwani LSH framework up to a constant factor.To bound the number of hash functions fromwe use that k_1 ≤ k/t ≤ k and k_2 < t.H = η (m_1 k_1 t + m_2 k_2) ≤ 8e t^t (k/t p_1^k/t + t-1/p_1^t-1). | http://arxiv.org/abs/1708.07586v2 | {
"authors": [
"Tobias Christiani"
],
"categories": [
"cs.DS",
"E.1; H.3.3"
],
"primary_category": "cs.DS",
"published": "20170825004738",
"title": "Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search"
} |
Cubulated moves for 2-knots.2010 Mathematics Subject Classification.Primary: 57M25. Secondary: 57M27, 57Q45. Key Words: Cubulated moves, Discrete knots, 2-knots.Juan Pablo DíazThis work was partially supported by CONACyT (México), FORDECYT 265667,Gabriela HinojosaThis work was partially supported by CONACyT (México), CB-2009-129939., Alberto Verjosvky, This work was partially supported by CONACyT (México), CB-2009-129280 andPAPIIT (Universidad Nacional Autónoma de México) #IN 106817. August 21, 2017 ================================================================================================================================================================================================================================================================================================================================================ In this paper, we prove that given two cubical links of dimension two in ℝ^4, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.§ INTRODUCTION In <cit.> it was shown that any smooth knot K^n:𝕊^n↪ℝ^n+2 can be deformed isotopically into the n-skeleton of the canonical cubulation ofℝ^n+2 and this isotopic copy is called cubical n-knot. In particular, every smooth 1-knot𝕊^1⊂ℝ^3 is isotopic to a cubical knot. There are two types of elementary “cubulated moves”. The first one (M1) is obtained by dividing each cube of the original cubulation of ℝ^3 into m^3 cubes,which means that each edge of the knot is subdivided into m equal segments. The second one (M2) consists in exchanging a connected set of edges in a face of thecubulation with the complementary edges in that face. If two cubical knots K_1 and K_2 are such that we can convert K_1 into K_2 using a finite sequence of cubulated moves then we say that they are equivalent via cubulated moves and is denoted byK_1c∼ K_2. In <cit.> it wasproved the following: Given two cubical knots K_1 and K_2in ℝ^3, K_1≅ K_2 ≅𝕊^1,they are isotopic if and only if K_1 is equivalent to K_2 by a finite sequence of cubulated moves; i.e., K_1∼K_2K_1c∼ K_2. Theorem 0 is analogous to the Reidemeister moves of classical tame knots for cubical knots. Notice that cubulated moves can be extended to cubical 2-knotsin a natural way:The first one (M1) is obtained by dividing each hypercube of the original cubulation ofℝ^4 into m^4 hypercubes, and the second one (M2) consists in exchanging a connected set of squared faces homeomorphic to adisk 𝔻^2 in a cube of the cubulation (or a subdivision of the cubulation) with the complementary faces in that cube.The study of 2-knots in ^4 has been considered by various authors, for instance in<cit.>, <cit.>, <cit.> and <cit.>. Our goal is to extend the Theorem 0 for cubical knots of dimension two: Given two cubical 2-knots K^2_1 and K^2_2in ℝ^4,then they are isotopic if and only if K^2_1 is equivalent to K^2_2 by cubulated moves; i.e.,K^2_1∼K^2_2K^2_1c∼K^2_2. § PRELIMINARIES§.§ Cubulations of ℝ^4The regular hypercubic honeycomb whose Schläfli symbol is {4,3,3,4}, is called a cubulation of ℝ^4. In other words, a cubulation ofℝ^4 is a decomposition into a collection of right-angled 4-dimensional hypercubes {4,3,3} calledcells such that any two are either disjoint or meet in one common k-face of some dimension k. This providesℝ^4 with the structure of a cubical complex whose category is similar to the simplicial category PL. The combinatorial structure of the regular Euclidean honeycomb {4,3,3,4} is the following: around each vertexthere are 8 edges, 24 squares, 32 cubes and 16 hypercubes. Around each edge there are 6 squares, 12 cubes and 8 hypercubes. Around each square there are 4 cubes and 4 hypercubes. Finally around each cube there are2 hypercubes.The canonical hypercubic honeycomb C of ℝ^4 is its decomposition into hypercubes which are the images of the unit hypercube:{4,3,3}=I^4=[0,1]^4={(x_1,x_2,x_3,x_4)∈ℝ^4| 0≤ x_i≤ 1}by translations by vectors with integer coefficients. Then all vertices of C have integer coordinates. Any regular hypercubic honeycomb {4,3,3,4} or cubulation of ℝ^4 is obtained from the canonical cubulation by applying a conformal transformationto the canonical cubulation. Remember that a conformal transformation is of the form x↦λA(x)+a, where λ≠0,a∈ℝ^4, A∈SO(4).The k-skeleton of C, denoted by 𝒮^k, consists of the union of the k-skeletons of the hypercubes inC, i.e., the union of all cubes of dimension k contained in the faces of the 4-cubes inC. We will call the 2-skeleton 𝒮^2 of C the canonical scaffolding of ℝ^4. Notice that all the previous definitions can be extendedin a natural way to ℝ^n+2. §.§ Cubical and smooth 2-knots In classical knot theory, a subset K of a space X is a knot if K is homeomorphic to ap-dimensional sphere𝕊^p embedded in either the Euclidean n-space ℝ^n or the n-sphere 𝕊^n=ℝ^n∪{∞}, where p<n. Two knots K_1, K_2 are equivalent or isotopic if there is a homeomorphism h:X↪ X such that h(K_1)=K_2; in other words (X,K_1)≅ (X,K_2). However, a knot K is sometimes defined to be an embedding K:𝕊^p↪𝕊^n≅ℝ^n∪{∞} (see <cit.>, <cit.>). We shall also find this convenient at times and will use the same symbol to denote either the map K or its image K(𝕊^p) in 𝕊^n. Let K^2 be a 2-dimensional knot in ℝ^4. If K^2 is contained in𝒮^2, we say that K^2 is a cubical knot.If K^2 is a smooth knot and K^2 denotes a cubical knot which is isotopic to K^2, i.e., K^2∼K^2, we say that K^2 is a cubical diagram of the knot K^2.Given two parametrized 2-dimensional smooth knots K^2_1, K^2_2:𝕊^2↪ℝ^4 we say they are smoothly isotopic if there exists asmooth isotopy H:𝕊^2×ℝ→ℝ^4 such that H(x,t)= {[ K^2_1(x),; K^2_2(x), ]. and H( · ,t) is an embedding of 𝕊^2 for all t∈ℝ. We will sayJ^3={(H(x,t),t)∈ℝ^5 | x∈𝕊^2,t∈ℝ} is the isotopic cylinder of K^2_1 and K^2_2.Note that J^3 is a smooth noncompact submanifold of codimension two in ℝ^5. Let p:ℝ^5↪ℝ be the projection onto the last coordinate. Let M be a connected subset of ℝ^5 such that p^-1(c)∩ M (or p|_M^-1(c)) is connected for all c∈ℝ, we say that Mis sliced by connected level sets of p.Observe that there is no restriction on the dimension of M. In<cit.> it was proved the following result. The isotopic cylinder J^3 can be cubulated. In other words, there exists an isotopic copy J^3 of J^3 contained in the 3-skeleton of the canonical cubulation of ℝ^5.Moreover J^3 can be chosen to be sliced by connected level sets of p.Consider J^3 as above; so J^3 is a cubical 3-manifold andthere exist integer numbersm_1 and m_2 and cubical knots K^2_1 and K^2_2 isotopic to K^2_1 and K^2_2, respectively; such that p|_J^3^-1(t)= K^2_1 for all t≤ m_1 and p|_J^3^-1(t)= K^2_2 for all t≥ m_2. §.§ Cubulated movesThe following are the allowed cubulated moves: M1 Subdivision: Given an integer m>1, consider the subcubulationC_m of C by subdividing each k-dimensional cell of C inm^k congruent k-cells in C_m, in particular each hypercube in C is subdivided in m^4 congruent hypercubes in C_m. Moreover as a cubical complex, each k-dimensional face of the 2-knot K^2 is subdivided into m^k congruentk-faces. Since C⊂C_m, then K^2 is contained in the scaffolding S^2_m (the 2-skeleton) of C_m. M2 Face Boundary Moves: Suppose that K^2 is contained in some subcubulation C_m of the canonical cubulation C of ℝ^4.Let Q^4∈C_m be a 4-cube such that A^2=K^2∩ Q^4 contains a 2-face. We can assume, up to applying the elementary (M1)-move if necessary, that A^2 consists of either one,two, or three squares such that it is a connected surface and it is contained in the boundary of a 3-cube F^3⊂ Q^4; in other words A^2 is a cubical disk contained in the boundary of F^3. The boundary ∂ F^3 is divided by ∂ A^2 into two cubulated surfaces, one of them is A^2 and we denote the other by B^2. Observe that bothcubulated surfaces share a common circle boundary. The face boundary moveconsists in replacing A^2 by B^2 (see Figure <ref>). There are three types of face boundary moves depending of the number of 2-faces in each A^2 and B^2. If A^2 has p squares then B^2 has 6-p squares. It can be easily shown that the (M2)-move can be extended to an ambient isotopy of ℝ^4 .Given two cubical 2-knots K^2_1 and K^2_2 in ℝ^4. We say that K^2_1 is equivalent to K^2_2 by cubulated moves, denoted byK^2_1c∼ K^2_2, if we can transform K^2_1 into K^2_2 by a finite number of cubulated moves. § MAIN THEOREM We are now ready to prove our main theorem. Notice that this proof is a generalization of the one given in <cit.> for the one dimensional case. Given two cubical 2-knots K^2_1 and K^2_2in ℝ^4,then they are isotopic if and only if K^2_1 is equivalent to K^2_2 by cubulated moves; i.e.,K^2_1∼K^2_2K^2_1c∼K^2_2.First, note that if K^2_1 and K^2_2 are equivalent by cubulated moves, then these 2-knots are isotopic by Remark <ref>.Hence, what remains to be proved is that two cubical 2-knots that are isotopic must also be equivalent by cubulated moves.Our strategy is like the one used for cubic knots of dimension one (see <cit.>).First, for i ∈{1,2}, we will smooth each K^2_i to obtain K^2_i, and then cubulate these two 2-knots to obtain K^2_i in such a way that A) K^2_i c∼K^2_i and B) K^2_1c∼K^2_2. In <cit.> was proved the following.Theorem. Any compact, closed, oriented, cubical surface M^2 in ℝ^4 is smoothable. More precisely, M^2 admits a global continuous transverse field of 2-planes and therefore by a theorem of J. H. C. Whitehead there is an arbitrarily small topological isotopy that moves M^2 onto a smooth surface M^2 in ^4 (see<cit.>, <cit.>). As a consequence, we have thatgiven a cubical knot K^2, there exists a smooth knot K^2 isotopic to K^2 such that K^2 is C^0-arbitrarily close to K^2.Let J^3 be the isotopic cylinder (see Definition <ref>) of K^2_1 and K^2_2. Then J^3 is a smooth submanifold of codimension two in ℝ^5.By Theorem <ref>, there exists an isotopic copy of J^3, say J^3, contained in the 3-skeleton of the canonicalcubulation C of ℝ^5. Recall that J^3 is sliced by connected level sets of p andthere exist integer numbersm_1 and m_2 such that p^-1(t)∩J^3 =K^2_1 for all t≤ m_1 and p^-1(t)∩J^3 = K^2_2 for all t≥ m_2, where K^2_1 and K^2_2 are cubical knots which are isotopic to K^2_1 and K^2_2, respectively.Now, we will use the following two results which will be proved in Sections <ref> and <ref>, respectively.Given a cubical 2-knot K^2, we can choose a small cubulation C_mfine enough so thatN^4_K={Q^4∈C_m | Q^4∩ K^2≠∅} is a closed tubular neighborhood of K^2 and Q^4∩ K^2 is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces).We can also choose K^2 isotopic to K^2 such that K^2 isC^0-arbitrarily close to K^2 and K^2⊂(N^4_K). LetK^2 be an isotopic copy of K^2 contained in ∂ N^4_K, then K^2c∼K^2; i.e., we can go from K^2 to K^2 by a finite sequence of cubulated moves.* The existence of K^2 is proved on Theorem 3.1 in <cit.> (see also the proof of Theorem <ref>).* We may assume, using a subdivision move if necessary, that the intersection Q^4∩ K^2 is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces). Given two cubical knots K^2_1 and K^2_2, we obtain K^2_1 and K^2_2 as in Lemma A. Then there exists a finite sequence of cubulated moves that carries K^2_1 into K^2_2. In other words, K^2_1 is equivalent to K^2_2 by cubulated moves: K^2_1c∼K^2_2 If we go back to the proof of the Theorem 1, we have by Lemma A that there exist a finite sequence of cubulated moves that carries K^2_1 into K^2_1 and also a finite sequence of cubulated moves that transforms K^2_2 into K^2_2. By Lemma B,there exists a finite sequence of cubulated moves that converts K^2_1 into K^2_2. As a consequence there exists a finite sequence of cubulated moves that carries K^2_1 onto K^2_2. §.§ K^2_i is equivalent to K^2_i by cubulated moves Our goal is to prove Lemma A.Given a cubical 2-knot K^2, we can choose a small cubulation C_mfine enough so thatN^4_K={Q^4∈C_m | Q^4∩ K^2≠∅} is a closed tubular neighborhood of K^2 and Q^4∩ K^2 is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces).We can also choose K^2 isotopic to K^2 such that K^2 isC^0-arbitrarily close to K^2 and K^2⊂(N^4_K). LetK^2 be an isotopic copy of K^2 contained in ∂ N^4_K, then K^2c∼K^2; i.e., we can go from K^2 to K^2 by a finite sequence of cubulated moves.Proof.The knots K^2 and K^2 are isotopic and both are contained in the 2-skeleton of N^4_K; however K^2⊂(N^4_K)and K^2⊂∂ N^4_K. Our goal is to construct a connected 3-manifold M^3such that it will be contained in the 3-skeleton of N^4_K and its boundarywill consist of two connected components, namely K^2 and K^2.Let B^4={Q^4⊂ N^4_K | Q^4∩K^2≠∅}. Since all hypercubes in N^4_K intersect K^2, B^4 consists of a finite number of hypercubes, say m, such that all of them also intersect K^2.By orienting K^2 we can enumerate the cubes in B^4 in such a way that consecutive numbers belong to neighboring hypercubes (hypercubes sharing a common 3-face). To construct the 3-manifold M^3, we will look at all possible cases of Q^4_j∈ B^4and find pieces M_j^3 which will be consist of the union of some3-facesof Q^4_j such that they are two by two neighboring faces. The union of all M_j^3 will be M^3.Notice that the boundary of these M_j^3's willintersect both K^2 and K^2, hence 3-faces corresponding to neighboring 4-cubes will share a common face. In order to describe the cubes M_j^3's in this proof, we will use the following notation for the 3-faces ofa given hypercube Q^4_j.Next, we will construct M^3 consideringall possible cases of both K^2∩ Q^4_j and K^2∩ Q^4_j.Claim 1: Let Q^4_j∈ B^4 and suppose that K^2∩ Q^4_j consists ofthree squares (two by two neighboring 2-faces).Then, using face boundary moves if necessary, we can assume thatK^2∩ Q^4_j is a connected 2-disk consisting of the union of at most three squares.Proof of Claim 1. Suppose that K^2∩ Q^4_j consists of three neighboring 2-faces F_1^2, F_2^2 and F_3^2. Observe thatF_1^2∪ F_2^2∪ F_3^2intersects fifteen distinct 2-faces of Q^4_j, hence K^2∩ Q^4_j must be contained in the remaining six 2-faces. Notice thatthese six 2-faces are contained in a 3-face C^3 of Q^4_j and applying(M2)-moves if necessary, we can assume that at most three 2-faces ofK^2∩ Q^4_j lie on C^3.Therefore K^2∩ Q^4_j consists of at most three 2-faces. See Figure <ref>.1. This proves claim 1. □ Case 1. Suppose that K^2∩ Q^4_j consists of three neighboring 2-faces F_1^2, F_2^2 and F_3^2. By the above claim, we have that K^2∩ Q^4_j is a connected 2-disk consisting of the union of at most three 2-faces; therefore K^2∩ Q^4_jconsists of at most three neighboring faces E_1^2, E_2^2 and E_3^2 such that E_i^2 is parallel to F_i^2 (i=1,2,3).Then the only possibility up to a face boundary moves (M2) is shown in Figure <ref>.2. Each K^2∩ Q^4_jandK^2∩ Q^4_j consist of three neighboring 2-faces. ThusM_j^3=M_3^3∪ M_-1^3∪ M_-4^3.Claim 2: Let Q^4_j∈ B^4 and suppose that K^2∩ Q^4_j consists oftwo squares sharing an edge (neighboring 2-faces). Then,using face boundary moves if necessary, we can assume that K^2∩ Q^4_j is a connected 2-disk consisting of the union of at most four faces.Proof of Claim 2. Suppose that K^2∩ Q^4_j consists of two neighboring 2-faces F_1 and F_2. Observe that the unionF_1∪ F_2 intersectsfifteen distinct 2-faces of Q^4_j, then K^2∩ Q^4_j must be contained in the remaining seven 2-faces. Notice that six of these seven 2-faces are contained in a 3-face C^3 of Q^4_j and applying(M2)-moves if necessary, we can assume that three 2-faces of K^2∩ Q^4_j lie on C^3. See Figure <ref>.1. Therefore K^2∩ Q^4_j consists of at most four 2-faces. This proves claim 2. □ Case 2. Suppose that K^2∩ Q^4_j=F_1^2∪ F_2^2 where F_1^2 and F_2^2 are two neighboring 2-faces. Since K^2∩ Q^4_j is a connected 2-disk consisting of the union of at most four 2-faces, and K^2∩K^2=∅; we have that K^2∩ Q^4_j is contained in the union of seven 2-faces of Q^4_j, such that F_1^2, F_2^2, …, F_6^2 belong to a 3-face C^3 and F_7^2belongs to the opposite 3-face C̅^̅3̅. See Figure <ref>.1. Since K^2 intersects someneighbor cubes of Q^4_j, we have three possibilities: (a) Suppose that K^2∩ Q^4_j consists of the union of four squares.Hence F_7^2 belongs to K^2∩ Q^4_j. Thus, there exist three 3-faces M_1^3, M_-1^3 and M_-4^3 that contained two 2-faces of (K ^2∪K^2)∩ Q^4_j (see Figure <ref>.2). Then M_j^3=M_1^3∪ M_-1^3∪ M_-4^3.(b) Suppose that K^2∩ Q^4_j consists of the union of three neighboring faces. Then considering all the possibilitiessatisfying that K^2∩ Q^4_j can be extended, we have thatK^2∩ Q^4_j consists of F_1^2, F_2^2 and F_3^2 such thatF_1^2 and F_3^2 are disjoint,F_1^2 and F_2^2 share an edge and so doesF_2^2 and F_3^2 (see Figure <ref>.3). So M_j^3=M_1^3∪ M_-1^3∪ M_-4^3. (c) Suppose that K^2∩ Q^4_j consists of the union of two neighboring faces. So considering all the options which K^2∩ Q^4_j can be extended, we have two possibilities for K^2∩ Q^4_j:(i) K^2∩ Q^4_j is the union of the two 2-faces F_7^2 and F_4^2 (see Figure <ref>.4). Then there exist two 3-faces M_4^3 and M_-2^3 containing two 2-faces of (K ^2∪K^2)∩ Q^4_j. So M_j^3=M_4^3∪ M_-2^3. (ii) K^2∩ Q^4_j is the union of the two 2-facesF_2^2 and F_3^2 (see Figure <ref>.5). Thenthere exist two 3-faces M_-1^3 and M_-4^3 containing two 2-faces: one from K ^2∩ Q^4_j and the other fromK^2∩ Q^4_j.So M_j^3=M_-1^3∪ M_-4^3.Claim 3: Let Q^4_j∈ B^4 and suppose that K^2∩ Q^4_j consists of one 2-face F^2. Then, using face boundary moves if necessary, we can assume thatK^2∩ Q^4_j is a connected 2-disk consisting of the union of at most five 2-faces.Proof of Claim 3. Observe that F^2 intersects twelve distinct 2-faces of Q^4_j, then K^2∩ Q^4_j must be contained in the remaining eleven 2-faces. These eleven 2-faces are distributed in the following way: six of them are contained in a 3-face C^3 and the remaining faces lie on a neighboring 3-face D^3 such that C^3∩ D^3 consists of one 2-face (see Figure <ref>.1). Since K^2∩ Q^4_j is a connected disk,then using (M2)-moves if necessary, we can assume that three 2-faces of K^2∩ Q^4_j lie on C^3 and two 2-faces lie on D^3. This proves claim 3. □ Case 3. Suppose that K^2∩ Q^4_j consists of one 2-face F^2. By claim 3, we can assume that at most three 2-facesF_1^2, F_2^2, F_3^2 of K^2∩ Q^4_j lie on C^3 and two 2-faces F_4^2, F_5^2 lie on D^3. We have five possibilities: (a) Suppose that K^2∩ Q^4_j consists of the union of five 2-faces. Then we have only one possibility for K^2∩ Q^4_j(see Figure <ref>.2). There exist three 3-faces M_1^3, M_-2^3 and M_-1^3 which contain two 2-faces of (K ^2∪K^2)∩ Q^4_j. Take M_j^3=M_1^3∪ M_-2^3∪ M_-1^3. (b) Suppose that K^2∩ Q^4_j consists of the union of four 2-faces. In this case, we have three possibilities: (i) K^2∩ Q^4_j is the union of the four 2-faces F_1^2, F_2^2, F_3^2 and F_4^2. Then we have only one possibility (see Figure <ref>.3).Let M_-2^3 be the 3-face of Q^4_j such that (K^2∩ Q^4_j)∪ F_2^2⊂ M_-2^3 andlet M_1^3 and M_-1^3 be the 3-faces of Q^4_j such that each of them is aneighboring 3-face of M_-2^3 andF_1^2⊂ M_1^3 and F_4^2⊂ M_-1^3. Then M_j^3=M_1^3∪ M_-1^3∪ M_-2^3. (ii) K^2∩ Q^4_j is the union of the 2-faces F_1^2, F_2^2 and F_4^2, F_5^2. Thus the only possible configuration is shown inFigure <ref>.4. Let M_1^3, M_-2^3 and M_-1^3 be as above and let M_-4^3 be the neighboring 3-face such that (K^2∩ Q^4_j)∪ F_5^2⊂ M_-4^3so M_j^3=M_1^3∪ M_-2^3∪ M_-1^3∪ M_-4^3. (iii) This configuration is similar to the one described in case 3b (i), but we exchange the cubes C^3 and D^3. Then, we get only one possibility(see Figure <ref>.5 ). Let M_j^3=M_1^3∪ M_-1^3∪ M_-4^3.(c) Suppose that K^2∩ Q^4_j consists of the union of three 2-faces. Let F_1^2, F_2^2, F_3^2 be 2-faces contained in a cube C^3 and let F_4^2, F_5^2, F_6^2 be 2-faces contained in the 3-cube D^3. Then we have three possibilities: (i) K^2∩ Q^4_j is the union of the 2-faces F_1^2, F_2^2 and F_3^2. Let M_1^3, M_-1^3 and M_-2^3 be as above, so M_j^3=M_1^3∪ M_-1^3∪ M_-2^3 (see Figure <ref>.6). (ii) K^2∩ Q^4_j is the union of the 2-faces F_2^2, F_3^2 and F_6^2. Thus M_j^3=M_1^3∪ M_-2^3 (see Figure <ref>.7). (iii) K^2∩ Q^4_j is the union of the 2-faces F_4^2, F_5^2 and F_6^2. Let M_2^3 be the 3-face which contains the three squares K^2∩ Q^4_j. Then M_j^3= M_2^3∪ M_-4^3(see Figure <ref>.8).(d) Suppose that K^2∩ Q^4_j consists of the union of two neighboring faces. Let F_1^2, F_2^2, F_3^2 be 2-faces contained in the cube C^3 and let F_4^2, F_5^2, F_6^2 be 2-faces contained in the cube D^3. We have two possibilities: (i) K^2∩ Q^4_j is the union of the 2-faces F_1^2 and F_2^2. ThusM_j^3=M_1^3∪ M_-2^3(see Figure <ref>.9). (ii) K^2∩ Q^4_j is the union of the 2-faces F_3^2 and F_4^2.ThusM_j^3=M_2^3∪ M_-4^3 (see Figure <ref>.10).(e) Suppose that K^2∩ Q^4_j consists ofone square. Then we have two possibilities: (i) K^2∩ Q^4_j is the square F_4^2. Thus M_j^3=M_-2^3∪ M_4^3 (see Figure <ref>.11). (ii) K^2∩ Q^4_jis the 2-face F_2^2. Thus M_j^3=M_-2^3 (see Figure <ref>.12).Claim 4: Let Q^4_j∈ B^4 and suppose that K^2∩ Q^4_j consists of an edge e. Then, using face boundary moves if necessary, we can assume thatK^2∩ Q^4_j is a connected 2-disk consisting of the union of at most six faces.Proof of Claim 4. Suppose that K^2∩ Q^4_j consists of an edge e. See Figure <ref>. Then K^2 must turn on Q^4_j; in other words, K^2∩ Q^4_j must contain two faces F_1^2 and F_2^2 such that F_1^2=v+{ae_i_1+be_j : 0≤ a, b≤ 1} and F_2^2=v+{ce_i_2+de_j : 0≤ c, d≤ 1}, where e_i_1, e_i_2 and e_j are distinct canonical vectors and v is a vector with integer coordinates. Notice that e_j is parallel to the edge e and onlynine 2-faces of Q^4_j satisfy both they do not intersect K^2 and they have an edge parallel to e_j;hence K^2∩ Q^4_j must be contained in the union of these nine 2-faces. Since K^2∩ Q^4_j is a connected disk it follows,up to applying face boundary moves (M2) if necessary, that it consists of the union of at most six faces.This proves claim 4. □ Case 4. Suppose that K^2∩ Q^4_j consists of an edge e. By claim 4,K^2∩ Q^4_j is contained in three 3-faces M_3^3, M_-2^3 and M_-4^3 such that any two of them share a2-face. (a) K^2∩ Q^4_j consists of the union of six 2-faces. Then, applying(M2)-moves if necessary, we have that K^2∩ Q^4_j consists of theunion of two 2-faces contained in M_3^3, three 2-facesin M_-3^3 and one 2-face in M_-4^3 (see Figure <ref>.2). Thus M_j^3=M_3^3∪ M_-3^3∪ M_-4^3. (b) K^2∩ Q^4_j consists of the union of five 2-faces. Then, applying(M2)-moves if necessary, as in the previous case we have that M_j^3=M_3^3∪ M_-3^3∪ M_-4^3. (see Figure <ref>.3).(c) K^2∩ Q^4_j consists of the union of four 2-faces. We have five possibilities: (i) K^2∩ Q^4_j consists of two 2-faces contained in M_4^3 and three 2-facesin M_-3^3.Thus M_j^3=M_-3^3∪ M_2^3∪ M_-4^3 (see Figure <ref>.4). (ii) K^2∩ Q^4_j consists of one 2-face contained in M_-4^3 and three 2-facesin M_-3^3. Then M_j^3=M_-3^3∪ M_-4^3 (see Figure <ref>.5). (iii) K^2∩ Q^4_j consists of one square contained in M_-4^3 and three 2-facesin M_-3^3 (see Figure <ref>.6). Then M_j^3=M_-3^3∪ M_-4^3. (iv) K^2∩ Q^4_j consists of two 2-faces contained in M_-3^3 and two 2-facesin M_2^3 (see Figure <ref>.7).Then M_j^3=M_-3^3∪ M_2^3∪ M_-4^3. (v) K^2∩ Q^4_j consists of three 2-faces contained in M_2^3 and one square in M_-4^3 (see Figure <ref>.8).So M_j^3= M_2^3∪ M_-4^3. (d) K^2∩ Q^4_j consists of the union of three 2-faces. We have two possibilities: (i) K^2∩ Q^4_j consists of one square contained in M_2^3 and two 2-facesin M_-3^3 (see Figure <ref>.9). Then M_j^3=M_2^3∪ M_-3^3∪ M_-4^3.(ii) K^2∩ Q^4_j consists of two 2-faces contained in M_-3^3 and one 2-facein M_-4^3 (see Figure <ref>.10). Then M_j^3=M_-3^3∪ M_-4^3. (e) K^2∩ Q^4_j consists of the union of two neighboring 2-faces. We have two possibilities: (i) K^2∩ Q^4_j consists of two 2-faces contained in M_4^3(see Figure <ref>.11). Then M_j^3=M_4^3∪ M_3^3. (ii) K^2∩ Q^4_j consists of two 2-faces contained in M_-4^3(see Figure <ref>.12).Thus M_j^3=M_-4^3. Claim 5: Let Q^4_j∈ B^4 and suppose that K^2∩ Q^4_j consists ofa vertex v. Then, using face boundary moves ifnecessary, we can assume thatK^2∩ Q^4_j is a connected 2-disk consisting of the union of at most six 2-faces.Proof of Claim 5. Since K^2∩ Q^4_j consists of a vertex v, then v must be a corner of K^2 i.e. v is a common vertex of three neighboring faces ofK^2.Hence K^2 must have a corner at some vertex of Q^4_j. Since K^2∩K^2 =∅, it follows that K^2∩ Q^4_j can be contained in either one, two or three 3-faces of Q^4_j such that these 3-faces do not contained v. Suppose that K^2∩ Q^4_j is a connected disk consisting of at least six 2-faces,then by a combinatorial analysis considering all the possible descriptions ofK^2, we should have that four of these 2-faces are contained in some 3-face of Q^4_j; hence applying a face boundary move (M2), we get thatK^2∩ Q^4_j can be reduced to at most six 2-faces. (see Figure <ref>.1.) This proves claim 5. □Case 5. Suppose that K^2∩ Q^4_j consists of a vertex v. By the above claim,K^2∩ Q^4_j is the union of at least two 2-faces of Q^4_j and its boundary must be contained in the union of the three neighboring 3-faces containing v. (a) K^2∩ Q^4_j is the union of six faces. Then K^2∩ Q^4_jis contained in three 3-faces M_-2^3, M_3^3 and M_4^3 of Q^4_j (see Figure <ref>.2).SinceK^2∩ Q^4_j is homeomorphic to a disk, we have thateach 3-faceM_i^3 must contain two 2-faces of it and two of these three 3-faces contain v.Then M_j^3=M_-2^3 ∪ M_3^3∪ M_4^3. (b) K^2∩ Q^4_j is the union of five 2-faces. Then K^2∩ Q^4_jis contained in the union of two 3-faces M_-4^3 andM_3^3 of Q^4_j (see Figure <ref>.3). So M_j^3= M_3^3∪ M_-4^3.(c) K^2∩ Q^4_j is the union of four 2-faces. Again, K^2∩ Q^4_jis contained in a 3-face M_4^3 of Q^4_j and its boundary must be contained in the union of the three neighboring 3-faces containing v. Hence, one 3-face M_3^3, M_-1^3, M_-2 must contain one 2-face.The only possibility is that four 2-faces lie in some 3-face M_4^3 ofQ^4_j (see Figure <ref>.4).In this case M_j^3=M_4^3. (d) K^2∩ Q^4_j is the union of three neighboring 2-faces. These three 2-faces must be contained in some 3-face M_-4^3.Then M_j^3=M_-4^3(see Figure <ref>.5). (e) K^2∩ Q^4_j is the union of two neighboring 2-faces. These two 2-faces must be contained in some 3-face M_-4^3.Then M_j^3=M_-4^3.Next, we will construct our 3-manifold M^3.LetM^3=∪M_j^3. By construction, each M_j^3 consists of the union of 3-faces contained in the3-skeleton of N_K^4, hence M^3 is contained in the3-skeleton of N_K^4. As we mention before,if we apply an (M2)-move on a hypercube Q^4_j, then this movement does not affect the corresponding choice of M_l^3 on its neighbor hypercube Q^4_l; in other words, the choice of M_j^3 depends only of the configuration of K^2∩ Q^4_j and K^2∩ Q^4_j in Q^4_j. Observe that the boundary of each M_j^3 is composed by2-faces belonging toK^2 and K^2andsome disjoint faces F^j_i (i=1,2,… ,s_j) that do not belong to neither K^2 nor K^2. Thus, if we take a neighbor hypercube of Q^4_j, say Q^4_j_l∈ B^4, and we construct M_j^3 and M_j_l^3, thenthe intersection M_j^3∩ M_j_l^3 consists of the union of some F^j_r_i; notice that each of these 2-faces F^j_i belongs to a neighbor hypercube of Q^4_j,Q^4_j_l_s∈ B. Hence M^3 is a 3-manifold whose boundary consists of two connected components, namely K^2 and K^2. Now, we will carry the knot K^2 onto the knot K^2 via a finite number of cubulated moves. Notice that M^3 isthe unionof m components M_1^3, … M_m^3 which are enumerated in such a way that M_n+1^3 is aneighbor of M_n^3 for all n, i.e., M_n+1^3∩ M_n^3 consists of a 2-face F_n^2.We will use induction on m. Consider M_1^3. We apply(M2)-moves on M_1^3 in such a way that the faces (of any dimension) belonging to K^2 are replaced bythose belonging to K^2 (by construction M_1^3∩ K^2≠∅ and M_1^3∩K^2≠∅). Next, we consider M_2^3. By the previous step, M_1^3 and M_2^3 share 2-faces belonging to K^2. Then we apply again (M2)-moves,the faces (of any dimension) belonging to K^2 are replaced by those belonging to K^2. We continue this finite process in this way.Notice that if l⊂K^2 then l⊂∂M^3, thus l is not a common 2-face of some pair M_i^3, M_j^3 belonging to M^3, hence ifl is replaced by a 2-face belonging to K^2, then this replacement will be kept in the following steps. Therefore, the result follows. □§.§ K^2_1 is equivalent to K^2_2 via cubulated moves In this section, we shall prove the Lemma B which says thatK^2_1 is equivalent to K^2_2 by cubulated moves.Consider again the projection p:ℝ^5↪ℝ on the last coordinate. We call horizontal hyperplane to an affine hyperplane parallel to the space ℝ^4×{0}.Thus p^-1(t)=ℝ^4_t is a horizontal hyperplane. Observe that each hyperplaneℝ^4_t has a canonical cubulation given by the restriction of the canonical cubulation of ℝ^5 to it.A 2-cell (2-face) F^2 of the canonical cubulation C of ℝ^5is called horizontal if p(F^2) is a constant number in ℕ. A 2-cell F^2 is vertical if p(F^2)=[m,m+1] for some m∈ℕ. Let Σ^3 be a cubulated 3-manifold and P^4 be a horizontal hyperplane in ℝ^5. We say that P^4intersects transversally to Σ^3, denoted byP^4⋔Σ^3, if P^4 intersects transversally each k-cube of Σ^3, k≥ 1.Let ℝ^4_t ⊂ℝ^5, t∉ℤ be an affine hyperplane. Thenℝ^4_t intersects J^3 transversally. Proof. By Theorem <ref>, ℝ^4_t∩J^3 is connected. Let x∈R^4_t∩J^3. Then x∈ Q^3_i, where Q^3_i is a 3-face of the cubulation of ℝ^5. Notice that Q^3_i is a vertical 3-face, since t∉ℤ.So, we have two possibilities: either x∈(Q^3_i) orx∈ F^2∖∂ F^2 where F^2 is a 2-face of Q^3_i, or x belongs to an edge. * If x∈(Q^3_i) then ℝ^4_t∩ Q^3_i=E^2, where E^2 is a square parallel to a 2-face and x∈ E^2.* If x belongs to F^2∖∂ F^2, then there exists another vertical 3-face Q^3_j such that x∈ Q^3_i∩ Q^3_j. Thus ℝ^4_t∩ Q^3_i must be a square E_i^2 such that it is parallel to a 2-face,and analogously ℝ^4_t∩ Q^3_j is also a square E_j^2, thus x∈ E_i^2∩ E_j^2. * If x belongs to an edge l, then there existvertical 3-faces Q^3_i_r r=1,2,3 such that x∈ Q^3_j∩_r=1^3 Q^3_i_r. As in the previous case, ℝ^4_t∩ Q^3_i_r must be a square E_i_r^2 parallel to a 2-face,and analogously ℝ^4_t∩ Q^3_j is also a square E_j^2, hencex∈ E_j^2∩_r=1^4 E_i_r^2.Therefore, the result follows. □ For x∉ℤ, the set p^-1(x)∩J^3 is a 2-knot. Proof. By the above, p^-1(x)∩J^3 is a cubulated compact connected surface. Since J^3 is homeomorphic to 𝕊^2×ℝ, it follows that p^-1(x)∩J^3 is homeomorphic to 𝕊^2. □Now, for each n∈ℕ we define K^2_n-:= p^-1(n-1/2)∩J^3 and K^2_n+:= p^-1(n+1/2)∩J^3.Observe that K^2_n- and K^2_n+ are cubical 2-knots. Let C^3 be the set of 3-cubes (3-cells) belonging to the canonical cubulation of ℝ^5. Consider the three spaces:M^3_n-:={Q^3∈𝒞 ^3| Q^3 ∩ K^2_n-≠∅},M^3_n+:={Q^3∈𝒞 ^3| Q^3 ∩ K^2_n+≠∅}andM^3_n:= p^-1(n)∩J^3. By construction M^3_n-= K^2_n-× [0,1] and M^3_n+= K^2_n+× [0,1].Let M^3:=M^3_n-∪ M^3_n∪ M^3_n+. Hence M^3=(p^-1(n-1,n+1)∩J^3), where Cl denotes closure.The space M^3 is homeomorphic to 𝕊^2× [0,1]. Proof. Since M^3 is a compact submanifold of J^3, then by Lemma <ref>, it is also connected. Now J^3 is homeomorphic to 𝕊^2×ℝ, and J^3-M^3 has two connected components, hence the result follows. □ M^3_n has the homotopy type of 𝕊^2. Proof. Consider the set M^3. Then by the previous Lemma, we have thatM^3≅𝕊^2× [0,1]. Notice that K^2_n-×{0}≅𝕊^2×{0} and K^2_n+×{0}≅𝕊^2×{1}.Hence M^3=M^3/(K^2_n-×{0})∪ (K^2_n+×{1}) is homeomorphic to 𝕊^3.By Alexander duality, we have that H_0(M^3-M^3_n,ℤ)=H^2 (M_n^3,ℤ), but M^3-M_n^3 has two simply connected components, so H_0 (M^3,ℤ)≅ℤ. Since M_n^3⊂ M^3≅𝕊^2× [0,1], we have that either π_2 (M_n^3)≅{0} orπ_2 (M_n^3)≅ℤ, but H̃^2(M_n^3,ℤ)≅ℤ, hence π_2 (M_n^3)≅ℤ. Again by Alexander duality, we have that H_1(M^3-M_n^3,ℤ)=H^1 (M_n^3,ℤ), but H_1(M^3-M_n^3,ℤ)≅{0} hence H^1 (M_n^3,ℤ)≅{0}. This implies thatπ_1 (M_n^3)≅{0}. Therefore, M_n^3 has the homotopy type of 𝕊^2. □ The space M^3 retracts strongly to M_n^3. Proof. Since M^3=M^3_n-∪ M^3_n∪ M^3_n+ is homeomorphic to 𝕊^2× [0,1],and M^3_n-= K^2_n-× [0,1] and M^3_n+= K^2_n+× [0,1],we have thatM^3_n-= K^2_n-× [0,1] retracts strongly to K^2_n-×{1} and M^3_n+= K^2_n+× [0,1] retracts strongly to K ^2_n+×{0}. Now ∂ M_n^3=(K^2_n-×{1})∪ (K^2_n+×{0}). Therefore, the result follows. □ Next, we are going to describe the subset M_n^3. Notice that the squares of M_n^3 are of four types, which we will denoteby T_-, T_+, T_± and T.* Asquare F^2⊂ M_n^3 belongs to T_- ifF^2⊂ M^3_n- but F^2⊄M^3_n+. * A square F^2⊂ M_n^3 belongs to T_+ if F^2⊂M^3_n+ but F^2⊄M^3_n-. * A square F^2⊂ M_n^3 belongs to T_± if F^2⊂M^3_n-∩ M^3_n+. * A square F^2⊂ M_n^3 belongs to T if F^2⊄M^3_n+∪ M^3_n-. By the above Lemma, there are copies of K^2_n- and K^2_n+ contained in ∂ M_n^3. By abuse of notation we will denote them in the same way. Notice that K^2 _n- is the union of squares of types T_- and T_±, and K^2_n+ is the union of squares of types T_+ and T_±. K^2_n-c∼K^2_n+: There exists a finite sequence of cubulated moves that carries the2-knot K^2_n- into the 2-knot K^2_n+. Proof. We will show it by cases. Case 1. Suppose that K^2_n-=K^2_n+.Clearly, the result is true.Case 2. Suppose that K^2_n-∩ K^2_n+=∅. In other words, K^2_n- andK^2_n+do not have squares of type T_±.Remember that M_n^3 is a cubical compact 3-manifold whose fundamental group is isomorphic to ℤ. Thus∂ M_n^3 has two connected components; namely K^2_n- and K^2_n+, such that their intersection is empty. Hence M_n^3 is the union of a finite number of cubes (3-faces) belonging to the 3-skeleton of the cubulation C, whose 2-faces are of any of the types T_-, T_+ and T.Next, we will carry the 2-knot K^2_n- onto the 2-knot K^2_n+ via a finite number of cubulated moves; i.e. we will carry the squares of type T_- onto the squares of type T_+. Let Q^3 be a cube contained in M_n^3. We can assume, up to (M1)-move, that if a square F^2⊂ Q^3 belongs to T_-, then Q^3∩ K^2_n+=∅and Q^3∩ K^2_n-consists of either a square, two neighboring 2-faces or three neighboring 2-faces. Analogously, ifF^2⊂ Q^3 belongs to T_+, then Q^3∩ K^2_n-=∅ and Q^3∩ K^2_n+ consists of a square, two neighboring 2-faces or three neighboring 2-faces.The compact space M_n^3 is the union of a finite number of cubes, say m. We will enumerate them by levels in the following way. Remember that M_n^3 is a cubical compact 3-manifold, so we may assume that M_n^3⊂ℝ_+^3.Let q:ℝ^3↪ℝ be the projection on the last coordinate. We definethe level k, M_k^3:=M_n^3∩ q^-1([k,k+1]), for k∈ℕ. Notice thatthere exists k_1 and k_2 positive integers such that M_k^3=∅ for k_1≤ k≤ k_2. Then we start enumerating the cubes Q^3∈ M^3 by levels. We start at the level k_1. The first cube Q_1^3 contains a 2-face of type T_-, and given the cube Q_n^3, the cubeQ_n+1^3 shares a 2-face F_n^2 withQ_n^3 and wheneverit is possible, we choose Q_n+1^3 in such a way that F_n^2 is parallel to F_n-1^2; otherwise we chooseQ_n+2^3 such that F^2_n+1 is parallel to F^2_n-1, at the end we continue on the level k_1+1 and so on.We will use induction on m. Consider the cube Q_1^3. We apply the (M2)-move to Q_1^3 replacing the 2-faces of type T by 2-faces of typeT_-. We consider Q_2^3. Observe that Q_1^3 and Q_2^3 share a 2-face of type T_-. Then we apply again the (M2)-move replacing the 2-faces of typeT by 2-faces of typeT_-. We continue inductively. Notice that if F^2⊂M_n^3 is a 2-face of type T_+ then F^2⊂∂ M_n^3; so F^2 is not a common 2-face of two cubes Q_i^3 and Q^3_j in M_n^3; hence ifF^2 is replaced by a 2-face of type T_- then this replacement is not modified inany other next step. Therefore, the result follows.Case 3. Suppose that the intersection K^2_n-∩ K^2_n+ contains a finite number of 2-faces. The 3-manifold M_n^3 consists of connected components C_i^3, i=1,…,r such that each C_i^3 is the union of cubes Q_i_1^3, …, Q_i_m_i^3∈C and the intersection C_i^3∩ C_j^3 is either empty or a square belonging toK^2_n-∩ K^2_n+. Therefore, we apply the previous argument to each C_i^3. Case 4. Suppose that the intersection K^2_n-∩ K^2_n+ contains a square of type T_±. The 3-manifold M_n^3 consists of 3-dimensional connected components C_i^3, i=1,…,rand cubical 2-disks γ_ij. As before, each C_i^3 is a union of cubes Q_i_1^3, …, Q_i_m_i^3∈C, and γ_ij is a cubical disk (or edge) joining the component C_i^3 with the component C_j^3. Observe that if γ_ij is the union of 2-faces of type T_±, hence γ_ij⊂ K^2_n-∩ K^2_n+. Moreover K^2_n-∩ K^2_n+= γ_ij and∂ M_n^3=K^2_n-∪ K^2_n+.Since π_2 (M_n^3)≅ℤ, thenC_i^3the homotopy type either the 2-sphere or the 3-ball.Suppose that C_i^3 has the homotopy type of the 2-sphere, then by hypothesis ∂ C_i^3=∂ M_n^3=K^2_n-∪ K^2_n+ contains a face F^2 of type T_±, but F^2 does not belong to any cube Q^3 of M_n^3; so F^2 does not belong to C_i^3. This is a contradiction, henceC_i^3 has the homotopy type of the 3-ball.By the above, ∂ C_i^3 is homeomorphic to 𝕊^2 and consists of 2-faces of type T_- and T_+. Moreover, ∂ C_i^3 consists of two disks D_-^2 and D_+^2, such that D_-^2 is the union of faces of type T_- and D_+^2 is the union of faces of type T_+. Now we apply the argument of the case 2, so D_+^2 is replaced by D_-^2. Since we have a finite number of components C_i^3, the result follows. □K^2_1c∼K^2_2: There exists a finite sequence of cubulated moves that carries K^2_1 intoK^2_2. In other words, K^2_1 is equivalent toK^2_2 by cubulated moves. Proof of Lemma B.Recall that there exist integer numbersm_1 and m_2 such that p^-1(t)∩J^3 = K^2_1 for all t≤ m_1 and p^-1(t)∩J^3 = K^2_2 for all t≥ m_2. Consider the integer m_1+1. ByLemma <ref> there exists a finite number of cubulated moves that carries the 2-knot K^2_1 into the 2-knot K^2_(m_1+1)-. We continue inductively, and again byLemma <ref> there exists a finite number of cubulated moves that carries the knot K^2_(m_2-1)+ into the knot K^2_2. Since, we have a finite number of integers contained in the interval [m_1,m_2], then there exists a finite sequence of cubulated moves that carries K^2_1 intoK^2_2. □99 artin E. Artin. Zur Isotopie zweidimensionalen Flächen im R_4. Abh. Math. Sem. Univ. Hamburg (1926), 174–177. BHV M. Boege, G. Hinojosa,and A. Verjovsky.Any smooth knot 𝕊^n↪ℝ^n+2 is isotopic to a cubic knot contained in the canonical scaffolding of ℝ^n+2.Rev. Mat Complutense (2011) 24: 1–13. DOI 10.1007/s13163-010-0037-4. DHVV J. P. Díaz,G. Hinojosa, R. Valdez, A. Verjovsky. Smoothing closed gridded surfaces embedded in^4. http://arxiv.org/abs/1702.05467CRS J.S. Carter, J. Rieger, M. Saito, A combinatorial descriptions of knotted surfaces and their isotopies, Adv. in Math 127 (1997), 1–51.CS J.S. Carter, M. Saito M, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, 55. American Mathematical Society, Providence, RI, 1998. dolbilin N. P. Dolbilin,M. A. Shtan ko, M. I.Shtogrin. Cubic manifolds in lattices. Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 93-107; translation in Russian Acad. Sci. Izv. Math. 44 (1995), no. 2, 301-313. funar L. Funar. Cubulations, immersions, mappability and a problem of Habegger.Ann.scient. Éc. Norm. Sup., 4e série, t. 32, 1999, pp. 681–700. HVV G. Hinojosa, A. Verjovsky, C. Verjovsky Marcotte. Cubulatedmoves and discrete knots. Journal of knot theory and its ramifications, Vol 22, No. 14 (2013) 1350079 (26 pages). DOI: 10.1142/S021821651350079XK S. Kamada, Surface-Knots in 4-Space. An Introduction Springer Monographs in Mathematics, Springer-Verlag 2017.mazur B. Mazur. The definition of equivalence of combinatorial imbeddings. Publications mathématiques de l'I.H.É.S., tome 3 (1959) p.5–17.matveev S. Matveev, M. Polyak. Finite-Type Invariants of Cubic Complexes. Acta Applicandae Mathematicae 75, pp. 125–132, 2003.Roseman D. Roseman, Reidemeister-type moves of surfaces in four-dimensional space, Knot Thoery, Banach Center Publications, Volume 42, Institute of Mathematics, Polish Academy of Sciences, Warszawa (1998), p. 347–380.RN Roice Nelson. https://plus.google.com/+RoiceNelson/postspugh Charles C. Pugh. Smoothing a topological manifold. Topology and its applications 124 (2002) 487–503. rokhlin Vladimir A. Rokhlin, New results in the theory of four-dimensional manifolds. Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. rolfsen D. Rolfsen. Knots and Links. Publish or Perish, Inc. 1976. whitehead J. H. C. Whitehead. Manifolds with transverse fields in Euclidean space. Annals of Mathematics vol. 73, no.1 (January, 1961) 154–212. zivaljevic Živaljević, Rade T. Combinatorial groupoids, cubical complexes, and the Lovász conjecture.Discrete Comput. Geom. 41 (2009), no. 1, 135–161. J. P. Díaz. Instituto de Matemáticas, Unidad Cuernavaca. Universidad Nacional Autónoma de México. Av. Universidad s/n, Col. Lomas de Chamilpa. Cuernavaca, Morelos, México, 62209.E-mail address: [email protected] .3cm G. Hinojosa. Centro de Investigación en Ciencias. Instituto de Investigación en Ciencias Básicas y Aplicadas. Universidad Autónoma del Estado de Morelos. Av. Universidad 1001, Col. Chamilpa. Cuernavaca, Morelos, México, 62209. E-mail address: [email protected] .3cm A. Verjovsky. Instituto de Matemáticas, Unidad Cuernavaca. Universidad Nacional Autónoma de México. Av. Universidad s/n, Col. Lomas de Chamilpa. Cuernavaca, Morelos, México, 62209.E-mail address: [email protected] | http://arxiv.org/abs/1708.07761v1 | {
"authors": [
"Juan Pablo Díaz",
"Gabriela Hinojosa",
"Alberto Verjovsky"
],
"categories": [
"math.GT"
],
"primary_category": "math.GT",
"published": "20170824163831",
"title": "Cubulated moves for 2-knots"
} |
§ INTRODUCTION The search for standard model particles resulting from the annihilation of dark matter provides an important complement to the efforts of direct searches for dark matter interactions and searches for dark matter production at particle accelerators. Among the theoretical candidates for the dark matter particle above a few GeV, Weakly Interacting Massive Particles (WIMPs) are well motivated <cit.> as they naturally provide the measured present day cold dark matter density <cit.>.In such models, the WIMPs either decay or annihilate into standard model particles that produce mono-energetic gamma-ray lines and/or a continuum of gamma rays with energies up to the dark matter particle mass. Good targets for searches with gamma-ray instruments are those with a large inferred dark matter (DM) density and low astrophysical backgrounds; i.e. those that would not produce gamma rays through conventional processes. Dwarf Spheroidal Galaxies (dSphs) fit these criteria since they are close by (tens to hundreds of kpc away) and with no theoretical basis for emitting very-high-energy (VHE) gamma rays. The center of our galaxy is also a tantalizing target since it is even closer, about ∼8 kpc away. However the center of the galaxy has many astrophysical backgrounds including diffuse emission along the galactic plane <cit.>; the radio source at the center of our galaxy, SgrA*, is also a very bright VHE emitter. The metric for a good DM target is the J-factor (or D factor in the case of decay) defined as: J = ∫∫ρ(l,Ω)^2dldΩD = ∫∫ρ(l,Ω)dldΩ, where ρ is the DM density and l is the direction along the line-of-sight. This work will focus on DM annihilations. We will also refer to the J-profile which is equal to J-factor per solid angle. The J-factor is directly proportional to the expected gamma-ray flux from DM annihilations, dΦ/dE = ⟨σν⟩/8π M^2J(ΔΩ)∑_iB_idN_γ,i/dE, where M is the WIMP particle mass, dN_γ/dE is the energy spectrum of a single annihilation with branching ratio B_i, and ⟨σν⟩ is the velocity-averaged annihilation cross section.VERITAS (Very Energetic Radiation Imaging Telescope Array System) is an array of four imaging atmospheric Cherenkov telescopes (IACTs), each 12 m in diameter, located at the Fred Lawrence Whipple Observatory in southern Arizona, USA.VERITAS is sensitive to gamma rays from approximately 85 GeV (after camera upgrade) to greater than 30 TeV with a typical energy resolution of 15-25%and an angular resolution (68% containment) of <0.1 degrees per event. The flux sensitivity of the standard analysis is such that a source with a flux of order of 1% of the Crab Nebula flux can be detected in approximately 25 hours of observation. § COMBINED ANALYSIS OF DWARF GALAXIES FROM 2007 TO 2013 From the beginning of four-telescope operations in 2007 to the summer of 2013, five dSphs in the northern hemisphere have been observed: Segue 1, Ursa Minor, Draco, Boötes and Willman 1. No significant gamma-ray excess was observed in the direction of any of the five dSphs. A dark matter search was performed with 216 hours of VERITAS data, including 92 hours on the dSph Segue 1. Table 1 summarizes the analysis results of each of the five dSphs. The 95% confidence level limits on the thermally-averaged annihilation cross section are shown in Figure 1. The search for DM annihilation and the limits on the cross section were derived from the Event Weighting method presented in Geringer Sameth et. al. <cit.>. Willman 1 was excluded from the combined search and limits since the J-factor can not be calculated reliably due to irregular kinematics <cit.>. Additionally, Bonnivard et. al. <cit.> have pointed out the possibility of contamination of the stellar samples used to perform the Jeans analysis which could bias J-profiles towards larger annihilation signals, possibly by orders of magnitude. Therefore, two different limits are shown in each panel in Figure 1: including Segue 1 and without. Each panel in Figure 1 represents a 100% branching ratio into a given standard model particle. The limits are presented as a band with the representing the ±1σ systematic uncertainty of the J-profile. § A DECADE OF DWARF GALAXY OBSERVATIONS: 2007 TO 2017 The indirect detection of dark matter is one of the collaboration's key science projects, and dwarf galaxies are an integral part of the program. Because VERITAS is not a survey instrument, we have to prioritize the DM targets within our observing budget. DSphs observed by VERITAS are subdivided into two categories: `Deep Exposure' dSphs and `Survey' dSphs. `Deep Exposure' dSphs are those with the best J-factors in the literature which get the majority of the observing time. This time is typically split between the ultra-faint and classical dSphs. This is done as a way to ensure that the overall DM program is not severely penalized in case one particular dSph falls out of favor as a good indirect DM target. Most of the dSphs that do not have deep exposures are in the survey for much shorter exposures, typically a few hours per year. The survey provides DM sensitivity to nearly all dSphs in the northern hemisphere as a form of insurance in case misunderstanding of velocity dispersion data leads to a dSph having a much larger J-factor than otherwise indicated. The survey and the deep exposure dSphs can be also be combined or `stacked' into a more sensitive limit than any one individual dSph as shown in the previous section. Table 2 lists all dSph observations since the beginning of four-telescope operations in 2007 to mid-2017. The exposure accumulated on for each dSph are divided into observing epochs: the original array configuration (v4), after the move of Telescope 1 (v5) and after the camera upgrade (v6). See <cit.> for details on the performance of the different epochs. The observation times in Table 2 do not include time intervals with bad weather or hardware malfunctions, but rather the total observing time so these numbers should be considered an upper limit to the possible exposure of each dSph. The analysis of many of these dSph data sets is still preliminary.§.§ Ursa Major II As shown in Table 2, a large amount of the total dSph observing time after the camera upgrade has been devoted to Ursa Major II.Ursa Major II was discovered in SDSS data with follow-up observations from Subaru <cit.>. It is one of the faintest known dSphs, with velocity dispersion measurements made with only 20 stars <cit.>, but with a large mass-to-luminosity ratio. J-factors for Ursa Major II typically indicate that it is one of the highest, often similar to or even exceeding Segue 1 (e.g. <cit.>. VERITAS observed Ursa Major II between late 2013 and spring 2017.After dead-time corrections and removal of time intervals with bad weather or hardware performance, a live time of 145 hours remain. Observation runs with fewer than all four telescopes were also removed from this analysis. The analysis of the data follows the same procedure as in <cit.>. The hFit algorithm is used for event reconstruction <cit.> and the crescent background method is used to subtract the remaining cosmic-ray background <cit.>. An excess of 1σ above the background was found at the Ursa Major II position. We report a flux upper limit of 1.4×10^-8m^-2s^-1 above 200 GeV at a 95% confidence level using the method of Rolke <cit.> assuming a powerlaw with index of -2.4. Figure 2 shows preliminary annihilation cross section limits for Ursa Major II. A version of the full maximum likelihood introduced by Aleksić et. al. <cit.> was used for the search for DM annihilations and for determining the limit to the velocity-averaged cross section.The J-profile is convolved with the VERITAS PSF, generated from monte carlo simulations has described in <cit.>. We do this because the DM profile is wider than the PSF, particularly at lower energies. The median and ±1σ J-profile was provided in the supplemental material of Geringer-Sameth et al. <cit.>. The annihilation spectrum is the one used from PPPC 4 DM ID <cit.>. As shown in Figure 2, the VERITAS limit for Ursa Major II is more constraining on the cross section than the 216 hrs combined dSph <cit.> limit at every DM mass by a factor of ∼30% to more than a factor of 2, depending on DM mass. § GALACTIC CENTERThe center of our galaxy harbors a 4×10^6 solar mass black hole, coincident with the radio source SgrA*. The region around the center of our galaxy is very complex in gamma rays. This region includes bright sources such as the supernova remnant G0.6+0.1 <cit.>, diffuse emission along the plane as observed by HESS <cit.> and Fermi-LAT as well as the extended Fermi bubbles <cit.>. The existence of several possible astrophysical backgrounds makes finding a DM signal from WIMP annihilation a complicated but not impossible prospect. Because VERITAS is a Northern Hemisphere instrument, the Galactic Center never transits above 30^∘ elevation. Observing at these elevations has the effect of raising the energy threshold and widening the PSF, but increasing sensitivity for higher energy showers. The degradation of the PSF is mostly offset by the displacement method of shower reconstruction <cit.>. VERITAS is therefore the most sensitive instrument for observing the Galactic Center at energies greater than 2 TeV, with a PSF of 0.12^∘ (68% containment). § DARK MATTER SUBHALOSCosmological N-body simulations <cit.> indicate that DM halos contain significant substructures <cit.>. Because of tidal disruption near the Galactic disk, most of the sub-halos are thought to survive at high Galactic latitude. The lack of material in these regions prevents the DM over-densities from attracting enough baryonic matter to trigger star formation. DM clumps would therefore be invisible to most astronomical observations from radio to X-rays. DM structures residing in the the Milky Way halo could potentially be nearby the Sun and therefore have a bright gamma-ray annihilation signal. These clumps would therefore only be visible at gamma-ray energies and not been in any other wavelength. The Second Fermi-LAT Catalog (2FGL) contains 576 high-energy gamma-ray sources detected by the LAT after the first 24 months of observations without any clear associations in any other wavelength. To identify the unassociated Fermi objects (UFOs) most likely to be DM clump candidates we adapt the selection criteria from <cit.>, requiring that they are located outside of the Galactic Plane, that they be non-variable, they exhibit power law spectra and that they have no known counterparts. The two best candidates were selected for VERITAS observations: 2FGL J0545.6+6018 and 2FGL J1115.0-0701. It should be noted that at the time these objects were proposed for observation, that the 2FGL catalog was the most up-to-date Fermi-LAT catalog available. 2FGL J1115.0-0701 was observed for 13.8 hours and 2FGL J0545.6+6018 was observed for 8.5 hours. No emission was detected by VERITAS, giving flux upper limits 0.5% and 0.6% of the Crab nebula flux above 250 GeV, respectively. Later analysis of 2FGL J1115.0-0701 in Fermi-LAT showed that the flux was variable, excluding the possibility of it being a DM subhalo. Updated Fermi-LAT analysis of 2FGL J0546.6+6018 shows that a curved power law is actually favored over a simple power law, but DM annihilation to b quarks or W bosons also provide a good spectral fit. Optical follow-up observations were performed within the 95% error region of 2FGL J0546.6+6018 by the 1.3m McGraw-Hill telescope located at Kitt Peak, AZ, USA. A previously undiscovered bright UV source was observed within the error circle, the analysis and interpretation of this is still ongoing. Swift-XRT observations show no new associations <cit.>. § SUMMARY AND FUTURE WORKVERITAS has an active Dark Matter and Astroparticle group that leads this key science project of indirect DM detection for the collaboration. Continued observations of dSphs and the Galactic Center region are a crucial part of the VERITAS long term plan <cit.>. Unfortunately, no potential subhalo candidates have been found in the 3FGL catalog. The Galactic Center region could potentially be the best indirect γ-ray source for dark matter discovery or stringent limits on annihilation cross sections at the highest masses. Future work for dwarf galaxies involve a new combined analysis with not only new data, but with other γ-ray instruments, namely Fermi-LAT and HAWC, which is still in the preliminary stages. New analysis techniques such as boosted decision trees (BDTs) <cit.> and template reconstruction <cit.> have potential for improving the overall dark matter sensitivity of the experiment. Additionally many dSphs are extended for IACTs, so an extended source analysis could give a boost to DM sensitivity by as much as a factor of two in some cases. § ACKNOWLEDGMENTSThis research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, and by NSERC in Canada. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. The VERITAS Collaboration is grateful to Trevor Weekes for his seminal contributions and leadership in the field of VHE gamma-ray astrophysics, which made this study possible.2 99 1996PhR...267..195J Jungman, G., Kamionkowski, M., & Griest, K. 1996, physrep, 267, 1952003NuPhB.650..391S Servant, G., & Tait, T. M. P. 2003, Nuclear Physics B, 650, 3912014A A...571A..16P Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, aap, 571, A161979ARNPS..29..313S Steigman, G. 1979, Annual Review of Nuclear and Particle Science, 29, 3131965PhL....17..164ZZeldovic, Y. B., Okun, L. B., & Pikelner, S. B. 1965, PhL, 17, 1641966PhRvL..17..712CChiu, H.-Y. 1966, Physical Review Letters, 17, 712zel1965advancesZel'dovich, Y. B. 1965, Advances in Astronomy and Astrophysics, ed. Z. Kopal, Vol. 3 (Academic Press), 2422006Natur.439..695A Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006, Nature, 439, 695 2015PhRvD..91h3535G Geringer-Sameth, A., Koushiappas, S. M., & Walker, M. G. 2015, Phys. Rev. D, 91, 083535 2015ApJ...801...74G Geringer-Sameth, A., Koushiappas, S. M., & Walker, M. 2015, apj, 801, 74 2015MNRAS.446.3002B Bonnivard, V., Combet, C., Maurin, D., & Walker, M. G. 2015, MNRAS, 446, 3002 1983ApJ...272..317LLi, T.-P., & Ma, Y.-Q. 1983, apj, 272, 3172005NIMPA.551..493R Rolke, W. A., López, A. M., & Conrad, J. 2005, Nuclear Instruments and Methods in Physics Research A, 551, 493 NaheeICRC2015 Park, N., & for the VERITAS Collaboration 2015, arXiv:1508.07070 2006ApJ...650L..41Z Zucker, D. B., Belokurov, V., Evans, N. W., et al. 2006, apjl, 650, L41 2012AJ....144....4M McConnachie, A. W. 2012, aj, 144, 4 2012AIPC.1505..709CChristiansen, J., & VERITAS Collaboration. 2012, in American Institute of Physics Conference Series, Vol. 1505, American Institute of Physics2014JCAP...02..008A Aleksić, J., Ansoldi, S., Antonelli, L. A., et al. 2014, jcap, 2, 008 VerDsph2017 Archambault et al. (The VERITAS Collaboration) Phys. Rev. D 95, 082001 2011JCAP...03..051C Cirelli, M., Corcella, G., Hektor, A., et al. 2011, jcap, 3, 051 2016ApJ...821..129A Archer, A., Benbow, W., Bird, R., et al. 2016, apj, 821, 129 2010ApJ...724.1044S Su, M., Slatyer, T. R., & Finkbeiner, D. P. 2010, apj, 724, 1044 springel Springel, V., et al. 2008, MNRAS, 391, 1685 diemand Diemand, J., et al. 2008, Nature, 454, 735 nieto Nieto, D., et al., 2011, 32nd ICRC, arXiV:1109.59352013ApJS..207...28S Stroh, M. C., & Falcone, A. D. 2013, apjs, 207, 28 2015ICRC...34..868S Staszak, D., & VERITAS Collaboration 2015, 34th International Cosmic Ray Conference (ICRC2015), 34, 868 2017APh....89....1KKrause, M., Pueschel, E., & Maier, G. 2017, Astroparticle Physics, 89, 1JodiTheseProceedingsChristiansen, J. & the VERITAS Collaboration, these proceedings | http://arxiv.org/abs/1708.07447v1 | {
"authors": [
"Benjamin Zitzer",
"VERITAS Collaboration"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170824150604",
"title": "The VERITAS Dark Matter Program"
} |
Department of Science and Technology, Linköping University, SE-60174 Norrköping, Sweden École Normale Supérieure, Lyon, CRAL, UMR CNRS 5574, Université de Lyon, France École Normale Supérieure, Lyon, CRAL, UMR CNRS 5574, Université de Lyon, France École Polytechnique, CNRS, LULI, F-91128 Palaiseau, France Univ Bordeaux, IMB, UMR 5251, F-33405 Talence, France ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real and Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario de Ciudad Real, 13071 Ciudad Real, Spain KTH Royal Inst Technol, Sch Elect Engn, Space & Plasma Phys, Stockholm, Sweden Department of Science and Technology, Linköping University, SE-60174 Norrköping, Sweden 52.35.Tc 52.65.Rr 52.35.SbThe expansion of a dense plasma into a dilute plasma across an initially uniform perpendicular magnetic field is followed with a one-dimensional particle-in-cell (PIC) simulation over MHD time scales. The dense plasma expands in the form of a fast rarefaction wave. The accelerated dilute plasma becomes separated from the dense plasma by a tangential discontinuity at its back. A fast magnetosonic shock with the Mach number 1.5 forms at its front. Our simulation demonstrates how wave dispersion widens the shock transition layer into a train of nonlinear fast magnetosonic waves. Emergence of MHD structures in a collisionless PIC simulation plasma A. Ynnerman December 30, 2023 ====================================================================A thermal pressure gradient in a magnetized plasma accelerates the dense plasma towards the dilute one and a rarefaction wave develops. The collision of the expanding plasma with the dilute plasma triggers shocks if the collision speed exceeds the phase velocity of the fastest ion wave. In the rest frame of the shock, the fast-moving upstream plasma is slowed down, compressed and heated as it crosses the shock and moves downstream. This net flux adds material to the downstream plasma, which lets the shock expand into the upstream direction. Shocks have been widely examined due to their key role in regulating the transfer of mass, momentum and energy in plasma. They are most easily described in a one-dimensional geometry. Shock tube experiments <cit.>, which enforce such a geometry, examined shocks in partially magnetized and collisional plasma. Particle collisions equilibrate the plasma and macroscopic quantities like flow speed and temperature are uniquely defined. The time-evolution of these quantities is described well by the equations of single-fluid magnetohydrodynamics (MHD) if collisions are frequent enough to establish a thermal equilibrium between electrons and ions on the time scales of interest. Numerical shock tube experiments investigating the thermal expansion of plasma have also been performed in order to test single-fluid MHD codes, since the important MHD shocks emerge under such conditions <cit.>. However, not all plasma shocks are collisional. The mean free path of the particles in the plasma, in which the Earth's bow shock <cit.> is immersed, is large compared to the thickness of its transition layer and it is sustained by electromagnetic fields. Collisionless plasmas support energetic structures that are not captured by a single-fluid MHD theory and that can play a vital role in the thermalization of plasma. Examples are magnetosonic solitons <cit.> and the beams of shock-reflected particles ahead of the bow shock <cit.>, which enforce a non-stationarity of the shock <cit.>. Single-fluid MHD simulations are nevertheless used to solve problems in collisionless plasma based on the argument that they can describe the plasma dynamics on a large enough scale. Here we examine with the particle-in-cell (PIC) code EPOCH <cit.> the relaxation of a thermal pressure gradient in the presence of a perpendicular magnetic field. We thus perform a numerical shock tube experiment with collisionless plasma to test the hypothesis that the plasma evolution will resemble its equivalent in MHD. The plasma parameters are within reach for laser-plasma experiments and our results can thus be tested experimentally. The expansion speed of our blast shell remains below that considered in Ref. <cit.> and no shock reformation takes place. We use the same setup as in Ref. <cit.>, where we investigated the initial evolution of the expanding plasma and observed a lower-hybrid wave shock at the front of the expanding plasma, which has no counterpart in a single-fluid theory. Here we show that this kinetic shock is transient and that the plasma dynamics is eventually regulated by structures that exist also in the single-fluid MHD model <cit.>. Our simulation setup is as follows: we resolve one spatial dimension and three particle velocity components. Open boundary conditions are used for the fields and reflecting boundary conditions are used for the computational particles (CPs). The simulation box is large enough to separate effects introduced by the boundaries from the area of interest. The length L_0 = 0.75 m of the simulation box is subdivided into evenly spaced grid cells with the length Δ_x = 5μ m. The particle dynamics is determined in PIC simulations exclusively by the charge-to-mass ratio. We consider here the fully ionized nitrogen that is frequently used in laser plasma experiments. The plasma in the interval 0 < x̂ < 2L_0/3 consists of ions with the number density n_0 and electrons with the number density 7n_0 = 2.75 × 10^20m^-3. The electron temperature is T_e = 2.32 × 10^7 K and the ion temperature T_i = T_e /12.5. We refer to this dilute plasma as ambient plasma. A denser plasma is located in the interval -L_0/3 ≤x̂≤ 0. It consists of ions with the density 10 n_0 and the temperature T_i. Its electrons have the density 70n_0 and the temperature 3T_e. All species are initially at rest. A spatially uniform background magnetic field with the strength B_0 = 0.85 T is aligned with z. We represent the electrons and ions of the ambient plasma by 3× 10^7 CPs each while the electrons and ions of the dense plasma are each resolved by 4.5 × 10^7 CPs.The values for the parameters of the ambient plasma are listed in Table <ref> (e, μ_0, m_e, m_i, k_B, γ_e=5/3, γ_i=3: elementary charge, permeability, electron mass, ion mass, Boltzmann constant and adiabatic constants for electrons and ions). These parameters are the electron plasma frequency ω_pe and gyro-frequency ω_ce, the electron thermal speed v_the and thermal gyroradius r_ge, the ion plasma frequency ω_pi and gyro-frequency ω_ci, the lower-hybrid frequency ω_lh, the ion acoustic speed c_s, the Alfvén speed v_A and the fast magnetosonic speed V_fms. The simulation box covers with x=x̂/r_ge the interval -2000 < x < 4000. Wave numbers are multiplied with r_ge. Unless stated otherwise, times are given in units of ω_lh^-1 and frequencies in units of ω_lh. The ion density n_ion is expressed in units of n_0. We examine the late times T_0 ≤ t ≤ T_max with T_0 = 461 (190 ns). We resolve T_max= 553 (227 ns) by 1.4 × 10^7 time steps, which exceeds that in Ref. <cit.> by the factor 100.Figure <ref> shows the ion phase space density, the ion density and the magnetic field at the time T_0.The front of the rarefaction wave has reached in Fig. <ref>(a) the position x≈ -1300. The ion density and the amplitude of B_z decrease and the ion speed increases with increasing x. This structure is a fast rarefaction wave. It expands up to x≈ -400 and ends in a precursor wave that is confined to the end of the rarefaction wave. The variation of the magnetic field amplitude and density across the precursor wave are in phase and it is a fast mode. It is spatially damped in the direction of larger x. The mean velocity, temperature and density of the ions remain approximately constant until x=625, where the ion density decreases, while the ion temperature and the amplitude of B_z increase. This structure is a tangential discontinuity.It is stable and long-lived. The ion distribution, density and the magnetic field remain unchanged in the interval 625 < x <1300. The oscillations within 1300<x<1950 correspond to the transition layer of a fast magnetosonic shock. We define the downstream region as the interval 625 < x < 1300 that is enclosed by the shock and the tangential discontinuity. A dilute population of hot ions is found to the left of the discontinuity and it is confined by it. These ions are gradually accelerated up to a velocity modulus ≈ 5× 10^5 m/s. They are accelerated by their interaction with electrostatic fluctuations, which are strong due to the large electron temperature and plasma density <cit.>. The denser population of hot ions in the interval -1000<x<-500 and v_x ≈ 0 in Fig. <ref>(a) has also been observed at the front of unmagnetized rarefaction waves <cit.> and is thus probably tied to ion acceleration by the spatially nonuniform electric field noise. The ions reach a peak speed of 2.5× 10^6 m/s (not shown). Some ions travel from the shock ahead of it. Their number density is too low to enforce a shock reformation.Figure <ref> examines the ion- and magnetic field distributions close to the tangential discontinuity at x≈ 625.Figure <ref>(a) shows that the ions move at the spatially uniform mean speed v_b ≈ 4.1 × 10^5 m/s or v_b ≈ V_fms/2. The simulation frame equals the upstream frame and v_b is thus the speed of the downstream plasma in the upstream frame. The ion phase space density, the value of B_z and that of n_ion change rapidly over 5r_ge and reach their respective downstream values B_z ≈ 1.3 T and n_ion≈ 1.5 at x = 640. The change in B_z is sustained by an electron drift along y and the electron temperature to the right of the discontinuity is about 100 eV below that to the left, which is about 3 keV (not shown). The density change at x≈ 625 yields a thermal pressure gradient force. The thermal pressure is P_th(x) ≈ n_e(x) k_B T_j and its change is Δ_P = P_th(x>625)-P_th(x<625). The gradient of the magnetic pressure P_B = B_z^2(x)/2μ_0 yields a force that points in the opposite direction. Figure <ref> suggests changes in the electron density and magnetic field amplitude of 25n_0 and B_0, which gives |P_B(x>625)-P_B(x<625)|≈ 0.6Δ_P. The moving magnetic field structure is exposed to the ram pressure of the upstream medium at x≈ 1950. The ram pressure P_R = n_0 m_i v_b^2 ≈ 0.5 Δ_P balances the difference between both pressures at x=625. The pressure balance implies that the boundary is stationary in the downstream frame. There is no net ion flow across this tangential discontinuity since the Larmor radius of the energetic ions, which move with a few 100 km/s in the downstream frame, is only about 100r_g. Figure <ref> shows the distributions of the ion phase space density, the ion density and the magnetic field at the front of the fast magnetosonic shock. The upstream ions are located in the interval x>1940.Figure <ref>(a) reveals ion velocity oscillations with an amplitude ≈ v_b. The distribution shows cusps at the maxima and the waves are not linear. This is confirmed by the non-sinusoidal oscillations of the ion density and the magnetic field distributions in Figs. <ref>(b,c). The magnetic field distribution is approximated well by a hyperbolic secant and the density follows approximately its square. The magnetic pressure ∝ B_z^2 follows the thermal pressure ∝ n_ion, which suggests that these waves are fast magnetosonic waves. The dispersion relation of the structures in the shock transition layer will determine the underlying wave modes. These are confined to the downstream plasma and hence we must evaluate their properties in the downstream frame. The ion density and magnetic field are 1.5n_ion and 1.3T in the downstream region, which give a lower-hybrid frequency ω_lh^* ≈ 1.5 ω_lh. We select x^* = x-v_bt^*-1920 and t^* = (t-T_0) as the transformation from the box frame into the downstream frame. Figure <ref>(a) depicts B_z(x^*,t^*) at the front of the expanding plasma. The speed of the wave front is v_f ≈ 8.5 × 10^5 m/s. This speed corresponds to the shock speed measured in the downstream frame x^* and it is thus well below the fast magnetosonic speed V^*_fms≈ 1.3 × 10^6 m/s in the downstream plasma. The speed of the wave front in the upstream medium is supersonic with v_f + v_b ≈ 1.5 V_fms.The power spectrum of B_z(x^*,ω) in Fig. <ref>(b) peaks below ω_lh^* and the wave frequency, at which the spectrum peaks, decreases with increasing x^*. A wave harmonic is observed close to x^* ≈ 0, where the amplitude of B_z is in the nonlinear regime (See Fig. <ref>(c)).Figure <ref>(c) compares the dispersion relation of the fast magnetosonic mode ω / k = V^*_fms in the electromagnetic limit k→ 0 to the power spectrum of the noise, which we measured in a separate PIC simulation. That simulation modelled a spatially uniform plasma in a thermal equilibrium with the plasma parameters of the downstream region in Fig. <ref>(a). The noise distribution in PIC simulations peaks at values (ω,k), which correspond to eigenmodes of the system <cit.>. The dispersion relation of the noise takes into account also the electrostatic component of the fast magnetosonic mode, which becomes important close to ω^*_lh. The wave power in Fig. <ref>(b) peaks in the frequency band 0.75 < ω / ω^*_lh < 1, where the phase speed of the fast magnetosonic wave is well below V^*_fms and where the phase speed decreases with increasing k. The dispersion relation thus explains firstly why the wave front in Fig. <ref>(a) moves at the speed v_f ≈ 0.65 V^*_fms and, secondly, why the wave frequency decreases with increasing x^* in Fig. <ref>(b). The steepening of the fast magnetosonic shock results in waves with a larger k that fall behind the shock due to their lower phase speed.In summary we have tracked with a 1D PIC simulation the expansion of a dense plasma into a magnetized ambient plasma over unprecedented time scales and we could observe the emergence of MHD structures. Their emergence was made possible by the low speed of the shock, which allowed it to dissipate the directed flow energy of the inflowing plasma without the need of reflecting many of its ions back upstream. We observed a fast rarefaction wave, which ended in a precursor wave, a tangential discontinuity and a fast magnetosonic shock. The shock involved large wave numbers, in which the fast magnetosonic wave branch is dispersive. We have shown for the first time that the dispersive nature of the fast magnetosonic wave branch transforms the fast magnetosonic shock into a train of non-linear oscillations, which gives rise to a broad shock transition layer. M. E. D. acknowledges financial support by a visiting fellowship of CRAL. The simulations were performed on resources provided by the Grand Equipement National de Calcul Intensif (GENCI) through grant x2016046960 and by the Swedish National Infrastructure for Computing (SNIC) at HPC2N (Umeå). Dolder60 K. Dolder, and R. Hide, Rev. Mod. Phys. 32 770 (1960). Borisov71 M. B. Borisov, S. G. Zaitsev, E. I. Chebotareva, and E. V. Lazareva, Fluid Dyn. 6 501 (1971). Brio88 M. Brio, and C. C. Wu, J. Comput. Phys. 75 400 (1988). Falle98 S. A. E. G. Falle, S. S. Komissarov, and P. Joarder, Mon. Not. R. Astron. Soc. 297 265 (1998). Balogh05 A. Balogh, S. J. Schwartz, S. D. Bale, M. A. Balikhin, D. Burgess, T. S. Horbury, V. V. Krasnoselskikh, H. Kucharek, B. Lembege, E. A. Lucek, E. Mobius, M. Scholer, M. F. Thomsen, and S. N. Walker, Space Sci. Rev. 118, 155 (2005). Stasiewicz03 K. Stasiewicz, M. Longmore, S. Buchert, P. K. Shukla, B. Lavraud, and J. Pickett, Geophys. Res. Lett. 30 2241 (2003). Guerolt17 R. Guerolt, Y. Ohsawa, and N. J. Fisch, Phys. Rev. Lett. 118 125101 (2017). Eastwood05 J. P. Eastwood, E. A. Lucek, C. Mazelle, K. Meziane, Y. Narita, J. Pickett, and R. A. Treumann, Space Sci. Rev. 118 41 (2005). Chapman05 S. C. Chapman, R. E. Lee, and R. O. Dendy, Space Sci. Rev. 121 5 (2005). Burgess07 D. Burgess, and M. Scholer, Phys. Plasmas 14 012108 (2007). Marcowith16 A. Marcowith, A. Bret, A. Bykov, M. E. Dieckman, L. O. Drury, B. Lembege, M. Lemoine, G. Morlino, G. Murphy, G. Pelletier, I. Plotnikov, B. Reville, M. Riquelme, L. Sironi, and A. S. Novo, Rep. Prog. Phys. 79 046901 (2016). Sundberg17 T. Sundberg, D. Burgess, M. Scholer, A. Masters, and A. H. Sulaiman, Astrophys. J. 836 L4 (2017). Arber15 T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, Plasma Phys. Controll. Fusion 57 113001 (2015). Dieckmann16 M. E. Dieckmann, G. Sarri, D. Doria, A. Ynnerman, and M. Borghesi, Phys. Plasmas 23 062111 (2016). Myong97 R. S. Myong, and P. L. Roe, J. Plasma Phys. 58 521 (1997). Dieckmann04 M. E. Dieckmann, A. Ynnerman, S. C. Chapman, G. Rowlands, and N. Andersson, Phys. Scripta 69 456 (2004). Sarri10 G. Sarri, M. E. Dieckmann, I. Kourakis, and M. Borghesi, Phys. Plasmas 17 082305 (2010). | http://arxiv.org/abs/1708.07993v1 | {
"authors": [
"M E Dieckmann",
"D Folini",
"R Walder",
"L Romagnani",
"E d'Humieres",
"A Bret",
"T Karlsson",
"A Ynnerman"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170826161811",
"title": "Emergence of MHD structures in a collisionless PIC simulation plasma"
} |
[email protected] Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, [email protected] Institute of Science and Technology (IST) Austria, 3400 Klosterneuburg, AustriaInstitute of Photonics and Electronics, The Czech Academy of Sciences, Chaberská 57, 182 00 Prague, Czech RepublicDepartment of Physics and Astronomy, University of Calgary, Calgary T2N 1N4, Alberta, Canada Institute for Quantum Science and Technology, University of Calgary, Calgary T2N 1N4, Alberta, CanadaBiomedical Optics Research Laboratory, Department of Neonatology, University Hospital Zurich, University of Zurich, CH-8091 Zurich, Switzerland Research Office for Complex Physical and Biological Systems (ROCoS), CH-8038 Zurich, SwitzerlandDepartment of Medical Physics, Isfahan University of Medical Sciences, Isfahan, IranDepartment of Oncology, University of Alberta, Cross Cancer Institute, Edmonton T6G 1Z2, Alberta, Canada Department of Physics, University of Alberta, Edmonton,AB T6G 2E1, Canada In this paper, we discuss biological effects of electromagnetic (EM) fields in the context of cancer biology. In particular, we review the nanomechanical properties of microtubules (MTs), the latter being one of the most successful targets for cancer therapy. We propose an investigation on the coupling of electromagnetic radiation to mechanical vibrations of MTs as an important basis for biological and medical applications. In our opinion optomechanical methods can accurately monitor and control the mechanical properties of isolated MTs in a liquid environment. Consequently, studying nanomechanical properties of MTs may give useful information for future applications to diagnostic and therapeutic technologies involving non-invasive externally applied physical fields. For example, electromagnetic fields or high intensity ultrasound can be used therapeutically avoiding harmful side effects of chemotherapeutic agents or classical radiation therapy.Towards non-invasive cancer diagnostics and treatment based on electromagnetic fields, optomechanics and microtubules J. A. Tuszynski December 30, 2023 ===================================================================================================================== § INTRODUCTION In spite of many efforts and important advances in cancer diagnostics and treatment, there are still millions of people dying each year throughout the world due to cancer, and the ”war on cancer” that was declared by the US President Richard Nixon in 1971 is still far from being won 46 years later. Conversely, the cancer incidence will most probably increase further <cit.> while the progress in cancer treatments has been stagnant for decades <cit.> with the exception of recent introduction of immmunotherapies. We should mention, however, that improvements in diagnostic imaging, especially using magnetic resonance imaging (MRI), and novel surgery approaches have led to progress in the field of oncology with improvements in clinical outcomes accruing gradually over the past few decades. The new surgery techniques for example enable a better delineation of the tumour area and the surrounding structures, allowing the medical oncologists to reduce the tumor margin and hence decrease damage to the normal tissue <cit.> leading to improved quality of life of the patentients. However, truly cutting-edge technologies are still to be discovered and implemented within the arsenal of cancer therapy modalities. In particular, the roles of biochemical signaling pathways, and biophysical aspects (nanomechanical states of the cellular- and subcellular structures, endogenous bioelectric current/potential and electromagnetic field) have not been fully understood and exploited in the field of oncology <cit.>.The aim of the present review article is to outline why it is worthwhile to focus on biophysical aspects with regard to future diagnosis and treatment approaches for cancer. In particular, we discuss why the bioelectric and mechanical properties of microtubules might play a significant role in cancer, whose better understanding could lead to the development of novel therapies. Biolectricity is a basic phenomenon associated with cellular and subcellular structures <cit.>. Most of the subcellular biomolecules (e.g. DNA, RNA, tubulin, actin, septin, etc.) are either charged and hence surrounded by counter-ions or endowed with high electric dipole moments that can enagage in dipole-dipole interactions and polarize electrically their local environment. Living organisms are replete with both moving and oscillating electric charges and can thus be regarded as complex electrochemical and mechanical systems. Complex patterns of direct current (DC) electric fields present within living organisms are key factors in morphogenesis and contain part of the information needed to produce a three-dimensional organism <cit.>. These factors, in our opinion, need to be addressed when finding novel methods of cancer diagnosis and treatment.§ ELECTROMAGNETIC FIELDS AFFECT CANCER CELLS It has been shown that extremely low-frequency (ELF), pulsed electromagnetic fields (PEMF) and sinusoidal electromagnetic fields (SEMF) can induce tumor cell apoptosis, inhibit angiogenesis, impede proliferation of neoplastic cells, and cause necrosis non-invasively, whereas human lymphocytes are negligibly affected <cit.>. Some studies describe the effects of intense (> 0.1 MV/m) nanosecond (10-300 ns) pulsed electric fields on mammalian cell structure and function. As the pulse durations decrease, effects on the plasma membrane decrease and effects on intracellular signal transduction mechanisms increase <cit.>. Low-power EMF within the range of 0.1-40 MHz may impair the DNA strand and cause inhibition of proliferation of the gallbladder cancer cells, and these effects are related to the frequency of the electromagnetic fields but not in a linear fashion <cit.>. Kirson et al. <cit.> have recently demonstrated that 100 KHz to 1 MHz AC fields have significant specific effects on dividing cells. The basis of these effects during cytokinesis was hypothesized to be the unidirectional dielectrophoretic forces induced by the inhomogeneous fields at the cleavage furrow separating the daughter cells that interfere with the orientation of spindle microtubules <cit.>. A review of other possible mechanisms involved in the interactions of these fields, dubbed TTFields (Tumor Treating Fields) and cancer cells has been recently published <cit.>. It is worth noting that in addtion to microtubules, actin filaments, DNA and even ion channels may be affected by TTFields and investigations into specific molecular mechanisms are on-going. As an additional electromagnetic mechanism with potential cancer treatment application, a study was undertaken to examine whether millimeter electromagnetic waves (MMWs) irradiation (42.2 GHz) can inhibit tumor metastasis enhanced by cyclophosphamide (CPA), an anticancer drug. <cit.>.Concerning high-power EMF, for example Elson (2009) focused on the potential of strong magnetic fields to play a role in cancer treatment. Results of this study show that pulsed magnetic field (PMF) in combination with ultraviolet C (UVC) have the ability to augment the cell killing effects of UVC radiation. In addition, the effects appear to be greater when PMF and UVC are applied at the same time <cit.>. Mitochondria are well known to play an important role in apoptosis. Steep pulsed electric fields (SPEF) could induce apoptosis markedly (P-value less than 0.01); SPEF with lower voltage (200V) and longer width (1.3 μ s) could induce apoptosis more effectively than SPEF with higher voltage (600V) and shorter width (100ns). These experimental results provide a possible mechanism and parameter selection basis for tumor treatment using SPEF <cit.>. There are many reports of enhanced transcription and replication in different cell culture systems exposed to electromagnetic fields, and reports of cytoreduction (necrosis and apoptosis) in tumors transplanted into animals exposed to similar, often much stronger electromagnetic fields, but where heating is negligible. Although the mechanism of inducing apoptosis has not been characterized yet, one major candidate for the initiation of such a process is the production of numerous breaks in DNA, and the inhibition of DNA repair processes, leading to the initiation of the apoptotic (programmed cell death) process <cit.>. Interestingly, electromagnetic frequencies at which cancer cells become sensitive appear to be tumor-specific and hence treatment with tumor-specific frequencies is feasible, well tolerated and may have biological efficacy in patients with advanced cancer. A study that examined a total of 163 patients diagnosed with various types of cancer has identified a total of 1524 distinct frequencies ranging from 0.1 Hz to 114 kHz. Most frequencies (57 to 92 percent) were specific for a single tumor.These observations suggest that electromagnetic fields, which are amplitude-modulated at tumor-specific frequencies, do not act solely on tumors but may have wide-ranging effects on tumor-host interactions, e.g. immune modulation <cit.>. In vitro effects of electromagnetic fields appear to be related to the type of electromagnetic field applied. It has been shown that human osteoblasts display effects of BEMER (Bio-Electro Magnetic Energy Regulation) type electromagnetic field (BTEMF) on gene regulation. Effects of BTEMF on gene expression in human mesenchymal stem cells and chondrocytes have been analyzed. Results indicate that BTEMF in human mesenchymal stem cells and chondrocytes provide the first indications to understanding therapeutic effects achieved with BTEMF stimulation <cit.>.Phenotypic changes in human breast cancer cells following low-level magnetic field (MF) exposure previously reported. Proteomic methods were used to investigate the biochemical effect of MF exposure in SF767 human glioma cells. Protein alterations were studied after exposure to 1.2 microTesla (microT) MF [12 milliGauss (mG), 60 Hertz (Hz)]. The results suggest that the analysis of differentially expressed proteins in SF767 cells may be useful as biomarkers for biological changes caused by exposure to magnetic fields <cit.>.Qutob et al. (2006) showed that there was no evidence that non-thermal RF fields can affect gene expression in cultured U87MG glioblastoma cells relative to the non-irradiated control groups, whereas exposure to heat shock at 43 degrees C for 1 h up-regulated a number of typical stress-responsive genes in the positive control group <cit.>. Gap junction genes are recognized as tumor suppressors <cit.> and effects on gap junctional communication also provide an appealing model for explaining tumor growth induced by exposure to weak magnetic fields. ELF exposure generally does not transmit nearly enough energy to cause mutagenesis of DNA, but has been shown to affect gap junction states and thus potentially to control proliferation and differentiation <cit.>. In the MHz region, several studies investigated the effect of the application of electromagnetic field in the MHz for cancer treatment. Normally, the intensity applied was high, inducing thermal effects. That the choice of the specific modulation of the EMF is important was shown by Andocs et al. <cit.>, demonstrating the the application of modulated EMF (13.56 MHz) causes a synergistically anticancer effect due to a hermal and a non-thermal mechanism triggered (modulated electrohyperthermia). Subsequent work showed that this treatment causes DNA-fragmentation <cit.> and up-regulation of heat-shock proteins <cit.> in the cancer cells. The superiority of using 13.56 MHz modulated electrohyperthermia in comparison to only classical hyperthermia to treat cancer was demonstrated recently <cit.>.In the GHz range, most of the studies used strong GHZ EMF for treating tumors by induce thermal effects <cit.>. However, EMF in the GHz range can also act as a co-cancerogen <cit.>, making the application of GHz EMF for cancer treatment not as an optimal approach.Terahertz (THz) radiation occupies a broad band of the EM spectrum between microwave and infrared frequencies, and is therefore non-ionizing. The THz region covers the frequency range from 0.1 to 10 THz and it offers non-invasive diagnostic capabilities that fill the “gaps between x-rays, MRI, and the isible range <cit.>. Applications of THz waves to the diagnosis of melanoma <cit.>, basal cell carcinoma <cit.>, and breast cancer <cit.>have already been demonstrated. THz technology has led to the development of commercially available diagnostic medical applications such as THz Pulsed Spectroscopy <cit.> and THz Pulsed Imaging <cit.>, which offer excellent contrast between diseased and healthy tissues. The first clinical trials of THz imaging as an intra-operative tool during cancer surgery are underway <cit.>. Studies on stem cells suggest that exposure to broad-spectrum THz pulses affects cell differentiation and gene expression <cit.> at both the transcript and protein levels. The mechanism by which THz radiation interacts with biological systems is fundamentally different to that of conventional ionizing therapies, due to its resonant effects on cell membranes <cit.>, proteins <cit.> and nucleic acids <cit.>. It has been recently demonstrated that intense, picosecond THz pulses induce changes in cellular functions<cit.>. Exposure of human tissue to intense THz pulses was found to activate the DNA damage response (DDR), and affect expression levels of many proteins, especially cell-cycle regulatory proteins offering a potential for therapeutic applications of this novel modality. In addition, a combination of this modality can be considered with standard chemotherapy since sub-μs pulsed electric fields applied to tumours through electrodes have been shown to permeabilize tumour cell membranes to cytotoxic agents<cit.>. Consequently, intense THz pulses may significantly lower the required therapeutic doses of cytotoxic drugs. However, the fundamental mechanisms of interaction of THz radiation with biological systems so far remain elusive. § MICROTUBULES It is well known that MTs, microfilaments and intermediate filaments are the main components of cytoskeleton of eukaryotic cells. MTs are the most rigid protein polymer among the three types of cytoskeletal filaments, which form the architecture of the cell. They exhibit unique physical behaviour and form special structures well suited for their own cellular functions <cit.>. The structure of MTs is cylindrical, and it typically involves 13 parallel protofilaments, which are connected laterally into hollow tubes. MTs have 25 nm external and 15 nm internal diameters. The length of MTs can vary from tens of nanometers to hundreds of microns <cit.>. MT biological functions rely on two essential properties. First, they are dynamic polymers that are assembled and disassembled rapidly in a fashion coordinated with motile reactions; second, they are relatively rigid structures able to resist the pico-Newton level forces exerted by kinesin and dynein motor proteins, and they provide the required mechanical stiffness for cilia and flagella <cit.>.Mechanical properties of MTs largely determine their functions. Quantifying the way they resist mechanical deformation by determining their Young's and shear modulus can lead to a better understanding of all the vital physiological mechanisms in which MTs are involved. For instance, it would be favorable for the stable MTs of the axon to be stiff and straight to support the extended structure required for long-distance axonal transport. Conversely, MTs in a proliferating cell should be dynamic and flexible to enable rapid redistribution during transitions between interphase and mitosis <cit.>. However, measuring and understanding MTs' mechanical properties is not a simple task. Two decades of measurements involving different techniques such as optical tweezers <cit.>, hydrodynamic flow <cit.>, atomic force microscope (AFM) <cit.>, and persistence length observations <cit.>, resulted in values of elastic (shear and Young's) modulus spanning a range of values between 1 MPa and 7 GPa <cit.>. For instance, experiments involving the MTs with lengths 24-68 nm yielding a value of 2 GPa for MTs assembled from pure bovine-brain tubulin <cit.>.Short MTs are flexible due to a low value of the shear modulus while longer tubes become more rigid, which is when the Young's modulus dominates the mechanical behaviour. Measurements on longer MTs would therefore provide better estimates of the Young's modulus, because neglecting the influence of shearing would introduce a smaller error. Since microtubules in biological conditions are often subject to a dynamic load, vibration analysis suggests itself as a method to study their dynamic response. Vibration normal modes describe the preferential pattern of structural dynamics of a microtubule, whereby its response to a time-varying force can be represented by a combination of these vibration modes. In addition, there are several hypotheses that ascribe biological relevance to the vibrations themselves. Furthermore, microtubule vibration mode patterns are reminiscent to buckling pattern, so the underlying mathematical apparatus is similar <cit.>.There are generally two modeling approaches that have been developed to study microtubule mechanics and vibrations. A continuum mechanics model <cit.> and a discrete model <cit.>. The gap between high-precision molecular modeling and continuum modeling can be bridged by an atomistic-continuum model; the first of such models to study the topic herein was presented by Liew et al. <cit.>. These theoretical treatments of the microtubule structure disclose their vibrational normal modes in a wide frequency range from acoustic to GHz frequencies <cit.>. For instance It has been shown that the microtubule lengths L in terms of their fundamental bending mechanical resonance frequencies between 100 and 200 kHz in vitro (20 ^∘C) <cit.>. Elastic wave propagation in MTs was analyzed in different works <cit.>. Dependence of frequency or velocity of propagation on the wave vector was evaluated for isotropic and orthotropic shell models with different parameters. In particular, the resonant condition for a 10 nm long MT corresponding to half of the wavelength at a frequency of about 460 MHz may occur for longitudinal oscillations with Young's modulus 1.7-2 GPa. These results hold for the orthotropic microtubule axisymmetric (n = 0) and nonaxisymmetric (n = 1) shell models <cit.>. Numerical calculations based on recently obtained experimental data for Young's modulus of MT, show that MT-water system supports interface elastic waves with maximal frequencies in a GHz range. In fact, <cit.> performed theoretical analysis for elastic vibrations of MT immersed in water and found that this system supports nonradiative elastic waves localised in the vicinity of the MT wall with maximal frequencies of order of tens of GHz. In the long wavelength limit, there exist three axisymmetric acoustic waves with propagation speed of approximately 200-600 m/s and an infinite set of helical waves with a parabolic dispersion law <cit.>. The role of mechanical vibrations of MT is not known in biology so far <cit.>. The fundamental issue for any biological relevance of MT vibrations or further phenomena assuming MT vibrations is the damping of MT vibrations. It is generally considered that protein and MT normal mode vibrations are overdamped <cit.>. Some works estimate that, depending on the type of the vibration mode and lowered coupling with the MT viscous environment, the quality factor Q of the vibration modes may be in the range of 0.01 - 10 <cit.>, hence reaching to underdamped regime. However, there are no solid experimental data on the damping of MT vibrations so far, only extensive theoretical works.Tubulin, a MT subunit, is a protein which has rather high charge and dipole moment compared to most other proteins <cit.>. Hence, it is natural to suggest that MT vibrations, if underdamped and excited, will be accompanied by an electrical field of the same frequency as was originally proposed by Pokorný et al. <cit.>. Several recent works developed this idea <cit.> including electromechanical vibrational models of whole cell microtubule network <cit.> and multi-mode vibration of single microtubule <cit.>. Charge and dipole moment of MT as well as of tubulin, other proteins and polar nanoobjects in general is also a key to coupling external electromagnetic field to vibrations of such objects; the coupling is significant only when vibrations are sufficiently underdamped <cit.>. Single protein normal vibration modes are in the range of cca. 0.03–3 THz <cit.> together with MT vibration band (kHz - GHz) can hypothetically enable interaction with electromagnetic field at frequencies across many orders of magnitude. It was hypothesized that vibrations of MT cytoskeleton generate coherent electromagnetic field which plays role in organization of processes in living cells and that this field is perturbed in cancer <cit.>. Within this hypothesis it is considered that the damping of MT vibrations is caused by the ambient medium, i.e., by the cytosol water. It is proposed that in cancer the changes in mitochondrial metabolism lead to change of the water structure around MT and to increased damping of Mt vibrations might cause a shift in the resonance frequency of oscillations in cells. Frequency changes in cancer cells were also predicted by Fröhlich <cit.>. A peculiar cancer diagnostic method developed by Vedruccio <cit.> claimed to exploit frequency selective effects of the interaction of the external electromagnetic field with cancer cells was interpreted using this hypothesis. Hypotheses of Pokorný and Fröhlich also inspired a number of experimental works aiming to directly electronically detect electromagnetic activity of living cells in radiofrequency and microwave bands <cit.>. However, solid evidence for such cellular electromagnetic activity remains elusive <cit.>. However, unless the possibility of underdamped MT vibrations and endogenous excitation is proved, these hypotheses of highly coherent biological electromagnetic field remain unrealistic. One important piece of puzzle could be brought by elucidation of one of the crucial assumptions in these hypotheses: low damping of microtubule vibrations. Knowledge of dynamic mechanical properties is also essential to assess effects of ns intense electric pulses which have been demonstrated to affect cytoskeleton <cit.>, thus opening a new avenue how to disrupt cell divison with potential cancer applications. However, exact mechanisms of action remains unclear.Thus, to enable new perspective diagnostic and therapeutic methods based on MT monitoring and manipulation, a rigorous experimental analysis of microtubule vibrations and dynamic mechanical properties is needed.§ MONITORING MECHANICAL VIBRATIONS OF MICROTUBULES VIA OPTOMECHANICAL COUPLINGThe emerging field of optomechanics is concerned with the study of the mechanical effects of light on mesoscopic and macroscopic mechanical oscillators. These phenomena have been realized in optomechanical systems consisting ofan optical cavity with a movable end-mirror or with a membrane-in-the middle. The radiation pressure exerted by the light inside the optical cavity couples the moving mirror or the membrane which acts as a mechanical oscillator to the optical field. This optomechanical coupling has been employed for a wide range of applications such as the cavity cooling of microlevers and nanomechanical resonators to their quantum mechanical ground state <cit.>, producing high precision detectors for measuring weak forces and small displacements and also for fundamental studies of the transition between the quantum and the classical world <cit.>.Optomechanical systems can also be applied for the sensitive detection of physical quantities such as spin <cit.>, atomic/molecular mass <cit.>, the concentration of biologically relevant molecules <cit.>, and thermal fluctuations <cit.>, as well as for frequency conversion <cit.>. Nanomechanical resonators (NMRs) with resonance frequencies in the GHz regime can be now fabricated <cit.> and they are suitable candidates for the study of the quantum behavior at the mesoscopic scale <cit.>. These GHz NMRs are characterized by reduced dimensions and therefore by very low masses, and at the same time, in this regime the nonlinear behavior of the mechanical systems becomes more relevant, consequently offering interesting theoretical <cit.> and experimental challenges <cit.>. These high-frequency resonators operating in the nonlinear regime open up new possibilities for the realization of novel devices and applications of NMR and nanoelectromechanical systems <cit.>. The optomechanical system can also be employed to observe the vibrations of isolated microtubules <cit.>. Information about the microtubule mechanical vibrations can be obtained by coupling the microtubule to an optical cavity. In fact, the optomechanical coupling between the microtubule and the optical field of the cavity modifies the response of the cavity field results in appearance of electromagnetically inducedtransparency peaks in the transmission feature of the optical probe field. The center frequency and linewidth of the transparency peak give the resonance frequency and damping rate of the vibrational mode of the microtubule. By properly selecting the parameters of the system, one can observe up to 1GHz vibration for the microtubule. The dielectric properties of the microtubule, however, raises the possibility to control the vibration of the microtubule by positioning tip electrodes close to surface of the microtubule.Applying voltage on the electrode plates creates an effective external force on the microtubule, modifies the resonance frequency of the microtubule vibration <cit.>.§ SUMMARYMicrotubules, key structures forming the cellular skeleton, have been among the most successful targets for anticancer therapy. Any interference with their functioning, especially during mitosis, can control the replication of a cancer cell. However, chemotherapeutic techniques to disrupt microtubules have several side effects on healthy cells, making chemotherapy a less than ideal modality to suppress cancer proliferation. It has been predicted that the mechanical properties of microtubules (such as their vibration frequencies) are different in cancer cells compared to healthy cells. Inspired by this, we believe that cancer treatment (and detection) may be possible based on the detection and control of microtubule mechanical vibrations in cells exposed to non-invasive external radiations (e.g. electromagnetic or ultrasound). We have proposed an optomechanical method to control and read out the vibrations of an isolated microtubule. This can help determine which frequencies can cause breakage in microtubules of cancer cells using resonance effects in microtubules from an external field. Moreover, this method may help to recognize cancer cells based on special frequencies in microtubules. So far, measuring a broad spectrum of mechanical vibrations of microtubules has not been an easy task, and there are only a small number of studies in this context. In order to improve our knowledge about the mechanical vibrations of microtubules, we proposed an optomechanical technique for measurement of microtubule dynamics at room temperature. Our approach is a step forward for monitoring mechanical frequencies of microtubules in a broad spectral range for a potential application in medical diagnosis and treatment. This may help scientists in the future to supplement standard cancer chemo- and radiotherapy approaches with non-ionizing physical fields to be used as therapeutic and possibly even diagnostic methods. It is a great goal to reach simple and non-invasive methods for diagnosis and treatment of diseases like cancer. We have proposed that an optomechanical setup can help us to monitor the all possible vibrations in a MT in vitro for a potential application in cancer diagnosis and treatment. In fact, if we know what frequency can destruct the MT structure it can beuseful for cancer treatment since cancer cells need MTs for cell division and if they are disrupted or destroyed their growth can be stopped or tumor can be vanished. This type of disruption can be done via an external weak EM signal or an ultrasound signal which can be focused on the cancerous cells in vivo and their growth can be controlled by these external signals with special frequencies.§ ACKNOWLEDGEMENTS The work of SB has been supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska Curie grant agreement No MSC-IF 707438 SUPEREOM. JAT gratefully acknowledges funding support from NSERC (Canada) for his research. MC acknowledges support from the Czech Science Foundation, projects no. 15-17102S and 17-11898S and he participates in COST Action BM1309, CA15211 and bilateral exchange project between Czech and Slovak Academies of Sciences, no. SAV-15-22. 10 plain url<#>1urlprefixURLSmith_2009 B. D. Smith, G. L. Smith, A. Hurria, G. N. Hortobagyi, and T. A. Buchholz, “Future of cancer incidence in the united states: burdens upon an aging, changing nation,” Journal of clinical oncology, vol. 27, no. 17, pp. 2758–2765, 2009.Jemal_2010 A. Jemal, M. M. Center, C. DeSantis, and E. M. Ward, “Global patterns of cancer incidence and mortality rates and trends,” Cancer Epidemiology and Prevention Biomarkers, vol. 19, no. 8, pp. 1893–1907, 2010.Soto C. Sonnenschein and A. M. Soto, “Why is it that despite signed capitulations, the war on cancer is still on?,” Organisms. Journal of Biological Sciences, vol. 1, no. 1, 2017.hanahan2014rethinking D. Hanahan, “Rethinking the war on cancer,” The Lancet, vol. 383, no. 9916, pp. 558–563, 2014.Alpuente L. A. Alpuente, A. M. López, and R. Y. Tur, “Glioblastoma: changing expectations?,” Clinical and Translational Oncology, vol. 13, no. 4, p. 240, 2011.Levin M. Levin, “Bioelectromagnetics in morphogenesis,” Bioelectromagnetics, vol. 24, no. 5, pp. 295–315, 2003.suresh2007biomechanics S. Suresh, “Biomechanics and biophysics of cancer cells,” Acta Materialia, vol. 55, no. 12, pp. 3989–4014, 2007.funk2015endogenous R. H. Funk, “Endogenous electric fields as guiding cue for cell migration,” Frontiers in physiology, vol. 6, 2015.levin2014molecular M. Levin, “Molecular bioelectricity: how endogenous voltage potentials control cell behavior and instruct pattern regulation in vivo,” Molecular biology of the cell, vol. 25, no. 24, pp. 3835–3850, 2014.levin2012molecular M. Levin, “Molecular bioelectricity in developmental biology: new tools and recent discoveries,” Bioessays, vol. 34, no. 3, pp. 205–217, 2012.Northrop H. S. Burr and F. S. C. Northrop, “Evidence for the existence of an electro-dynamic field in living organisms,” Proceedings of the National Academy of Sciences, vol. 25, no. 6, pp. 284–288, 1939.Berg H. Berg, B. GÜnther, I. Hilger, M. Radeva, N. Traitcheva, and L. Wollweber, “Bioelectromagnetic field effects on cancer cells and mice tumors,” Electromagnetic biology and medicine, vol. 29, no. 4, pp. 132–143, 2010.Beebe S. J. Beebe, P. F. Blackmore, J. White, R. P. Joshi, and K. H. Schoenbach, “Nanosecond pulsed electric fields modulate cell function through intracellular signal transduction mechanisms,” Physiological measurement, vol. 25, no. 4, p. 1077, 2004.Chen P. Chen, Y. Yang, H. Tao, and H. Yang, “Effects of electromagnetic fields of different frequencies on proliferation and dna damage of gallbladder cancer cells,” Nan fang yi ke da xue xue bao= Journal of Southern Medical University, vol. 26, no. 3, pp. 328–330, 2006.Kirson1 E. D. Kirson, V. Dbalỳ, F. Tovaryš, J. Vymazal, J. F. Soustiel, A. Itzhaki, D. Mordechovich, S. Steinberg-Shapira, Z. Gurvich, R. Schneiderman, et al., “Alternating electric fields arrest cell proliferation in animal tumor models and human brain tumors,” Proceedings of the National Academy of Sciences, vol. 104, no. 24, pp. 10152–10157, 2007.Kirson2 E. D. Kirson, Z. Gurvich, R. Schneiderman, E. Dekel, A. Itzhaki, Y. Wasserman, R. Schatzberger, and Y. Palti, “Disruption of cancer cell replication by alternating electric fields,” Cancer research, vol. 64, no. 9, pp. 3288–3295, 2004.Tuszynski J. A. Tuszynski, C. Wenger, D. E. Friesen, and J. Preto, “An overview of sub-cellular mechanisms involved in the action of ttfields,” International journal of environmental research and public health, vol. 13, no. 11, p. 1128, 2016.Logani M. K. Logani, I. Szabo, V. Makar, A. Bhanushali, S. Alekseev, and M. C. Ziskin, “Effect of millimeter wave irradiation on tumor metastasis,” Bioelectromagnetics, vol. 27, no. 4, pp. 258–264, 2006.Elson E. Elson, “I. the little explored efficacy of magnetic fields in cancer treatment and postulation of the mechanism of action,” Electromagnetic biology and medicine, vol. 28, no. 3, pp. 275–282, 2009.Mi Y. Mi, C. Sun, C. Yao, C. Li, D. Mo, L. Tang, and H. Liu, “Effects of steep pulsed electric fields (spef) on mitochondrial transmembrane potential of human liver cancer cell,” in Engineering in Medicine and Biology Society, 2007. EMBS 2007. 29th Annual International Conference of the IEEE, pp. 5814–5817, IEEE, 2007.Barbault A. Barbault, F. P. Costa, B. Bottger, R. F. Munden, F. Bomholt, N. Kuster, and B. Pasche, “Amplitude-modulated electromagnetic fields for the treatment of cancer: discovery of tumor-specific frequencies and assessment of a novel therapeutic approach,” Journal of Experimental & Clinical Cancer Research, vol. 28, no. 1, p. 51, 2009.Walther M. Walther, F. Mayer, W. Kafka, and N. Schütze, “Effects of weak, low-frequency pulsed electromagnetic fields (bemer type) on gene expression of human mesenchymal stem cells and chondrocytes: an in vitro study,” Electromagnetic biology and medicine, vol. 26, no. 3, pp. 179–190, 2007.Kanitz M. Kanitz, F. Witzmann, W. Lotz, D. Conover, and R. Savage, “Investigation of protein expression in magnetic field-treated human glioma cells,” Bioelectromagnetics, vol. 28, no. 7, pp. 546–552, 2007.Qutob S. Qutob, V. Chauhan, P. Bellier, C. Yauk, G. Douglas, L. Berndt, A. Williams, G. Gajda, E. Lemay, A. Thansandote, et al., “Microarray gene expression profiling of a human glioblastoma cell line exposed in vitro to a 1.9 ghz pulse-modulated radiofrequency field,” Radiation research, vol. 165, no. 6, pp. 636–644, 2006.Omori Y. Omori, M. L. Z. Dagli, K. Yamakage, and H. Yamasaki, “Involvement of gap junctions in tumor suppression: analysis of genetically-manipulated mice,” Mutation Research/Fundamental and Molecular Mechanisms of Mutagenesis, vol. 477, no. 1, pp. 191–196, 2001.Mesnil M. Mesnil, V. Krutovskikh, Y. Omori, and H. Yamasaki, “Role of blocked gap junctional intercellular communication in non-genotoxic carcinogenesis,” Toxicology letters, vol. 82, pp. 701–706, 1995.Yamasaki H. Yamasaki, V. Krutovskikh, M. Mesnil, T. Tanaka, M. L. Zaidan-Dagli, and Y. Omori, “Role of connexin (gap junction) genes in cell growth control and carcinogenesis,” Comptes Rendus de l'Académie des Sciences-Series III-Sciences de la Vie, vol. 322, no. 2, pp. 151–159, 1999.Hu G. Hu, H. Chiang, Q. Zeng, and Y. Fu, “Elf magnetic field inhibits gap junctional intercellular communication and induces hyperphosphorylation of connexin43 in nih3t3 cells,” Bioelectromagnetics, vol. 22, no. 8, pp. 568–573, 2001.Yamaguchi D. T. Yamaguchi, J. Huang, D. Ma, and P. K. Wang, “Inhibition of gap junction intercellular communication by extremely low-frequency electromagnetic fields in osteoblast-like models is dependent on cell differentiation,” Journal of cellular physiology, vol. 190, no. 2, pp. 180–188, 2002.Huang X. Huang, X. Feng, C. Zorman, M. Mehregany, and M. Roukes, “Vhf, uhf and microwave frequency nanomechanical resonators,” New Journal of Physics, vol. 7, no. 1, p. 247, 2005.andocs2009strong G. Andocs, H. Renner, L. Balogh, L. Fonyad, C. Jakab, and A. Szasz, “Strong synergy of heat and modulated electromagnetic field in tumor cell killing,” Strahlentherapie und Onkologie, vol. 185, no. 2, pp. 120–126, 2009.meggyeshazi2014dna N. Meggyesházi, G. Andocs, L. Balogh, P. Balla, G. Kiszner, I. Teleki, A. Jeney, and T. Krenács, “Dna fragmentation and caspase-independent programmed cell death by modulated electrohyperthermia,” Strahlentherapie und Onkologie, vol. 190, no. 9, pp. 815–822, 2014.andocs2015upregulation G. Andocs, N. Meggyeshazi, L. Balogh, S. Spisak, M. E. Maros, P. Balla, G. Kiszner, I. Teleki, C. Kovago, and T. Krenacs, “Upregulation of heat shock proteins and the promotion of damage-associated molecular pattern signals in a colorectal cancer model by modulated electrohyperthermia,” Cell Stress and Chaperones, vol. 20, no. 1, pp. 37–46, 2015.andocs2016comparison G. Andocs, M. Rehman, Q. Zhao, Y. Tabuchi, M. Kanamori, and T. Kondo, “Comparison of biological effects of modulated electro-hyperthermia and conventional heat treatment in human lymphoma u937 cells,” Cell death discovery, vol. 2, 2016.converse2006computational M. Converse, E. J. Bond, B. Veen, and C. Hagness, “A computational study of ultra-wideband versus narrowband microwave hyperthermia for breast cancer treatment,” IEEE transactions on microwave theory and techniques, vol. 54, no. 5, pp. 2169–2180, 2006.nguyen2017three P. T. Nguyen, A. Abbosh, and S. Crozier, “Three-dimensional microwave hyperthermia for breast cancer treatment in a realistic environment using particle swarm optimization,” IEEE Transactions on Biomedical Engineering, vol. 64, no. 6, pp. 1335–1344, 2017.mendecki1978microwave J. Mendecki, E. Friedenthal, C. Botstein, F. Sterzer, R. Paglione, M. Nowogrodzki, and E. Beck, “Microwave-induced hyperthermia in cancer treatment: apparatus and preliminary results,” International Journal of Radiation Oncology* Biology* Physics, vol. 4, no. 11-12, pp. 1095–1103, 1978.lerchl2015tumor A. Lerchl, M. Klose, K. Grote, A. F. Wilhelm, O. Spathmann, T. Fiedler, J. Streckert, V. Hansen, and M. Clemens, “Tumor promotion by exposure to radiofrequency electromagnetic fields below exposure limits for humans,” Biochemical and biophysical research communications, vol. 459, no. 4, pp. 585–590, 2015.hardell2015increasing L. Hardell and M. Carlberg, “Increasing rates of brain tumours in the swedish national inpatient register and the causes of death register,” International journal of environmental research and public health, vol. 12, no. 4, pp. 3793–3813, 2015.Savage G. Huang, “10 emerging technologies that will change your world,” Technology Review, vol. 107, no. 1, pp. 32–+, 2004.fitzgerald A. Fitzgerald, E. Berry, N. Zinovev, G. Walker, M. Smith, and J. Chamberlain, “An introduction to medical imaging with coherent terahertz frequency radiation,” Physics in Medicine and biology, vol. 47, no. 7, p. R67, 2002.woodward R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” Journal of Investigative Dermatology, vol. 120, no. 1, pp. 72–78, 2003.ashworth P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, “Terahertz pulsed spectroscopy of freshly excised human breast cancer,” Optics express, vol. 17, no. 15, pp. 12444–12454, 2009.fitzgerald2006 A. J. Fitzgerald, V. P. Wallace, M. Jimenez-Linan, L. Bobrow, R. J. Pye, A. D. Purushotham, and D. D. Arnone, “Terahertz pulsed imaging of human breast tumors,” Radiology, vol. 239, no. 2, pp. 533–540, 2006.chen2011 H. Chen, T.-H. Chen, T.-F. Tseng, J.-T. Lu, C.-C. Kuo, S.-C. Fu, W.-J. Lee, Y.-F. Tsai, Y.-Y. Huang, E. Y. Chuang, et al., “High-sensitivity in vivo thz transmission imaging of early human breast cancer in a subcutaneous xenograft mouse model,” Optics express, vol. 19, no. 22, pp. 21552–21562, 2011.wilmink G. J. Wilmink and J. E. Grundt, “Invited review article: current state of research on biological effects of terahertz radiation,” Journal of Infrared, Millimeter, and Terahertz Waves, vol. 32, no. 10, pp. 1074–1122, 2011.Titova2013 L. V. Titova, A. K. Ayesheshim, A. Golubov, R. Rodriguez-Juarez, A. Kovalchuk, F. A. Hegmann, and O. Kovalchuk, “Intense picosecond thz pulses alter gene expression in human skin tissue in vivo,” in SPIE BiOS, pp. 85850Q–85850Q, International Society for Optics and Photonics, 2013.bock J. Bock, Y. Fukuyo, S. Kang, M. L. Phipps, L. B. Alexandrov, K. Ø. Rasmussen, A. R. Bishop, E. D. Rosen, J. S. Martinez, H.-T. Chen, et al., “Mammalian stem cells reprogramming in response to terahertz radiation,” PloS one, vol. 5, no. 12, p. e15806, 2010.alexandrov2013 B. S. Alexandrov, M. L. Phipps, L. B. Alexandrov, L. G. Booshehri, A. Erat, J. Zabolotny, C. H. Mielke, H.-T. Chen, G. Rodriguez, K. Ø. Rasmussen, et al., “Specificity and heterogeneity of terahertz radiation effect on gene expression in mouse mesenchymal stem cells,” Scientific reports, vol. 3, p. 1184, 2013.doria A. Doria, G. Gallerano, E. Giovenale, G. Messina, A. Lai, A. Ramundo-Orlando, V. Sposato, M. D'Arienzo, A. Perrotta, M. Romano, et al., “Thz radiation studies on biological systems at the enea fel facility,” Infrared Physics & Technology, vol. 45, no. 5, pp. 339–347, 2004.falconer R. J. Falconer and A. G. Markelz, “Terahertz spectroscopic analysis of peptides and proteins,” Journal of Infrared, Millimeter, and Terahertz Waves, vol. 33, no. 10, pp. 973–988, 2012.alexandrov2010 B. Alexandrov, V. Gelev, A. Bishop, A. Usheva, and K. Rasmussen, “Dna breathing dynamics in the presence of a terahertz field,” Physics Letters A, vol. 374, no. 10, pp. 1214–1217, 2010.fischerr B. Fischer, M. Walther, and P. U. Jepsen, “Far-infrared vibrational modes of dna components studied by terahertz time-domain spectroscopy,” Physics in medicine and biology, vol. 47, no. 21, p. 3807, 2002.fischer2005 B. M. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, and P. U. Jepsen, “Terahertz time-domain spectroscopy and imaging of artificial rna,” Optics Express, vol. 13, no. 14, pp. 5205–5215, 2005.titova L. V. Titova, A. K. Ayesheshim, A. Golubov, D. Fogen, R. Rodriguez-Juarez, F. A. Hegmann, and O. Kovalchuk, “Intense thz pulses cause h2ax phosphorylation and activate dna damage response in human skin tissue,” Biomedical optics express, vol. 4, no. 4, pp. 559–568, 2013.titova2013intense L. V. Titova, A. K. Ayesheshim, A. Golubov, R. Rodriguez-Juarez, A. Kovalchuk, F. A. Hegmann, and O. Kovalchuk, “Intense picosecond thz pulses alter gene expression in human skin tissue in vivo,” in SPIE BiOS, pp. 85850Q–85850Q, International Society for Optics and Photonics, 2013.titova3 L. V. Titova, A. K. Ayesheshim, A. Golubov, R. Rodriguez-Juarez, R. Woycicki, F. A. Hegmann, and O. Kovalchuk, “Intense thz pulses down-regulate genes associated with skin cancer and psoriasis: a new therapeutic avenue?,” Scientific reports, vol. 3, 2013.titova4 L. V. Titova, A. K. Ayesheshim, D. Purschke, A. Golubov, R. Rodriguez-Juarez, R. Woycicki, F. A. Hegmann, and O. Kovalchuk, “Effect of intense thz pulses on expression of genes associated with skin cancer and inflammatory skin conditions,” in SPIE BiOS, pp. 89411G–89411G, International Society for Optics and Photonics, 2014.lucas M. L. Lucas and R. Heller, “Il-12 gene therapy using an electrically mediated nonviral approach reduces metastatic growth of melanoma,” DNA and cell biology, vol. 22, no. 12, pp. 755–763, 2003.kubota Y. Kubota, Y. Tomita, M. Tsukigi, H. Kurachi, T. Motoyama, and L. M. Mir, “A case of perineal malignant melanoma successfully treated with electrochemotherapy,” Melanoma research, vol. 15, no. 2, pp. 133–134, 2005.gothelf A. Gothelf, L. M. Mir, and J. Gehl, “Electrochemotherapy: results of cancer treatment using enhanced delivery of bleomycin by electroporation,” Cancer treatment reviews, vol. 29, no. 5, pp. 371–387, 2003.Alberts B. Alberts, “Vesicular traffic in the secretory and endocytic pathways,” Molecular biology of the cell, pp. 599–651, 1994.Civalek Ö. Civalek and Ç. Demir, “Bending analysis of microtubules using nonlocal euler–bernoulli beam theory,” Applied Mathematical Modelling, vol. 35, no. 5, pp. 2053–2067, 2011.Venier P. Venier, A. C. Maggs, M.-F. Carlier, and D. Pantaloni, “Analysis of microtubule rigidity using hydrodynamic flow and thermal fluctuations.,” Journal of biological chemistry, vol. 269, no. 18, pp. 13353–13360, 1994.Hawkins T. L. Hawkins, D. Sept, B. Mogessie, A. Straube, and J. L. Ross, “Mechanical properties of doubly stabilized microtubule filaments,” Biophysical journal, vol. 104, no. 7, pp. 1517–1528, 2013.Kurachi M. Kurachi, M. Hoshi, and H. Tashiro, “Buckling of a single microtubule by optical trapping forces: direct measurement of microtubule rigidity,” Cytoskeleton, vol. 30, no. 3, pp. 221–228, 1995.Vinckier A. Vinckier, C. Dumortier, Y. Engelborghs, and L. Hellemans, “Dynamical and mechanical study of immobilized microtubules with atomic force microscopy,” Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena, vol. 14, no. 2, pp. 1427–1431, 1996.Kis A. Kis, S. Kasas, B. Babić, A. Kulik, W. Benoit, G. Briggs, C. Schönenberger, S. Catsicas, and L. Forro, “Nanomechanics of microtubules,” Physical review letters, vol. 89, no. 24, p. 248101, 2002.Mizushima J. Mizushima-Sugano, T. Maeda, and T. Miki-Noumura, “Flexural rigidity of singlet microtubules estimated from statistical analysis of their contour lengths and end-to-end distances,” Biochimica et Biophysica Acta (BBA)-General Subjects, vol. 755, no. 2, pp. 257–262, 1983.Jack1 J. Tuszyński, T. Luchko, S. Portet, and J. Dixon, “Anisotropic elastic properties of microtubules,” The European Physical Journal E: Soft Matter and Biological Physics, vol. 17, no. 1, pp. 29–35, 2005.Kasas S. Kasas, A. Kis, B. M. Riederer, L. Forró, G. Dietler, and S. Catsicas, “Mechanical properties of microtubules explored using the finite elements method,” ChemPhysChem, vol. 5, no. 2, pp. 252–257, 2004.Sirenko Y. M. Sirenko, M. A. Stroscio, and K. Kim, “Elastic vibrations of microtubules in a fluid,” Physical Review E, vol. 53, no. 1, p. 1003, 1996.portet2005elastic S. Portet, J. Tuszyński, C. Hogue, and J. Dixon, “Elastic vibrations in seamless microtubules,” European Biophysics Journal, vol. 34, no. 7, pp. 912–920, 2005.farajpour2014surface A. Farajpour, A. Rastgoo, and M. Mohammadi, “Surface effects on the mechanical characteristics of microtubule networks in living cells,” Mechanics Research Communications, vol. 57, pp. 18–26, 2014.arani2013nonlinear A. G. Arani, A. Shirali, M. N. Farahani, S. Amir, and A. Loghman, “Nonlinear vibration analysis of protein microtubules in cytosol conveying fluid based on nonlocal elasticity theory using differential quadrature method,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 227, no. 1, pp. 137–145, 2013.mustapha2016torsional K. B. Mustapha and B. T. Wong, “Torsional frequency analyses of microtubules with end attachments,” ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 96, no. 7, pp. 824–842, 2016.Wang C. Wang, C. Ru, and A. Mioduchowski, “Vibration of microtubules as orthotropic elastic shells,” Physica E: Low-dimensional Systems and Nanostructures, vol. 35, no. 1, pp. 48–56, 2006.wang2009dynamic C. Wang, C. Li, and S. Adhikari, “Dynamic behaviors of microtubules in cytosol,” Journal of Biomechanics, vol. 42, no. 9, pp. 1270–1274, 2009.daneshmand2012coupled F. Daneshmand and M. Amabili, “Coupled oscillations of a protein microtubule immersed in cytoplasm: an orthotropic elastic shell modeling,” Journal of biological physics, vol. 38, no. 3, pp. 429–448, 2012.pokorny1997vibrations J. Pokorný, F. Jelínek, V. Trkal, I. Lamprecht, and R. Hölzel, “Vibrations in microtubules,” Journal of Biological Physics, vol. 23, pp. 171–179, 1997.deriu2010anisotropic M. A. Deriu, M. Soncini, M. Orsi, M. Patel, J. W. Essex, F. M. Montevecchi, and A. Redaelli, “Anisotropic elastic network modeling of entire microtubules,” Biophysical journal, vol. 99, no. 7, pp. 2190–2199, 2010.havelka2017deformation D. Havelka, M. A. Deriu, M. Cifra, and O. Kučera, “Deformation pattern in vibrating microtubule: Structural mechanics study based on an atomistic approach,” Scientific Reports, vol. 7, 2017.Xiang P. Xiang and K. M. Liew, “Dynamic behaviors of long and curved microtubules based on an atomistic-continuum model,” Computer Methods in Applied Mechanics and Engineering, vol. 223, pp. 123–132, 2012.liew2015mechanical K. Liew, P. Xiang, and L. Zhang, “Mechanical properties and characteristics of microtubules: a review,” Composite Structures, vol. 123, pp. 98–108, 2015.Qian X. Qian, J. Zhang, and C. Ru, “Wave propagation in orthotropic microtubules,” Journal of Applied Physics, vol. 101, no. 8, p. 084702, 2007.kucera2016 O. Kučera, D. Havelka, and M. Cifra, “Vibrations of microtubules: Physics that has not met biology yet,” Wave Motion, 2016.Dubost G. DUBOST, A. HOLLAND, J. BARE, and F. BELLOSSI, “Morphological transformations of human cancer cells and microtubules caused by frequency specific pulsed electric fields broadcast by an enclosed gas plasma antenna,” in Proceedings-7th International Workshop on Biological Effects of EMF–October, 2012.Pokorny J. Pokornỳ, C. Vedruccio, M. , and O. Kučera, “Cancer physics: diagnostics based on damped cellular elastoelectrical vibrations in microtubules,” European Biophysics Journal, vol. 40, no. 6, pp. 747–759, 2011.howard2001 J. Howard et al., “Mechanics of motor proteins and the cytoskeleton,” 2001.tuszynski2003evolution J. A. Tuszynski, E. J. Carpenter, J. T. Huzil, W. Malinski, T. Luchko, and R. F. Luduena, “The evolution of the structure of tubulin and its potential consequences for the role and function of microtubules in cells and embryos,” International Journal of Developmental Biology, vol. 50, no. 2-3, pp. 341–358, 2003.pokorny1998electric J. Pokornỳ, F. Jelınek, and V. Trkal, “Electric field around microtubules,” Bioelectrochemistry and Bioenergetics, vol. 45, no. 2, pp. 239–245, 1998.cifra_electric_2010 M. Cifra, J. Pokorný, D. Havelka, and O. Kučera, “Electric field generated by axial longitudinal vibration modes of microtubule,” Biosystems, vol. 100, pp. 122–131, May 2010.kucera_mechano-electrical_2012 O. Kučera and D. Havelka, “Mechano-electrical vibrations of microtubules—Link to subcellular morphology,” Biosystems, vol. 109, no. 3, pp. 346–355, 2012.havelka_high-frequency_2011 D. Havelka, M. Cifra, O. Kučera, J. Pokornỳ, and J. Vrba, “High-frequency electric field and radiation characteristics of cellular microtubule network,” Journal of theoretical biology, vol. 286, pp. 31–40, 2011.havelka_electro-acoustic_2014 D. Havelka, O. Kučera, M. A. Deriu, and M. Cifra, “Electro-Acoustic Behavior of the Mitotic Spindle: A Semi-Classical Coarse-Grained Model,” PLoS ONE, vol. 9, p. e86501, Jan. 2014.havelka_multi-mode_2014 D. Havelka, M. Cifra, and O. Kučera, “Multi-mode electro-mechanical vibrations of a microtubule: In silico demonstration of electric pulse moving along a microtubule,” Applied Physics Letters, vol. 104, p. 243702, June 2014.krivosudsky2016microwave O. Krivosudskỳ and M. Cifra, “Microwave absorption by nanoresonator vibrations tuned with surface modification,” EPL (Europhysics Letters), vol. 115, no. 4, p. 44003, 2016.karplus1981internal M. Karplus, J. A. McCammon, and W. L. Peticolas, “The internal dynamics of globular protein,” Critical Reviews in Biochemistry, vol. 9, no. 4, pp. 293–349, 1981.turton2014terahertz D. A. Turton, H. M. Senn, T. Harwood, A. J. Lapthorn, E. M. Ellis, and K. Wynne, “Terahertz underdamped vibrational motion governs protein-ligand binding in solution,” Nature communications, vol. 5, p. 3999, 2014.wheaton2015probing S. Wheaton, R. M. Gelfand, and R. Gordon, “Probing the raman-active acoustic vibrations of nanoparticles with extraordinary spectral resolution,” Nature Photonics, vol. 9, no. 1, pp. 68–72, 2015.pokorny2013postulates J. Pokornỳ, J. Pokornỳ, and J. Kobilková, “Postulates on electromagnetic activity in biological systems and cancer,” Integrative Biology, vol. 5, no. 12, pp. 1439–1446, 2013.Froehlich H. Frohlich, “Coherent electric vibrations in biological systems and the cancer problem,” IEEE Transactions on Microwave Theory and Techniques, vol. 26, no. 8, pp. 613–618, 1978.Vedruccio C. Vedruccio and A. Meessen, “Em cancer detection by means of non-linear resonance interaction,” Proc. and Extended Papers book. PIERS, pp. 28–31, 2004.holzel2001electric R. Hölzel, “Electric activity of non-excitable biological cells radio frequencies,” Electro-and magnetobiology, vol. 20, no. 1, pp. 1–13, 2001.jelinek2007measurement F. Jelinek, J. Saroch, O. Kucera, J. Hasek, J. Pokorny, N. Jaffrezic-Renault, and L. Ponsonnet, “Measurement of electromagnetic activity of yeast cells at 42 ghz,” RADIOENGINEERING, vol. 16, no. 1, p. 36, 2007.jelinek_measurement_2009 F. Jelínek, M. Cifra, J. Pokorný, J. Vaniš, J. Šimša, J. Hašek, and I. Frýdlová, “Measurement of Electrical Oscillations and Mechanical Vibrations of Yeast Cells Membrane Around 1 kHz,” Electromagnetic Biology and Medicine, vol. 28, pp. 223–232, Jan. 2009.kucera_spectral_2015 O. Kucera, K. Cervinkova, M. Nerudova, and M. Cifra, “Spectral perspective on the electromagnetic activity of cells,” Current topics in medicinal chemistry, vol. 15, no. 6, pp. 513–522, 2015.carr2017calcium L. Carr, S. M. Bardet, R. C. Burke, D. Arnaud-Cormos, P. Lévêque, and R. P. O’connor, “Calcium-independent disruption of microtubule dynamics by nanosecond pulsed electric fields in u87 human glioblastoma cells,” Scientific reports, vol. 7, 2017.Genes2008 C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Physical Review A, vol. 77, no. 3, p. 033804, 2008.Teufel1 J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. D. Whittaker, K. Lehnert, and R. W. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” arXiv preprint arXiv:1103.2144, 2011.Liberato S. De Liberato, N. Lambert, and F. Nori, “Quantum noise in photothermal cooling,” Physical Review A, vol. 83, no. 3, p. 033809, 2011.Ludwig C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Physical review letters, vol. 101, no. 13, p. 133903, 2008.Barzanjeh2 S. Barzanjeh, M. Naderi, and M. Soltanolkotabi, “Back-action ground-state cooling of a micromechanical membrane via intensity-dependent interaction,” Physical Review A, vol. 84, no. 2, p. 023803, 2011.Bradaschia C. Bradaschia, R. Del Fabbro, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, A. Brillet, O. Cregut, P. Hello, et al., “The virgo project: a wide band antenna for gravitational wave detection,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 289, no. 3, pp. 518–525, 1990.LaHaye M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science, vol. 304, no. 5667, pp. 74–77, 2004.Kippenberg T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Optics Express, vol. 15, no. 25, pp. 17172–17205, 2007.O'Connell A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, et al., “Quantum ground state and single-phonon control of a mechanical resonator,” Nature, vol. 464, no. 7289, pp. 697–703, 2010.Poot M. Poot and H. S. van der Zant, “Mechanical systems in the quantum regime,” Physics Reports, vol. 511, no. 5, pp. 273–335, 2012.Pikovski I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. Kim, and C. Brukner, “Probing planck-scale physics with quantum optics,” arXiv preprint arXiv:1111.1979, 2011.Bawaj M. Bawaj, C. Biancofiore, M. Bonaldi, F. Bonfigli, A. Borrielli, G. Di Giuseppe, L. Marconi, F. Marino, R. Natali, A. Pontin, et al., “Probing deformed commutators with macroscopic harmonic oscillators,” Nature communications, vol. 6, 2015.Liberati2 A. Belenchia, D. M. Benincasa, S. Liberati, F. Marin, F. Marino, and A. Ortolan, “Testing quantum gravity induced nonlocality via optomechanical quantum oscillators,” Physical review letters, vol. 116, no. 16, p. 161303, 2016.Barzanjehdis S. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Physical Review A, vol. 88, no. 4, p. 042331, 2013.spin1 D. Rugar, J. Sidles, and A. Hero, “Single-spin magnetic resonance force microscopy,” tech. rep., IBM ALMADEN RESEARCH CENTER SAN JOSE CA, 2005.spin2 R. Budakian, H. Mamin, and D. Rugar, “Spin manipulation using fast cantilever phase reversals,” Applied physics letters, vol. 89, no. 11, p. 113113, 2006.mol1 Y. Yang, C. Callegari, X. Feng, K. Ekinci, and M. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano letters, vol. 6, no. 4, pp. 583–586, 2006.mol2 M. Li, H. X. Tang, and M. L. Roukes, “Ultra-sensitive nems-based cantilevers for sensing, scanned probe and very high-frequency applications,” Nature nanotechnology, vol. 2, no. 2, pp. 114–120, 2007.mol3 K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nature nanotechnology, vol. 3, no. 9, pp. 533–537, 2008.Shekhawat F. Huber, H. Lang, N. Backmann, D. Rimoldi, and C. Gerber, “Direct detection of a braf mutation in total rna from melanoma cells using cantilever arrays,” Nature nanotechnology, vol. 8, no. 2, pp. 125–129, 2013.Badzey R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” arXiv preprint cond-mat/0603108, 2006.Paul M. Paul, M. Clark, and M. Cross, “The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation,” Nanotechnology, vol. 17, no. 17, p. 4502, 2006.Tamayo J. Tamayo, M. Calleja, D. Ramos, and J. Mertens, “Underlying mechanisms of the self-sustained oscillation of a nanomechanical stochastic resonator in a liquid,” Physical review B, vol. 76, no. 18, p. 180201, 2007.Barzanjeh2011 S. Barzanjeh, D. Vitali, P. Tombesi, and G. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Physical Review A, vol. 84, no. 4, p. 042342, 2011.Barzanjeh1 S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Physical Review Letters, vol. 109, no. 13, p. 130503, 2012.Andrews R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” arXiv preprint arXiv:1310.5276, 2013.Barzanjehconf S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Quantum interface between optics and microwaves with optomechanics,” in 2013 Conference on Lasers Electro-Optics Europe International Quantum Electronics Conference CLEO EUROPE/IQEC, pp. 1–1, 2013.Barzanjeh2015 S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Physical review letters, vol. 114, no. 8, p. 080503, 2015.Peng H. Peng, C. Chang, S. Aloni, T. Yuzvinsky, and A. Zettl, “Ultrahigh frequency nanotube resonators,” Physical review letters, vol. 97, no. 8, p. 087203, 2006.Painter1 J. Chan, T. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” arXiv preprint arXiv:1106.3614, 2011.Painter2 S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Physical Review X, vol. 5, no. 4, p. 041002, 2015.Lifshitz R. Lifshitz and M. Cross, “Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays,” Physical Review B, vol. 67, no. 13, p. 134302, 2003.Cross M. Cross, A. Zumdieck, R. Lifshitz, and J. Rogers, “Synchronization by nonlinear frequency pulling,” Physical review letters, vol. 93, no. 22, p. 224101, 2004.Katz I. Katz, A. Retzker, R. Straub, and R. Lifshitz, “Signatures for a classical to quantum transition of a driven nonlinear nanomechanical resonator,” Physical review letters, vol. 99, no. 4, p. 040404, 2007.Barzanjehnon S. Barzanjeh, M. Naderi, and M. Soltanolkotabi, “Generation of motional nonlinear coherent states and their superpositions via an intensity-dependent coupling of a cavity field to a micromechanical membrane,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 44, no. 10, p. 105504, 2011.Rips S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative wigner functions of nonlinear nanomechanical oscillators,” New Journal of Physics, vol. 14, no. 2, p. 023042, 2012.Rips1 S. Rips, I. Wilson-Rae, and M. Hartmann, “Nonlinear nanomechanical resonators for quantum optoelectromechanics,” Physical Review A, vol. 89, no. 1, p. 013854, 2014.Shah S. Shahidani, M. Naderi, M. Soltanolkotabi, and S. Barzanjeh, “Steady-state entanglement, cooling, and tristability in a nonlinear optomechanical cavity,” JOSA B, vol. 31, no. 5, pp. 1087–1095, 2014.BarzanjehJJ S. Barzanjeh and D. Vitali, “Phonon josephson junction with nanomechanical resonators,” Physical Review A, vol. 93, no. 3, p. 033846, 2016.Yu M.-F. Yu, G. J. Wagner, R. S. Ruoff, and M. J. Dyer, “Realization of parametric resonances in a nanowire mechanical system with nanomanipulation inside a scanning electron microscope,” Physical Review B, vol. 66, no. 7, p. 073406, 2002.Aldridge J. Aldridge and A. Cleland, “Noise-enabled precision measurements of a duffing nanomechanical resonator,” Physical review letters, vol. 94, no. 15, p. 156403, 2005.Kozinsky I. Kozinsky, H. C. Postma, I. Bargatin, and M. Roukes, “Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators,” Applied Physics Letters, vol. 88, no. 25, p. 253101, 2006.Lupascu A. Lupaşcu, E. Driessen, L. Roschier, C. Harmans, and J. Mooij, “High-contrast dispersive readout of a superconducting flux qubit using a nonlinear resonator,” Physical review letters, vol. 96, no. 12, p. 127003, 2006.Woolley M. Woolley, A. Doherty, G. Milburn, and K. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Physical Review A, vol. 78, no. 6, p. 062303, 2008.ShabirOPTO S. Barzanjeh, V. Salari, J. Tuszynski, M. Cifra, and C. Simon, “Optomechanical proposal for monitoring microtubule mechanical vibrations,” Physical Review E, vol. 96, no. 1, p. 012404, 2017. | http://arxiv.org/abs/1708.08339v1 | {
"authors": [
"V. Salari",
"Sh. Barzanjeh",
"M. Cifra",
"C. Simon",
"F. Scholkmann",
"Z. Alirezaei",
"J. A. Tuszynski"
],
"categories": [
"physics.med-ph",
"physics.bio-ph",
"quant-ph"
],
"primary_category": "physics.med-ph",
"published": "20170824201640",
"title": "Towards non-invasive cancer diagnostics and treatment based on electromagnetic fields, optomechanics and microtubules"
} |
1SISSA, Via Bonomea 265, 34136 Trieste, Italy2INFN-Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy3INAF-Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131 Trieste, Italy 4INAF-IRA, Via P. Gobetti 101, 40129 Bologna, ItalyThe continuity equation is developed for the stellar mass content of galaxies, and exploited to derive the stellar mass function of active and quiescent galaxies over the redshift range z∼ 0-8. The continuity equation requires two specific inputs gauged on observations: (i) the star formation rate functions determined on the basis of the latest UV+far-IR/sub-mm/radio measurements; (ii) average star-formation histories for individual galaxies, with different prescriptions for discs and spheroids. The continuity equation also includes a source term taking into account (dry) mergers, based on recent numerical simulations and consistent with observations. The stellar mass function derived from the continuity equation is coupled with the halo mass function and with the SFR functions to derive the star formation efficiency and the main sequence of star-forming galaxies via the abundance matching technique. A remarkable agreement of the resulting stellar mass function for active and quiescent galaxies, of the galaxy main sequence and of the star-formation efficiency with current observations is found; the comparison with data also allows to robustly constrain the characteristic timescales for star formation and quiescence of massive galaxies, the star formation history of their progenitors, and the amount of stellar mass added by in-situ star formation vs. that contributed by external merger events. The continuity equation is shown to yield quantitative outcomes that must be complied by detailed physical models, that can provide a basis to improve the (sub-grid) physical recipes implemented in theoretical approaches and numerical simulations, and that can offer a benchmark for forecasts on future observations with multi-band coverage, as it will become routinely achievable in the era of JWST. § INTRODUCTION Some recent findings have significantly rekindled interest in the field of galaxy formation and evolution. The first concerns the discovery of an abundant population of dusty star-forming galaxies at redshifts z 1, that has been shown to be responsible for the bulk of the cosmic star formation history, in particular around the crucial redshifts z≈ 2-3 where it peaks (e.g., Gruppioni et al. 2013; Rowan-Robinson et al. 2016; Lapi et al. 2017; Bourne et al. 2017; Dunlop et al. 2017; Novak et al. 2017), and to be present even out to z 6 (e.g., Cooray et al. 2014; Riechers et al. 2017; Zavala et al. 2017). Such achievement has become feasible only recently thanks to wide-area far-IR/sub-mm surveys conducted by Herschel, ASTE/AzTEC, APEX/LABOCA, JCMT/SCUBA2, and ALMA-SPT (e.g., Gruppioni et al. 2013, 2015; Lapi et al. 2011; Weiss et al. 2013; Strandet et al. 2016; Koprowski et al. 2014, 2016), in many instances eased by gravitational lensing from foreground objects (e.g., Negrello et al. 2014, 2017; Nayyeri et al. 2016). In fact, galaxies endowed with star formation rates Ṁ_⋆ a few tens M_⊙ yr^-1 at redshift z 2 were largely missed by rest-frame optical/UV surveys because of heavy dust obscuration, difficult to correct for with standard techniques based only on UV spectral data (e.g., Bouwens et al. 2016, 2017; Mancuso et al. 2016a; Pope et al. 2017; Ikarashi et al. 2017; Simpson et al. 2017).High-resolution, follow-up observations of these galaxies in the far-IR/sub-mm/radio band via ground-based interferometers, such as SMA, VLA, PdBI, and recently ALMA, have revealed star formation to occur in a few collapsing clumps distributed over spatial scales smaller than a few kpcs (see Simpson et al. 2015; Ikarashi et al. 2015; Straatman et al. 2015; Spilker et al. 2016; Barro et al. 2016; Tadaki et al. 2017). A strongly baryon-dominated stellar core with high ongoing SFR is often surrounded out to 15 kpc by a clumpy, unstable gaseous disk in nearly keplerian rotation (e.g., Genzel et al. 2017; Swinbank et al. 2017).Observations of dusty star-forming galaxies in the optical and near/mid-IR band from Spitzer, WISE, and HST have allowed to characterize their stellar mass content. The vast majority feature stellar masses strongly correlated to the SFR, in the way of an almost linear relationship dubbed 'Main Sequence', with a normalization steadily increasing as a function of redshift and a limited scatter around 0.25 dex (see Daddi et al. 2007; Rodighiero et al. 2011, 2015; Speagle et al. 2014; Whitaker et al. 2014; Renzini & Peng 2015; Salmon et al. 2015; Tasca et al. 2015; Kurczynski et al. 2016; Tomczak et al. 2016; Bourne et al. 2017; Dunlop et al. 2017; Schreiber et al. 2017).Another relevant piece of news concerns the discovery by deep near-IR surveys of an increasing number of massive galaxies M_⋆ several 10^10M_⊙ at high redshift z 2 (see Bernardi et al. 2013, 2017; Ilbert et al. 2013; Duncan et al. 2014; Tomczak et al. 2014; Caputi et al. 2015; Grazian et al. 2015; Thanjavur et al. 2016; Song et al. 2016; Davidzon et al. 2017). Even more interestingly, some of them are found to be already in passive evolution at z 2-3, and to feature chemical properties similar to local early-type galaxies, including a (super)solar metallicity and a pronounced α-enhancement. There is the intriguing yet still debated possibility that the dusty star-forming objects seen in the far-IR/sub-mm band constitute the progenitors of the massive (quiescent) galaxies increasingly detected at high redshifts via deep near-IR surveys (Straatman et al. 2014, 2016; Lonoce et al. 2015; Kriek et al. 2016; Mawatari et al. 2016; Michalowski et al. 2016; Davidzon et al. 2017; Glazebrook et al. 2017).Relevant model-independent information on the cosmic star formation and mass growth history can be inferred by comparing the observed SFR function, stellar mass function, and main sequence for active and quiescent galaxies (e.g., Leja et al. 2015; Contini et al. 2016; Tomczak et al. 2016; Mancuso et al. 2016a,b; Steinhardt et al. 2017). This procedure can provide stringent constraints, e.g., on the typical timescales for star formation and quiescence, on the overall star formation efficiency, on the initial mass function (IMF), and on the amount of stellar mass added by in-situ star formation vs. that contributed by external merger events. Such outcomes can also be helpful to improve the (sub-grid) physical recipes implemented in theoretical models and numerical simulations, that currently face some difficulties in reproducing the observed abundances of strongly star-forming and massive quiescent galaxies at z 2-3 (e.g., Wellons et al. 2015; Behroozi & Silk 2017; Dave et al. 2017; Furlong et al. 2017; Rong et al. 2017; Hopkins et al. 2017).In the present paper we pursue the above strategy, for the first time in a quantitative way, by exploiting the specific tool constituted by the `continuity equation'. Being originally devised to connect quasar statistics to the demographics of supermassive black hole relics (Cavaliere et al. 1971; Soltan 1982; Small & Blandford 1992; Salucci et al. 1999; Yu & Lu 2004, 2008; Marconi et al. 2004; Merloni & Heinz 2008; Shankar et al. 2009, 2013; Aversa et al. 2015), here we develop it for the stellar mass content of galaxies, in order to derive the stellar mass function of active and passive galaxies at different redshifts from the SFR functions and average star formation histories for individual objects. Our approach includes in the continuity equation a source term taking into account dry mergers and tidal stripping effects, gauged on observations and on state-of-the-art numerical simulations. With the term dry mergers we refer to events adding the whole mass content in stars of merging objects without contributing significantly to in-situ star formation; starbursts triggered by wet mergers, although included as star-forming objects populating the SFR functions, are expected to contribute little to the final stellar mass, and especially so for massive galaxies.Moreover, we will exploit the abundance matching technique to derive the star formation efficiency and the main sequence of star-forming galaxies, and compare the outcome to recent observational determinations. Specifically, we will demonstrate via the continuity equation that the dusty, strongly star-forming galaxies at z 2 are indeed the progenitors of massive quiescent galaxies, and that the latter's mass growth is dominated by in-situ star formation with an overall efficiency of less than 20%.The plan of the paper is straightforward. In <ref> we describe the basic ingredients of our analysis: the SFR functions and the adopted star formation histories for individual galaxies; in <ref> we solve the continuity equation for the stellar mass function of active and passive galaxies, and describe how to derive from those the star formation efficiency and the main sequence of star-forming galaxies; in <ref> we present our results and compare them to observations, discussing the relevant implications for galaxy formation and evolution; in <ref> we summarize our findings.Throughout the work we adopt the standard flat cosmology (Planck Collaboration XIII 2016) with round parameter values: matter density Ω_M = 0.32, baryon density Ω_b = 0.05, Hubble constant H_0 = 100h km s^-1 Mpc^-1 with h = 0.67, and mass variance σ_8 = 0.83 on a scale of 8h^-1 Mpc. Stellar masses and SFRs (or luminosities) of galaxies are evaluated assuming the Chabrier's (2003) IMF.§ BASIC INGREDIENTS Our analysis relies on two basic ingredients: (i) an observational determination of the SFR function at different redshifts; (ii) deterministic evolutionary tracks describing the average star formation history of individual galaxies. In this section we recall the notions relevant for the investigation of the stellar mass function, deferring the reader to the papers by Mancuso et al. (2016a,b) and Lapi et al. (2017) for more details. §.§ SFR functions and cosmic SFR density The first ingredient is constituted by the intrinsic SFR function dN/ dlogṀ_⋆, namely the number density of galaxies per logarithmic bin of SFR [logṀ_⋆,logṀ_⋆+ dlogṀ_⋆] at given redshift z. This has been accurately determined by Mancuso et al. (2016a,b) and Lapi et al. (2017) by exploiting the most recent determinations of the evolving galaxy luminosity functions from (dust-corrected) UV, far-IR, sub-mm, and radio data.The SFR function can be described as a smooth Schechter functiondN dlogṀ_⋆(Ṁ_⋆,z) = 𝒩(z) [Ṁ_⋆Ṁ_⋆, c(z)]^1-α(z) e^-Ṁ_⋆/Ṁ_⋆, c(z) ,with three parameters: the overall normalization 𝒩, the characteristic SFR Ṁ_⋆, c and the faint end slope α. The redshift evolution of each parameter has been measured via an educated fit to the observed data in unitary redshift bins by Mancuso et al. (2016a,b). As extensively discussed by the latter authors, the SFR function is mainly determined by (dust-corrected) UV data for SFR Ṁ_⋆ 30M_⊙ yr^-1 since in this range dust emission is mainly due to the diffuse (cirrus) dust component and standard UV dust-corrections based on the UV slope are reliable (see Meurer et al. 1999; Calzetti 2000; Bouwens et al. 2015, 2016, 2017); here we use the Meurer/Calzetti extinction law, but note that switching to a Small Magellanic Cloud (SMC) extinction law affects mildly the SFR function at the faint end (see also Sect. <ref> and Fig. <ref>). On the other hand, the SFR function is mainly determined by far-IR/sub-mm/radio data for SFRs Ṁ_⋆ 10^2M_⊙ yr^-1 since in this range dust emission is largely dominated by molecular clouds, and UV corrections are wildly dispersed and statistically fail (see Silva et al. 1998; Efstathiou et al. 2000; Coppin et al. 2015; Reddy et al. 2015; Fudamoto et al. 2017).The resulting SFR functions at representative redshifts are illustrated along with the relevant data collection in Fig. <ref>. In Mancuso et al. (2016a,b; 2017) and Lapi et al. (2017) we have validated them against independent datasets, including integrated galaxy number counts at significative far-IR/sub-mm/radio wavelengths, counts/redshift distributions of strongly gravitationally-lensed galaxies, main sequence of star-forming galaxies and AGNs, redshift evolution of the cosmic SFR, and high-redshift observables including the history of cosmic reionization.All in all, our determination of the SFR functions implies a significant number density of dusty star-forming galaxies with SFR Ṁ_⋆ 10^2M_⊙ yr^-1, currently missed by (dust-corrected) UV data. To highlight more clearly this point, in Fig. <ref> we also report at z 1 the SFR function that would have been inferred basing solely on UV data, dust corrected via the UV slope. The UV data considerably underestimate the SFR function for SFRs Ṁ_⋆ 30M_⊙ yr^-1, because of strong dust extinction. Interestingly, the shape of the SFR function for Ṁ_⋆ 10^2M_⊙ yr^-1, which so far has been probed only indirectly at z 4 due to sensitivity limits in current wide-areas far-IR surveys, is found to agree out to z 6 with the constraints from the recent VLA-COSMOS radio survey (Novak et al. 2017) and from the few individual galaxies detected at z 5 with ALMA and SMA (e.g., Riechers et al. 2017; Zavala et al. 2017). We shall demonstrate via the continuity equation that a robust probe on the bright end of the SFR function at high-redshift z 4 is provided by the galaxy stellar mass function.For the analysis in the present paper, we shall assume that at z 1 active galaxies populate the total (UV+far-IR/sub-mm/radio) SFR function, and feature a spheroid-like star-formation history; the latter envisages a nearly constant beahvior of the SFR as a function of galaxy age, with a timescale of 1 Gyr at high SFR Ṁ_⋆ 30M_⊙ yr^-1, increasing to a few Gyrs for lower SFRs (see Sect. <ref> for details). At high SFRs such a population comprises dusty starforming, far-IR/sub-mm selected galaxies, which will turn out to be the progenitors of local massive dead spheroids with masses M_⋆ a few 10^10M_⊙; at low SFRs, it comprises mildly obscured, UV selected galaxies (e.g., Lyman break galaxies), that will end up in objects with stellar masses M_⋆ 10^10M_⊙.At z 1 we will show that a bimodal star-formation history is required. On the one hand, the bright end of the total (UV+far-IR/sub-mm/radio) SFR function is assumed to be populated by galaxies with the same spheroid-like star-formation history sketched above; since the knee of the SFR function toward z∼ 0 recedes a lot, the star-formation timescales are on average appreciably longer than at z∼ 1, attaining up to a few Gyrs (possibly splitted in many recurrent, shorter bursts); this population comprises a mixed bag of objects, including low-mass spheroids (e.g., bulges), irregulars, and reactivations of massive galaxies. On the other hand, the UV-inferred SFR function is assumed to be populated by galaxies with a disc-like star-formation history, i.e., exponentially declining SFR as a function of galaxy age with long timescales of several Gyrs (see also Cai et al. 2013, 2014); these objects will end up in disc-dominated galaxies with stellar masses M_⋆ several 10^10M_⊙. In Sect. <ref> we will describe in detail the adopted spheroid-like or disc-like star-formation histories for individual galaxies.From the SFR function, we can straightforwardly compute the cosmic SFR density asρ_ SFR(z) = ∫ dlogṀ_⋆dN dlogṀ_⋆ Ṁ_⋆ ,integrated down to a limit Ṁ_⋆ 10^-1M_⊙ yr^-1 for fair comparison to observational data, in particular with current blank-field UV surveys at high z 4; the outcome is illustrated in Fig. <ref>. The result from the (dust-corrected) UV-inferred SFR functions is in good agreement with the UV data by Schiminovich et al. (2005) at z≲ 4 and by Bouwens et al. (2015, 2016, 2017) at z≳ 4. It also agrees with the estimate by ALMA observations of UV-selected galaxies in the HUDF (see Dunlop et al. 2017); this is because the rather small area of the HUDF survey allows to pick up only moderately star forming galaxies with mild dust obscuration, on which the UV slope-based corrections still work pretty well.However, the cosmic SFR density from (dust-corrected) UV data is inconsistent with other datasets both at low and high redshift. Specifically, at redshifts z 4 it falls short with respect to the multiwavelength determination by Hopkins & Beacom (2006) based on UV/optical, radio, Hα and mid-IR 24 μm data, to the far-IR measurements from Herschel by Magnelli et al. (2013) and Gruppioni et al. (2013), and to the recent estimate from deblended data from Herschel, JCMT/AzTEC and JCMT/SCUBA-2 in the GOODS field by Liu et al. (2017). At redshifts z 4 it underestimates the determinations based on stacking of far-IR data from Herschel by Rowan-Robinson et al. (2016), the measurements from radio data by Novak et al. (2017), and the estimates based on long GRB rates from Swift by Kistler et al. (2009; 2013). This mostly reflects the fact, already mentioned above, that the UV-inferred SFR functions (even corrected for dust extinction via the UV slope) appreciably underestimate the number density of dusty galaxies with Ṁ_⋆ 30M_⊙ yr^-1.The agreement with all these datasets is substantially improved when basing on the cosmic SFR density computed from the UV+far-IR/sub-mm/radio SFR functions. We also illustrate the contribution to the total density from objects with spheroid-like and disc-like star-formation histories. We remark that at z 1 most of the SFR density is contributed by dusty starforming progenitors of local massive quiescent spheroids, while at z 1 it is contributed both by disc-dominated galaxies and by low-mass spheroids, irregulars, and reactivated massive galaxies. §.§ Star-formation history of individual galaxies The second ingredient of our analysis is constituted by deterministic evolutionary tracks for the history of star formation in individual galaxies. The relevant quantity Ṁ_⋆(τ|M_⋆,t) is the behavior of the SFR as a function of the internal galactic age τ (i.e., the time since the beginning of significant star formation activity) for a galaxy with relic stellar mass M_⋆ at cosmological time t (corresponding to redshift z).For high z 1 strongly star-forming galaxies (that will turn out to be the progenitors of local dead massive spheroids), we base on the indications emerging from many SED-modeling studies (e.g., Papovich et al. 2011; Smit et al. 2012; Moustakas et al. 2013; Steinhardt et al. 2014; Citro et al. 2016; Cassará et al. 2016); these suggest a slow power-law increase of the SFR Ṁ_⋆∝τ^κ with κ 1 over a timescale τ_ sphe Gyr, then followed by a rapid quenching, at least for massive objects. For the sake of simplicity (cf. Sect. <ref>), here we adopt the lawṀ_⋆(τ|M_⋆,t) = κ+1 1-ℛ M_⋆τ_ sphe (τ/τ_ sphe)^κ Θ_ H[τ≤τ_ sphe] ,where Θ(·) is the Heaviside step function. The quantity ℛ is the fraction of mass restituted to the interstellar medium by massive stars, computed in the instantaneous recycling approximation; for a Chabrier IMF and star-formation timescalesGyr, ℛ≈ 0.4 applies.As to the parameters involved in the above expressions, recent observations by ALMA have shown that in high-redshift galaxies the star formation occurred within a compact regiona few kpcs over timescales τ_ sphe 0.5-1 Gyr at violent rates Ṁ_⋆ a few 10^2M_⊙ yr^-1 under heavily dust-enshrouded conditions (e.g., Scoville et al. 2014, 2016; Simpson et al. 2015; Ikarashi et al. 2015; Straatman et al. 2015; Spilker et al. 2016; Tadaki et al. 2017). A duration of the main star formation episode τ_ sphe 0.5-1 Gyr in high-redshift dusty star-forming galaxies, which are the candidate progenitors of massive spheroids, is also confirmed by local observations of the α-enhancement, i.e., iron underabundance compared to α elements. This occurs because star formation is stopped, presumably by some form of energetic feedback (e.g., due to the central supermassive black hole), before type Ia surpernova explosions can pollute the interstellar medium with substantial iron amounts (e.g., Romano et al. 2002; Thomas et al. 2005; Gallazzi et al. 2006; for a review see Renzini 2006). Contrariwise, in low-mass spheroidal galaxies with M_⋆ 10^10M_⊙ data on the age of stellar population and on chemical abundances indicate that star formation has proceeded for longer times, mainly regulated by supernova feedback and stellar winds (see review by Conroy 2013).On this basis, we parameterize the timescale for the duration of the main SFR episode in objects with spheroid-like star-formation history as a function of the peak SFR value Ṁ_⋆(τ_ sphe|M_⋆,t)=(κ+1)M_⋆/(1-ℛ) τ_ sphe via the implicit equationτ_ sphe = τ_ sphe^+ +τ_ sphe^- 2+τ_ sphe^+ -τ_ sphe^- 2× × tanh[Ṁ_⋆(τ_ sphe|M_⋆,t) 5M_⊙yr^-1] ,τ_ sphe^+=0.6Gyr (1+z 3)^-3/2 , τ_ sphe^- = t_z ;this has to be solved on a grid of M_⋆ and t (or z; see details in Sect. <ref>). The tanh(·) function interpolates smoothly between the short timescale τ_ sphe^+ 1 Gyr for high star-forming galaxies, and a long timescale τ_ sphe^-∼ t_z of the order of the cosmic time for galaxies with low SFRs. Note that in τ_ sphe^+ the dependence on redshift matches that of the dynamical time ∝ 1/√(G ρ)∝ (1+z)^-3/2, in turn following the increase in average density ρ∝ (1+z)^3 of the ambient medium. Our results will be insensitive to the specific shape of the smoothing function. With a similar parameterization, Mancuso et al. (2016b) have been able to reproduce the main sequence of star-forming galaxies at z≈ 2. We recall that at z 1, since the knee of the SFR functions recedes a lot, most of the objects with spheroid-like star-formation histories are characterized by moderate SFR Ṁ_⋆ 10M_⊙ yr^-1, hence rather long star-formation timescales up to a few Gyrs (cf. Fig. <ref>, bottom panel).As to the quenching timescale, the observed fraction of far-IR detected host galaxies in X-ray (e.g., Mullaney et al. 2012; Page et al. 2012; Rosario et al. 2012; Barger et al. 2015; Stanley et al. 2015; Harrison et al. 2016) and optically selected AGNs (e.g., Mor et al. 2012; Wang et al. 2013; Willott et al. 2015; Xu et al. 2015; Netzer et al. 2016; Harris et al. 2016) points toward a SFR abruptly stopping, at least in massive galaxies, after τ_ sphe over a short timescale 10^8 yr due to the action of feedbacks (see Lapi et al. 2014). To avoid introducing an additional parameter, in Eq. (<ref>) we truncate the SFR abruptly after τ_ sphe^+; we checked that an exponential quenching over a short timescale τ_ sphe^+/ζ with ζ a few will add an additional, small delay and produce very similar outcomes on the stellar mass function.On the other hand, in low redshift z 1 disc-dominated galaxies, it is well known that on average star formation declines exponentially as a function of the galactic age, with a long characteristic timescale of several Gyrs; for example, for our Milky Way it amounts to ≈ 6-7 Gyrs (see Chiappini et al. 1997; Courteau et al. 2014; Pezzulli & Fraternali 2016). In view of these classic evidences, we adoptṀ_⋆(τ|M_⋆,t)= 1 1-ℛ M_⋆τ_ disce^-τ/τ_ disc ,τ_ disc =6Gyr (1+z 2)^-3/2 .In Fig. <ref> (top panel) we illustrate an example of the resulting spheroid-like and disk-like star-formation histories; the relative star-formation timescales as a function of SFR and cosmic time are also shown (bottom panel). For both disc- and spheroid-like histories, we assume a dispersion of 0.25 dex around the average star-formation timescales; this value is inspired by the scatter observed in the specific SFRs Ṁ_⋆/M_⋆ (the inverse of a mass doubling time) of active galaxies at different redshift (e.g., Madau & Dickinson 2014), and it will turn out to produce the observed dispersion in the resulting star formation main sequence (cf. Sect. <ref> and Fig. <ref>).We caveat the reader that in the literature there have been attempts to parameterize with a unique shape the average star-formation history of galaxies. A classic way involves the so-called 'delayed exponential' model Ṁ_⋆(τ)∝τ^κe^-τ/τ_⋆, with two parameters κ and τ_⋆ controlling the early powerlaw rise and the late exponential decline. More recently, analogy with the behavior of the cosmic SFR density (see Gladders et al. 2013) and indications from numerical simulations (see Diemer et al. 2017) have suggested a lognormal shape Ṁ_⋆(τ)∝ e^-(lnτ/τ_⋆)^2/2σ_⋆^2/√(2πσ_⋆^2) τ withthe parameters τ_⋆ and σ_⋆ controlling peak time and width. Other descriptions with more complex parametric form have also been proposed based on observations (e.g., Leitner & Kravtsov 2011) or empirical models (e.g., Behroozi et al. 2013; Moster et al. 2013). All these shapes can be useful to describe the star-formation history averaged over the entire population of a galaxy survey; however, chemical and photometric data require to differentiate between disc-like and spheroid-like star-formation histories, making the parametric models for each class (e.g., see Fig. 2 in Diemer et al. 2017) essentially indistinguishable from our simple adopted shapes. For example, to describe the history of a starforming disc the timescale of the early rise has to be much faster than that of the late decline, to mirror the exponential model of Eq. (<ref>); contrariwise, in a massive spheroid progenitor the SFR must be nearly constant and then abruptly quenched, to mirror the power-law truncated model of Eq. (<ref>).§ THE CONTINUITY EQUATION The continuity equation has been originally devised for connecting the AGN statistics to the demographics of both active and dormant supermassive black holes (Cavaliere et al. 1971; Soltan 1982; Small & Blandford 1992; Salucci et al. 1999; Yu & Lu 2004, 2008; Marconi et al. 2004; Merloni & Heinz 2008; Shankar et al. 2009, 2013). Aversa et al. (2015) have been the first to show that it can be also applied to the stellar component in galaxies, to link the evolution across cosmic times of the SFR function to the stellar mass functions. The continuity equation in integral formulation is written dN dlogṀ_⋆(Ṁ_⋆,t) = ∫ dlog M_⋆ ∂_t [ dN dlog M_⋆(M_⋆,t)-S(M_⋆,t)]dτ dlogṀ_⋆(Ṁ_⋆|M_⋆,t) ; here the term on the l.h.s. is the (known) SFR function, while under the integral on the r.h.s. the first factor is the cosmic time derivative of the (unknown) stellar mass function minus a source term due to dry mergers (i.e., adding the whole mass content in stars of merging objects without contributing significantly to in-situ star formation), and the second factor is the overall time spent by a galaxy in a bin of SFR obtained from the star formation history. The interested reader can find in Aversa et al. (2015) an extended discussion of how and under which hypothesis the standard differential form of the continuity equation is recovered.In general, the continuity equation above is integro-differential and has to be solved numerically. If the source term due to dry merging is negligible (as it turns out to be indeed for z 1 according to simulations, see Sect. <ref> for details) and the star formation histories have simple shapes like in Eqs. (<ref>) and (<ref>), the continuity equation can be solved analytically along the following lines (see Aversa et al. 2015). First, the time lapses spent by the galaxy in a logarithmic bin of SFR read dτ dlogṀ_⋆(Ṁ_⋆|M_⋆,t)= 1κ (κ+1)^1/κ Ṁ_⋆^1/κ M_⋆^1/κ τ_ sphe^1+1/κ ln(10) Θ_ H[Ṁ_⋆≤(κ+1) M_⋆ (1-ℛ) τ_ sphe] , = τ_ disc ln(10) Θ_ H[Ṁ_⋆≤M_⋆ (1-ℛ) τ_ disc] , for galaxies with spheroid- and disc-like star-formation histories, respectively. In both expressions the Heaviside step function Θ_ H(·) specifies the maximum SFR contributing to a given final stellar mass.Inserting these expressions in the continuity equation Eq. (<ref>), differentiating with respect to Ṁ_⋆ and then integrating over cosmic time yield the closed form solutions dN(log M_⋆,t) dlog M_⋆ =-κ (1+κ)^1/κM_⋆^1/κ ∫_0^t dt' ∂_lnṀ_⋆ f_τ_ sphe [Ṁ_⋆^-1/κ τ_ sphe^-1-1/κdN dlogṀ_⋆(Ṁ_⋆,t')]_|Ṁ_⋆ = (1+κ) M_⋆ (1-ℛ) τ_ sphe ; =-∫_0^t dt' ∂_lnṀ_⋆ f_τ_ disc [τ_ disc^-1dN dlogṀ_⋆(Ṁ_⋆,t')]_|Ṁ_⋆ = M_⋆ (1-ℛ) τ_ disc , again for galaxies with spheroid- and disc-like star-formation histories, respectively; in both expression we have used the shorthand f_τ≡ 1+∂_logṀ_⋆logτ, which is not trivially equal to one when τ depends explicitly on the SFR (as in Eq. <ref>). The above equation is numerically solved on a grid in M_⋆ and z (or cosmic time t_z). We use a grid of 100 equally-spaced points in log M_⋆ [M_⊙] within the range [8,13] and a grid of 1000 equally-spaced points in redshift z within the range [0,20]; for optimal interpolation, the SFR functions and the star-formation timescales have been defined on the same grid of redshift and on a grid of 100 equally-spaced points in logṀ_⋆ [M_⊙ yr^-1] within the range [-2,4]. §.§ Dry merging In presence of mergers, the source S(M_⋆,t)=S_+-S_- actually includes the difference between a creation S_+ and a destruction S_- term. The former depends on the merger rate of objects with smaller masses into the descendant mass M_⋆, while the latter depends on the merger rates of the mass M_⋆ into more massive objects. Given the merger rate dN_ merg/ dlog M_⋆dμdt for the production of a descendant mass M_⋆ by the merging of two progenitors with (smaller to higher) mass ratio μ, the creation term readsS_+(M_⋆,t) = 1 2 ∫_μ_ min^1 dμ dN_ merg dlog M_⋆dμdt(M_⋆,μ,t)while the destruction term is writtenS_-(M_⋆,t)= 1 2∫_μ_ min^1 dμ [ dN_ merg dlog M_⋆dμdt(M_⋆(1+μ)/μ,μ,t)+. + .dN_ merg dlog M_⋆dμdt(M_⋆ (1+μ),μ,t)] .In the above μ_ min is the minimum progenitors' mass ratio; typically μ_ min=0.3 includes only 'major mergers', 0.1 includes major and minor mergers, 0.1 practically includes all mergers. We take μ_ min=0.01 in the following.We base on the outcomes of the Illustris simulations by Rodriguez-Gomez et al. (2015, 2016), who provide a handy fitting function for the merger rate per descendant galaxydn_ merg dμdt(M_⋆,μ,t) = A(z) (M_⋆ 10^10M_⊙)^ω(z)××[1+(M_⋆ 2× 10^11M_⊙)^δ(z)] μ^β(z)+γ log(M_⋆/10^10M_⊙)whereA(z) = A_0 (1+z)^η , ω(z)=ω_0 (1+z)^ω_1 ,β(z) = β_0 (1+z)^β_1, δ(z)=δ_0 (1+z)^δ_1with A_0≈ 10^-2.2287 Gyr^-1, η≈ 2.4644, ω_0≈ 0.2241, ω_1≈-1.1759, β_0≈-1.2595, β_1≈ 0.0611, γ≈ -0.0477, δ_0≈ 0.7668, δ_1≈ -0.4695. The above authors have validated this expression against various datasets, including observations of galaxy pairs. Two remarks are in order here. First, we caveat that at z 1 the Illustris simulation does not perfectly reproduce the observed galaxy stellar mass function; however, this does not concern much the merger rates, since as we shall demonstrate the growth in stellar mass at high redshift is mainly dominated by in situ star formation. Second, our results will turn out to be robust against other choices of the merger rate; we checked this by exploiting the galaxy merger rates extracted from the hydrodynamic simulations by Stewart et al. (2009), and the halo merger rates based on the N-body simulation by Fakhouri et al. (2010) coupled with empirical relationships connecting halo and stellar mass (e.g., Moster et al. 2013).Multiplying the merger rates above by the stellar mass function yields the quantitydN_ merg dlog M_⋆dμdt(M_⋆,μ,t) =dn_ merg dμdt(M_⋆,μ,t)× ×dN dlog M_⋆dt(M_⋆,t)entering the source terms in Eqs. (<ref>) and (<ref>). As discussed by Rodriguez-Gomez et al. (2015, 2016) the above expression also takes into account stellar mass stripping from satellites prior to dry mergers. Plainly, when merging is introduced, the continuity equation becomes fully integro-differential and must be solved numerically. We have computed the full solution and found that, to a good approximation, one can solve the problem iteratively, using Eq. (<ref>) as the zero-th order solution and then updating it with the correction due to the merging terms. §.§ Cosmic stellar mass density and Soltan argument Once the redshift-dependent stellar mass function is known, the cosmic stellar mass density is obtained asρ_M_⋆(t) ≡∫ dlog M_⋆M_⋆dN dlog M_⋆(M_⋆,t) ,where the integration is typically performed over stellar masses above 10^8M_⊙ for fair comparison with observational determinations (see discussion by Madau & Dickinson 2014).Interestingly, if τ_ sphe,disc is independent of, or only weakly dependent on Ṁ_⋆, a Soltan (1982) argument holds for the stellar content of galaxies (see Aversa et al. 2015); classically, this connects the cosmic luminosity density to the relic mass density of a population via an average conversion efficiency. In the present context, the Soltan argument can be easily found by multiplying both sides of Eq. (<ref>) by Ṁ_⋆ and integrating over it and over cosmic time, to obtainρ_M_⋆ = (1-ℛ) ∫_0^tdt' ∫ dlogṀ_⋆ Ṁ_⋆dN dlogṀ_⋆(Ṁ_⋆,t') .We highlight that here the IMF-dependent factor 1-ℛ plays the role of the radiative efficiency in the classic Soltan argument for black holes. Note that for conventional IMFs most of the stellar mass in galaxies resides in stars with mass 1M_⊙; since these stars emit most of their luminosity in the near-IR, the galaxy stellar mass M_⋆ can be inferred by the near-IR luminosity functions. On the other hand, the SFR function is determined from UV and far-IR/sub-mm/radio observations, as discussed in Sect. <ref>. Thus in principle accurate determinations of the SFR and stellar mass functions in these independent manners could be exploited via the Soltan argument above to constrain the average galaxy IMF at different redshifts. In practice, however, the dependence on the IMF is weak, and current observational uncertainties do not allow to fulfill this program now. §.§ Star formation efficiency We now connect the stellar mass function to the underlying, gravitationally dominant DM component, with the aim of deriving the star formation efficiency f_⋆≡ M_⋆/f_ bM_ H. This represents the fraction of the baryonic mass f_ bM_ H≈ 0.16M_ H initially associated to a DM halo of mass M_ H that has been eventually converted into stars. To this purpose, we exploit the abundance matching technique, a standard way of deriving a monotonic relationship between galaxy and halo properties by matching the corresponding integrated number densities (e.g., Vale & Ostriker 2004; Shankar et al. 2006; Moster et al. 2013; Behroozi et al. 2013).For fair comparison with the determination of the star formation efficiency f_⋆≡ M_⋆/f_ b ⟨ M_ H⟩ via weak gravitational lensing (e.g., Velander et al. 2014; Hudson et al. 2015; Mandelbaum et al. 2016) and galaxy kinematics (e.g., More et al. 2011; Wojtak & Mamon 2013), that are based on galaxy samples selected by stellar mass, we aim at deriving the average halo mass ⟨ M_ H⟩ (M_⋆,z) associated to a given M_⋆. In the abundance matching formalism, this relationship is obtained via the equation (see Aversa et al. 2015 for details)∫_log⟨ M_ H⟩ (M_⋆,z)^∞ dlog M_ H'dN dlog M_ H(M_ H',z) == ∫_-∞^+∞ dlog M_⋆'dN dlog M_⋆(M_⋆',z) 1 2erfc{log[M_⋆/M_⋆']√(2) σ_log M_ H} ,holding when a lognormal distribution of M_ H at given M_⋆ with dispersion σ_log M_ H is assumed.We follow previous studies based on various semi-empirical methods of galaxy and halo connection (see Rodriguez-Puebla et al. 2015, their Fig. 10) and adopt σ_log M_ H≈max[0.05,0.05+0.15 (log M_⋆ [M_⊙]-10)] for log M_⋆ [M_⊙] within the range [8.5,12.5]. In Eq. (<ref>) the quantity dN/ dlog M_ H is usually taken as the halo mass function from N-body simulations (e.g., Tinker et al. 2008; Watson et al. 2013; Bocquet et al. 2016; Comparat et al. 2017), that includes galaxy groups and clusters. This is particularly suitable when comparing with observational determinations of the star formation efficiency based on weak gravitational lensing (see references above), that integrate all the DM mass along the line of sight, including that associated to the surrounding galaxy environment.However, in order to infer the star formation efficiency of individual galaxies, and not of a galaxy system like a group or a cluster, it would be more appropriate to use the galaxy halo mass function, i.e., the mass function of halos hosting one individual galaxy. This can be built up from the overall halo mass function by adding to it the contribution of subhalos, and by probabilistically removing from it the contribution of halos corresponding to galaxy systems via halo occupation distribution modeling. We defer the reader to Appendix A of Aversa et al. (2015) for details on such a procedure. §.§ Galaxy main sequence The vast majority of galaxies is endowed with stellar masses strongly correlated to the ongoing SFR, in the way of an almost linear relationship dubbed 'Main Sequence', with a normalization steadily increasing as a function of redshift, and with a limited scatter around 0.25 dex (see Daddi et al. 2007; Rodighiero et al. 2011, 2015; Speagle et al. 2014; Whitaker et al. 2014; Renzini & Peng 2015; Salmon et al. 2015; Tasca et al. 2015; Kurczynski et al. 2016; Tomczak et al. 2016; Bourne et al. 2017; Dunlop et al. 2017; Schreiber et al. 2017).We exploit the abundance matching between the SFR functions (Fig. <ref>) and the stellar mass functions (Fig. <ref>) self-consistently derived from the continuity equation (cf. Eq. <ref>) to compute the average SFR ⟨Ṁ_⋆⟩(M_⋆,z) associated to a given stellar mass M_⋆. This reads∫_log⟨Ṁ_⋆⟩ (M_⋆,z)^∞ dlogṀ_⋆'dN dlogṀ_⋆(Ṁ_⋆',z) = = ∫_-∞^+∞ dlog M_⋆'dN dlog M_⋆(M_⋆',z) 1 2erfc{log[M_⋆/M_⋆']√(2) σ_logṀ_⋆} ,holding when a lognormal distribution of Ṁ_⋆ at given M_⋆ with dispersion σ_logṀ_⋆≈ 0.15 dex is adopted (see Aversa et al. 2015).The comparison of the resulting main sequence with the observational data will actually constitute an additional constraint on the assumed star formation histories for individual galaxies (see Eqs. <ref> and <ref>), on the star formation timescales and the associated scatter, and on the robustness of our results to other aside assumptions discussed in previous sections.§ RESULTS In Fig. <ref> we present the stellar mass function at different redshifts obtained via the continuity equation, including both in situ star formation and (dry) mergers. We highlight the average result as solid lines, and the 1σ dispersion expected from the scatter in the star formation timescales and merging histories as shaded areas (see Sect. <ref> for details). We compare our results to recent observational data (Moffett et al. 2016; Thanjavur et al. 2016; Bernardi et al. 2017; Davidzon et al. 2017; Tomczak et al. 2014; Grazian et al. 2015; Song et al. 2016), finding an excellent agreement.We stress that in-situ star formation within galaxies dominates over dry mergers in building up the stellar mass function at high redshifts, all the way down to z∼ 1, while at lower redshifts z 1 dry mergers can contribute appreciably to the stellar mass growth.This is highlighted on comparing the solid and dashed lines in Fig. <ref>, which illustrate the mass function at different redshifts when including or not dry mergers, respectively (actually the two sets of curves are superimposed for z 1.5). The effect of dry mergers on the stellar mass function is twofold: the number of low mass galaxies is decreased appreciably because of the merging into larger units, whereas the high-mass end of the stellar mass function is boosted toward larger masses because of mass additions from smaller objects; dry mergers mainly affect the most massive galaxies that are typically dominated by the spheroidal component. Such a picture is in agreement with what is recently emerging from state-of-the-art numerical simulations (see Schaye et al. 2015; Rodriguez-Gomez et al. 2016), semiempirical models (see Behroozi et al. 2013) and analysis of observations based on density-matching arguments (see Hill et al. 2017).The dotted lines show the contribution to the stellar mass function from galaxies with spheroid-like star-formation history that featured SFRs Ṁ_⋆ 100M_⊙ yr^-1; this corresponds to the limiting value currently sampled in wide-area far-IR surveys out to z 4 (e.g., the Herschel-ATLAS, see Lapi et al. 2011). It is seen that the descendants of these galaxies populate the high-mass end of the local stellar mass function, and thus are mainly present-day massive dead spheroids (e.g., Moffett et al. 2016). This demonstrates on a statistical basis that strongly starforming galaxies observed in the far-IR/(sub-)mm band constitute the progenitors of massive spheroids. By the same token, we stress that to test at z 4 the outcomes of the continuity equation, and better constrain the input SFR functions and the parameters of the star-formation history for spheroid progenitors, it will be extremely relevant to improve the accuracy in the determination of the stellar mass function at the high-mass end for M_⋆ a few 10^10M_⊙ out to z 6 via wide-area near-IR surveys.In Fig. <ref> we focus on the stellar mass function of galaxies with disc-like star-formation histories at z≈ 0. Our result is in excellent agreement with the observed stellar mass function of disc-dominated galaxies from decomposed data (Moffett et al. 2016; Bernardi et al. 2017); thus we find a good correspondence between objects populating the UV-inferred SFR function, to which we assigned disc-like star formation histories, and galaxies with observed disc-dominated morphology in the stellar mass function. We highlight that discs contribute considerably to the total stellar mass function for stellar masses M_⋆ a few 10^10M_⊙, and that the effects of mergers on their stellar mass function are negligible. It is seen from Eq. (<ref>) that the observed steepness for M_⋆ a few 10^10M_⊙ in the local stellar mass function of disc-dominated galaxies mirrors that in the input UV-inferred SFR functions at z 1. We caveat that assigning a disc-like star-formation history (with long star-formation timescales) even to objects populating the UV+far-IR/sub-mm/radio SFR functions at z 1 would considerably overproduce the number of massive discs; this is because the UV+far-IR/sub-mm/radio functions are much higher than the UV-inferred ones at given SFR. In fact, the UV+far-IR/sub-mm/radio SFR functions at z 1must be populated by objects with spheroid-like star-formation history; the continuity equation shows these star formation events to change little the total stellar mass function at z∼ 0 with respect to that at z∼ 1, mildly affecting the number density of galaxies with M_⋆ 10^10M_⊙.In Fig. <ref> we show how our resulting stellar mass function depend on the input SFR function and on the parameters of the star formation history. To highlight such dependencies in simple terms it is convenient to assume a piecewise powerlaw shape of the SFR function dN/ dlogṀ_⋆∝Ṁ_⋆^-χ, with χ 1 at the faint and χ>1 at the bright end. Then it is easily seen from Eq. (<ref>) that the resulting stellar mass function (in absence of mergers) behaves asdN dlog M_⋆ ∝ 1+κ χ (1+κ)^χ(1-ℛ)^χ+1/κ M_⋆^-χ τ_ sphe^χ-1 ∝ χ(1-ℛ)^χM_⋆^-χ τ_ disc^χ-1 ,for galaxies with spheroid-like and disc-like star formation histories, respectively. Thus, the stellar mass function features an almost direct dependence on the star-formation timescales τ_ sphe,disc at the high-mass end, which is mostly contributed by high SFRs where χ> 1; on the other hand, the dependence is inverse but mild at the low-mass end, mainly contributed by low-SFR galaxies with χ 1; note, however, that the value of the SFR where χ appreciably exceeds unity is much lower for the UV-inferred than for the UV+far-IR/sub-mm/radio SFR functions. The dependence on the parameter κ entering the star-formation history Ṁ_⋆(τ)∝τ^κ is mild, direct at the low-mass and inverse at the high-mass end. The dependence on the IMF is encapsulated in the restituted fraction 1 - ℛ, and in the factor used to convert the observed UV+far-IR/sub-mm/radio luminosity function into the SFR function; e.g., passing from the Chabrier to the Salpeter (1955) IMF, the high mass end of the stellar mass function is increased somewhat, while a strong suppression is originated when basing on a top-heavy IMF (e.g., Lacey et al. 2010). Finally, adopting a SMC extinction law in place of the Calzetti for the determination of the input SFR function amounts to alter somewhat the exponent χ and thus changes little the final outcome on the stellar mass function; on the other hand, adopting the UV-inferred SFR function at any redshift (i.e., neglecting far-IR/sub-mm/radio data) would imply to strongly underestimate the stellar mass function for large stellar masses (see discussion by Mancuso et al. 2016a,b) that are indeed built up in dusty star-forming galaxies with violent SFRs.In Fig. <ref> we focus on the stellar mass function of quiescent (passively evolving) galaxies; these systems have been increasingly observed with appreciable number density out to high redshift z 4 after selection via color-color diagrams in deep near-IR surveys (see Tomczak et al. 2014; Davidzon et al. 2017; Lonoce et al 2017; Glazebrook et al. 2017). Typically, these selections tend to pick up galaxies that have been quenched since, and then passively evolving over, a quiescence time interval Δ t_ qui∼ 250-500 Myr. Thus we compute the associated stellar mass function from the continuity equation by replacing the upper limit of integration in Eq. (<ref>) with τ_ sphe-Δ t_ qui. The result for two different values of Δ t_ qui≈ 250 and 500 Myr encompasses very well with the observational determinations out to z 4. Plainly, higher quiescence time Δ t_ qui imply a lower mass function, especially toward higher redshift where the cosmic time is smaller and progressively closer to τ_ sphe. The decrease of the mass function at the low mass end is due to the fact that small galaxies are still actively forming stars, since they feature longer star formation timescales; as a consequence, the fraction of galaxies in passive evolution decreases rapidly with stellar mass. The downturn shifts toward larger masses toward higher z, passing from 10^10 to a few 10^10M_⊙ from z≈ 0 to z 3.In Fig. <ref> we show the cosmic stellar mass density, obtained according to Eq. (<ref>). Our result from integrating the overall stellar mass function from the continuity equation is compared with the data collection by Madau & Dickinson (2014) and with the recent estimates by Song et al. (2016) and Davidzon et al. (2017) at high redshift. The agreement between our results and the data is remarkably good. We also highlight the contribution to the total stellar mass density from galaxies with disc-like and spheroid-like star-formation histories; the latter dominates the overall mass density at any redshift, though at z 1 discs brings an appreciable contribution around 40%. Note that the fraction of galaxies that are quiescent (here we use Δ t_ qui≈ 250 Myr, see above discussion), which are basically massive spheroids, constitute only a fraction 50% of the total mass density (contributed also by small spheroids/irregulars and discs that are still active) in the local Universe and rapidly declines to values 10% at higher redshift z 2. The overall shape agrees well with the estimates by Muzzin et al. (2013), Straatman et al. (2014) and Davidzon et al. (2017).For reference, in the figure we also report the mass density (scaled down by a factor 10^-2) of galaxy halos with mass M_ H 10^8.5M_⊙, the minimal threshold for efficient star formation required to solve the missingsatellite problem (see Boylan-Kolchin et al. 2014; Wetzel et al. 2016; Lapi et al. 2017). The evolution in halos and in the stellar mass content of galaxies differs both in shape and in normalization; these differences stem from: the inefficiency of galaxy formation due to feedback processes (e.g., supernovae, stellar winds, active galactic nuclei); the decrease at high z in the number density of massive halos, that are the hosts of the most massive galaxies; the inability to grow massive galaxies at high redshift since the growth timescales become comparable to the age of the Universe.In Fig. <ref> we show the star-formation efficiency, computed according to Eq. (<ref>). The green solid line and shaded area illustrate the outcome when matching the total stellar mass function from the continuity equation to the overall halo mass function at z≈ 0; note that the shaded area takes into account the uncertainty in the determination of the local stellar mass function from the continuity equation and that arising from the rather flat shape of the average ⟨ M_ H⟩ (M_⋆) correlation at the high mass end. Our result is compared with the local data for early and late type galaxies by various authors, determined via weak lensing (see Mandelbaum et al. 2016; Velander et al. 2014; Rodriguez-Puebla et al. 2015; Hudson et al. 2015) and satellite kinematics (see More et al. 2011; Wojtak & Mamon 2013). We stress that the abundance matching results must be confronted with the data of spheroidal galaxies for stellar masses above, and with the data of disc-dominated galaxies below, a few 10^10M_⊙; this is because spheroids and discs mostly contribute to the local stellar mass function in such stellar mass ranges (see Moffett et al. 2016). Provided that, we find a very good agreement, within the uncertainties of the respective datasets.The dotted green line is instead the outcome when matching the local stellar mass function with the galactic halo mass function. This highlights that the decrease in star-formation efficiency at large stellar masses is somewhat spurious, being related to the fact that the most massive galaxies tend to live at the center of group/cluster halos, which contain a lot of DM. Considering instead only the DM mass belonging to individual galactic halos would imply the efficiency to stay almost constant or increase somewhat at large masses out to M_⋆∼ 10^12M_⋆.The resulting values and shape of the star formation efficiency as a function of stellar mass is easily understood in terms of feedback processes. It is apparent that, because of feedbacks, galaxy formation is a very inefficient process: at most 20-30% of the original baryonic content of halos is converted into stars; this occurs for galaxies with final stellar mass around a few 10^10M_⊙ (corresponding to halos with mass M_ H≈ 10^12M_⊙). At small stellar masses, the action of supernova feedback is predominant, while for large stellar masses AGN feedback is likely more relevant; the mass of maximum efficiency corresponds approximately to the transition between supernova and AGN feedbacks (see Shankar et al. 2006; Moster et al. 2013; Aversa et al. 2015).The green dashed line is the outcome of matching the stellar mass function and the overall halo mass function at z≈ 2, and compares well with the efficiencies measured at the same redshift from Hα observation by Burkert et al. (2016). The outcome is also similar, within a factor of 2, to the determination via abundance matching by Moster et al. (2013), Behroozi et al. (2013), and Aversa et al. (2015). The similarity of the efficiency at z≈ 2 to the local value is indicative that star formation is mainly an in-situ process (see Lilly et al. 2013; Moster et al. 2013; Aversa et al. 2015; Mancuso et al. 2016a).We also present as a red solid line the outcome of matching the overall halo mass function with the stellar mass function of passively evolving galaxies, again finding a pleasingly agreement with the local data for spheroidal galaxies. It is extremely interesting to notice that the global efficiency at z≈ 2 can be brought on the efficiency of quiescent galaxies at z≈ 0 by allowing:(i) an evolution of the stellar mass by a factor 50% due to late star formation or dry mergers (see Rodriguez-Puebla et al. 2017; also Sect. <ref>); (ii) an halo mass evolution by a factor 4 (M_ H/10^14M_⊙)^0.12 due to late smooth accretion or tidal stripping (see McBride et al. 2009; Fakhouri et al. 2010; Lapi et al. 2013). We stress that such result is again indicative of the in-situ nature of the star formation in spheroid progenitors, and is also extremely relevant for understanding the evolution of the specific angular momentum in galaxies (see Shi et al. 2017).In Fig. <ref> we show the main sequence of star-forming galaxies at different redshifts, as obtained by matching the SFR function and the stellar mass functions from the continuity equation after Eq. (<ref>). The outcome at z≈ 2 is in pleasing agreement with the observational determination from large statistics of mass-selected galaxy samples by Rodighiero et al. (2015). This further substantiate our assumed star formation histories for individual galaxies, that are illustrated on three representative cases by red dotted lines; their shape is dictated by the slowly increasing SFR Ṁ_⋆∝τ^1/2 and appreciably rising stellar mass M_⋆∝τ^3/2, which imply Ṁ_⋆∝ M_⋆^1/3. Then the main sequence corresponds to the portions of such tracks where galaxies spend most of their lifetime in logarithmic bins of M_⋆.To highlight the relevance of observational selections different from that based on stellar mass, in Fig. <ref> we also report data points for individual, far-IR selected galaxies by Koprowski et al. (2016), Ma et al. (2015b), Negrello et al. (2014), along with Dye et al. (2015), da Cunha et al. (2015), and Dunlop et al. (2017) mainly at redshifts z∼ 1-4. An appreciable fraction of the individual, far-IR selected galaxies around z≈ 2 (highlighted in red) lie above the main sequence, i.e., at SFR values higher than expected on the basis of the average relationship at given M_⋆. These off-main-sequence objects can be simply interpreted (see Mancuso et al. 2016b) as galaxies caught in an early evolutionary stage, and still accumulating their stellar mass. Thus young star-forming galaxies are found to be preferentially located above the main sequence or, better, to the left of it. As time goes by and stellar mass increases, the galaxy moves toward the average main sequence relationship, around which it will spend most of its lifetime. Afterwards, the SFR is quenched by feedbacks and the galaxy will then evolve passively to become a local early-type; then it will populate a region of the SFR versus stellar mass diagram substantially below the main sequence. These loci of 'red and dead' galaxies are indeed observed locally (see Renzini & Peng 2015), and start to be pinpointed even at high redshift (see Man et al. 2016).§ SUMMARY We have developed the continuity equation for the stellar mass content of galaxies, and have exploited it to derive the stellar mass function of active and quiescent galaxies at redshifts z∼ 0-8 from the observed SFR functions and disc-like or spheroid-like star-formation histories for individual galaxies. Our approach based on the continuity equation includes a source term due to dry merging gauged on state-of-the-art numerical simulations and consistent with observations. We have then used the abundance matching technique to investigate the star formation efficiency and the main sequence of star-forming and quiescent galaxies. By comparing these outcomes to current observational estimates, we have inferred constraints on the characteristic timescales for star formation and quiescence, on the overall star formation efficiency, and on the amount of stellar mass added by in-situ star formation vs. that contributed by external (dry) merger events.Our main findings are the following: * We have found that the stellar mass function computed from the continuity equation is in excellent agreement with current observational constraints in the extended redshift range z∼ 0-8.At high redshift z 1 the mass function is produced by galaxies with spheroid-like star-formation histories, featuring an approximately constant (or slowly increasing) behavior of the SFR as a function of galactic age; the SFR must last for a time τ_ sphe≈ fraction of Gyr in strongly star-forming galaxies, while it can proceed over a longer time interval up to a few Gyrs for less massive objects: this reflects the differential action of supernova and AGN feedbacks in systems with different mass. We stressed the relevance of using as input of the continuity equation the SFR function estimated from far-IR/sub-mm/radio, in addition to UV, observations. This is because strongly star-forming galaxies are heavily dust-enshrouded, and as such their intrinsic SFR is considerably underestimated by UV observations, even when corrected for dust extinction according to standard prescriptions based on the UV slope. We have highlighted that the mass growth of spheroids is dominated by in-situ star formation for z 1, while at lower redshift dry mergers contribute a mass budget 50% especially in the most massive objects. * At low redshift z 1, we have shown that the stellar mass function of disc-dominated galaxies is well reproduced in our approach when using as input the UV-inferred SFR functions and an exponentially declining SFR history with a long characteristic timescales τ_ disc≈ several Gyrs. On the other hand, we have noted that assigning such a disc-like star-formation history to the UV+far-IR/sub-mm/radio SFR functions would considerably overproduce the number of massive discs; this is because obscuration is mild in starforming discs, so the SFR function from dust-corrected UV data must be effectively exploited as input of the continuity equation. The effects of mergers on the stellar mass function of discs are negligible. * We have found that the stellar mass function of quiescent galaxies from the continuity equation is in excellent agreement with current observational constraints for z 4. We thus have demonstrated quantitatively via the continuity equation that the dusty, strongly star-forming galaxies recently discovered thanks to wide area far-IR/sub-mm surveys at z 1 are indeed the progenitors of the massive quiescent galaxies increasingly detected out to high redshift z 4 via deep near-IR surveys. We have estimated that the typical time of quiescence (i.e., with absent or negligibly small SFR) for these galaxies is around Δ t_ qui≈ 250-500 Myr. To further test the outcomes of the continuity equation, and better constrain the input SFR functions and the parameters of the star-formation history for individual galaxies, it will be crucial to improve the accuracy in the determination of the stellar mass function at the high-mass end M_⋆ 10^11M_⊙ out to z 6 via wide areas near-IR surveys. * We have determined the cosmic mass density, finding it in excellent agreement with observational determination out to z∼ 0-8, both for active and quiescent galaxies. The continuity equation implies an analogue of the Soltan argument for the stellar component, in such a way that the cosmic stellar mass density is by construction consistent with the cosmic time-integrated star formation history, besides a factor depending on the IMF. * We have determined the star-formation efficiency of galaxies as a function of the stellar mass in the local Universe, finding it in good agreement with diverse observations. We have found, in line with previous studies, that the efficiency of star-formation is lower than f_⋆≈ 20-30%, with the maximum value being attained around a characteristic stellar mass of a few 10^10M_⊙. The behavior as a function of stellar mass canbe ascribed to different form of feedbacks regulating star formation in galaxies, with supernovae and stellar winds dominating for stellar masses below the characteristic one, and AGN feedback dominating above. We have also pointed out that the decline of the efficiency for large masses is somewhat spurious, being related to the fact that the most massive galaxies tend to live at the center of group/cluster halos, which contain a lot of DM; considering instead only the DM mass belonging to individual galactic halos would imply the efficiency to stay almost constant or increase somewhat at large stellar masses. Finally, we have stressed that the similarity of the efficiency at z 2 to the local one is indicative of the early, in-situ nature of the star formation process, at least for massive spheroidal galaxies; in fact, we have noted that the global efficiency at z≈ 2 can be brought on that observed locally for quiescent galaxies by letting the stellar mass to evolve of a modest factor 50% due to late star formation or dry mergers and the halo mass to evolve by a factor of a few due to late smooth accretion and/or tidal stripping. * We have computed the main sequence of star forming galaxies via abundance matching of the input SFR function and of the stellar mass function self-consistently derived from the continuity equation. We have found a remarkable agreement with the observational determinations at different redshifts, so further constraining our input star-formation histories and timescales. We have highlighted how off-main sequence galaxies (located above the average relation) can be simply interpreted in the light of our star formation histories as young objects, caught when their stellar mass is still to be accumulated; they will then progressively move onto the main sequence, where they will spend most of their lifetime as active galaxies, before being quenched.Finally, we conclude by stressing that the added value of the continuity equation, developed here on the stellar component of galaxies, is to provide quantitative, yet largely model-independent outcomes which must be complied by detailed physical models. In particular, the continuity equation allows a full exploitation of the redshift-dependent SFR functions, stellar mass functions, and galaxy main sequence, in order to determine the average star-formation histories and timescales of individual galaxies. Our analysis highlights that a bimodal star-formation history is required for spheroids and discs: the former must be characterized by a nearly constant SFR over short timescalesGyr (increasing somewhat for less star-forming objects), and the latter must feature a SFR exponentially declining over long timescales of several Gyrs. Such outcomes of the continuity equation can provide inspiring hints on ways to improve the (sub-grid) physical recipes implemented in theoretical models and numerical simulations; moreover, they can offer a benchmark for forecasts on future observations at very high redshift with multi-band coverage on medium and wide areas, as it will become routinely achievable with the advent of the JWST. We are grateful to J. Beacom, K. Glazebrook, M. Massardi, F. Pozzi, and P. Salucci for stimulating discussions. We thank the referee for a constructive report. Work partially supported by PRIN MIUR 2015 `Cosmology and Fundamental Physics: illuminating the Dark Universe with Euclid' and PRIN INAF 2014 `Probing the AGN/galaxy co-evolution through ultra-deep and ultra-high-resolution radio surveys'. AL acknowledges the RADIOFOREGROUNDS grant (COMPET-05-2015, agreement number 687312) of the European Union Horizon 2020 research and innovation programme. Alavi, A., Siana, B., Richard, J., et al. 2016, ApJ, 832, 56Aversa, R., Lapi, A., De Zotti, G., Shankar, F., & Danese, L. 2015, ApJ, 810, 74Barger, A. J., Cowie, L. L., Owen, F. N., et al. 2015, ApJ, 801, 87Barro, G., Kriek, M., Pérez-González, P. G., et al. 2016, ApJ, 827, L32Behroozi, P., & Silk, J. 2017, MNRAS, in press [arXiv:1609.04402]Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013, ApJ, 770, 57Bernardi, M., Meert, A., Sheth, R. K., et al. 2017, MNRAS, 467, 2217Bernardi, M., Meert, A., Sheth, R. K., et al. 2013, MNRAS, 436, 697Bocquet, S., Saro, A., Dolag, K., & Mohr, J.J. 2016, MNRAS, 456, 2361Bourne, N., Dunlop, J. S., Merlin, E., et al. 2017, MNRAS, 467, 1360Bouwens, R.J., Oesch, P.A., Illingworth, G.D., Ellis, R.S., & Stefanon, M. 2017, ApJ, 843, 129Bouwens, R. J., Aravena, M., De Carli, R., et al. 2016, ApJ, 833, 72Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2015, ApJ, 803, 34Burkert, A., Forster Schreiber, N. M., Genzel, R., et al. 2016, ApJ, 826, 214Cai, Z.-Y., Lapi, A., Bressan, A., et al. 2014, ApJ, 785, 65Cai, Z.-Y., Lapi, A., Xia, J.-Q., et al. 2013, ApJ, 768, 21Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682Caputi, K. I., Ilbert, O., Laigle, C., et al. 2015, ApJ, 810, 73Cassará, L. P., Maccagni, D., Garilli, B., et al. 2016, A&A, 593, A9Cavaliere, A., Morrison, P., & Wood, K. 1971, ApJ, 170, 223Chabrier G. 2003, ApJ, 586, L133Chiappini, C., Matteucci, F., & Gratton, R. 1997, ApJ, 477, 765Citro, A., Pozzetti, L., Moresco, M., & Cimatti, A. 2016, A&A, 592, A19Comparat, J., Prada, F., Yepes, G., & Klypin, A. 2017, MNRAS, 469, 4157Conroy, C. 2013, ARA&A, 51, 393Contini, E., Kang, Xi, Romeo, A. D., & Xia, Q. 2017, ApJ, 837, 27Cooray, A., Calanog, J., Wardlow, J. L., et al. 2014, ApJ, 790, 40Coppin, K. E. K., Geach, J. E., Almaini, O., et al. 2015, MNRAS, 446, 1293Courteau, S., Cappellari, M., de Jong, R. S., et al. 2014, RvMP, 86, 47Cucciati, O., Tresse, L., Ilbert, O., et al. 2012, A&A, 539, A31da Cunha, E., Walter, F., Smail, I. R., et al. 2015, ApJ, 806, 110Daddi, E., Alexander, D. M., Dickinson, M., et al. 2007, ApJ, 670, 173Davé, R., Rafieferantsoa, M.H., Thompson, R.J., & Hopkins, P.F. 2017, MNRAS, 467, 115Davidzon, I., Ilbert, O., Laigle, C., et al. 2017, ApJ, in press [arXiv:1701.02734]Diemer, B., Sparre, M., & Torrey, P. 2017, ApJ, 839, 26Duncan, K., Conselice, C. J., Mortlock, A., et al. 2014, MNRAS, 444, 2960Dunlop, J. S., McLure, R. J., Biggs, A. D., et al. 2017, MNRAS, 466, 861Dye, S., Furlanetto, C., Swinbank, A. M., et al. 2015, MNRAS, 452, 2258Efstathiou, A., Rowan-Robinson, M., & Siebenmorgen, R. 2000, MNRAS, 313, 734 Finkelstein, S. L., Ryan, R. E., Jr., Papovich, C., et al. 2015, ApJ, 810, 71Fudamoto, Y., Oesch, P.A., Schinnerer, E., et al. 2017, MNRAS, in press [arXiv:1705.01559]Furlong, M., Bower, R. G., Crain, R. A., et al. 2017, MNRAS, 465, 722Gallazzi, A., Charlot, S., Brinchmann, J., & White, S. D. M. 2006, MNRAS, 370, 1106Genzel, R., Forster Schreiber, N. M., Ubler, H., et al. 2017, Natur, 543, 397Gladders, M.D., Oemler, A., Dressler, A., Poggianti, B., Vulcani, B., & Abramson, L. 2013, ApJ, 770, 64Glazebrook, K., Schreiber, C., Labbé, I., et al. 2017, Natur, 544, 71Grazian, A., Fontana, A., Santini, P., et al. 2015, A&A, 575, A96Gruppioni, C., Calura, F., Pozzi, F., et al. 2015, MNRAS, 451, 3419Gruppioni, C., Pozzi, F., Rodighiero, G., et al. 2013, MNRAS, 432, 23Harris, K., Farrah, D., Schulz, B., et al. 2016, MNRAS, 457, 4179Harrison, C. M., Simpson, J. M., Stanley, F., et al. 2016, MNRAS, 457, L122Hill, A.R., Muzzin, A., Franx, M., et al. 2017, ApJ, 837, 147Hopkins, P.F., Wetzel, A., Keres, D., et al. 2017, MNRAS, in press [arXiv:1702.06148]Hudson, M. J., Gillis, B. R., Coupon, J., et al. 2015, MNRAS, 447, 298Ikarashi, S., Ivison, R. J., Caputi, K.I., et al. 2017, ApJ, 835, 286Ikarashi, S., Ivison, R. J., Caputi, K.I. 2015, ApJ, 810, 133Ilbert, O., McCracken, H. J., le Fevre, O., et al. 2013, A&A, 556, A55Kennicutt, R. C. 1983, ApJ, 272, 54Koprowski, M., Dunlop, J. S., Michalowski, M. J., et al. 2016, MNRAS, 458, 4321Koprowski, M. P., Dunlop, J. S., Michalowski, M. J., Cirasuolo, M., & Bowler, R. A. A. 2014, MNRAS, 444, 117Kriek, M., Conroy, C., van Dokkum, P.G., et al. 2016, Natur, 540, 248Kurczynski, P., Gawiser, E., Acquaviva, V., et al. 2016, ApJL, 820, L1Lacey, C. G., Baugh, C. M., Frenk, C. S., et al. 2010, MNRAS, 405, 2Lapi, A., Mancuso, C., Celotti, A., & Danese, L. 2017, ApJ, 835, 37Lapi, A., Salucci, P., & Danese, L. 2013, ApJ, 772, 85Lapi, A., Gonzalez-Nuevo, J., Fan, L., et al. 2011, ApJ, 742, 24Lee, S.-K., Idzi, R., Ferguson, H. C., et al. 2009, ApJS, 184, 100Leitner, S.N., & Kravtsov, A.V. 2011, ApJ, 734, 48Leja, J., van Dokkum, P. G., Franx, M., & Whitaker, K. E. 2015, ApJ, 798, 115Lilly, S. J., Carollo, C. M., Pipino, A., et al. 2013, ApJ, 772, 119Lonoce, I., Longhetti, M., Maraston, C., et al. 2015, MNRAS, 454, 3912Ma, J., Gonzalez, A. H., Viera, J. D., et al. 2016, ApJ, 832, 114Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415Magnelli, B., Popesso, P., Berta, S., et al. 2013, A&A, 553, A132Man, A.W.S., Greve, T.R., Toft, S., et al. 2016, ApJ, 820, 11Mancuso, C., Lapi, A., Shi, J., et al. 2016a, ApJ, 823, 128Mancuso, C., Lapi, A., Shi, J., et al. 2016b, ApJ, 833, 152Mandelbaum, R., Wang, W., Zu, Y., et al. 2016, MNRAS, 457, 3200Marconi, A., Risaliti, G., Gilli, R., et al. 2004, MNRAS, 351, 169Mawatari, K., Yamada, T., Fazio, G. G., Huang, J.-S., & Ashby, M. L. N. 2016, PASJ, 68, 46Mendel, J. T., Simard, L., Palmer, M., Ellison, S.L., & Patton, D.R. 2014, ApJS, 210, 3Merloni, A., & Heinz, S. 2008, MNRAS, 388, 1011Meurer, G. R., Heckman, T. M., & Calzetti, D. 1999, ApJ, 521, 64Michalowski, M.J., Dunlop, J.S., Koprowski, M.P., et al. 2016, MNRAS, 469, 492Moffett, A.J., Lange, R., Driver, S.P., et al. 2016, MNRAS, 462, 4336Mor, R., Netzer, H., Trakhtenbrot, B., Shemmer, O., & Lira, P. 2012, ApJL, 749, L25More, S., van den Bosch, F. C., Cacciato, M., et al. 2011, MNRAS, 410, 210Moster, B. P., Naab, T., & White, S. D. M. 2013, MNRAS, 428, 3121Moustakas, J., Coil, A. L., Aird, J., et al. 2013, ApJ, 767, 50Mullaney, J. R., Daddi, E., Bethermin, M., et al. 2012, ApJL, 753, L30Nayyeri, H., Keele, M., Cooray, A., et al. 2016, ApJ, 823, 17Negrello, M., Amber, S., Amvrosiadis, A., et al. 2017, MNRAS, 465, 3558Negrello, M., Hopwood, R., Dye, S., et al. 2014, MNRAS, 440, 1999Netzer, H., Lani, C., Nordon, R., et al. 2016, ApJ, 819, 123Novak, M., Smolcic, V., Delhaize, J., et al. 2017, A&A, 602, A5Oesch, P. A., Bouwens, R. J., Carollo, C. M., et al. 2010, ApJL, 725, L150Page, M. J., Symeonidis, M., Vieira, J., et al. 2012, Natur, 485, 213Papovich, C., Finkelstein, S. L., Ferguson, H. C., Lotz, J. M., & Giavalisco, M. 2011, MNRAS, 412, 1123Pezzulli, G., & Fraternali, F. 2016, MNRAS, 455, 2308Planck Collaboration XIII 2016, A&A, 594, A13Pope, A., Montana, A., Battisti, A., et al. 2017, ApJ, 838, 137Reddy, N. A., Kriek, M., Shapley, A. E., et al. 2015, ApJ, 806, 259Renzini, A. 2006, ARA&A, 44, 141Renzini, A., & Peng, Y.-J. 2015, ApJL, 801, L29Riechers, D.A., Daisy Leung, T.K., Ivison, R., et al. 2017, ApJ, submitted [preprint arXiv:1705.09660]Rodighiero, G., Brusa, M, Daddi, E., et al. 2015, ApJL, 800, L10Rodighiero, G., Daddi, E., Baronchelli, I., et al. 2011, ApJL, 739, L40Rodriguez-Gomez, V., Pillepich, A., Sales, L.V., et al. 2016, MNRAS, 458, 2371Rodriguez-Gomez, V., Genel, S., Vogelsberger, M., et al. 2015, MNRAS, 449, 49Rodriguez-Puebla, A., Avila-Reese, V., Yang, X., et al. 2015, ApJ, 799, 130Romano, D., Silva, L., Matteucci, F., & Danese, L. 2002, MNRAS, 334, 444Rong, Y., Jing, Y., Gao, L., et al. 2017, MNRAS, 471, L36Rosario, D. J., Santini, P., Lutz, D., et al. 2012, A&A, 545, 45Rowan-Robinson, M., Oliver, S., Wang, L., et al. 2016, MNRAS, 461, 1100Salmon, B., Papovich, C., Finkelstein, S. L., et al. 2015, ApJ, 799, 183Salpeter, E. E. 1955, ApJ, 121, 161Salucci, P., Szuszkiewicz, E., Monaco, P., & Danese, L. 1999, MNRAS, 307, 637Schaye, J., Crain, R. A., Bower, R. G., et al. 2015, MNRAS, 446, 521Schreiber, C., Pannella, M., Leiton, R., et al. 2017, A&A, 599, A134Scoville, N., Sheth, K., Aussel, H., et al. 2016, ApJ, 820, 83Scoville, N., Aussel, H., Sheth, K., et al. 2014, ApJ, 783, 84Shankar, F., Weinberg, D. H., & Miralda-Escude, J. 2013, MNRAS, 428, 421Shankar, F., Weinberg, D. H., & Miralda-Escude, J. 2009, ApJ, 690, 20Shankar, F., Lapi, A., Salucci, P., de Zotti, G., & Danese, L. 2006, ApJ, 643, 14Shi, J., Lapi, A., Mancuso, C., Wang, H., & Danese, L. 2017, ApJ, submittedSilva, L., Granato, G. L., Bressan, A., & Danese, L. 1998, ApJ, 509, 103Simpson, J. M., Smail, I., Wang, W.-H., et al. 2017, ApJ, 844, L10Simpson, J. M., Smail, I., Swinbank, A. M., et al. 2015, ApJ, 799, 81Small, T. A., & Blandford, R. D. 1992, MNRAS, 259, 725Smit, R., Bouwens, R. J., Franx, M., et al. 2012, ApJ, 756, 14Soltan, A. 1982, MNRAS, 200, 115Song, M., Finkelstein, S. L., Ashby, M. L. N., et al. 2016, ApJ, 825, 5Speagle, J. S., Steinhardt, C. L., Capak, P. L., & Silverman, J. 2014, ApJS, 214, 15Spilker, J. S., Bezanson, R., Marrone, D. P., et al. 2016, ApJ, 832, 19Stanley, F., Harrison, C. M., Alexander, D. M., et al. 2015, MNRAS, 453, 591Steinhardt, C.L., Yurk, D., & Capak, P., 2017, MNRAS, 468, 849Steinhardt, C. L., Speagle, J. S., & Capak, P. 2014, ApJL, 791, L25Straatman, C. M. S., Spitler, L. R., Quadri, R. F., et al. 2016, ApJ, 830, 51Straatman, C. M. S., Labbé, I., Spitler, L. R., et al. 2015, ApJ, 808, L29Straatman, C.M.S., Labbé, I., Spitler, L.R., et al. 2014, ApJ, 783, L14Strandet, M. L., Weiss, A., Vieira, J. D., et al. 2016, ApJ, 822, 80Swinbank, M., Harrison, C., Trayford, J., et al. 2017, ApJ, 467, 3140Tadaki, K.-I., Genzel, R., Kodama, T., et al. 2017, ApJ, 834, 135Tasca, L. A. M., Le Févre, O., Hathi, N. P., et al. 2015, A&A, 581, A54Thanjavur, K., Simard, L., Bluck, A. F. L., & Mendel, T. 2016, MNRAS, 459, 44Thomas, D., Maraston, C., Bender, R., & Mendes de Oliveira, C. 2005, ApJ, 621, 673Tinker, J. L., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709Tomczak, A. R., Quadri, R. F., Tran, K. H., et al. 2016, ApJ, 817, 118Tomczak, A. R., Quadri, R. F., Tran, K.-V. H., et al. 2014, ApJ, 783, 85Vale, A., & Ostriker, J. P. 2004, MNRAS, 353, 189van der Burg, R. F. J., Hildebrandt, H., & Erben, T. 2010, A&A, 523, A74Velander, M., van Uitert, E., Hoekstra, H., et al. 2014, MNRAS, 437, 2111Wang, R., Wagg, J., & Carilli, C. L. 2013, ApJ, 773, 44Watson, W.A., Iliev, I.T., D'Aloisio, A., et al. 2013, MNRAS, 433, 1230Weiss, A., De Breuck, C., Marrone, D. P., et al. 2013, ApJ, 767, 88Wellons, S., Torrey, P., Ma, C.-P., et al. 2015, MNRAS, 449, 361Whitaker, K. E., Franx, M., Leja, J., et al. 2014, ApJ, 795, 104 Willott, C. J., Bergeron, J., & Omont, A. 2015, ApJ, 801, 123Wojtak, R., & Mamon, G. A. 2013, MNRAS, 428, 2407Wyder, T. K., Treyer, M. A., Milliard, B., et al. 2005, ApJL, 619, L15Xu, L., Rieke, G. H., Egami, E., et al. 2015, ApJ, 808, 159Yu, Q., & Lu, Y. 2004, ApJ, 602, 603Yu, Q., & Lu, Y. 2008, ApJ, 689, 732Zavala, J.A., Montana, A., Hughes, D.H., et al. 2017, Natur, submitted | http://arxiv.org/abs/1708.07643v1 | {
"authors": [
"A. Lapi",
"Claudia Mancuso",
"A. Bressan",
"L. Danese"
],
"categories": [
"astro-ph.GA",
"astro-ph.CO"
],
"primary_category": "astro-ph.GA",
"published": "20170825081303",
"title": "Stellar Mass Function of Active and Quiescent Galaxies via the Continuity Equation"
} |
[pages=1-last]arXiv_Acrobat.pdf | http://arxiv.org/abs/1708.07605v5 | {
"authors": [
"Bolei Wang",
"Feifei Gao",
"Shi Jin",
"Hai Lin",
"Geoffrey Ye Li"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170825034155",
"title": "Spatial- and Frequency-Wideband Effects in Millimeter-Wave Massive MIMO Systems"
} |
plain abbrv | http://arxiv.org/abs/1708.07366v1 | {
"authors": [
"Martin Sulzmann",
"Peter Thiemann"
],
"categories": [
"cs.FL",
"cs.LO",
"cs.PL"
],
"primary_category": "cs.FL",
"published": "20170824120228",
"title": "A Computational Interpretation of Context-Free Expressions"
} |
firstpage–lastpage 2013Decoherence of two coupled singlet-triplet spin qubits S. Das Sarma Accepted ?. Received ?; in original form ?. =========================================================== We report the discovery of a radio quiet type 2 quasar (SDSS J165315.06+234943.0 nicknamed the “Beetle” at z=0.103)with unambiguousevidence for active galactic nucleus (AGN) radio induced feedback acting acrossa total extension of ∼46 kpc andup to∼26 kpc from the AGN.To the best of our knowledge, this is the first radio quiet system where radio induced feedback has been securely identified at≫several kpc from the AGN. The morphological, ionization and kinematic properties of the extended ionized gasare correlated with the radio structures. We find along the radio axis (a) enhancement of theoptical line emission at the location of the radio hot spots (b) turbulent gas kinematics (FWHM∼380-470 km s^-1)across the entire spatial range circumscribed by them (c) ionization minima for the turbulent gasat the location of thehot spots, (d)hightemperatureT_e1.9×10^4 K at the NEhot spot. Turbulent gas is alsofound far from the radio axis, ∼25 kpc in the perpendicular direction. We propose a scenario in which theradio structures have perforated the interstellar medium of the galaxy and escaped into the circumgalactic medium.While advancing, they have interacted with in-situ gasmodifying its properties. Our results show that jets of modest power can be the dominant feedback mechanism acting across huge volumes in radio quiet systems, including highly accreting luminous AGN, whereradiative mode feedback may be expected. galaxies: active - galaxies: evolution - quasars: individual: SDSS J1653+23 § INTRODUCTIONFeedback induced by the activity of supermassive black holes (SMBH) in massive galaxies is thought to play a critical role in their evolution by means of regulating the amount of gas available for star formation and black hole growth.This type of AGN (active galactic nuclei) feedback refers to the processes of interaction between the energy and radiation generated by accretion onto the massive black hole and the gas in the host galaxy (Fabian ).This may give answers to some fundamental questions in cosmology (Silk & Rees ,King , Silk & Mamon ), such as the discrepancy between the predicted and observed high mass end of the galaxy luminosity function and the observed correlationsbetween the black hole mass and someproperties of the spheroidal component in galaxies (Ferrarese & Merritt , McConnell & Ma ). Hydrodynamic simulations show that the energy output of these outflows can indeed regulate the growth and activity of black holes and their host galaxies (di Matteo et al. ). For this,the intense flux of photons and particles produced by the AGN must sweep the galaxy clean of interstellar gas and terminate star formation and the activity of the SMBH (Fabian ). The most powerful outflows with potentially the most extreme effects on the environment are expected in quasars, the most powerful AGN (Page et al. , Woo et al. ). Observational evidence for such dramatic an impact ishowever controversial.Since their recent discovery, optically selected type 2quasars (QSO2,Zakamska et al. , Reyes et al. ) have been unique systems for investigatingthe wayfeedback works in the most powerful AGN. The activenucleus isobscured and thisallows a detailed study of many propertiesof the surrounding medium without the overwhelming glare of the quasar, which strongly affects related studies of their unobscured type 1counterparts (QSO1). During recent years it has become clear that ionized outflows areubiquitousin QSO2 at different z (Villar Martín et al. , , Greene et al. ,Harrison et al. , Liu et al. ,Harrison et al.,Forster-Schreiber et al. , McElroy et al. ). Extreme motions are often measured, with FWHM>1000 km s^-1 and typical velocity shifts V_S∼several×100 km s^-1.The outflows are triggered by AGN related processes (e.g. Villar Martín et al.2014, hereafter <cit.>, Zakamska & Greene ). The case for effective gas ejection and star formation truncationexerted by the ionized outflows is less clear.Recent integral field spectroscopic studies ofQSO2 at z0.7 have suggested that large scale, wide angle AGN driven ionizedoutflows are prevalent in these systems and their actioncan be exerted across many kpc, even the entire galaxy (Liu et al. ,Harrison et al., McElroy et al. ). Thekinetic energies, Ė_ o, outflow masses, M_ o, and mass outflow rates, Ṁ_ o, are estimated to be large enoughfor a significant impact on their hosts. In the proposed scenario, processes related to the accretion disk(thermal, radiation and magnetic driving) launch a relativistic wide angle windwhich then shocksthe surrounding gas and drives the outflow(e.g. Proga , Zubovas & King ). On the other hand, the above observationalresults have been questioned by different authors (<cit.>, Karouzos et al. , Villar Martín et al. , Husemann et al. ; see also Husemann et al. for QSO1) who find that the ionized outflows areconstrained in general within R_ o1-2 kpc.This raises doubts about their impact on the host galaxies, since their effectiveness as a feedback mechanism would be significantly reduced. Another mechanism of AGN feedback is that induced by relativistic jets. The jets originate in the vicinity of the SMBH, on scales of a few to 100 times the gravitational radius. Three dimensional relativistic hydrodynamic simulations of interactions of AGN jets with a dense turbulent two-phase interstellar medium, show that the originally well-directed jets forman energy-driven almostsphericalbubble that can affect the gas to distances up to several kpc from the injection region, depending on the jet power. The shocks resulting from such interactions create a multiphase ISM and radial outflows (e.g. Mukeherjee et al. ). Observational studies of powerful radio galaxies and radio loud QSO1at different z show that the interaction between the radio structures and the ambient gas canindeed induce outflows that can extend across many kpc,sometimes well outside the galaxy boundaries (Tadhunter et al. , McCarthy et al. , Villar Martín et al. ,, Solórzano Iñarrea et al. , Humphrey et al. , Fu & Stockton ).Theymay have enormous energies sufficient to eject a large fraction of the gaseous content out of the galaxy and/or quench star formation, thus having an important impact on the evolution of the host galaxies(Morganti et al. , Nesvadba et al. , ). This feedback mechanism (hereafter radio induced feedback)has been proposed to explain the gas cooling problem at the centre of galaxy clusters (see Fabianfor a review). Itmay also prevent the formation of extremely bright galaxies at the centre of clusters (e.g.Sijacki & Springel ). Such scenarios involve a central dominant elliptical of a cluster going through a powerful radio loud phase. Simulations by <cit.>show that radio induced feedback can have significant effectsalso in more modest environments, including isolated ellipticals.It could explain the evidence for reduced cooling also in elliptical galaxiesand the related absence of cold gas in these systems (Mathew & Baker , Best et al. ).The role of radio induced feedback has not been sufficiently explored inradio quiet quasars. Only ∼15±5% of QSO2 are thought to beradio loud(Lal & Ho ). The same percentage applies to QSO1 (Kellermann et al. ). However, this does not discard radio induced outflowsas a potentially significant feedback mechanism.Radio-gas interactions[By “radio-gas interactions” we will refer to the interactionsbetween any AGN related radio structure (jets, lobes, hot spots) and the ambient gas] and the associated outflows have been observed in radio quiet AGN for several decades (e.g. Wilson & Ulvestad , Whittle ,). The most extremeionized outflows are often found in objects (includingQSO2) with some degree of radio activity, even if they are not radio loud (e.g. Mullaney et al. , <cit.>, Husemann et al. , , Zakamska & Greene ). Itssignificant role as a feedback mechanism has been quantified inseveral low luminosity AGN (Seyfert galaxies) (Tadhunter et al. , Alatalo et al. ). Although the effects are obvious in a small region (R1 kpc from the SMBH), theoutflows can disturb and heat the molecular gas, thus having the potential toquench star formation (Guillard et al. ). Large scale effects (≫ several kpc) due to radio induced feedbackin radio quiet AGN are usually not expected, nor are they usually observed.Traditionally, it is assumed that radio quiet AGN contain low power jets which will decelerate and lose their power after a few kpc. However, different results suggest that radio induced feedback may indeed occur across many kpc in some systems. On one hand, radio-gas interaction simulations show that while ahigh power jet (P_ jet10^45erg s^-1) can remainrelativistic to large scales and drillwith relative ease through tens of kpc, a jet with low mechanical power (P_ jet10^43erg s^-1), although less efficient in accelerating clouds, is trapped in the interstellar medium (ISM) for a longer time and hence can affect the ISM over a larger volume (Bicknell , Mukherjee et al. ). On the other hand,some radio quiet quasarsassociated with modest or low power jets show large radio sources with sizes in the range of tens or even hundreds of kpc (Kellerman et al. ). Thus, it is clear thatthese radio sources canescape the galaxy boundaries and expand into the circumgalactic medium (CGM).[Following Tumlinson et al. , we consider the CGM as the gas surrounding galaxies outside their disks or ISM and inside their virial radii.]However, to the best of our knowledge, the effects of radio-gas interactions have not been directly observedbeyond scales of ∼several kpc from the AGN in such systems. The “Teacup”QSO2 (z=0.085), which shows∼10-12 kpc radio/optical bubbles, may be the radio quiet AGN where the effects of radio-gas interactions have been identified acrossthe largest spatial scale so far (Harrison et al. , Ramos Almeida et al. ).It is not clear, however,whether the bubbleshave been inflatedinstead by a wide angle large scale AGN driven wind (Harrison et al. ). We report here unambiguous evidence for radio induced feedback working across a total extension of ∼46 kpc and up to ∼26 kpc from the AGN, well into the CGM, in a radio quiet QSO2.We present a detailed study ofSDSS J165315.06+234943.0 (nicknamed the “Beetle” hereafter) at z=0.103 based on GTC optical images andlong slit spectroscopy and VLA radio maps. The data reveal thatradio induced feedbackis modifying the morphological, kinematic and physical properties of the gaseous environment on scales from (possibly) ∼1 kpc up to the CGM.Previous knowledge on the “Beetle”is summarized in Sect. <ref>. Theobservations,reduction and analysis methods of the different data sets (optical imaging and spectroscopy, radio maps) are described inSect.<ref>.Analysis and results are presented inSect. <ref> and discussed inSect. <ref>. The summary and conclusions are in Sect. <ref>. We adopt H_0=71 km s^-1 Mpc^-1, Ω_Λ=0.73 and Ω_m=0.27.This gives an arcsec to kpc conversion of 1.87 kpc arcsec^-1 at z=0.103.§.§ The radio quiet QSO2 SDSS J165315.06+234943.0The “Beetle”was originally selected by<cit.> for their Sloan Digital Sky Survey (SDSS, York et al. ) catalogue of optically selected QSO2 at z0.8. Its [OIII]λ5007 luminosity, log(L_ [OIII])=42.6,corresponds to a bolometric luminosity log(L_ bol)∼46.0 (Stern & Laor ), in the range of quasars (Woo & Urry ). The information and calculations that follow are provided because they will be relevant in later sections. * Radio loudness.The “Beetle”has been detected at 1.4 GHz in the VLA-FIRST survey with a peak flux of 6.4 mJy, while its peak fluxin the VLA NVSS surveyis 9.4 mJy. In order to classify this QSO2 according to the radio-loudness, we have used the rest-frame radio power at 5 GHz P_ 5GHz and log(L_ [OIII]) (see Fig. 7 in Lal & Ho ; see also Xu et al. ). These two luminosities separate objects clearly intothe two families of radio loud and radio quiet AGN with a significant gap between them (Fig. <ref>). We calculate P_ 5GHz = 4 π D_ L^2 S_ 5GHz (1+z)^-1-α whereD_ L is the luminosity distance, S_ 5GHz is the observed flux density at 5 GHz and α is the spectral index such that S_ν∝ν^α.We computeS_ 5GHz from the available S_ 1.4GHz. The index α is unknown. Forα =+0.094 (Lal & Ho ), log(P_ 5GHz)=30.4 (erg s^-1 Hz^-1). Considering the range -1.1≤α≤+0.85spanned by QSO2s (Lal & Ho ), we obtain log(P_5GHz)=30.4^+0.4_-0.6 erg s^-1 Hz^-1. Fig. <ref> shows that the “Beetle” is a radio quiet QSO2, and this conclusion is not affected by the uncertainty in α.* Infrared luminosity and its origin.The IRAS fluxes for the “Beetle" are S_12μ m=0.10, S_25μ m=0.15, S_60μ m=0.49 and S_100μ m1.01 Jy. Following <cit.>, thetotal infrared luminosity is 4.3×10^11 L_⊙ < L_IR(8-1000 μ m) 5.5×10^11 L_⊙[k correction is negligible for z=0.103]implying that the object isin the regime of luminous infrared galaxies (LIRGs, 10^11≤ L_ IR/L_⊙<10^12). AssumingS_60μ m/S_100μ m≥1.4 as observed in LIRGs (e.g. Lu et al. , Pearson et al. ) then L_IR(8-1000 μ m)=(5.0±0.4)×10^11 L_⊙.The far infrared luminosity is L_FIR(40-500 μ m)=(1.7±0.3)×10^11 L_⊙.If L_ IR were to dominated by a starburst contribution, thiswould imply a star forming rate SFR=87±7 M_ ⊙ yr^-1 (Kennicutt ).However, this is a gross upper limit sinceL_ IRiscontaminated (maybe dominated) by AGN heated dust. This is based on two arguments. Following <cit.>, we have calculated q =log [F_ FIR/3.75 × 10^12 Hz/S_ 1.4 GHz]=1.87±0.09, where F_FIR is the far infrared flux in erg s^-1 cm^-2 and S_ 1.4 GHz is in units oferg s^-1 cm^-2 Hz^-1. For the “Beetle" q is within the range measured for radio emitting AGN (<q_ AGN>=2.0 with rms scatter σ_AGN=0.59) and outside the range measured for star forming galaxies (<q_ SF>=2.3 with rms scatter σ_AGN=0.18).The IR colour α_IR=log(S_ 25 μ m/S_ 60 μ m)/log(60/25)=-1.3is alsoconsistent with AGN values (α_IR>-1.5, in comparison withα_IR<-1.5 forstar forming galaxies, Mauch & Sadler ). * Stellar velocity dispersion and stellar mass Two components of the MgI triplet at5169Å and 5185 Åare clearly detected in the nuclear GTC spectrum. They have σ=210±25 km s^-1 and 220±33 km s^-1 respectively, which isin reasonable agreement with σ_ *=169±8 km s^-1 provided by the the SDSS Data Release 10 (DR10) kinematic fits. We will assume σ_*∼200 km s^-1. Estimated stellar masses for the “Beetle”are in the range 10.64log(M_ */M_⊙)11.34, with a median value 11.2. The different values have been retrieved from the SDSS DR10 and the Vizier catalogue(Ochsenbein al. )based on <cit.>.According to these authors (see alsoSimmard et al. ), the spheroidal component contains a masslog(M_ sph/M_⊙)=11.0. * Black hole mass and Eddington luminosity. We have constrained the black hole mass M_BH using the correlation between stellar velocity dispersion σ_* and M_BHfor early type galaxies(McConnell & Ma ): log(M_BH/M_⊙)= 8.39+5.20 log(σ_*/200 km s^-1)=8.4, where σ_*∼200 km s^-1. This is consistent with the value log(M_BH/M_⊙)= 8.5 obtained from the M_BH vs. bulge (or spheroidal component) massM_ sph correlation log(M_ BH/M_⊙)=8.46+1.05 log(M_ sph/10^11 M_⊙). The implied Eddington luminosityis L_ Edd =1.26 ×10^38 (M_BH/M_⊙) erg s^-1∼3.2 × 10^46 erg s^-1. * A nuclear ionized outflow driven by the nuclear activity was identified in the SDSSoptical spectrum of this QSO (<cit.>),withFWHM∼970 km s^-1 and a velocity blueshift V_ s∼ -90 km s^-1.§ OBSERVATIONS, DATA REDUCTION AND ANALYSIS §.§ GTC imaging§.§.§ Observations Hα Tunable Filter (TF) imaging observations were taken in service mode on July 25th 2014 (programmeGTC37-14A) for the “Beetle” using the optical imager and long slit spectrograph OSIRIS[http://www.gtc.iac.es/en/pages/instrumentation/osiris.php] mounted on the 10.4m Gran Telescopio Canarias (GTC).OSIRIS consists of a mosaic of two 2048 × 4096 Marconi CCD42-82 (with a 9.4 arcsec gap between them) and covers the wavelength range from 0.365 to 1.05 μm with a field of view of 7.8 x 7.8 arcmin and a pixel size of 0.127 arcsec. However, the OSIRIS standard observation modes use 2×2 binning, hence the effective pixel size during our observations was 0.254 arcsec.Our goal is to detect extended ionized gas (Hα) associated with the QSO2. For this,a filter FWHM of 20 Å was used (∼830 km s^-1 at the QSO2 z) centred on the redshifted Hα and taking into account the dependence of the wavelengthobserved with the red TF with distance relative to the optical centre. The order sorter filter f723/45 was also used. The FWHM is adequate to cover the velocity range expected to be spanned by any plausible extended ionized features. Contamination by [NII]λλ6548,6583 is expected to be negligible, since the lines are located at-16.5 Å and +22 Årelative to Hα, and Hα is the strongest of the three lines at all locations (our long spectroscopy confirms this). The total exposure time on source was 2700 seconds distributed in 3×900 second exposures. To be able to correct for ghost images and cosmic rays we dithered between three positions, moving the telescope ±15 arcsec in RA and Dec. To sample the continuum near the Hα+[NII] complex we took one continuum image to the blue of Hα using directly the OSIRIS Red Order Sorter (OS) Filter f666/36, which covers the 6490 - 6845 Å spectral window (5884 - 6206 Å rest frame). This filter was used to avoid emission line contamination (by, for instance, [OI]λ6300). The disadvantage is that objects with a steep continuum slope (for instance, very red objects) will leave prominentresiduals due to the significant shift in λ relative to Hα. Thanks to the complementary spectroscopic data, we can confirm that this has no impact on our analysis and interpretation. The total exposure time of the continuum image was 600 seconds split into 3×200 second exposure. The same dithering pattern was applied as for the Hα image.§.§.§ Reduction processThe TF and the OS filter OSIRIS images were bias and flat-field corrected as usual, using a set of bias frames and dome flats. We observed a photometricstandard star for flux calibration using the exact same set up as the one used for the science object. After correcting thecorresponding frames for bias and flat-field we applied aperture photometry using the routine PHOT in pyIRAF. The PHOT output gives us the integrated number of counts within the aperture for the standard star observed. We then used a customized Python routine that generates a TF transmission curve for a given central wavelength and FWHM. The resulting TF transmission curve was normalized and multiplied by the spectra of the standard star in units of erg s^-1 cm^-2 Å^-1. This gives us the expected flux of the standard star in the corresponding TF and OS filter in units of erg s^-1 cm^-2. We then multiply these numbers by the exposure time used for the standard star observations and divide the result by the output from PHOT. The result will be a flux calibration factor (F) in units of erg cm^-2 counts^-1. Finally, to calibrate in flux our science frames we divide them by their corresponding exposure times and multiply them by F, which transforms the units of the frames after bias and flat field correction (counts) into erg s^-1 cm^-2.Once the images are calibrated in flux, we need to subtract the sky emission. When using a TF imaging technique the wavelength observed changes relative to the optical centre. In the case of OSIRIS, this change is given by<cit.>: λ = λ_0-5.04*r^2 Where λ_0 is the wavelength in Åat the optical centre, located at pixel (2098, 1952) of CCD1 in the case of OSIRIS, and r is the distance from that centre in arcmin. Therefore, the different sky lines around the wavelength range selected for the observations are observed in the form of rings of emission spread over the CCDs. This effect becomes important if the sample objects are relatively extended anda careful sky emission subtraction must be performed.With this aim we first cut a rectangular region of the CCD that is large enougt to contain a sizeable region of the sky, and small enough so that the effects of the sky emission rings are reduced to the minimum possible. Then we used FIT1D in pyIRAF to fit a 1-dimensional polynomial function to the overall structure in the x and y direction respectively. The first x/y 1-dimensional fit is performed only on sky regions free from emission from stars in the frame and the target observation, and it is then extended through the entire rectangular cut. The following y/x is applied to all the rectangular cut at once. The result of this process is a synthetic image containing only the sky emission. Then this image is directly subtracted from the original flux calibrated image of the source resulting in an image cleaned from any background and sky emission.The next step during the reduction process is subtracting the continuum from the emission line image. Prior to this they were geometrically aligned. Based onthe centroid of the stars in the aligned images andthe position of some important features in the galaxy, we estimate that the alignment accuracy is better than 0.5 pixels. The continuum emission was then subtracted from the emission line image in order to produce the final “pure” Hα image. §.§ GTC long slit spectroscopyLong slit spectroscopic observations were performed in service mode withOSIRIS on GTC between June 26th and July 23rd 2015 (programme GTC1-15A). The R2500V and R2500R volume-phased holographic gratings (VPHs) were used, that provide a spectral coverage of 4500- 6000 Å and 5575 - 7685 Å, respectively, with dispersions of 0.8 and 1.04 Å/pixel. A 1.23 arcsec slit width was used at three different orientations (position angle PA 0^∘, PA 48^∘ and PA 83^∘)[All position angles throughout the text are quoted N to E] in order to map the emission around the quasar. The spectral resolution FWHM_inst measured from several prominent sky lines is 3.37±0.13 and 4.41±0.06Å for R2500V and R2500R respectively. Several 300 second spectra were obtained at each orientation (Table <ref>), with shifts in the slit direction of 20" between consecutive exposures for both a better fringing correction and a more accurate background subtraction. Additionally, a series of 3 x 900 second R2500V spectra were taken on 07/03/2016 under GTC Directors Discretionary Time (programme GTC01-16ADDT). The same slit width was usedwithPA 140^∘. Those spectra where obtained avoiding the quasar nucleus and roughly perpendicular to the main radio axis (PA 48^∘) . §.§.§ Reduction process and spectral fitting The reduction of the spectra was done using standard procedures and IRAF tasks. Images were first bias and flat-field corrected, by using lamp flats. The 2-dimensional spectra were wavelength calibrated using Xe+Ne+HgAr lamps, with a resulting error consistent with the nominal spectral resolutions of the VPHs. After the wavelength calibration, sky background was subtracted and a two dimensional spectrum was obtained. Flux calibration was done using observations of spectrophotometric standard stars, white dwarfs, obtained the same nights as the scientific spectra. Finally, the diffferent individual spectra were averaged, obtaining a mean spectrum for each of the orientations.Like the images, the spectra were corrected for atmospheric and foreground galactic extinction (A_ V=0.153).The spectral profiles of the nuclear emission lines are complex (several kinematic components). They were fitted following the multi-Gaussian procedure described in detail in <cit.>. In order to have a more realistic estimation of the uncertainties, whenever possible, the lines were fitted in two different ways: a) with kinematic constraints based on the results of the [OIII]λλ4949,5007fits and b) without prior kinematic constraints. When both methods provided fits of similar quality, the fitted parametervalues (fluxes, FWHM and V_ s) are the averageof all fits. The error for each parameter is chosen as the largest value between the fit errors and the standard deviation of all valid fits (see Villar Martín et al.for more details). The FWHM values were corrected for instrumental broadening by subtracting the instrumental profile FWHM_ inst in quadrature. When physically meaningful or mathematically valid fits (e.g. for faint emission lines) could not be obtained without kinematic constraints, the [OIII] doublet lines were used as reference to force the individual kinematic components to have the same velocity shift and/or FWHM. §.§ VLA radio maps Observations with the Karl G. Jansky Very Large Array (VLA) were performed on 13 and 18 July 2015 with the A-configuration (programme 15A-237) and on 01 June 2016 with the B-configuration (programme 16A-088). The on-source exposure time in A-configuration was 0.7 h and in the B-configuration 3 h. The correlator was set to produce a 1 GHz bandwidth, ranging from 1 - 2 GHz, with 1 MHz channels.We used 3C 286 for flux calibration, and obtained a 5 min scan on the strong (∼5 Jy) source J1609+2641 every ∼50 min for both phase and bandpass calibration. Flagging of radio-frequency interference (RFI), which affected ∼40% of our data, as well as a standard bandpass, phase and flux calibration was performed using the VLA data reduction pipeline in the Common Astronomy Software Applications (CASA) v. 4.6 (McMullin et al. ). We manually inspected the pipeline-calibrated data to ensure that the quality of the data products was sufficient for further reduction. We subsequently applied one self-calibration to the A-configuration data and two iterative self-calibrations to the B-configuration data to improve the phases. The data were imaged by applying a Fourier transform using a multi-frequency synthesis technique, and subsequently cleaning the continuum signal. We imaged the A-configuration data using uniform weighting, which resulted in a beam of 0.80 × 0.74 arcsec (PA = 41.8^∘) and rms noise level of 0.066 mJy beam^-1. We also imaged the A-configuration data using robust +0.3 weighting (Briggs ), which resulted in a beam of 1.05 × 0.97 (PA = 47.7^∘) arcsec and rms noise level of 0.018 mJy beam^-1. The B-configuration data were imaged using robust +0.3 weighting (Briggs 1995), which resulted in a beam of 3.52 × 3.25 arcsec (PA = 53.8^∘) and rms noise level of 0.013 mJy beam^-1. The B-configuration data revealed the radio continuum from star formation in two galaxies, the companion galaxy ∼11 arcsec or ∼20 kpc south of the “Beetle”and a galaxy ∼42 arcsec or ∼78 kpc NE. This allowed us to overlay the radio and optical images to within an estimated accuracy of ∼0.6 arcsec.§ ANALYSIS AND RESULTS §.§ Radio vs. optical morphology The continuum image of the “Beetle” (Fig. <ref>, left) reveals an elliptical galaxy with clear signatures of a merger/interaction event, such as shells and filaments. The shells do not completely encircle the central galaxy and their main axis is aligned with the galaxy major axis. This has been observed in otherelliptical galaxieswith high ellipticity (e.g. Athanassoula &Bosma , Duc et al. ) as is the case for the “Beetle”(e=b/a=0.78, Vizier catalogue based on Hakobian et al. ). Simulations of galactic interactions show that such shells may result from (nearly) head-on collisions with smaller disk galaxies (e.g. Quinn , Struck et al. ). The interacting galaxy is probablythe smallergalaxy located ∼11 arcsec or ∼20 kpc South of the QSO2[All quoted distances and sizes will be projected values unless otherwise specfied] (Fig. <ref>). The GTC-OSIRIS optical spectra of this object show that it is at the same z=0.10351±0.00008 asthe “Beetle”(Fig. <ref>). Based on the [OIII]/Hβ=0.70±0.05, [NII]/Hα=0.34±0.03, [SII]/Hα=0.46±0.03 and [OI]/Hα=0.06±0.01 ratios, it is classified as a star forming galaxy (Baldwin et al. , Kewley et al. ).The Hα+continuum narrow band imagerevealsa spectacular structure of knots and arcs(Fig. <ref>, right),with a main axis that runs perpendicular to the main galaxy axis and the complex stellar tidal debris seenin the continuum image. Three arcs are identified, which are not detected in the continuum image. This already suggests that they are emission line dominated features, as confirmed by the continuum subtracted Hα image (Fig. <ref>).One arc is located at ∼13.5 arcsec (25 kpc) to the NE from the QSO2 centroid. Two more are at ∼10.2 arcsec (19 kpc)and ∼13.7 arcsec (26 kpc) SWof the QSO2. Thus, these features are in the circumgalactic environment.The morphology of the arcs is reminiscent of optical bow shocks seen in some powerful radio galaxies, where the extended radio structures are interacting with the environment (e.g. Coma A, Tadhunter et al. ; see also Harrison et al.for a related example of a radio quiet QSO2: the “Teacup). The “Beetle”is certainly not associated with such a luminous radio source as Coma A (it is ∼300 times fainter) and is classified as radio quiet. However, based on this resemblance a scenario ofradio jets/lobes being responsible for the observed optical ionized gas features appears plausible. This prompted us to obtain a deeper radio map which has confirmed the existence of a previously unknownextended radio source. The VLA observations reveal a radio source with a total extent of ∼24.5 arcsec or ∼46 kpc (Fig. <ref>). We detected two outer hot-spots, at ∼14 arcsec NE and ∼10.5 arcsec SW from the radio core, respectively. Thereforethe two hot spotsoverlap with the NE and inner SW arcs.Fig. <ref> (middle panel)shows our VLA high resolution (A-configuration) radio map of theBeetle. The inner radio jet is detected across a total extent of∼2.3 arcsec or∼4.3 kpc (right panel). It consists of a bright hot spot ∼0.8 arcsec or ∼1.5 kpc towards the SW and fainter emission towards the NE. If this is a result of Doppler-boosting, it suggests that the inner jet is approaching on the SW side and receding on the NE side. This is consistent with the outer lobe shown in the low-resolution map (B-configuration,Fig. <ref>, left).Only on the SW side, we detect an outer radio lobe that connects the inner radio emission to the outer SW hot-spot. This indicates that the SW side of the radio source is Doppler boosted and approaching also on large scales. Interestingly, while the modest radio power of the “Beetle" is typical ofFR-I sources, the hot-spot morphologyresembles more closely that of the Fanaroff-Riley type II (FR-II) sources(Fanaroff & Riley ).Our current data do not reveal whether the radio core represents re-started activity, or forms a continuous structure with the outer radio lobes and hot-spots. However, the innerjet hasPA 68. If we extrapolate its axis towards the NE,it seems to cross the northern Hα arcat a location where this appears broken. This may suggest that the radio core is actively feeding the outer lobes. The angle between the inner jet and the outer radio axis is 19, which would indicate precession of the radio jet or bending as it propagates outward. A deeper radio map might reveal an fainter, so far unseen extended radio structure thatmay have perforated the northern arc.The radio and emission line morphologies areclosely correlated. The large-scale radio axis is the same as the axis defined by thebrightest Hα knots. In fact, the NE radio hot spot overlaps with the brightest circumgalactic Hα feature.The centre of the southern arc closer to the galaxy overlaps with the SW radio hotspot. The close correlation between the radio and Hα morphologies suggests that the interaction of the radio structures with the circumgalactic medium shapes themorphology of thelarge scale ionized gas, possibly also of the radio source (alternative, less convincing scenarios will be discussed in Sect. <ref>).While the NEradio hotspotseems to define roughlythe edge of the NE ionized structures,the optical line emission extends well beyondthe SW radio hot spot. This will be further discussed below based on the spectroscopic data. §.§ The nuclear outflow§.§.§ Physical properties and excitation mechanismThe results of the fits of the nuclear emission lines are shown inTable <ref> and Fig. <ref>. Three kinematic components are isolated in the[OIII]λλ4959,5007 lines with FWHM=144±3, 398±36 and 973±25 km s^-1 respectively. The broadest component traces the ionized outflow. It is blueshifted by -144±6 km s^-1,relative to the narrowest component. These values are inagreement with those obtained by<cit.> based on the SDSS optical spectrum. The fits to the rest of the main optical lines produceconsistent results within the errors. As already pointed out by VM14, the line ratios place the three kinematic components, including the outflow, in the AGN area of the diagnostic diagrams (Baldwin et al. , Kewley et al. ).The reddening E(B-V), electron density n_ e and temperature T_ e have been inferredfor the three kinematic components using Hα/Hβ (Hγ/Hβ is affected by large uncertainties especially for the outflowing gas; see Fig. <ref>),[SII]λ6716/λ6731 and reddening corrected [OIII]λλ4959,5007/λ4363 respectively (Table <ref>). Inferring n_ e for the outflow is not trivial, given the complexity of the [SII] doublet (3 components per line) and the partial blend of both lines (Fig.<ref>). It is found that the fits producing kinematically reasonable results (in comparison with the other emission lines) and physically meaningful results (as opposite to unphysical, such as [SII]λ6716/λ6731>1.4 for any of the individual components) result in[SII]λ6716/λ6731=0.69±0.20, as indicated inTable <ref>.This implies n_ e=2571^+14181_-1461cm^-3 for the outflowing gas. The minimum possible value of the [SII] ratio (0.49)is actually in the regime where the ratio saturates and becomes insensitive to increasing densities (Osterbrock ). Thus,the “best” density n_ e=2571 cm^-3 is effectively almost a lower limit. The outflowing gas density issubstantially higher than the densities of the narrow (n_ e=730^+634_-365cm^-3)and the intermediate(n_ e=425^+89_-78cm^-3) components respectively. Different works suggest that such high densities are frequent in the nuclear ionized outflows of luminous type 2 AGN (e.g. Holt et al. , <cit.>, Villar Martín et al. ).We have tested whether a good fit can also be obtained by forcing the broadest component to havelow density ([SII]λ6716/λ6731=1.4 or n_ e100 cm^-3, Osterbrock ). However, we get reasonable fitsonlyunder very strict constraints on the kinematics and individual line ratios. We thus consider that the higher densities are more realistic.This is supported also by the high [OIII]/Hβ=17.8±3.3 ratio, which points to n_ e∼few×1000-10^4 cm^-3 (<cit.>).High densities in the narrow line region (NLR) of the “Beetle”, wherethe gas emitting the broadest component is likely to be located, are also suggested by [ArIV]λ4711/λ4740=1.10±0.08 which implies n_ e=3450^+1280_-1127, i.e. at least n_ e∼2300 cm^-3. The nuclear outflow showsthe highest reddeningof the three kinematic components with Hα/Hβ=5.8±1.2, as often found for the ionized outflows in QSO2 (<cit.>). As proposed by <cit.>, the trend of the outflowing gas to have the highest reddening and densitycan be naturally explained if the outflow is concentrated in a smaller region, closer to the the AGN than the intermediate and narrow components(Bennert et al. ,).It also shows significantly higher T_e4 = T_e/ 10^4=2.2^+4.6_-0.5 K, compared with 1.38^+0.08_-0.07and 1.56^+0.19_-0.13Kfor the narrow and intermediate components respectively. The additional heating may be produced by shocks induced by the outflow (e.g. Villar Martín et al. ). §.§.§ Radial sizeWe have derived the radial sizeR_o of the nuclear outflow along the three different slit PA. Upper limits or actual sizes are constrained by comparing the spatial distribution of the broadest kinematic componentand the seeing profile.Lower limits are obtained by applying the spectroastrometric method (seeCarniani et al.and Villar Martín et al.for a detailed description and discussion on the uncertainties of both methods).Themethod of spectroastrometry, which gives lower limits, consists of measuring the relative position of thecentroid of the [OIII] spatial profile as a function of velocity.We assume as spatial zero the continuum centroid. Although this does not necessarily mark the exact location of the AGN,it is a reasonable assumption for our purposes, since the associated uncertainty on R_ o is expected to be few×100 pc. We consider as zero velocity that of the narrow core of the [OIII] nuclear line.The results are shown in Fig. <ref>. We infer R_o1.2 kpc, 1.3 kpc and 0.5 kpc along PA48, PA83 and PA0 respectively. To compare theseeing profile with the spatial distribution of the outflow, a spatial profile of the [OIII] line (continuum subtracted) was extracted froma spectral (velocity) window clearly dominated by the outflow. As discussed in <cit.>, it is essential to use an accurate seeing profile for a reliable assessment of any possible excess above the seeing distribution, especially the wings. Following that work, the seeing profiles along PA48, PA83 and PA0 were specifically builtusing non-saturated stars with well detected wings in the acquisition images obtained at similar time as the spectra. The seeing profiles were extracted from apertures of the same width in arcsec as the narrow slit of the spectroscopic observations. The outflow spatial distribution is found to be resolved(FWHM_obs=1.25±0.05) relative tothe seeing(FWHM=0.85±0.04) only along PA48 (Fig. <ref>).We obtain R_o=0.86±0.07 kpc after correcting for seeing broadening. The error accounts for the uncertainty on FWHM_ obs and the seeing FWHM. All uncertainties considered, this R_o is in good agreement with the spectroastrometric result (R_o1.2 kpc). The same method demonstrates that the outflow spatial distribution is consistent with the seeing disk along both PA83 (seeing FWHM=1.25±0.07) andPA0 (seeing FWHM=1.1±0.1). This impliesR_o0.7 and 0.8 kpc respectively, also in reasonableagreement with the spectroastrometric lower limits taking into account all uncertainties. §.§.§ Mass and energetics Following standard procedures (see Villar Martín et al.for a discussion on the method and all related uncertainties) we have calculated the outflow mass M_ o, mass outflow rate Ṁ_ o, energy Ė_ o and momentum ṗ_ o injection rates. We assume n_e=2571 cm^-3, V_ o = |V_ s|+ FWHM/2=630 km s^-1 (see Tables <ref> and<ref>; Arribas et al. ) and R_o=0.86 kpc. The Hβ flux of the outflowing component isolated in the nuclear spectrum is F(Hβ_ o)=(1.3±0.2)×10^-15 erg s^-1. Slit losses are expected to be small (F(Hβ_ o)=(1.5±0.2)×10^-15 erg s^-1 as measured from the SDSS 3" fibre spectrum, <cit.>). Correctingfor reddening (as implied by Hα/Hβ∼5.8), the outflow luminosity is L(Hβ_ o)∼3.2×10^41 erg s^-1.Therefore, M_ o=3.5×10^6 M_⊙ and Ṁ _ o=2.6 M_ ⊙ yr^-1. In addition,Ė_ o=3.2×10^41 erg s^-1 (or ∼10^-5× L_ Edd) and ṗ_ o=1.0×10^34 dyne (or ∼0.01 L_ Edd/c), where L_ Edd is the Eddington luminosity (Sect. <ref>). As discussed in <cit.>, the outflowing gas is likely to have a density gradient with increasing densities at decreasing distances from theAGN. If such is the case, the assumption of higher densities would result in even lower values.§.§ The circumgalactic gasWe have studied the spatial distribution and properties of the ionized gas along the fourslit PAdescribed in Sect. <ref>.1-dimensional spectra were extracted from different apertures centred on prominent emission line features detected along each slit and low surface brightness regions in between. The slit locations, the 2-dimensional spectra of the [OIII] doublet and the apertures used for our study are shown in Figs. <ref> and <ref>. §.§.§ Spatial extension of the emission lines Extended ionized gas is detected along the 4 slit PA (Table <ref>). The maximum extensionis measured along PA48 (i.e. the radio hot spots axis, Fig. <ref>), with line emissiondetected acrossD_ max∼38 arcsec or 71 kpc.Towards the NE, as already seen in the Hα image,the brightest [OIII] emission overlaps with the NE hot spot. Towards the SW, line emission is detected well beyond thehotspot.A compact (along the slit) emission line feature is detected at ∼10.5 arcsecor ∼20 kpc beyondthe hot spot and R_ max∼41 kpc from the QSO2 centroid (aperture (ap.) G in Fig.<ref>). Line emission is detected also between the high surface brightness features, at least from ap. A toF. Along PA83 the spatial distribution of the emission lines is highly asymmetric. D_max∼ R_max∼23 kpc towards the E,up to the Eastern edge of the Hα bubble identified in the narrow band image (ap. H, Fig.<ref>). Towards the W, the lines are just barely resolved compared with the seeing disk.Along PA0,extended line emission associated with the QSO2 is tentatively detected only towards the N,up to R_ max∼29 kpc. This detection is confirmed by the PA140 spectrum (Fig.<ref>). Along this PA, which did not cross the galactic nucleus, the line emission extends at both sides of the radio axis across D_max∼23 arcsec or 43 kpc.This suggests thatline emission is detected across the entire area delineated bythe NW edge-brightened bubble.§.§.§ KinematicsThe visual inspection of the2-dimensional [OIII]spectra reveals variations on the kinematic properties of the gas (FWHM and V_s) correlated with the radio structures. This is particularly striking along the radio axis (PA48) and PA140. AlongPA48(Fig. <ref>) the lines are clearly broader at the location of the NE hot spot. Moreover, a broad blueshiftedwing isdetected at the location of the SW hot spot. A sharp, continuous change in V_ s of ∼550 km s^-1 occurs in aperture C (Fig. <ref>, right panel) in a region of very low surface brightness. Along PA140the lines appear clearly split in two components at the location crossing the radioaxis (see aperture K in Fig. <ref>, bottom right panel).We show in Fig. <ref> (left panels) how the kinematics of the ionized gasvaries spatially along PA48.The lines were fitted with 1 or 2 Gaussians, as required, using the 1-dimensional spectra extracted from each aperture (Fig. <ref>). FWHM and V_s are plotted at each location for the individual kinematic components.The threecomponents identified in the nuclear spectrum (see Sect. <ref>) are also plotted at the spatial zero. The variation of the [OIII]/Hβ ratio isshown whenmeasurable.Two kinematic components are required to fit [OIII]across the spatial range circumscribed by the radio hot spots (a single component is required beyond).The narrowest component has a FWHM150 km s^-1 and extends all the way along the slit, including beyond the maximum observed spatial extend of the radio emission. In contrast, a broader component of FWHM∼380-470 km s^-1 is detected across the entire spatial range circumscribed by the radio hot spots.Such broad lines reveal turbulent kinematics. For comparison, the extended ionized gas in mergers show typical FWHM<250 km s^-1 even in the most dynamically disturbed systems with signs of AGN activity (Bellocchi et al. , Arribas et al. ).The split components are confirmed and clearly seen in the 2-dimensional PA 140 extranuclear spectrum (Fig. <ref>) at the location where the slit intersects the radio axis (see also Fig. <ref>, right panels).Turbulent kinematics aredetected not only along the radio axis but also at locations far from it. As an example, FWHM=392±27and344±74km s^-1 for apertures L and M along PA140, which trace circumgalactic gas at ∼8 and 25 kpc form the radio axis in the perpendicular direction. Higher S/N spectra may reveal that the large FWHM are due to two split kinematic components. The velocity curve of the turbulent gas along the radio axis (Fig. <ref>, left) shows a change in V_ s, |Δ V_ s|=119±23 km s^-1, from blueshifted in the SW to redshifted in the NE. As discussed in Sect. <ref>, the radio orientation places the inner jet and SW radio hot spot closer to the observer. The kinematic behaviour of the circumgalactic turbulent gas suggests that it is expanding in an outflow, with the NEgasmovingaway and the SW gas moving towards the observer.The narrow component presents the opposite behaviour (from more redshifted in the SW to more blueshifted in the NE). The scenario where this component is emitted by gas on the far side of the expanding bubbles while the broad component is emitted by the near side seems unlikely.It is not clear why both sides would emit lines with different FWHM and ionization level (see below). A more natural explanation is that the narrow component isemitted by a gas reservoir unaffected by the radio structures. This is supported by the lack of any apparent correlation in either the spatial distribution or the kinematics of this narrow component with the radio structures. In this scenario, its velocity curve suggests that it may be infalling gas. The ionization level as traced by [OIII]/Hβ presents minimum values for the turbulent gasat the hot spots (Fig. <ref>, left), where [OIII]/Hβ∼3.7-5.7 compared with ∼9.2-9.8 for the narrow component.Along PA83, the eastern bubble (aperture H, Fig. <ref>) has FWHM=197±16 km s^-1 and [OIII]/Hβ=6.8±0.7. The profiles are slightly asymmetric with a red excess.The inner knot(aperture I) has similar FWHM=219±15 km s^-1, also with a slight red excess, and [OIII]/Hβ=8.4±1.2. The similarity ofthe FWHM and [OIII]/Hβ ratio compared with aperture A along PA48are consistent with these regions being part of the same structure, as shown by the Hα image.We shall discuss in Sect. <ref> how all the above results are natural consequences of the interaction between the radio structures with the ambient circumgalactic gas, which is compressed and kinematically perturbed in the process. §.§.§ Physical properties and excitation mechanism Different line ratios measured with the 1-dimensional spectra extracted from prominent features along PA48 and PA83were used to investigate the excitation mechanism and physical properties of the circumgalactic gas. In addition tothe NE hot spot (PA48)and the Eastern bubble (apertureH, PA83, Fig. <ref>), a spectrum was extracted along PA48 from a large aperture covering the location of the SW radio hot spot and apertures E and F. Relevantline ratios are shown in Table <ref>.The[SII] doublet implies n_ e=184^+60_-54 cm^-3 at the NE hot spot location (Table <ref>). Interestingly, the temperature T_e implied by the [OIII] lines is very high. The [OIII]4959,5007/[OIII]4363 ratio corrected for reddening with Hα/HβimpliesT_ e=24,000^+27,800_-5,600 K.The observed, uncorrected ratio sets a lower limit T_ e=19,000^+1,500_-1,200, also rather high.We show in Fig. <ref> threestandard BPT (Baldwin et al. ) diagnostic diagrams often used to discriminate between different types of galaxies according to <cit.>: HII-region-type, LINERS, Seyferts and composite AGN-HII. For the “Beetle”, the different apertures are in general located in the “Seyfert” area of the diagram, implying a dominant contribution of an excitation mechanism harder than stellar photoionization. Photoionization by the quasar continuum and shock related mechanisms (Dopita & Sutherland ) are natural possibilities. This is confirmed by the detection of strong HeIIλ4686 at the location of the NE hot spot with HeII/Hβ=0.17±0.03, which is inconsistent with stellar photoionization and typical of AGN.We can check whether the AGN can provide sufficient photons to explain the line luminosities of the circumgalactic gas. The total Hα luminosity of the NE bubble as measured from the narrow band image is L_ Hα∼1.7×10^41 erg s^-1. Correctingfor reddening,assuming an average Hα/Hβ∼4.1(Table <ref>), an intrinsic L_ Hβ∼1.3×10^41 erg s^-1 is derived. The ionizing photon luminosity required to power it is Q_ ion^ abs= L_ Hβ/h ν_ Hβα_ B/α^eff_Hβ=5.1 × 10^53 s^-1(Osterbrock ) whereQ_ ion^ abs is the ionizing photon luminosity absorbed by the gas, α_ B=2.5×10^-13 cm^3 s^-1, α^eff_Hβ=1.6×10^-14 cm^3 s^-1 for T_e∼2.0×10^4 Kas implied by the [OIII] lines and h ν_ Hβ is the energy of a Hβ photon. The NE bubble subtends an angle of ∼90^∘on the plane of the sky around the AGN. Assuming its shape is roughly symmetricalong and perpendicular to the line of sight, the necessary isotropic output from the AGN is Q_ ion^ AGN'=3.5×10^54 s^-1.The AGN ionizing luminositycan be estimated from L^ AGN_ ion∼ 0.35 × L_ bol =3.9 × 10 ^45 erg s^-1 (Stern et al. ). Assuming L_ ν∝ν^ξ, with ξ=-1.5, givesQ^ AGN_ ion∼5.0±10^55 s^-1. This is thus sufficient to explain theHα luminosity of the circumgalactic structures (including the SW ones, which are significantly fainter), provided they havelow opacity within their volume. On the other hand,the clear correlation between the optical and the radio morphologies suggests thatshocks induced by the interaction with the radio structures contribute to the emission, at least at certain locations. This may also explain the very high T_ e at the NE hot spot location. § DISCUSSIONWe discuss in this section the nature of the large scale outflow mechanismand the origin of the circumgalactic gas around the “Beetle”. §.§ Giant bubbles inflated by a nuclear AGN wind We inferred in Sect. <ref>, M_ o=3.5×10^6 M_⊙, Ṁ _ o=2.6 M_ ⊙ yr^-1,Ė_ o=3.2×10^41 erg s^-1 (or ∼10^-5× L_ Edd) and ṗ_ o=1.0×10^34 dyne (or ∼0.01 L_ Edd/c) for the nuclear outflow.If this is powered by an AGN wind, both Ė_ o and ṗ_ oare well below the minimum values required to have a substantial impact on the host galaxy(Ė_ o0.05 L_ Edd and ṗ_ o20 L_ Edd/c, Zubovas & King ). The outflow high density (n_ e=2571^+14181_-1461 cm^-2) and small size (R_ o1 kpc)are consistent with <cit.>, who showed that such high densities would be sufficient to quench an AGN powered wind prior to reaching radii 1 kpc. Thus, if the nuclear ionized outflow has been generated by an AGN wind, we find no evidence for a significant impact on the host galaxy, in agreement with <cit.> who found this to be a common situation inQSO2 at z0.7. The arguments above are based on the assumption that the AGN driven wind responsible for the nuclear outflow expands in ahigh densitymedium.Ifon the contrary it encounterspaths of less resistancewith densities significantly less than the normal volume averaged gas densities, it could advance more freely andexpand to much larger distances(Nims et al. , Mukherjee et al. ).Such scenario has been proposed, for instance, to explainthe large scale radio and optical bubbles associated with the “Teacup”QSO2 (Harrison et al. ). We investigate next whether such scenario could explain the properties (size, velocity, mass) of the circumgalactic optical arcs of the “Beetle”.Following<cit.> the kinetic energy of the wind can be approximated as L_ kin∼0.05×L_ bol∼5×10^44 erg s^-1. We use equations 6 and 7 in their paper to calculate the radius R_ s of the forward shock of the wind and its velocity v_s. We use the same fiducial values for R_ 0=100 pc, n_ H,0=10 cm^-3, v_ in=0.1 c and the density varying as a power law of index β=1.0 (see Nims et al.for details). In 30 Myr, which is a reasonable quasar life time, the wind could create a bubble of radial size R_ s∼25 kpc as observed for the “Beetle”. At that time, itsexpansionvelocity will be v_ s∼535 km s^-1. This is in reasonable agreement with the V_s values measured for the circumgalactic gas (Figs. <ref>), taking into account that these are line of sight projected velocities. It thus seems plausible that an AGN wind expanding through paths of low resistance can create the huge bubbles. Can this mechanism eject sufficient mass? We have estimated what fraction of the gas draggedby the nuclear outflow can escapethe gravitational potential of the galaxy. For this, we constrain the gas fractionthat moves with velocity v higher than the escape velocity v_esc. v_esc for an isothermal sphere at a distance r is given by (Rupke et al. ) v_ esc( r) = √(2) v_c [1 +ln(r_ max/r)]^1/2 where v_c is the circular velocity which scales with the stellar velocity dispersion σ_*as v_c = √(2)σ_ *, where σ_*∼200 km s^-1 (Sect. <ref>).r_ max is unknown. We have considered 3 possible truncation values r_max=3, 10, 100 kpc. We assume r=R_o∼0.86 kpc, which is the radial size of the outflow (Sect. <ref>). v_ esc at R_ o is in the range ∼633-960 km s^-1 for r_max in the range 3-100 kpc. Only ∼3% ofthe total nuclear L_ Hβ is emitted bygas moving at v_ esc> 633 km s^-1.Thus, the nuclear outflow can eject only ∼0.03× M_o=10^5 M_ ⊙ outside the galaxy at a rate of ∼0.03×Ṁ_o=0.08 M_⊙ yr^-1(Sect. <ref>). These are gross upper limits, since higher v_esc are possible.The total Hα intrinsic (reddening corrected) luminosity of the NE bubble isL_ Hα∼4.0×10^41 erg s^-1.Assuming n_ e∼184 cm^-3as measuredatthe location of the NE hot spot, the total NE bubble mass of the Hα emitting gasis ∼2.0×10^7 M_ ⊙.It follows that >3.0×10^8yr of continuous mass ejection would have been required to create the NE bubble, whichis longer than any realisticquasar episode.The inconsistency gets worse if the mass of the SW arcs are also taken into account. Because it will be relevant in Sect. <ref>, we highlight here that the same conclusion applies if the nuclear outflow was triggered by the small scale radio jet instead of an AGN wind, since the necessary timeof continuous mass ejection islonger than realistic radio source ages. §.§ Large scale radio-gas interactionsThe above scenario is also difficult to reconcile with the clear correlation between the morphological, kinematic and ionization properties of the ionized gas and the large scale radio structures (Sect. <ref>). Amore natural scenario is one in whichthe radio structures have been originated inthe neighbourhood of the SMBH and have escaped out of the galaxyboundaries. Themorphological features(distanthotspots, a diffuse lobe-like component andthe inner jet-like structurein the central kpc region) are well knownconstituents of the radio structures in many active galaxies. No convincing alternative explanations (see, for instance, Harrison et al. ) for their origin seem likely. These structures have interacted in their advance with the in-situ (see below) circumgalacticgas. Our preferred scenario isstrongly supported by studies of radio galaxies and radio loud quasarswith clear signs of radio-gas interactions(e.g. Clark et al. , Villar Martín et al. , Tadhunter et al. , Best et al. , Humphrey et al. ). We have seen that (a) the line emission is enhanced at the location of the radio hot spots, especially the NE one, (b) the gas kinematics are turbulent across the spatial range circumscribed by the radio hot spots (not beyond), (c) the turbulent gasshows minima in the ionization level at the location of the radio hot spots, (d) the gas overlapping with the NE hot spot is very hotT_e1.9×10^4 K and (e) there is turbulent gasnot only along the radio axis (up to ∼26 kpc from the AGN), but also far away from it (∼25 kpc from the radio axis, in the direction perpendicular to it) (Sect. <ref>).Powerful radio galaxies with radio-gas interactions at different redshifts often show close radio-optical associations, emission line flux enhancement, ionization minima and kinematic turbulence (split lines, broad components) coincident with the radiostructures, anti-correlation between line FWHM and gas ionization level, high T_ehave beenseen in such systems. They can all be naturally explained by the interaction between the radio structures and the ambient gas.The shocks resulting from such interactions create radial outflows that drag, heat and compress the multiphase gaseous medium (Bicknell , Mukherjee et al. ). The presence of turbulent gas in the “Beetle”up to ∼26kpcalong the radio axisand ∼25 kpc away from it reveals the action of the shocks across a huge volume. This has been observed also in some powerful radio galaxies(Villar Martín et al. , Tadhunter et al. ). Turbulent kinematics up to ∼6 kpc from the AGNhave also been reported for the “Teacup” radio quiet QSO2 (Keel et al. , Ramos Almeida et al. ). Although a scenario where the outflows have been triggered by an AGN wind (rather than by the radio structures) cannot be rejected, it is possible that a similar mechanism as the one inferred for the “Beetle” is at work (Harrison et al. ). <cit.>have shown that a radio source of similar power as the “Beetle” cancreatea wealth ofobservable features around ellipticals in poor environments, such as X-ray cavities surroundedby weak shocks, large buoyant bubbles, extended filamentary multiphase gas features (including cold gas with T_e20,000 K gas) and subsonic chaotic turbulence.The effects can be visible up toseveral kpc from the SMBH. Whether this can occur up to ∼26 kpc is not clearfrom the point of view of the simulations.Powerful radio galaxies with large volumes of gas affected by radio induced outflows(Villar Martín et al. , Tadhunter et al. ), show wide area radio features (e.g. radio lobes)overlapping with the optical bubbles or bow shocks. This is not the case in the “Beetle”, but this is not an inconsistency of our proposed scenario.Although a relativistic jet may appear to be collimated in radio,it will in fact launch a bubble defined by the forward shock (Bicknell, Mukherkee et al. ).Hence although the radio contours may trace predominantly the central axis, where the electrons are relativistic andemit intensesynchrotron radiation, the outer shock will be larger,more dilute anddifficult to detect. This may explain why broad radio lobes overlapping with the Hα bubble edges have not been detected in the “Beetle”. A deeper radio scan may reveal them. We can apply the methodology of <cit.> (see Sect. <ref>) to investigate whether a radio source of the same mechanical power as that of the “Beetle” can inflate the huge observed bubbles. The equations were originally derived to model the expansion of a homogeneous spherically symmetric wind. While strong fast jets produce asymmetric bubbles with the jet head progressing much faster, weak jets (as that in the “Beetle”) evolve more spherically (Mukherjee et al. ). We thusconsiderthe equations valid as well in the scenario where the outflow has been induced by theradiosource. The mechanical power of the radio source can be calculated as P_ jet∼5.8×10^43(P_ radio/10^40)^0.70 erg s^-1(Cavagnolo et al. ). For the “Beetle”, we infer P_ radio = 9.1^+45.3_-5.9× 10^40 erg s^-1 using the radio power at 1.4 GHz, integrating between 10 MHz and 30 GHz, considering the range of expected spectral index α=0.094^+0.76_-1.01 and F_ ν∝ν^α. Hence, P_ jet= 2.7^+6.8_-1.4× 10^44erg s^-1.In 35 Myr, which is a reasonable radio source age (Parma et al. ), the jet could therefore create a bubble of radial size R_ s∼25 kpc. At that time, itsexpansionvelocity will be v_ s∼442 km s^-1. §.§ The origin of the circumgalactic gas As we saw in Sect. <ref>, it is unlikely that the circumgalactic features consist of gas ejected from the galaxy by the nuclear outflow, independentlyof whether this has been triggered by and AGN wind or the small scale radio jet.It appears more likely that they consist of in-situ gas, maybe tidal remnants redistributed duringgalactic interactions across∼71 kpc (Sect. <ref>), as observed.Such events are expected to populate the CGM with tidal debris. The circumgalaticgas around the “Beetle” has been shaped and rendered visible due to the illumination by the QSO and the action of the radio induced outflows. The possible detection of an infalling circumgalacticreservoir of gas (Sect. <ref>) could pointto a scenarioin which part ofthe CGMis raining back, falling towards the gravitational centre probably defined bythe QSO2 host. §.§ Is the “Beetle” related to high excitation radio galaxies? The novelty of our results lies on the fact thatthe “Beetle” is radio quiet.As explainedis Sect. <ref> radio-induced feedback has been rarely considered as potentially relevant in radio quiet luminous AGN, especiallyacross large spatial scales (≫ several kpc). In this section we analyze in more depth the implications of our results, by comparing the “Beetle” with radio galaxies.Radio galaxies can be classified into two distinct classes according to the optical emission line ratios and the [OIII]λ5007 equivalent width:high (HERG) and low excitation (LERG) radio sources, whichshow profound differences (e.g. Laing et al. , Tadhunter et al. , Best & Heckman ). Contrary to LERG, HERG have high luminosity accretion disks (+ X ray corona), bright line emission, NIR/sub-mm evidence of dusty obscuring torus and orientation-dependent properties. Some fundamental differences have been interpreted in terms of differences on the underlying accretion modes. In HERG, accretion at high Eddington ratios proceeds via a geometrically thin, optically thinaccretion disk with a high radiative efficiency, while LERG are powered by advection-dominated accretion flows (ADAF, Narayan & Yi ) with low radiative efficiency. <cit.> show that LERG are predominantly low-power Fanaroff-Riley class I (FR-I) sources, while the HERG population consists of both FR-I and the more powerful FR-II sources, whose relativistic jets end in bright hot-spots. The dominant feedback mechanism is also thought to be clearly distinct between LERG and HERG. LERG are dominated by radio-mode feedback, where the bulk of the energy is ejected in kinetic form through radio jets (e.g., Croton et al. ) and efficiently coupled to the galaxies' gaseous environment (Best et al. ). The radiative (or quasar) modeis thought to dominate in HERG. How does the “Beetle”fit into this picture? Based on its optical spectroscopic properties,the “Beetle” is indistinguishable from HERG (Fig. <ref>).It has a very high excitation index (Best & Heckman ) EI = log_10([OIII]/Hβ) -1/3 [log_10([NII]/Hα) + log_10 ([SII]/Hα)+ log_10([OI]/Hα)]=1.70,significantly higher than the limiting value 0.95 proposed by <cit.> to separate LERG and HERG. Indeed, it is at the high end of EW_ [OIII]λ 5007=430 Å(rest frame value) and EI values spanned by the radio galaxy sample studied by these authors. It has an Eddington ratioλ= L_ bol/ L_ Edd=0.33 (Sect. <ref>). The transition between the radiatively efficientaccretion regime and the radiatively inefficient accretion regime typically occurs around λ∼0.01, i.e., 1% of the Eddington rate, below which, the kinetic output dominates over the radiative output. The “Beetle” is accreting at a very high rate. In fact, few radio galaxiesaccrete atsuch high rates (Best &Heckman , Ishibisahi et al. ,Sikora et al. ). The rest frame radio power of the “Beetle”(log(P_1.4GHz)=30.2 in erg s^-1 Hz^-1), although modest, is within the range measured in radio galaxies. A critical difference is that itsvery high bolometric luminosity makes it a radio quiet system. HERG of similar L_[OIII] show several×100 timesmore powerful radio sources (Fig. <ref>,left). Following <cit.>we define the radio loudness parameter R'= L_ 1.4GHz/ L_ [OIII], where L_ 1.4GHz is the rest frame luminosity at 1.4GHz. For the “Beetle”log(R')=-2.78below the range spanned by radio galaxies (both HERG and LERG) which show log(R')-2.5 (Ishibashi et al. , Best & Heckman ).LERG and HERG show an anti-correlation between log(R') andlog(λ') where the new accretion rate λ'=3500×L_[OIII]/L_ Edd=0.42 iscalculated in coherence with <cit.>[The authors assume L_ bol=3500 × L_[OIII] following<cit.>]. This is likelya result of either an enhancement of the radio jet emission at low accretion rate and/or a decrease in the optical emission due to the decline of the radiative efficiency in the radiatively inefficient accretion flow (RIAF) mode. The best fit is different for HERGand LERG (Buttiglione et al. , Ishibashi et al. ), being much steeper for the former. For the “Beetle”, log(λ')=-0.37 and log(R')=-3.28. All uncertainties considered, including the scatter of the correlations, these values are consistent withthe HERG fit (expected log(R')=-2.6)and inconsistent with the LERG fit (expected log(R')=-0.65). Interestingly, the “Teacup” is very similar to the “Beetle”in numerous aspects. It is also a radio quiet luminous QSO2 hosted by a bulge dominated system. It shows prominent shells indicative of previous merger activity. The axis of the radio and optical bubbles (∼10-12 kpc in size) runs roughly perpendicular to the main shells axis, as in the “Beetle” (Keel et al. , Harrison et al. , Ramos Almeida et al. )and the large scale outflow may also have been triggered by the radio structures (Harrison et al. ). We have used the following information published in the literature: the optical nuclear line ratios (Keel et al. ), L_ [OIII]=3.8×10^43 erg s^-1, log(P_ 1.4GHz)=30.2 in erg s^-1 Hz^-1 (Harrison et al. ),log(M_BH)=8.4, indirectly inferred from the bulge stellar mass log(M_ sph/M_⊙)=10.8 (Vizier catalogue, based on Simard et al. ). We obtain EI=1.52, log(λ')=-0.10 and log(R')=-3.16, i.e. very high excitation index,very high accretion rateand very low radio loudness parameter, demonstrative of its radio quietness (Fig. <ref>). As for the “Beetle”, these values are consistent with the extrapolation of thelog(R') vs.log(λ')correlation of HERGinto the radio quiet regime and inconsistent with the LERG correlation. The properties of the “Beetle” and the “Teacup" suggest that they are high excitation radio quiet QSO2 which are similar to the population of HERG,extended into the regime of very high accretion efficiencies and modest radio luminosities, i.e.,the radio quiet regime. This is consistent with <cit.> who proposed that there is a continuous variation of radio luminosity in quasars and there is no evidence of a “switch” at some set of critical parameter values that turns on powerful radio jets. Based on the very high accretion rate and the radio quietness (or very low R') radiative feedback would be expected to play a dominant role in both objects. We have seen, however, that although this mechanism cannot be discarded in the central region near the AGN (1 kpc), radio induced feedback dominates in the “Beetle” (maybe also the “Teacup”)across a huge volume of many 10s of kpc in size, reaching regions well outside the galaxy boundary and into the circumgalactic medium. This opens the possibility that radio mode feedback may contribute to the re-heating of the cooling gas in massive galaxies not only in radio loud systems (Best et al. ), but also in at least some radio quiet AGN.§ SUMMARY AND CONCLUSIONS We have presented a detailed study of the “Beetle”, a radio quiet QSO2 at z=0.103, basedon GTC optical images andlong slit spectroscopy as well as VLA radio maps.* The “Beetle”showsclear morphological signs of a tidal interaction (shells, filaments), probably with a star forming companion galaxy located at ∼20 kpc. The system is associated witha spectacular set of circumgalactic ionized knots and three arcsreminiscent of optical bow shocks or bubbles, with a main axis aligned with the radio axis. They are located at at 19, 25 and 26 kpc from the AGN.The maximum total extension of the ionized gas is measured along the radio axis (∼71 kpc), with maximum extension from the QSO2∼41 kpc. These circumgalactic features are ionized by shocks and/or the quasar continuum. * The radio map has revealed the existence of a previously unknownextended radio source, with acentral jet ∼4.3 kpc long and two hot spots separated by ∼46 kpc, well outside the galaxy boundaries.The innerjet and the outer radio axis form a 19 angle, which would indicate a substantial precession of the radio jetor its bending as it propagates outward.* The morphological, ionization and kinematic properties of the large scale ionized gas are correlated with the radio structures. We have seen that (a) the line emission is enhanced at the location of the radio hot spots, especially the NE one, (b) gas kinematicturbulence (FWHM∼380-470 km s^-1) is detected across the spatial range circumscribed by the radio hot spots (not beyond), (c) the turbulent gasshows minima in the ionization level at the location of the radio hot spots, (d) the gas overlapping with the NE hot spot is very hotT_e1.9×10^4 K and (e) there is turbulent gasnot only along the radio axis (up to ∼26 kpc from the AGN), but also far away from it (∼25 kpc from the radio axis, in the direction perpendicular to it). In addition, a reservoir of more quiescent gas has also been detected whose kinematic pattern suggests infall.* Wepropose the following scenario: the interaction between the QSO2 host and the nearby companion galaxy has redistributed galactic material (stars and gas) across ∼70 kpc, at opposite sides of bothgalaxies. Part of the gas is in the process of fallingtowards the gravitational centre, probably defined by the QSO2 host. The radio structures have escaped from the vicinity of the active nucleus out into the circumgalactic medium. They interact with the gas located in their path. Shocks drag, heat and compress the multiphase medium within a huge volumeenhancing the line emission, perturbing the kinematics and changing the physical properties up to ∼25 kpc along the radio axis and perpendicular to it. Our preferred scenario isstrongly supported by studies of radio galaxies and radio loud quasars at different redshifts with clear signs of radio-gas interactions. * Specific 3D hydrodynamic simulations of interactions between AGNjets and the multiphase ISM adapted to the “Beetle”would be of great valueto quantify accurately how radio induced feedback is affecting its large scale environment in terms of energy injection, heating and mass ejection. Ultimately, this would show whether the radio induced feedback can have a significant impact on the evolution of the QSO2 host by means of quenching star formation and/or ejecting a significant fraction of the galaxy ISM. * Large scale effects (≫ several kpc) of radio induced feedbackin radio quiet AGN are usually not expected. Our results demonstrate that this idea needs to be reconsidered. We have found that jets of modest power can remain strong enough in radio quiet objectsfor length scales of 10s of kpc to interact with the galactic and circumgalacticenvironment and provide a feedback mechanismthat can affect huge volumes in and around galaxies. * The radio and optical properties of the “Beetle” imply a very high accretion index, very high accretion rate and a very low radio loudness parameter compared with high excitation radio galaxies (HERG).We propose that the “Beetle”(possibly also the famous “Teacup" radio quiet QSO2 at z∼0.085,Keel et al. , Gagne et al. )are high excitation radio quiet QSO2 which are similar to the population of HERG, extended into theradio quiet regime of objects withhigh bolometric luminosities and low or moderate radio luminosities. The “Beetle” demonstrates that jets of modest power can be the dominant feedback mechanism acting across huge volumes in highly accreting luminous radio quiet AGN, where radiative (or quasar) mode feedback would be expected to dominate.§ ACKNOWLEDGMENTS Partly based on observations made with the GTCtelescope, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, under Director's Discretionary Time.Wethank the GTC staff for their support with the optical imaging and spectroscopic observations and the VLA staff for executing the radio observations. Thanks to Gustaaf van Moorsel for his valuable advice about the radio observations.MVM, AC and BE acknowledgesupportfrom the Spanish Ministerio de Economía y Competitividad through the grants AYA2015-64346-C2-2-P and AYA2012-32295. AH acknowledges Fundação para a Ciência e a Tecnologia (FCT) support through UID/FIS/04434/2013, and through project FCOMP-01-0124-FEDER-029170 (Reference FCT PTDC/FIS-AST/3214/2012) funded by FCT-MEC (PIDDAC) and FEDER (COMPETE), in addition to FP7 project PIRSES-GA-2013-612701. AH also acknowledges a Marie Curie Fellowship co-funded by the FP7 and the FCT (DFRH/WIIA/57/2011) and FP7 / FCT Complementary Support grant SFRH/BI/52155/2013. BE acknowledges support fromthe European Union 7th Framework Programme (FP7-PEOPLE-2013-IEF) under grant 624351.CRA acknowledges the Ramón y Cajal Program of the Spanish Ministry of Economy and Competitiveness through project RYC-2014-15779 and the Spanish Plan Nacional de Astronomía y Astrofisíca under grant AYA2016-76682-C3-2-P.This research has made use of: 1) the VizieR catalogue access tool, CDS,Strasbourg, France. The original description of the VizieR service waspublished in Ochsenbein et al. A&AS, 143, 23; 2)the NASA/IPAC circumgalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration; 3) data from Sloan Digital Sky Survey. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. [Alatalo et al.2015]ala15Alatalo et al. 2014, ApJ, 798, 31 [Allen et al.2008]allen08Allen M. G., Groves B. A., Dopita M. A., Sutherland Ralph S., Kewley L., 2008, ApJS, 178, 20 [Arribas et al.2014]arr14Arribas S., Colina L., Bellocchi E., Maiolino R., Villar Martín M., 2014, A&A, 568, 14 [Athanassoula & Bosma1985]ath85 Athanassoula E.,Bosma A., 1985, Annu. Rev. Astron. Astrophys., 23, 147 [Baldwin, Philips & Televich1981]bal81Baldwin J., Philips M., Televich R., 1981, PASP, 93, 5[Bellocchi et al.2013]bel13Bellocchi E., Arribas S., Colina L., Miralles-Caballero D., 2013, A&A, 557, 59 [Bennert et al.2006a]ben06aBennert N., Jungwiert B., Komossa S., HaasM., Chini R., 2006a, A&A, 456, 953 [Bennert et al.2006b]ben06bBennert N., Jungwiert B., Komossa S., HaasM., Chini R., 2006b, A&A, 459, 55 [Bessiere et al.2012]bes12Bessiere P.S.,Tadhunter C.N.,Ramos Almeida C.,Villar MartínM.,2012, MNRAS, 426, 276[Best et al.2006]best06Best P.N., Kaiser C.R., Heckman T.M., Kauffmann G., 2006, MNRAS, 368, L67 [Best & Heckman2012]best12Best P. N., Heckman T. M., 2012, MNRAS, 421, 1569 [Best et al.2001]best01Best P. N., Röttgering H. J. A., Longair M. S., 2001, MNRAS, 311, 23 [Bicknell1994]bic94Bicknell G.V., 1994, ApJ, 422, 542 [Briggs1995]bri95Briggs D. S., 1995, PhD thesis, New Mexico Institute of Mining and Technology [Buttigione et al.2010]but10Buttiglione S., Capetti A., Celotti A., Axon D. J., Chiaberge M., Macchetto F. D.. Sparks W. B., 2010, A&A, 509, 6 [Carniani et al.2015]car15Carniani S., Marconi A., Maiolino R. et al. 2015, A&A, 598, 102 [Cavagnolo et al.2010]cav10Cavagnolo K. W., McNamara B. R., Nulsen P. E. J., Carilli C. L., Jones C.,Bîrzan L., 2010, ApJ, 720, 1066 [Clark et al.1998]cla98Clark N. E., Axon D. J., Tadhunter C. N., Robinson A., O'Brien P., 1998, ApJ, 494, 546 [Croton et al.2006]cro06Croton D.J., Springel V., White S.D.M. et al. 2006, MNRAS, 365, 11 [di Matteo et al.2005]dim05Di Matteo T., Springel V. Hernquist L., 2005, Nature, 433, 604 [Dopita & Sutherland1996]dop96Dopita M., Sutherland R., 1996, ApJS, 102, 161[Duc et al.2015]duc15Duc P.A. et al. 2015, MNRAS, 446, 120[Fabian et al.2002]fab02Fabian A.C., Celotti A., Blundell K. M., Kassim N. E., Perley R.A., 2002, MNRAS, 331, 369 [Fabian2012]fab12Fabian A.C., 2012, ARA&A, 50, 455[Falcke & Biermann1995]fal95Falcke H., Biermann P.L., 1995, A&A, 293, 665[Fanaroff & Riley1974]fan74Fanaroff B.L.,Riley J.M., 1974, MNRAS, 167, 31 [Fauchier-Giguére & Quataert2012]fau12Fauchier-Giguére C.A., Quataert E., 2012, MNRAS, 425, 605[Ferrarese & Merritt2000]fer00Ferrarese L., Merritt D., 2000, ApJ, 539, L9 [Ferruit et al.1997]fer97Ferruit P., Binette L., Sutherland R. S., PecontalE., 1997, A&A, 322, 73 [Förster Schreiberet al.2014]for14 Förster Schreiber N.M. et al. 2014, ApJ, 787, 38 [Fu & Stockton2009]fu09Fu H., Stockton A., 2009, ApJ, 690, 953 [Gagne et al.2014]gag14Gagne J. P., Crenshaw, D. M., Kraemer, S. B. et al., 2014, ApJ, 792, 72 [Gaspari et al.2012]gas12 Gaspari M., Brighenti F., Temi P., 2012, MNRAS, 424, 190[Ghisellini et al.2004]ghi04Ghisellini G., Haardt F., Matt G., A&A, 413, 535 [González et al.2014]gon14González J.J., Cepa J., González Serrano J.I., Sánchez Portal M., 2014, MNRAS, 443, 3289 [Greene et al.2011]gre11Greene J. E., Zakamska N., Ho L., Barth A.,2011, ApJ, 732, 9 [Guillard et al.2012]gui12Guillard P., Ogle P. M., Emonts B.H.C. et al. 2012, ApJ, 747. 95 [Hakobian et al.2009]hak09Hakobian A.A. et al. 2009, Ap, 52, 40[Hardcastle et al.2007]har07Hardcastle M. J., Evans D. A.,Croston J.H.,2007, MNRAS, 376, 1849 [Harrisonet al.2012]har12Harrison C. M. et al., 2012, MNRAS, 426, 1073 [Harrison et al.2014]har14Harrison C. M., Alexander D. M., Mullaney J. R., Swinbank A. M., 2014, MNRAS, 441, 3306[Harrison et al.2015]har15Harrison C. M., Thomson A.P., Alexander D. M., Bauer F.E., Edge A.C., Hogan M.T.,Mullaney J. R., Swinbank A. M., 2015, ApJ, 800, 45 [Heckman et al.2004]hec04Heckman T., Kauffmann G., Brinchmann J., Charlot S., Tremonti S., White S.D.M., 2004, ApJ, 613, 109 [Helou et al.1985]hel85Helou G., Soifer B.T. Rowan-Robinson M., 1985, ApJ, 298, L7 [Holt et al.2011]holt11Holt J., Tadhunter C. N., Morganti R., Emonts B., 2011, MNRAS, 410, 1527 [Humphrey et al.2006]hum06 HumphreyA., Villar MartínM., Fosbury R., Vernet J., di Serego Alighieri S., 2006, MNRAS, 369, 1103 [Humphrey et al.2010]hum10 Humphrey A. Villar Martín M., Sánchez S. F., Martínez-Sansigre A., González Delgado R., Pérez E.,Tadhunter C., Pérez-Torres M. A., 2010, MNRAS, 408, L1 [Husemann et al.2013]hus13Husemann B., Wisotzki L., Sánchez S. F., Jahnke K., 2013, A&A, 549, 43[Husemann et al.2016]hus16Husemann B.,ScharwaechterJ.,Bennert V. N.,MainieriB., Woo J.-H.,Kakkad D., 2016, A&A, 594, 44[Ishibashi et al.2014]ish14 Ishibashi W., Auger M. W., Zhang D., Fabian A. C., 2014, MNRAS, 443, 1339[Karouzos et al.2016]kar16Karouzos M., Woo J.H., Bae H.J., 2016, ApJ, 819, 148 [Keel et al.2012]keel12 Keel W.C., Chojnowski S.D.,Bennert V. N. et al., 2012, MNRAS, 420,878[Keel et al.2014]keel14 Keel W.C., Maksym W.P., Bennert V.N., 2014, AJ, 149, 155[Keel et al.2017]keel17 Keel W.C. et al. 2017, ApJ, 835, 256[Kellermannet al.1994]kel94Kellermann K. I., Sramek R. A., Schmidt M., Green R. F., Shaffer D. B, 1994, AJ, 108, 1163 [Kennicutt1998]ken98Kennicutt R.C. Jr., 1998, Annu. Rev. A&A, 36, 189 [Kewley et al.2001]kew01Kewley L., Dopita M., Sutherland R., Heisler C. Trevena J., 2001, AJ, 556, 121 [Kewley et al.2006]kew06Kewley L., Groves B., Kauffmann G., Heckman T., 2006, NRAS, 372, 961 [King2003]king03King A., 2003, ApJ, 596, L27[Lacy et al.2001]lacy01Lacy M., Laurent-Muehleisen S. A., Ridgway S. E., Becker R. H., White R. L., 2011, ApJL, 551, 17[Laing et al.1994]lai94Laing R. A., Jenkins C. R., Wall J. V., Unger S. W., 1994, in Bicknell G.V.,Dopita M.A., Quinn P.J. Eds., The First Stromlo Symposium: The Physics of Active Galaxies.ASP Conference Series, Cambridge University Press, Vol. 54, 201 [Lal & Ho2010]lal10Lal D. V., Ho L. C., 2010, A&A, 139, 1089[Lehner et al.2015]leh15Lehner N., Howk J.C., Wakker B., 2015, ApJ, 904, 79[Liu et al.2013]liu13Liu G., Zakamska N. L.,Greene J. E., Nesvadba N., Liu X., 2013, MNRAS, 436, 2576[Lu et al.2014]lu14Lu N., Zhao Y., Xu C.K. et al., 2014,ApJL, 787, L23 [Mathews& Baker1971]mat71 Mathews W.G., Baker J.C., 1971, ApH, 170, 241 [Mauch & Sadler2007]mau07 Mauch T., Sadler M.S., 2007, MNRAS, 375, 931 [McCarthy et al.1996]mcc96McCarthy P. J., Baum S. A.,Spinrad H., 1996, ApJSS, 106, 281 [McElroy et al.2015]mce15 McElroy R., Croom S., Pracy M., Sharp R.,Ho, I.T., Medling A., 2015, MNRAS, 446, 2186[McMullin et al.2007]mcm07 McMullin J. P., Waters B., Schiebel D., Young W., Golap K., 2007, ASP Conf. Ser. 376, 127[McNamara & Nulsen2007]mcn07McNamara B.R., Nulsen P.E.J., 2007, ARAA, 45, 17[McConnell & Ma2013]mcc13McConnell N.J., Ma C.P., 2013, ApJ, 764, 184 [Mendel et al.2014]men14Mendel J. T., Simard L., Palmer M., Ellison S. Patton D., 2014, ApJS, 210, 3 [Morganti et al.2005]mor05Morganti R., Tadhunter C. N., Oosterloo T. A., 2005, A&A, [Mullaney et al.2013]mul13Mullaney J.R., Alexandeer D.M., Fine S., Goulding A.D., Harrison C.M., Hickox R.C., 2013, MNRAS, 433, 622 [Mukherjee et al.2016]muk16Mukherjee D., Bicknell G. V.,Sutherland R., Wagner A., 2016, MNRAS, 461, 967 [Narayan & Yi1994]nar94 Narayan R., Yi I., 1994, ApJL, 428, 13[Nesvadba et al.2006]nes06 Nesvadba N. P. H., Lehnert, M. D., Eisenhauer F., Gilbert A., Tecza M., Abuter R., 2006, ApJ, 650, 693 [Nesvadb et al.2008]nes08 Nesvadba N. P. H., Lehnert M. D., De Breuck C., Gilbert A. M., van Breugel W., 2008, A&A, 491, 407[Nims et al.2015]nims15Nims J., Quataert E., Faucher-Giguére C.A., 2015, MNRAS, 447, 3612[Ochsenbein al.2000]och00 Ochsenbein F., Bauer P., Marcout J., 2000, A&AS, 143, 23 [Osterbrock1989]ost89 Osterbrock D.E., 1989,Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. University Science Books. ISBN 0-935702-22-9 [Page et al.2012]page12Page et al., 2012, Nature, 485, 2012 [Page et al.2012]par99Parma P., Murgia M., Morganti R., Capetti A., de Ruiter H. R., Fanti R., 1999, A&A, 344, 7 [Pearson et al.2016]pea16Pearson C., Rogopoulou D, Hurley P. et al., 2016, ApJSS, 227, 9 [Proga2007]prog07Proga D., 2007, in Ho L. C., Wang J.-W., eds, ASP Conf. Ser. Vol. 373, The Central Engine of Active Galactic Nuclei. Astron. Soc. Pac., San Francisco, p. 267 [Quinn1984]quinn84Quinn P., 1984, ApJ, 279, 596[Ramos Almeida et al.2012]ram12Ramos Almeida C. et al., 2012, MNRAS,419, 687 [Ramos Almeida et al.2017]ram17Ramos Almeida C., Piqueras López J., Villar Martín M., Bessiere P., 2017, MNRAS, in press (arXiv:1705.07631)[Reyes et al.2008]rey08Reyes R.et al. 2008, AJ, 136, 2373 [Rupke et al.2002]rup02Rupke D. S., Veilleux S., Sanders D. B., 2002, ApJ, 570, 588 [Sanders & Mirabel1996]san96Sanders D.B., Mirabel I.F., 1996, Annu. Rev. A&A, 34, 749 [ Sijacki & Springel2006]sij06 Sijacki D.,Springel V., 2006, MNRAS, 371, 1025 [Sikora et al.2013]sik13Sikora M., Stasińska G., Koziel-Wierzbowska D., Madejski G.M.,Asari N.V., 2013, ApJ, 765, 62[Silk & Rees1998]silk98Silk J., Rees M., 1998, A&A, 331, L1[Silk & Rees2012]silk12Silk J., MamonG., 2012, RAA,12, 917[Simard et al.2011]sim11Simard L., Mendel J., Trevor P., David R., Ellison S., McConnachie A., 2011, ApJS, 196, 11 [Solórzano-Iñarrea et al.2002]sol01Solórzano-Iñarrea C., Tadhunter C. N., Axon D. J., 2002, MNRAS, 323, 965 [Stern & Laor2012]ste12Stern J., Laor A., 2012, MNRAS, 426, 2703[Stern et al.2014]ste14Stern J., Laor A., Baskin A., 2014, MNRAS, 438, 901[Struck1999]str99Struck C., 1999, PhR, 321, 1[Tadhunter etal.1994]tad94Tadhunter C., Shaw M., Clark N., Morganti R., 1994, A&A, 288, L21 [Tadhunter et al.1998]tadh98Tadhunter C. N., Morganti R., Robinson A., Dickson R., Villar Martín M., Fosbury R. A. E., 1998, MNRAS, 298, 1035 [Tadhunter et al.2000]tadh00Tadhunter C. N., Villa -Martín M., Morganti R., Bland-Hawthorn J., Axon D., 2000, MNRAS, 314, 849[Tadhunter et al.2014]tadh14Tadhunter C., Morganti R., Rose M., Oonk J. B. R., Oosterloo T., 2014, Nature 511, 440[Tumlison et al.2017]tum17Tumlinson J., Peeples M., Werk J., 2017, ARA&A, in press[Villar Martín et al.1999]vm99Villar Martín M., Tadhunter C., Morganti R., Axon D., Koekemoer A., 1999, MNRAS, 307, 24[Villar Martín et al.2003]vm03Villar Martín M., Vernet J., di Serego Alighieri S., Fosbury R., Humphrey A., Pentericci L., 2003, MNRAS, 346, 273 [Villar Martín et al.2011]vm11Villar Martín M., Humphrey A., González Delgado R., Colina L., Arribas S., 2011b, MNRAS, 418, 2032 [VM14]vm14Villar Martín M., Emonts B., Humphrey A., Cabrera Lavers A., BinetteL., 2014, MNRAS, 440, 3202 (VM14)[Villar Martín et al.2015]vm15Villar Martín M.,BellocchiE., Stern J.,Tadhunter C.,González Delgado R., 2015, MNRAS, 454, 439[Villar Martín et al.2016]vm16Villar Martín M., Arribas S., Emonts B., Humphrey A., Tadhunter C., Bessiere P., Cabrera Lavers A., Ramos Almeida C., 2016, MNRAS, 460, 130[Whittle1985]whi85Whittle M., 1985, MNRAS, 213, 33[Whittle1992]whi92Whittle M., 1992, ApJ, 387, 109[Wilson & Ulvestad1983]wil83Wilson A. S., Ulvestad J. S., 1983, ApJ, 275, 8 [Woo & Urry2002]woo02Woo J.H., Urry C.M., 2002, ApJ, 579, 530[Woo et al.2017]woo17Woo J.H., Son D., Bae H.J., 2017,ApJ, in press (arXiv: 1702.06681)[Whittle1992]whi92Whittle M., 1992, ApJSS, 79, 49 [Xu et al.1999]xu99Xu C., Livio M., Baum S., 1999, AJ, 118, 1169[York et al.2000]york00York D. G. et al., 2000, AJ, 120, 1579[Zakamska et al.2003]zak03Zakamska N. L. et al. 2003, AJ, 126, 2125[Zakamska & Greene2014]zak14Zakamska N., Greene J., 2014, MNRAS, 442, 784[Zakamska et al.2016]zak16Zakamska et al. 2016, MNRAS, 455, 4191[Zubovas & King2012]zub12Zubovas K., King A., 2012, ApJL, 745, 34[Zubovas & King2014]zub14Zubovas K., King A., 2014, MNRAS, 439, 400 | http://arxiv.org/abs/1708.07530v1 | {
"authors": [
"Montse Villar Martin",
"Bjorn Emonts",
"Antonio Cabrera Lavers",
"Clive Tadhunter",
"Dipanjan Mukherjee",
"Andrew Humphrey",
"Javier Rodriguez Zaurin",
"Cristina Ramos Almeida",
"Miguel Perez Torres",
"Patricia Bessiere"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170824192125",
"title": "Galaxy wide radio induced feedback in a radio quiet quasar"
} |
Multi-Agent Q-Learning for Minimizing Demand-Supply Power Deficit in Microgrids Raghuram Bharadwaj D. IISc, Bangalore, India [email protected] D. Sai Koti Reddy IISc, Bangalore, India [email protected] Shalabh Bhatnagar IISc, Bangalore, India [email protected] December 30, 2023 ==============================================================================================================================================================================================================================================We consider the problem of minimizing the difference in the demand and the supply of power using microgrids. We setup multiple microgrids, that provide electricity to a village. They have access to the batteries that can store renewable power and also the electrical lines from the main grid. During each time period, these microgrids need to take decision on the amount of renewable power to be used from the batteries as well as the amount of power needed from the main grid. We formulate this problem in the framework of Markov Decision Process (MDP), similar to the one discussed in <cit.>. The power allotment to the village from the main grid is fixed and bounded, whereas the renewable energy generation is uncertain in nature. Therefore we adapt a distributed version of the popular Reinforcement learning technique, Multi-Agent Q-Learning to the problem. Finally, we also consider a variant of this problem where the cost of power production at the main site is taken into consideration. In this scenario the microgrids need to minimize the demand-supply deficit, while maintaining the desired average cost of the power production.§ INTRODUCTION Electricity is one of the most important components of the modern life. According to a recent survey, there are a total of 18,452 un-electrified villages in India <cit.>. Providing electricity to these villages is difficult for a number of reasons. The village may be situated very far away from the main grid and it would be difficult to establish a direct electrical line between the main grid and the village. Also, due to increasing global warming, we want to make less use of fossil fuels for the generation of power. Our objective in this work is to provide a solution to electrifying these villages.The concept of smart grid <cit.>, is aimed at improving traditional power grid operations.It is a distributed energy network composed of intelligent nodes (or agents) that can either operate autonomously or communicate and share energy. Thepower grid is facing wide variety of challenges due to incorporation of renewable and sustainable energy power generation sources. The aim of smart grid is to effectively deliver energy to consumers and maintain grid stability.A microgrid is a distributed networked group consisting of renewable energy generation sources with the aim of providing energy to small areas. This scenario is being envisaged as an important alternative to the conventional scheme with large power stations transmitting energy over long distances. The microgrid technology is useful particularly in the Indian context where extending power supply from the main grids to remote villages is a challenge. While the main power stations are highly connected, microgrids with local power generation, storage and conversion capabilities, act locally or share power with a few neighboring microgrid nodes <cit.>. Integrating microgrids into smartgrid poses several technical challenges. These challenges need to be addressed in order to maintain the reliable and stable operation of electric grid.Research on smartgrids can be classified into two areas -Demand-side management and Supply-side management. Demand side management (DSM) (<cit.>) deals with techniques developed to efficiently use the power by bringing the customers into the play. The main idea is to reduce the consumption of power during peak time and shifting it during the other times. This is done by dynamically changing the price of power and sharing this information with the customers. Key techniques to address DSM problem in smart grid are peak clipping, valley filling,load shifting <cit.>. In <cit.>, Reinforcement Learning (RL) is used in smart grids for pricing mechanism so as to improve the profits of broker agents who procure energy from power generation sources and sell it to consumers.Supply-side management deals with developing techniques to efficiently make use of renewable and non-renewable energy at the supply side. In this paper, we consider one such problem of minimizing Demand-Supply deficit in microgrids. In <cit.>, authors developed an MDP and applied Dynamic optimization methods to this problem. But when the model information (the renewable energy generation in this case) is not known, we cannot apply these techniques.In our current work, we setup microgrids closer to the village. These microgrids has power connections from the main grid and also with the batteries that can store renewable energy. Owing to their cost, these batteries will have limited storage capacities. Each microgrid needs to take decision on amount of renewable energy that needs to be used at every time slot and the amount of power that needs to be drawn from the main grid. Consider a scenario where, microgrids will use the renewable energy as it is generated. That is, they do not store the energy. Then, during the peak demand, if the amount of renewable energy generated is low and power obtained from the main grid is also low, it leads to huge blackout. Thus, it is important to intelligently store and use the renewable energy. In this work, we apply Multi-agent Q-learning algorithm to solve this problem. § ORGANIZATION OF THE PAPER The rest of the paper is organized as follows. The next section describes the important problems associated with the microgrids and solution techniques to solve them. Section III presents the results of experiments of our algorithms. Section IV provides the concluding remarks and Section V discusses the future research directions.§ OUR WORKIn this section, we discuss two important problems associated with the microgrids. We first formulate the problem in the framework of Markov Decision Process (MDP), similar to the one described in <cit.>. We then apply cooperative Multi-Agent Q-Learning algorithm to solve the problems. §.§ Problem 1 - Minimizing Demand-Supply Deficit Consider a village that has not been electrified yet. We setup microgrids close to the village, and provides electrical connections to the village. It also has power connections from the main grid. Decision on power supply is generally taken in time slots, for example every two hours. Microgrids do not have non-renewable power generation capabilities. They have access to the renewable power and needs to obtain the excess power from the main grid. For ease of explanation, we consider two renewable sources - solar and wind. At the beginning of each time slot, microgrids obtain the power demand from customers with the help of the smart meters <cit.>. These microgrids need to take decisions on amount of renewable power to be used so that the expected long term demand-supply deficit is minimized.One natural solution without the use of storage batteries is to fully use the solar and wind power generated in each time slot. The excess demand will be then requested from the main grid. If the requested power is less than or equal to the maximum allotted power, the main grid transfers this to the microgrids. Other wise, it transfers the maximum allotted power. This idea is described in Algorithm 1.Our objective in the current work is to do better than the above described solution. Microgrids are equipped with batteries that can store the renewable power. That is, we deploy the microgrids at the sites where there is availability of renewable resources. The goal is to intelligently use the stored power in batteries and take the optimal action at every time slot to minimize the demand-supply deficit.We formulate the problem in the framework of Markov Decision Process (MDP). MDP <cit.> is the most popular mathematical framework for modeling optimal sequential decision making problems under uncertainty. A Markov decision process is defined via tuple < S; A; P;R >, where S is the set of states. A is the set of actions. P respectively the probability transition matrix. P_a(s,s^') is the probability that by taking action a in the state s, the system moves into the state s^'. R is the reward function where R_a(s,s^') is the reward obtained by taking action a in the state s and the s^' being the next state. In infinite horizon discounted setting <cit.>, the goal is to obtain a stationary policy π, which is a mapping from state space to action space that maximizes the following objective: ∑_t = 0^∞γ^tR_a_t(s_t,s_t+1),where a_t = π(s_t), and γ∈ [0,1] is the discount factor.We model Power Demand as a Markov process. That is, the current Demand depends only on the previous value. The Demand value, and the amount of power left in the batteries form the state of the MDP. That is, state = [Demand, solar_batterylevel, wind_batterylevel]. Based on the current state, the microgrids need to take decision on the number of units of power to be used from their batteries, and the number of units of power that is needed from the main grid. Note that the power from the main grid is fixed and bounded during all the time periods, while power from therenewable sources is uncertain in nature.Action = [solar_power, wind_power, main_power], The state evolves as follows:solar_batterylevel= solar_batterylevel - solar_power + solar(),andwind_batterylevel= wind_batterylevel - wind_power+ wind(),where solar() and wind() are the solar and wind power generation policies.Our objective is to minimize the expected long term discounted difference in the demand and supply of the power. So, the single stage Reward function is as follows: R = - (Demand - (solar_power+wind_power+main_power))^2When there is uncertainty in the system, like the renewal energy generation in this case, traditional solution techniques like value iteration and policy iteration cannot be applied. Reinforcement Learning provides us with the algorithms which can be applied when the model information is not completely known. One such popular model-free algorithm is Q-learning <cit.>. However, microgrids applying Q-Learning independently is not a feasible solution, as they are working towards a common goal. Hence we apply co-operative Multi-Agent Q-Learning algorithm <cit.> to solve the problem. At every time slot, the microgrids exchange the battery level information among themselves. Then they apply Q-Learning on the joint state and obtain the joint action. Each microgrid then selects its respective action. Let the joint state at time k be s_k. We select a joint action, based on an ϵ-greedy policy. An ϵ-greedy policy is one where we select the action that gives maximum reward with probability 1-ϵ and a random action with probability ϵ. Let it be a_k and so we move to a new state s_k+1. Let the single stage reward obtained be r_k Then the Q-values are updated as follows:Q(s_k,a_k) = Q(s_k,a_k) +α[r_k+γ max_aQ(s_k+1,a) - Q(s_k,a_k)],where α∈ [0,1] is the learning parameter and γ∈ [0,1] is the discount factor.We update this Q-values until convergence. At the end of this process, we will have an optimal policy that gives for each state, the optimal action to be taken. The idea is described inAlgorithm 2. §.§ Problem 2 - Balancing Demand-Supply Deficit and cost of Power ProductionIn the above formulation, cost of the non-renewable power production at the main grid site is not taken into consideration. Therefore, it is natural to use the maximum alloted power from the main grid and the optimization is done on the amount of power drawn from the solar and the wind batteries. In this formulation, we put a cost on amount of power that can be obtained from the main grid. That is, we modify our single stage Reward as follows:-c*(Demand - (solar_power + wind_power+main_power))^2+(1-c)*(main_power)^2. c ∈[0,1],where c is the parameter that controls the demand-supply objective and the main grid production cost. Note here that when c =1, it is same as the problem 1. In this formulation, our objective is not only to minimize the demand-supply deficit, but also maintain a desired average production of power at the main site. Similar to the above problem, we apply multi-agent Q-Learning to obtain the optimal solution. § EXPERIMENTSWe consider two microgrids operating on the solar and the wind power batteries respectively. Demand values are taken to be 8,10 and 12. The probability transition matrix for the Demand is given below.P = [[ 0.1 0.6 0.3; 0.3 0.1 0.6; 0.6 0.3 0.1; ]]. The maximum storage capacity of the batteries is set to 5. The renewable energy generation process for simulation purposes is taken to be Poisson with mean 2. Discount factor of the Q-learning algorithm is set to 0.9.We begin our simulations with the state [8,5,5]. First we run Algorithm 1 for 10^6 iterations and compute the average deficit in demand and supply. Then we run Algorithm 2 for 10^8 iterations with ϵ-greedy policy with ϵ = 0.85 and obtain an optimal policy. We then compute the average deficit using optimal Q-values for 10^4 iterations. We compare both the algorithms in Figure 1. With regard to the problem 2, we plot the values of average deficit in the power and the average production at the main grid obtained by different values of c in (<ref>). The maximum power allocation at the main site is taken to be 8. This is shown in Figure 2.§.§ Observations * In Figure 1, we see that the Algorithm 1 performs better than the Algorithm 2, when the Maximum Power Allocation (MPA) at the main site is less than 3. This is because, when the MPA is very less compared to the minimum demand value, storing the power in the batteries doesn't provide any advantage. * On the other hand, if the MPA is not very small and comparable to the minimum demand, the Algorithm 2 outperforms Algorithm 1. Thus, in this case it is useful to store the power in the batteries. This is the main result of our paper. * In Figure 2, we observe that as the value of c increases, the average demand-supply deficit decreases and the average power obtained from the main grid increases.§ CONCLUSIONWe considered the problem of electrifying a village by setting up microgrids close to the village. These microgrids have access to the renewable energy storage batteries and also electrical connections from the main grid. We identified two problems associated with the microgrid. First problem is to minimize the expected long-run discounted demand-supply deficit. We model this problem in the framework of MDP <cit.>. This formulation doesn't take into consideration the cost of power production at the main grid site. Finally, we formulated MDP taking the cost of power production into consideration. We applied Multi-Agent Q-Learning algorithm to solve these problems. Simulations show that, when maximum power allocation at the main site is not very less, storing the power in the batteries and using them intelligently is the better solution compared to not using the storage batteries. § FUTURE WORKAs future work, we would like to consider the possibility of power sharing between the microgrids. In this case, along with decision on amount of power to be used from stored batteries, microgrids also have to make decision on the amount of power that can be shared with others. We would also like to consider theheterogeneous power price system. In this scenario, the price of power production at the main grid will vary from time to time. § ACKNOWLEDGMENT The authors would like to thank Robert Bosch Centre for Cyber-Physical Systems, IISc, Bangalore, India for supporting part of this work.IEEEtran | http://arxiv.org/abs/1708.07732v2 | {
"authors": [
"Raghuram Bharadwaj Diddigi",
"D. Sai Koti Reddy",
"Shalabh Bhatnagar"
],
"categories": [
"cs.SY",
"cs.AI"
],
"primary_category": "cs.SY",
"published": "20170825133641",
"title": "Multi-Agent Q-Learning for Minimizing Demand-Supply Power Deficit in Microgrids"
} |
[email protected] 1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan 2Division of Physics, Faculty of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan 3Center for Integrated Research in Fundamental Science and Engineering (CiRfSE), Faculty of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan 4RIKEN, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan 5Institut de Radioastronomie Millimétrique (IRAM) 38406, Saint Martin d'Héres, France 6Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan 7National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588, Japan 8Academia Sinica, Institute of Astronomy and Astrophysics, PO Box 23-141, Taipei 106, Taiwan 9Research Center for the Early Universe, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, JapanWe have conducted an interferometric line survey in the 0.8 mm band toward the young high-mass protostar candidate NGC 2264 CMM3 with ALMA.CMM3 is resolved into the two continuum peaks, CMM3A and CMM3B, at an angular separation of 0.9”. Thus, CMM3 is found to be a binary system candidate.We have detected molecular outflows associated with CMM3A and CMM3B each, indicating active star formation.In addition to the two peaks, six faint continuum peaks are detected around CMM3A and CMM3B, most of which are thought to be evolved low-mass protostars.CMM3A is found to be rich in molecular line emission including complex organic molecules such as HCOOCH_3 and CH_3OCH_3.The emission of complex organic molecules is distributed within a compact region around the continuum peak of CMM3A.Hence, CMM3A apparently harbors a hot core.On the other hand, CMM3B is deficient in molecular line emission, although its continuum flux is almost comparable to that of CMM3A.Possible origins of the striking difference between CMM3A and CMM3B are discussed. § INTRODUCTIONNGC 2264 C is a nearby <cit.> cluster forming region.Many millimeter and submillimeter sources have so far been identified in this region <cit.>.<cit.> found eleven outflow lobes in the NGC 2264 C region by an extensive mapping observation of CO.Recently, <cit.> found the SiO emission with narrow line width in the periphery of NGC 2264 C, and suggested that it reveals a remnant of past star formation activities.In such a complex structure of this region, NGC 2264 CMM3 is known to be the most massive core <cit.> located almost at its geometrical center.According to the theoretical model, CMM3 is predicted to evolve into a main sequence star with a mass of 8M_⊙ <cit.>.The protostar of CMM3 is deeply embedded in an envelope, and hence, no mid-infrared source is identified even in the 24 μm band of Spitzer.A definitive evidence of star formation in CMM3 was first found by <cit.>, who identified a compact bipolar outflow with a dynamical age of 140–2000 year by the CO and CH_3OH observations with the Submillimeter Array (SMA).<cit.> also identified the bipolar outflows in the CO, H_2CO, SO, and SiO lines.Judging from the non-detection of mid-infrared sources and the association of dynamically young outflows, <cit.> suggested that CMM3 is a candidate of a high mass protostar in the early evolutionary phase.Toward CMM3, a chemical composition has been studied by a spectral line survey observation with single dish telescopes in the 4 mm, 3 mm, and 0.8 mm bands <cit.>.In their survey, carbon-chain molecules such as C_4H, HC_5N, and C_3S are found to be relatively abundant in CMM3 compared with the representative high-mass star forming region Orion KL, while complex organic molecules are deficient.This result suggests chemical youth of the CMM3 envelope. Recently, we carried out a spectral line survey toward CMM3 at a high angular resolution with ALMA in the 0.8 mm band.Here we report the results on the continuum emission and the distributions of the emission of complex organic molecules.§ OBSERVATION AND REDUCTIONThe observations were carried out with ALMA on 19 July 2014 and 20 May 2015, where the numbers of the 12 m antenna used in those observations were 31 and 39, respectively.Minimum and maximum baseline lengths were 14 m and 649 m on the ground, respectively.Twelve spectral windows were employed to cover the frequency ranges of 336.0–347.3 GHz and 348.0–359.3 GHz at a frequency resolution of 0.98 MHz.The phase-center coordinate is (α_ J2000, δ_ J2000) = (6:41:12.3, +9:29:8.0).The quasars J0510+1800 and J0725-0054 were observed for the bandpass calibration in the 2014 and 2015 sessions, respectively.The quasars J0643+0857 and J0700+1709 were observed for the amplitude and phase gain calibrations in the 2014 and 2015 sessions, respectively.The absolute flux was calibrated by using the quasar J0750+125.The uncertainty of the absolute flux calibration is 10 % according to ALMA Cycle 2 Technical Handbook (Lundgren 2013).In these observations, any structure extended over ∼ 13” or larger is subject to the resolving-out effect, judging from the minimum baseline length of 14 m.Data reduction was carried out with the Common Astronomy Software Applications package (CASA; Version 4.5.3).A continuum image was obtained by averaging line-free channels of all the spectral windows.Self-calibration was performed by using the continuum data, and then the solutions were applied to both the continuum and spectral line data.After the self-calibration procedures, the data were imaged and CLEANed by using the Briggs weighting with the robustness parameter of 0.5.The resulting synthesized beam and the root-mean-square (rms) noise of the continuum image are 0.37”× 0.30” (P.A.=-30.7^∘) and 0.35 mJy beam^-1, respectively.The signal-to-noise ratio in the continuum image was improved from 190 to 723 by applying the self-calibration procedure.The synthesized beam and the rms noise of spectral images range from 0.28” to 0.49” and from 4.2 mJy beam^-1 to 7.0 mJy beam^-1 at the frequency resolution of 0.98 MHz, respectively, depending on the frequency.§ RESULTS §.§ Detection of a Potential Binary System in the CMM3Figure <ref>a shows the 0.8 mm continuum image of NGC 2264 CMM3 observed with ALMA.The beam size of 0.37”× 0.30” (P.A.=-30.7^∘) corresponds to a linear scale of ∼270 au at the distance of 738 pc.CMM 3 is well resolved into the two continuum peaks CMM3A and CMM3B with comparable flux densities of 431.8 mJy and 313.0 mJy, respectively (Table <ref>).The angular separation between the two peaks is ∼ 0.9”, which corresponds to ∼660 au.The line of sight velocities of molecules associated with CMM3A (∼ 7.4 km s^-1) are similar to those of CMM3B (∼ 7.8 km s^-1).Therefore, the two protostars are most likely gravitationally bound to each other.In the previous studies, CMM3 has been recognized as a single source because of insufficient angular resolutions <cit.>.It should be noted that the potential binary components, CMM3A and CMM3B, are surrounded by an extended envelope. The continuum emission shows asymmetric elongation toward the south-eastern and northern directions from CMM3B and slightly toward the north-western direction from CMM3A.This emission would likely trace the circumbinary envelope.In addition to the two bright continuum peaks, six compact continuum sources (CMM3C, CMM3D, CMM3E, CMM3F, CMM3G, and CMM3H) are detected around CMM3A/B, as shown in Figure <ref>a.Here, we have identified local intensity peaks detected with more than 10σ as the continuum peaks (Table <ref>).CMM3C, CMM3D, and CMM3E are point sources, because the FWHM evaluated by the 2D-Gaussian fitting is similar to the synthesized beam of the observation.In the near-infrared images (Figures <ref>c–f), faint sources seem to be associated with CMM3C, CMM3D, and CMM3E.Therefore, these three continuum sources would be low-mass Class I or Class II protostars.On the other hand, CMM3F is marginally resolved, and is associated with no infrared source.It may represent a low-mass protostellar/prestellar source.For CMM3G and CMM3H, no parameter was derived by the 2D Gaussinan fitting, because the fitting did not converge.Note that these two peaks are close to the outflow lobe B1 reported by <cit.>. §.§ Chemical Difference between CMM3A and CMM3BAlthough CMM3A and CMM3B have comparable flux densities in the 0.8 mm continuum emission, their spectral patterns are found to be much different from each other. Figures <ref>a and <ref>b show the spectra of CMM3A and CMM3B, respectively, in the frequency range from 338.3 GHz to 338.8 GHz.The spectrum of CMM3A shows many emission lines of CH_3OH and SO_2, and those of complex organic molecules such as HCOOCH_3, CH_3CH_2OH, and c-C_2H_4O even in this narrow frequency range.The full spectra (336.0–347.3 GHz and 348.0–359.3 GHz) and line lists of this spectral line survey will be available in a forthcoming paper (Watanabe et al. in prep).In the previous observation with the single-dish telescope ASTE in the 0.8 mm band (Figure <ref>c), the lines of these complex organic molecules were not seen at all except for CH_3OH in this frequency range <cit.>, although HCOOCH_3 and CH_3OCH_3 have been weakly detected in the 3 mm band <cit.>.This means that the hot core was not traced by thesingle-dish telescope observation even in the sub-mm band, since its contribution is heavily diluted.The single-dish spectrum is dominated by the contribution from the surrounding envelope.For example, only 4 % of flux density of the CH_3OH (13_1 - 13_0A^-+) line observed with ASTE is calculated to come from CMM3A and CMM3B.The high angular resolution observation with ALMA allows us to detect the hot core associated with CMM3A for the first time.Figures <ref>a, b and c show integrated intensity maps of CH_3OCH_3, HCOOCH_3, and CH_3OH, respectively.These transition lines are selected as examples to demonstrate the distributions of complex organic molecules and that of CH_3OH with the very high upper-state energy (451 K), because they seem to be less contaminated from other molecular lines.As expected from their spectra (Figure <ref>a,b), the complex organic molecules and the very high excitation CH_3OH are associated only with CMM3A.Furthermore, they are concentrated in a compact region (<300 au) around the continuum peak.High excitation lines (E_ u>100 K) of complex organic molecules are detected, and hence, a hot core is thought to have already been formed in CMM3A.The existence of the hot core is indeed supported by the high rotation temperature (385 ± 58 K) of CH_3OH estimated by using the rotation diagram method under the assumption of optically thin conditions and local thermodynamic equilibrium (LTE) (Figure <ref>).The uncertainty of the derived rotation temperature is relatively large because of the non-LTE effects, optical depths of lines, blending with the other lines, and the resolving-out effect for low-excitation lines.Nevertheless, the high temperature condition is evident.In contrast to the richness of emission lines in CMM2A, only CH_3OH is detected marginally in CMM3B.Although the HCOOCH_3 may marginally be detected, it could be blended with the CH_3OH line.Since no complex organic molecules are definitively detected (Figure <ref>a, b), a hot core has not yet been formed in CMM3B.We here evaluate column densities of CH_3OH, CH_3OCH_3, and HCOOCH_3, and their fractional abundances relative to H_2.The column densities are derived under the LTE approximation with the rotation temperatures of 100 K and 300 K.Because CH_3OCH_3 and HCOOCH_3 are not definitively detected in CMM3B, their column densities cannot be determined for CMM3B.In order to compare the chemical compositions between CMM3A and CMM3B on the same basis, we derived the column densities by using the same transitions.As mentioned above, the rotation temperature evaluated from the rotation diagram of CH_3OH (Figure <ref>) suffers from various systematic uncertainties.Therefore, we employ the rotation temperatures of 100 K and 300 K for both CMM3A and CMM3B.We use a few transitions without contamination from other molecular lines as listed in the captions of Table <ref>, and evaluate the column densities by taking the averages of the results from the transition lines.The results are summarized in Table <ref>. The column densities of H_2 (N_ H_2) are calculated from the continuum peak flux of the 0.8 mm band by using the formula used by <cit.> with the dust temperatures of 100 K and 300 K.Here, we adopt the mass absorption coefficient of the dust of κ_ν=0.0182 cm^2g^-1 derived by <cit.>, which is an interpolated value of the numerical simulation <cit.>.The gas-to-dust mass ratio is assumed to be 100.The derived H_2 column densities and fractional abundances of molecule relative to H_2 are summarized in Table <ref>.The fractional abundances of CH_3OH and CH_3OCH_3 are found to be higher in CMM3A than those in CMM3B by factor of 5, while the upper limit of HCOOCH_3 is not stringent.This chemical difference will be discussed in the latter section. The HCOOCH_3/CH_3OH and CH_3OCH_3/CH_3OH ratios in CMM3A are roughly comparable to or even higher than the averaged values in the low-mass and high-mass protostars <cit.>.Therefore, CMM3A possesses analogous chemical compositions of complex organic molecules to other protostars. §.§ OutflowsFigures <ref>a and c show the 0th and the 1st moment maps of HCN, respectively.We find a prominent bipolar outflow blowing from CMM3A toward northern and southern directions, which are blue-shifted and red-shifted, respectively (Figure <ref>c).The blue-shifted and red-shifted outflows can also be seen in the channel maps of HCN (Figure <ref>).This bipolar outflow system corresponds to that found by <cit.> in the CO(J=2-1) line with SMA.Their B1 and R1 lobes are seen in Figure <ref>, while B2 and R2 lobes are out of the field of view in the present ALMA observation.As seen in the expanded figure of the 0th HCN moment map (Figure <ref>b), this bipolar outflows system is launched from CMM3A.It is also visible in other molecular lines including CO (J=3-2), CS (J=7-6), SO (N_J=8_7-7_6), and CH_3OH (e.g., 13_1-13_0,A^-+) (Figure <ref>a-d). It should be noted that the continuum peak CMM3G is close to the B1 lobe (Figure <ref>).This indicates that the blue-shifted outflow is interacting with a dense material traced by the continuum emission (CMM3G) and the B1 lobe is induced by the shocks caused by the interaction.Moreover, we find that the outflow is elongated toward the northwest direction from the B1 lobe (Figure <ref>a).This elongation can be seen from 3.3 km s^-1 to -16.6 km s^-1 in the channel maps (Figure <ref>).Therefore, the direction of outflow may be changed by the dynamical interaction with the dense clump of CMM3G. In addition to the prominent bipolar outflow, we find another bipolar outflow system along the northeast to southwest direction (Figures <ref>a).The second outflow is also seen from 16.5 km s^-1 to -5.0 km s^-1 in the channel maps of HCN (Figure <ref>).This outflow system is well-collimated, and is also visible in other molecular lines including CO, CS, SO, and CH_3OH (Figure <ref> e-h).The velocity-integrated emission from the second outflow is much fainter than that from the prominent outflow, because the velocity range of the second outflow is narrower (∼ 20 km s^-1 than that of the prominent outflow (∼ 70 km s^-1).Moreover, a part of the second outflow is strongly affected by the resolving-out effect at the system velocity (8.2 km s^-1 and 6.6 km s^-1 in Figure <ref>).The southwestern lobe of this outflow can be seen in the CH_3OH (5_0 - 4_0 A^+) line in Figure 4a of <cit.>.This collimated outflow system seems to be launched from CMM3B, since the outflow axis just passes through the peak of CMM3B (Figure <ref>b).The velocity field map (Figure <ref>c) showssmall velocity gradient along the bipolar outflow axis.This result indicates that the outflow blows almost in parallel to the plane of sky.The detection of this outflow provides us with a strong evidence of the presence of a protostar in CMM3B.Its well collimated feature suggests that the outflow system would most likely be in the early phase <cit.>.§ DISCUSSIONIn this study, we have found that CMM3 is a potential binary protostellar source.More importantly, the binary components, CMM3A and CMM3B, show different spectral appearance in spite of their comparable continuum intensities.It is interesting to note that a similar feature has recently been reported for the young low-mass binary source NGC 1333 IRAS4A <cit.>.We here discuss possible origins of the difference in CMM3. The first possibility is the difference of the physical evolutionary stage between the two sources.Considering the prominent outflow feature and the association of the hot core, CMM3A seems to harbor young high-mass (or intermediate-mass) protostar.In contrast, deficient molecular emission in CMM3B might originate from its youth.Namely, the protostar may be so young that the hot core would not have been developed yet or would be still tiny around the protostar.In this case, most of organic molecules are still frozen out on dust grains, and their gas phase abundances are expected to be low.This possibility is supported by the well collimated outflow from CMM3B, which is usually interpreted as a sign of a young dynamical age <cit.>.The envelope mass is derived based on the formula given by <cit.>.For CMM3A, it is evaluated to be 18 M_⊙ and 5.7 M_⊙, assuming the dust temperature of 100 K and 300 K, respectively.For CMM3B, it is 13 M_⊙ and 4.1 M_⊙, respectively.Thus, the envelope mass of CMM3B could be higher than CMM3A, if the dust temperature is lower in CMM3B because of its youth.The second possibility is the dust opacity effect.The dust emission at 0.8 mm is not always optically thin at a scale of a few tens of au.If the dust opacity is so high that the emission of various molecules originated from the innermost part of the dust emitting region is heavily absorbed by the dust, the molecular emission could be very weak.This may be the situation for CMM3B.Since the dust temperature of CMM3A is expected to be higher than that of CMM3B, the dust opacity would be lower in CMM3A, and this effect is less significant.As mentioned before, it is most likely that the outflow from CMM3B is almost on the plane of sky.In this case, the disk/envelope system of CMM3B is almost edge-on, which could make the dust opacity high.The third possibility is the difference of masses of the protostars between CMM3A and CMM3B.If CMM3B is less massive than CMM3A, a size of the hot region where complex organic molecules can be sublimated is very small due to low luminosity.Since the hot region is smaller in CMM3B than CMM3A, the dust temperature is also expected to be lower in CMM3B than CMM3A.Therefore, a larger envelope gas would still be remaining in CMM3B, as discussed in the first possibility.Apparently, these three possibilities are interrelated to one another.The scenario of a less massive protostar for CMM3B is related to the evolutionary scenario, because the protostar evolution would strongly depend on its mass.It is also related to the dust opacity scenario, because a young protostar is deeply embedded in the parent cloud.In order to disentangle these three possibilities, we need to define the spectral energy distributions for the two sources.Then, we can derive the disk/envelope mass accurately, and estimate the evolutionary stage of the protostar reasonably.High angular resolution observations resolving CMM3A and CMM3B at various wavelength from cm wave to sub-mm wave are awaited.We thank the anonymous reviewer for helpful comments and suggestions to improve this paper.This paper makes use of the ALMA data set ADS/JAO.ALMA#2013.1.01192.S.ALMA is a partnership of the ESO (representing its member states), the NSF (USA) and NINS (Japan), together with the NRC (Canada) and the NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by the ESO, the AUI/NRAO and the NAOJ. The authors are grateful to the ALMA staff for their excellent support.This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, funded by the National Aeronautics and Space Administration and the National Science Foundation.This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.This study is supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (No. 25108005 and 16K17657). Facilities: ALMA. [Arce & Sargent(2006)]Arce2006 Arce, H. G., & Sargent, A. I. 2006, , 646, 1070[Cunningham et al.(2016)]Cunningham2016 Cunningham, N., Lumsden, S. L., Cyganowski, C. J., Maud, L. T., & Purcell, C. 2016, , 458, 1742 [Jørgensen et al.(2016)]Jorgensen2016 Jørgensen, J. K., van der Wiel, M. H. D., Coutens, A., et al. 2016, , 595, A117[Kamezaki et al.(2014)]Kamezaki2014 Kamezaki, T., Imura, K., Omodaka, T., et al. 2014, , 211, 18 [Imai et al.(2016)]Imai2016 Imai, M., Sakai, N., Oya, Y., et al. 2016, , 830, L37[López-Sepulcre et al.(2016)]Ana2016 López-Sepulcre, A., Watanabe, Y., Sakai, N., et al. 2016, , 822, 85[López-Sepulcre et al.(2017)]Ana2017 López-Sepulcre, A., Sakai, N., Neri, R., et al. 2017, , accepted, <https://doi.org/10.1051/0004-6361/201630334>[Lundgren(2013)]Lundgren2013 Lundgren, A. 2013, ALMA Cycle 2 Technical Handbook Version 1.1, ALMA [Maury et al.(2009)]Maury2009 Maury, A. J., André, P., & Li, Z.-Y. 2009, , 499, 175 [Ossenkopf & Henning(1994)]Ossenkopf1994 Ossenkopf, V., & Henning, T. 1994, , 291, 943[Peretto et al.(2006)]Peretto2006 Peretto, N., André, P., & Belloche, A. 2006, , 445, 979[Peretto et al.(2007)]Peretto2007 Peretto, N., Hennebelle, P., & André, P. 2007, , 464, 983[Sakai et al.(2007)]Sakai2007 Sakai, N., Sakai, T., & Yamamoto, S. 2007, , 660, 363[Saruwatari et al.(2011)]Saruwatari2011 Saruwatari, O., Sakai, N., Liu, S.-Y., et al. 2011, , 729, 147 [Skrutskie et al.(2006)]Skrutskie2006 Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, , 131, 1163[Taquet et al.(2015)]Taquet2015 Taquet, V., López-Sepulcre, A., Ceccarelli, C., et al. 2015, , 804, 81[Ward-Thompson et al.(2000)]Ward-Thompson2000 Ward-Thompson, D., Zylka, R., Mezger, P. G., & Sievers, A. W. 2000, , 355, 1122[Watanabe et al.(2015)]Watanabe2015 Watanabe, Y., Sakai, N., López-Sepulcre, A., et al. 2015, , 809, 162 | http://arxiv.org/abs/1708.07582v1 | {
"authors": [
"Yoshimasa Watanabe",
"Nami Sakai",
"Ana Lopez-Sepulcre",
"Takeshi Sakai",
"Tomoya Hirota",
"Sheng-Yuan Liu",
"Yu-Nung Su",
"Satoshi Yamamoto"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170825002110",
"title": "Discovery of Striking Difference of Molecular-Emission-Line Richness in the Potential Proto-Binary System NGC 2264 CMM3"
} |
§ BACKGROUND AND OPEN QUESTIONSGalaxy clusters are the largest gravitationally bound systems and are believed to be formed via mergers of smaller systems. They have masses of the order of 10^14-10^15 M_⊙, with 15-20% in the form of a hot (10^8 K) gas that pervades the cluster volume, emitting X–rays via the Bremsstrahlung mechanism and mm-wave radiation via the Sunyaev–Zeldovich (SZ) effect. The presence of a non–thermal (i.e. relativistic particles and magnetic fields) component emitting synchrotron radiation has been revealed by a variety of radio observations over the last few decades. In particular, there are four different sources of radio emission found in galaxy clusters (see <cit.> for a recent observational review): * discrete radio sources associated with cluster galaxies;* radio halos (RHs): Mpc–scale diffuse radio sources with steep spectrum and low surface brightness that are found in the central regions of a number of merging clusters;* mini halos: central, diffuse radio sources extending over ∼ 100 kpc scales, typical of dynamically relaxed systems;* radio relics: diffuse radio sources with elongated morphology, significantly polarized, mostly located at the outskirts of a small number of merging clusters.In this paper we will mainly focus on the study of RHs and what they can tell us about the formation and evolution of galaxy clusters.The particle life time due to radiative losses is much shorter than the RH crossing time (e.g. <cit.>), therefore a re–acceleration mechanism is required to explain the presence of Mpc–size structures in galaxy clusters. Theoretical efforts over the last two decades have provided a scenario for the formation of RHs based on the in–situ re–acceleration of relativistic particles due to merger–driven turbulence (e.g., <cit.> for a review). Radio observations of large samples of clusters carried out over the last ten years in the Extended GMRT Radio Halo Survey (EGRHS, <cit.>, <cit.>, <cit.> and <cit.>) have established two key properties: * RHs are not ubiquitous, but are found only in the 20-30% of clusters that are X–ray luminous (e.g., <cit.>, <cit.>);* RHs are connected to the cluster dynamical state, being found only in merging clusters but not in relaxed systems (<cit.>). This behaviour appears as a bimodal distribution between the 1.4 GHz RH power P_1.4 and the X–ray luminosity L_ X, where clusters with a RH follow the P_1.4 - L_ X correlation and clusters without a RH are significantly below such correlation (e.g. <cit.>);With the advent of SZ cluster surveys (e.g., <cit.>, <cit.>, <cit.>), RHs could be directly linked to the cluster mass and their bimodality has been confirmed in the P_1.4 - M_500[where M_500 is the total cluster mass within the radius R_500, defined as the radius corresponding to a total density contrast 500ρ_c(z), where ρ_c(z) is the critical density of the Universe atthe cluster redshift.] plane too, with RH clusters following the correlation, whereas clusters without RHs appear well below the correlation (<cit.>, <cit.>). In addition, it has been proven that the fraction of clusters with RH is larger in surveys of mass–selected clusters, being ∼ 50% for clusters with M_500 > 6× 10^14 M_⊙ (<cit.>).These observational results support the current scenario for the formation of giant RHs whose formation history depends on the interplay between the galaxy cluster merging rate throughout the cosmic epochs and the process of particle acceleration.Despite the success of the turbulent re–acceleration model in explaining current RH observations, several questions related to the formation and evolution of diffuse radio emission in galaxy clusters are still open, in particular: * what is the fraction of giant radio halos in less massive clusters?* how does the fraction of clusters with giant radio halos evolve with cosmic epoch?§ KAT–7 OBSERVATIONS OF GALAXY CLUSTERSWe recently began to address the first open question by selecting a mass-limited (M_ 500 > 4 × 10^14 M_⊙) sample of nearby (z < 0.1) clusters from the Planck SZ cluster catalogue (<cit.>) and observed it with the 7–element Karoo Array Telescope (KAT–7, <cit.>) at 1.86 GHz. The sample, consisting of 15 clusters observed with a 2.3 - 2.9 arcmin angular resolution (depending upon declination), is described in <cit.>, to which we refer the reader for details. We will summarize here the main results presented in the light of the future opportunities offered by the upcoming MeerKAT telescope described in Section <ref>.The results of KAT–7 observations can be divided in three groups:* six clusters which host radio emission. For these systems, the limited angular resolution prevented us from reliably disentangling compact sources blended in a possible diffuse halo, therefore those targets were conservatively discarded from the final statistical analysis;* three candidate RHs, i.e. clusters hosting diffuse radio emission that spatially correlates with a highly disturbed X–ray morphology. The two most interesting cases are shown in Figure <ref>: residual, ∼ 800 kpc–wide radio emission is still visible in the PSZ1G 018.75+23.57 cluster after removing two point sources identified in the Northern VLA Sky Survey (NVSS, <cit.>). As no spectral index information is available for these sources, they were subtracted by assuming a fiducial spectral index α = 0.7. The errors introduced by the subtraction are ∼ 20% of the thermal noise error estimates, as the proximity between the two observing frequencies makes spectral index uncertainties negligible. Including uncertainties in source subtraction, the flux density of the residual diffuse emission at 1.86 GHz is S_1.86 = 48.3 ± 3.1 mJy.The second case in Figure <ref> referes to the Triangulum Australis cluster (<cit.>), which shows ∼Mpc scale diffuse radio emission offset, but clearly spatially correlated, with the X–ray brightness distribution. The comparison between higher angular resolution observations from the 843 MHz Sydney University Molonglo Sky Survey (SUMSS, <cit.>) and the ESO Digitized Sky Survey optical image of the field reveals a spatial correspondence between some of the patchy 843 MHz radio emission and bright optical galaxies (see Figure 6 in <cit.>). Although the point source contamination to the RH flux density measured at 1.86 GHz may not be negligible, its accurate estimate is not straightforward because of the uncertainty in the frequency extrapolation from 843 MHz and the still somewhat limited angular resolution of SUMSS, requiring further, dedicated observations; * seven clusters for which no radio emission is detected down to the 0.3 - 0.8 mJy beam^-1 sensitivity limit and for which P_ 1.4 < 10^24.4 Watt Hz^-1 upper limits to the RH power could be set.Upper limits to the RH power provide the most stringent test of the P_1.4 - M_500 correlation and are plotted in Figure <ref> together with literature measurements and the best estimates of the RH power for the PSZ1G 018.75+23.57 and Triangulum Australis cases. Although the KAT–7 upper limits are compatible with the current correlation at the 95% confidence level and, therefore, do not provide evidence for the presence of bimodality at low masses, they still show that bright RHs are rare in small clusters.§ MEERKAT'S UNIQUE CONTRIBUTIONWe are now in the position to show that the upcoming MeerKAT radio telescope is optimally placed to perfom crucial tests of current models of radio emission in clusters. With its 64, pseudo–randomly distributed antennas with a very dense 1 km core, MeerKAT offers a unique combination of excellent uv–coverage that is crucial in order to sample all the spatial scales spanned by RHs and the ∼ 5 arcsec angular resolution necessary to identify and subtract discrete radio sources that may contaminate the RH emission. Moreover, with a 22 K system temperature[http://public.ska.ac.za/meerkat/meerkat-schedule] at 1.4 GHz, MeerKAT's pointed sensitivity is unmatched in the Southern Hemisphere.MeerKAT is likely to be the only instrument in the Southern Hemisphere able to perform two critical tests: * follow up observations of the KAT–7 sample. Given the current MeerKAT specifications, a 10 fold improvement in brightness sensitivity over the KAT–7 survey can be achieved in 10 hours per target, while avoiding the confusion limit. Such a survey will be able to either detect RHs or place upper limits that will be one order of magnitude below the current ones, i.e. well below the 95% confidence level of the correlation. Given its angular resolution, it will also be able to identify, model and subtract point–like emission in the seven clusters where our current observations were not conclusive. Such a survey will provide the first statistically complete test of the current RH model in small systems, assessing the presence/absence of a bimodal distribution;* observations of high redshift systems. Very little is known about RHs in high redshift clusters, as statistically complete samples are observed only up to z ∼ 0.3 (<cit.>, <cit.>). Due to its very good angular resolution and uv coverage, MeerKAT is again very well suited to observe higher redshift clusters whose RHs are naturally expected to be smaller.We selected 55 clusters from the South Pole Telescope SZ catalogue (<cit.>) with M_500 > 5 × 10^14 M_⊙ (80% complete at z > 0.4). Assuming that the current P_1.4 - M_500 correlation holds at higher redshift, Figure <ref> shows the parameter space that this possible MeerKAT survey would constrain. Assuming a 5 hour observation per individual target, the survey would detect all the clusters at z > 1 with P_ 1.4≳ 6 × 10^23 Watt Hz^-1, z > 0.8 with P_ 1.4≳ 4 × 10^23 Watt Hz^-1 and z > 0.5 with P_ 1.4≳ 2 × 10^23 Watt Hz^-1 respectively. Such a survey will complement the EGRHS (which provides the deepest and largest sample available to date) up to z ∼ 0.8 at the same sensitivity level. Such a survey which will not only provide a critical test of current models, but also provide an independent probe of the merging history of clusters (<cit.>). § ACKNOWLEDGMENTSThis work is based on research supported in part by the National Research Foundation of South Africa (Grant Numbers 103424) and by the National Research Foundation under grant 92725. Any opinion, finding and conclusion or recommendation expressed in this material is that of the author(s) and the NRF does not accept any liability in this regard. This work was also partly supported by the Executive Programme of Scientific and Technological Co-operation between the Italian Republic and the Republic of South Africa 2014–2016 and by the National Research Foundation of South Africa (Grant Numbers 103424). The KAT–7 is supported by SKA South Africa and by the National Science Foundation of South Africa.99 feretti12 Feretti, L., Giovannini, G., Govoni, F. & Murgia, M., 2012, A&AR, 20, 54 jaffe97 Jaffe, W.J., 1997 ApJ, 212, 1 brunetti14 Brunetti, G., & Jones, T.W., 2014, Intern. Journal of Modern Phys. D, Vol. 23, issue 4 venturi07 Venturi, T., Giacintucci, S., Brunetti, G., et al., 2007, A&A, 463, 937 venturi08 Venturi, T., Giacintucci, S., Dallacasa, D., et al., 2008, A&A, 484 327 kale13 Kale, R., Venturi, T., Giacintucci, S., et al., 2013, A&A, 557, 99 kale15 Kale, R., Venturi, T., Giacintucci, S., et al., 2015, A&A, 597, 92 brunetti07 Brunetti, G., Venturi, T., Dallacasa, D., et al., 2007, ApJ, 670, L5 cassano08 Cassano, R., Brunetti, G., Venturi, T., et al., 2008, A&A 480, 687 cassano10 Cassano, R., Ettori, S., Giacintucci, S., et al., 2010, ApJ, 721L, 82 brunetti09 Brunetti, G., Cassano, R., Dolag, K. and Setti, G., 2009, A&A, 507, 661 marriage11 Marriage, T., Acquaviva, V., Ade, P.A.R. et al., 2011, ApJ, 737, 61 reichardt13 Reichardt, C.L., Stalder, B., Bleem, L.E. et al., 2013, ApJ, 763, 127 planck14 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, AA29 basu12 Basu, K., 2012, MNRAS, 421L, 11 cassano13 Cassano, R., Ettori, S., Brunetti, G., et al., 2013, ApJ, 777, 141 cuciti15 Cuciti, V., Cassano, R., Brunetti, Dallacasa, D., G., Kale, R., Venturi, T. & Ettori, S., 2015, A&A, 580, 97 foley16 Foley, A.R., Alberts, T., Armstrong, R.P., et al., 2016, MNRAS, 460, 1664 bernardi16 Bernardi, G., Venturi, T., Cassano, R. et al., 2016, MNRAS, 456, 1259 condon98 Condon, J.J., Cotton, W.D., Greisen E.W. et al., 1998, AJ, 115, 1693 scaife15 Scaife, A.M.M., Oozeer N., de Gasperin, F., et al., 2015, MNRAS, 451, 4021 bock99 Bock, D., Large, M.I. & Sadler, E.M., 1999, AJ, 117, 1578 cassano16 Cassano, R., Brunetti, G., Giocoli, C., Ettori, S., 2016, A&A, 593, A81 | http://arxiv.org/abs/1708.07728v1 | {
"authors": [
"G. Bernardi",
"T. Venturi",
"R. Cassano",
"G. Brunetti",
"D. Dallacasa",
"B. Fanaroff",
"B. Hugo",
"S. Makhatini",
"N. Oozeer",
"O. M. Smirnov",
"J. T. L. Zwart"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170825132927",
"title": "A MeerKAT view on galaxy clusters"
} |
Hamiltonian MechanicsD. Karapetyan and A. Vernitsky.Institute for Analytics and Data Science, University of Essex, Essex, [email protected] Department of Mathematical Sciences, University of Essex, Essex, [email protected] Adaptive Implementation of the Serial Schedule Generation Scheme using Preprocessing and Bloom Filters Daniel Karapetyan1 Alexei Vernitski2 December 30, 2023 ================================================================================================================The majority of scheduling metaheuristics use indirect representation of solutions as a way to efficiently explore the search space.Thus, a crucial part of such metaheuristics is a “schedule generation scheme” – procedure translating the indirect solution representation into a schedule.Schedule generation scheme is used every time a new candidate solution needs to be evaluated.Being relatively slow, it eats up most of the running time of the metaheuristic and, thus, its speed plays significant role in performance of the metaheuristic.Despite its importance, little attention has been paid in the literature to efficient implementation of schedule generation schemes.We give detailed description of serial schedule generation scheme, including new improvements, and propose a new approach for speeding it up, by using Bloom filters.The results are further strengthened by automated control of parameters.Finally, we employ online algorithm selection to dynamically choose which of the two implementations to use.This hybrid approach significantly outperforms conventional implementation on a wide range of instances. § INTRODUCTIONResource Constrained Project Scheduling Problem (RCPSP) is to schedule a set of jobs J subject to precedence relationships and resource constraints.RCPSP is a powerful model generalising several classic scheduling problems such as job shop scheduling, flow shop scheduling and parallel machine scheduling.In RCPSP, we are given a set of resources R and their capacities c_r, r ∈ R.In each time slot, c_r units of resource r are available and can be shared between jobs.Each job j ∈ J has a prescribed consumption v_j,r of each resource r ∈ R.We are also given the duration d_j of a job j ∈ J.A job consumes v_j,r units of resource r in every time slot that it occupies.Once started, a job cannot be interrupted (no preemption is allowed).Finally, each resource is assigned a set 𝑝𝑟𝑒𝑑_j ⊂ J of jobs that need to be completed before j can start.There exist multiple extensions of RCPSP.In the multi-mode extension, each job can be executed in one of several modes, and then resource consumption and duration depend on the selected mode.In some applications, resource availability may vary with time.There could be set-up times associated with certain jobs.In multi-project extension, several projects run in parallel sharing some but not all resources.In this paper we focus on the basic version of RCPSP, however some of our results can be easily generalised to many of its extensions and variations. Most of the real-world scheduling problems, including RCPSP, are NP-hard, and hence only problems of small size can be solved to optimality, whereas for larger problems (meta)heuristics are commonly used.Metaheuristics usually search in the space of feasible solutions; with a highly constrained problem such as RCPSP, browsing the space of feasible solutions is hard.Indeed, if a schedule is represented as a vector of job start times, then changing the start time of a single job is likely to cause constraint violations.Usual approach is to use indirect solution representation that could be conveniently handled by the metaheuristic but could also be efficiently translated into the direct representation.Two translation procedures widely used in scheduling are serial schedule generation scheme (SSGS) and parallel schedule generation scheme <cit.>.Some studies conclude that SSGS gives better performance <cit.>, while others suggest to employ both procedures within a metaheuristic <cit.>.Our research focuses on SSGS. With SSGS, the indirect solution representation is a permutation π of jobs.The metaheuristic handles candidate solutions in indirect (permutation) form.Every time a candidate solution needs to be evaluated, SSGS is executed to translate the solution into a schedule (t_j, j ∈ J), and only then the objective value can be computed, see Figure <ref>.SSGS is a simple procedure that iterates through J in the order given by π, and schedules one job at a time, choosing the earliest feasible slot for each job.The only requirement for π is to respect the precedence relations; otherwise SSGS produces a feasible schedule for any permutation of jobs.The pseudo-code of SSGS is given in Algorithm <ref>, and its two subroutines 𝑓𝑖𝑛𝑑 and 𝑢𝑝𝑑𝑎𝑡𝑒 in Algorithms <ref> and <ref>.Commonly, the objective of RCPSP is to find a schedule that minimises the makespan, i.e. the time required to complete all jobs; however other objective functions are also considered in the literature.We say that an objective function of a scheduling problem is regular if advancing the start time of a job cannot worsen the solution's objective value.Typical objective functions of RCPSP, including makespan, are regular.If the scheduling problem has a regular objective function, then SSGS guarantees to produce active solutions, i.e. solutions that cannot be improved by changing t_j for a single j ∈ J.Moreover, it was shown <cit.> that for any active schedule S there exists a permutation π for which SSGS will generate S.Since any optimal solution S is active, searching in the space of feasible permutations π is sufficient to solve the problem.This is an important property of SSGS; the parallel schedule generation scheme, mentioned above, does not provide this guarantee <cit.> and, hence, may not in some circumstances allow a metaheuristic finding optimal or near-optimal solutions.The runtime of a metaheuristic is divided between its search control mechanism that modifies solutions and makes decisions such as accepting or rejecting new solutions, and SSGS.While SSGS is a polynomial algorithm, in practice it eats up the majority of the metaheuristic runtime (over 98% as reported in <cit.>).In other words, by improving the speed of SSGS twofold, one will (almost) double the number of iterations a metaheuristic performs within the same time budget, and this increase in the number of iterations is likely to have a major effect on the quality of obtained solutions.In our opinion, not enough attention was paid to SSGS – a crucial component of many scheduling algorithms, and this study is to close this gap.In this paper we discuss approaches to speed up the conventional implementation of SSGS.Main contributions of our paper are: * A detailed description of SSGS including old and new speed-ups (Section <ref>). * New implementation of SSGS employing Bloom filters for quick testing of resource availability (Section <ref>). * A hybrid control mechanism that employs intelligent learning to dynamically select the best performing SSGS implementation (Section <ref>). Empirical evaluation in Section <ref> confirms that both of our implementations of SSGS perform significantly better than the conventional SSGS, and the hybrid control mechanism is capable of correctly choosing the best implementation while generating only negligible overheads.§ SSGS IMPLEMENTATION DETAILS Before we proceed to introducing our main new contributions in Sections <ref> and <ref>, we describe what state-of-the-art implementation of SSGS we use, including some previously unpublished improvements.§.§ Initialisation of A The initialisation of A in line <ref> of Algorithm <ref> iterates through T slots, where T is the upper bound on the makespan.It was noted in <cit.> that instead of initialising A at every execution of SSGS, one can reuse this data structure between the executions. To correctly initialise A, at the end of SSGS we restore A_t, r for each r ∈ R and each slot where some job was scheduled: A_t, r c_r for r ∈ R and t = 1, 2, …, M, where M is the makespan of the solution.Since M ≤ T and usually M ≪ T, this notably improves the performance of SSGS <cit.>. §.§ Efficient search of the earliest feasible slot for a job The function 𝑓𝑖𝑛𝑑(j, t^0, I, A)finds the earliest slot feasible for scheduling job j. Its conventional implementation (Algorithm <ref>) takes O(T |R|) time, where T is the upper bound of the time horizon.Our enhanced implementation of 𝑓𝑖𝑛𝑑 (Algorithm <ref>), first proposed in <cit.>, has the same worst case complexity but is more efficient in practice.It is inspired by the Knuth-Morris-Pratt substring search algorithm.Let t_j be the assumed starting time of job J.To verify if it is feasible, we need to test sufficiency of resources in slots t_j, t_j + 1, …, t_j + d_j - 1.Unlike the conventional implementation, our enhanced version tests these slots in the reversed order.The order makes no difference if the slot is feasible, but otherwise testing in reversed order allows us to skip some slots; in particular, if slot t is found to have insufficient resources then we know that feasible t_j is at least t + 1.A further speed up, which was not discussed in the literature before, is to avoid re-testing of slots with sufficient resources.Consider the point when we find that the resources in slot t are insufficient.By that time we know that the resources in t + 1, t + 2, …, t_j + d_j - 1 are sufficient.Our heuristic is to remember that the earliest slot t^test to be tested in future iterations is t_j + d_j. cell base/.style=minimum width=1cm, minimum height=1cm, draw=none, transform shape, unknown/.style=cell base, fill=white!90!black, label=[font=]center:?, available/.style=cell base, fill=none, label=[font=]center:+, unavailable/.style=cell base, fill=none, label=[font=]center:-, known available/.style=cell base, pattern=north east lines, pattern color=gray,label=[font=]center:+§.§ Preprocessing and Automated Parameter Control We observe that in many applications, jobs are likely to require only a subset of resources.For example, in construction works, to dig a hole one does not need cranes or electricians, hence the `dig a hole' job will not consume those resources.To exploit this observation, we pre-compute vector R_j of resources used by job j, and then iterate only through resources in R_j when testing resource sufficiency in 𝑓𝑖𝑛𝑑 (Algorithm <ref>, line <ref>) and updating resource availability in 𝑢𝑝𝑑𝑎𝑡𝑒 (Algorithm <ref>, line <ref>).Despite the simplicity of this idea, we are not aware of anyone using or mentioning it before.We further observe that some jobs may consume only one resource.By creating specialised implementations of 𝑓𝑖𝑛𝑑 and 𝑢𝑝𝑑𝑎𝑡𝑒, we can reduce the depth of nested loops.While this makes no difference from the theoretical point of view, in practice this leads to considerable improvement of performance.Correct implementations of 𝑓𝑖𝑛𝑑 and 𝑢𝑝𝑑𝑎𝑡𝑒 are identified during preprocessing and do not cause overheads during executions of SSGS.Having individual vectors R_j of consumed resources for each job, we can also intelligently learn the order in which resource availability is tested (Algorithm <ref>, line <ref>).By doing this, we are aiming at minimising the expected number of iterations within the resource availability test.For example, if resource r is scarce and job j requires significant amount of r, then we are likely to place r at the beginning of R_j.More formally, we sort R_j in descending order of probability that the resource is found to be insufficient during the search.This probability is obtained empirically by a special implementation of SSGS which we call . is used once to count how many times each resource turned out to be insufficient during scheduling of a job.(To avoid bias,tests every resource in R_j even if the test could be terminated early.)After a single execution of , vectors R_j are optimised, and in further executions default implementation of SSGS is used.One may notice that the data collected in the first execution of SSGS may get outdated after some time; this problem in addressed in Section <ref>. Ordering of R_j is likely to be particularly effective on instances with asymmetric use of resources, i.e. on real instances.Nevertheless, we observed improvement of runtime even on pseudo-random instances as reported in Section <ref>.§ SSGS IMPLEMENTATION USING BLOOM FILTERSPerformance bottleneck of an algorithm is usually its innermost loop.Observe that the innermost loop of the 𝑓𝑖𝑛𝑑 function is the test of resource sufficiency in a slot, see Algorithm <ref>, line <ref>.In this section we try to reduce average runtime of this test from O(|R|) to O(1) time.For this, we propose a novel way of using a data structure known as Bloom filter. Bloom filters were introduced in <cit.> as a way of optimising dictionary lookups, and found many applications in computer science and electronic system engineering <cit.>. Bloom filters usually utilise pseudo-random hash functions to encode data, but in some applications <cit.> non-hash-based approaches are used.In our paper, we also use a non-hash-based approach, and to our knowledge, our paper is the first in which the structure of Bloom filters is chosen dynamically according to the statistical properties of the data, with the purpose of improving the speed of an optimisation algorithm.In general, Bloom filters can be defined as a way of using data, and they are characterised by two aspects: first, all data is represented by short binary arrays of a fixed length (perhaps with a loss of accuracy); second, the process of querying data involves only bitwise comparison of binary arrays (which makes querying data very fast). We represent both each job's resource consumption and resource availability at each time slot, by binary arrays of a fixed length; we call these binary arrays Bloom filters.Our Bloom filters will consist of bits which we call resource level bits.Each resource bit, denoted by u_r,k, r ∈ R, k ∈{ 1, 2, …, c_r }, means “k units of resource r” (see details below). Let U be the set of all possible resource bits.A Bloom filter structure isan ordered subset L ⊆ U, see Figure <ref> for an example.Suppose that a certain Bloom filter structure L is fixed.Then we can introduce B^L(j), the Bloom filter of job j, and B^L(t), the Bloom filter of time slot t, for each j and t.Each B^L(j) and B^L(t) consists of |L| bits defined as follows: if u_r,k is the ith element of L then B^L(j)_i = 1ifv_j,r≥ k, 0otherwise,andB^L(t)_i = 1ifA_t,r≥ k, 0otherwise. bf bit/.style=minimum width=1cm, minimum height=0.5cm, draw=black, transform shape, font=, brace/.style=decorate,decoration=brace,amplitude=5pt,raise=1pt To query if a job j can be scheduled in a time slot t, we compare Bloom filters B^L(j) and B^L(t) bitwise; then one of three situations is possible, as the following examples show.Consider a job j and three slots, t, t' and t”, with the following resource consumption/availabilities, and Bloom filter structure as in Figure <ref>:v_j,1= 3 v_j,2= 2 v_j,3= 0 B^L(j)= (110 10 000)A_t,1= 2 A_t,2= 3 A_t,3= 4 B^L(t)= (100 11 111)A_t',1= 3 A_t',2= 1 A_t',3= 4 B^L(t')= (110 10 111)A_t”,1= 3 A_t”,2= 2 A_t”,3= 2 B^L(t”)= (110 10 100)For two bit arrays of the same length, let notation `≤' mean bitwise less or equal. By observing that B^L(j) ≰B^L(t), we conclude that resources in slot t are insufficient for j; this conclusion is guaranteed to be correct.By observing that B^L(j) ≤ B^L(t'), we conclude tentatively that resources in slot t' may be sufficient for j; however, further verification of the complete data related to j and t' (that is, v_j,· and A_t',·) is required to get a precise answer; one can see that v_j,2≥ A_t',2, hence this is what is called a false positive.Finally, we observe that B^L(j) ≤ B^L(t”), and a further test (comparing v_j,· and A_t”,·) confirms that resources in t” are indeed sufficient for j. Values of B^L(j), j ∈ J, are pre-computed when L is constructed, and B^L(t), t = 1, 2, …, T, are maintained by the algorithm.§.§ Optimisation of Bloom Filter Structure The length |L| of a Bloom filter is limited to reduce space requirements and, more importantly for our application, speed up Bloom filter tests. Note that if |L| is small (such as 32 or 64 bits) then we can exploit efficient bitwise operators implemented by all modern CPUs; then each Bloom filter test takes only one CPU operation.We set |L| = 32 in our implementation.While obeying this constraint, we aim at minimising the number of false positives, because false positives slow down the implementation. Our L building algorithm is as follows: * Start with L=U, such as in Figure <ref>. * If |L| is within the prescribed limit, stop. * Otherwise select u_r,k∈ L that is least important and delete it. Go to step <ref>. By `least important' we mean that the deletion of it is expected to have minimal impact of the expected number of false positives.Let L = (…, u_r,q, u_r,k, u_r, m, …).Consider a job j such that k ≤ v_j,r < m and a slot t such that q ≤ A_t,r < k.With L as defined above, Bloom filters correctly identify that resources in slot t are insufficient for job j: B^L(j) ≰B^L(t).However, without the resource level bit u_r,k we get a false positive: B^L(j) ≤ B^L(t).Thus, the probability of false positives caused by deleting u_r,k from L in is as follows: ( ∑_k = i^m - 1 D^r_k ) ·( ∑_k = q^i - 1 E^r_k ),where D^r_k is the probability that a randomly chosen job needs exactly k units of resource r, and E^r_k is the probability that a certain slot, when we examine it for scheduling a job, has exactly k units of resource r available.The probability distribution D^r is produced from the RCPSP instance data during pre-processing.[In multi-mode extension of RCPSP, this distribution depends on selected modes and hence needs to be obtained empirically, similarly to how we obtain E^r.]The probability distribution E^r is obtained empirically during the run of(see Section <ref>); each time resource sufficiency is tested within , its availability is recorded.§.§ Additional Speed-ups While positive result of a Bloom filter test generally requires further verification using full data, in some circumstances its correctness can be guaranteed.In particular, if for some r ∈ R and j ∈ J we have u_r,k∈ L and v_j,r = k, then the Bloom filter result, whether positive or negative, does not require verification.Another observation is that updating B^L(t) in 𝑢𝑝𝑑𝑎𝑡𝑒 can be done in O(|R_j|) operations instead of O(|R|) operations.Indeed, instead of computing B^L(t) from scratch, we can exploit our structure of Bloom filters.We update each bit related to resources r ∈ R_j, but we keep intact other bits.With some trivial pre-processing, this requires only O(|R_j|) CPU operations.We also note that if |R_j| = 1, i.e. job j uses only one resource, then Bloom filters will not speed up the 𝑓𝑖𝑛𝑑 function for that job and, hence, in such cases we use the standard 𝑓𝑖𝑛𝑑 function specialised for one resource (see Section <ref>). Performance bottleneck of an algorithm is usually its innermost loop.Observe that the innermost loop of the 𝑓𝑖𝑛𝑑 function is the test of resource sufficiency in a slot, see Algorithm <ref>, line <ref>.In this section we try to reduce expected runtime of this loop from O(|R|) to O(1) time.We propose a novel way of using a data structure known as a Bloom filter; we shall start with explaining how Bloom filters can be applied to the problem we study, and provide a general definition of Bloom filters later in this subsection. We represent both each job's resource consumption and the availability of resources at each time slot by binary arrays of a fixed length; we call these binary arrays Bloom filters. Let us start with an example. Suppose we have three resources, each with capacity 10. We can represent a job's resource consumption as a binary array of length 6 whose bits contain answers to the following questions: * Does the job need at least 1 unit of resource 1?* Does the job need at least 6 units of resource 1?* Does the job need at least 1 unit of resource 2?* Does the job need at least 6 units of resource 2?* Does the job need at least 1 unit of resource 3?* Does the job need at least 6 units of resource 3? (As you are looking at this simple example, the choice of precisely two levels 1 and 6 for each of three resources seems arbitrary, and in this example, actually, it is arbitrary. Further in this section we propose how to make an optimal choice of the number and values of levels of each resource individually when designing the structure of Bloom filters representing jobs and slots.)On the other hand, for each time slot we can represent availability of resources during that slot as an array of the same length, that is: * Is at least 1 unit of resource 1 available?* Are at least 6 units of resource 1 available?* Is at least 1 unit of resource 2 available?* Are at least 6 units of resource 2 available?* Is at least 1 unit of resource 3 available?* Are at least 6 units of resource 3 available?For each job and each slot, checking whether the job's resource consumption can be satisfied in this time slot is done simply by comparing these two short binary arrays bitwise; this is faster than comparing several pairs of numbers representing the job's resource consumption and availability of resources. For example, suppose a job j requires 1, 5, 10 units of resources 1, 2, 3, respectively, and in a slot t_1 we have 10, 5, 10 units of resources 1, 2, 3, respectively, available. Then the Bloom filter of j is 101011, and the Bloom filter of t_1 is 111011. Comparing these two arrays bitwise, we observe that the latter one is greater[a bit array a_1 … a_n is greater than a bit array b_1 … b_n if a_i ≥ b_i for each i = 1, …, n.] than the former one; hence, we can tentatively conclude that j can be scheduled in t_1.Now consider a slot t_2 in which we have 10, 5, 5 units of resources 1, 2, 3, respectively, available. The Bloom filter of t_2 is 111010. Comparing the Bloom filter of j with the Bloom filter of t_2, we observe that the latter one is not greater than the former one (due to 0 in the 6th bit of the latter one and 1 in the 6th bit of the former one); thus, we conclude correctly that j cannot be scheduled in t_1.Now consider a slot t_3 in which we have 10, 5, 6 units of resources 1, 2, 3, respectively, available. The Bloom filter of t_3 is 111011. Comparing the Bloom filter of j with the Bloom filter of t_3 we conclude, incorrectly, that j can be scheduled in t_3. Thus, if Bloom filters give us a positive answer (like in the examples with t_1 and t_3), it needs to be checked against the complete data, that is, the job's resource consumption and availability of resources. If Bloom filters frequently give us an incorrect positive answer (called a false positive), this slows down the algorithm, and we want to minimise the number of such errors. In the example above, the range of capacity of each resource was split into three subranges: (0), (1, 2, …, 5) and (6, 7, …, 10). In the rest of this section we discuss how we can split the capacity of each resource into subranges in a way which minimises the probability of false positives produced by Bloom filters, thus making the algorithm faster.In general, Bloom filters can be defined as a way of using data characterised by two aspects: first, all data is represented by short binary arrays of a fixed length (perhaps with a loss of accuracy); second, the process of querying data involves only bitwise comparison of binary arrays (which makes querying data very fast). Bloom filters were introduced in <cit.> as a way of optimising dictionary lookups, and found many applications in computer science and electronic system engineering <cit.>. To our knowledge, our paper is the first in which the structure of Bloom filters is chosen dynamically according to the statistical properties of the data, with the purpose of improving the speed of an optimisation algorithm. §.§ Distributions needed for the algorithm To make the description simpler, we shall first assume that we only have one resource with capacity c. A generalisation to several resources is below in Subsection <ref>.Our method of minimising the number of false positives uses two empirically found probability distributions, D and E. By D we denote the distribution of the probability that a randomly chosen job needs exactly k units of the resource. This distribution can be easily produced in advance from the list of jobs. (In case of a multi-mode extension of RCPSP, D also depends on probability of each mode being selected and can be obtained empirically, similarly to how we obtain distribution E below.)By E we denote the distribution of the probability that when we examine a certain time slot (in order to decide whether a job can be scheduled at this time slot), exactly k units of the resource are available.E is obtained empirically within the 𝑓𝑖𝑛𝑑^data(j, t^0, I, A) functions (see Section <ref>).Every time resource sufficiency is tested within 𝑓𝑖𝑛𝑑^data(j, t^0, I, A), its availability is recorded.§.§ Building the Bloom filterLike in the example above, the whole range of levels of a resource 1, …, c will be split into several subranges 1, …, p_1; p_1+1, …, p_2; … ; p_L-1, …, p_L = c. For each job, its Bloom filter will consist of L bits, and the i-th bit will be 1 if and only if the number of units of the resource required by the job is greater than or equal to p_i. For each slot, its Bloom filter will also consist of L bits, and the i-th bit will be 1 if and only if the number of units of the resource available is greater than or equal to p_i.Denote the k-th entry in distribution D (or E) by d_k (or e_k); that is, d_k is the probability that a randomly chosen job needs exactly k units of the resource, and e_k is the probability that a certain slot, when we examine it to schedule a job, has exactly k units of the resource available. If two levels k and k+1 of a resource are represented by the same bit in Bloom filters (that is, k and k+1 are within the same subrange p_i+1, …, p_i+1), there will be false positives if a certain job requires exactly k+1 units of the resource, but there are only k units of the resource available.Since we want to minimise the probability of false positives, when we allocate resource levels to Bloom filters, we want to minimise the products d_k+1e_k whenever levels of a resource k and k+1 are represented by the same bit of a Bloom filter. More generally, we want to minimise the products d_k+ie_k whenever levels of a resource k and k+i are represented by the same bit of a Bloom filter. In our code we use a fast greedy algorithm which aims to minimise these products, as described below. Consider a dictionary Δ whose index consists of intervals within 1,2,…,c, and whose two entries for each value of the index is the corresponding probability in D and E. To start, we let Δ be as follows: [ 1 2 … c; d_1 d_2 … d_c; e_1 e_2 … e_c ]For each k=1,…,r-1 calculate the product d_k+1e_k; it expresses the probability of false positives if we denote k and k+1 by the same bit of the Bloom filter. Choose a value of k for which this product is minimal. Recalculate Δ by merging the two columns, as follows:[ 1 … (k,k+1) … c; d_1 … d_k+d_k+1 … d_c; e_1 … e_k+e_k+1 … e_c ]Repeat this process until the size of the index is equal to the desired length L of the Bloom filter.If there are several resources, to decide on the structure of the Bloom filter, proceed like above, but instead of considering one pair of distributions D and E, build a dictionary with a longer index, which will correspond to all possible numbers of units of the first resource, then all possible numbers of units of the second resource, etc., with entries reflecting the corresponding two distributions D^r and E^r for each resource r ∈ R. Thus, is we have a set R of resources with capacities c_r, r ∈ R, we build a dictionary whose index consists of all pairs (r, k), where r ∈ R and k ∈ 1, …, c_r, and whose two entries for each value of index are d_k^r and e_k^r. For each r ∈ R and for each k ∈ 1, …, c_r, calculate the product d_k+1^r e_k^r; this is the probability of false positives if levels k and k+1 of resource r are represented by the same bit of a Bloom filter. Choose the minimal one among these products and merge the corresponding two indices and entries of the dictionary. Repeat this process until the size of the index is equal to the desired length L of the Bloom filter.One can be tempted to conjecture that some resources are, in some way, more important than others, and this should be somehow reflected in the algorithm; actually, our algorithm implicitly takes this information into account, because it aims to minimise false positives across all resources, thus comparing each resource against others. § HYBRID CONTROL MECHANISMSo far we have proposed two improved implementations of SSGS: one using Bloom filters (which we denote ), and the other one not using Bloom filters (which we denote ).While it may look likeshould always be superior to , in practiceis often faster.Indeed, Bloom filters usually speed up the 𝑓𝑖𝑛𝑑 function, but they also slow down 𝑢𝑝𝑑𝑎𝑡𝑒, as inwe need to update not only the values A_t,r but also the Bloom filters B^L(t) encoding resource availability.If, for example, the RCPSP instance has tight precedence relations and loose resource constraints then 𝑓𝑖𝑛𝑑 may take only a few iterations, and then the gain in the speed of 𝑓𝑖𝑛𝑑 may be less than the loss of speed of 𝑢𝑝𝑑𝑎𝑡𝑒.In such casesis likely to be slower then . In short, either of the two SSGS implementations can be superior in certain circumstances, and which one is faster mostly depends on the problem instance.In this section we discuss how to adaptively select the best SSGS implementation.Automated algorithm selection is commonly used in areas such as sorting, where multiple algorithms exist.A typical approach is then to extract easy to compute input data features and then apply off-line learning to develop a predictor of which algorithm is likely to perform best, see e.g. <cit.>.With sorting, this seems to be the most appropriate approach as the input data may vary significantly between executions of the algorithm.Our case is different in that the most crucial input data (the RCPSP instance) does not change between executions of SSGS.Thus, during the first few executions of SSGS, we can test how each implementation performs, and then select the faster one.This is a simple yet effective control mechanism which we call . cell base/.style=minimum width=1cm, minimum height=1cm, draw=black, transform shape, BF/.style=cell base, pattern=north east lines, pattern color=white!60!black, label=[font=]center:BF, NBF/.style=cell base, pattern=north west lines, pattern color=white!80!black, label=[font=]center:NBF, data/.style=cell base, pattern=grid, pattern color=white!80!black, label=[font=]center:data is entirely transparent for the metaheuristic; the metaheuristic simply calls SSGS whenever it needs to evaluate a candidate solution and/or generate a schedule.Thecontrol mechanism is then intelligently deciding each time which implementation of SSGS to use based on information learnt during previous runs.An example of howperforms is illustrated in Figure <ref>.In the first execution, it usesto collect data required for bothand .For the next few executions, it alternates betweenand , measuring the time each of them takes.During this stage,counts how many timeswas faster than the next execution of .Then we use the sign test <cit.> to compare the implementations.If the difference is significant (we use a 5% significance level for the sign test) then we stop alternating the implementations and in future use only the faster one.Otherwise we continue alternating the implementations, but for at most 100 executions.(Without such a limitation, there is a danger that the alternation will never stop – if the implementations perform similarly; since there are overheads associated with the alternation and time measurement, it is better to pick one of the implementations and use it in future executions.)Our decision to use the sign test is based on two considerations: first, it is very fast, and second, it works for distributions which are not normal.This makes our approach different from <cit.> where the distributions of runtimes are assumed to be normal.(Note that in our experiments we observed that the distribution of running times of an SSGS implementation resembles Poisson distribution.) As pointed out in this and previous sections, optimal choices of parameters of the SSGS implementations mostly depend on the RCPSP instance – which does not change throughout the metaheuristic run; however solution π also affects the performance.It should be noted though that metaheuristics usually apply only small changes to the solution at each iteration, and hence solution properties tend to change relatively slowly over time.Consequently, we assume that parameters chosen in one execution of SSGS are likely to remain efficient for some further executions.Thus,`restarts' every 10,000 executions, by which we mean that all the internal data collected by SSGS is erased, and learning starts from scratch, see Figure <ref>.This periodicity of restarts is a compromise between accuracy of choices and overheads, and it was shown to be practical in our experiments. § EMPIRICAL EVALUATION In this Section we evaluate the implementations of SSGS discussed above.To replicate conditions within a metaheuristic, we designed a simplified version of Simulated Annealing.In each iteration of our metaheuristic, current solution is modified by moving a randomly selected job into a new randomly selected position (within the feasible range).If the new solution is not worse than the previous one, it is accepted.Otherwise the new solution is accepted with 50% probability.Our metaheuristic performs 1,000,000 iterations before terminating.We evaluate ,and .These implementations are compared to `conventional' SSGS, denoted by , which does not employ any preprocessing or Bloom filters and uses conventional implementation of 𝑓𝑖𝑛𝑑 (Algorithm <ref>). We found that instances in the standard RCPSP benchmark set PSPLIB <cit.> occupy a relatively small area of the feature space.For example, all the RCPSP instances in PSPLIB have exactly four resources, and the maximum job duration is always set to 10.Thus, we chose to use the PSPLIB instance generator, but with a wider range of settings.Note that for this study, we modified the PSPLIB instance generator by allowing jobs not to have any precedence relations.This was necessary to extend the range of network complexity parameter (to include instances with scarce precedence relations), and to speed up the generator, as the original implementation would not allow us to generate large instances within reasonable time. All the experiments are conducted on a Windows Server 2012 machine based on Intel Xeon E5-2690 v4 2.60 GHz CPU.No concurrency is employed in any of the implementations or tests.To see the effect of various instance features on SSGS performance, we select one feature at a time and plot average SSGS performance against the values of that feature.The rest of the features (or generator parameters) are then set as follows: number of jobs 120, number of resources 4, maximum duration of job 10, network complexity 1, resource factor 0.75, and resource strength 0.1.These values correspond to some typical settings used in PSPLIB.For formal definitions of the parameters we refer to <cit.>.For each combination of the instance generator settings, we produce 50 instances using different random generator seed values, and in each of our experiments the metaheuristic solves each instance once.Then the runtime of SSGS is said to be the overall time spent on solving those 50 instances, over 50,000,000 (which is the number of SSGS executions).The metaheuristic overheads are relatively small and are ignored. Hybrid line/.append style=line width=5pt, black, opacity=0.3 BF line/.append style=black, mark=x, mark size=3pt Non-BF line/.append style=thick, black, mark=*From the results reported in Figure <ref> one can see that our implementations of SSGS are generally significantly faster than , but performance of each implementation varies with the instance features.In some regions of the instance spaceoutperforms , whereas in other regionsoutperforms .The difference in running times is significant, up to a factor of two in our experiments.At the same time,is always close to the best ofand , which shows efficiency of our algorithm selection approach.In fact, whenandperform similarly,sometimes outperforms both; this behaviour is discussed below. Another observation is thatis always faster than(always below the 100% mark) which is not surprising; indeed,improves the performance of both 𝑓𝑖𝑛𝑑 and 𝑢𝑝𝑑𝑎𝑡𝑒.In contrast,is sometimes slower than ; on some instances, the speed-up of the 𝑓𝑖𝑛𝑑 function is overweighed by overheads in both 𝑓𝑖𝑛𝑑 and 𝑢𝑝𝑑𝑎𝑡𝑒.Most important though is thatoutperformsin each of our experiments by 8 to 68%, averaging at 43%.In other words, within a fixed time budget, an RCPSP metaheuristic employingwill be able to run around 1.8 times more iterations than if it used . To verify thatexhibits the adaptive behaviour and does not just stick to whichever implementation has been chosen initially, we recorded the implementation it used in every execution for several problems, see Figure <ref>. For this experiment, we produced three instances: first instance has standard parameters except Resource Strength is 0.2; second instance has standard parameters except Resource Factor is 0.45; third instance has standard parameters except Maximum Job Duration is 20.These parameter values were selected such that the two SSGS implementations would be competitive and, therefore, switching between them would be a reasonable strategy. One can see that the switches occur several times throughout the run of the metaheuristic, indicating thatadapts to the changes of solution.For comparison, we disabled the adaptiveness and measured the performance if only implementation chosen initially is used throughout all iterations; the results are shown on Figure <ref>.We conclude thatbenefits from its adaptiveness.§ CONCLUSIONS AND FUTURE WORKIn this paper we discussed the crucial component of many scheduling metaheuristics, the serial schedule generation scheme (SSGS).SSGS eats up most of the runtime in a typical scheduling metaheuristic, therefore performance of SSGS is critical to the performance of the entire metaheuristic, and thus each speed-up of SSGS has significant impact.We described existing and some new speed-ups to SSGS, including preprocessing and automated parameter control.This implementation clearly outperformed the `conventional' SSGS in our experiments.We further proposed a new implementation that uses Bloom filters, particularly efficient in certain regions of the instance space.To exploit strengths of both implementations, we proposed a hybrid control mechanism that learns the performance of each implementation and then adaptively chooses the SSGS version that is best for a particular problem instance and phase of the search.Experiments showed that this online algorithm selection mechanism is effective and makesour clear choice.Note thatis entirely transparent for the metaheuristic which uses it as if it would be simple SSGS; all the learning and adaptation is hidden inside the implementation. The idea behind online algorithm selection used in this project can be further developed by making the number of executions between restarts adaptable.To determine the point when the established relation between the SSGS implementations may have got outdated, we could treat the dynamics of the implementations' performance change as two random walks, and use the properties of these two random walks to predict when they may intersect.All the implementations discussed in the paper, in C++, are available for downloading from <http://csee.essex.ac.uk/staff/dkarap/rcpsp-ssgs.zip>.The implementations are transparent and straightforward to use.While we have only discussed SSGS for the simple RCPSP, our ideas can easily be applied in RCPSP extensions.We expect some of these ideas to work particularly well in multi-project RCPSP, where the overall number of resources is typically large but only a few of them are used by each job. Acknowledgements We would like to thank Prof. Rainer Kolisch for providing us with a C++ implementation of the PSPLIB generator.It should be noted that, although the C++ implementation was developed to reproduce the original Pascal implementation, the exact equivalence cannot be guaranteed; also, in our experiments we used a modification of the provided C++ code. plain | http://arxiv.org/abs/1708.07786v1 | {
"authors": [
"Daniel Karapetyan",
"Alexei Vernitski"
],
"categories": [
"cs.DS",
"cs.PF"
],
"primary_category": "cs.DS",
"published": "20170825154657",
"title": "Efficient Adaptive Implementation of the Serial Schedule Generation Scheme using Preprocessing and Bloom Filters"
} |
Brasselet number and Newton polygons]Brasselet number and Newton polygonsDepartment of Mathematics, Universidade Federal de São Carlos (UFSCar), Brazil [email protected], [email protected] Department of Mathematics, Universidade Federal de São Carlos (UFSCar), Brazil [email protected] http://www.dm.ufscar.br/profs/hartmann Luiz Hartmann is Partially support by FAPESP: 2016/16949-8 and bothauthors are partially supported by CAPES/PVE:88881.068165/2014-01 [2010]Primary: 14M25,55S35; Secondary: 14B05, 32S05; 58K45 We present a formula to compute the Brasselet number of f:(Y,0)→ (, 0) where Y⊂ X is a non-degenerate complete intersection in a toric variety X. As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when (X,0) = (ℂ^n,0) we derive sufficient conditions to obtain the invariance of the Euler obstruction for families of complete intersections with an isolated singularity which are contained in X. [ Luiz Hartmann December 30, 2023 ===================== § INTRODUCTION Given a germ of an analytic functionf:(ℂ^n,0) → (ℂ,0) with an isolated critical point attheorigin, an important invariant of this germ is its Milnor number <cit.>,denoted by μ(f). The Milnor number is considered as a central invariant,sinceit provides algebraic, topological and geometric information about the germf. For instance, the Milnor number coincides with the number of Morse pointsof a morsefication of f.Initially the Milnor number was associated to germs of analytic functions f: (ℂ^n, 0) → (ℂ, 0) with an isolated critical point, andconsequently it was used to study isolated hypersurface singularities. However this invariant is well defined in many others contexts, for example curves <cit.>, isolated complete intersection singularities (ICIS) <cit.>,and determinantal varieties of codimension two <cit.>, to name just a few.The local Euler obstruction was defined by MacPherson in <cit.> forthe construction of characteristic classes of singular complex algebraic varieties.Thereafter, it has been deeply investigated by many authors such as Brasselet and Schwartz <cit.>,Dutertre <cit.>, Gaffney, Grulha and Ruas <cit.>, Gonzalez-Sprinberg<cit.>, Lê and Teissier <cit.>,Matsui and Takeuchi <cit.>, among others. We denote by (X,0) a germ of an analytic singular space embedded inℂ^n and by f:(X,0) → (ℂ,0) a germ of an analyticfunctionwith an isolated critical point at the origin. Brasselet,Massey, Parameswaran and Seade <cit.> introduced an invariant associatedto fcalled the Euler obstruction of fand denoted by Eu_f,X(0). Roughly speaking, Eu_f,X(0) is the obstruction toextending a lifting of the conjugate of the gradient vector field of f as asection of the Nash bundle of (X , 0). This invariant is closed relatedwith the local Euler obstruction of X, what explains its name.An important consequence of the definition given by MacPherson, is that the local Euler obstruction is a constructible function, which means that, it is constant along the strata of a Whitney stratificationof X. This is essentially a consequence of the topological triviality of X on the Whitney strata. As a consequence, the local Eulerobstructiondoes not depend on the Whitney stratification of X. The followingLefschetz-type formula was proved byBrasselet, Lê and Seade <cit.>. Let (X,0) ⊂ (ℂ^n,0) be an equidimensional complex analytic singularity germ with a Whitney stratification {V_i}, then given a generic linear form L, there exists ε_0 such that for any ε with 0<ε< ε_0, we have Eu_X(0)=∑_iχ(V_i∩ B_ε∩ L^-1(δ) ) · Eu_X(V_i), where χ is the Euler-Poincaré characteristic, B_ε:=B_ε(0) is the ball with center the origin and radius ε, Eu_X(V_i) is the value of the local Euler obstruction of X at any point of the stratum V_i,and 0 < |δ|≪ε≪ 1. The previous theorem says that the local Euler obstruction, as aconstructiblefunction on X, satisfies the Euler condition relatively to a generic linear function.For the Euler obstruction of an analytic function f:(X,0) →(ℂ,0) with an isolated critical point at the origin, there is alsoaLefschetz-type formula. This formula was proved in <cit.>. The purposeof the authors was to understand what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the origin. Let (X,0) ⊂ (ℂ^n,0) be an equidimensional complex analytic singularity germ with a Whitney stratification {V_i}, and let f : (X,0)→ (ℂ,0) be a function with an isolated singularity at 0. Then, Eu_f,X(0)= Eu_X(0)-(∑_iχ(V_i∩ B_ε∩ f^-1(δ) ) · Eu_X(V_i) ), where 0 < |δ|≪ε≪ 1.The last equation presents the relation between the localEulerobstruction of X andthe Euler obstruction of f.Seade, Tibăr and Verjovsky continued the study of the properties of Eu_f,X(0) in <cit.>. The authors proved that the Euler obstruction of f is closely related to the number of Morse points of a morsefication of f, asfollows. Let (X,0) be an equidimensional complex analytic singularity germ of dimension d and f: (X,0) → (ℂ,0) a germ of an analytic function with an isolated critical point at the origin. Then Eu_f,X(0) = (-1)^d n_ reg, where n_ reg is the number of Morse points in the regular part of X appearing in a stratified morsefication of f.Therefore, the Euler obstruction of f is the number of Morse pointsof a morsefication of f on the regular part of X, up to the sign. Hencethis invariant can be seen as a generalization of the Milnor number of f.Another invariant associated with a germ of an analytic function f:(X,0) → (ℂ,0) is the Brasselet number introduced by Dutertreand Grulha in <cit.>. We will denote this number by B_f,X(0). Iffhas an isolated critical point, then the Brasselet number satisfies theequality B_f,X(0) =Eu_X(0) -Eu_f,X(0).If f is linear and generic, then B_f,X(0)=Eu_X(0), hence it can be viewed as ageneralization of the local Euler obstruction. Moreover,even if f has a non-isolated singularity it provides interesting results.For example, the Brasselet number has a Lê-Greuel type formula(see <cit.> or Theorem <ref> below), the difference of the Brasselet numbers B_f,X(0) and B_f,X^g(0) is measure by the number of Morse critical points on the topstratum of the Milnor fiber of f appearing in a morsefication of g, whereg: (X,0) → (ℂ,0) is a prepolar function (see Definition<ref>) and X^g=X ∩ g^-1(0).Although they are important, the invariants mentioned above are not easilycomputed using their definition. In the literature there are formulas whichmake the computation easier, see <cit.>. Some authors workedonmore specific situations. In the special case of toric surfaces, an interesting formula for the local Euler obstruction was proved byGonzalez-Sprinberg <cit.>. This formula was generalized by Matsuiand Takeuchi <cit.> for normal toric varieties of any dimension. Toric varieties are particularly interesting objects, since they have astrong relation with elementary convex geometry. On these varieties we havean action of the algebraic torus (ℂ^*)^n that induces a finitedecomposition of the variety into orbits, all of which are homeomorphic to a torus.In <cit.>, Varchenko described the topology of the Milnor fiber of afunction f:(ℂ^n,0) → (ℂ,0) using the geometry of theNewton polygon of f, and consequently, the Milnor number can be expressedby volumes of polytopes related to the Newton polygon of f. In his proof,he constructed a toric modification of ℂ^n on which the pull-backof f defines a hypersurface with only normal crossing singularities. Whileℂ^n is a very special smooth toric variety, it seems natural togeneralize his formula to Milnor fibers over general singular toricvarieties. This was done by Matsui and Takeuchi in <cit.>.We use <cit.> to establish several combinatorial formulas for thecomputation of the Brasselet number of f:(Y,0)→ (, 0) where Y⊂ Xis a non-degenerate complete intersection in a toric variety X. Theseformulas will be given in terms of volumes of Newton polygons associated tof.This paper is organized as follows. In Section 2, we present some backgroundmaterial concerning the Brasselet number and toric varieties, which will be used in theentire work. In Section 3, we compute the Brasselet number of a polynomialfunction f : (X, 0) → (ℂ, 0), where X ⊂ℂ^n is atoric variety. Moreover, we compute this invariant for functions defined onX^g = X∩ g^-1(0), where g: X →ℂ^k is a non-degeneratecomplete intersection. As a consequence, assuming that g has an isolatedcritical point on X and on X^f, we also obtain a formula for the number ofstratified Morse critical points on the top stratum of the Milnor fiber of fappearing in a morsefication of g: X ∩ f^-1(δ) ∩B_ε→ℂ. As applications we establish severalresults concerning constancy of these invariants. In Section 4, weconsider the case where (X,0) = (ℂ^n,0) and we derive sufficientconditions to obtain the constancy of the Euler obstruction for families ofcomplete intersections with an isolated singularity which are contained on X.Weuse this result to study the invariance of the Bruce-Roberts' Milnor numberforfamilies of functions defined on hypersurfaces. In Section 5, we work in thecase of surfaces,in the case where X is a 2-dimensional toric variety. In this situation, we present acharacterization of a polynomial function g: X →ℂ whichhas a stratified isolated singularity at the origin. We use this characterization to present some examples for a class of toric surfacethat is also determinantal. In Section <ref>, westudy the GSV-index on non-degenerated complete intersection in toric varieties.§ GENERALITIES: STRATIFICATIONS, BRASSELET NUMBER AND TORIC VARIETIESFor the convenience of the reader and to fix the notation we present somegeneral factsin order to establish our results.§.§ Stratifications and Brasselet numberIn order to introduce the definition and the properties of the Brasseletnumber,we need some notions about stratifications. For moredetails, we refer to Massey <cit.>. Let A ⊂^n,B⊂^m and f: A →B be a function with n, m∈. We willfix the notation, A^f:=A∩ f^-1(0). Consider X⊂^n areduced complex analytic set of dimension d which is included in an openset U. Let F : U →ℂ be a holomorphic function and f : X →ℂ be the restrictionof F to X, f:=F|_X. A good stratification of X relative to f is a stratification 𝒱 of X which is adapted to X^f, such that {V_i ∈𝒱; V_i ⊄X^f} is a Whitney stratification of X ∖ X^f, and for any pair of strata (V_α,V_β) such that V_α⊄X^f and V_β⊂ X^f, the (a_f)-Thom condition is satisfied. We call the strata included in X^f the good strata. By <cit.>, given a stratification 𝒮 of X one can refine𝒮 to obtain a Whitney stratification 𝒱 of X whichis adapted to X^f. By <cit.> the refinement𝒱 satisfies the(a_f)-Thom condition, goodstratifications always exist. Consider X and f as before. Let 𝒱={V_i} be a stratificationof X. The critical locus of frelative to 𝒱, denoted by Σ_𝒱 f, is the unionof thecritical locus of f restricted to each of the strata,Σ_𝒱 f = ⋃_iΣ(f|_V_i).A critical point of f relative to 𝒱 is a point p ∈Σ_𝒱 f. If the stratification 𝒱 is clear, we will refer to the elements of Σ_𝒱 f simply as stratifiedcritical points of f. If p is an isolated point of Σ_𝒱f, we call p astratified isolated critical point of f (with respect to 𝒱). If 𝒱 is a Whitney stratification ofX and f : X →ℂ has a stratified isolated critical point at the origin, then {V_α∖ X^f,V_α∩ X^f∖{0 }, { 0};V_α∈𝒱},is a good stratification for f. We call it the goodstratification induced by f. Suppose that X is equidimensional. Let 𝒱 = {V_i }_i=0^q be a good stratification of X relative to f. The Brasselet number is defined by B_f,X(0) := ∑_i =1^qχ(V_i∩ B_ε(0) ∩ f^-1(δ) ) · Eu_X(V_i), where 0< | δ| ≪ε≪ 1. If f has a stratified isolated critical point at the origin and Xis equidimensional, Theorem <ref> implies thatB_f,X(0) =Eu_X(0) -Eu_f,X(0).The Brasselet number has many interesting properties,it satisfies several multiplicity formulas, which enable the authors to establish in <cit.> a relative version of the local index formula and a Gauss-Bonnet formula for B_f,X(0). However, oneof the most important propertiesof this invariant is the Lê-Greuel type formula (Theorem <ref>). To present this result we needto impose some conditions on the functions to ensure that X^g meetsX^f in a nice way. So it is necessary to define: Let 𝒱 be a good stratification of X relative to f. We say that g : (X,0) → (ℂ, 0) is prepolar with respect to 𝒱 at the origin if the origin is an isolated critical point of g. The condition that g is prepolar means that g has an isolated criticalpoint (in the stratified sense), both on X and on X^f, and that X^gintersects transversely each stratum of 𝒱 in a neighborhood ofthe origin, except perhaps at the origin itself. However, it is important to note that, whileX^g meets X^f in a nice way, X^f may have arbitrarily bad singularities when restricted to X^g. The (a_f)-Thomcondition in Definition<ref> together with the hypothesis that g is prepolar ensurethat g:X ∩ f^-1(δ) ∩ B_ε→ℂ has nocritical points on g^-1(0) <cit.>.Therefore, the number of stratified Morse critical points on the topstratum V_q ∩ f^-1(δ)∩ B_ε(0), in a morseficationof g : X ∩ f^-1(δ) ∩ B_ε(0) →ℂ, does not depend on the morsefication. The next theorem shows that the Brasselet number satisfies a Lê-Greueltype formula <cit.>. Suppose that X is equidimensional and that g is prepolar with respect to 𝒱 at the origin. Then, B_f,X(0) -B_f,X^g(0)= (-1)^d-1n_q, where n_q is the number of stratified Morse critical points on the top stratum V_q ∩ f^-1(δ)∩ B_ε(0) appearing in a morsefication of g : X ∩ f^-1(δ) ∩ B_ε(0) →ℂ, and 0 < |δ| ≪ε≪ 1. In particular, this number is independent on the morsefication. §.§ Toric varietiesThe theory of toric varieties can be seen as a cornerstone for the interaction between combinatorics and algebraic geometry,which relates the combinatorial study of convex polytopes to algebraictorus actions. Moreover, for polynomial functions defined on such varieties,it is possible to obtain a combinatorial description of the topology of theirMilnor fibers in terms of Newton polygons (see <cit.>). The readermay consult <cit.> for an overview about toricvarieties.Let N ≅ℤ^d be a ℤ-lattice of rank d and σa strongly convex rational polyhedral cone in N_ℝ = ℝ⊗_ℤ N. We denote by M the dual lattice of N andthe polar cone σ̌ of σ in M_ℝ = ℝ⊗_ℤ M by σ̌ = {v ∈ M_ℝ; ⟨ u,v⟩≥ 0 for anyu ∈σ},where ⟨· , ·⟩ is the usual inner product in ^d. Then the dimension of σ̌ is d and we obtaina semigroup S_σ:= σ̌∩ M. A d-dimensional affine toric variety X_σ is defined by the spectrum of ℂ[S_σ],X=Spec(ℂ[S_σ]). The algebraic torus T = Spec(ℂ[M])≅(ℂ^*)^d acts naturally on X_σ and the T-orbits inX_σ are indexed by the faces Δ ofσ̌ (Δ≺σ̌). We denote by𝕃(Δ) the smallest linearsubspace of M_ℝ containing Δ. For a face Δ ofσ̌, denote by T_Δ the T-orbit inSpec(ℂ[M ∩𝕃(Δ)]) which corresponds toΔ. We observe that thed-dimensional affine toric varieties are exactly those d-dimensional affine, normal varieties admitting a (ℂ^*)^d-action with an open, dense orbit homeomorphic to (ℂ^*)^d. Moreover, each T-orbit T_Δ is homeomorphic to (ℂ^*)^r, where r is the dimension of 𝕃(Δ). Therefore we obtain a decomposition X_σ = _Δ≺σ̌ T_Δ into T-orbits, which are homeomorphic toalgebraic torus (ℂ^*)^r. Due to this fact, and also theinformations coming from the combinatorial residing in these varieties, manyquestions that were originally studied for functions defined onℂ^dcan be extended to functions defined on toric varieties.Consider f: X_σ→ℂ a polynomial function onX_σ,a function that corresponds to an element f=∑_v∈S_σa_v · v of ℂ[S_σ], where a_v ∈ℂ. Let f=∑_v∈ S_σa_v · v be a polynomial function on X_σ. * (a) The set {v ∈ S_σ; a_v ≠ 0 }⊂ S_σ is called the support of f and we denote it by f; * (b) The Newton polygon Γ_+(f) of f is the convex hull of ⋃_v ∈ f (v+ σ̌) ⊂σ̌. Now let us fix a function f ∈ℂ[S_σ] such that 0 ∉ f,f: X_σ→ℂ vanishes at the T-fixed point0. Considering M(S_σ) the ℤ-sublattice of rank d inM generated by S_σ, we have that each element v of S_σ⊂ M(S_σ) is identified with a ℤ-vectorv=(v_1,…,v_d). Then for g = ∑_v ∈ S_σ b_v · v ∈ℂ[S_σ] we can associate a Laurent polynomial L(g) = ∑_v∈ S_σ b_v · x^v on T = (ℂ^*)^d, wherex^v:=x_1^v_1· x_2^v_2…x_d^v_d. We say that f=∑_v∈ S_σa_v· v∈ℂ[S_σ] is non-degenerate if for any compact face γ ofΓ_+(f) the complex hypersurface x=(x_1,…,x_d) ∈(ℂ^*)^dL(f_γ)(x) = 0in (ℂ^*)^dis smooth and reduced, where f_γ := ∑_v ∈γ∩S_σ a_v · v. We can also study non-degeneracy in case of complete intersections defined onX_σ. Let f_1,f_2, …, f_k ∈ℂ[S_σ] (1 ≤ k≤ d =X_σ) and consider the following subvarieties ofX_σ: V:= {f_1=…=f_k-1 = f_k =0 }⊂ W:={f_1=…=f_k-1 =0 }.Assume that 0 ∈ V. For each faceΔ≺σ̌ such that Γ_+(f_k)∩Δ≠∅, we set I(Δ)=j=1,2,…, k-1Γ_+(f_j) ∩Δ≠∅⊂{1,2,…,k-1 } and m(Δ)= # I(Δ) + 1.Let 𝕃(Δ) and M(S_σ∩Δ) be as before and𝕃(Δ)^* the dual vector space of 𝕃(Δ). ThenM(S_σ∩Δ)^* is naturally identified with a subset of𝕃(Δ)^* and the polar cone Δ̌=u∈𝕃(Δ)^*⟨ u, v⟩≥ 0 v∈Δ of Δ in𝕃(Δ)^* is a rational polyhedral convex cone with respect tothe lattice M(S_σ∩Δ)^* in 𝕃(Δ)^*. (i) For a function f = ∑_v∈Γ_+(f) a_v · v ∈ℂ[S_σ] on X_σ and u ∈Δ̌, we set f|_Δ = ∑_v∈Γ_+(f) ∩Δ a_v · v ∈ℂ[S_σ∩Δ] and Γ(f|_Δ;u) = v ∈Γ_+(f)∩Δ⟨ u, v⟩ =min⟨ u,w⟩, forw ∈Γ_+(f) ∩Δ. We call Γ(f|_Δ;u) the supporting face of u in Γ_+(f) ∩Δ. (ii) For j ∈ I(Δ) ∪{k } and u ∈Δ̌, we define the u-part f_j^u∈ℂ[S_σ∩Δ] of f_j by f_j^u = ∑_v∈Γ(f_j|_Δ;u) a_v · v∈ℂ[S_σ∩Δ], where f_j = ∑_v∈Γ_+(f_j) a_v · v∈ℂ[S_σ]. By taking a ℤ-basis of M(S_σ) and identifying theu-parts f_j^u with Laurent polynomials L(f_j^u) on T =(ℂ^*)^d as before, we have that the following definition doesnot depend on the choice of the ℤ-basis of M(S_σ). We say that (f_1,…,f_k) is non-degenerate if for any face Δ≺σ̌ such that Γ_+(f_k) ∩Δ≠∅ (including the case where Δ = σ̌) and any u ∈ Int(Δ̌) ∩ M(S_σ∩Δ)^* the following two subvarieties of (ℂ^*)^d are non-degenerate complete intersections x ∈ (ℂ^*)^dL(f_j^u)(x)=0,∀j ∈ I(Δ) ; x ∈ (ℂ^*)^dL(f_j^u)(x)=0, ∀ j ∈ I(Δ) ∪{k},i.e., if they are reduced smooth complete intersections varieties in the torus (ℂ^*)^d.This last definition is presented in <cit.> and it is ageneralization ofthe one in <cit.>. For these non-degenerate singularities, it is possible to describetheir geometrical and topological properties using the combinatorics of theNewton polygon. This isdone in <cit.> using mixed volume as follows. For each face Δ≺σ̌ of σ̌ such thatΓ_+(f_k) ∩Δ≠∅, we set f_Δ =∏_j∈ I(Δ) f_j· f_k ∈ℂ[S_σ] and consider its Newton polygon Γ_+(f_Δ) = {∑_j ∈ I(Δ)Γ_+(f_j) } + Γ_+(f_k) ⊂σ̌. Let γ_1^Δ,…,γ_ν(Δ)^Δ be the compact faces ofΓ_+(f_Δ) ∩Δ (≠∅) such thatγ_i^Δ = Δ -1. Then for each 1≤ i ≤ν(Δ) there exists a unique primitive vector u_i^Δ∈ Int(Δ̌) ∩ M(S_σ∩Δ)^* which takes itsminimal value in Γ_+(f_Δ) ∩Δ exactly onγ_i^Δ.For j ∈ I(Δ) ∪{k }, set γ(f_j)_i^Δ:= Γ(f_j|_Δ;u_i^Δ) and d_i^Δ:= min_w∈Γ_+(f_k) ∩Δ⟨u_i^Δ,w⟩. Note that we haveγ_i^Δ =∑_j ∈ I(Δ) ∪{k }γ(f_j)_i^Δfor any face Δ≺σ̌ such thatΓ_+(f_k) ∩Δ≠∅ and 1 ≤ i ≤ν(Δ).For each face Δ≺σ̌ such that Γ_+(f_k) ∩Δ≠∅, Δ≥ m(Δ) and 1 ≤ i≤ν(Δ), we set I(Δ) ∪{k} ={j_1,j_2,…,j_ m(Δ)-1,k=j_ m(Δ)}andK_i^Δ := ∑_α_1+…+α_ m(Δ)= Δ -1α_q ≥ 1 forq≤ m(Δ)-1α_ m(Δ)≥ 0^ Vol_ℤ(γ(f_j_1)_i^Δ,…, γ(f_j_1)_i^Δ_α_1-times,…,γ(f_j_ m(Δ))_i^Δ,…,γ(f_j_ m(Δ))_i^Δ_α_ m(Δ)-times).Here Vol_ℤ(γ(f_j_1)_i^Δ,…, γ(f_j_1)_i^Δ_α_1-times,…,γ(f_j_ m(Δ))_i^Δ,…,γ(f_j_ m(Δ))_i^Δ_α_ m(Δ)-times)is thenormalized (Δ -1)-dimensional mixed volume with respect tothe lattice M(S_σ∩Δ) ∩ L(γ_i^Δ) (seeDefinition 2.6, pg 205 from <cit.>). For Δ such thatΔ -1 =0, we setK_i^Δ = Vol_ℤ(γ(f_k)_i^Δ,…, γ(f_k)_i^Δ_0-times) := 1(in this case γ(f_k)_i^Δ is a point).§ THE BRASSELET NUMBER AND TORUS ACTIONSWe present formulas for the computation of the Brasselet number ofa function defined on a non-degenerate complete intersection contained intoric variety using Newtonpolygons.As applications we establish several results concerningits invariance for families of non-degenerate completeintersections.Let X_σ⊂ℂ^n be a d-dimensional toric variety and(f_1,…,f_k): (X_σ,0) → (ℂ^k,0) a non-degeneratecomplete intersection, with 1≤ k ≤ d. From now on we will denote by gthe complete intersection (f_1,…,f_k-1) and by f the function f_k.Let 𝒯 be the decomposition of X_σ =_Δ≺σ̌ T_Δinto T-orbits, and𝒯_g the decomposition of X_σ^g =_Δ≺σ̌ T_Δ∩ X_σ^g. Sinceg is a non-degenerate complete intersection, 𝒯_g is a decomposition of X_σ^g into smooth subvarieties <cit.>.We are interest inthe situation that𝒯_g is a Whitney stratification. Let S_σ = ℤ_+^3 and let X_σ = ℂ^3 be the smooth 3-dimensional toric variety. Consider (g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection, where g(z_1,z_2,z_3) = z_2^2 - z_1^3 -z_1^2z_3^2. The decomposition 𝒱 = {V_0={0}, V_1={axis - z_3}∖{0}, V_2= (X_σ^g)_reg} is a Whitney stratification of the non-degenerate complete intersection X_σ^g, where (X_σ^g)_reg is the regular part ofX_σ^g (see Figure (<ref>)). For a subset I ⊂{1,2,3} we denote the face ∑_i ∈ Iℝ_+e_i^* of σ̌ = (ℝ_+^3)^* by Δ_I and by Δ_0 the face {0}, where e_i^* denotes the elements of the canonical basis of (ℝ^3)^*. The Whitney stratification 𝒯 of X_σ is given by thefollowingT-orbits: T_Δ_0 = {0}, T_Δ_1 = ℂ^*×{0}×{0}, T_Δ_2 = {0}×ℂ^*×{0}, T_Δ_3 = {0}×{0}×ℂ^*, T_Δ_12 = ℂ^*×ℂ^*×{0}, T_Δ_13 = ℂ^*×{0}×ℂ^*, T_Δ_23 = {0}×ℂ^*×ℂ^*, T_Δ_123 = ℂ^*×ℂ^*×ℂ^*. Now, observe that T_Δ_3∩ X_σ^g = T_Δ_3 = V_1, then the stratification 𝒯_g = { T_Δ_I∩ X_σ^g, Δ_0}_Δ_I≺σ̌ is a Whitney stratification of X_σ^g. In Section <ref> we will provide others examples wherethe stratification 𝒯_g is Whitney (see <cit.>for more examples). We observe that in general 𝒯_g is not a Whitneystratification see <cit.>. Let X_σ⊂ℂ^n be a d-dimensional toric variety and (g,f): (X_σ,0) → (ℂ^k,0) a non-degenerate complete intersection. If 𝒯_g is a Whitney stratification, then B_f,X_σ^g(0) =∑_Γ_+ (f) ∩Δ≠∅Δ ≥ m(Δ) (-1)^Δ -m(Δ)( ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ^g(T_Δ∩ X_σ^g). Let (X_σ^g)^f := X^g_σ∩ X^f_σ be the zero set of f in X^g_σ. By <cit.> or <cit.>, there exists a refinement of 𝒯_g, denoted by 𝒯_(g,f) = {W_i}, such that 𝒯_(g,f) is adapted to (X_σ^g)^f and it is a Whitney stratification. Moreover, this stratification satisfies the (a_f)-Thom condition (see <cit.>), and then it is a good stratification of X^g_σ with relation to f. By definition, the Brasselet number reads B_f,X_σ^g(0) = ∑_iχ(W_i∩ B_ε(0) ∩ f^-1(δ) ) · Eu_X_σ^g(W_i), where 0< | δ| ≪ε≪ 1. As 𝒯_(g,f) is a refinement of 𝒯_g, each T_Δ∩ X^g_σ is a disjoint union of W_i,T_Δ∩ X^g_σ = _W_i ∩ T_Δ∩ X^g_σ≠∅ W_i, and since 𝒯_g is a Whitney stratification, Eu_X_σ^g(T_Δ∩ X^g_σ) = Eu_X_σ^g(W_i), for any W_i that meets T_Δ∩ X^g_σ. Then B_f,X_σ^g(0) = ∑_Δχ(T_Δ∩ X^g_σ∩ B_ε∩ f^-1(δ) ) · Eu_X_σ^g(T_Δ∩ X^g_σ). Note that f:X_σ→ℂ has the isolated stratified critical value 0 ∈ℂ by <cit.>. Thus the integers χ (T_Δ∩ X^g_σ∩ B_ε∩ f^-1(δ) ) can be calculated by the nearby cycle functor ψ_f <cit.>, χ( T_Δ∩ X^g_σ∩ B_ε∩ f^-1(δ) ) = χ(ψ_f(ℂ_T_Δ∩ X_σ^g)_0), since X_σ^g∩ B_ε∩ f^-1(δ)= _Δ≺σ̌ T_Δ∩ B_ε∩ X_σ^g∩ f^-1(δ) is a Whitney stratification of the Milnor fiber X_σ^g∩ B_ε∩ f^-1(δ). By the proof of <cit.> equations (52), (77), (78), we have that χ(ψ_f(ℂ_T_Δ∩ X_σ^g)_0) = (-1)^Δ -m(Δ)( ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ). To complete the proof, we observe that f≡ 0 on T_Δ for any face Δ such that Γ_+(f) ∩Δ = ∅, then we can neglect those faces.If f: X_σ^g→ℂ has a stratified isolated critical point, then Equation (<ref>) holds. Altogether, we have: Let X_σ⊂ℂ^n be a d-dimensional toric variety and (g,f): (X_σ,0) → (ℂ^k,0) a non-degenerate complete intersection. If 𝒯_g is a Whitney stratification and f: X_σ^g→ℂ has an isolatedsingularity at the origin, then Eu_f,X_σ^g(0) = Eu_X_σ^g(0) - ∑_Γ_+ (f) ∩Δ≠∅Δ ≥ m(Δ) (-1)^Δ -m(Δ)( ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ^g(T_Δ∩ X_σ^g).When k=1, using a similar argument used in Theorem <ref>, weobtain B_f,X_σ(0). In fact, using again <cit.> we canpass to a refinement of 𝒯 to obtain a good stratification ofX_σ relative to f. Thus we can use the same argument asin the proof of Theorem <ref> to obtainB_f,X_σ(0) = ∑_Δχ(T_Δ∩B_ε(0)∩ f^-1(δ) ) · Eu_X_σ(T_Δ).Now for each face Δ≺σ̌ such thatΓ_+(f) ∩Δ≠∅, letβ_1^Δ, β_2^Δ,…,β_μ(Δ)^Δ bethe compact faces of Γ_+(f) ∩Δ such thatdimβ_i^Δ = Δ -1. Let Γ_i^Δbe the convex hull of β_i^Δ⊔{0} in𝕃(Δ) and consider the normalized (Δ)-dimensionalvolume ofΓ_i^Δ, Vol_ℤ(Γ_i^Δ) ∈ℤ,with respect tothe lattice M(S_σ∩Δ). Here M(S_σ∩Δ) denotes the sublattice ofM(S_σ) generated by S_σ∩Δ. Then we have thefollowing result. Assume that f = ∑_v∈ S_σ a_v · v ∈ℂ[S_σ] is non-degenerate. ThenB_f,X_σ(0)= ∑_Γ_+ (f) ∩Δ≠∅(-1)^Δ - 1( ∑_i=1^μ(Δ)Vol_ℤ(Γ_i^Δ) )·Eu_X_σ(T_Δ). For each (Δ -1)-compact face β_i^Δ of Γ_+(f) ∩Δ≠∅, with 1≤ i ≤μ(Δ) we have K_i^Δ := Vol_ℤ(β_i^Δ,…,β_i^Δ_( dimΔ -1)-times), and from <cit.> we now that Vol_ℤ(β_i^Δ,…, β_i^Δ_(Δ - 1)-times) = Vol_ℤ(β_i^Δ). Therefore, the result follows from the fact that Vol_ℤ(Γ_i^Δ) = d_i^Δ· K_i^Δ, for 1≤ i ≤μ(Δ). We will apply Theorem <ref> in order to show that the Brasseletnumber is invariant for some families of complete intersections.We needsome new concepts and notations. A familia é para todo t∈?OkA deformation of a map germ f : (X, 0) → (ℂ^k, 0) is another map germ F : (ℂ× X) → (ℂ^k, 0) such that F(0,x) = f(x), for all x ∈ X. We assume that F is origin preserving, that is, F(t, 0) = 0 for all t ∈ℂ, so we have a 1-parameter family of map germs f_t : (X, 0) → (ℂ^k, 0) given by f_t(x) = F(t, x).Moreover, associated to the family f_t : (X, 0) → (ℂ^k, 0) wehave the family X^f_t = X ∩ f_t^-1(0) of subvarieties of X. In the particular case of a polynomial function f : (X, 0) → (ℂ,0), any polynomial deformation f_t can be written as: f_t(x) = f(x)+ ∑_i=1^rθ_i(t) ·h_i(x)for some polynomials h_i: (X,0) → (ℂ,0) and θ_i:(ℂ,0) → (ℂ,0), where θ_i(0)=0, for all i=1,…,r.Suppose that f: X_σ→ℂ is a polynomial function defined on a toric variety X_σ. Consider a family as infamily1 with Γ_+(h_i) ⊂Γ_+(f), for alli=1,…,r. Let γ_i_1^Δ,γ_i_2^Δ,…, γ_i_ν(Δ)^Δand β_1^Δ,β_2^Δ,…,β_μ(Δ)^Δbe the compact faces of Γ_+(h_i) ∩Δ and of Γ_+(f) ∩Δ, respectively, such thatγ_l^Δ=β_j^Δ = Δ -1. If for each face Δ≺σ̌, with Γ_+(h_i) ∩Δ≠∅,we have that γ_i_l^Δ∩β_j^Δ = ∅, foralll=1,…,i_ν(Δ) and j=1,…,μ(Δ), thenΓ_+(f_t) = Γ_+(f), for allt∈ℂ.In this case, we fix the notation Γ_+(h_i) ⫋Γ_+(f),for all i=1,…,r. In the sequel, we present some applications of Theorem <ref>. Let X_σ⊂ℂ^n be a d-dimensional toric variety. Suppose that (g,f_t): (X_σ,0) → (ℂ^k,0) is a non-degenerate complete intersection for each t∈ℂ with f_t(x) = f(x)+∑_i=1^rθ_i(t) · h_i(x) being a polynomial function on X_σ with h_i satisfying the condition (<ref>) for all i=1,…,r. If 𝒯_g is a Whitney stratification and (g,f_t) is a family of non-degenerate complete intersections, then B_f_t,X_σ^g(0) is constant for the family. As we have already noted, for each face Δ≺σ̌ such that Γ_+(f) ∩Δ≠∅, the Newton polygon Γ_+(f_Δ) of the function f_Δ = ∏_j∈ I(Δ) f_j· f ∈ℂ[S_σ] is {∑_j ∈ I(Δ)Γ_+(f_j) } + Γ_+(f) ⊂σ̌. Therefore, Γ_+(f_Δ) = Γ_+((f_t)_Δ), for all t ∈ℂ, where (f_t)_Δ = ∏_j∈ I(Δ) f_j· f_t,since Γ_+(f) = Γ_+(f_t). Then the result follows by Theorem <ref>. Roughly speaking the Brasselet number depends only on themonomials of smallest degree in each variable.Given f: (X_σ,0) → (ℂ,0) a non-degenerate function, asbefore, we denote by 𝒯_f the decomposition X_σ^f= _Δ≺σ̌ T_Δ∩X_σ^fof X_σ^f, and by 𝒯^f thedecomposition { T_Δ∩ X_σ^f,T_Δ∖ X_σ^f, {0}}_Δ≺σ̌ of X_σ. If 𝒯_fis a Whitney stratification of X_σ^f, then 𝒯^f is aWhitney stratification of X_σ which is adapted to X_σ^f.In fact, this follows from the fact that 𝒯 is a Whitneystratification of X_σ. Consequently, 𝒯^f isagood stratification of X_σ relative to f.Let g and f be non-degenerate polynomial functions on X_σ. Ingeneral we have no way to relatethe local Euler obstructionsEu_X_σ^g(T_Δ∩X_σ^g)toEu_X_σ(T_Δ). However, if we assumetheadditional hypothesis that g has an isolated critical point at 0 both inX_σ and in X_σ^f (in stratified sense), thefollowing result holds. Let X_σ⊂ℂ^n be a d-dimensional toric variety and (g,f): (X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection. Suppose that 𝒯^f is a good stratification of X_σ relative to f and that g is prepolar with respect to 𝒯^f, then B_f,X_σ^g(0) =∑_Γ_+ (f) ∩Δ≠∅Δ ≥ 2 (-1)^Δ -2( ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ(T_Δ). Consider 𝒯_g = { T_Δ∩X_σ^g}_Δ≺σ̌ the stratificationof X_σ^g. Using the same argument as in the proof of Theorem<ref> we can pass to a refinement 𝒯_(g,f) and we obtain a good stratification of X_σ^g relative to f. Then B_f,X_σ^g(0) =∑_Γ_+(f)∩Δ≠∅χ(W_Δ∩ B_ε∩ f^-1(δ) ) · Eu_X_σ^g(W_Δ), where W_Δ are the strata of 𝒯_(g,f) which are not contained in {f = 0 }, and 0 < |δ| ≪ε≪ 1. Moreover, for Δ≺σ̌, we have Eu_X_σ(T_Δ) =Eu_X_σ^g(W_Δ), since X_σ^g intersects the strata of 𝒯^f transversally (see <cit.>). Hence, B_f,X_σ^g(0)= ∑_Γ_+(f)∩Δ≠∅χ(X_σ^g∩ T_Δ∩ B_ε∩ f^-1(δ) ) · Eu_X_σ(T_Δ). Finally, as Γ_+(g) ∩Δ≠∅ for any face 0⪵Δ≺σ̌, then m(Δ) = 2, for all face Δ≺σ̌ such that Γ_+(f) ∩Δ≠∅. Since g is prepolar, X_σ^g∩ B_ε∩ f^-1(δ)= _Δ≺σ̌ T_Δ∩ B_ε∩ X_σ^g∩ f^-1(δ) is a Whitney stratification of the Milnor fiber X_σ^g∩ B_ε∩ f^-1(δ). Therefore, we obtain the result by nearbycicle1 and (<ref>). Therefore, if 𝒯^f is a good stratification of X_σ relative to f and g is prepolar with respect to 𝒯^f, we can obtain a more general version of Corollary <ref>, since we can relatethe local Euler obstructionsEu_X_σ^g(T_Δ∩X_σ^g) to theEuler local obstructions Eu_X_σ(T_Δ). Let X_σ⊂ℂ^n be a d-dimensional toric variety and (g,f): (X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection. Suppose that g_s(x),f_t(x)=g(x) + ∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x) is a family of non-degenerate complete intersections with l_i and h_j satisfying the condition (<ref>) for all i=1,…,m and j=1,…,r. If 𝒯^f_t is a good stratification of X_σ relative to f_t and g_s is prepolar with respect to 𝒯^f_t at the origin, for all s,t ∈ℂ, then, B_f_t,X_σ^g_s(0) is constant for all t,s ∈ℂ.Since Γ_+(l_i) ⊂Γ_+(g) andΓ_+(h_j) ⊂Γ_+(f), for each face Δ≺σ̌ such that Γ_+(f) ∩Δ≠∅, theNewton polygon Γ_+(f_Δ) of the functionf_Δ =∏_Γ_+(f) ∩Δ≠∅g · f ∈ℂ[S_σ]is equals toΓ_+((f_t)^s_Δ ), where(f_t)^s_Δ = ∏_Γ_+(f_t) ∩Δ≠∅ g_s · f_t ∈ℂ[S_σ].Then by nearbycicle1 and(<ref>), we can concludethat the Euler characteristic χ X_σ^g_s∩ T_Δ∩ B_ε∩ f_t^-1(δ) is constant for all s,t ∈ℂ. Moreover, as g_s is prepolar with respect to 𝒯^f_t, then we can proceed exactly in the same way as in Theorem <ref> to obtain Eu_X_σ(T_Δ) = Eu_X_σ^g_s(T_Δ∩ X_σ^g_s), and this concludes the proof. As a consequence of Proposition<ref> and Theorem <ref>, we have that if𝒯^fis a good stratification of X_σ relative to f and if g:X_σ→ℂ is prepolar with respect to 𝒯^f, we can express thenumber of stratified Morse critical points on the stratum of maximum dimensionappearing in a morsefication of g : X_σ∩ f^-1(δ) ∩B_ε→ℂ in terms of volumes of convex polytopes.More precisely, by Theorem <ref>, we have[ (-1)^d-1n_d= ∑_Γ_+ (f) ∩Δ≠∅(-1)^Δ - 1(∑_i=1^μ(Δ)Vol_ℤ(Γ_i^Δ))·Eu_X_σ(T_Δ) ;- ∑_Γ_+ (f) ∩Δ≠∅Δ≥ 2 (-1)^Δ - 2(∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ(T_Δ) ;] where n_d is the number of stratified Morse critical points on the topstratum T_Δ_d∩ f^-1(δ)∩ B_ε appearing in amorsefication of g : X_σ∩ f^-1(δ) ∩ B_ε→ℂ. Therefore, if f_t(x)= f(x)+∑_j=1^rθ_j(t) · h_j(x) is afamily of non-degenerate polynomial functions on X_σ and if(g_s,f_t): (X_σ,0) → (ℂ^2,0) is a family of non-degeneratecomplete intersections which satisfy the same hypotheses as Corollary<ref>, then (-1)^d-1n_d = B_f_t,X_σ(0) -B_f_t,X_σ^g_s(0) is constant for all s,t ∈ℂ. More precisely, we can state the following result. Let X_σ⊂ℂ^n be a d-dimensional toric variety and (g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection. Suppose that g_s(x),f_t(x)=g(x) + ∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x) is a family of non-degenerate complete intersections with l_i and h_j satisfying the condition (<ref>) for all i=1,…,m and j=1,…,r. If 𝒯^f_t is a good stratification of X_σ relative to f_t and g_s is prepolar with respect to 𝒯^f_t at the origin, for all s,t ∈ℂ, then (-1)^d-1n_d is constant for all s, t ∈ℂ. We will present some applications of the results presented before in the nextsection and some examples in Section <ref>.§ THE LOCAL EULER OBSTRUCTION AND BRUCE-ROBERTS' MILNOR NUMBER GSV index vai para a seção 5 junto com a parte finalA introducao da secao esta estranha, talvez colocar na ordem em queé a secao.+ -In this section, we derive sufficientconditions to obtain the invariance of the local Euler obstruction for familiesofcomplete intersectionswhich arecontained in X_σ⊂^n. As an application, we study theinvariance of the Bruce-Roberts' Milnor number for families of functionsdefined on hypersurfaces. §.§ Local Euler obstruction of non-degenerate complete intersections As observed in <cit.> the local Euler obstruction is not atopologicalinvariant. However, for non-degenerate complete intersections we have the following result. LetX_σ⊂ℂ^n be a d-dimensional toric variety and k a positive integer in {2,…,d}. Consider g=(f_1,…,f_k-1): (X_σ,0) → (ℂ^k-1,0) a non-degenerate complete intersection, such that 𝒯_g is a Whitney stratification. Suppose that g_s(x) = f_1(x) + ∑_i_1=1^m_1θ_i_1(s) · h_i_1(x),…,f_k-1(x) + ∑_i_k-1=1^m_k-1θ_i_k-1(s) · h_i_k-1(x) is a family of non-degenerate complete intersections, such that 𝒯_g_s is a Whitney stratification and h_i_p satisfies the condition (<ref>) for all p ∈{1,…,k-1 } and i_p ∈{1, …, m_p}. If L: ℂ^n →ℂ is a linear formwhich is generic with respect to X_σ^g_s for all s ∈ℂ and (g_s,L) is a non-degenerate complete intersection, then Eu_X_σ^g_s(0) is invariant for the family {g_s}_s∈. As L: ℂ^n →ℂ is a linear formwhich is generic with respect to X_σ^g_s, for all s ∈ℂ, Eu_X_σ^g_s(0) =B_L,X_σ^g_s(0). Moreover, T_Δ∩ X^g_s_σ∩ B_ε∩ L^-1(δ) induces a Whitney stratification on the Milnor fiber X^g_s_σ∩ B_ε∩ L^-1(δ). Therefore, the result follows from Theorem <ref>, and from the fact that for all face Δ≠{0} of σ̌, the Newton polygon Γ_+(L_Δ) of the function L_Δ = ∏_j∈ I(Δ) f_j · L ∈ℂ[S_σ]is equal to Γ_+(L_Δ^s). Here, L_Δ^s = ∏_p∈ I(Δ) f_p + ∑_i_p=1^m_pθ_i_p· h_i_p·L ∈ℂ[S_σ]. Let S_σ = ℤ_+^n and let X_σ = ℂ^n be the smooth n-dimensional toric variety. Consider g=(f_1,…,f_k-1): ℂ^n →ℂ^k-1 a non-degenerate complete intersection with an isolated singularity at 0, where 2≤ k ≤ n. Suppose that g_s(x) = f_1(x) + ∑_i_1=1^m_1θ_i_1(s) · h_i_1(x),…,f_k-1(x) + ∑_i_k-1=1^m_k-1θ_i_k-1(s) · h_i_k-1(x) is a family of non-degenerate complete intersections with an isolated singularity at 0, where h_i_p satisfies the condition (<ref>) for all p ∈{1,…,k-1 } and i_p ∈{1, …, m_p}. If L: ℂ^n →ℂ is a linear formwhich is generic with respect to X_σ^g_s, for all s ∈ℂ and (g_s,L) is a non-degenerate complete intersection, then Eu_X_σ^g_s(0) is invariant for the family {g_s}_s∈. As g_s is a family of complete intersections with an isolated singularity at 0, the decomposition 𝒯_g_s of X_σ^g_s = _Δ≺σ̌ T_Δ∩ X_σ^g_s is a Whitney stratification, once 𝒯 is a Whitney stratification of X_σ. Therefore, the result follows from Theorem <ref>. With the same assumptions as in Theorem <ref>, suppose that f_t(x) =f_k(x)+∑_i_k=1^m_kθ_i_k(t)· h_i_k(x) is a family ofpolynomial functions such that (g_s,f_t) is a family of non-degeneratecompleteintersections where h_i_k satisfies the condition (<ref>), Γ_+(h_i_k) ⫋Γ_+(f_k), for alli_k=1,…,m_k.Moreover, assume that f_t: X_σ^g_s→ℂ has a stratifiedisolated critical point at 0. For each face Δ≺σ̌satisfying Γ_+(f_k) ∩Δ≠∅, the Newton polygonΓ_+(f_Δ) of the function f_Δ = (∏_j∈ I(Δ) f_j )· f_k ∈ℂ[S_σ]is equals to Γ_+((f_t)^s_Δ ), where****conferir as trocas de indices e ξ por θ (f_t)^s_Δ = ∏_p∈ I(Δ) f_p + ∑_i_p=1^m_pθ_i_p· h_i_p· f_t = f_k+∑_i_k=1^m_kθ_i_k· h_i_k∈ℂ[S_σ].By Theorem <ref> we conclude that theEuler characteristic of the Milnor fiber of f_t: X_σ^g_s→ℂ is invariant for all s,t∈. Therefore, Eu_f_t,X_σ^g_s(0) is invariant for the family.§.§ Bruce-Roberts' Milnor numberIn <cit.>, Bruce and Roberts introduced a Milnor number forfunctions germs on singular varieties. Let X be asufficiently small representative of the germ (X,0) and let I(X) denotethe ideal in 𝒪_n,0 consisting of the germs of functionsvanishing on X. We say that two germs f and g in 𝒪_n,0are ℛ_X- equivalent if there exists a germ of diffeomorphismϕ: (ℂ^n,0)→ (ℂ^n,0) such that ϕ(X)=Xand f∘ϕ=g. Let θ_n denote the 𝒪_n,0- module ofgerms of vector fieldson (ℂ^n,0). Each vector field ξ∈θ_n can be seen as aderivation ξ:𝒪_n,0→𝒪_n,0. We denote byθ_X those vector fields that are tangent to X, θ_X:=ξ∈θ_ndg(ξ)=ξ g∈ I(X), ∀ g∈I(X). Let f be a function in 𝒪_n,0 Acho que aqui está faltando um be alguma coisa and let df(θ_X) be the ideal {ξ f: ξ∈θ_X} in 𝒪_n,0. The number μ_BR(X,f)=_ℂ𝒪_n,0/df(θ_X) is called the Bruce-Roberts number of f with respect to X. We refer to <cit.> for more details about μ_BR(X,f).In particular, μ_BR(X,f) is finite if and only if f is ℛ_X-finitely determined.An interesting open problem is to know whether the Bruce-Roberts number is atopological invariant or not. In <cit.>Grulhagave a partial answer. The author proved that, if (X,0) is a hypersurface whose logarithmic characteristic variety LC(X)<cit.>colocar onde está a definicao no artigo ok, isCohen-Macaulay and iff_t is a C^0- R_X-trivial deformation of f, then μ_BR(f_t,X) isconstant. For any hypersurface X the problem of LC(X) being Cohen-Macaulayremains open. When X is a weighted homogeneous hypersurface with an isolatedsingularity, LC(X) is Cohen-Macaulay by <cit.>.Let us recall that μ(f) denotes the Milnor number <cit.> of agerm of an analytic function f : (ℂ^n,0) → (ℂ, 0) withan isolated critical point at the origin and it is defined as μ(f) = _ℂ𝒪_n,0/J (f ),where 𝒪_n,0 is the ring of germs ofanalytic functions at the origin, and J(f ) is the Jacobian ideal of f. Using <cit.>, and assuming that L satisfies the samehypotheses as in Corollary <ref> we prove the following result. Let S_σ = ℤ_+^n and let X_σ = ℂ^n be the smooth n-dimensional toric variety. Consider (g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection, and g_s(x),f_t(x)=g(x) + ∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x) a family of non-degenerate complete intersections with h_j and l_i satisfying the condition (<ref>). Suppose that, for all s,t ∈ℂ,X_σ^g_s⊂ℂ^n is a weighted homogeneous hypersurface with an isolated singularity at the origin. If f_t: ℂ^n →ℂ is a polynomial function with an isolated singularity at the origin such that f_t: X_σ^g_s→ℂ has also a stratified isolated singularity at the origin, then μ_BR(f_t,X_σ^g_s) is constant to all s,t ∈ℂ. From <cit.> we have μ_BR(f_t,X_σ^g_s) = μ(f_t) +Eu_X_σ^g_s(0) + (-1)^n-1( Eu_f_t,X_σ^g_s(0) + 1).By the hypothesis X_σ = ^n, then L(g^μ)(x) = g_μ(x), for allμ∈ Int(Δ̌) ∩ M(S_σ∩Δ)^*. From definitions <ref> and <ref>, we can conclude that f_t:ℂ^n →ℂ is a family of non-degenerate polynomialfunctions. Furthermore, Γ_+(f) = Γ_+(f_t), for allt ∈ℂ, since h_j satisfies the condition (<ref>). Then, by <cit.> χ(f^-1(δ) ∩ B_ε) = χ(f_t^-1(δ) ∩ B_ε), for allt ∈ℂ, where 0 < | δ| ≪ε≪ 1. Consequently, μ(f_t) is constant, once χ(f^-1(δ) ∩ B_ε)= 1 + (-1)^n-1μ(f). Therefore the result follows from Corollary <ref> and the remark that follows Corollary <ref>.We observe that this result is a kind of generalization of<cit.>. § TORIC SURFACESLet f be a polynomial function defined on a 2-dimensional toric varietyX_σ⊂ℂ^n. In this section, we present a characterizationof the polynomial functions g: X_σ→ℂ which areprepolar with respect to 𝒯^f at the origin. Using this characterization and the results of the last sections wepresent some examples of computation of the Brasselet number B_f,X_σ, for a class of toric surfaces X_σ that are also determinantal.Let us remember that a strongly convex cone in ℝ^2 has thefollowing normal form. Let σ⊂ℝ^2 be a strongly convex cone, then σ is isomorphic to the cone generated by the vectors v_1 = pe_1 - qe_2 and v_2 = e_2, for some integers p,q ∈ℤ_>0 such that 0<q<p and p,q are coprime. Given a cone σ⊂ℝ^2, Riemenschneider <cit.>provedthat the binomials which generate the ideal I_σ aregiven by the quasiminors of a quasimatrix, where X_σ =V(I_σ). In the following we recall the definition ofquasimatrix. Given A_i, B_i, C_l,l+1∈ℂ with i=1,…,n and l=1,…,n-1, a quasimatrix with entries A_i, B_i, C_l,l+1 is written as A= [ A_1 A_2 ⋯ A_n-1 A_n; B_1 B_2 ⋯ B_n-1 B_n; C_1,2 ⋯ C_n-1,n ]. The quasiminors of the quasimatrix A are defined by A_i · B_j - B_i· (C_i,i+1· C_i+1,i_2⋯ C_j-1,j)· A_jAdicionei pontos de multiplicacao na equacao, estao certos?ok for 1 ≤ i < j ≤ n. Given σ⊂ℝ^2 generated by v_1 = p e_1 - q e_2and v_2 = e_2, with p and q as above, let us consider theHirzebruch-Jung continued fractionp/p-q = a_2 - 1/a_3 -1/… - 1/a_n-1 = [[a_2, a_3, …,a_n-1]]wherethe integers a_2, …, a_n-1 satisfies a_i ≥ 2, fori=2,…,n-1. By <cit.> we have:completar...Esse acho que não da pra colocar pq eu encontrei essa configuração em uma traducao na verdade The ideal I_σ is generated by the quasiminors of the quasimatrix [ z_1 z_2 z_3 ⋯ z_n-2 z_n-1; z_2 z_3 z_4 ⋯ z_n-1 z_n; z_2^a_2 - 2 z_3^a_3 - 2 ⋯ z_n-1^a_n-1-2; ], where the a_i are given by the Hirzebruch-Jung continued fraction of p/p-q. Moreover, this set of generators is minimal. Then, if a_i = 2 for i=3,…, n-2, we have that X_σ is adeterminantal surface <cit.>, in particular if the minimal dimension ofembedding of X_σ is 4,ifp/p-q = a_2 -1/a_3then X_σ is always determinantal and the idealI_σ is generated by the 2 × 2 minors of the matrix [z_1 z_2z_3^a_3 -1; z_2^a_2 -1 z_3 z_4;]. We will consider σ as in Proposition <ref>. Takea_2,…, a_n-1 the integers coming from the Hirzebruch-Jungcontinued fraction of p/p-q, onde entram os a's? ok we willdenote by μ_1 =(μ_1^1,μ_1^2)=(1,0), μ_2 = (μ_2^1,μ_2^2)=(1,1), μ_i+1^j=a_i ·μ_i^j - μ_i-1^j,the minimal set ofgenerators of S_σ, with i=2,…,n-1; j=1,2. We note that it ispossible to show that μ_n =(μ_n^1,μ_n^2)=(q,p) (see <cit.>). Thus φ: (ℂ^*)^2 × X_σ→ X_σgiven by φ((t_1,t_2),(z_1,…,z_n)) = (t_1 · z_1,t_1 · t_2 ·z_2,t_1^μ_3^1· t_2^μ_3^2· z_3,…, t_1^q·t_2^p· z_n)is an action of (ℂ^*)^2 in X_σ. Each orbit of φis an embedding of a d-dimensional torus, 0≤ d≤ 2, in X_σ. The action φ has 4 orbits, that are[ T_Δ_0 ={(0,…,0) }; T_Δ_1 =(t_1,0,…,0)t_1 ∈ℂ^* ≅ℂ^*; T_Δ_2 =(0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*≅ℂ^*; T_Δ_3 = (t_1,t_1 · t_2,t_1^μ_3^1· t_2^μ_3^2,…,t_1^q·t_2^p)t_1,t_2 ∈ℂ^*≅ (ℂ^*)^2 ]. Moreover, as in Section 3,X_σ = _Δ_i≺σ̌ T_Δ_i,with i=0,1,2,3, is a decomposition ofX_σ in strata satisfying the Whitney conditions. The toric surfaces obtained in Proposition <ref> are normaltoric surfaces, then they are smooth or they have isolated singularity at theorigin. Therefore, if f =∑_v∈ S_σ a_v · v ∈ℂ[S_σ] is a non-degenerate polynomial function on X_σ, then 𝒯_f=T_Δ_i∩ X_σ^fi=0,1,2,3, is a Whitney stratification of X_σ^f, since T_Δ_1 and T_Δ_2 are smooth subvarieties of T_Δ_3 = X_σ which satisfy T_Δ_1∩T_Δ_2 = { (0,…,0)}. As a consequence, 𝒯^f=T_Δ_i∖ X_σ^f, T_Δ_i∩ X_σ^f, {0 } i=0,1,2,3,is a good stratification of X_σ relative to f. Next, we characterize the polynomial functionswhich have a stratified isolated singularity at the origin. Let σ⊂ℝ^2 be a strongly convex cone and 𝒯 the Whitney stratification of X_σ⊂ℂ^n whose the strata are T_Δ_0, T_Δ_1, T_Δ_2 and T_Δ_3. Then, a non-degenerate polynomial function g: (X_σ,0) → (ℂ,0) has an isolated singularity at the origin (in the stratified sense) if, and only if,g(z_1,…,z_n) = c_1z_1^p_1 + h(z_1,…,z_n) + c_nz_n^p_n,where h is a polynomial function on X_σ, c_1,c_n ∈ℂ^* and p_1,p_n ∈^*_+. Let us write g as followsg(z_1,…,z_n) = ∑_l=1^m c_lz_1^p_1^l z_2^p_2^l… z_n^p_n^l,where l=1,…,m, p_i^l∈^*_+ and c_l ∈ℂ. Suppose that g has a stratified isolated singularity at the origin 0 ∈ℂ^n, with respect to the stratification 𝒯, then there must be l_1,l_n ∈{1,…,m } such that [ c_l_1∈ℂ^*, p_1^l_1≠ 0 and p_i^l_1 = 0, fori ∈{2,…,n}; c_l_n∈ℂ^*, p_n^l_n≠ 0 and p_i^l_n = 0, fori ∈{1,…,n-1} ], otherwise T_Δ_1, T_Δ_2⊂Σ_𝒯 g, since T_Δ_1=(t_1,0,…,0)t_1 ∈ℂ^*, T_Δ_2= (0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*. In other words, g must contain monomials of the form c_1z_1^p_1 and c_nz_n^p_n. Now, suppose that g has the form mentioned above, thenΓ_+(g) ∩Δ_1 ≠∅, Γ_+(g) ∩Δ_2≠∅ and Γ_+(g) ∩Δ_3 ≠∅. By the proof of <cit.> we can conclude that g|_T_Δ_1: T_Δ_1→ℂ and g|_T_Δ_2: T_Δ_2→ℂ are non-degenerate polynomial functions. Moreover, [T_Δ_1=(t_1,0,…,0)t_1 ∈ℂ^* ≅ℂ^*,;T_Δ_2=(0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*≅ℂ^*,;T_Δ_3= (t_1,t_1 · t_2,t_1^μ_3^1· t_2^μ_3^2,…,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*≅ (ℂ^*)^2. ] Then by <cit.> there exists an ε>0 such that g|_T_Δ_1, g|_T_Δ_2 and g|_T_Δ_3 have no singularities in T_Δ_1∩ B_ε, T_Δ_2∩ B_ε and T_Δ_3∩ B_ε, respectively. As a consequence of Lemma <ref> we obtain information about thesingular set of g by just looking at its Newton polygon Γ_+(g). More precisely, anon-degenerate polynomial function g: (X_σ,0) → (ℂ,0) hasan isolatedsingularity at the origin (in the stratified sense) if, and only if, Γ_+(g) intersectsΔ_1 and Δ_2, exactly inthe same way as the classic case,in the case where X_σ =ℂ^2.Let (g,f): (X_σ,0) → (ℂ^2,0) be a non-degenerate complete intersection, such that f and g have no irreducible components in common. The polynomial function g is prepolar with respect to 𝒯^f if, and only if, g(z_1,…,z_n) = c_1z_1^p_1 + h(z_1,…,z_n) + c_nz_n^p_n,whereh is a polynomial function on X_σ, c_1,c_n ∈ℂ^*and p_1,p_n ∈_+^*. Consider the good stratification 𝒯^f = T_Δ_i∖ X_σ^f, T_Δ_i∩ X_σ^f, {0 }i=0,1,2,3, of X_σ relative to f. The sets T_Δ_1 and T_Δ_2 are given by T_Δ_1=(t_1,0,…,0)t_1 ∈ℂ^* T_Δ_2= (0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*. From the fact that f is a non-degenerate polynomial function, there are only two possibilities for each stratum T_Δ_i∩ X_σ^f. Either T_Δ_i∩ X_σ^f is a finite set or T_Δ_i∩ X_σ^f = T_Δ_i. Therefore, if g is prepolar with respect to 𝒯^f, then Γ_+(g) ∩Δ_1≠∅ and Γ_+(g) ∩Δ_2≠∅, otherwise T_Δ_i∩ X_σ^f⊂Σ_𝒯^f g or T_Δ_i∖ X_σ^f⊂Σ_𝒯^f g, for i=1,2. Now, suppose that g have the form mentioned above, then the result follows from Lemma <ref> and from the fact that, as f and g have no irreducible components in common,X_σ^g∩ X_σ^f is a finite set. Let σ⊂ℝ^2 be the cone generated by the vectors v_1 = e_2 and v_2 = ne_1 - e_2. The toric surface associated to σ is X_σ = V(I_σ) ⊂ℂ^n+1, where I_σ is the ideal generated by the 2 × 2 minors of the matrix [ z_1 z_2 z_3 … z_n-1 z_n; z_2 z_3 z_4 … z_n z_n+1 ], i.e., X_σ is a determinantal surface with codimension n-1. Consider f: X_σ→ℂ the function given by f(z_1,…,z_n+1)=z_1^d + z_n+1^d + tg(z_1,…,z_n+1), where g(z_1,…,z_n+1) = ∑_l=1^m z_1^p_1^l z_2^p_2^l… z_n+1^p_n+1^l is a polynomial function on X_σ satisfying p_1^l + p_2^l + … + p_n+1^l > d for every l=1,…,m. If f is a non-degenerate polynomial function then B_f,X_σ(0)= 2d -nd^2. Indeed, consider h:X_σ→ℂ the function given by h(z_1,…,z_n+1) = z_1^d + z_n+1^d. The Newton polygon Γ_+(h) has an unique 1-dimensional compact face β_1, that is the straight line segment connecting the points (d,0) and (d,nd) in σ̌. Using the same notation of Proposition <ref>, we have that Γ_1^Δ_1 is the straight line segment connecting the points (0,0) and (d,0), Γ_1^Δ_2 is the straight line segment connecting the points (0,0) and (d,nd) and Γ_1^Δ_3 is the triangle with vertices (0,0), (d,0) and (d,nd). Therefore, Vol_ℤ(Γ_1^Δ_1)=Vol_ℤ(Γ_1^Δ_2) = d andVol_ℤ(Γ_1^Δ_3)=nd^2. By Proposition <ref>, B_h,X_σ(0)=2d-nd^2, since X_σ has an isolated singularity at the origin, and consequently Eu_X_σ(T_Δ_1) = Eu_X_σ(T_Δ_2) = Eu_X_σ(T_Δ_3) = 1. Now, S_σ is the semigroup generated by {(1,0),(1,1),(1,2)…,(1,n) }, and then Γ_+(g) ⊂Γ_+(h). Moreover, by Lemma <ref>, f has an isolated singularity at the origin, thus B_f,X_σ(0)=Eu_X_σ(0) - Eu_f,X_σ(0). However, González-Sprinberg <cit.> proved that Eu_X_σ(0) = 3-(n+1). Hence, Eu_f,X_σ(0) = 3-(n+1)-2d+nd^2. Therefore, a morsefication of f has 3-(n+1)-2d+nd^2 Morse points on the regular part of X_σ. colocar uma familia em g tambem e usar esse exemplo no final da secao 3 e tb no final da secao 5 Let σ⊂ℝ^2 be the cone generated by the vectors v_1 = e_2 and v_2 = 2e_1 - e_2. The toric surface associated to σ is X_σ = V(I_σ) ⊂ℂ^3, with I_σ the ideal generated by z_1z_3 - z_2^2. Consider f: X_σ→ℂ the function given by f(z_1,z_2,z_3)=z_2^2 - z_1^3, which is a non-degenerate polynomial function, whose singular set is Σ f = (0,0,z_3)z_3 ∈ℂ⊂ X_σ. Moreover, Γ_+(f) has a unique 1-dimensional compact face β_1, which is the straight line segment connecting the points (3,0) and (2,2) in σ̌. Thus, Γ_1^Δ_1 is the straight line segment connecting the points (0,0) and (3,0), Γ_1^Δ_3 is the triangle with vertices (0,0), (3,0) and (2,2), and Γ_1^Δ_2 = ∅. Therefore, by Proposition <ref>, B_f,X_σ(0)=3-6=-3 since X_σ has an isolated singularity at the origin, and consequently Eu_X_σ(T_Δ_1) =Eu_X_σ(T_Δ_2) = Eu_X_σ(T_Δ_3) = 1. Now let g: X_σ→ℂ be the non-degenerate polynomial function given by g(z_1,z_2,z_3)= z_1 - z_3^2, which is prepolar with respect to 𝒯^f. Moreover, (g,f) is a non-degenerated complete intersection. The Newton polygon Γ_+(g· f) has two 1-dimensional compact faces γ_1 and γ_2, which are the straight line segment connecting the points (4,0) and (3,2) and the straight line segment connecting the points (3,2) and (4,6), respectively. Thus, the primitive vectors u_1^Δ_3, u_2^Δ_3∈ Int(Δ̌_̌3̌) ∩ M(S_σ∩Δ_3)^* which take their minimal value in Γ_+(g· f) ∩Δ_3 exactly on γ_1 and γ_2, respectively, are u_1^Δ_3=(2,1) and u_2^Δ_3=(4,-1). Let us observe that [γ(g)_1^Δ_3:= Γ(g|_Δ_3;u_1^Δ_3) ={(1,0) };γ(g)_2^Δ_3:= Γ(g|_Δ_3;u_2^Δ_3) = α_1; d_1^Δ_3:= d_2^Δ_3:= 6; K_1^Δ_3 = K_2^Δ_3 = 1 ], where α_1 is the 1-dimensional compact face of Γ_+(g). Applying Theorem <ref>, we have B_f,X_σ^g(0)= 12. Therefore, we obtain the following equality B_f,X_σ(0) - B_f,X_σ^g(0) = -3 - 12 =-15, which means that the number of stratified Morse critical points on the top stratum T_Δ_3∩ f^-1(δ)∩ B_ε(0) appearing in a morsefication of g : X_σ∩ f^-1(δ) ∩ B_ε(0) →ℂ is 15. Moreover, if we consider h, l: X_σ→ℂ the polynomial functions given by h(z_1,z_2,z_3)=-z_1^2 z_3^2, l(z_1,z_2,z_3)=z_3^3, and observe that Γ_+(h)⫋Γ_+(f), Γ_+(l)⫋Γ_+(g), by Corollary <ref> we have B_f_t,X_σ(0) =B_f,X_σ(0) =-3,B_f_t,X_σ^g_s(0)= B_f,X_σ^g(0) =12, where f_t(x)=f(x)+t · h(x) is a deformation of the cusp f_0(z_1,z_2,z_3) = z_2^2 - z_1^3 (see Figure (<ref>)) and g_s(x)=g(x)+ s · l(x). Consequently, B_f_t,X_σ(0) - B_f_t,X_σ^g_s(0) = -3 - 12 =-15, for all t, s ∈ℂ. § INDICES OF VECTOR FIELDSA toric surface X_σ, which is a cyclic quotient singularity, always possesses a smoothing <cit.>. Therefore, when we consider a radial continuous vector field v on X_σ with an isolated singularity at 0, we can relate the Eulercharacteristic of a fiber of this smoothing with the GSV index of v in X_σ. The definition of this index forsmoothable isolated singularity can be found in <cit.>.In the particular, for the case of toric surfaceswhich are also isolated determinantal singularities, we have the following result concerning GSV index.Let X_σ⊂ℂ^n be a toric surface that is also anisolated determinantal singularity,σ is generated by thevectors v_1 = p e_1 - q e_2 and v_2 = e_2, where 0 < q < p, p,q are coprime, and whose the Hirzebruch-Jung continued fraction is p/p-q = [[a_2, 2, 2, …, 2, a_n-1]].Consider f_t(x) = f(x)+∑_j=1^rθ_j(t)· h_j(x) a family ofnon-degenerate polynomialfunctions on X_σ, which satisfies the conditions Γ_+(h_j) ⫋Γ_+(f), for allj=0,…,r.If this family has an isolated singularity at the origin, then thefollowing result holds. Let v_t be the vector field given by the gradient of the function f_t. Then, the following are equivalent: * (a) Eu_f_t,X_σ(0) is constant for the family; * (b) Ind_GSV(v_t,X_σ,F) is constant for the family, where F is the flat map associated to the smoothing of X_σ. By <cit.> the determinantal Milnor number of the function f on the Isolated Determinantal Singularity X_σ is μ(f|_X_σ) = #Σ(f̃|_X_σ_s), where X_σ_s is a fiber of a smoothing of X_σ, f̃|_X_σ_s is a morsefication of f and #Σ(f̃|_X_σ_s) denote the number of Morse points of f̃ on X_σ_s. From the definition of the GSV index in the case of smoothable varieties (see <cit.>) we have μ(f|_X_σ) =Ind_GSV(v,X_σ,F).Then the proof follows by <cit.>, where it is proved that Eu_f_t,X_σ(0) is constant for the family if and only if μ(f_t|_X_σ) is constant for the family. In <cit.>, the authors extended the concept of GSV index and proved a Lê-Greuel formula (see <cit.>) which holds with the same hypotheses of Theorem <ref>.However, in <cit.> the authors worked with the constructible functiongiven by the characteristic function, while in <cit.> is considered thelocal Euler obstruction. Hence the GSV index and the Brasselet numberare not related in general.Assuming that f_t is generically a submersion, for non-degenerate complete intersections, we have the following result. Troquei o corolario por proposicao. ok Let S_σ = ℤ_+^n and X_σ = ℂ^n be the smooth n-dimensional toric variety. Let (g,f): (X_σ,0) → (ℂ^2,0) be a non-degenerate complete intersection, such that 𝒯_g is a Whitney stratification of X_σ^g. If g_s(x),f_t(x)=g(x) + ∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x) is a family of non-degenerate complete intersections with h_j and l_i satisfying the condition (<ref>) for all i=1,…,m and j=1,…,r, such that 𝒯_g_s is a Whitney stratification of X_σ^g_s and g_s is prepolar with respect to 𝒯^f_t at the origin. Then, Ind_GSV(g_s, 0; f_t ) is invariant to the family. In <cit.>, the authors used <cit.> (considering the Euler characteristic as constructible function) to provide the following interpretation to the GSV index,∑_Δ≺σ̌( χ(T_Δ∩ X_σ∩ B_ε(0) ∩ f^-1(δ) ) - χ(T_Δ∩ X_σ^g∩ B_ε(0) ∩ f^-1(δ) ) ) =Ind_GSV(g, 0; f ), where Ind_GSV(g, 0; f ) is the GSV-index of g on X^f relative to the function f (see <cit.>). Moreover, f_t is a family of non-degenerate polynomial functions, since (g_s,f_t) is non-degenerate complete intersections, for all s,t ∈ℂ. Then, to compute Ind_GSV(g, 0; f ) we apply <cit.> to the first term of the equality above andto the second term of the equality above we used nearbycicle1, (<ref>). Therefore, the result follows from the fact that h_j and l_i satisfying the condition (<ref>) for all i=1,…,m and j=1,…,r. Consider the toric surface X_σ = V(I_σ) ⊂ℂ^3, with I_σ the ideal generated by z_1z_3 - z_2^2. Let f_t and g_s be the same families of functions from Example <ref>, then Ind_GSV(g_s, 0; f_t ) = -15 for all t, s ∈ℂ. § ACKNOWLEDGMENTS The authors are grateful to Nivaldo de Góes Grulha Jr. from ICMC-USP for helpful conversations in developing this paper and to Bruna Oréfice Okamoto fromDM-UFSCar for helpful conversations about the Bruce-Roberts' Milnor number. Through the project CAPES/PVE Grant88881.068165/2014-01 of the program Science without borders,Professor Mauro Spreafico visited the DM-UFSCar in São Carlos providinguseful discussions with the authors. Moreover, the authors were partially supported by this project, therefore we are gratefulto this program. We would like to thank the referee formany valuable suggestions which improved this paper.amsalpha-lmp | http://arxiv.org/abs/1708.08338v3 | {
"authors": [
"Thais M. Dalbelo",
"Luiz Hartmann"
],
"categories": [
"math.AT"
],
"primary_category": "math.AT",
"published": "20170824190924",
"title": "Brasselet number and Newton polygons"
} |
Two Dimensional Discrete Dynamics of Integral Value Transformations Pabitra Pal Choudhury December 30, 2023 =================================================================== We examine recursive monotonic functions on the Lindenbaum algebra of 𝖤𝖠. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ∧𝖢𝗈𝗇(φ)). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of 𝖢𝗈𝗇 that bounds f everywhere, then f must be somewhere equal to an iterate of 𝖢𝗈𝗇. § INTRODUCTION It is a well-known empirical phenomenon that natural axiomatic theories are well-ordered by their consistency strength. However, without a precise mathematical definition of “natural,” it is difficult to explain this observation in a strictly mathematical way. One expression of this phenomenon comes from ordinal analysis, a research program whereby recursive ordinals are assigned to theories as a measurement of their consistency strength. One method for calculating the proof-theoretic ordinal of a theory T involves demonstrating that T can be approximated over a weak base theory by a class of formulas that are well understood. In particular, the Π^0_1 fragments of natural theories are often proof-theoretically equivalent to iterated consistency statements over a weak base theory, making these theories amenable to ordinal analysis. For discussion, see, e.g., Beklemishev <cit.> and Joosten <cit.>. Why are the Π^0_1 fragments of natural theories proof-theoretically equivalent to iterated consistency statements? Our approach to this question is inspired by Martin's approach to another famous question from mathematical logic: why are natural Turing degrees well-ordered by Turing reducibility? Martin conjectured that (i) the non-constant degree invariant functions meeting a certain simplicity condition (f∈ L(ℝ))[Martin's Conjecture is stated under the hypothesis 𝖹𝖥+𝖠𝖣+𝖣𝖢, which is satisfied by L(ℝ) assuming that there are ω many Woodin cardinals with a measurable above them all.] are pre-well-ordered by the relation “f(a)≤_Tg(a) on a cone in the Turing degrees” and (ii) the successor for this well-ordering is induced by the Turing jump. Martin's conjecture is meant to capture the idea that the Turing jump and its iterates into the transfinite are the only natural non-trivial degree invariant functions.In this paper we investigate analogous hypotheses concerning jumps on consistent axiomatic theories, namely, consistency statements. We fix elementary arithmetic 𝖤𝖠 as our base theory. 𝖤𝖠 is a subsystem of 𝖯𝖠 that is often used as a base theory in ordinal analysis and in which standard approaches to arithmetization of syntax can be carried out without substantial changes; see <cit.> for details. We write [φ] to denote the equivalence class of φ modulo 𝖤𝖠-provable equivalence. We write φ⊢ψ if 𝖤𝖠⊢φ→ψ and say that φ implies ψ. If φ⊢ψ but ψ⊬φ we say that φ strictly implies ψ. The Lindenbaum algebra of 𝖤𝖠 is the set of equivalence classes of sentences ordered by ⊢. We focus on recursive functions f that are monotonic, i.e.,if φ⊢ψ, then f(φ)⊢ f(ψ). We note that (i) a function f is monotonic just in case f preserves implication over 𝖤𝖠 and (ii) all monotonic functions induce functions on the Lindenbaum algebra of 𝖤𝖠. We adopt the convention that all functions named “f” in this paper are recursive.Our goal is to demonstrate that φ↦(φ∧𝖢𝗈𝗇(φ)) and its iterates into the transfinite are canonical among monotonic functions. Our first theorem to this end is the following.Let f be monotonic. Suppose that for all consistent φ,(i) φ∧𝖢𝗈𝗇(φ) implies f(φ) and(ii) f(φ) strictly implies φ. Then for every true φ, there is a true ψ such that ψ⊢φ and [f(ψ)] = [ψ∧𝖢𝗈𝗇(ψ)]. There is no monotonic function f such that for all consistent φ, (i) φ∧𝖢𝗈𝗇(φ) strictly implies f(φ) and (ii) f(φ) strictly implies φ. We note that this result depends essentially on the condition of monotonicity. Shavrukov and Visser <cit.> studied recursive functions f that are extensional over the Lindenbaum algebra of 𝖯𝖠, i.e., if 𝖯𝖠⊢(φ↔ψ) , then 𝖯𝖠⊢(f(φ)↔ f(ψ)), and proved the following theorem.(Shavrukov–Visser) There is a recursive extensional function f such that for all consistent φ, (i) φ∧𝖢𝗈𝗇(φ) strictly implies f(φ) and (ii) f(φ) strictly implies φ.In particular, Shavrukov and Visser proved that for any consistent φ, the sentenceφ^⋆:=φ∧∀ x(𝖢𝗈𝗇(IΣ_x+φ)→𝖢𝗈𝗇(IΣ_x+φ+𝖢𝗈𝗇(IΣ_x+φ)))has deductive strength strictly between φ and φ∧𝖢𝗈𝗇(φ), and that the map φ↦φ^⋆ is extensional. By a theorem of Kripke and Pour-El <cit.>, the Lindenbaum algebras of 𝖯𝖠 and 𝖤𝖠 are effectively isomorphic, whence Theorem <ref> also applies to 𝖤𝖠. Thus, Corollary <ref> cannot be strengthened by weakening the hypothesis of monotonicity to the hypothesis of extensionality.We also note that Friedman, Rathjen, and Weiermann <cit.> introduced a notion of slow consistency with which they produced a Π^0_1 sentence 𝖲𝗅𝗈𝗐𝖢𝗈𝗇(𝖯𝖠) with deductive strength strictly between 𝖯𝖠 and 𝖯𝖠+𝖢𝗈𝗇(𝖯𝖠). In general, the statement 𝖲𝗅𝗈𝗐𝖢𝗈𝗇(φ) has the form∀ x (F_ϵ_0(x)↓→𝖢𝗈𝗇(IΣ_x+φ))where F_ϵ_0 is a standard representation of a recursive function that is not provably total in 𝖯𝖠. This is not in conflict with Corollary <ref>, however, since φ∧𝖢𝗈𝗇(φ) and φ∧𝖲𝗅𝗈𝗐𝖢𝗈𝗇(φ) are provably equivalent for all φ such that φ⊢∀ x F_ϵ_0(x)↓. On the other hand, changing the definition of the 𝖲𝗅𝗈𝗐𝖢𝗈𝗇(φ) so that the function in the antecedent varies with the input φ results in a map that is not monotonic.Theorem <ref> generalizes to the iterates of 𝖢𝗈𝗇 into the effective transfinite. For an elementary presentation α of a recursive well-ordering (see Definition <ref>) and a sentence φ, we define sentences 𝖢𝗈𝗇^β(φ) for every β<α.𝖢𝗈𝗇^0(φ):= ⊤𝖢𝗈𝗇^β+1(φ):= 𝖢𝗈𝗇(φ∧𝖢𝗈𝗇^β(φ))𝖢𝗈𝗇^λ(φ):= ∀β<λ(𝖢𝗈𝗇^β(φ))For a precise definition using Gödel's fixed point lemma, see Definition <ref>. Note that for every φ, [𝖢𝗈𝗇^1(φ)]=[𝖢𝗈𝗇(φ)].We warn the reader that there is some discrepancy between our notation and the notation used by other authors. Our iteration scheme 𝖢𝗈𝗇^α+1(φ)≡𝖢𝗈𝗇(φ∧𝖢𝗈𝗇^α(φ)) is sometimes denoted 𝖢𝗈𝗇((𝖤𝖠+φ)_α), e.g., <cit.>. Moreoever, the notation 𝖢𝗈𝗇^α+1(φ) is sometimes used to denote 𝖢𝗈𝗇(𝖢𝗈𝗇^α(φ)), e.g., <cit.>. With each predicate 𝖢𝗈𝗇^α we associate a function φ↦(φ∧𝖢𝗈𝗇^α(φ)).Theorem <ref> then generalizes into the effective transfinite as follows. Let f be monotonic. Suppose that for all φ, (i) φ∧𝖢𝗈𝗇^α(φ) implies f(φ),(ii) if [f(φ)]≠[], then f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. Then for every true φ, there is a true ψ such that ψ⊢φ and [f(ψ)] = [ψ∧𝖢𝗈𝗇^α(ψ)].There is no monotonic f such that for all φ, if [φ∧𝖢𝗈𝗇^α(φ)]≠[], then both(i) φ∧𝖢𝗈𝗇^α(φ) strictly implies f(φ) and(ii) f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. Thus, if the range of a monotonic function f is sufficiently constrained, then for some φ and some α, [f(φ)]=[φ∧𝖢𝗈𝗇^α(φ)]≠[].This property still holds even when these constraints on the range of f are relaxed considerably. More precisely, if a monotonic function is everywhere bounded by a finite iterate of 𝖢𝗈𝗇, then it must be somewhere equivalent to an iterate of 𝖢𝗈𝗇. Let n∈ℕ. Let f be a monotonic function such that for every φ, (i) φ∧𝖢𝗈𝗇^n(φ) implies f(φ) and (ii) f(φ) implies φ. Then for some φ and some k≤ n, [f(φ)]=[φ∧𝖢𝗈𝗇^k(φ)]≠[]. To generalize this result into the effective transfinite, we focus on a particular class of monotonic functions that we call Π^0_1.A function f is Π^0_1 if f(φ)∈Π^0_1 for all φ. Our main theorem is the following: if a monotonic function is everywhere bounded by a transfinite iterate of 𝖢𝗈𝗇, then it must be somewhere equivalent to an iterate of 𝖢𝗈𝗇. This to say that the iterates of the consistency operator are inevitable; no monotonic function that is everywhere bounded by some iterate of 𝖢𝗈𝗇 can avoid all of the iterates of 𝖢𝗈𝗇.Let φ↦ f(φ) be a monotonic Π^0_1 function Then either (i) for some β≤α and some φ, [φ∧ f(φ)]=[φ∧𝖢𝗈𝗇^β(φ)]≠[] or(ii) for some φ, (φ∧𝖢𝗈𝗇^α(φ))⊬ f(φ). The main theorem bears a striking similarity to the following theorem of Slaman and Steel <cit.>.(Slaman–Steel) Suppose f: 2^ω→2^ω is Borel, order-preserving with respect to ≤_T, and increasing on a cone. Then for any α<ω_1 either(i) for some β≤α, f(x)≡_Tx^(β) cofinally or(ii) (x^(α)<_Tf(x)) cofinally. There are two notable disanalogies between Theorem <ref> and Theorem <ref>. First, Theorem <ref> guarantees only that sufficiently constrained functions are somewhere equivalent to an iterate of 𝖢𝗈𝗇, whereas Theorem <ref> guarantees cofinal equivalence with an iterate of the Turing jump. Second, by assuming 𝖠𝖣, Slaman and Steel inferred that this behavior happens not only cofinally but also on a cone in the Turing degrees. There is no obvious analogue of 𝖠𝖣 from which one can infer that if cofinally many Lindenbaum degrees have a property then every element in some non-trivial ideal of Lindenbaum degrees has that property.We then turn our attention to a generalization of consistency, namely, 1-consistency. Recall that a theory T is 1-consistent if T is consistent with the true Π^0_1 theory of arithmetic. Just as the Π^0_1 fragments of natural theories are often proof-theoretically equivalent to iterated consistency statements over a weak base theory, the Π^0_2 fragments of natural theories are often proof-theoretically equivalent to iterated 1-consistency statements over a weak base theoryConservativity theorems relating 1-consistency and iterated consistency play an important role in the proof-theoretic analysis of arithmetic theories. For instance, it is a consequence of Beklemshev's reduction principle <cit.> that for any Π^0_1 φ, 𝖤𝖠 + 1𝖢𝗈𝗇(𝖤𝖠) ⊢φ if and only if 𝖤𝖠 + {𝖢𝗈𝗇^k(𝖤𝖠):k<ω}⊢φ. This fact plays an integral role in Beklemishev's <cit.> consistency proof of 𝖯𝖠. We show that this conservativity result is drastically violated in the limit. For functions f and g, we say that f majorizes g if there is a consistent φ such that for all ψ, if ψ⊢φ then f(ψ)⊢ g(ψ); if in addition φ is true then we say that f majorizes g on a true ideal. For any elementary presentation α of a recursive well-ordering, 1𝖢𝗈𝗇 majorizes 𝖢𝗈𝗇^α on a true ideal. It is tempting to conjecture on the basis of this result that 1𝖢𝗈𝗇 is the weakest monotonic function majorizing each 𝖢𝗈𝗇^α for α a recursive well-ordering. We prove that this is not the case. There are infinitely many monotonic functions f such that for every recursive ordinal α, there is an elementary presentation a of α such that f majorizes 𝖢𝗈𝗇^a on a true ideal but also 1𝖢𝗈𝗇 majorizes f on a true ideal. Theorem <ref> demonstrates that for any monotonic f with a sufficiently constrained range, f must agree cofinally with 𝖢𝗈𝗇. We would like to strengthen cofinally to on a true ideal. One strategy for establishing this claim would be to show that every set that is closed under 𝖤𝖠 provable equivalence and that contains cofinally many true sentences also contains every sentence in some true ideal. We show that this strategy fails. There is a recursively enumerable set 𝒜 that contains arbitrarily strong true sentences and that is closed under 𝖤𝖠 provable equivalence but does not contain any true ideals. It is not clear whether Theorem <ref> can be strengthened in the desired manner.§ NO MONOTONIC FUNCTION IS STRICTLY BETWEEN THE IDENTITY AND 𝖢𝗈𝗇 In this section we prove that no monotonic function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ∧𝖢𝗈𝗇(φ)). Most of the work is contained in the proof of the following lemma.Let f be a monotonic function such that for all consistent φ, f(φ) strictly implies φ. Then for every true sentence φ there is a true sentence θ such that θ⊢φ and f(θ)⊢ (θ∧𝖢𝗈𝗇(θ)).Let f be as in the statement of the theorem. By assumption the following statement is true. χ:=∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))) Let φ be a true sentence. Then the sentence ψ:=φ∧χ is true. Letθ:=(ψ∧ (f(ψ)→𝖢𝗈𝗇(ψ))).Note that θ⊢φ. f(θ)⊢ (θ∧ f(ψ)). Clearly θ⊢ψ. So f(θ)⊢ f(ψ) since f is monotonic. Also f(θ)⊢θ by assumption. (θ∧ f(ψ))⊢ (ψ∧𝖢𝗈𝗇(ψ)). Immediate from the definition of θ. (ψ∧𝖢𝗈𝗇(ψ))⊢ (θ∧𝖢𝗈𝗇(θ)). Clearly (ψ∧𝖢𝗈𝗇(ψ))⊢θ. It suffices to show that(ψ∧𝖢𝗈𝗇(ψ))⊢𝖢𝗈𝗇(θ).We reason as follows.(ψ∧𝖢𝗈𝗇(ψ)) ⊢∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧f(ζ)))by choice of ψ. ⊢𝖢𝗈𝗇(ψ)→𝖢𝗈𝗇(ψ∧f(ψ))by instantiation.⊢𝖢𝗈𝗇(ψ∧f(ψ))by logic. ⊢𝖢𝗈𝗇(θ)by the definition of θ. It is immediate from the preceding claims that f(θ)⊢ (θ∧𝖢𝗈𝗇(θ)). A number of results follow immediately from the lemma.Let f be monotonic. Suppose that for all consistent φ,(i) φ∧𝖢𝗈𝗇(φ) implies f(φ) and(ii) f(φ) strictly implies φ. Then for every true φ, there is a true ψ such that ψ⊢φ and [f(ψ)] = [ψ∧𝖢𝗈𝗇(ψ)].By the lemma, for every true φ there is a true ψ such that ψ⊢φ and f(ψ)⊢ (ψ∧𝖢𝗈𝗇(ψ)). Since we are assuming that (ψ∧𝖢𝗈𝗇(ψ))⊢ f(ψ), it follows that [f(ψ)]=[ψ∧𝖢𝗈𝗇(ψ)]. We note that this theorem applies to a number of previously studied operators. For instance, the theorem applies to the notion of cut-free consistency, i.e., consistency with respect to cut-free proofs. 𝖤𝖠 does not prove the cut-elimination theorem, which is equivalent to the totality of super-exponentiation (over 𝖤𝖠), and does not prove the equivalence of cut-free consistency and consistency. Another such operator is the Friedman-Rathjen-Weiermann slow consistency operator discussed in §1. Theorem <ref> implies that these operators exhibit the same behavior as the consistency operator “in the limit.” Indeed, for any φ such that φ proves the cut-elimination theorem, φ∧𝖢𝗈𝗇(φ) and φ∧𝖢𝗈𝗇_𝖢𝖥(φ) are 𝖤𝖠-provably equivalent. Likewise, for any φ that proves the totality of F_ϵ_0, φ∧𝖢𝗈𝗇(φ) and φ∧𝖲𝗅𝗈𝗐𝖢𝗈𝗇(φ) are 𝖤𝖠-provably equivalent.As a corollary of Theorem <ref> we note that no monotonic function reliably produces sentences strictly between those produced by the identity and by 𝖢𝗈𝗇. There is no monotonic function f such that for all consistent φ, (i) φ∧𝖢𝗈𝗇(φ) strictly implies f(φ) and (ii) f(φ) strictly implies φ. Shavrukov and Visser <cit.> studied functions over Lindenbaum algberas and discovered a recursive extensional uniform density function g for the Lindenbaum algebra of 𝖤𝖠, i.e., (i) for any φ and ψ such that ψ strictly implies φ, g(⟨φ,ψ⟩) is a sentence with deductive strength strictly between φ and ψ and (ii) if 𝖤𝖠⊢(φ↔ψ) then, for any θ, [g(⟨φ,θ⟩)]=[g(⟨ψ,θ⟩)] and [g(⟨θ,φ⟩)]=[g(⟨θ,ψ⟩)]. They asked whether this result could be strengthened by exhibiting a recursive uniform density function that is monotonic in both its coordinates. As a corollary of our theorem we answer their question negatively. There is no monotonic uniform density function for the Lindenbaum algebra of 𝖤𝖠.Suppose there were such a function g over the Lindenbaum algebra of 𝖤𝖠. Then given any input of the form ⟨φ,(φ∧𝖢𝗈𝗇(φ))⟩, g would produce a sentence with deductive strength strictly between φ and (φ∧𝖢𝗈𝗇(φ)). We then note that f:φ↦ g(⟨φ,(φ∧𝖢𝗈𝗇(φ))⟩) is monotonic, but that for every consistent φ, φ∧𝖢𝗈𝗇(φ) strictly implies f(φ) and f(φ) strictly implies φ, contradicting the previous theorem. Our negative answer to the question raised by Shavrukov and Visser makes use of a Π^0_2 sentence ∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))). Shavrukov and Visser raised the following question in private communication. Is there a recursive uniform density function for the lattice of Π^0_1 sentences over 𝖤𝖠 that is monotonic in both its coordinates? It is clear from the proof of the lemma that any monotonic f meeting the hypotheses of Theorem <ref> is not only cofinally equivalent to 𝖢𝗈𝗇; for every true ψ that impliesχ:=∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))),there is a true θ such that θ⊢ψ and [ψ∧𝖢𝗈𝗇(ψ)]=[θ∧𝖢𝗈𝗇(θ)]=[f(θ)]. This observation points the way toward a corollary of our theorem; namely that any monotonic function strictly meeting the hypotheses of the theorem must have the same range as φ↦(φ∧𝖢𝗈𝗇(φ)) in the limit. To prove this, we first prove a version of jump inversion—φ↦ (φ∧𝖢𝗈𝗇(φ)) inversion—for Lindenbaum algebras. This is to say that the range of 𝖢𝗈𝗇 contains a true ideal in the Lindenbaum algebra. A similar result is established for true Π^0_2 sentences in <cit.>. Suppose φ⊢𝖢𝗈𝗇(⊤). Then for some ψ, [φ]=[(ψ∧𝖢𝗈𝗇(ψ))]. Let ψ:=𝖢𝗈𝗇(⊤)→φ. φ⊢(ψ∧𝖢𝗈𝗇(ψ)). Trivially, φ⊢ψ. Since φ⊢𝖢𝗈𝗇(⊤), it follows that from the formalized second incompleteness theorem, i.e., 𝖢𝗈𝗇(⊤)⊢𝖢𝗈𝗇(𝖢𝗈𝗇(⊤)), that φ⊢𝖢𝗈𝗇(𝖢𝗈𝗇(⊤)). But 𝖢𝗈𝗇(⊤) is the first disjunct of ψ, so φ⊢𝖢𝗈𝗇(ψ). (ψ∧𝖢𝗈𝗇(ψ))⊢φ. Note that 𝖢𝗈𝗇(ψ)⊢𝖢𝗈𝗇(⊤). The claim then follows since clearly (ψ∧𝖢𝗈𝗇(⊤))⊢φ.Let f be monotonic. Suppose that for all consistent φ,(i) φ∧𝖢𝗈𝗇(φ) implies f(φ) and(ii) f(φ) strictly implies φ.Then the intersection of the ranges of f and 𝖢𝗈𝗇 in the Lindenbaum algebra contains a true ideal.Let φ be a sentence such that φ⊢𝖢𝗈𝗇(⊤) and φ⊢∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))). Note that both of these sentences are true, and hence φ is in an element of a true ideal. By the previous proposition, there is a ψ such that [ψ∧𝖢𝗈𝗇(ψ)]=[φ]. By Remark <ref> there is a θ such that [f(θ)]=[ψ∧𝖢𝗈𝗇(ψ)], that is, φ is in the range of f. § ITERATING 𝖢𝗈𝗇 INTO THE TRANSFINITE By analogy with Martin's Conjecture, we would like to show that there is a natural well-ordered hierarchy of monotonic functions and that the successor for this well-ordering is induced by 𝖢𝗈𝗇. Thus, we define the iterates of 𝖢𝗈𝗇 along elementary presentations of well-orderings.By an elementary presentation of a recursive well-ordering we mean a pair (𝒟,<) of elementary formulas, such that (i) the relation < well-orders 𝒟 in the standard model of arithmetic and (ii) 𝖤𝖠 proves that < linearly orders the elements satisfying 𝒟, (iii) it is elementarily calculable whether an element represents zero or a successor or a limit and (iv) the elementary formulas defining the set of limit ordinals and the successor relation provably in 𝖤𝖠 satisfy their corresponding first order definitions in terms of <. Given an elementary presentation ⟨α,<⟩ of a recursive well-ordering and a sentence φ, we use Gödel's fixed point lemma to define sentences 𝐂𝐨𝐧^⋆(φ,β) for β<α as follows.𝖤𝖠⊢𝐂𝐨𝐧^⋆(φ,β)↔∀γ<β, 𝖢𝗈𝗇(φ∧𝐂𝐨𝐧^⋆(φ,γ)).We use the notation 𝖢𝗈𝗇^β(φ) for 𝐂𝐨𝐧^⋆(φ,β).Note that, since the following clauses are provable in 𝖤𝖠. * 𝖢𝗈𝗇^0(φ) ↔⊤* 𝖢𝗈𝗇^γ+1(φ) ↔𝖢𝗈𝗇(φ∧𝖢𝗈𝗇^γ(φ))* 𝖢𝗈𝗇^λ(φ) ↔∀γ<λ, 𝖢𝗈𝗇^γ(φ) for λ a limit.Note that this hierarchy is proper for true φ by Gödel's second incompleteness theorem. We need to prove that for transfinite α, 𝖢𝗈𝗇^α is monotonic over the Lindenbaum algebra of 𝖤𝖠. Before proving this claim we recall Schmerl's <cit.> technique of reflexive transfinite induction. Note that “𝖯𝗋(φ)” means that φ is provable in 𝖤𝖠.(Schmerl) Suppose that < is an elementary linear order and that 𝖤𝖠⊢∀α(𝖯𝗋(∀β<α ,A(β))→ A(α)). Then 𝖤𝖠⊢∀α A(α).From 𝖤𝖠⊢∀α(𝖯𝗋(∀β<α ,A(β))→ A(α)) we infer𝖤𝖠⊢𝖯𝗋(∀αA(α)) →∀α𝖯𝗋(∀β<α,A(β)) →∀αA(α).Löb's theorem, i.e., if 𝖤𝖠⊢𝖯𝗋(ζ)→ζ, then 𝖤𝖠⊢ζ, then yields 𝖤𝖠⊢∀α A(α). If φ⊢ψ, then 𝖢𝗈𝗇^α(φ)⊢𝖢𝗈𝗇^α(ψ).Let 𝒜(β) denote the claim that 𝖢𝗈𝗇^β(φ)⊢𝖢𝗈𝗇^β(ψ).We want to prove that 𝒜(α), without placing any restrictions on α. We prove the equivalent claim that 𝖤𝖠⊢𝒜(α). By Proposition <ref>, it suffices to show that𝖤𝖠⊢∀α(𝖯𝗋(∀β<α,𝒜(β))→𝒜(α)).Reason within 𝖤𝖠. Suppose that 𝖯𝗋(∀β<α,𝒜(β)), which is to say that 𝖯𝗋(∀β<α,𝖯𝗋 (𝖢𝗈𝗇^β(φ) →𝖢𝗈𝗇^β(ψ))).Since 𝖢𝗈𝗇^α(φ) contains 𝖤𝖠, we infer that𝖢𝗈𝗇^α(φ)⊢∀β<α𝖯𝗋(𝖢𝗈𝗇^β(φ) →𝖢𝗈𝗇^β(ψ)).Since 𝖢𝗈𝗇^α(φ) proves that for all β<α, 𝖤𝖠⊬𝖢𝗈𝗇^β(φ) we infer that𝖢𝗈𝗇^α(φ)⊢∀β<α𝖢𝗈𝗇(𝖢𝗈𝗇^β(ψ)).Thus, 𝖢𝗈𝗇^α(φ)⊢∀β<α(𝖢𝗈𝗇^β(ψ)).This concludes the proof of the proposition. Thus, for each predicate 𝖢𝗈𝗇^α the function φ↦(φ∧𝖢𝗈𝗇^α(φ)) is monotonic over the Lindenbaum algebra of 𝖤𝖠. In this section we show that the functions given by iterated consistency are minimal with respect to each other. We fix an elementary presentation α of a recursive well-ordering. We assume that f is a monotonic function such that for every consistent φ, f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. We would like to relativize the proof of Lemma <ref> to 𝖢𝗈𝗇^β. However, the proof of Lemma <ref> relied on the truth of the principle ∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))).It is not in general clear that 𝖢𝗈𝗇^α(φ) implies 𝖢𝗈𝗇^α(φ∧ f(φ)). To solve this problem, we define a sequence of true sentences (θ_β)_β≤α such that for every sentence φ, if φ⊢θ_β then 𝖢𝗈𝗇^β(φ) implies 𝖢𝗈𝗇^β(φ∧ f(φ)). Thus, we are able to relativize the proof of Lemma <ref> for 𝖢𝗈𝗇^β to sentences that imply θ_β. Given an elementary presentation α of a recursive well-ordering, we use Gödel's fixed point lemma to define sentences θ^⋆(β) for β<α as follows.𝖤𝖠⊢θ^⋆(β)↔∀γ<β(𝖳𝗋𝗎𝖾_Π_3(θ^⋆(γ))) ∧∀ζ((∀γ<β𝖯𝗋(ζ→θ^⋆(γ))) →(𝖢𝗈𝗇^β(ζ) →𝖢𝗈𝗇^β(ζ∧f(ζ)))). We use the notation θ_β for θ^⋆(β). Note that every sentence in the sequence (θ_β)_β≤α has complexity Π^0_3. Note moreover that for a successor β+1, θ_β+1 is equivalent to θ_β∧∀ζ(𝖯𝗋(ζ→θ_β) →(𝖢𝗈𝗇^β+1(ζ)→𝖢𝗈𝗇^β+1(ζ∧ f(ζ)))). Let f be monotonic such that, for all φ, (i) φ∧𝖢𝗈𝗇^α(φ) implies f(φ),(ii) if [f(φ)]≠[], then f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. Then for each β≤α, the sentence θ_β is true.Let f be as in the statement of the lemma. We prove the claim by induction on β≤α. The base case β=0 is trivial. For the successor case we assume that β<α and that θ_β is true; we want to show that θ_β+1 is true. So let ζ be a sentence such that ζ⊢θ_β. We want to show that 𝖢𝗈𝗇^β+1(ζ) implies 𝖢𝗈𝗇^β+1(ζ∧ f(ζ)). We prove the contrapositive, that 𝖢𝗈𝗇^β+1(ζ∧ f(ζ)) implies 𝖢𝗈𝗇^β+1(ζ).So suppose 𝖢𝗈𝗇^β+1(ζ∧ f(ζ)), i.e., †ζ∧ f(ζ)⊢𝖢𝗈𝗇^β(ζ∧ f(ζ)). We reason as follows.Since ζ⊢θ_β, ζ⊢∀γ<β, 𝖳𝗋𝗎𝖾_Π_3(θ_γ). From this we infer⋆ζ⊢𝖯𝗋(ζ→∀γ<β, 𝖳𝗋𝗎𝖾_Π_3(θ_γ)) by Σ^0_1 completeness. Moreover, since ζ⊢θ_β,ζ ⊢∀φ((∀γ<β𝖯𝗋(φ→θ_γ)) →(𝖢𝗈𝗇^β(φ) →𝖢𝗈𝗇^β(φ∧f(φ)) ) )by the definition of θ_β.⊢∀γ<β𝖯𝗋 (ζ→θ_γ) →(𝖢𝗈𝗇^β(ζ) →𝖢𝗈𝗇^β(ζ∧f(ζ)))by instantiation.⊢𝖢𝗈𝗇^β(ζ)→𝖢𝗈𝗇^β(ζ∧f(ζ))by (<ref>).ζ∧f(ζ) ⊢𝖢𝗈𝗇^β(ζ∧f(ζ)) by (<ref>). ⊢𝖢𝗈𝗇^β(ζ) by logic.ζ ⊢𝖢𝗈𝗇^β(ζ)→f(ζ)by logic.Thus, (ζ∧𝖢𝗈𝗇^β(ζ))⊢ f(ζ). Since f(φ) always strictly implies φ∧𝖢𝗈𝗇^β(φ), we infer that [ζ∧𝖢𝗈𝗇^β(ζ)]=[]. This is to say that 𝖢𝗈𝗇^β+1(ζ). For the limit case we let β be a limit ordinal and assume that for every γ<β, θ_γ is true. We want to show that θ_β is true. Let ζ be a sentence such that for every γ<β, ζ⊢θ_γ. We want to show that 𝖢𝗈𝗇^β(ζ) implies 𝖢𝗈𝗇^β(ζ∧ f(ζ)). So assume that 𝖢𝗈𝗇^β(ζ), i.e., for every γ<β, 𝖢𝗈𝗇^γ(ζ). Let γ<β. Since β is a limit ordinal, γ+1<β. So by the inductive hypothesis θ_γ+1 is true. That is, by the definition of θ_γ+1,∀φ(𝖯𝗋(φ→θ_γ)→ (𝖢𝗈𝗇^γ(φ)→𝖢𝗈𝗇^γ(φ∧ f(φ)))). By instantiation, we infer that 𝖯𝗋(ζ→θ_γ)→ ( 𝖢𝗈𝗇^γ(ζ)→𝖢𝗈𝗇^γ(ζ∧ f(ζ))). Since ζ⊢θ_γ and 𝖢𝗈𝗇^γ(ζ), this means that 𝖢𝗈𝗇^γ(ζ∧ f(ζ)). Since γ was a generic ordinal less than β, we get that∀γ<β, 𝖢𝗈𝗇^γ(ζ∧ f(ζ)),i.e., 𝖢𝗈𝗇^β(ζ). This completes the proof of the lemma.Let f be monotonic. Suppose that for all φ, (i) φ∧𝖢𝗈𝗇^α(φ) implies f(φ),(ii) if [f(φ)]≠[], then f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. Then for every true χ, there is a true ψ such that ψ⊢χ and [f(ψ)] = [ψ∧𝖢𝗈𝗇^α(ψ)].Let χ be a true sentence. By the lemma, θ_α is true. Soφ:=χ∧θ_αis true. We letψ:=φ∧(f(φ)→𝖢𝗈𝗇^α(φ)).Note that ψ⊢χ. We now show that [ψ∧𝖢𝗈𝗇^α(ψ)]=[f(ψ)]. f(ψ)⊢ (ψ∧ f(φ)). Since f is monotonic. (ψ∧ f(φ))⊢ (φ∧𝖢𝗈𝗇^α(φ)). By the definition of ψ. (φ∧𝖢𝗈𝗇^α(φ))⊢ (ψ∧𝖢𝗈𝗇^α(ψ)). It is clear from the definition of ψ that (φ∧𝖢𝗈𝗇^α(φ))⊢ψ. So it suffices to show that (φ∧𝖢𝗈𝗇^α(φ))⊢𝖢𝗈𝗇^α(ψ).φ∧𝖢𝗈𝗇^α(φ) ⊢∀ζ((∀β<α𝖯𝗋(ζ→θ_β)) →(𝖢𝗈𝗇^α(ζ)→𝖢𝗈𝗇^α(ζ∧f(ζ)))) by choice of φ. ⊢∀β<α𝖯𝗋(φ→θ_β) →(𝖢𝗈𝗇^α(φ)→𝖢𝗈𝗇^α(φ∧f(φ))) by instantiation. ⊢∀β<α𝖯𝗋(φ→θ_β)→𝖢𝗈𝗇^α(φ∧f(φ)) by logic.Since 𝖢𝗈𝗇^α(φ∧ f(φ))⊢𝖢𝗈𝗇^α(ψ), to prove the desired claim it suffices to show thatφ∧𝖢𝗈𝗇^α(φ)⊢∀β<α𝖯𝗋(φ→θ_β). We reason as follows.φ ⊢θ_α by choice of φ. ⊢∀β<α(𝖳𝗋𝗎𝖾_Π_3θ_β)by definition of θ_α. ⊢𝖯𝗋(φ→∀β<α(𝖳𝗋𝗎𝖾_Π_3θ_β))by Σ^0_1 completeness. ⊢∀β<α𝖯𝗋(φ→𝖳𝗋𝗎𝖾_Π_3 θ_β) ⊢∀β<α𝖯𝗋(φ→θ_β)It is immediate from the preceding claims that f(ψ)⊢ψ∧𝖢𝗈𝗇^α(ψ). By assumption, ψ + 𝖢𝗈𝗇^α(ψ) ⊢ f(ψ), so it follows that [f(ψ)]=[ψ∧𝖢𝗈𝗇^α(ψ)].There is no monotonic f such that for all φ, if [φ∧𝖢𝗈𝗇^α(φ)]≠[], then both (i) φ∧𝖢𝗈𝗇^α(φ) strictly implies f(φ) and(ii) f(φ) strictly implies φ∧𝖢𝗈𝗇^β(φ) for all β<α. § FINITE ITERATES OF 𝖢𝗈𝗇 ARE INEVITABLE In this section and the next section we prove that the iterates of 𝖢𝗈𝗇 are, in a sense, inevitable. First we show that, for every natural number n, if a monotonic function f is always bounded by 𝖢𝗈𝗇^n, then it is somewhere equivalent to 𝖢𝗈𝗇^k for some k≤ n. In §5, we turn to generalizations of this result into the effective transfinite.Let n∈ℕ. Let f be a monotonic function such that for every φ, (i) φ∧𝖢𝗈𝗇^n(φ) implies f(φ) and (ii) f(φ) implies φ. Then for some φ and some k≤ n, [f(φ)]=[φ∧𝖢𝗈𝗇^k(φ)]≠[].We suppose, towards a contradiction, that there is no ψ and no k≤ n such that [f(ψ)]=[ψ∧𝖢𝗈𝗇^k(ψ)]≠[]. We then let φ_1 be a true statement such thatφ_1⊢∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧f(ζ))) φ_1 ⊢∀k∀ζ(𝖢𝗈𝗇^k+1(ζ)→𝖯𝗋((ζ∧𝖢𝗈𝗇^k(ζ))↔f(ζ))).The first condition is that φ_1 proves that for every consistent φ, f(φ) strictly implies φ. The second condition is that φ_1 proves that f(ζ) never coincides with ζ∧𝖢𝗈𝗇^k(ζ), unless [ζ∧𝖢𝗈𝗇^k(ζ)]=[].We define a sequence of statements, starting with φ_1, as follows:φ_k+1:=φ_k∧(f(φ_k)→𝖢𝗈𝗇^k(φ_k)).We will use our assumption to show that, for all k, φ_k∧𝖢𝗈𝗇^k(φ_k)⊢𝖢𝗈𝗇^k(φ_k+1). From this we will deduce that [f(φ_n+1)]=[φ_n+1∧𝖢𝗈𝗇^n(φ_n+1)]≠[], contradicting the assumption that f and 𝖢𝗈𝗇^n never coincide. Most of the work is contained in the proof of the following lemma.For all k, for all j≥ k, (φ_k∧𝖢𝗈𝗇^k(φ_k)) ⊢𝖢𝗈𝗇^k(φ_j).We prove the claim by a double induction. The primary induction is on k. For the base case k=1, we prove the claim by induction on j. The base case j=1 follows trivially. For the inductive step we assume that (φ_1∧𝖢𝗈𝗇(φ_1))⊢𝖢𝗈𝗇(φ_j) and show that (φ_1∧𝖢𝗈𝗇(φ_1))⊢𝖢𝗈𝗇(φ_j+1).φ_1∧𝖢𝗈𝗇(φ_1) ⊢∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧f(ζ))) by choice of φ_1. ⊢𝖢𝗈𝗇(φ_j)→𝖢𝗈𝗇(φ_j∧f(φ_j))by instantiation.φ_1∧𝖢𝗈𝗇(φ_1) ⊢𝖢𝗈𝗇(φ_j) by the inductive hypothesis. ⊢𝖢𝗈𝗇(φ_j∧f(φ_j))by logic. ⊢𝖢𝗈𝗇(φ_j+1)by definition of φ_j+1.For the inductive step we assume that the claim is true of k-1, i.e., ∀ j≥ k-1((φ_k-1∧𝖢𝗈𝗇^k-1(φ_k-1))⊢(𝖢𝗈𝗇^k-1(φ_j))). We prove the claim for k. Once again, we prove the claim by induction on j. The base case j=k follows trivially. For the inductive step we assume that φ_k∧𝖢𝗈𝗇^k(φ_k)⊢𝖢𝗈𝗇^k(φ_j). We want to prove that φ_k∧𝖢𝗈𝗇^k(φ_k)⊢𝖢𝗈𝗇^k(φ_j+1).φ_k∧𝖢𝗈𝗇^k(φ_k) ⊢∀x∀ζ(𝖢𝗈𝗇^x+1(ζ)→𝖯𝗋((ζ∧𝖢𝗈𝗇^x(ζ))↔f(ζ)))by choice of φ_1. ⊢𝖢𝗈𝗇^k(φ_j)→𝖯𝗋((φ_j∧𝖢𝗈𝗇^k-1(φ_j))↔f(φ_j))by instantiation.φ_k∧𝖢𝗈𝗇^k(φ_k) ⊢𝖢𝗈𝗇^k(φ_j)by the inner inductive hypothesis. ⊢𝖯𝗋((φ_j∧𝖢𝗈𝗇^k-1(φ_j))↔f(φ_j))by logic.Thus, φ_k∧𝖢𝗈𝗇^k(φ_k) proves that one of the following cases holds.(φ_j∧𝖢𝗈𝗇^k-1(φ_j))⊬ f(φ_j)f(φ_j)⊬(φ_j∧𝖢𝗈𝗇^k-1φ_j) We now show that φ_k∧𝖢𝗈𝗇^k(φ_k) refutes the second option. φ_k∧𝖢𝗈𝗇^k(φ_k)⊢𝖯𝗋(f(φ_j)→(φ_j∧𝖢𝗈𝗇^k-1φ_j)). By the outer inductive hypothesis, 𝖤𝖠 proves the following conditional: θ:=((φ_j-1∧𝖢𝗈𝗇^k-1(φ_j-1))→(𝖢𝗈𝗇^k-1(φ_j))). Thus, f(φ_j) (which contains 𝖤𝖠) also proves θ.We now show that f(φ_j)⊢𝖢𝗈𝗇^k-1(φ_j).f(φ_j) ⊢φ_j∧f(φ_j-1)since f is monotonic. ⊢(φ_j-1∧(f(φ_j-1)→𝖢𝗈𝗇^j-1(φ_j-1)))∧f(φ_j-1)by the definition of φ_j. ⊢φ_j-1∧𝖢𝗈𝗇^j-1(φ_j-1)by logic. ⊢φ_j-1∧𝖢𝗈𝗇^k-1(φ_j-1)since j≥ k. ⊢𝖢𝗈𝗇^k-1(φ_j)since f(φ_j) proves θ.By Σ^0_1 completeness, (φ_k∧𝖢𝗈𝗇^k(φ_k))⊢𝖯𝗋(f(φ_j) →𝖢𝗈𝗇^k-1(φ_j)). (φ_k∧𝖢𝗈𝗇^k(φ_k)) ⊢𝖢𝗈𝗇^k (φ_j+1). We reason as follows.(φ_k∧𝖢𝗈𝗇^k(φ_k)) ⊢𝖯𝗋((φ_j ∧𝖢𝗈𝗇^k-1(φ_j))→f(φ_j))by the previous claim.⊢𝖢𝗈𝗇(φ_j∧f(φ_j)∧𝖢𝗈𝗇^k-1(φ_j)).⊢𝖢𝗈𝗇(φ_j+1∧𝖢𝗈𝗇^k-1(φ_j))by the definition of φ_j+1. ⊢𝖢𝗈𝗇(φ_j+1∧𝖢𝗈𝗇^k-1(φ_j+1))by the outer inductive hypothesis. ⊢𝖢𝗈𝗇^k(φ_j+1)by definition of 𝖢𝗈𝗇^k.This concludes the proof of the lemma. As an instance of the lemma, we get that (φ_n∧𝖢𝗈𝗇^n(φ_n)) ⊢𝖢𝗈𝗇^n(φ_n+1). We reason as follows.f(φ_n+1) ⊢φ_n∧(f(φ_n)→𝖢𝗈𝗇^n(φ_n)) by the definition of φ_n+1.f(φ_n+1) ⊢f(φ_n) since f is monotonic. ⊢𝖢𝗈𝗇^n(φ_n) by logic. ⊢𝖢𝗈𝗇^n(φ_n+1) by the lemma.On the other hand, φ_n+1∧𝖢𝗈𝗇^n(φ_n+1)⊢ f(φ_n+1) since f is everywhere bounded by 𝖢𝗈𝗇^n. Thus, [f(φ_n+1)]=[φ_n+1∧𝖢𝗈𝗇^n(φ_n+1)], contradicting the assumption that there is no ψ and no k≤ n such that [f(ψ)]=[ψ∧𝖢𝗈𝗇^k(ψ)]≠[].§ TRANSFINITE ITERATES OF 𝖢𝗈𝗇 ARE INEVITABLE. Generalizing the proof of Theorem <ref> into the transfinite poses the following difficulty. Recall that the proof of Theorem <ref> makes use of a sequence of sentences starting with φ_0:=⊤ whereφ_k+1:=φ_k∧(f(φ_k)→𝖢𝗈𝗇^k(φ_k)).It is not clear what the ωth sentence in the sequence should be. A natural idea is that for a limit ordinal λ the corresponding “limit sentence” should quantify over the sentences in the sequence beneath it and express, roughly, ∀γ<λ(𝖳𝗋𝗎𝖾(φ_γ) ∧ (𝖳𝗋𝗎𝖾(f(φ_γ))→𝖢𝗈𝗇^γ(φ_γ))). However, if the sentences in the sequence (φ_γ)_γ<λ have unbounded syntactic complexity, then we are not guaranteed to have a truth-predicate with which we can quantify over them.Nevertheless, we show that Theorem <ref> generalizes into the transfinite given an additional assumption on complexity. Note that φ↦(φ∧𝖢𝗈𝗇(φ)) can be factored into two functions—the identity and φ↦𝖢𝗈𝗇(φ)—the latter of which always produces a Π^0_1 sentence. For the rest of this section, we will focus on monotonic functions φ↦φ∧ f(φ) where f is monotonic and also f(φ)∈Π^0_1 for all φ. A function f is Π^0_1 if f(φ)∈Π^0_1 for all φ. For the next theorem we fix an elementary presentation Γ of a recursive well-ordering. In the statement of the theorem and throughout the proof α, β, γ, δ, etc. are names of ordinals from the notation system Γ. Let f be a monotonic Π^0_1 function. Then either (i) for some β≤α and some φ, [φ∧ f(φ)]=[φ∧𝖢𝗈𝗇^β(φ)]≠[] or(ii) for some φ, (φ∧𝖢𝗈𝗇^α(φ))⊬ f(φ).Let f be a monotonic Π^0_1 function such that for every φ, (φ∧𝖢𝗈𝗇^α(φ))⊢ (φ∧ f(φ)). We assume, for the sake of contradiction, that there is no sentence ζ and no β≤α such that [ζ∧𝖢𝗈𝗇^β(ζ)]=[ζ∧ f(ζ)]≠[]. We then let φ be the conjunction of the following four sentences.∀ζ(𝖢𝗈𝗇(ζ)→𝖢𝗈𝗇(ζ∧ f(ζ))) ∀β≤α∀ζ(𝖢𝗈𝗇^β(ζ)→∀δ<β, 𝖯𝗋((ζ∧𝖢𝗈𝗇^δ(ζ))↔ (ζ∧ f(ζ)))) ∀ζ∀η( 𝖯𝗋(ζ→η) →𝖯𝗋(f(ζ)→ f(η)) ) ∀ x (𝖯𝗋(𝖳𝗋𝗎𝖾_Π^0_2(x))→𝖳𝗋𝗎𝖾_Π^0_2(x)) The first expresses that for every consistent φ, f(φ) strictly implies φ. The second sentence expresses that if β<α, then f(ζ) and ζ∧𝖢𝗈𝗇^β(ζ) never coincide, unless [ζ∧𝖢𝗈𝗇^β(ζ)]=[] . The third sentence expresses the monotonicity of f. The fourth sentence expresses the Π^0_2 soundness of 𝖤𝖠. Note that each of these sentences is true, so their conjunction φ is also true. Each of the four sentences is Π^0_2, whence so is φ. We are interested in the following sequence (φ_β)_β≤Γ. Note that the sentences in the sequence (φ_β)_β≤Γ all have complexity Π^0_2. Note moreover that since φ_1 is true, so is φ_β for every β.φ_1 :=φ.φ_γ := φ_1∧∀δ<γ(𝖳𝗋𝗎𝖾_Π_1(f(φ_δ))→𝖢𝗈𝗇^δ(φ_δ))for γ>1. Formally, we define the sequence (φ_β)_β≤Γ by Gödel's fixed point lemma as in Definition <ref>.We may assume that the ordinal notation system Γ is provably linear in 𝖤𝖠. Thus, 𝖤𝖠⊢∀β≤α,∀γ<β(𝖳𝗋𝗎𝖾_Π_2(φ_β)→𝖳𝗋𝗎𝖾_Π_2(φ_γ)). Our goal is to show that [φ_α+1∧𝖢𝗈𝗇^α(φ_α+1)]=[φ_α+1∧ f(φ_α+1)]contradicting the assumption that f and 𝖢𝗈𝗇^α never coincide. The main lemmas needed to prove this result are the following.𝖤𝖠⊢∀γ≤α𝖯𝗋( (φ_γ∧ f(φ_γ)) →φ_α).𝖤𝖠⊢∀β≤α∀γ≤β𝖯𝗋((φ_β∧𝖢𝗈𝗇^γ(φ_β) )→𝖢𝗈𝗇^γ(φ_β∧ f(φ_β))). Lemma <ref> is needed to derive Lemma <ref>. We now show how we use Lemma <ref> to derive Theorem <ref>. As an instance of Lemma <ref>, letting α=β=γ, we infer that𝖤𝖠⊢𝖯𝗋((φ_α∧𝖢𝗈𝗇^α(φ_α)) →𝖢𝗈𝗇^α(φ_α∧ f(φ_α))).From the soundness of 𝖤𝖠, we infer that∓φ_α + 𝖢𝗈𝗇^α(φ_α) ⊢𝖢𝗈𝗇^α(φ_α∧ f(φ_α)).We then reason as follows.φ_α+1 ⊢φ_α∧(f(φ_α)→𝖢𝗈𝗇^α(φ_α)) by the definition of φ_α+1. f(φ_α+1) ⊢f(φ_α) since f is monotonic.φ_α+1 + f(φ_α+1) ⊢φ_α∧𝖢𝗈𝗇^α(φ_α) by logic. ⊢𝖢𝗈𝗇^α(φ_α+1) by ∓.On the other hand, φ_α+1 + 𝖢𝗈𝗇^α(φ_α+1)⊢ f(φ_α+1) since f is everywhere bounded by 𝖢𝗈𝗇^α. Since φ_1 is true, so too is φ_α+1, whence we infer that [φ_α+1∧𝖢𝗈𝗇^α(φ_α+1)]=[φ_α+1∧ f(φ_α+1)]≠[], contradicting the claim that there is no sentence ζ and no β≤α such that [ζ∧𝖢𝗈𝗇^β(ζ)]=[ζ∧ f(ζ)]≠[]. It remains to prove Lemma <ref> and Lemma <ref>. We devote one subsection to each. §.§ Proof of Lemma <ref> In this subsection we prove Lemma <ref>. First we recall the statement of the lemma.𝖤𝖠⊢∀γ≤α(𝖯𝗋(φ_γ∧ f(φ_γ)) →φ_α).We reason in 𝖤𝖠. Let γ≤α. We assume that η𝖳𝗋𝗎𝖾_Π_2(φ_γ)∧𝖳𝗋𝗎𝖾_Π_1(f(φ_γ)). We we want to derive φ_α, i.e.φ_1∧∀σ<α(𝖳𝗋𝗎𝖾_Π_1(f(φ_σ))→𝖢𝗈𝗇^σ(φ_σ)).The first conjunct follows trivially from the assumption that 𝖳𝗋𝗎𝖾_Π_2(φ_γ). We now prove the second conjunct of φ_α in two parts, first for all σ such that α>σ≥γ and then for all σ<γ. α>σ≥γ: From the assumption that 𝖳𝗋𝗎𝖾_Π^0_2(φ_γ) we infer that φ_1, whence we infer that f is monotonic. Thus, for all δ≥γ, f(φ_δ)⊢ f(φ_γ), i.e., 𝖤𝖠⊢(f(φ_δ)→ f(φ_γ)). From φ_1 we also infer that 𝖤𝖠 is Π^0_2 sound, and so we infer that for all δ≥γ, 𝖳𝗋𝗎𝖾_Π_1(f(φ_δ))→𝖳𝗋𝗎𝖾_Π_1(f(φ_γ)). From the assumption that 𝖳𝗋𝗎𝖾_Π_1(f(φ_γ)) we then infer that for all δ≥γ, 𝖳𝗋𝗎𝖾_Π_1(f(φ_δ)), whence for all δ≥γ, 𝖳𝗋𝗎𝖾_Π_1(f(φ_δ))→𝖢𝗈𝗇^δ(φ_δ).σ< γ: By Remark <ref>, <ref> implies that ∀σ<γ (𝖳𝗋𝗎𝖾_Π_1(f(φ_σ))→𝖢𝗈𝗇^σ(φ_σ)).This completes the proof of Lemma <ref>.§.§ Proof of Lemma <ref> In this subsection we prove Lemma <ref>. We recall the statement of Lemma <ref>.𝖤𝖠⊢∀β≤α∀γ≤β𝖯𝗋(φ_β +𝖢𝗈𝗇^γ(φ_β) →𝖢𝗈𝗇^γ(φ_β∧ f(φ_β))).The proof of this lemma is importantly different from the proof of Lemma <ref>. In particular, to push the induction through limit stages we need to know not only that the inductive hypothesis is true but also that it is provable in 𝖤𝖠. We resolve this issue by using Schmerl's technique of reflexive transfinite induction (see Proposition <ref>).In the proof of the lemma, we let 𝒞(γ,δ) abbreviate the claim that φ_δ+𝖢𝗈𝗇^γ(φ_δ)⊢𝖢𝗈𝗇^γ(φ_δ∧ f(φ_δ)).We want to show that 𝖤𝖠⊢∀β≤α(∀γ≤β(𝒞(γ,β))).By Proposition <ref> it suffices to show that 𝖤𝖠⊢∀α(𝖯𝗋(∀β≤α∀γ≤β𝒞(γ,β))→∀γ≤α𝒞(γ,α)).[The reader might expect that we need to write “β<α” instead of “β≤α” in the antecedent for this to match the statement of Proposition <ref>. However, it is clear from the proof of Proposition <ref> that this suffices.]Thus, we reason in 𝖤𝖠 and fix α. We assume that△𝖯𝗋(∀β≤α, ∀γ≤β, 𝒞(γ,β)).We let γ≤α and we want to show that 𝒞(γ,α).Since φ_α⊢φ we infer that ♯φ_α+𝖢𝗈𝗇^γ(φ_α)⊢∀δ<γ, 𝖯𝗋((φ_α∧𝖢𝗈𝗇^δ(φ_α)) ↔ (φ_α∧ f(φ_α))). We first note that bothφ_α ⊢∀δ<γ(𝖳𝗋𝗎𝖾_Π_1(f(φ_δ))→𝖢𝗈𝗇^δ(φ_δ))by the definition of φ_α and alsoφ_α+ f(φ_α) ⊢∀δ<γ𝖯𝗋(f(φ_α) →f(φ_δ))since φ_1 proves the monotonicity of f.⊢∀δ<γ(f(φ_α)→𝖳𝗋𝗎𝖾_Π_1(f(φ_δ)))since φ_1 proves the Π^0_2 soundness of 𝖤𝖠. ⊢∀δ<γ, 𝖳𝗋𝗎𝖾_Π_1(f(φ_δ))by logic.Thus, we may reason as follows.φ_α+f(φ_α) ⊢∀δ<γ, 𝖢𝗈𝗇^δ(φ_δ) ⊢∀δ<γ, 𝖢𝗈𝗇^δ(φ_δ∧f(φ_δ)))since (<ref>) delivers 𝒞(δ,δ). ⊢∀δ<γ, 𝖢𝗈𝗇^δ(φ_α)by Lemma <ref>.Thus, by Σ^0_1 completeness,𝖤𝖠⊢∀δ<γ𝖯𝗋( (φ_α∧ f(φ_α)) →𝖢𝗈𝗇^δ(φ_α)).Combined with (<ref>), this deliversφ_α+𝖢𝗈𝗇^γ(φ_α) ⊢∀δ<γ𝖯𝗋 ( (φ_α ∧𝖢𝗈𝗇^δ(φ_α)) →f(φ_α)). ⊢∀δ<γ, 𝖢𝗈𝗇(φ_α ∧f(φ_α) ∧𝖢𝗈𝗇^δ(φ_α)). ⊢∀δ<γ, 𝖢𝗈𝗇(φ_α ∧f(φ_α) ∧𝖢𝗈𝗇^δ(φ_α∧f(φ_α)))since (<ref>) delivers 𝒞(δ,α). ⊢𝖢𝗈𝗇^γ(φ_α∧f(φ_α)).This completes the proof of Lemma <ref>. Theorem <ref> shows the inevitability of the consistency operator. For a sufficiently constrained monotonic function f, f must coincide with an iterate of 𝖢𝗈𝗇 on some non-trivial sentence. However, it is not clear from the proofs of Theorem <ref> or Theorem <ref> that f must coincide with 𝖢𝗈𝗇 on a true sentence. Let f be a monotonic Π^0_1 function. Suppose that for every φ,(φ∧𝖢𝗈𝗇^α(φ))⊢ f(φ).Must there be some β≤α and some true φ such that[φ∧ f(φ)]=[φ∧𝖢𝗈𝗇^β(φ)]?§ 1-CONSISTENCY AND ITERATED CONSISTENCY Just as the Π^0_1 fragments of natural theories can often be approximated by iterated consistency statements, the Π^0_2 fragments of natural theories can often be approximated by iterated 1-consistency statements. A theory T is 1-consistent if T+Th_Π^0_1(ℕ) is consistent. The 1-consistency of 𝖤𝖠+φ can be expressed by the following Π^0_2 sentence, 1𝖢𝗈𝗇(φ):∀ x (𝖳𝗋𝗎𝖾_Π^0_1(x)→𝖢𝗈𝗇(φ∧𝖳𝗋𝗎𝖾_Π^0_1(x))). In this section, we investigate the relationship between 1-consistency and iterated consistency. First, we show that 1𝖢𝗈𝗇 majorizes every iterate of 𝖢𝗈𝗇^α.For any elementary presentation α of a recursive well ordering, there is a true sentence φ such that for every ψ, if ψ⊢φ, then (ψ∧ 1𝖢𝗈𝗇(ψ)) implies (ψ∧𝖢𝗈𝗇^α(ψ)). Moreover, if [ψ∧𝖢𝗈𝗇^α(ψ)]≠[] then (ψ∧ 1𝖢𝗈𝗇(ψ)) strictly implies (ψ∧𝖢𝗈𝗇^α(ψ)).Let α be an elementary presentation of a recursive well-ordering. Let φ be a true sentence such that φ⊢𝖳𝖨^α_Π^0_1, i.e., φ implies the validity of transfinite induction along α for Π^0_1 predicates. We prove that (φ∧ 1𝖢𝗈𝗇(φ))⊢𝖢𝗈𝗇^α+1(φ). Since φ∧ 1𝖢𝗈𝗇(φ)⊢𝖳𝖨^α_Π^0_1, it suffices to show that:Base case: (φ∧ 1𝖢𝗈𝗇(φ))⊢𝖢𝗈𝗇(φ)Successor case: (φ∧ 1𝖢𝗈𝗇(φ))⊢∀β<α( 𝖢𝗈𝗇^β(φ)→𝖢𝗈𝗇^β+1(φ))Limit case: (φ∧ 1𝖢𝗈𝗇(φ))⊢∀λ(lim(λ)→((∀β<λ𝖢𝗈𝗇^β(φ))→𝖢𝗈𝗇^λ(φ)))The base case and the limit case are both trivial. For the successor case we first note that by the definition of 1𝖢𝗈𝗇(φ),1𝖢𝗈𝗇(φ)⊢∀ x (𝖳𝗋𝗎𝖾_Π^0_1(x)→𝖢𝗈𝗇(φ∧𝖳𝗋𝗎𝖾_Π^0_1(x))),and so by substituting 𝖢𝗈𝗇^β(φ) in for x,⊕ 1𝖢𝗈𝗇(φ)⊢𝖳𝗋𝗎𝖾_Π^0_1(𝖢𝗈𝗇^β(φ))→𝖢𝗈𝗇(φ∧𝖳𝗋𝗎𝖾_Π^0_1(𝖢𝗈𝗇^β(φ))).Thus, we reason as follows.1𝖢𝗈𝗇(φ)⊢𝖢𝗈𝗇^β(φ) →𝖢𝗈𝗇(φ∧𝖳𝗋𝗎𝖾_Π^0_1(𝖢𝗈𝗇^β(φ)))by (<ref>). →𝖢𝗈𝗇(φ∧𝖢𝗈𝗇^β(φ)). →𝖢𝗈𝗇^β+1(φ)by the definition of 𝖢𝗈𝗇^β+1.It is clear that the implication φ∧1𝖢𝗈𝗇(φ)⊢φ∧𝖢𝗈𝗇^α(φ) is strict as long as [φ∧𝖢𝗈𝗇^α(φ)]≠[]. This completes the proof of the proposition. In light of the previous proposition, one might conjecture that 1𝖢𝗈𝗇 is the weakest monotonic function majorizing every function of the form 𝖢𝗈𝗇^α for some recursive well-ordering α on true sentences. However, this is not so. To demonstrate this, we use a recursive linear order that has no hyperarithmetic infinite descending sequences. Harrison <cit.> introduced such an ordering with order-type ω_1^CK×(1+ℚ); see also Feferman and Spector <cit.> who consider such orderings in the context of iterated reflection principles. We use a presentation ℋ of Harrison's ordering such satisfying the conditions explicated in Definition <ref>. We note that since ℋ has no hyperarithmetic descending sequences, transfinite induction along ℋ for Π^0_1 properties is valid. Our idea is to produce a function stronger than each 𝖢𝗈𝗇^α but weaker than 1𝖢𝗈𝗇 by iterating 𝖢𝗈𝗇 along the Harrison linear order.There are infinitely many monotonic functions f such that for every recursive ordinal α, there is an elementary presentation a of α such that f majorizes 𝖢𝗈𝗇^a on a true ideal but also 1𝖢𝗈𝗇 majorizes f on a true ideal.In Definition <ref>, we used Gödel's fixed point lemma to produce iterates of 𝖢𝗈𝗇 along an elementary well-ordering. We similarly use Gödel's fixed point lemma to define sentences 𝐂𝐨𝐧^⋆(φ,β) for β∈ℋ as follows.𝖤𝖠⊢𝐂𝐨𝐧^⋆(φ,β)↔∀γ<_ℋβ, 𝖢𝗈𝗇(φ∧𝐂𝐨𝐧^⋆(φ,γ)).We use the notation 𝖢𝗈𝗇^β(φ) for 𝐂𝐨𝐧^⋆(φ,β). Recall that we are assuming that it is elementarily calculable whether an element of ℋ is zero or a successor or a limit. Thus, the following clauses are provable in 𝖤𝖠. * 𝖢𝗈𝗇^0(φ) ↔⊤* 𝖢𝗈𝗇^γ+1(φ) ↔𝖢𝗈𝗇(φ∧𝖢𝗈𝗇^γ(φ))* 𝖢𝗈𝗇^λ(φ) ↔∀γ<_ℋλ, 𝖢𝗈𝗇^γ(φ) for λ a limit.For γ∈ℋ, the function φ↦Con^γ(φ) is monotonic. This follows immediately from Proposition <ref>. Note that in the statement of Lemma <ref> we assume only that < is an elementary linear ordering, not a well-ordering.There are infinitely many monotonic functions f such that for every recursive well-ordering α, there is an elementary presentation a of α such that f majorizes 𝖢𝗈𝗇^a on true sentences. If x<_ℋy then 𝖢𝗈𝗇^y(φ) strictly implies 𝖢𝗈𝗇^x(φ) for every φ such that 𝖢𝗈𝗇^x(φ)≠[]. Given the order type of ℋ, this means that for infinitely many γ, for every recursive well-ordering α, 𝖢𝗈𝗇^γ majorizes 𝖢𝗈𝗇^a where a represents α in ℋ. 1𝖢𝗈𝗇 majorizes 𝖢𝗈𝗇^a on true sentences for each a∈ℋ.Since every Π^0_1 definable subset of ω has an ℋ-least element, the sentence 𝖳𝖨^ℋ_Π^0_1, which expresses the validity of transfinite induction along ℋ for Π^0_1 predicates, is true. But then if φ⊢𝖳𝖨^ℋ_Π^0_1, then for any γ∈ℋ, (φ∧ 1𝖢𝗈𝗇(φ)) strictly implies (φ∧𝖢𝗈𝗇^γ(φ)) as long as [(φ∧𝖢𝗈𝗇^γ(φ))]≠[], as in Proposition <ref>. § AN UNBOUNDED RECURSIVELY ENUMERABLE SET THAT CONTAINS NO TRUE IDEALS In this section we prove a limitative result. Theorem <ref> demonstrates that if f is monotonic and that for all consistent φ, (i) φ∧𝖢𝗈𝗇(φ) implies f(φ) and (ii) f(φ) strictly implies φ, then for cofinally many true φ, [f(φ)] = [φ∧𝖢𝗈𝗇(φ)]. It is natural to conjecture that cofinal equivalence with 𝖢𝗈𝗇 be strengthened to equivalence to 𝖢𝗈𝗇 in the limit, i.e., on a true ideal. One strategy to strengthen Theorem <ref> in this way would be to show that every recursively enumerable set that contains arbitrarily strong true sentences and that is closed under provable equivalence contains a true ideal.We now show that the aforementioned strategy fails. To this end, we define a recursively enumerable set 𝒜 that contains arbitrarily strong true sentences and that is closed under provable equivalence but does not contain any true ideals. We are grateful to Matthew Harrison-Trainor for simplifying the proof of the following proposition.There is a recursively enumerable set 𝒜 that contains arbitrarily strong true sentences and that is closed under 𝖤𝖠 provable equivalence but does not contain any true ideals.Let {φ_0,φ_1,...} be an effective Gödel numbering of the language of arithmetic. We describe the construction of 𝒜 in stages. During a stage n we may activate a sentence ψ, in which case we say that ψ is active until it is deactivated at some later stage n+k. After describing the construction of 𝒜 we verify that A has the desired properties.Stage 0: Numerate φ_0 and φ_0 into 𝒜. Activate the sentences (φ_0∧𝖢𝗈𝗇(φ_0)) and (φ_0∧𝖢𝗈𝗇(φ_0)).Stage n+1: There are finitely many active sentences. For each such sentence ψ, numerate θ_0:=(ψ∧φ_n+1) and θ_1:=(ψ∧φ_n+1) into 𝒜. Deactivate the sentence ψ and activate the sentences (θ_0∧𝖢𝗈𝗇(θ_0)) and (θ_1∧𝖢𝗈𝗇(θ_1)).We dovetail the construction with a search through 𝖤𝖠 proofs. If we ever see that 𝖤𝖠⊢φ↔ψ for some φ that we have already numerated into 𝒜, then we numerate ψ into 𝒜.Now we check that 𝒜 has the desired properties. It is clear that 𝒜 is recursively enumerable and that 𝒜 is closed under 𝖤𝖠 provable equivalence. 𝒜 contains arbitrarily strong true sentences. That is, for each true sentence φ, there is a true sentence ψ such that ψ⊢φ and ψ∈𝒜. At any stage in the construction of 𝒜, there are finitely many active sentences, ψ_0, ..., ψ_k. An easy induction shows that exactly one of ψ_0,...,ψ_k is true. Indeed, exactly one of φ_0 or φ_0 is true, and hence so is exactly one of φ_0∧𝖢𝗈𝗇(φ_0) and φ_0∧𝖢𝗈𝗇(φ_0). And if θ is true, then so is exactly one of ζ_0:=θ∧φ_k and ζ_1:=θ∧φ_k, and hence so too is exactly one of ζ_0∧𝖢𝗈𝗇(ζ_0) and ζ_1∧𝖢𝗈𝗇(ζ_1).Let φ_k be a true sentence. At stage k in the construction of 𝒜 there are only finitely many active sentences ψ_0,...,ψ_n. We have already seen that exactly one of ψ_i is true. But then φ_k∧ψ_i is true, (φ_k∧ψ_i⊢φ_k), and (φ_k∧ψ_i) is numerated into 𝒜. 𝒜 contains no true ideals. An easy induction shows that if ψ_0 and ψ_1 are both active at the same stage, then for any θ, if θ implies both ψ_0 and ψ_1 then θ∈[].Let φ be a true sentence in 𝒜. By the previous remark, the only sentences in 𝒜 that strictly imply φ are (i) 𝖤𝖠 refutable sentences and (ii) sentences that imply φ∧𝖢𝗈𝗇(φ). Since the Lindenbaum algebra of 𝖤𝖠 is dense, this means there is some ψ such that (φ∧𝖢𝗈𝗇(φ)) strictly implies ψ strictly implies φ but ψ∉𝒜.The following questions remain. Is the relation of cofinal agreement on true sentences an equivalence relation on recursive monotonic operators?Let f be recursive and monotonic. Suppose that for all consistent φ,(i) φ∧𝖢𝗈𝗇(φ) implies f(φ) and(ii) f(φ) implies φ. Must f be equivalent to the identity or to 𝖢𝗈𝗇 on a true ideal? plain | http://arxiv.org/abs/1708.07615v3 | {
"authors": [
"Antonio Montalbán",
"James Walsh"
],
"categories": [
"math.LO",
"03F03"
],
"primary_category": "math.LO",
"published": "20170825052101",
"title": "On the inevitability of the consistency operator"
} |
On Secure Communication using RF Energy Harvesting Two-Way Untrusted Relay Vipul Gupta Dept. of EECSUniversity of California, Berkeley, CA, USAE-mail: [email protected] Sanket S. Kalamkar Dept. of EEUniversity of Notre Dame, IN, USAE-mail: [email protected] Adrish Banerjee Dept. of EEIIT Kanpur, IndiaE-mail: [email protected] December 30, 2023 ================================================================================================================================================================================================================================================================================= We focus on a scenario where two wireless source nodes wish to exchange confidential information via an RF energy harvesting untrusted two-way relay. Despite its cooperation in forwarding the information, the relay is considered untrusted out of the concern that it might attempt to decode the confidential information that is being relayed. To discourage the eavesdropping intention of the relay, we use a friendly jammer. Under the total power constraint, to maximize the sum-secrecy rate, we allocate the power among the sources and the jammer optimally and calculate the optimal power splitting ratio to balance between the energy harvesting and the information processing at the relay. We further examine the effect of imperfect channel state information at both sources on the sum-secrecy rate. Numerical results highlight the role of the jammer in achieving the secure communication under channel estimation errors. We have shown that, as the channel estimation error on any of the channels increases, the power allocated to the jammer decreases to abate the interference caused to the confidential information reception due to the imperfect cancellation of jammer's signal. Energy harvesting, imperfect channel state information, physical layer security, two-way relay, untrusted relay§ INTRODUCTION The demand for higher data rates has led to a shift towards higher frequency bands, resulting in higher path loss. Thus relays have become important for reliable long distance wireless transmissions. The two-way relay has received attention in the past few years due to its ability to make communications more spectral efficient <cit.>. In a two-way relay-assisted communication, the relay receives the information from two nodes simultaneously, which it broadcasts in the next slot.§.§ Motivation To improve the energy efficiency, harvesting energy from the surrounding environment has become a promising approach, which can prolong the lifetime of energy-constrained nodes and avoid frequent recharging and replacement of batteries. In <cit.> and <cit.>, authors have proposed the concept of energy harvesting using radio-frequency (RF) signals that carry information as a viable source of energy. Simultaneous wireless information and power transfer has applications in cooperative relaying. The works in <cit.> study throughput maximization problems when the cooperative relays harvest energy from incoming RF signals to forward the information, where references <cit.> have focused on two-way relaying. Though the open wireless medium has facilitated cooperative relaying, it has also allowed unintended nodes to eavesdrop the communication between two legitimate nodes. Traditional ways to achieve secure communication rely on upper-layer cryptographic methods that involve intensive key distribution. Unlike this technique, the physical layer security aims to achieve secure communication by exploiting the random nature of the wireless channel. In this regard, Wyner introduced the idea of secrecy rate for the wiretap channel, where the secure communication between two nodes was obtained without private keys <cit.>. For cooperative relaying with energy harvesting, <cit.> investigate relay-assisted secure communication in the presence of an external eavesdropper. The security of the confidential message may still be a concern when the source and the destination wish to keep the message secret from the relay, despite its help in forwarding the information <cit.>. Hence the relay is trusted in forwarding the information, but untrusted out of the concern that the relay might attempt to decode the confidential information that is being relayed.[In this case, the decode-and-forward relay is no longer suitable to forward the confidential information.] In practice, such a scenario may occur in heterogeneous networks, where all nodes do not possess the same right to access the confidential information. For example, if two nodes having the access to confidential information wish to exchange that information but do not have the direct link due to severe fading and shadowing, they might require to take the help from an intermediate node that does not have the privilege to access the confidential information. §.§ Related WorkIn <cit.>, authors show that the cooperation by an untrusted relay can be beneficial and can achieve higher secrecy rate than just treating the untrusted relay as a pure eavesdropper. In <cit.>, authors investigate the secure communication in untrusted two-way relay systems with the help of external friendly jammers and show that, though it is possible to achieve a non-zero secrecy rate without the friendly jammers, the secrecy rate at both sources can effectively be improved with the help from an external friendly jammer. In <cit.>, authors have focused on improving the energy efficiency while achieving the minimum secrecy rate for the untrusted two-way relay. The works in <cit.> assume that the relay is a conventional node and has a stable power supply.As to energy harvesting untrusted relaying, the works in <cit.> analyze the effect of untrusted energy harvesting one-way relay on the secure communication between two legitimate nodes. To the best of our knowledge, for energy harvesting two-way untrusted relay, the problem of achieving the secure communication has not been yet studied in the literature. §.§ ContributionsThe contributions and main results of this paper are as follows:* First, assuming the perfect channel state information (CSI) at source nodes, we extend the notion of secure communication via an untrusted relay for the two-way wireless-powered relay, as shown in Fig. <ref>. To discourage the eavesdropping intentions of the relay, a friendly jammer sends a jamming signal during relay's reception of signals from source nodes.* To harvest energy, the relay uses a part of the received RF signals which consist of two sources' transmissions and the jamming signal. Hence we utilize the jamming signal effectively as a source of extra energy in addition to its original purpose of degrading relay's eavesdropping channel.* Under the total power constraint, we exploit the structure of the original optimization problem and make use of the signomial geometric programming technique <cit.> to jointly find the optimal power splitting ratio for energy harvesting and the optimal power allocation among sources and the jammer that maximize the sum-secrecy rate for two source nodes.* Finally, with the imperfect CSI at source nodes, we study the joint effects of the energy harvesting nature of an untrusted relay and channel estimation errors on the sum-secrecy rate and the power allocated to the jammer. We particularly focus on the role of jammer in achieving the secure communication, where we show that the power allocated to the jammer decreases as the estimation error on any of the channels increases, in order to subside the detrimental effects of the imperfect cancellation of the jamming signal at source nodes.§ SECURE COMMUNICATION WITH PERFECT CSI §.§ System ModelFig. <ref> shows the communication protocol between two legitimate source nodes S_1 and S_2—lacking the direct link between them—via an untrusted two-way relay R. All nodes are half-duplex and have a single antenna <cit.>. To discourage eavesdropping by the relay, a friendly jammer J sends the jamming signal during relay's reception of sources' signals. The communication of a secret message between S_1 and S_2 happens over two slots of equal duration T/2. In the first slot, the nodes S_1 and S_2 jointly send their information to the relay with powers P_1 and P_2, respectively, and the jammer J sends the jamming signal with power P_J. The powers P_1, P_2, and P_J are restricted by the power budget P such that P_1 + P_2 + P_J ≤ P. This constraint may arise, for instance, when the sources and the jammer belong to the same network, and the network has a limited power budget to cater transmission requirements of sources and the jammer. The relay uses a part of the received power to harvest energy. In the second slot, using the harvested energy, the relay broadcasts the received signal in an amplify-and-forward manner.Let h_1, h_2, and h_J denote the channel coefficients of the reciprocal channels from the relay to S_1, S_2, and jammer J, respectively. In this section, we assume that both sources have the perfect CSI for all channels, which can be obtained from the classical channel training, estimation, and feedback from the relay. But if there are errors in the estimation and/or feedback, the sources will have imperfect CSI, which is the focus of Section <ref>. Hence the relay is basically trusted when it comes to providing the services like feeding CSI back to transmitters and forwarding the information but untrusted in the sense that it is not supposed to decode the confidential information that is being relayed <cit.>. Both sources have the perfect knowledge of the jamming signal <cit.>.[Jammer can use pseudo-random codes as the jamming signals that are known to both sources beforehand but not to the untrusted relay.]§.§ RF Energy Harvesting at RelayThe relay is an energy-starved node. It harvests energy from incoming RF signals which include information signals from nodes S_1 and S_2 and the jamming signal from the jammer. To harvest energy from received RF signals, the relay uses power splitting (PS) policy <cit.>. In PS policy, the relay uses a fraction β of the total received power for energy harvesting.Under PS policy, the energy harvested by the relay is[For the exposition, we assume that the incident power on the energy harvesting circuitry of the relay is sufficient to activate it.]E_H = βη(P_1|h_1|^2 + P_2|h_2|^2 + P_J|h_J|^2)(T/2), where η is the energy conversion efficiency factor with 0 < η < 1. The transmit power of the relay in the second slot is P_H = E_H/T/2 = βη(P_1|h_1|^2 + P_2|h_2|^2 + P_J|h_J|^2).§.§ Information Processing and Relaying ProtocolIn the first slot, the relay receives the signaly_R =√((1-β))(√(P_1)h_1x_1 + √(P_2)h_2x_2 + √(P_J)h_Jx_J) + n_R,where x_1 and x_2 are the messages of S_1 and S_2, respectively, with 𝔼[|x_1|^2] = 𝔼[|x_1|^2] = 1. Also x_J is the artificial noise by the jammer with 𝔼[|x_J|^2] = 1, and n_R is the additive white Gaussian noise (AWGN) at relay with mean zero and variance N_0. Using the received signal y_R, the relay may attempt to decode the confidential messages x_1 and x_2. To shield the confidential messages x_1 and x_2 from relay's eavesdropping, we assume that the physical layer security coding like stochastic encoding and nested code structure can be used (see <cit.> and <cit.>). The relay can decode one of the sources' confidential messages, i.e., either x_1 or x_2, if its rate is such that it can be decoded by considering other source's message as noise <cit.>. In this case, at relay, the signal-to-noise ratio (SNR) corresponding to x_1, i.e., the message intended for S_2, is given bySNR_R_2 = βP_1|h_1|^2/βP_2|h_2|^2 + βP_J|h_J|^2 + N_0,where β= 1-β. Accordingly the achievable throughput of S_1-R link is C_2^R = (1/2)log(1 + SNR_R_2). In (<ref>), the term βP_2|h_2|^2, corresponding to S_2's message for S_1 indirectly serves as an artificial noise for the relay in addition to the signal βP_J|h_J|^2 from the jammer. Similarly, the SNR corresponding to x_2, i.e., the message intended for S_1, is given bySNR_R_1 = βP_2|h_2|^2/βP_1|h_1|^2 + βP_J|h_J|^2 + N_0,where βP_1|h_1|^2 serves as an artificial noise for the relay. Thus the achievable throughput of S_2-R link is C_1^R = (1/2)log(1 + SNR_R_1). Let γ_i = P_i |h_i|^2/N_0, where i ∈{ 1,2,J}. It follows thatSNR_R_2 = βγ_1/βγ_2 + βγ_J + 1, SNR_R_1 = βγ_2/βγ_1 + βγ_J + 1.The relay amplifies the received signal y_R given by (<ref>) by a factor α based on its harvested power P_H. Accordingly,α = √(P_H/βP_1|h_1|^2 + βP_2|h_2|^2 + βP_J|h_J|^2 + N_0)= √(βη(γ_1 + γ_2 + γ_J)/βγ_1 + βγ_2 + βγ_J + 1).The received signal at S_2 in the second slot is given byy_2 = h_2(α y_R) + n_2,where n_2 is AWGN with power N_0. We assume that S_1 and S_2 know x_J beforehand. Hence after cancelling the terms that are known to S_2, i.e., the terms corresponding to x_2 and x_J, the resultant received signal at S_2 isy_2 = h_2α√(βP_1)h_1x_1_desired signal + h_2α n_R + n_2_noise.The perfect CSI allows S_2 to cancel unwanted components of the signal. Substituting α from (<ref>) in (<ref>), we can express the SNR at node S_2 asSNR_S_2 = γ_1|h_2|^2ββη(γ_1 + γ_2 + γ_J)/(|h_2|^2βη + β)(γ_1 + γ_2 + γ_J) + 1,and the corresponding achievable throughput of link R-S_2 is C_2^S = (1/2)log(1 + SNR_S_2). Similarly the received signal at S_1 isy_1 = h_1α√(βP_2)h_2x_2_desired signal + h_1α n_R + n_1_noise. The SNR at node S_1 isSNR_S_1 = γ_2|h_1|^2ββη(γ_1 + γ_2 + γ_J)/(|h_1|^2βη + β)(γ_1 + γ_2 + γ_J) + 1,and the corresponding achievable throughput of link R-S_1 is C_1^S = (1/2)log(1 + SNR_S_1). §.§ Secrecy Rate and Problem FormulationFor the communication via two-way untrusted relay, the sum-secrecy rate is given byC_S= [C_1^S - C_1^R ]^+ + [C_2^S - C_2^R ]^+ = [1/2log_2(1+ SNR_S_1) - 1/2log_2(1+ SNR_R_1)]^+ + [1/2log_2(1+ SNR_S_2) - 1/2log_2(1+ SNR_R_2)]^+,where [x]^+ ≜max(x,0). Given the total power budget P, we have a constraint on transmit powers, i.e., P_1 + P_2 + P_J ≤ P. To maximize the sum-secrecy rate, we optimally allocate powers P_1, P_2, and P_J to S_1, S_2, and J, respectively, and find the optimal power splitting ratio β. We can formulate the optimization problem asmaximize_β, β, P_1, P_2,P_J C_S subject to P_1 + P_2 + P_J ≤ P, β + β= 1, β, β≤ 1, β, β, P_1, P_2, P_J≥ 0. Based on the non-negativeness of two terms in the secrecy rate expression given by (<ref>), we need to investigate four cases. We calculate the sum-secrecy rate in all four cases, with the best case being the one that gives the maximum sum-secrecy rate.§.§ Case I: C_1^S - C_1^R ≥ 0 and C_2^S - C_2^R ≥ 0Substituting γ_i = P_i|h_i|^2/N_0 and simplifying the problem in (<ref>), it follows that minimize_β, β, P_1, P_2,P_J 1/2log_2f(β,β, γ_1,γ_2, γ_J)/g(β,β,γ_1,γ_2, γ_J) subject to γ_1N_0/|h_1|^2 + γ_2N_0/|h_2|^2 + γ_JN_0/|h_J|^2≤ P , β + β= 1, β, β≤ 1, β, β, P_1, P_2, P_J≥ 0, wheref(β,β,γ_1,γ_2, γ_J)= [β(γ_1 + γ_2 + γ_J) + 1]^2× 1 + (γ_1 + γ_2 + γ_J)(|h_2|^2βη + β)× 1 + (γ_1 + γ_2 + γ_J)(|h_1|^2βη + β), andg(β,β,γ_1,γ_2, γ_J) = (β(γ_2 + γ_J) + 1)(β(γ_1 + γ_J) + 1)×(γ_1 + γ_2 + γ_J)(β+ |h_2|^2βη(βγ_1 + 1))+1 ×(γ_1 + γ_2 + γ_J)(β+ |h_1|^2βη(βγ_2 + 1))+1.We can drop the logarithm from the objective (<ref>) as it retains the monotonicity and yields the same optimal solution. We introduce an auxiliary variable t and do the following transformation. minimize_β, β, P_1, P_2,P_J f(β,β,γ_1,γ_2, γ_J)/t subject to t ≤ g(β,β,γ_1,γ_2, γ_J), γ_1N_0/|h_1|^2 + γ_2N_0/|h_2|^2 + γ_JN_0/|h_J|^2≤ P , β + β≤ 1, β, β≤ 1, t, β, β, P_1, P_2, P_J≥ 0 .The above transformation is valid for t>0 because, to minimize the objective f(β,β,γ_1,γ_2, γ_J)/t, we need to maximize t, and it happens when t = g(β,γ_1,γ_2, γ_J). Hence under the optimal condition, we have t = g(β,γ_1,γ_2, γ_J), and the problems (<ref>) and (<ref>) are equivalent. Further we can replace the constraint (<ref>) byβ + β≤ 1.The substitution of (<ref>) by (<ref>) in problem (<ref>) yields an equivalent problem because β + β= 1 under the optimal condition, i.e., if β + β< 1, we can always increase the value of β so that β + β= 1. The increase in β leads to more harvested energy, which in turn increases the transmit power of the relay and the sum-secrecy rate. The objective (<ref>) is a posynomial function and (<ref>), (<ref>), and (<ref>) are posynomial constraints <cit.>. When the objective and constraints are of posynomial form, the problem can be transformed into a Geometric Programming (GP) form and converted into a convex problem <cit.>. Also, as the domain of GP problem includes only real positive variables, the constraint (<ref>) is implicit. But the constraint (<ref>) is not posynomial as it contains a posynomial function g which is bounded from below and GP cannot handle such constraints. We can solve this problem if the right-hand side of (<ref>), i.e., g(β,β,γ_1,γ_2, γ_J), can be approximated by a monomial. Then the problem (<ref>) reduces to a class of problems that can be solved by Signomial Geometric Programming (SGP) <cit.>. To find a monomial approximation of the form g(𝐱) = c∏_i=1^5x_i^a_i of a function g(𝐱) where 𝐱 = [β,β,γ_1,γ_2, γ_J]^T is the vector containing all variables, it would suffice if we find an affine approximation of h(𝐲) = log g(𝐲) with ith element of 𝐲 given by y_i = log x_i <cit.>. Let the affine approximation of h(𝐲) be h(𝐲) = logg(𝐱) = log c + 𝐚^T𝐲. Using Taylor's approximation of h(𝐲) around the point 𝐲_0 in the feasible region and equating it with h(𝐲), it follows thath(𝐲) ≈ h(𝐲_0) + ∇ h(𝐲_0)^T(𝐲 - 𝐲_0) = log c + 𝐚^T𝐲,for 𝐲≈𝐲_0. From (<ref>), we have 𝐚 = ∇ h(𝐲_0), i.e.,a_i = .x_i/g(𝐱)∂ g/∂ x_i|_𝐱 = 𝐱_0,and c = exp(h(𝐲_0) - ∇ h(𝐲_0)^T𝐲_0) = g(𝐱_0)∏_i=1^5 x_0,i^a_i,where x_0,i is an ith element of 𝐱_0.We substitute the monomial approximation g(x) of g(x) in (<ref>) and use GP technique to solve (<ref>). The aforementioned affine approximation is, however, imprecise if the optimal solution lies far from the initial guess 𝐱_0 as the Taylor's approximation would be inaccurate. To overcome this problem, we take an iterative approach, where, if the current guess is 𝐱_k, we obtain the Taylor's approximation about 𝐱_k and solve a GP again. Let the current solution of GP be 𝐱_k+1. In the next iteration, we take Taylor's approximation around 𝐱_k+1 and solve a GP again. We keep iterating in this fashion until the convergence. Since the problem (<ref>) is close to GP (as we have only one constraint in (<ref>) that is not a posynomial), the aforementioned iterative approach works well in our case and yields the optimal solution <cit.>. If the obtained optimal solution contradicts with our initial assumption that C_1^S - C_1^R ≥ 0 and C_2^S - C_2^R ≥ 0, we move to other three cases discussed below. §.§ Case II: C_1^S - C_1^R ≥ 0 and C_2^S - C_2^R < 0In this case, the secrecy rate is given by C_S = (C_1^S - C_1^R)^+, and we need to solve the problem (<ref>) with the following expressions for f(β,β,γ_1,γ_2, γ_J) and g(β,β,γ_1,γ_2, γ_J):f(β,β,γ_1,γ_2, γ_J)=[β(γ_1 + γ_2 + γ_J) + 1]× 1 + (γ_1 + γ_2 + γ_J)(|h_2|^2βη + β),g(β,β,γ_1,γ_2, γ_J)= (β(γ_2 + γ_J) + 1) × 1 +(γ_1 + γ_2 + γ_J)(β+ |h_2|^2βη(βγ_1 + 1)).We again check if the assumption C_1^S - C_1^R ≥ 0 and C_2^S - C_2^R < 0 is valid; if not, we move to the remaining two cases. §.§ Case III: C_1^S - C_1^R < 0 and C_2^S - C_2^R ≥ 0 This case is similar to Case II, and only the subscripts 1 and 2 need to be interchanged in the expressions of f(β,β,γ_1,γ_2, γ_J) and g(β,β,γ_1,γ_2, γ_J). If the solution obtained does not satisfy the initial assumptions, we move to Case IV. §.§ Case IV: C_1^S - C_1^R < 0 and C_2^S - C_2^R < 0 In this case, the sum-secrecy rate is zero.Algorithm <ref> summarizes the aforementioned process of obtaining the optimal sum-secrecy rate and power allocation by solving (<ref>). § SECURE COMMUNICATION WITH IMPERFECT CSIWe now investigate the effect of imperfect CSI on sum-secrecy rate. We model the imperfection in channel knowledge as in <cit.>, where the channel coefficients are given ash_i = ĥ_i + Δ h_i,for i ∈{ 1,2, J}. Here ĥ_i is the estimated channel coefficient and Δ h_i is the error in estimation which is bounded as |Δ h_i| ≤ϵ_i. ϵ_i is the maximum possible error in estimating h_i with respect to S_1 and S_2. We consider the worst case scenario where the relay knows all channel coefficients perfectly, while legitimate nodes S_1 and S_2 concede estimation errors according to (<ref>). In this case, SNRs at the relay corresponding to the messages x_1 and x_2 remain the same as in (<ref>). The signal received at S_2 in the second slot isy_2= h_2α(√(βP_1)h_1x_1 + √(βP_2)h_2x_2 + √(βP_J)h_Jx_J + n_R ) + n_2 = (ĥ_2 + Δ h_2)α(√(βP_1)(ĥ_1 + Δ h_1)x_1 + √(βP_2)(ĥ_2 + Δ h_2)x_2 + √(βP_J)(ĥ_J + Δ h_J)x_J) + n_R) + n_2,where ĥ_1, ĥ_2, and ĥ_J are the channel coefficients estimated by node S_2. Using these imperfect channel estimates, the node S_2 tries to cancel the self-interference and the known jammer's signal in the following manner:y_2= (ĥ_2 + Δ h_2)α(√(βP_1)(ĥ_1 + Δ h_1)x_1 + √(βP_2)(ĥ_2 + Δ h_2)x_2 + √(βP_J)(ĥ_J + Δ h_J)x_J) + n_R) + n_2 - ĥ_2α(√(βP_2)ĥ_2x_2 + √(βP_J)ĥ_Jx_J)_imperfect interference cancellation.It follows thaty_2 = ĥ_2α√(βP_1)ĥ_1x_1 + (ĥ_2 + Δ h_2)α n_R + n_2 + Δ h_2α(√(βP_1)ĥ_1x_1 + √(βP_2)ĥ_2x_2 + √(βP_J)ĥ_Jx_J) + ĥ_2α(√(βP_1)Δ h_1x_1 + √(βP_2)Δ h_2x_2 + √(βP_J)Δ h_Jx_J). As (<ref>) shows, due to the imperfect CSI, S_2 cannot cancel the jamming signal and the self-interference completely. Here we ignore the smaller terms of the form Δ h_i Δ h_j as they will be negligible compared to other terms. The received SNR at S_2 is thus given by (<ref>) at the top of the next page.Using the triangle inequality, it follows that|ĥ_i| - |Δ h_i| ≤ h_i ≤ |ĥ_i| + |Δ h_i|, ∀ i ∈{ 1,2,J }.The worst case secrecy rate will occur whenh_i = |ĥ_i| + |Δ h_i| = |ĥ_i| + ϵ_i, ∀ i ∈{ 1,2,J }, and this will happen when the phase of h_i and Δ h_i are the same and Δ h_i concedes maximum error, i.e., |Δ h_i| = ϵ_i. Then the worst case SNR (denoted by SNR_S_2^wc) at node S_2 is given by (<ref>) at the top of the next page. Similarly the worst case SNR (denoted by SNR_S_1^wc) at S_1 is given by (<ref>) at the top of the next page. In (<ref>), we again denote estimated channels by ĥ_1, ĥ_2, and ĥ_J for brevity, but these values may be different from those estimated by S_2.Using these worst case SNRs, we maximize the worst case sum-secrecy rate and solve for the corresponding optimal power allocation and β using SGP as done for problem in (<ref>), i.e., for the case of perfect CSI. § NUMERICAL RESULTS AND DISCUSSIONS§.§ Effect of Power Splitting Ratio β Fig. <ref> shows the sum-secrecy rate (left y-axis) and the harvested energy (right y-axis) versus the total power budget for a random channel realization: |h_1|^2 = 0.6647, |h_2|^2 = 2.9152, and |h_J|^2 = 1.3289. We set η = 0.5 and N_0 = 1. Higher β (= 0.85) than the optimal β (the solution of the problem (<ref>)) results in higher harvested energy, which increases relay's transmit power, but the reduced strength of the received information signal at the relay (thus at nodes S_1 and S_2) due to higher β dominates the secrecy performance of the system. A lower β (= 0.15) ensures more power for the information processing at relay, but this reduces the harvested energy (reducing its transmit power to forward the information) and increases the chances of relay eavesdropping the secret message. As a result, the sum-secrecy rate reduces.§.§ Effect of Power AllocationFor different values of maximum channel estimation errors, Fig. <ref> compares the sum-secrecy rate when the total power is allocated optimally (obtained by solving the problem (<ref>)) and equally among nodes S_1, S_2, and jammer J for the same system parameters used to obtain Fig. <ref>. For exposition, we consider ϵ_1 = ϵ_2 = ϵ_J = ϵ in numerical results. The case ϵ = 0 corresponds to the perfect CSI at S_1 and S_2. Since the equal power allocation does not use channel conditions optimally, it suffers a loss in sum-secrecy rate as expected. Due to the error in channel estimation, the nodes S_1 and S_2 cannot cancel the self-interference (information signals sent to the relay in the first slot) and the jamming signal perfectly from the received signal in the second slot. This reduces the SNR at legitimate nodes S_1 and S_2, which further reduces the sum-secrecy rate.§.§ Effect of Imperfect CSIFig. <ref> shows three cases based on the knowledge of channel conditions at S_1 and S_2.[These three cases in Fig. <ref> should not be confused with four cases considered in Section <ref>.]The sum-secrecy rate in Case II is slightly better than that in Case I, because in Case II, a higher fraction of the total power is allocated to the jammer (see the right y-axis of Fig. <ref>) to use the perfect channel knowledge about h_J. But this has a side-effect: the imperfect CSI about h_1 and h_2 leads to higher interference from the jammer to S_1 and S_2. As a result, Case II does not gain much compared to Case I in terms of the sum-secrecy rate. Under Case III, the sum-secrecy rate is the highest, because S_1 and S_2 can cancel the jamming signal more effectively as they have imperfect CSI about only one channel. When ϵ is small enough (less than 0.06 in this case), the power allocated to the jammer in Case III is higher than that in Cases I and II. This is because when ϵ is small, if we allocate the power to S_1 and S_2 instead of jammer, it increases relay's chances of eavesdropping the information due to the increased received power, which dominates the detrimental effect incurred due to imperfect cancellation of jammer's signal at S_1 and S_2. But if ϵ goes beyond a threshold, the loss in the secrecy rate due to the imperfect cancellation of jammer's interference dominates, and the system is better off by allocating more power to S_1 and S_2 and using each other's signals to confuse the relay. Hence the power allocated to jammer in Case III is smaller than that in Cases I and II at higher ϵ. In Case III, the redistribution of the power from jammer to S_1 and S_2 with the increase in ϵ keeps the sum-secrecy rate almost the same.§ CONCLUDING REMARKS AND FUTURE DIRECTIONSIn a two-way untrusted relay scenario, though the signal from one source can indirectly serve as an artificial noise to the relay while processing other source's signal, the non-zero power allocated to the jammer implies that the assistance from an external jammer can still be useful to achieve a better secrecy rate. But the knowledge of two sources about channel conditions decides the contribution of the jammer in achieving the secure communication. For example, as the channel estimation error on any of the channel increases, the power allocated to the jammer decreases to subside the interference caused at the sources due to the imperfect cancellation of the jamming signal. The optimal power splitting factor balances between the energy harvesting and the information processing at relay. Hence the joint allocation of the total power and the selection of the power splitting factor are necessary to maximize the sum-secrecy rate. Future directions: There are several interesting future directions that are worth investigating. First the proposed model can be extended to general setups such as multiple antennas at nodes and multiple relays. Another interesting future direction is to investigate the effect of the placement of the jammer and the relay, which also incorporates the effect of path loss. Third we have considered the bounded uncertainty model to characterize the imperfect CSI. Extension to other models of imperfect CSI such as the model where only channel statistics are known is also possible.IEEEtran | http://arxiv.org/abs/1708.07989v1 | {
"authors": [
"Vipul Gupta",
"Sanket S Kalamkar",
"Adrish Banerjee"
],
"categories": [
"cs.IT",
"cs.NI",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170826155910",
"title": "On Secure Communication using RF Energy Harvesting Two-Way Untrusted Relay"
} |
The Study of the Bloch Transformed Fields Scattered by Locally Perturbed Periodic Surfaces Ruming ZhangCenter for Industrial Mathematics, University of Bremen ; ========================================================================================== Scattering problems from locally perturbed periodic surfaces have been studied both theoretically and numerically in recent years. In this paper, we will discuss more details of the structure of the Bloch transformed total fields. The idea is inspired by Theorem a in <cit.>, which considered how the total field depends on the incident plane waves, from a smooth enough periodic surface. We will show that when the incident field satisfies some certain conditions, the Bloch transform of the total field depends analytically on the quasi-periodicities in one periodic cell except for a countable number of points, while in the neighborhood of each point, a square-root like singularity exists. We also give some examples to show that the conditions are satisfied by a large number of commonly used incident fields. This result also provides a probability to improve the numerical solution of this kind of problems, which is expected to be discussed in our future papers. § INTRODUCTION The problems that quasi-periodic incident fields scattered by periodic surfaces have been well studied in the past few years. A classical way to handle this kind of problems, which are defined in unbounded domains, is to reduce the problems into one periodic cell,see <cit.>. However, when the incident field is no longer (quasi-) periodic, or the surface is no longer periodic, the scattered field is no longer (quasi-)periodic,the classical way fails to work, so newmethods are required for these more difficult problems. Another way is to treat this kind of problems as incident fields scattered by rough surfaces, see <cit.>. In recent years, the Floquet-Bloch based method was applied to analyze the well-posedness of this kind of problems,see <cit.>.The Bloch transform build up a "bridge" between the non-periodic scattering problem and a family of quasi-periodic scattering problems, which also provided a computational method for numerical solutions. In these papers, the properties of the Bloch transformed total fields are important both theoretically and numerically, thus we will discuss some more properties in this paper. This Floquet-Bloch based method was studied theoretically in <cit.> and <cit.>, for non-periodic incident fields, e.g. Herglotz wave functions or Green's functions, scattered by (locally perturbed) periodic surfaces. Based on the theoretical results, a numerical method was established to solve the scattering problems with a non-periodic incident field with a periodic surface, for cases in 2D space, see <cit.> and for3D space, see <cit.>. Later on, the numerical method was extended for scattering problems with a locally perturbed surface in 2D space in <cit.>. The method was also applied to locally perturbed periodic waveguide problems in <cit.>. In the numerical solution of the scattering problems with (locally perturbed) periodic surfaces, for a chosen discrete space, the convergence rate depends on the regularity of the Bloch transformed total field. To improve the numerical method, a more detailed study of the regularity is required. In the paper <cit.>, Kirsch has studied how the total fields depend on the incident angle and wave number of incident plane waves, from smooth enough periodic surfaces. The method adopted in this paper is the perturbation theory. Inspired by this idea, we may extend the result to the dependence of the quasi-periodicity of the Bloch transformed total field, when the incident field and its Bloch transform satisfy some certain conditions, and the surface is locally perturbed. In future, it is hopefully that the result could be helpful to improve the numerical method to solve such kind of scattering problems, or might be extended to solve problems in higher dimensional spaces.The rest of the paper is organized as follows. In Section 2,the known results of the locally perturbed periodic problems will be shown. In Section 3, we will study extend the result in <cit.> to the non-perturbed cases, and finally, the result will be extended to the locally perturbed case in Section 4. For a more general case, we will show some results in Section 5. The definition and some properties of the Bloch transform will be shown in the Appendix. § LOCALLY PERTURBED PERIODIC PROBLEMS§.§ Notations and domains Let ζ be a Lipschitz continuous periodic function with period Λ, and the periodic surface is defined byΓ:={(x_1,ζ(x_1)): x_1∈}. Let Λ^*=2π/Λ, define the periodic cells by=(-Λ/2,Λ/2],=(-Λ^*/2,Λ^*/2]=(-π/Λ,π/Λ].Let ζ_p be a Lipschitz local perturbation of ζ such that supp(ζ_p-ζ) is a compact domain in . For simplicity, assume that supp(ζ_p-ζ)⊂, then define the locally perturbed surface byΓ_p:={(x_1,ζ_p(x_1)): x_1∈}.We can define the space above the surfaces byΩ:={(x_1,x_2): x_1∈, x_2>ζ(x_1)} Ω_p:={(x_1,x_2): x_1∈, x_2>ζ_p(x_1)}.For simplicity, in this paper, we assume that Γ and Γ_p are all above the straight line {x_2=0}, and H be a positive number such that H>max{ζ_∞,ζ_p_∞}. Define the line Γ_H=×{H} and the spaces with between the surfaces Γ or Γ_p and Γ_HΩ_H:={(x_1,x_2): x_1∈, ζ(x_1)<x_2<H} Ω^p_H:={(x_1,x_2): x_1∈, ζ_p(x_1)<x_2<H}.We can also define the curves and domains in one periodic cell: Γ^Λ=Γ∩, Γ^Λ_H=Γ_H∩; Ω^Λ=Ω∩×,Ω^Λ_H=Ω_H∩×. §.§ Introduction to scattering problems Given an incident field u^i that satisfies Δ u^i+k^2 u^i=0inΩ_p,then it is scattered by the surface Γ_p, so there is a scattered field u^s such that it satisfies the Helmholtz equation Δ u^s+ k^2 u^s=0 in Ω_pand the Dirichlet boundary conditionu^s=-u^i on Γ_p. In this paper, we only take the sound-soft surface as an example. In fact, the results could be extended to problems with some other different settings, for example, the impedance boundary conditions, or inhomogeneous mediums.To guarantee that u^s is an out-going wave, i.e., u^s is propagating upwards, it is required that u^s satisfies the Upward Propagating Radiation Condition (UPRC) in the domain above Γ_Hu^s(x)=1/2π∫_ e^ı x_1 ξ+ı√(k^2-|ξ|^2)(x_2-H)û^s(ξ,H)ξ̣ x_2≥ H,where û^s(ξ,H) is the Fourier transform of u^s(·,H). The UPRC is equivalent to ∂ u^s/∂ x_2=T^+ u^s on Γ_H,where T^+ is the Dirichlet-to-Neumann map defined by(T^+ϕ)(x_1)=ı/2π∫_√(k^2-|ξ|^2)e^ı x_1ξϕ̂(ξ)ξ̣,where ϕ̂ is the Fourier transform of the function ϕ. It has been prove in <cit.> that the operator T^+ is bounded from H^1/2_r(Γ_H) to H^-1/2_r(Γ_H) for any |r|<1. Define the total field u=u^i+u^s, then it satisfies the following equationsΔ u+k^2 u = 0 in Ω_pu = 0 on Γ_p ∂ u/∂ x_2 = T^+u+[∂ u^i/∂ x_2-T^+u^i] on Γ_H.We can formulate the variational problem, i.e., for any u^i∈ H_r^1(Ω^p_H), seek for the weak solution u∈H^1_r(Ω_H^p) such that for any test function ϕ∈ H^1(Ω_H^p) with a compact support, satisfies the variational equation∫_Ω_H^p[∇ u·∇ϕ-k^2uϕ]x̣-∫_Γ_HT^+(u|_Γ_H) ϕṣ=∫_Γ_H[∂ u^i/∂ x_2-T^+(u^i|_Γ_H)]ϕṣ. The space H^1_r(Ω_H^p) with a tilde is defined as a subspace of H^1_r(Ω_H^p), i.e., H^1_r(Ω_H^p):={ϕ∈ H^1_r(Ω_H^p): ϕ|_Γ_p=0}. In the rest of this paper, we will always use the tilde to indicate the space of functions that satisfies homogeneous Dirichlet boundary conditions. In <cit.>, the uniquely solubility of the variational form (<ref>) was proved.For any |r|<1, given any u^i∈ H^1_r(Ω_H^p), there is a unique u∈H^1_r(Ω_H^p) of the variational problem (<ref>). §.§ Bloch transformed fieldsAs the Bloch transform only works on periodic domains, we have to transform the variational problem (<ref>) defined in a locally perturbed periodic domain Ω_H^p into the one defined in a periodic domain. Following <cit.>, define adiffeomorphism Φ_p such that maps the periodic domain Ω_H_0 to Ω_H_0^p, where max{ζ_∞,ζ_p_∞}<H_0<H, byΦ_p(x_1,x_2)=(x_1,x_2+(x_2-H)^3/(ζ(x_1)-H)^3(ζ_p(x_1)-ζ_(x_1)), (x_1,x_2)^⊤∈Ω_H_0.The operator Φ_p-I_2 (where I_2 is the identity operator in ^2) has a support which is a subset of Ω^Λ_H_0⊂Ω^Λ_H. Then let u_T=u∘Φ_p, then it satisfies the following variational problem∫_Ω_H[A_p∇ u_T·∇ϕ_T-k^2 c_p u_Tϕ_T]x̣-∫_Γ_HT^+(u_T|_Γ_H)ϕ_Tṣ =∫_Γ_H[∂ u^i/∂ x_2-T^+(u^i|_Γ_H)]ϕ_Tṣfor all ϕ_T∈H^1(Ω_H) with compact support, whereA_p(x)=|∇Φ_p(x)|[(∇Φ_p(x))^-1((∇Φ_p(x))^-1)^T]∈ L^∞(Ω_H,^2× 2), c_p(x)=|∇Φ_p(x)|∈ L^∞(Ω_H).From the definition of Φ_p, the supports of A_p-I and c_p-1 are both subsets of Ω^Λ_H.Suppose r≥ 0, let w=_Ω u_T, from the mapping property of the Bloch transform and u_T∈ H_r^1(Ω_H)⊂ H^1(Ω_H), w∈ L^2(;H^1_α(Ω^Λ_H)), and satisfies the following variational problem for any ϕ∈ L^2(;H^1_α(Ω^Λ_H))∫_ a_α(w(α,·),ϕ(α,·))α̣+b(^-1_Ω w,^-1_Ωϕ)=∫_∫_Γ^Λ_Hf(α,·)ϕ(α,·)ṣα̣,where a_α(w(α,·),ϕ(α,·)) =∫_Ω^Λ_H[∇ w(α,·)·∇ϕ(α,·)-k^2w(α,·)ϕ(α,·)]x̣-∫_Γ_H^ΛT^+_α[w(α,·)|_Γ_H^Λ]ϕ(α,·)ṣ b(ξ,ψ)=[Λ/2π]^1/2∫_Ω^Λ_H(A_p-I)∇ξ·∇ψṣ-k^2[Λ/2π]^1/2∫_Ω^Λ_H(c_p-1)ξψx̣, f(α,·)=∂_Ω u^i(α,·)/∂ x_2-T^+_α[(_Ω u^i)(α,·)|_Γ^Λ_H],where the α-quasi-periodic DtN map T^+_α is defined byT^+_α(ψ)=ı∑_j∈√(k^2-|Λ^* j-α|^2)ψ̂(j)e^ı(Λ^*j-α)x_1,ψ=∑_j∈ψ̂(j)e^ı(Λ^*j-α)x_1. At the end of this subsection, we will list some ofthe useful results in <cit.>. The first one is the equivalence between the original problem (<ref>) and the one with Bloch transform (<ref>). Assume u^i∈ H^1_r(Ω_H^p) for some r∈[0,1), then u_T∈H^1_r(Ω_H) satisfies (<ref>) if and only if w=_Ω u_T∈ H_0^r(;H^1_α(Ω^Λ_H)) satisfies (<ref>). Then the variational problem is uniquely solvable for a Liptschiz interface Γ_p. If ζ_p is Lipschitz continuous, then (<ref>) is uniquely solvable in H^r_0(;H^1_α(Ω^Λ_H)) for all u^i∈ H^1_r(Ω_H^p), r∈[0,1).There is an equivalent form of (<ref>) if the u^i∈ H_r^1(Ω_H^p) for r∈(1/2,1). If ζ_p is Lipschitz continuous and u^i∈ H^1_r(Ω_H^p) for r∈(1/2,1), then w∈ H_0^r(;H^1_α(Ω^Λ_H)) equivalently satisfies for all α∈ and ϕ_α∈ H^1_α(Ω^Λ_H) that a_α(w(α,·),ϕ_α)+b(^-1_Ω w,ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ,§ SCATTERING FROM NON-PERTURBED PERIODIC SURFACESIn Section <ref>, we have shown some known results for the solvabilityof the scattering problems from locally perturbed periodic surfaces. The regularity of the Bloch transform of the total field w(α,·)depends on the decaying rate of the incident field. In this section, we will turn to a further study of the structure of the Bloch transformed field, for non-perturbed periodic surfaces. Based on the results in this section, the discussion for perturbed surfaces will be carried out in the next section.The paper <cit.> provided a very good inspiration for the problem in this section. Firstly, we will define some sets and functions spaces. Define the set of all singularities with a fixed wave number k and a period Λ by:={α∈: ∃n∈, s.t., |Λ^*n-α|=k},which is, definitely, a countable set located periodically on the real line. Let ⊂ be an interval (the case that = is included), W⊂^2 be a bounded periodic cell of a periodic domain, and define the space of functions defined in the domain ×Ω^Λ_H that depends analytically on the first variableC^ω(;S(W)):={f∈𝒟'(× W): ∀ α_0∈, ∃ δ>0, s.t., ∀α ∈ (α_0-δ,α_0+δ)∩,. . ∃C>0,f_n∈ S(W) s.t., f(α,x)=∑_n=0^∞ (α-α_0)^n f_n(x), f_n_ S(W)≤ C^n},where S(W) is a Sobolev space defined on W. In this paper, S(W) is either H^n(Ω^Λ_H) or H^n(Ω^Λ_H), n=0,1,2.Define the set of functions defined in the domain × W that depends C^n-continuously on the first variableC^n(;S(W)):={f∈𝒟'(× W): , ∀α∈, j=0,…,n, ∂^j f(α,·)/∂α^j∈ S(W),. .∂^j f(α,·)/∂α^j_S(W) is uniformly bounded for α∈},thus we can define the space C^∞(;S(W)) in the same wayC^∞(;S(W)):={f∈𝒟'(× W): , ∀α∈, j=0,1,…,∞, ∂^j f(α,·)/∂α^j∈ S(W),. .∂^j f(α,·)/∂α^j_S(W) is uniformly bounded for α∈}, With the definitions of the new spaces and sets, Theorem a in <cit.> could be rewritten in the following form with a fixed wave number k.Suppose the incident field u^i(α,x) is the plain wave, i.e., u^i(α,x):=e^ıα x_1-ı√(k^2-α^2)x_2, and it is scattered by a smooth enough sound soft surface.The total field, denoted by u(α,x), belongs to the space C^0((-k,k);H^1(Ω^Λ_H)).For any interval ⊂(-k,k)∖, the total field u belongs to C^ω(;H^1(Ω^Λ_H)). Moreover, for any α_0∈∩(-k,k), i.e., there is an n_0∈ such that |Λ^*n_0-α|=k, there is a neighborhood U of α_0 and quasi-periodic functions v,w∈ C^ω(U∩(-k,k); H^1(Ω^Λ_H)) such that u=v+β_n_0(α)w, where β_n_0=√(k^2-|Λ^* n_0-α|^2) with non negative real and imaginary parts. We will just show the main idea of the proof in <cit.>. The proof is based on the variational form of the function v(α,x):=e^-ıα x_1u(α,x), i.e., for any ϕ∈H_0^1(Ω^Λ_H),a_α(v(α,·),ϕ)=-2ı√(k^2-α^2)∫_Γ^Λ_He^-ı√(k^2-α^2)x_2ϕṣ,where a_α(ψ,ϕ):=∫_Ω^Λ_H[_αψ·_αϕ-k^2 v(α,·)ϕ]x̣-∫_Γ_H^ΛT^+_α[ψ|_Γ^Λ_H]ϕṣ,_α=+ıα(1,0)^⊤, and the operator T^+_α is the DtN map defined on periodic functions in Γ^Λ_H byT^+_α(ψ)=ı∑_j∈√(k^2-|Λ^*j-α|^2)ψ̂(j)e^ıΛ^* j x_1,ψ=∑_j∈ψ̂(j)e^ıΛ^* j x_1. Each term in the variational form depends analytically on α∈(-k,k) except for the term ∫_Γ_H^ΛT^+_α[v(α,·)|_Γ^Λ_H]ϕṣ. The operator T^+_α depends analytically on α∈(-k,k)∖, thus the solution v(α,·) depends analytically in α∈(-k,k)∖. For each α_0∈, there is a neighbourhood of U such that T^+_α could be split into one operator depends analytically on α∈ U and the one with a √(k^2-|Λ^*n_0-α|^2)-singularity. Thus the singularity of v(α,·) with respect to α∈ U could be deduced from the Neumann series.For any α_0∈, there is either 1) an n_0∈ such that |Λ^*n_0-α_0|=k or 2) two n_1, n_2∈ such that |Λ^*n_1-α_0|=|Λ^*n_2-α_0|=k. Take Case 1) for example, i.e., |Λ^*n_0-α_0|=k, then either Λ^*n_0-k=α_0 or Λ^*n_0+k=α_0. If Λ^*n_0-k=α_0, β_n_0=√(α-α_0)·√(2k+α_0-α) where the second term, which is independent of n_0, is analytic for α in a small enough neighborhood of α_0. If k+Λ^*n_0=α_0,β_n_0=ı√(2k-α_0+α)·√(α-α_0), where the first term, which is independent in n_0, is analytic for α in a small enough neighborhood of α_0 as well. Then for α in a small neighborhood U of α_0, the total field u in Theorem <ref> has the form of u=v+√(α-α_0) wwhere v, w are bothfunctions in C^ω(U∩(-k,k);H^1(Ω^Λ_H)). Similar conclusion could be obtained for the second case. Thus we get a simpler form (<ref>) of the representation of the regularity form near the singularity α_0∈. The result in Theorem <ref> can be extended to a wider family of incident fields with the form ϕ(α,x)=e^ıα x_1-ı√(k^2-α^2)x_2 for α∈, where both the plane waves and the evanescent waves are included.If the incident field ϕ(α,x)=e^ıα x_1-ı√(k^2-α^2)x_2, where α∈, then the total field u(α,·)belongs to the space C^0(;H^1(Ω^Λ_H)) and for any interval ⊂∖, u∈ C^ω(;H^1_α(Ω^Λ_H)). Moreover, for any α_0∈, there is a small neighbourhood U of α_0 and a pair of functions v(α,·), w(α,·)∈ C^ω(U∩; H^1(Ω^Λ_H)) such that u(α,·)=v(α,·)+√(α-α_0) w(α,·).It is easy to extend the result of Theorem <ref> to the case that α∈ (-∞,-k)∪(k,+∞). The only case that may make problem is when α in the neighbourhood of ± k, for the plain wave is no longer analytic in (-k-δ,-k+δ) or (k-δ,k+δ), where δ>0 is a small enough positive number. Let α∈ (k-δ,k+δ), and δ is small enough such that ∩(k-δ,k+δ)={k} then the incident waveϕ(α,x)=exp(ıα x_1-ı√(k^2-α^2)x_2). As e^-ıα x_1ϕ(α,x)=e^-ı√(k^2-α^2)x_2 has the forme^-ıα x_1ϕ(α,x)=cosh(-ı√(k^2-α^2)x_2)+sinh(-ı√(k^2-α^2)x_2).Define the functionsϕ_1(α, x)=e^ıα x_1cosh(-ı√(k^2-α^2)x_2), ϕ_2(α, x)=- e^ıα x_2sinc(-ı√(k^2-α^2)x_2)√(α+k)x_2,where sinc is a function defined inbysinc(z)=sinh(z)/zz≠ 01z=0 .As cosh and sinc have Taylor's series at z=0cosh(z)=∑_n=0^∞z^2n/(2n)!,sinc(z)=∑_n=0^∞z^2n/(2n+1)!,the functions cosh(-ı√(k^2-α^2)x_2)and sinc(-ı√(k^2-α^2)x_2) has the Taylor's expansioncosh(-ı√(k^2-α^2)x_2)=∑_n=0^∞(α^2-k^2)^n x_2^2n/(2n)! sinc(-ı√(k^2-α^2)x_2)=∑_n=0^∞(α^2-k^2)^n x_2^2n/(2n+1)!,thus they are both analytic functions in α in a small neighbourhood of k. Then the incident field is written into the form of ϕ(α,x)=ϕ_1(α,x)+√(α-k) ϕ_2(α,x).From the proof of Theorem <ref> in <cit.>, the total field with the incident field ϕ_j∈ C^ω((k-δ,k+δ);H^1(Ω^Λ_H)), j=1,2, u_j has the decomposition thatu_j=v_j+√(α-k)w_jwhere v_j, w_j∈ C^ω((k-δ,k+δ);H^1_α(Ω^Λ_H)) are all analytic functions in α. Then the total field u satisfiesu =u_1+u_2=v_1+√(α-k)w_1+√(α-k) [v_2+√(α-k)w_2]=[v_1+(α-k)w_2]+√(α-k) [w_1+v_2].The terms v_1+(α-k)w_2 and w_1+v_2 are both in the space C^ω((k-δ,k+δ);H^1(Ω^Λ_H)), let v=v_1+(α-k)w_2 and w=w_1+v_2, thenu=v+√(α-k)w,the case that α_0=k is proved. The case that α_0=-k could be proved similarly. The proof is finished.In the rest of this section, we will consider the scattering problems with non-perturbed periodic surfaces. Let u∈H_r^1(Ω^H) be the total field with the incident u^i∈ H_r^1(Ω^H) and w=_Ω u∈ H_0^r(;H^1_α(Ω^Λ_H)). As w is Λ^*-periodic in α, we only consider the regularity in one periodic cell . For any α_0∈, α_0+Λ^*∈, thusdistributes periodically in . Let α_1∈ be a fixed number, there are finite number of elements ofthat lies in [α_1,α_1+Λ^*]. Let S be the number, then α_1<α_2<⋯<α_S=α_1+Λ^* where [α_1,α_S]∩={α_1,…,α_S}. We will define the space of the functions defined in × W that satisfy the property of u(α,·) inCorollary <ref> as follows. ^ω(;S(W);):={u∈ C^0(;S(W)):for anysubinterval _0⊂∖, .u∈ C^ω(_0;S(W)); ∀α_j∈∩, there is a neighbourhood U of α_j.and a pair v,w∈ C^ω(U∩,S(W)) such that u=v+√(α-α_j)w}.We assume that the incident field u^i∈ H_r^1(Ω_H) satisfies that the Bloch transform(_Ω u^i)(α,·)∈^ω(;H^1(Ω^Λ_H);). Assumption <ref> is satisfied by a large number of incident waves.In fact, the Green's functions in a half-space satisfies Assumption <ref>. We will take this for example. In the following examples, we set Λ=2π thus Λ^*=1. The half-space Green's function has the form ofu^i(x,y)=ı/4[H_0^(1)(k|x-y|)-H_0^(1)(k|x-y'|)]where y=(y_1,y_2)^T lies above Γ_H. Bloch transform of the Green's function has the representation of( u^i)(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)+ıβ_j y_2sinc(β_j x_2)x_2.As the number of elements in S_N is finite, suppose the number to be M, then for each j=1,…,M, defineu_j(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)e^ıβ_j y_2sinc(β_j x_2)x_2, u^1_j(α,x)=ı/2π∑_j∈e^ıα_j (x_2-y_2)sinc(β_j y_2)y_2sinc(β_j x_2)x_2, u^2_j(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)cos(β_j y_2)sinc(β_j x_2)x_2.then u_j=β_j u_j^1+u_j^2. As sinc and cos has the Taylor's expansions thatsinc(x)=∑_n=0^∞(-1)^n x^2n/(2n+1)!,cos(x)=∑_n=0^∞(-1)^n x^2n/(2n)!,u_j^1, u_j^2 are analytic functions in both α and x.Note that (_Ω u^i)(α,x)-∑_j=1^M u_j(α,x) is the convergent sum of analytic functions, _Ω u^i∈^ω(;H^1(Ω^Λ_H);).With Assumption <ref>, the regularity result for the Bloch transform of the total field comes directly from Corollary <ref>.Let r>1/2. Suppose the incident filed u^i∈ H_r^1(Ω_H) satisfies Assumption <ref>, with a total field u∈H_r^1(Ω_H). Then the Bloch transform of the total field w(α,x)=(_Ω u)(α,x)∈^ω(;H^1(Ω^Λ_H);). When r>1/2, from Theorem <ref>, the variational form is equivalent to the decoupled system a_α(w(α,·),ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ.Then the periodic function v(α,x):=e^ıα x_1w(α,x) satisfies the variational form for any periodic function ϕ∈H^1_0(Ω^Λ_H)a_α(v(α,·),ϕ)=∫_Γ^Λ_He^ıα x_1f(α,·)ϕṣ.Thus the conclusion could be obtained from the same argument of the proof of Corollary <ref>.With the regularity results for non-perturbed periodic surfaces, we will discuss the locally perturbed cases in the next section. § SCATTERING FROM LOCALLY PERTURBED PERIODIC SURFACES In this section, we still hold the assumption that r>1/2, then from Theorem <ref>, the variational form is equivalent to a_α(w(α,·),ϕ_α)+b(u_T,ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ.Let v(α,x)=e^ıα x_1w(α,·), ϕ=e^ıα x_1ϕ_α(x), then v is the solution of the following systema_α(v(α,·),ϕ)=∫_Γ^Λ_He^ıα x_1f(α,·)ϕṣ-b_α(u_T,ϕ),where the sesquilinear form b_α is defined byb_α(ξ,ψ)=[Λ/2π]^1/2∫_Ω^Λ_He^ıα x_1[(A_p-I)ξ·_αψ-k^2(c_p-1)ξψ]x̣,which depends analytically on α, thus the system (<ref>) has a right hand side that depends analytically on α. So from similar arguments in Theorem <ref>. Suppose u^i∈ H_r^1(Ω_H^p) for some r>1/2 satisfies Assumption <ref>, then the Bloch transformw=_Ω u_T∈ H_0^r(;H^1(Ω^Λ_H)) belongs to the space ^ω(;H^1(Ω^Λ_H);). A direct corollary of Theorem <ref> is described as follows.Suppose α_1,…,α_s∈∩, there are S+1 functions defined in ×Ω^Λ_H, i.e.,w_1(α,·),…,w_S(α,·),w_S+1(α,·) that belong to the space C^∞(;H^1(Ω^Λ_H)) such thatw(α,·)=w_S+1(α,·)+∑_n=1^S√(α-α_n) w_n(α,·). § FURTHER REGULARITY RESULTS OF SCATTERING PROBLEMS In this section, we will consider a more generalized class of incident fields. In the following part, we only want to show the regularity results without proving them, for the proofs are quite similar. Although the Green's function u^i(x,y) satisfies Assumption <ref>, there is a large class of incident fields that does not satisfy this assumption. We will firstly study the regularity property of the Bloch transform of the Herglotz wave function. Some of the results are from <cit.>. Define the weighted Hilbert space L^2_cos(-π/2,π/2) as the closure of C_0^∞(-π/2,π/2) in the normϕ_L^2_cos(-π/2,π)=[∫_-π/2^π/2|ϕ(θ)|^2/cosθθ̣]^1/2. The downward propagating Herglotz wave function with the density in L^2_cos(-π/2,π/2) is defined by(Hg)(x)=∫__-e^ı k x· dg(d)ṣ(d)=∫_π/2^π/2e^ı k(sinθ x_1-cosθ x_2)ϕ(θ)θ̣,where ϕ∈ L^2_cos(-π/2,π/2), _- is the lower half of one unit circle in ^2. From <cit.>, the Bloch transform of the Herglotz wave functions has the representation of (_Ω Hg)(α,x)=√(Λ^*)∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2ϕ(arcsin(Λ^*j-α)/k)/√(k^2-|Λ^*j-α|^2)For some choice of the function ϕ, i.e., ifϕ(t)=h[cos(t)]cos(t), where h is an analytic function defined in [0,1] with h(0)=0, then(_Ω Hg)(α,x)=√(Λ^*)/k∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2h[√(k^2-|Λ^*j-α|^2)/k].By differentiating _Ω Hg with respect to α,∂/∂α(_Ω Hg)(α,x)= -ı√(Λ^*)/k x_1∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2 h[√(k^2-|Λ^*j-α|^2)/k]-ı√(Λ^*)/kx_2∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2Λ^*j-α/√(k^2-|Λ^*j-α|^2)h[√(k^2-|Λ^*j-α|^2)/k]+√(Λ^*)/k^2∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2Λ^*j-α/√(k^2-|Λ^*j-α|^2)h'[√(k^2-|Λ^*j-α|^2)/k],then ∂/∂α(_Ω Hg)(α,x)∈ L^p(;H^1(Ω^Λ_H)) for any 1≤ p<2, thus the function _Ω Hg∈ W^1,p_0(;H^1(Ω^Λ_H)). From the Sobolev embedding theorem, _Ω Hg∈ H^r_0(;H^1(Ω^Λ_H)) for some r∈[1/2,1). Thus from the property of the inverse Bloch transform, Hg∈ H_r^1(Ω_H).As in each open interval ⊂∖, the terms in the finite sum do not change, and each term is an analytic function in both α and x. Suppose α_0∈ is a singularity point, then when δ>0 is small enough such that (α_0-δ,α_0+δ)∩={α_0}. Consider the domain (α_0-δ,α_0], it is easy to prove that (_Ω Hg)(α,x)=v(α,x)+√(α-α_0) w(α,x) where v,w∈ C^ω([α_0-δ,α_0];H^1(Ω^Λ_H)). Similar result could be obtained in the interval [α_0,α_0+δ). We will define the following space.^ω_c(;S(W);):={u∈ C^0(;S(W)): for any subinterval _0⊂∖,.u∈ C^ω(_0;S(W)); ∀α_j∈∩, there is a small enough δ>0 and two pairs v_1,w_1∈ C^ω([α_j-δ,α_j];S(W)) and v_2,w_2∈ C^ω([α_j,α_j+δ];S(W)) such that.u=v_1+√(α-α_j) w_1 for α∈(α_j-δ,α_j]; u=v_2+√(α-α_j) w_2 for α∈[α_j,α_j+δ).},then the Herglotz wave function discussed above belongs to this space. We assume that the incident field u^i∈ H_r^1(Ω^H_p) for r>1/2 satisfies that the Bloch transform _ω u^i∈^ω_c(;H^1(Ω^Λ_H);). With the incident field satisfying Assumption <ref>, we can obtain the following corollary of Theorem <ref>. Suppose u^i satisfies Assumption <ref>, then the total field u with the Bloch transform w=_Ω u∈^ω_c(;H^1(Ω^Λ_H);). Recall the definition of the setof singularities, and let α_j, j∈ be an ascending series of , then we can get the corollary of the regularity result in each interval (α_j,α_j+1). Suppose u^i satisfies Assumption <ref>, then the Bloch transform w=_Ω u belongs to the space C^0(;H^1(Ω^Λ_H)). Moreover, for each j∈, for α in the interval [α_j,α_j+1], there are three functionsw^j_0, w^j_1, w^j_2∈ C^∞([α_j,α_j+1];H^1(Ω^Λ_H)) such thatw=w^j_0+√(α-α_j) w^j_1+√(α-α_j+1)w^j_2.§ APPENDIX: BLOCH TRANSFORM A useful tool to handle the (locally perturbed) periodic scattering problems is the Bloch transform, see <cit.>, which build up a "bridge" between the infinitely defined problem and a family ofcoupled quasi-periodic problems defined in onesingle periodic cell. For any function ϕ∈ C_0^∞(Ω), the one dimensional Bloch transform is defined as(_Ωϕ)(α,x)=[Λ/2π]^1/2∑_j∈ϕ(x+(Λ j 0 ))e^-ıαΛ j,α∈,x_1∈,where Ω is a Λ-periodic (where Λ>0) stripor curve in x_1-direction in ^2, which satisfies that if (x_1,x_2)^T∈Ω, then (x_1±Λ,x_2)^T∈Ω. Though Ω does not always have the same definition throughout this paper, we all denote the one dimensional Bloch transforms by _Ω. Thus _Ωϕ(α,x) is α-quasiperiodic in x_1 for period Λ, for a fixed α, i.e.,(_Ωϕ)(α,x+(Λ 0 ))=e^ıαΛ(_Ωϕ)(α,x), and 2π/Λ-periodic in α∈ for a fixed x, i.e.(_Ωϕ)(α+2π/Λ,x)=(_Ωϕ)(α,x). Let Λ^*=2π/Λ and set=(-Λ/2,Λ/2],=(-Λ^*/2,Λ^*/2]=(-π/Λ,π/Λ], and let Ω^Λ=×∩Ω. Then _Ω maps a function in C_0^∞(Ω) into a function that belongs to the space C^∞(×Ω^Λ). To introduce more about the properties of the Bloch transform _Ω, we need some notations and spaces. In the rest of this section, Ω is assumed to be bounded in x_2-direction. Firstly, for any s,r∈, define the weighted spaceH_r^s(Ω)={ϕ∈𝒟'(Ω): x→(1+|x_1|^2)^r/2ϕ(x)∈ H^s(Ω)}with the normϕ_H_r^s(Ω)=(1+|x_1|^2)^r/2ϕ(x)_H^s(Ω).We can also define the function space in ×Ω^Λ. For some integers r∈ and some real number s∈, define the spaceH^r_0(;H^s_α(Ω^Λ))={ϕ∈𝒟'(×Ω^Λ): ∑_ℓ=0^r∫_∂^ℓ_αϕ(α,·)^2_H^s()α̣}with the norm defined asϕ_H_0^r(;H^s_α(Ω^Λ))=[∑_ℓ=0^r∫_∂^ℓ_αϕ(α,·)^2_H^s()α̣]^1/2.With the interpolation in r, the definition could be extended to all r≥ 0. We can also define the spaces with a negative r by duality arguments, thus the space H^r_0(;H^s_α(Ω^Λ)) is well defined for any r, s∈. With these definitions, the following properties for Bloch transform holds. For proofs see Theorem 8 in <cit.>, and we will omit them here. The Bloch transform extends to an isomorphism between H^s_r(Ω) and H^r_p(;H^s_α(Ω^Λ)) for all s,r∈. When s=r=0, _Ω is an isometry. The inverse operator (_Ω^-1w)(x+(Λ j 0 ))=[Λ/2π]^1/2∫_ w(α,x)e^ıΛ jαα̣, x∈Ω^Λ, j∈.Moreover, the L^2-adjoint operator _Ω^* equals to its inverse _Ω^-1. § ACKNOWLEDGEMENTS.The author was supported by the University of Bremen and the European Union FP7 COFUND under grant agreement n^∘600411. alpha | http://arxiv.org/abs/1708.07560v3 | {
"authors": [
"Ruming Zhang"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170824212812",
"title": "The Study of the Bloch Transformed Fields Scattered by Locally Perturbed Periodic Surfaces"
} |
[][email protected] Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland [][email protected] Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, SwitzerlandInstitute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland Paul Scherrer Institute, 5232 Villigen, Switzerland Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland University of Warwick, Coventry, CV4 7AL, United Kingdom University of Warwick, Coventry, CV4 7AL, United Kingdom University of Warwick, Coventry, CV4 7AL, United Kingdom University of Warwick, Coventry, CV4 7AL, United Kingdom Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, SwitzerlandWe probe ultra-strong light matter coupling between metallic terahertz metasurfaces and Landau-level transitions in high mobility 2D electron and hole gases. We utilize heavy-hole cyclotron resonances in strained Ge and electron cyclotron resonances in InSb quantum wells, both within highly non-parabolic bands, and compare our results to well known parabolic AlGaAs/GaAs quantum well (QW) systems. Tuning the coupling strength of the system by two methods, lithographically and by optical pumping, we observe a novel behavior clearly deviating from the standard Hopfield model previously verified in cavity quantum electrodynamics: an opening of a lower polaritonic gap.Landau polaritons in highly non-parabolic 2D gases in the ultra-strong coupling regime Jérôme Faist December 30, 2023 ======================================================================================§ I. INTRODUCTION Light-matter interaction phenomena are the driving force of quantum optics, and have been investigated in platforms ranging from atoms <cit.> to solid state systems <cit.>. The creation of quasi-particles called cavity polaritons in solid-state-based systems enables strong, non-linear, photon-photon interactions. This allowed the observation of Bose-Einstein condensation in solids <cit.>, superfluidity, quantized vortices and dark solitons, forming the fascinating field ofquantum fluids of light <cit.>. A cavity polariton exists when the light-matter coupling strength is larger than the dephasing rates of the individual components, which are then in strong coupling with reversible energy exchange between light and matter. The vacuum Rabi frequency Ω_R quantifies this coupling strength and by tuning the magnitude of Ω_R, different physical regimes can be explored. An increase of the interaction strength towards the transition frequency ω_ij leads to the so-called ultra-strong coupling regime <cit.>. Usually negligible terms, such as the polarization self-interaction and counter-rotating terms, then have to be included in the Hamiltonian representing the system. This impacts the physics of the resulting quasiparticles: the ground state of an ultra-strongly coupled system contains virtual photons <cit.>, with a number proportional to the normalized light-matter coupling ratio Ω_R/ω_ij <cit.>. One way to enhanceΩ_R and thus drive the system into the ultra-strong coupling regime is to exploitthe collective enhancement due to the simultaneous coupling of N material excitations to the same cavity mode, yielding an increased Rabi frequency √(N) Ω_R. The collective radiative coupling of material excitations in a reduced volume (V < λ^3) is also the basis of the phenomenon of Dicke superradiance and of the superradiant phase transition <cit.>. Dicke superradiance has been observed in atomic systems <cit.> and more recently the superradiant phase transition has been realized in a driven-dissipative system of cold atoms<cit.>. Such phenomena have attracted great attention lately also in the solid-state community, with observation of superradiant-related physics in different experimental platforms <cit.>, for example semiconductor quantum dots <cit.> or quantum wells <cit.>.In cavity quantum electrodynamics (QED) within the dipolar approximation, in solid-state systems and in semiconductor intersubband systems such a phase transition is prevented by the so-called `no-go' theorem <cit.>. The term containing the squared vector potential (A^2 term, also called the diamagnetic term) in the minimal-coupling Hamiltonian shifts the energy dispersion towards higher energies, such that the lower branch can never become gapless, and accordingly can never reach the ground state and exhibit the critical point associated to theDicke quantum phase transition <cit.>.Recently, ultra-strong coupling has been obtained with the so-called Landau polaritons<cit.>, where the cyclotron transition of a semiconductor-based system is optically coupled to a cavity in the THz range. Using meta material-based resonators<cit.> or Fabry-Pérot cavities <cit.>it has been possible to explicitly distinguish the A^2-term from the vacuum Bloch-Siegert shift related to the counter-rotating terms of the interaction.Suggestions to circumvent the no-go theorem in cavity QED include the use of multi-level atomic systems <cit.> or by considering systems with linear dispersion relation, like in graphene, although this question is still under debate <cit.>. Other works showed, that also by including Coulomb dipole-dipole interaction <cit.> one could possibly restore a Dicke-like model. Other investigations by D. De Bernardis et al. <cit.> deal with the breakdown of the gauge invariance, which is also subject of other recent investigations <cit.>. Very recently, P. Nataf et al. <cit.> included the Rashba spin-orbit interaction under a static magnetic field and found a magnetostatic instability which opens a promising way towards Dicke superradiant phases.Depending on the shape of the confining potential of an intersubband transition, an additional higher order resonance which reduces the oscillator strength of the main mode, could lead to renormalized energies in the dipolar gauge (in contrast to the Coulomb gauge) but not to a Dicke phase transition <cit.>.In experiments on Landau polaritons with two dimensional electron gases (2DEGs) in AlGaAs/GaAs QWs, which exhibit parabolic in-plane band dispersion, a coupling ratio beyond unity <cit.> has been achieved and a very good agreement of the polaritonic dispersion with the Hopfield-like Hamiltonian <cit.> including all counter-rotating and diamagnetic terms has been verified <cit.>. It has to be noted that the model developed in Ref.<cit.> that was adopted in the previously cited papers and in this work has been originally developed for Fabry-Pérot resonators. Here, striving to engineer ultra-strong coupling deviating from a standard Hopfield model in a purely ground state system, we utilize two QW systems with a non-parabolic 2D electron or hole gas: a strained germanium quantum well (s-Ge QW) with an occupied non-parabolic heavy-hole band and indium antimonide quantum wells (InSb QW) with extremely light mass electrons. We couple the 2DHG/2DEG Landau-level transitions (ω_ij = ω_cyc) to a terahertz (THz) metamaterial resonator with subwavelength electric field confinement. In the present work with s-Ge and InSb QWs, we systematically scale our cavity frequency f_LC = ω_LC/2π lithographically to control the coupling rate, and observe an opening of apolaritonic gap below the cold cavity frequency.The energy of the lower polariton branch no longer reaches the cold cavity frequency at large detunings(high magnetic fields) as in the Hopfield model <cit.>. A similar effect is observedwhen optically pumping an intentionally undoped s-Ge QW creating more carriers in the QW.We modify the standard Hopfield Hamiltonian including an effective reduction of the diamagnetic terms to capture the purely experimentally observed change, obtaining a very good agreement to our experimental polariton branches.§ II. NON-PARABOLIC QUANTUM WELLSThe matter part of our coupled system, the inter-Landau level transition, is tunable in energy by an external static magnetic field as the cyclotron resonance scales as ω_cyc = eB/m^*. The cyclotron resonance is directly accessible in transmission THz time domain spectroscopy (as shown in Fig. <ref> (c), (d)), and the effective mass of the carriers can be deduced by a linear fit to the measured resonance. In contrast to a standard and well known AlGaAs/GaAs QW, the s-Ge QW and InSb QWs exhibit additional and more complex properties, appealing to conduct ultra-strong coupling experiments, which we compare then to the standard AlGaAs/GaAs QW. The s-Ge and InSb QWs are showing e.g. heavy non-parabolicity, strain and spin-orbit interaction <cit.>, with a heavier and lighter cyclotron effective mass than the standard AlGaAs/GaAs quantum wells, for which the cyclotron mass, due to the electron confinement, is found to be m^*_GaAs = 0.071 m_e <cit.>.The s-Ge QW has a thickness L = 20 nm, heavy-hole (HH) density1.3×10^12 cm^-2, effective mass m^*_HH = 0.118 m_e, g-factor 5.0, and mobility 1.5×10^6 cm^2V^-1s^-1 <cit.>. In this system the 1.3 % biaxial compressive strain, provided by the Si_0.3Ge_0.7 barriers, lifts the degeneracy of the heavy-hole and light-hole band at the Γ-point. The band structure of this structure was calculated using the 6×6 k · p method following Ref. <cit.>, including the heavy-hole, light-hole and split-off bands, and a wavevector k_z = π/L to include the influence of quantum confinement. As shown in Fig. <ref>(a), the heavy-hole band is strongly non-parabolic due to the applied strain. The effective mass at the Γ-point is m^*_HH = 0.0675 m_e, as obtained from a parabolic fit at low wavevectors (green line). The gray shaded area indicates the range of the oscillatory chemical potential μ_F, which lies clearly in the region that is no longer within the parabolic approximation. Note that within this range the in-plane dispersion remains essentially isotropic.The InSb QWs have a strongly non-parabolic conduction band with a very small bandgap (≈ 180meV <cit.>), featuring a very light effective mass electron which we determined by cyclotron resonance measurements to be m^*_e = 0.0248 m_e for a double side doped (DSD) quantum well and m^*_e = 0.0243 m_e for a single side doped (SSD) QW. The effective mass of the InSb DSD QW at the Γ-point is m^*_e = 0.020 m_e, as obtained from a parabolic fit at low wavevectors (Fig. <ref> (b) green line). The DSD QW has a thickness of L = 23 nm, an electron density of 4.9×10^11 cm^-2 with a mobility of 3.49×10^5 cm^2V^-1s^-1, and the SSD QW has a thickness of L = 21 nm, an electron density of 3.65×10^11 cm^-2 with a mobility of 2.03×10^5 cm^2V^-1s^-1. Details on the growth of such QWs are published in Ref. <cit.>.§ III. LITHOGRAPHIC TUNING For the cavity we chose complementary split ring resonators (cSRRs), which can be described by a lumped element electric circuit model with a characteristic LC-resonance where the vacuum electric field fluctuations are greatly enhanced due to the strongly sub-wavelength cavity volume <cit.>. We design a cavity with a LC-resonance f_LC = ω_LC/2π and then scale the geometry of the resonator by a linear factor a (from a = 0.5 to a = 2.3/2.4 on all QWs) with constant metal thickness (4 nm Ti and 200 nm Au). The lithographic tuning was experimentally verified and measured on a bare Si substrate and on GaAs in our previous work <cit.>, revealing a linear frequency scaling with the inverse of the geometrical scaling factor f_LC∝ a^-1, as shown in Fig. <ref> (a),with an optical micrograph of one cavity of the full array with scaling a = 2 shown in the inset (b). As the frequency of the cavity depends on the dielectric environment in close vicinity, the cold cavity frequencies, i.e. the frequencies withoutfree carrier contributions, for the s-Ge and InSb QW samples were further determined by depositing three cavity arrays chosen from across the frequency range of f ≈ 200GHz to f ≈ 900 GHz (= different scalings) on buffer layer structures. The buffer layers have the same growth structure as the QW sample, thus the same refractive index, but do not contain a QW. From a linear fit to the measured cavity frequencies (Fig. <ref> (c), bright green) on the reference structures the expected bare cavity frequencies for all arrays of cSRR deposited on the QWs are deduced (see Fig. <ref> (c), dark green rectangles). Additionally, the lithographic accuracy (electron beam lithography for s-Ge QW and photolithography for InSb and GaAs QWs) is verified using scanning electron microscopy and the frequencies of the expected cold cavity are corrected accordingly (this correction remained very small < 2%). Probing the coupled samples with THz time domain spectroscopy (see S.M. for details <cit.>) in transmission, we measure the polariton dispersion of each frequency and resonator array on the s-Ge, InSb and GaAs QWs at a temperature of 3K. One example of such measurement of a s-Ge QW at high filling factors, thus at low frequencies respectively, is shown in Fig. <ref>. The bare cavity frequency for the shown scaling factor a = 2 is at f_LC = 208 GHz (solid cyan line), which lies between the frequencies of the polariton branches at high and low magnetic field, with f_LP = 165 GHz (lower polariton (LP)) and f_UP = 292 GHz (upper polariton (UP)), respectively. This is a very striking and peculiar feature, as in the standard Hopfield-like Hamiltonian <cit.> used to describe the ultra-strong coupling, one asymptotically recovers the cold cavity frequency at high magnetic fields.§ IV. TUNING BY OPTICAL PUMPING Additionally to the lithographic tuning, it is an attractive option to use an optical pump to manipulate in-situ the polariton state. It has been shown that organic molecules can be photochemically changed reversibly by UV and VIS radiation to switch from the weak to ultra-strong coupling regime <cit.>. Recently, exploiting the alignment of the light polarization in respect to carbon nano-tubes enabled such continuous tuning from weak and ultra-strong coupling<cit.>. In the work of G. Günter et al., <cit.>, the authors were able to show an ultra-fast switch-on of ultra-strong coupling by optically exciting carriers in a QW to enable the intersubband transition. For the current study, we use an optical pump to change the carrier concentration in a s-Ge QW.As a pump we used part of our Ti:Sapphire beam at 800 nm with an optical fluence of approximately 7 μJ/cm^2 (the beam area onto the sample was ∼ 3 mm^2.) No time-delay-dependent carrier density can be observed (see Supplementary Material), suggesting a lifetime much longer than the time lapse between the laser pulses at our laser repetition rate of 80 MHz. As a result, no special care had to be taken to adjust the delay between the optical pump and THz probe pulses: our optical pumping is in fact a way to change "statically" the carrier density of the sample employing the same cavity.Working with a s-Ge QW which has the same growth structure as the s-Ge layer used for the lithographic tuning but no intentional doping, we still observe a cyclotron resonance from the residual carriers in the QW with a mass of m^* = 0.076 m_0. Working in a non-parabolic QW as s-Ge, the cyclotron mass is carrier-density-dependent, thus optically pumping the QW leads to an increase of the cyclotron mass to m^* = 0.08 m_0 (see supplementary material for details). The employed cavity is the same as scaling a =1 from the lithographic tuning study with a frequency of f = 415. THz transmission spectra as function of magnetic field are taken without additional optical pump in Fig. <ref> (a) and with optical pump in Fig. <ref> (b). The coupling strength increases from Ω/ω = 0.17 without pumping to Ω/ω = 0.25 with additional optical pump. In the Hopfield model, the increased coupling strength would predict an increase of the upper polariton branch frequency, leading to a larger upper polariton gap while far detuned the lower polariton branch still reaches the unchanged cavity frequency in the Hopfield model. In Fig. <ref> (c) where the maxima of the polariton branches with and without optical pump are extracted and plotted together, it isclearly visible that the lower polariton branch does not reach the cold cavity frequency far detuned, thus opening a lower polariton gap, as also observed in the measurements that use lithographically tuning.§ V. ANALYSIS AND DISCUSSION The deviation of the lower polariton frequency f_LP at high magnetic fields from the cold cavity frequency f_LC is extracted for all lithographically tuned cavities on GaAs, s-Ge and InSb QWs and displayed in Fig. <ref> (a) as function of the inverse scaling factor a. In the case of the GaAs QW (Fig. <ref> (a), dark blue circles), we verify again that the deviation for all frequencies (f ≈ 200-900GHz) is less than 5% (and less than 15 GHz in absolute terms). For the s-Ge and InSb QWs instead, we observe an increasing deviation with increasing coupling strength, up to 20% for the s-Ge QW (Fig. <ref> (a), red squares, 43 GHz absolute deviation) and 10% for the InSb QWs (Fig. <ref> (a), yellow and orange stars). Interestingly, one cavity array (f_LC = 160GHz) of the s-Ge QWs (Fig. <ref> (a), bright green triangle), which shows a characteristic crosshatch pattern of strain relaxation (see S. M. <cit.>), has a polariton frequency of the lower branch very close to the cold cavity frequency again, with a deviation of less than 5% as for the GaAs QW. The Hopfield-like Hamiltonian can be written as the sum of different contributions <cit.>: H = H_mat+H_int+H_dia+H_cavity,with the material excitation H_mat, the bare cavity electromagnetic field H_cavity, the interaction term H_intand the diamagnetic term H_dia arising from the self-interaction of the light, including all counter-rotating terms. The diamagnetic term is <cit.> H_dia = ħ∑_k(D_k(a^†_ka_k+a_ka^†_k)+D_k(a_k a_-k+a^†_k a^†_-k)) and leads to a renormalization of the polariton energies. The value D of the diamagnetic terms is in the case of parabolic dispersion approximated as D_k = D ≈Ω_R^2/ω_cyc <cit.> by evaluating the Thomas-Reiche-Kuhn sum rule (also known as the f-sum rule) <cit.>. To capture the observed different renormalization of the polariton energies, we introduce here a parameter d which we use to reduce effectively the strength of the diamagnetic term D = d Ω^2_R/ω_cyc. In the effective model, a polaritonic gap opening with respect to the bare cavity frequency for both polariton branches is predicted (see details in S. M. <cit.>), just as observed in Fig. <ref>.We fit the measured polariton branches to extract the normalized coupling ratio Ω_R/ω_cyc following the procedure described in Ref. <cit.>, but implementing the modified Hopfield model with the prefactor d as an additional fitting parameter for the effective diamagnetic term D = dΩ^2_R/ω_cyc. The fit (solid green line) is in very good agreement with the measured polariton branch dispersion, yielding a prefactor d = 0.7 (normalized root mean square (RMS) deviation below ≈ 3%) for the shown cavity frequency. The normalized coupling rate is as large as Ω_R/ω_cyc = 0.57. If, instead, one keeps the prefactor at d = 1 for the standard parabolic Hopfield model, the fit (dashed blue line) does not describe the measured data very accurately (normalized RMS deviation rises to above 15%). One efficient way to plot<cit.> the polariton dispersion is to show the normalized polariton frequencies as function of normalized coupling strength, as in Fig. <ref> (b). Thus, for each anti-crossing, which we measure as transmission spectra as function of magnetic field, we extract the upper and lower polariton frequency ω_LP/UP at the point of the minimal splitting of the two branches (= vacuum Rabi frequency) and normalize to the cold cavity frequency ω_LC (exact step by step procedure can be found in the S.M. <cit.>). In the Hopfield model, the point of minimal splitting corresponds strictly to the resonant condition ω_LC = ω_cyc, whereas in an effective model with d<1, this point of minimally splitted branches shift towards lower magnetic fields (as for a pure Dicke model with D ≡ 0, see S. M. <cit.>). The solid blue line in Fig. <ref> (b) corresponds to the calculated standard Hopfield model, which agrees well with our previous experiments on AlGaAs/GaAs QWs <cit.>. Full dark blue circles show our data from Refs. <cit.>, nicely agreeing with the Hopfield model. For the scaling study on s-Ge QWs (Fig. <ref> (b), red squares and diamonds) instead, we observe a clear deviation of our measured polariton frequencies from the calculated Hopfield dispersion. The upper and the lower polariton frequencies are at lower frequencies than expected by the Hopfield model at high normalized couplings Ω_R/ω_cyc. The prefactor d and the Rabi frequency serve as fitting parameters and it is especially notable, that the obtained value for d is dependent on the coupling strength and does not represent a constant value in our fitting result (see also data tables in the S.M. <cit.>). The deviation from the Hopfield model increases with increasing coupling strength, which we achieve by scaling the cavity frequency to lower values and thus corresponds to an anticrossing a lower magnetic fields. The measurement with largest deviation shown is the same as in Fig. <ref> as described before with d = 0.7 for Ω_R/ω_cyc = 0.57. Further measurements with even lower cavity frequencies result in only the observation of the upper branch, as the lower polariton branch is lower than the experimentally observable frequency region (THz-TDS bandwidth > 0.1 THz). In the supplementary material, for these measurements where only the upper branch is measured experimentally, we also include in the same plot the frequency of the lower polariton branch that we extrapolate by assuming that the normalized light-matter coupling strength scales linearly with the size of the resonator <cit.>.Additionally, in Fig. <ref> (b) we also include a measurement at f_LC = 160 with a partially relaxed s-Ge QW sample (as in Fig. <ref> (a)), where the semiconductor material shows a crosshatch pattern, characteristic of strain relaxation (see S. M. <cit.>). The best fit of the polariton branches yields a prefactor d = 0.95, with the normalized polariton frequencies (Fig. <ref>, green triangles) close to the Hopfield model again. This result suggests that the strain plays a critical role for the observed renormalization of the polariton energies. In conclusion, we report a solid-state system in the ultra-strong coupling regime that exhibits a mode softening of the polariton branches compared to the standard Hopfield model, where the lower polariton never reaches the ground state, regardless of how high the normalized coupling rate is. We found again that the Hopfield model<cit.>, fits our data within the experimental errors for the GaAs/AlGaAs 2DEGs.In contrast, in the s-Ge QW and InSb QW systems the polariton branches are indeed experimentally observed at lower frequencies. We capture this change by an effective model including a reduced diamagnetic term, which at high filling factor is about 30% less than in the Hopfield model. Key features of our system, which might lead to a theoretical model to predict this observed change, include the strain in the systems, non-parabolic band dispersions and (Rashba)<cit.> spin-orbit coupling effects <cit.>. Which physical effect leads to the observed deviation from the Hopfield and whether this could lead to a Dicke quantum phase transition remains an open question and needs to be investigated in the future. However, with our results we clearly enter an uncharted regime of the ultra-strong coupling with a purely ground state, solid state system that does not fall withinthe validity of a Hopfield model. Furthermore, contributing effect of the very tight electric field confinement in the metamaterial resonator cannot be ruled out, as most theoretical models <cit.> still consider Fabry-Perot type cavities with optical confinement over a wavelength of light. The strong electric field gradients that occur at deep subwavelength dimensions in our metamaterials will enhance, through Maxwell's equation, the vector potential and therefore magnetic coupling to the system.Exploring the nature of the ground state and its excitations in this parameter region would be highly relevant and could answer fundamental questions concerning the possibility of a Dicke superradiant transition outside of driven systems <cit.>. Experimentally, this study would benefit from further increasing in the light-matter coupling strength by, for example, increasing the number of quantum wells, the hole density, or further down-scaling of the resonator frequencies. Investigating other complex solid state material system might also be able to help to disentangle the possible causes and shed light on the origins of the observed opening of a lower polariton gap and its implications.We thank Elena Mavrona for supporting measurements and Curdin Maissen for discussions. We acknowledge financial support from the ERC grant 340975-MUSiC and the EPSRC (UK). We also acknowledge financial support from the Swiss National Science Foundation (SNF) through the National Centre of Competence in Research Quantum Science and Technology (NCCR QSIT) and Molecular Ultrafast Science and Technology (NCCR MUST).51 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Raimond et al.(2001)Raimond, Brune, and Haroche]RaimondRMP2001 author author J. Raimond, author M. Brune, and author S. Haroche, 10.1103/RevModPhys.73.565 journal journal Reviews of Modern Physics volume 73,pages 565 (year 2001)NoStop [Devoret et al.(2007)Devoret, Girvin, and Schoelkopf]devoret2007 author author M. H. Devoret, author S. M. Girvin,and author R. J. Schoelkopf,@noopjournal journal Annalen der Physik volume 16, pages 767 (year 2007)NoStop [Kasprzak et al.(2006)Kasprzak, Richard, Kundermann, Baas, Jeambrun, Keeling, Marchetti, Szymanska, Andre, Staehli, Savona, Littlewood, Deveaud, and Dang]KasprzakNature2006 author author J. Kasprzak, author M. Richard, author S. Kundermann, author A. Baas, author P. Jeambrun, author J. M. J. Keeling, author F. M. Marchetti, author M. H. Szymanska, author R. Andre, author J. L. Staehli, author V. Savona, author P. B.Littlewood, author B. Deveaud,and author L. S. Dang, @noopjournal journal Nature volume 443, pages 409 (year 2006)NoStop [Carusotto and Ciuti(2013)]carusottoRevModPhys85299 author author I. Carusotto and author C. Ciuti, 10.1103/RevModPhys.85.299 journal journal Rev. Mod. Phys. volume 85,pages 299 (year 2013)NoStop [Ciuti et al.(2005)Ciuti, Bastard, and Carusotto]Ciuti2005 author author C. Ciuti, author G. Bastard, and author I. Carusotto,@noopjournal journal Phys. Rev. Bvolume 72 (year 2005)NoStop [Forn-Díaz et al.(2019)Forn-Díaz, Lamata, Rico, Kono, and Solano]FornDiaz:2018vs author author P. Forn-Díaz, author L. Lamata, author E. Rico, author J. Kono,and author E. Solano, @noopjournal journal Rev. Mod. Phys(year 2019), http://arxiv.org/abs/1 1 NoStop [Kockum et al.(2019)Kockum, Miranowicz, De Liberato, Savasta, and Nori]Kockum:2019 author author A. F. Kockum, author A. Miranowicz, author S. De Liberato, author S. Savasta,and author F. Nori, @noopjournal journal Nature Reviews Physics , pages 1 (year 2019)NoStop [Hagenmüller et al.(2010)Hagenmüller, De Liberato, and Ciuti]Hagenmueller2010 author author D. Hagenmüller, author S. De Liberato,and author C. Ciuti, 10.1103/PhysRevB.81.235303 journal journal Phys. Rev. B volume 81,pages 235303 (year 2010)NoStop [Ciuti and Carusotto(2006)]Ciuti:PRA:06:033811 author author C. Ciuti and author I. Carusotto, @noopjournal journal Phys. Rev. A volume 74, pages 033811 (year 2006)NoStop [Dicke(1954)]Dicke1954 author author R. H. Dicke, 10.1103/PhysRev.93.99 journal journal Phys. Rev. volume 93,pages 99 (year 1954)NoStop [Hepp and Lieb(1973)]Hepp1973 author author K. Hepp and author E. H. Lieb, http://dx.doi.org/10.1016/0003-4916(73)90039-0 journal journal Annals of Physics volume 76, pages 360(year 1973)NoStop [Wang and Hioe(1973)]Wang1973 author author Y. K. Wang and author F. T. Hioe, 10.1103/PhysRevA.7.831 journal journal Phys. Rev. A volume 7,pages 831 (year 1973)NoStop [Skribanowitz et al.(1973)Skribanowitz, Herman, MacGillivray, andFeld]Skribanowitz1973 author author N. Skribanowitz, author I. P. Herman, author J. C. MacGillivray,and author M. S.Feld, 10.1103/PhysRevLett.30.309 journal journal Phys. Rev. Lett. volume 30, pages 309 (year 1973)NoStop [Baumann et al.(2010)Baumann, Guerlin, Brennecke, andEsslinger]Baumann2010 author author K. Baumann, author C. Guerlin, author F. Brennecke,andauthor T. Esslinger, http://dx.doi.org/10.1038/nature09009 journal journal Nature volume 464, pages 1301 (year 2010)NoStop [Cong et al.(2016)Cong, Zhang, Wang, Noe, Belyanin, and Kono]Cong2016 author author K. Cong, author Q. Zhang, author Y. Wang, author G. T. Noe, author A. Belyanin,and author J. Kono, 10.1364/JOSAB.33.000C80 journal journal J. Opt. Soc. Am. B volume 33, pages C80 (year 2016)NoStop [Scheibner et al.(2007)Scheibner, Schmidt, Worschech, Forchel, Bacher, Passow,and Hommel]Scheibner2007 author author M. Scheibner, author T. Schmidt, author L. Worschech, author A. Forchel, author G. Bacher, author T. Passow,and author D. Hommel, @noopjournal journal Nat. Phys. volume 3, pages 106 (year 2007)NoStop [Laurent et al.(2015)Laurent, Todorov, Vasanelli, Delteil, Sirtori, Sagnes, andBeaudoin]PhysRevLettLaurent2015 author author T. Laurent, author Y. Todorov, author A. Vasanelli, author A. Delteil, author C. Sirtori, author I. Sagnes,and author G. Beaudoin, 10.1103/PhysRevLett.115.187402 journal journal Phys. Rev. Lett. volume 115, pages 187402 (year 2015)NoStop [Bialynicki-Birula and Rza żżewski(1979)]Bialynicki1979 author author I. Bialynicki-Birula and author K. Rza żżewski, 10.1103/PhysRevA.19.301 journal journal Phys. Rev. A volume 19, pages 301 (year 1979)NoStop [Nataf and Ciuti(2010)]Nataf2010 author author P. Nataf and author C. Ciuti,http://dx.doi.org/10.1038/ncomms1069 journal journal Nat. Commun. volume 1, pages 72 (year 2010)NoStop [Todorov and Sirtori(2012)]Todorov2012 author author Y. Todorov and author C. Sirtori, 10.1103/PhysRevB.85.045304 journal journal Phys. Rev. B volume 85, pages 045304 (year 2012)NoStop [Emary and Brandes(2003)]Emary2003 author author C. Emary and author T. Brandes, 10.1103/PhysRevE.67.066203 journal journal Phys. Rev. E volume 67, pages 066203 (year 2003)NoStop [Scalari et al.(2012)Scalari, Maissen, Turcinkova, Hagenmüller, De Liberato, Ciuti, Reichl, Schuh, Wegscheider, Beck, and Faist]Scalari2012 author author G. Scalari, author C. Maissen, author D. Turcinkova, author D. Hagenmüller, author S. De Liberato, author C. Ciuti, author C. Reichl, author D. Schuh, author W. Wegscheider, author M. Beck,and author J. Faist, 10.1126/science.1216022 journal journal Science volume 335, pages 1323 (year 2012)NoStop [Maissen et al.(2014)Maissen, Scalari, Valmorra, Beck, Cibella, Leoni, Reichl, Charpentier, Wegscheider, andFaist]Maissen2014 author author C. Maissen, author G. Scalari, author F. Valmorra, author M. Beck, author S. Cibella, author R. Leoni, author C. Reichl, author C. Charpentier, author W. Wegscheider,and author J. Faist, 10.1103/PhysRevB.90.205309 journal journal Phys. Rev. B volume 90, pages 205309 (year 2014)NoStop [Li et al.(2018)Li, Bamba, Zhang, Fallahi, Gardner, Gao, Lou, Yoshioka, Manfra, and Kono]Li2018 author author X. Li, author M. Bamba, author Q. Zhang, author S. Fallahi, author G. C. Gardner, author W. Gao, author M. Lou, author K. Yoshioka, author M. J. Manfra,andauthor J. Kono, https://doi.org/10.1038/s41566-018-0153-0 journal journal Nat. Photon. volume 12, pages 324 (year 2018)NoStop [Baksic et al.(2013)Baksic, Nataf, and Ciuti]Baksic2013 author author A. Baksic, author P. Nataf, and author C. Ciuti, 10.1103/PhysRevA.87.023813 journal journal Phys. Rev. A volume 87, pages 023813 (year 2013)NoStop [Hagenmüller and Ciuti(2012)]Hagenmuller2012 author author D. Hagenmüller and author C. Ciuti, 10.1103/PhysRevLett.109.267403 journal journal Phys. Rev. Lett. volume 109, pages 267403 (year 2012)NoStop [Chirolli et al.(2012)Chirolli, Polini, Giovannetti, andMacDonald]Chirolli2012 author author L. Chirolli, author M. Polini, author V. Giovannetti,andauthor A. MacDonald, 10.1103/PhysRevLett.109.267404 journal journal Phys. Rev. Lett. volume 109, pages 267404 (year 2012)NoStop [Keeling(2007)]Keeling2007 author author J. Keeling, http://stacks.iop.org/0953-8984/19/i=29/a=295213 journal journal J. Phys.: Condens. Mat.volume 19, pages 295213 (year 2007)NoStop [Vukics et al.(2014)Vukics, Grießer, and Domokos]Vukics2014 author author A. Vukics, author T. Grießer,and author P. Domokos, 10.1103/PhysRevLett.112.073601 journal journal Phys. Rev. Lett. volume 112, pages 073601 (year 2014)NoStop [De Bernardis et al.(2018a)De Bernardis, Jaako, and Rabl]DeBernardis2018 author author D. De Bernardis, author T. Jaako,and author P. Rabl, 10.1103/PhysRevA.97.043820 journal journal Phys. Rev. A volume 97, pages 043820 (year 2018a)NoStop [De Bernardis et al.(2018b)De Bernardis, Pilar, Jaako, De Liberato, andRabl]PhysRevA.98.053819 author author D. De Bernardis, author P. Pilar, author T. Jaako, author S. De Liberato,and author P. Rabl, 10.1103/PhysRevA.98.053819 journal journal Phys. Rev. A volume 98, pages 053819 (year 2018b)NoStop [Malekakhlagh et al.(2017)Malekakhlagh, Petrescu, and Türeci]Malekakhlagh2017 author author M. Malekakhlagh, author A. Petrescu,and author H. E. Türeci, @noopjournal journal Phys. Rev. Lett. volume 119 (year 2017)NoStop [Di Stefano et al.(2018)Di Stefano, Settineri, Macri, Garziano, Stassi, Savasta, andNori]DiStefano2018 author author O. Di Stefano, author A. Settineri, author V. Macri, author L. Garziano, author R. Stassi, author S. Savasta,and author F. Nori, @noopjournal journal arXiv:1809.08749(year 2018)NoStop [Nataf et al.(2019)Nataf, Champel, Blatter, and Basko]Nataf_PhysRevLett2019 author author P. Nataf, author T. Champel, author G. Blatter,andauthor D. M. Basko, 10.1103/PhysRevLett.123.207402 journal journal Phys. Rev. Lett. volume 123, pages 207402 (year 2019)NoStop [Bayer et al.(2017)Bayer, Pozimski, Schambeck, Schuh, Huber, Bougeard, and Lange]Bayer2017 author author A. Bayer, author M. Pozimski, author S. Schambeck, author D. Schuh, author R. Huber, author D. Bougeard,and author C. Lange, @noopjournal journal Nano Letters volume 17, pages 6340 (year 2017)NoStop [Scalari et al.(2013)Scalari, Maissen, Hagenmüller, De Liberato, Ciuti, Reichl, Wegscheider, Schuh, Beck,and Faist]Scalari2013 author author G. Scalari, author C. Maissen, author D. Hagenmüller, author S. De Liberato, author C. Ciuti, author C. Reichl, author W. Wegscheider, author D. Schuh, author M. Beck,and author J. Faist, http://dx.doi.org/10.1063/1.4795543 journal journal J. Appl. Phys. volume 113, eid 136510 (year 2013),http://dx.doi.org/10.1063/1.4795543NoStop [Scalari et al.(2014)Scalari, Maissen, Cibella, Leoni, Carelli, Valmorra, Beck, and Faist]Scalari2014 author author G. Scalari, author C. Maissen, author S. Cibella, author R. Leoni, author P. Carelli, author F. Valmorra, author M. Beck,and author J. Faist, http://stacks.iop.org/1367-2630/16/i=3/a=033005 journal journal New J. Phys. volume 16,pages 033005 (year 2014)NoStop [Maissen et al.(2017)Maissen, Scalari, Beck, and Faist]Maissen2017 author author C. Maissen, author G. Scalari, author M. Beck,and author J. Faist, http://stacks.iop.org/1367-2630/19/i=4/a=043022 journal journal New J. Phys. volume 19,pages 043022 (year 2017)NoStop [Geiser et al.(2012)Geiser, Castellano, Scalari, Beck, Nevou, and Faist]Geiser2012 author author M. Geiser, author F. Castellano, author G. Scalari, author M. Beck, author L. Nevou,and author J. Faist, 10.1103/PhysRevLett.108.106402 journal journal Phys. Rev. Lett. volume 108, pages 106402 (year 2012)NoStop [Zhang et al.(2016)Zhang, Lou, Li, Reno, Pan, Watson, Manfra, and Kono]Zhang2016 author author Q. Zhang, author M. Lou, author X. Li, author J. L. Reno, author W. Pan, author J. D. Watson, author M. J.Manfra,and author J. Kono, http://dx.doi.org/10.1038/nphys3850 journal journal Nat. Phys. volume 12, pages 1005 (year 2016)NoStop [Failla et al.(2016)Failla, Keller, Scalari, Maissen, Faist, Reichl, Wegscheider, Newell, Leadley, Myronov,and Lloyd-Hughes]Failla2016 author author M. Failla, author J. Keller, author G. Scalari, author C. Maissen, author J. Faist, author C. Reichl, author W. Wegscheider, author O. J.Newell, author D. R.Leadley, author M. Myronov,and author J. Lloyd-Hughes, http://stacks.iop.org/1367-2630/18/i=11/a=113036 journal journal New Journal of Physics volume 18, pages 113036 (year 2016)NoStop [Myronov et al.(2014)Myronov, Morrison, Halpin, Rhead, Casteleiro, Foronda, Shah, and Leadley]Myronov2014 author author M. Myronov, author C. Morrison, author J. Halpin, author S. Rhead, author C. Casteleiro, author J. Foronda, author V. A. Shah,and author D. Leadley, http://stacks.iop.org/1347-4065/53/i=4S/a=04EH02 journal journal Jpn. J. Appl. Phys. volume 53, pages 04EH02 (year 2014)NoStop [Failla et al.(2015)Failla, Myronov, Morrison, Leadley,and Lloyd-Hughes]Failla2015 author author M. Failla, author M. Myronov, author C. Morrison, author D. R. Leadley,and author J. Lloyd-Hughes, 10.1103/PhysRevB.92.045303 journal journal Phys. Rev. B volume 92, pages 045303 (year 2015)NoStop [Lehner et al.(2018)Lehner, Tschirky, Ihn, Dietsche, Keller, Fält, and Wegscheider]Lehner2018 author author C. A. Lehner, author T. Tschirky, author T. Ihn, author W. Dietsche, author J. Keller, author S. Fält,and author W. Wegscheider, 10.1103/PhysRevMaterials.2.054601 journal journal Phys. Rev. Materials volume 2, pages 054601 (year 2018)NoStop [Khodaparast et al.(2004)Khodaparast, Doezema, Chung, Goldammer, and Santos]Khodaparast2004 author author G. A. Khodaparast, author R. E. Doezema, author S. J. Chung, author K. J. Goldammer,andauthor M. B. Santos, 10.1103/PhysRevB.70.155322 journal journal Phys. Rev. B volume 70, pages 155322 (year 2004)NoStop [Sun et al.(2007)Sun, Thompson, and Nishida]Sun2007 author author Y. Sun, author S. E. Thompson, and author T. Nishida,@noopjournal journal Journal of Applied Physics volume 101, pages 104503 (year 2007)NoStop [Adachi(1989)]Adachi1989 author author S. Adachi, @noopjournal journal J. Appl. Phys. volume 66, pages 6030 (year 1989)NoStop [of cavity cyclotron polariton modes in strained germanium 2D hole gas in the ultra-strong coupling regime"(2017)]Supplementary author author S. of cavity cyclotron polariton modes in strained germanium 2D hole gas in the ultra-strong coupling regime", @noopjournal journal Supplemental Material(year 2017)NoStop [Schwartz et al.(2011)Schwartz, Hutchison, Genet, andEbbesen]Schwartz2011 author author T. Schwartz, author J. A. Hutchison, author C. Genet, and author T. W. Ebbesen,10.1103/PhysRevLett.106.196405 journal journal Phys. Rev. Lett. volume 106,pages 196405 (year 2011)NoStop [Gao et al.(2018)Gao, Li, Bamba, and Kono]GaoNatPhys2018nanotubes author author W. Gao, author X. Li, author M. Bamba,and author J. Kono, 10.1038/s41566-018-0157-9 journal journal NATURE PHOTONICS volume 12, pages 362+ (year 2018)NoStop [Günter et al.(2009)Günter, Anappara, Hees, Sell, Biasiol, Sorba, De Liberato, Ciuti, Tredicucci, Leitenstorfer, and Huber]Gunter2009 author author G. Günter, author A. A. Anappara, author J. Hees, author A. Sell, author G. Biasiol, author L. Sorba, author S. De Liberato, author C. Ciuti, author A. Tredicucci, author A. Leitenstorfer,and author R. Huber, http://dx.doi.org/10.1038/nature07838 journal journal Nature volume 458, pages 178 (year 2009)NoStop | http://arxiv.org/abs/1708.07773v4 | {
"authors": [
"Janine Keller",
"Giacomo Scalari",
"Felice Appugliese",
"Shima Rajabali",
"Mattias Beck",
"Johannes Haase",
"Christian A. Lehner",
"Werner Wegscheider",
"Michele Failla",
"Maksym Myronov",
"David R. Leadley",
"James Lloyd-Hughes",
"Pierre Nataf",
"Jerome Faist"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170825152053",
"title": "Landau polaritons in highly non-parabolic 2D gases in the ultra-strong coupling regime"
} |
III. BRITE and SMEI satellite photometry of 28 Cygni Based in part on data collected by the BRITE-Constellation satellite mission, built, launched and operated thanks to support from the Austrian Aeronautics and Space Agency and the University of Vienna, the Canadian Space Agency (CSA), and the Foundation for Polish Science & Technology (FNiTP MNiSW) and National Science Centre (NCN). European Organisation for Astronomical Research in theSouthern Hemisphere (ESO), Karl-Schwarzschild-Str. 2,85748 Garching b. München, Germany;[email protected] Astronomical Institute, Wrocław University, Kopernika 11,51-622 Wrocław, Poland European Organisation for Astronomical Research in theSouthern Hemisphere (ESO), Casilla 19001, Santiago 19, ChileInstituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo,Rua do Matão 1226, Cidade Universitária, 05508-900 São Paulo, SP, BrazilObservatório Nacional, Rua General José Cristino 77, São Cristóvão RJ-20921-400, Rio de Janeiro, Brazil Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),Maragha, P.O. Box 55134-441, Iran Nicolaus Copernicus Astronomical Center, ul. Bartycka 18, 00-716Warsaw, Poland Institut für Kommunnikationsnetze und Satellitenkommunikation, TechnicalUniversity Graz, Inffeldgasse 12, 8010 Graz, Austria Institute of Astrophysics, University of Vienna, Universitätsring 1,1010 Vienna, Austria Département de physique and Centre de Recherche en Astrophysique duQuébec (CRAQ), Université de Montréal, C.P. 6128,Succ. Centre-Ville, Montréal, Québec, H3C 3J7, CanadaInstitute of Automatic Control, Silesian University of Technology, Akademicka 16, Gliwice, Poland Department of Physics, Royal Military College of Canada, PO Box 17000, Stn Forces, Kingston, Ontario K7K 7B4, Canada Universität Innsbruck, Institut für Astro- und Teilchenphysik,Technikerstrasse 25, 6020 Innsbruck, Austria Be stars are important reference laboratories for the investigation of viscous Keplerian discs.In some cases, the disc feeder mechanism involves a combination of nonradial pulsation (NRP) modes.Can high-cadence photometry shed further light on the role of NRP modesin facilitating rotation-supported mass loss? The BRITE Constellation of nanosatellites obtained mmag photometry of 28 Cygni for 11 months in 2014-2016.Observations with the Solar Mass Ejection Imager (SMEI) in 2003-2010 and 118 Hα line profiles, half of them from 2016, were added. For decades, 28 Cyg has exhibited four large-amplitudefrequencies: two closely spaced frequencies of spectroscopically confirmed g modes near 1.5 c/d, one slightly lower exophotospheric (Štefl) frequency, and at 0.05 c/d the difference (Δ) frequency between the two g modes.This top-level framework is indistinguishable from η Cen (Paper I), which is also very similar in spectral type, rotation rate, and viewing angle.The circumstellar (Štefl) frequency is the only one that does not seem to be affected by the Δ frequency.The amplitude of the Δ frequency undergoes large variations; aroundmaximum the amount of near-circumstellar matter is increased, and the amplitude of the Štefl frequency grows by some factor. During such brightenings dozens of transient spikes appear in the frequency spectrum, concentrated in three groups.Only eleven frequencies were common to all years of BRITE observations. Be stars seem to be controlled by several coupled clocks, most of which are not very regular on timescales of weeks to months but function for decades. The combination of g modes to the slow Δ variability and/or the atmospheric response to it appears significantly nonlinear.Like in η Cen, the Δ variability seems the main responsible for the modulation of the star-to-disc mass transfer in 28Cyg.A hierarchical set of Δ frequencies may reach the longest timescales known of the Be phenomenon.Satellite photometry of Be star 28 Cygni D. Baade et al.Short-term variability and mass loss in Be starsD. Baade1 A. Pigulski2 Th. Rivinius3 A.C. Carciofi4 D. Panoglou5 M. Ghoreyshi 4, 6 G. Handler7 R. Kuschnig8, 9 A.F.J. Moffat10 H. Pablo10 A. Popowicz11 G.A. Wade12 W.W. Weiss9 K. Zwintz13Received:; accepted: ================================================================================================================================================================================================ § INTRODUCTION§.§ Be stars Be stars owe their designation to the occurrence of emission lines forming in a self-ejected Keplerian disc.The latest broad review of the physics of Be stars is available from <cit.>. This so-called Be phenomenon poses two core challenges:a) What enables Be stars to toss up sizeable amounts of mass? Next to rapid rotation <cit.>, nonradial pulsations (NRPs) seem to be the most common property known of Be stars <cit.>.Following earlier suspicions, long series of echelle spectra of μ Cen finally enabled <cit.> to demonstrate that mass-loss events in Be stars can be triggered by the temporary combination of several NRP modes.Paper I in this series <cit.> reported the second such example, based on space photometry with BRITE-Constellation.Also in some other Be stars, rotation-assisted multi-mode NRP seems a sufficient condition for the occurrence of discrete or repetitive mass-loss events <cit.>.Spectroscopically strictly single-periodic stars like 28 ω CMa <cit.> with large-amplitude disc variability caution that the condition of multiple modes may not be necessary. But space photometry of 28 ω CMa conveys a much more complex picture <cit.>.Models for NRP-driven mass loss from rapidly rotating stars <cit.> do not yet take into account strong interactions between a small number of modes. Magnetic toy models are faced with non-detections <cit.>.b) What governs the interface between star and disc?The interplay between mass injection and viscosity rules the life cycle of Be-star discs. The main effect of the viscosity is a redistribution of the specific angular momentum of the ejecta.Only a small fraction of the gas attains Keplerian velocities while the rest falls back to the star.During phases of replenishment, a Keplerian disc can form <cit.>.When the supply of new gas is shut off, the inner disc quickly switches from decretion to accretion. During the further course of time, this transition zone slowly grows, and eventually most of the disc becomes an accretion disc <cit.>.The disc is cleared from the inside out as also observations suggest <cit.>.This somewhat violent coalescence of mass ejection and reaccretion is probably not further complicated by a significant wind of immediate stellar origin <cit.>.Instead of directly from the star, winds in Be stars may obtain a fair part of their supply of matter through radiative ablation of the disc <cit.>.Near-stellar matter can be approximated as a pseudo-photosphere <cit.> that reemits stellar radiation.At optical wavelengths, the region sampled is within very few radii of the stellar photosphere.The observable signature depends on the viewing angle <cit.> and is largest for near face-on orientations.Circumstellar gas in the line of sight absorbs light (see Fig. <ref> for some illustrationof the aspect-angle dependency).Precision photometry can extract much of the information encoded in the short- and mid-term variability of Be stars about both the central star and the circumstellar disc.It requires careful distinction between genuine pulsations and more volatile frequencies that may serve as diagnostics of circumstellar gas dynamics; seeand <cit.>.A prominent part of the latter seem to be the so-calledŠtefl frequencies <cit.>.For unknown reasons they are roughly 10% lower than the dominant spectroscopic frequency.The relevance of Be stars also has dual, stellar and circumstellar, roots.Be stars combine near-critical rotation with pulsation, making them a testbed of evolution models, and the optical angular diameters of Be discs are the largest known of any viscous (accretion or decretion) discs.For instance, quasar accretion discs measure a few lightdays across in microlensing events <cit.>.AGN in Seyfert galaxies are less powerful so that 0.01 pc is an upper limit for them.The most nearby Seyfert galaxies such as the Circinus Galaxy, NGC 1068, and NGC 4725 are at 5-20 Mpc <cit.>, bringing the angular sizes of their AGN to about 0.2 mas.Most accreting stars are compact objects with small accretion discs. Only systems with high mass-exchange rates and consisting of two main-sequence (MS) stars or an MS star and a giant develop sizeable discs.Such discs have diameters of no more than a few times that of the MS mass gainers, which typically are cooler and smaller than early-type Be stars.For instance, <cit.> measured the dimensions of the accretion disk of β Lyrae in the H band as 1 × 0.6 mas.In their Table 2, <cit.> compiled diameters from optical long-baseline interferometry at different wavelengths of 22 Be stars. They span approximate ranges of 0.5-4 mas and 1.5-15 stellar radii. In addition to being more extended, Be discs are also quite bright.These properties permit accretion-disc physics to be studied at high resolution in space, velocity, and time and at excellent signal-to-noise ratio.For the realisation of this diagnostic value, advanced radiative transfer tools have been developed: e.g., the radiatively-driven-wind model SIMECA <cit.>, HDUST <cit.>, and BEDISK <cit.>.HDUST has also been combined with a 1D hydrodynamics grid code <cit.> and a 3D smoothed particle hydrodynamics code <cit.> to form a Viscous Decretion Disc (VDD) model as employed by, e.g., <cit.> and <cit.>.Some of the diagnostically most valuable observations of Be stars are delivered by space-borne photometers.MOST <cit.>, CoRoT <cit.>, and Kepler <cit.> have pioneered the field, and the Solar Mass Ejection Imager <cit.> produced stellar photometry as a byproduct for nearly eight years.For an overview see <cit.>.Currently, BRITE Constellation (Sect. <ref>) is accumulating a large database of naked-eye stars.A first cursory preview of the broad range of variabilities observed by BRITE in Be stars was already published <cit.>.Similarly to , the present work uses BRITE space photometry to study the variability of a well-known single object. §.§ 28 Cygni28 Cyg (= HR 7708 = HD 191610 = HIP 99303) is a seemingly single B2 IV(e) star <cit.> with v sin i = 320 km/s <cit.> and no record of shell absorption lines.<cit.> list six shell stars between B1 and B3 and an average v sin i of 335 km/s on the Slettebak scale <cit.> with a maximum of 400 km/s.For a disc with opening angle ≤20^∘, the inclination angle, i, of 28 Cyg should, therefore, not exceed 75^∘.Comparison to the sample of <cit.>, which does not include shell stars, suggests a lower limit around 40^∘.The VDD model predicts <cit.> anticorrelations between colour indices U-B and B-V as observed by <cit.> in 28 Cyg as well as between V and B-V. A quantitative comparison is impossible because the calculations use a no-disc state for reference, which probably did not occur in 28 Cyg, and some maximum state, which the observations cannot be related to. The observed U-B vs. B-V slope is compatible with 30^∘≤ i ≤ 70^∘ (<cit.> do not illustrate intermediate aspect angles).28 Cyg was among the first Be stars in which short-term variability was observed <cit.>. Spectroscopy followed suit and detected periods near 0.64 d (frequency: 1.56 c/d) <cit.> and 0.7 d (1.4 c/d) <cit.>. <cit.> established a photometric peak-to-valley (PTV) amplitude of about 0.1 mag in V, and variations in B-V and U-B amounted to ∼0.07 mag PTV in both colours without obvious relation to the V magnitude.From a combination of a subset of these data with other observations, <cit.> inferred frequencies of 1.45 c/d and 0.09 c/d.<cit.> found it difficult to derive a single-periodic light curve and noted a remarkably low variability of only a few 0.01 mag on a timescale of hundreds of days.The spectroscopic studies presented signatures of nonradial pulsation. Most intriguing are reports by <cit.> and <cit.> of synchronous variations in photospheric and UV wind lines, i.e., a hypothetical link between pulsations and mass loss (see Sect. <ref>).From 209 echelle spectra obtained in 1997 and 1998, <cit.> confirmed and refined the 1.56-c/d frequency but also deduced a second one at 1.60 c/d.Both carried the signature of low-order nonradial g-mode pulsation that is typical of Be stars <cit.>.Inspired by the example of μ Cen <cit.>, the authors searched for indications of enhanced mass loss repeating with the beat frequency of 0.055 c/d.Only one outburst was clearly found to coincide with a moment when both variations reached their maximal amplitude.§ OBSERVATIONS, DATA REDUCTION, AND ANALYSIS METHODS §.§ BRIght Target Explorer (BRITE) BRITE-Constellation consists of five nanosatellites as described by <cit.>.<cit.> report on pre-launch and in-orbit tests.The pipeline processing from the raw images to the instrumental magnitudes delivered to users is elaborated by <cit.> who also report per-orbit photometric errors of 2-6 mmag for stars of comparable brightness as 28 Cyg.At V = 4.9 mag <cit.>, the star is nearly one magnitude brighter than the normal faint end of the capabilities for low-amplitude variabilities with BRITE <cit.>.After a proprietary period of typically one year, pipeline-processed BRITE data are made public[URL: http://brite.craq-astro.ca/doku.php].The datasets for 28 Cyg are summarized in Table <ref>. BRITE-Constellation members Lem (BLb) and Toronto (BTr) observed 28 Cyg for 15 and 40 d, respectively, in 2014.In a second visit in 2015, Lem and Toronto collected data for 13 and 156 d, respectively, and UniBRITE (UBr) contributed an additional 18 days.BTr and BLb revisited 28 Cyg in 2016 for another 152 and 18 d, respectively.A `b' (`r') in the abbreviation of a satellite refers to the blue 390-460 nm (red 550-700 nm) passband.The observations in 2015 and 2016 were obtained in the so-called chopping mode, in which the satellites nod between two positions slightly farther apart than the width of the point spread function.The difference between the on- and off-position aperture photometry is less affected by the progressing radiation damage of the detectors than stare-mode data are <cit.>.The point-spread function (PSF) of BRITE varies strongly across the field.Its two nodding positions are enclosed in a window of dimensions 11' × 24' <cit.>.At any one instant, the PSF covers about one-third of one nodding halve of the window.However, pointing and tracking errors introduce considerable exposure-to-exposure jitter, thereby reducing the scope of background removal.For the flux contribution from other discrete sources in the field see Fig. <ref>.The input data used are from data releases <cit.> DR2 for the observations in 2014, DR3 for 2015, and DR5 in 2016. Their preparation for time-series analysis (TSA) consisted of detrending them for dependencies on position (X/Y) and detector temperature.The procedure was the same as inexcept that a first very conservative detrending for these three parameters of the data was already applied to the 1-s exposure data.A second such decorrelation was performed with orbit-averaged data mostly with a view towards identifying and eliminating obvious outliers and correcting for any residual trends. For perfect detrending a priori knowledge about the intrinsic variability is required.Because this information does not exist, the data were reduced several times to find the best possible compromise between the removal (mostly by clipping) of artifacts and the preservation of ephemeral events.The detrending was applied separately to each so-called setup (data strings obtained with the same satellite attitude, etc.).Before the merger of data from different setups, small constant offsets were applied if shifts in zeropoint seemed to require compensation.The (pseudo-)Nyquist frequency of the results is f_ orbit/2 ≈ 7.2 c/d.The resulting light curves are plotted in Fig. <ref>.Because BRITE does not perform differential photometry, it is not really possible to determine the intrinsic accuracy beyond the reference values provided by <cit.>.Of course, one could subtract all variabilities found by the TSA and, then, compute some statistics.However, if the purpose is to assess the significance of the frequencies, this and similar methods lead to circular reasoning.As a substitute, Fig. <ref> offers the histogram of the magnitude differences between immediately consecutive orbits of BTr in 2015 and in 2016 which produced the bulk of the observations (see Table <ref>).BTr has an orbital frequency of 14.66 c/d and a duty cycle of 5-27% (Table <ref>) so that for the observed multiple frequencies of up to 3 c/d (Sect. <ref>) the inter-orbit variability is not negligible.Therefore, Fig. <ref> only enables a worst-case estimate.The TSA was performed after conversion of the fluxes to instrumental magnitudes and subtraction of the mean.Three datasets were fully examined, namely the BTr observations from 2015 as well as 2016 and their combination. The much smaller datasets from BLb are noisier and only cover a fraction of the BTr time intervals so that they were omitted from the analysis.Because potentially disturbing low-frequency variability was not seen, no further corrections were attempted. §.§ Solar Mass Ejection Imager (SMEI)The Solar Mass Ejection Imager <cit.> was a secondary payload onboard the Coriolis spacecraft and tasked with monitoring space weather in the inner solar system.The signal recorded for this purpose was the Thomson-scattered solar light from ejected interplanetary electrons. Coriolis was launched in 2003 January into an orbit with a period of 101.6 minutes.The goal design lifetime was 5 years, but SMEI was eventually powered off only in 2012 September.SMEI delivered nearly 4π surface brightness measurements with three cameras with a field of view of 3 × 120 deg^2 each and a basic angular sampling of 0.2^∘ (0.1^∘ for Camera 3).An inner zone of avoidance of 20^∘ radius enabled some overlap between cameras.Camera 3 observed the region closest to the Sun, Camera 1 the zone farthest away from it.Point sources drifted through the combined field of view within ∼1.5 min, and continuous 4-s exposures were obtained with one frame-transfer CCD per camera.The core unfiltered photometric passband ranged from 450 nm to 950 nm with a maximum near 700 nm.After an electronics failure in 2006, the temperature of Camera 3 rose, and its performance gradually degraded substantially.The point spread function of the deliberately defocussed cameras measures ∼1^∘ across, is highly asymmetric, and varies strongly over the path along which an object moved through the field of view <cit.>.With its aperture of 1.76 cm^2, SMEI could detect point sources of ∼10^ m. Surface-brightness measurements with a precision of 0.1% required careful removal of stars brighter than ∼6^ m from the images <cit.>.In addition to direct and scattered light from discrete solar-system objects, the strongest background signal by far was the zodiacal light, which in some directions exceeded the signal from solar mass ejections by two orders of magnitude.It was removed by careful modelling.However, apart from any systematic issues, the stellar SMEI data inherited much photon noise from this background, to which the South Atlantic Anomaly made additional contributions.Radiation damage of the CCDs caused a loss in sensitivity of 1.6% per year and cosmetic problems.Stellar fluxes <cit.> extracted by the UCSD pipeline <cit.> are available on the Web[http://smei.ucsd.edu/new_smei/data&images/stars/timeseries.html]. The only accompanying telemetry are observing dates and times so that it is not possible to analyze the observations separately for each camera.This has major repercussions as demonstrated below.So many years after the termination of the SMEI mission, the contact address provided at the UCSD SMEI Web site seems no longer actively supported. An example of observations with SMEI of relatively faint objects (novae) is available from <cit.> who provide overall absolute errors around maximum (3.5-5.4 mag) of 0.01-0.02 mag.SIMBAD <cit.> lists 443 sources within 0.6^∘ from 28 Cyg.Only relatively few have magnitudes in the R band, the best match of the sensitivity of SMEI. B and V fluxes for 323 and 314 sources, respectively in SIMBAD account for 1.2 and 2.2 times as much flux as 28 Cyg alone does.The distribution of these fluxes in annuli of 3 arcmin width is shown in Fig. <ref>.The SMEI observations of 28 Cyg extended over 7.9 years from Feb. 4, 2003 to Dec. 30, 2010 so that the nominal frequency resolution is ≤0.001 c/d.This dataset comprises a total of 34,335 data points.Since each data point results from one 101.6-minute orbit, more than 80% of all orbits contributed, and the nominal Nyquist frequency is ∼7.1 c/d.After removal of extreme outliers, the raw light curve (Fig. <ref>) revealed a strong need for correction for instrumental annual and other long-term variability. In addition, at some phases, the distribution of data points is bimodal, which is possibly due to the combination of data from different cameras.After prewhitening for the annual variability by subtracting means in phase intervals of 0.01, a considerable annual pattern was still left because of season-dependent excess noise. These annual phase intervals were removed from the dataset which thereby shrunk to 26,157 data points.Another 1,184 presumed outliers were eliminated after its inspection. Next, the spacing in time of the remaining 24,973 observations was studied.It can vary if the posted times correspond to the mean time of the 4-s exposures co-added for each orbit and the number of the latter varies because of rejected exposures.It was hypothesized that the best orbital data points are those that result from orbits without rejected measurements and that they can be identified by their spacing corresponding closely to the orbital period of 101.6 minutes.This selection reduced the number of data points to 23,708.Before this scheme was adopted, various others were explored, including permutations of the above steps.When the filtering according to the time elapsed between consecutive data points was performed at the beginning, it eliminated far more measurements than when done at the end.This demonstrates that the filtering of the annual light curve, which contains some subjective elements, has an objective background.In addition to the annual variability (Fig. <ref>), SMEI data suffer instrumental variabilities faster than a year but slower than those measured in the BRITE observations of 28 Cyg (Sect. <ref> and Table <ref>). Figure <ref> depicts the full light curve with all corrections applied as described above.For the observations with SMEI of α Eri (B6 Ve), ζ Pup, and ζ Oph (O9.5 V), <cit.>, <cit.>, and <cit.>, respectively, used a boxcar filter of width 10 d to remove spurious non-stellar signals.In spite of the low accuracy of the individual SMEI measurements, their large number and long time span permitted these studies to detect significant variabilities at frequencies 0.7-7.2 c/d with amplitudes down to a few mmag.Experiments with filter widths between 6 d and 15 d showed that in the range above 0.5 c/d, where most of the BRITE frequencies occur, the results of the TSA are not very sensitive to the choice of the filter width, and it was decided to not apply any such filtering.The frequencies in common with both BRITE observing runs and their amplitudes are included in Table <ref>.All values therein are averages over the full range in time.Figure <ref> indicates the presence of some very slow regular variability. In the frequency spectrum, there is a strong, well-isolated peak at f_f = 0.00471 c/d.The associated semi-amplitude is 16.7 mmag, i.e., far higher than any of the variations seen with BRITE.A sine fit with this frequency is overplotted in Fig. <ref>.Between phases 0.7 and 1.0, a similar bi-modal distribution of values as in the annual light curve (Fig. <ref>) is seen.The variability may, therefore, be due to an imperfect relative calibration of the cameras.For comparison, the SMEI data of some stars with similar positions in the sky (λ Cyg [B5 V], υ Cyg [B2 Vne], QR Vul [B3 Ve]) were analyzed.Very strong variability with 0.00411 c/d was found in υ Cyg and in QR Vul with 0.00473 c/d.Accordingly, f = 0.00471 c/d is not thought to be intrinisic to 28 Cyg.The statistical properties of the SMEI data at their high-frequency end are illustrated by Fig. <ref> which presents a histogram of the magnitude differences between immediately consecutive orbits.Comparison to the analogous Fig. <ref> for BTr shows that the stellar-to-instrumental ratio of the contributions to these differences is smaller for SMEI (which has about the same orbital period as the BRITE satellites have).This difference is owed to the systematics of the SMEI observations and the large noise from the bright background due to zodiacal and scattered solar light. § ANALYSIS RESULTS§.§ BRITE §.§.§ Common periodic variations in 2015 and 2016Figure <ref> presents the frequency spectra of the BTr observations in 2015 and 2016 (Fig. <ref>).Beyond 3.5 c/d, no significant features can be found above the general noise floor. Neither can obvious evidence for 1 c/d and related peaks be seen.A global pictorial overview of the variability of 28 Cyg as observed by BRITE is provided by the two time-frequency diagrammes in Fig. <ref>.In the BRITE frequency spectra, the three highest peaks above 0.5 c/d occur at f_1 = 1.381 c/d, f_2 = 1.545 c/d, and f_3 = 1.597 c/d (here and in the following, all frequencies, etc.are from the combined analysis of the BTr observations in 2014, 2015, and 2016 unless explicitly identified otherwise).The strongest peak below 0.5 c/d appears at f_Δ32 = 0.052 c/d.This slower variation is thought to be of initial relevance because the preparation of the raw data did not include any correction for slow variations.Growing evidence for the significance of f_Δ32 will emerge during the course of this subsection.Despite of the much shorter duration of the observations in 2014, f_Δ32, f_1, f_2, and f_3 were clearly detected also in this dataset alone.Between the above frequencies, the following relation exists:0.0517 c/d = f_Δ32≈ f_3 - f_2 = 0.0521 c/dThe deviation from full equality amounts to 0.0004 c/d which is of the order of the nominal single-season errors.Figure <ref> presents the light curve associated with f_Δ32 in 2015 and 2016. In 2015, 28 Cyg did not undergo any obvious outburst signalling mass-loss (Fig. <ref>).However, Fig. <ref> suggests small enhancements of the photometric scatter during the maxima of the variability with f_Δ32 = 0.051 c/d.The phase dependency of the scatter became very visible (Fig. <ref>) in 2016.The main contribution to this was made by the last three f_Δ32 cycles in 2016, the amplitudes of which were up to an order of magnitude higher than in the five preceding cycles (Fig. <ref>).At the time of these brightenings, the light curve also exhibited interspersed short fadings well below the mean flux level.Figure <ref> shows also that the variability is clustered in three frequency groups with approximate ranges 0.1-0.5 c/d (g_0), 1.0-1.7 c/d (g_1), and 2.2-3.0 c/d (g_2).The intervals between them are not perfectly devoid of power spikes.Contrary to g_1, the frequencies in g_2 common to 2015 and 2016 do not belong to the most prominent features of the group.The large number of temporary frequencies prompted a search for frequency variations of f_1, f_2, f_3 and other frequencies in single-season data.For each frequency, a sine curve was fitted to 40-d intervals of the dataset concerned, and this time window was shifted in steps of 1 d.No frequency appeared constant (Fig. <ref>).However, lower-amplitude frequencies exhibited higher scatter, and the shorter time interval of 40 d implies larger uncertainties in general.Except for f_1 to f_3, which are discussed in the following, the results are, therefore, mostly inconclusive.In both 2015 and 2016, f_2 and f_3 were modulated with their Δ frequency, f_Δ32.The semi-amplitudes amounted to 0.005-0.007 c/d, and f_2 and f_3 occurred nearly in antiphase (Figs. <ref> and <ref>).There is no signature of the large increase in amplitude of f_Δ32 in the last two months of the 2016 observations. To test the robustness of the result, the time interval was varied between 20 d and 60 d in steps of 5 d. There was no elementary change, and the antiphased modulation of f_2 and f_3 was confirmed in all cases.But at 60 d the amplitude was much reduced, and towards 20 d f_3 and f_2 were no longer properly resolved because, then, the length of the interval corresponds to 1/(f_3 - f_2). A similar frequency modulation does not affect the nearby f_1. f_1 only suffered a major perturbation (Fig. <ref>) before the strong increase in amplitude of the f_Δ32 variability around mJD = 610 (Fig. <ref>; mJD denotes a modified Julian Date: HJD - 2,457,000).However, during that phase of increased overall photometric amplitude, the amplitude of f_1 displayed semi-regular large-amplitude variations with a cycle length comparable to, but clearly different from, that of f_Δ32 (Fig. <ref>).There was similar activity at the end of the observations in 2015 (Fig. <ref>). By contrast, the amplitude variations of f_2 and f_3 (Fig. <ref>) were an order of magnitude lower than those of f_1.The difference between (i) the stellar variabilities with f_2 and f_3 and (ii) the exophotospheric one with f_1 is very well visible in Fig. <ref>.§.§.§ Ephemeral variationsFigure <ref> gives rise to the suspicion that ephemeral elements form a major part of the variability of 28 Cyg. Motivated by the strong changes around mJD 600, the 2016 data were split into two sets, mJD≤600 and mJD>600. The differences between their frequency spectra are stunning (Fig. <ref>). The extra features during the outburst may be mixtures of (i) combination frequencies, (ii) harmonics, (iii) side lobes due to the large amplitude variation of f_Δ32, (iv) circumstellar variations and/or (v) stellar pulsations.Since none of them was detected in two or more independent datasets, their large number makes it next to impossible to identify the nature of any such single feature, and their short lifespans prevent their characterization as periodic.Temporarily enhanced visibility is not necessarily the same as increased amplitude or power.In very rapidly rotating stars, the angular parts of NRP eigenfunctions are described by a series of spherical harmonics <cit.>.If the decomposition of eigenmodes into spherical harmonics changes during an outburst/brightening, this could also modify their visibility.In that case, such modes would not add to the cause of outbursts/brightenings.At the current level of analysis, only f_Δ32 can be considered as causing mass loss.The variability of Be stars is not erratic or even chaotic: The clear signature of low-order g-mode pulsations in high-resolution spectra <cit.> demonstrates coherent, and coherently varying, large-scale structures in the photosphere.Moreover, for several dozen spikes in the frequency spectrum the data were convolved with them, and in most cases the resulting light curve had an approximately sinusoidal shape.Finally, the frequency spectra of the 2015/16 BTr data were searched for patterns by calculating histograms of nearest-neighbour differences, frequency spectra (of the frequency spectra), and cross-correlation functions.The results were all negative, except for a possible overabundance of differences close to 0.01 c/d (e.g., 1.369 c/d - 1.360 c/d and 2.742 c/d - 2.732 c/d in Table <ref>; 1.369 and 2.742 c/d are harmonics).In the SMEI frequency spectrum, a feature exists at 0.00995 c/d; at 8.6 mmag, the amplitude is large.A similar light curve arises from the BTr observations in 2016. §.§ SMEI The three main BRITE frequencies above 0.5 c/d stand out also in the SMEI data (Fig. <ref>).Peaks at f_a = 1.3799 c/d, f_b = 1.5447 c/d, and f_c = 1.5970 c/d can be identified with f_1, f_2, and f_3 (Table <ref>).That is, the mean frequencies in 2003-2010 (SMEI) as well as in 2014-2016 (BRITE) were the same.The long duration of the SMEI observations permits the constancy of the frequencies to be examined on shorter timescales. The results of single sine curves fitted to the data (Fig. <ref>: amplitudes; Fig. <ref>: frequencies) exhibit some commonalities of the amplitude variations of f_2 and f_3 whereas the behaviour of f_1 is different and more pronounced. With a factor of ∼3, the largest relative amplitude variation was associated with f_Δ32.In the SMEI observations, a similar relation exists betweenf_a, f_b, and f_c as between f_1, f_2, and f_3 with BRITE:0.05093 c/d = f_Δcb≈ f_c - f_b = 0.05225 c/dThe light curve after prewhitening for 0.00471 c/d (Sect. <ref>) suggests that SMEI did not capture outbursts larger than 0.05 mag and longer than 100 d.This is in agreement with, but less stringent than, the BRITE observations (Sect. <ref>) and the findings of <cit.>. Even after this correction, no hint of a photometric counterpart of the fading of the Hα emission from 2005/2007 to 2013 (Sect. <ref>) is present.§.§ Spectroscopy <cit.> (see also Sect. <ref>) found two frequencies, ν_1 = 1.54562 c/d and ν_2 = 1.60033 c/d with radial-velocity amplitudes of ∼20 and ∼10 km/s, respectively.If corrected for a likely 1-c/yr alias, ν_2 becomes 1.59726 c/d, and ν_1 and ν_2 can be perfectly identified with BRITE frequencies f_2 = 1.5453 c/d and f_3 = 1.5973 c/d, respectively.The associated line-profile variability seems to be of the same ℓ = -m = 2 g-mode variety that is typical of classical Be stars <cit.>.Accordingly, f_2 and f_3 are due to nonradial pulsation g modes.As of February 2017, the BeSS database <cit.> contained 118 wavelength-calibrated Hα line profiles contributed by more than a dozen amateurs between 1995 July and 2017 January, about half of them in 2016. Those observed since August 1997 were downloaded from the BeSS Web site.The equivalent widths derived from them are presented in Fig. <ref>.Like the photometry, the contemporaneous Hα spectroscopy conveys the picture of 28 Cyg as one of the more quiet early-type Be stars.The Hα line emission was at a crudely constant high level through the end of 2002.From mid-2007 until early 2012 it was weak but again fairly constant.Thereafter it started to rise but did not reach the previous high level.The BRITE photometry started nearly 3 years after the onset of this partial recovery.During the BRITE observations the line emission stayed at some plateau with little variability.The overall timescale is at the long end of such cycles <cit.>. There are no BeSS spectra from 1998 when <cit.> observed an outburst with Heros.A single Hα profile 20 d before the amplitude maxima of f_a, f_b, and f_c seen by SMEI (Sect. <ref>) is free of anomalies.Of the 62 BeSS spectra from 2016, 57 were obtained between mJD 630 and 740 in response to an alert when the online control of the BRITE observations had recognized the increased activity starting around mJD 600.The spectra were manually normalized with fitted splines, resampled from their original wavelength bins between 0.03 Å and 0.18 Å to a common 0.1 Å, and are plotted in Fig. <ref>.The two Hα emission peaks are well, albeit not perfectly, approximated by Gaussians so that differential measurements from fitted Gaussians make sense.Figure <ref> presents the variability in total equivalent width, peak-height (V/R) ratio, and peak separation measured in the BeSS spectra; because of the inhomogeneous provenience of the observations only rough error estimates can be given.After mJD 650, there was a gentle increase by 25% in equivalent width.At about the same time, a fairly marked drop by more than 10% occurred in the peak separation.No attempt was made to search for repetitive variability on any timescale.From 44 ground-based spectra <cit.> derived an NRP period of 16.5 h (f: 1.45 c/d), which is a blend of f_1-f_3, and found it co-phased with the equivalent width of C IV λλ 1548,1551 in 17 R ≈ 10,000 spectra from the International Ultraviolet Explorer (IUE) <cit.>.In 25 IUE spectra obtained within 56 h, <cit.> found a period of 0.646±0.019 d (1.55±0.05 c/d, i.e., very close to f_2 = 1.545 c/d and still consistent with f_3 = 1.600 c/d).Because the flux amplitude increased towards lower wavelengths, it was attributed to pulsational temperature variations.The C IV λ 1551 wind line exhibited correlated variability in equivalent width while the photospheric C III λ 124.7 nm and Si II λ 126.5 nm lines did not.However, Peters & Gies did not state whether the variability in line strength was above the escape velocity so that the relation to genuine mass loss is not clear.§.§ Caveats and numerical limitsThe analysis of the photometry of 28 Cyg is hitting a fundamental mathematical limit: To measure a frequency f, a conservative rule of thumb is that the data should cover a time interval T ≥ 2/f. In the presence of noise, intervals larger by an order of magnitude are desirable.If the case of closely spaced power peaks, which, moreover, undergo independent variations, this minimal duration will often not suffice.The timescales of frequency and amplitude variations in 28 Cyg are comparable to this safe width of time windows.This touches upon the definition of a frequency.Figure <ref> illustrates the complexity of the variabilities, their variations in both frequency and power, in a single picture.For the numerous secondary frequencies (Sect. <ref>), error and significance analyses attain a very limited meaning in the presence of the numerous interrelations.The remainder of the paper only discusses frequencies f_1, f_2, f_3, and f_Δ32 (Table <ref>) all four of which were detected by BRITE in 2015 as well as in 2016 (Sect. <ref>) and confirmed with SMEI (Sect. <ref>).Moreover, f_2 and f_3 were first spectroscopically discovered.The maximal differences between the photometric values of these four frequencies are 0.0001, 0.0002, 0.0007, and 0.0013 c/d, respectively.Figure <ref> suggests a frequency jitter above that level.Therefore, Table <ref> only contains frequencies detected in both BRITE datasets (from 2015 and 2016) and agreeing to better than ∼0.002 c/d. As will become amply clear below, only very few of the many processes in 28 Cyg, if any, seem to be running on clocks with that precision.Moreover, the entries in Table <ref>are seasonal averages.There are only eleven such seemingly persistent frequencies; they are marked in Fig. <ref>.A low-threshold search between 0 and 3.5 c/d at a sampling of 0.001 c/d found ≥60 frequency peaks in 2015 and ≥100 in 2016.For randomly distributed frequencies, the probabability of chance coincidences within 0.002 c/d is, then, less than 6%.It is lower for the much fewer high-amplitude variabilities of Table <ref>.For a set of n lines in two independent observations (BRITE and SMEI), the single-frequency probability is raised to the nth power.The dashed lines in Fig. <ref> mark 3-σ noise levels after subtraction of these variabilities. This rigorous approach does not take into account the large amplitude variability of f_Δ32 especially at the end of the observations from 2016 (Fig. <ref>).Moreover, it attributes all other variability completely to stellar or instrumental noise. Such a classification system may be too crude for the variability of 28 Cyg (Sect. <ref>).For reference, we mention the work of <cit.> who determined a total of 22 frequencies with errors between 0.9 and 92 10^-5 c/d and amplitudes from 0.6 to 17 mmag from BRITE observations of the SPB star HD 201433.SMEI and BRITE frequencies agreed at the 0.8 and 1.8 σ level, respectively.In the longer SMEI observing sequences, 9 rotationally split frequency pairs could be resolved.Given the large PSFs of SMEI and BRITE, there is the risk that the main frequencies f_1, f_2, f_3, and _Δ32 are not intrinsic to 28 Cyg but are contributed by one or more other stars.Any such star would need to have fallen into the apertures of both SMEI and BRITE.The radius of a circle including the PSF at both nodding positions is ∼12 arcmin.As of August 2017, the International Variable Star Index <cit.> in VizieR <cit.> contains 2 variable stars within 15 arcmin from 28 Cyg.One (VSX J200906.4+365552) is an M5 V star that is listed as varying in R between 8.4 and 8.8 mag with unknown period.For the other one (VSX J201000.6+365111), too, a period is not known, and the stated R magnitude range is 11.8-12.4 mag.In principle, this still leaves dozens of other objects as hypothetical real sources of the observed frequencies.However, since f_Δ32 is the difference between f_3 and f_2, these three frequencies must be from one and the same star.Moreover, f_2 and f_3 were detected spectroscopically so that the common source of these three frequencies was in the aperture of the spectrograph.Only f_1 could be from a second star; however, there are plausible reasons (Sect. <ref>) to identify it as the Štefl frequency of this Be star. § DISCUSSION§.§ Conceptual frameworkMost Be stars undergo large-scale variations of their discs on timescales of years.Changes in total disc mass manifest themselves in varying Hα emission-line strengths and photometric amplitudes of several 0.1 mag <cit.>. Evidence of more nearly periodic slow photometric variability in Be stars is still restricted to shorter timescales. <cit.> present light curves of 11 Be stars with apparent periods >80 d.However, it is impossible to recognize any Be star by specific properties of this variability.Perhaps, this informational shallowness is the reason why the key to the understanding of the Be phenomenon has not been found therein.Superimposed on this slow variability are outbursts.Their spectroscopic record is particularly rich and multifacetted for μ Cen, reaching from pulsationally triggered mass ejections <cit.> via new matter settling in the circumstellar disc <cit.> to gas moving towards the star at high velocity <cit.>.<cit.> also captured an outburst in spectra of ω Ori.The build-up of line emission, i.e., a disc, as well as super-equatorial velocities associated with Štefl frequencies <cit.> are evidence of star-disc mass transfer. There may also be numerous failed mass-loss events, during which matter is tossed up but does not reach an orbit as suggested by rapid variability in broad-band polarimetry <cit.>. Such cases are not easily diagnosed by spectroscopy but may still emit additional light or remove light from the line of sight, or both. Therefore, the following will speak of `brightenings' and leave it open from which level on they may be photometric equivalents of spectroscopic mass-loss events.Inspection of the light curve (Fig. <ref>) shows that 28 Cyg quite regularly experiences brightenings with amplitudes well above the higher-frequency variability floor.For the interpretation of apparent exophotospheric photometric variability, the model calculations by <cit.> are used as the foundation.They imply that at an inclination angle near 70^∘ the photometric variability due to emission from variable amounts of near-stellar matter vanishes.On this basis, the estimates of the inclination angle of 28 Cyg, 40^∘ < i < 75^∘ (Sect. <ref>), suggest that matter ejected into the equatorial plane produces extra emission that ought to reveal itself as brightenings (see also Fig. <ref>, top panel). However, the sensitivity may be small.In the wavelength region covered by BRITE Constellation, the matter involved is expected within about two radii of (one radius above) the stellar photosphere <cit.>.Photometry alone cannot determine whether this matter is already part of the disc, drifting towards it, or falling back to the star.It may form one large contiguous structure or be fractionated into numerous small blobs.Therefore, the following will use the term `exophotospheric emission regions' (EERs) to allude to such situations.The aspect-angle dependence may be different if matter were also lost at higher stellar latitudes <cit.>.The ejecta may initially subtend an angle comparable to or smaller than the photosphere so that they absorb photospheric radiation as they quickly cool radiatively below the photospheric temperature.With time, the material spreads out, becoming much less dense, and partly reaches a Keplerian orbit.Most of this matter will not be seen projected on the stellar disc but, while close to the star, produce a brightening resulting from the reprocessing of stellar radiation.To date, in the BRITE sample of Be stars, only η Cen has been examinedin similar detail as 28 Cyg.In μ Censtrong emission from irregularly variable EERs seen at high inclination prevented detection of the stellar variability.The top-level variability of η Cen comprises the following: (i) Two spectroscopically confirmed g modes, (ii) their Δ frequency with a much larger amplitude than the sum of the two g-mode amplitudes, and (iii) a so-called Štefl frequency which is exophotospheric in nature (cf. Sect. <ref>), ∼10% slower than the g-modes, and has a very large amplitude.has argued that the Δ frequency modulates the star-to-disc mass transfer and the amplitude of the Štefl frequency traces the degree of orbital circularisation of matter in the inner disc.A transient 0.5-c/d variability that <cit.> attribute to a cloud revolving ω CMa for a few orbits may be of a similar nature.The structural similarity of the variabilities of η Cen and 28 Cyg is easily recognizable.28 Cyg, too, features two g modes (f_2 and f_3), their Δ frequency f_Δ32, whose average amplitude is not much smaller than those of f_2 and f_3 together, and f_1, which is ∼10% lower than f_2 and f_3 but has the largest mean amplitude (see Fig. <ref>).The structure of this section follows the structure of the variability. The stellar variability is evaluated in Sect. <ref>. Section <ref> compiles the evidence that f_Δ32 regulates the mass loss, and Sect. <ref> examines f_1 as a circumstellar Štefl frequency.The discussion begins with the slow irregular variability.§.§ Slow and irregular variabilityThe clearest indicator of structural variations of Be discs is the violet-to-red ratio of emission peaks, V/R.It is a robust estimator as it only requires a differential measurement.In high-inclination stars like 28 Cyg, the large rotational peak separation makes V/R variations easily detectable.It can be induced on very different timescales as global disc oscillations by the quadrupole moment of a rotationally distorted central star <cit.>, dynamically by any sufficiently massive and close companion <cit.>, radiatively by a hot companion <cit.>, and by non-axisymmetric mass injections.Also in this respect, 28 Cyg is at the far quiet end of the activity scale of early-type Be stars.The persistent high degree of symmetry of the Hα line emission suggests that any undetected companion star is too ineffective in affecting the disc.Therefore, the explanation of the slow equivalent-width variations of the Hα profiles (Fig. <ref>), which leave no doubt that the disc around 28 Cyg was eroded for some years and was replenished around 2013, must be sought in a single-star context.Contrary to many years of ground-based photometric monitoring, <cit.> recorded a discrete mass-loss event in 1998, when the absolute value of the Hα equivalent width dropped by ∼30%.The event lasted 10-15 d and was the only one observed in two seasons.The low level of disc activity is confirmed by the 5-months BeSS snapshot (Figs. <ref> and <ref>), during which no such event was observed (i.e., after the brightening starting around mJD 600).The slow variability is probably due to oscillatory disc instabilities <cit.>.The low variability in line emission is noteworthy because in the VDD model, without replenishment, Be discs are re-accreted within months to years <cit.>.Radiative ablation may accelerate this process <cit.>. Therefore, one might expect the Hα line emission to be fairly variable.The simplest resolution of this puzzle may consist of quasi-continuous star-to-disc mass transfer, which could also include small outbursts every couple of weeks. §.§ The g modes f_2 and f_3 and other stellarvariabilitiesThe most remarkable fact about the variability of 28 Cyg is the tight link, via f_Δ32, between the two g modes with f_2 and f_3.Neither the reason nor the geometric depth of the combination is known.It could be just a linear superposition and merely appear as a coupling due to the extra emission from the mass ejections it causes, perhaps amplified by a nonlinear response of the atmosphere to this slow and possibly non-adiabatic variability.There could also be non-linear mode coupling deep inside the star <cit.>.Two closely spaced frequencies also form part of an anchor variability pattern (cf. Sect. <ref>) in many other (early-type) Be stars.Examples include η Cen , ψ Per, 25 ψ^1 Ori <cit.>, 10 CMa, and 27 CMa <cit.>. However, in pole-on stars, the light curves are dominated by emission from varying amounts of near-stellar matter that blanket any underlying stellar variability. μ Cen , κ CMa, and ω CMa <cit.> are such cases.Therefore, the fraction of (early-type) Be stars with observed photometric Δ frequencies will always be lower than the genuine incidence of Δ frequencies.If many Be stars exhibit large-amplitude Δ frequencies, the conditions for their formation should not be too special.f_2 and f_3 are the most stable among the frequencies investigated (Figs. <ref>, <ref>, and <ref>; Table <ref>).However, they undergo a subtle modulation with f_Δ32, i.e., their own difference.Not any less mysterious is that f_2 and f_3 vary in antiphase (Figs. <ref> and <ref>).This may harbour some hint as to how and where the coupling of these two NRP modes takes place.While it is still conceivable that the Δ frequency modulates the position of the layer where the waves associated with the g modes are reflected and so modulates their apparent frequencies, it is not at all clear why two very closely spaced frequencies are affected in antiphase.Also the constancy in amplitude of f_2 and f_3 stands out above much of the rest (Figs. <ref>, <ref>, and <ref>).If the mass-loss modulation and apparent driving with f_Δ32 (Sect. <ref>) were ultimately powered by these two g modes, one might not expect this.Comparison to groundbased studies shows that most of them found some blend of f_1, f_2, and f_3 but could not resolve it.The implied apparentvariability of the frequencies contributed to the notion prevailing at that time that the rapid variability of Be stars is mostly semi-regular.Thanks to BRITE and SMEI it is now clearthat stellar variabilities can persist for decades. §.§ f_Δ32: The brightening in 2016 and other mass-loss indicators f_Δ32 was not detected by ground-based observations; 0.09 c/d reported by <cit.> is close to the first harmonic.f_Δ32 is the only peak in group g_0 that occurred in both 2015 and 2016. It is a variability in its own right as plain beat frequencies do not appear in power spectra.Figure <ref> illustrates the difference well: The rise in amplitude of f_Δ32 at the end of the observations in 2016 sets in rather suddenly and after an extended period of quiescence.Similarly rapid rises and drops in the amplitude of Δ frequencies have also been seen in 25 ψ^1 Ori <cit.>.A simple beating of two frequencies looks different.During its maximum, the amplitude of f_Δ32 reaches a PTV value of 100 mmag.This is well outside the domain of nonradial pulsations in other B-type stars and exceeds the (time-independent, see Fig. <ref>) amplitude sum of its parent frequencies f_2 and f_3 by a factor of 4-5.Other stars with Δ amplitudes above the sum of the parent amplitudes include η Cen , 10 CMa, and 25 ψ^1 Ori <cit.>.Moreover, the variability with f_Δ32 is highly non-sinusoidal with the range above the inflection points of the light curve being much larger than that below (Fig. <ref>). This asymmetry and the large amplitude can be understood as near-circumstellar matter emitting extra light <cit.>.Such large-amplitude asymmetries have also been observed by BRITE Constellation in 25 ψ^1 Ori <cit.>. Because of the EER effects, it is not possible to infer the real photospheric amplitude associated with f_Δ32, and the above comparison of Δ-frequency amplitude and parent g-mode amplitudes has no energetic meaning.Figure <ref> reveals another important detail: In the f_Δ32 cycle around mJD 630, several measurements appear up to ∼100 mmag below the upper envelope of the light curve.They reach well below the mean brightness so that there is a temporary deficit of light.These fadings should not be instrumental because both UBr and BTr independently recorded a similar, slightly weaker, event around mJD 198 (Fig. <ref>).The reality of this dimming is undoubtable.But the absence of simultaneous observations at all other times makes it impossible to assess the reality of any of the other, mostly much sparser, groups of bright and faint points. However, Figs. <ref> and <ref> do point to increased scatter around extrema of the f_Δ32 light curve.Comparable rapid switching between increased and attenuated brightness during phases of maximal Δ amplitude has been observed by BRITE in 25 ψ^1 Ori <cit.>.This star has a spectral type and v sin i similar to 28 Cyg so that it is also viewed from a similar perspective.One possible interpretation (cf.Sect. <ref>) is, therefore, that ejecta span some range in stellar latitude.The emission by the lifted-up matter is permanent (but not constant) whereas the only temporary removal of light implies strong variability in the amount of matter along the line of sight to the photosphere.This corroborates the conclusion that at the end of the BRITE observations in 2016 28 Cyg suffered a (small) series of mass ejections.That is, these brightenings probably were real outbursts.With this behaviour of 28 Cyg (and 25 ψ^1 Ori) there are three ways how a Δ frequency can sculpt mass-loss from Be stars:1.) Spectroscopy of μ Cen <cit.> has shown that major enhancements of the Hα line emission happen every time when the vectorial sum of at least two out of three specific NRP modes exceeds a threshold in amplitude.2.) In η Cen , the mass loss is basically permanent as is suggested by the continuous presence of the Štefl frequency in emission lines, which seems to be the response by the disc to mass transfer.The latter is continually modulated by f_Δ32 with minor apparently stochastic deviations superimposed.3.) The same may be at work in 28 Cyg but at a lower level (Fig. <ref>, Sect. <ref>).Much more prominent are single events spaced with the Δ period.But the series of such events starts suddenly and, by inference from 25 ψ^1 Ori, also declines quickly (the observations in 2016 of 28 Cyg were terminated before the end of the active phase because the Sun was too close).This confirms the report by <cit.> that they saw only one Hα line-emission outburst and could rule out repetitions with f_Δ32 = 0.05 c/d during their 100-d and 65-d observing windows.It remains to be seen whether these three categories are genuine or the artifact of incomplete temporal sampling.There is no explanation of the timing of the brightenings in 28 Cyg.They may involve a third clock (in addition to f_2 and f_3).§.§ f_1 as a Štefl frequency and other circumstellarvariabilities called the high-amplitude 1.56-c/d frequency of η Cen a Štefl frequency after the discoverer of this kind of variability <cit.>. In η Cen, the Štefl frequency was spectroscopically detected <cit.>.Typical Štefl frequencies occur during outbursts, are best seen in emission lines and in absorption lines formed high in the atmosphere, and - for unknwon or accidental reasons - appear to have values ∼10% below the main spectroscopic frequency.Condensed to the capabilities of photometry, this is a good initial description of f_1 in 28 Cyg.In η Cen, both amplitude and frequency of the Štefl frequency are strongly variable and appear modulated with the Δ frequency .In 28 Cyg, the variations are also large but there is no regularity in the frequency, only a major perturbation just before the series of brightenings in 2016 (Figs. <ref> and <ref>).This large frequency jitter is also reminiscent of Achernar, in which <cit.> meticulously documented phase shifts of one of the two detected frequencies.The maxima of the brightenings are roughly matched by strong amplitude maxima of f_1 (compare Figs. <ref>, <ref>, and <ref>).This is compatible with the spectroscopically established notion of : When an outburst/brightening occurs, the distribution of matter in the inner disc is not yet homogeneous <cit.>.The level of inhomogeneity reveals itself by the amplitude of the Štefl frequency.As in η Cen (and other Be stars observed by BRITE), the f_1 is embedded in a frequency group (g_1; Figs. <ref>, <ref>, and <ref>) the strength of which is crudely related to that of the f_1.In 28 Cyg, a fair part of the neighbourhood of f_1 changes during a major brightening (Fig. <ref>).Both observations probably strengthen the notion that these apparent companion frequencies are circumstellar.On all these the grounds, the classification of f_1 as the Štefl frequency of 28 Cyg appears safe.At 9 M_⊙, the model of <cit.> should be a good match of a B2 star like 28 Cyg <cit.>.At the assumed typical fractional critical rotation rate of 80% <cit.> its equatorial radius is increased from 5.7 R_⊙ to 6.5 R_⊙.The orbital radius of matter circling this star with the Štefl frequency, 1.4 c/d, amounts to 6.9 R_⊙.Comparison of the two radii not only supports the exophotospheric interpretation of the Štefl frequency but is also consistent with the model prediction that, at optical wavelengths, the emission from exophotospheric matter arises from a region measuring less than about two equatorial radii.In spite of all these details, the net effect of the brightenings in 2016 remains uncertain.The event reported by <cit.> impacted a fair part of the disc.But the cadence of the spectroscopy did not trace accompanying star-disc connections.The BRITE photometry has the necessary sampling but cannot equally clearly distinguish between photospheric and exophotospheric variability.BRITE photometry and Heros spectroscopy can be reconciled if the 1998 drop in Hα emission strength was caused by a short increase of the continuum flux similar to that recorded by BRITE in 2016.This would be in good agreement also with the models of <cit.>.The contemporaneous BeSS spectroscopy (Fig. <ref>) exhibits no direct counterpart of the photometric brightenings.A most sensitive indicator could be the wings of the Hα line emission.The effects of the high orbital velocity of, and Thomson scattering by, inner disc matter should both strengthen the wings <cit.>, but only in dense discs.At the time of the BeSS spectroscopy in 2016 the disc of 28 Cyg was only moderately developed.This probably explains why enhanced Hα wings wer not detected (Fig. <ref>) in the wake of thebrightening around mJD 630.Figure <ref> could be read as follows: At some moment, fresh matter was injected into the disc.With the help of viscosity, part of it drifted outward but most of it lost angular momentum and fell back to the star.Because the disc is optically thick in Hα, the equivalent width does not change immediately after the outburst but only starts to increase when the outward drifting matter reaches a domain that was not optically thick before.Since this region is farther away from the star and the disc is Keplerian, the separation of the emission peaks decreases.This description is consistent with VDD hydrodynamic simulations coupled with radiative transfer calculations <cit.>.Fig. <ref> illustrates a periodic model <cit.> that alternates between one year of disc build-up at a fixed disc-feeding rate with one year of disc dissipation when the mass loss is shut off.At t=8 yr, the outburst begins, replenishing the remnant of the disc from the previous two-year cycle with fresh material. Initially the line equivalent width drops slightly, as a result of the increased EER continuum emission, but then raises steadily and continues to grow past the cessation of the disk feeding at t=9 yr owing to the quick decrease in the EER continuum emission. As observed in 28 Cyg (Fig.<ref>), the peak separation anticorrelates with line emission, but this is no longer true during disc dissipation.Although f_1 exhibits large amplitude variations, it never disappears completely.If Štefl frequencies are indeed close tracers of star-to-disc mass transfer, the implication is that this mass transfer was variable but permanent during the BRITE observations of 28 Cyg.Such continuous, albeit variable, process may make larger contributions to the regular disc replenishment needed to overcome the demolishing effect of the viscosity than outbursts do.This is another important commonality with η Cen .It seems supported also by the persistence, for five months, of fairly strong wings of the not so strong Hα line emission (Fig. <ref>).Even in a late-type Be star, whose disc may additionally be truncated by a companion star, the Hα line emission forms over many stellar radii <cit.>.Therefore, the inner disc volume, when well filled, acts as a buffer that is insensitive to minor variations of the mass inflow.The amplitude of f_1 was also increased at the end of the observations in 2015 (Figs. <ref> and <ref>) when there was no hint of a major brightening (Fig. <ref>).This might indicate that the steady mass-loss component can also lead to inhomogeneous mass distributions close to the star (unless due to intrinsic local instabilities), or the outburst was not strong enough to develop a disc.Unless winds of Be stars originate from the circumstellar discs <cit.>, in which case a direct link to pulsation is not expected, it would be interesting to repeat the simultaneous UV and optical spectroscopy by <cit.> to identify which of f_1, f_2, and f_3, if any, correlates with UV flux and wind.To date, in no Be star, including 28 Cyg, has precision photometry or spectroscopy found mass loss clearly modulated with the frequency of a single NRP mode.Instead, timescales 20-50 times longer (occurring as Δ frequencies) have been inferred.It would be useful to extend UV observations to this range and also to check for the distribution of the variability between sub- and super-escape velocities.The sampling should permit detection of the circumstellar f_1. § CONCLUSIONS AND OUTLOOK In the photometric variability of early-type Be stars some incipient commonalities emerge as a set of frequencies that follow a common pattern whose numerical values form the specific fingerprint of every star.The top-level variability of 28 Cygni is structurally the same as that of the very similar star η Cen :(i) Most frequencies are concentrated in groups (Fig. <ref>).(ii) Four frequencies dominate the short- and medium-term variability. They fall into three different categories:(iia) Two spectroscopically confirmed g modes <cit.>.(iib) One Štefl frequency (f_1), which originates from the star-disc transition zone and is about 10% lower in value than f_2 and f_3.It may occasionally become undetecable, and it can drastically increase around brightenings but does not contribute to them (Fig. <ref>).Its frequency is much less stable than the frequencies of the g modes (Figs. <ref> and <ref>).But its avarage value is roughly constant at all times. (iic) One Δ frequency (f_Δ32) of the two g modes. It has the largest amplitude.During amplitude maxima, star-to-disc mass loss seems increased as indicated by the enhancement of the Štefl variability of emission lines in η Cen (Paper I; for 28 Cyg, see Figs. <ref>, <ref>, and <ref>).This process may also be persistently modulated with the Δ frequency.At this level, η Cen and 28 Cyg only differ in the numerical values of the parameters quantifying the frequencies and their amplitudes.Both stars also share the long-term persistence of their basic variability patterns, including considerable amplitude variations especially of f_1 and f_Δ32.The existence of a second star like η Cen lends considerable support to the description of the variability given in .An inner `engine' clocked by the g modes and powered by the Δ variability lifts matter above the photosphere.In the inner disc, a viscosity-ruled outer engine sorts the matter delivered to it into two categories, namely super- and sub-Keplerian specific angular momentum.Part of this process is observed as Štefl frequency, the amplitude of which increases with the total amount of inner-disc matter and its deviation from circular symmetry.It is not yet possible to guess whether or not η Cen and 28 Cyg are representative of a significant fraction of all early-type Be stars.<cit.>, <cit.>, <cit.>, <cit.>, and others have argued that the myriads of frequencies in Be stars are due to pulsations.With so much support, this conclusion is not to be dismissed lightly. However, Papers I and II of this series have suggested on several grounds that a circumstellar origin may be a fairly reasonable alternative for a good part of this variability.The coming and going of many features in the frequency spectrum also of 28 Cyg (Sect. <ref> and Fig. <ref>) may also point at the Štefl frequency as the tip of the iceberg.Therefore, it makes sense to search for the key to the understanding of the Be phenomenon in the small set of anchor frequencies instead of transient variations.Only for them is there any plausible direct observed link to mass loss (Figs. <ref>, <ref>, and <ref>).Ultimately, this requires that undulations of the short-term variability can also explain disc life cycles of up to a decade inferred from the Hα emission-line strength (taken as a proxy of the total amount of matter in the circumstellar disc). In 28 Cyg, the anchor frequencies have persisted for several decades which is long enough to let them enter into the explanation.The simplest extrapolation in timescale to nearly a decade (0.1 c/yr) would be a combination of modes with Δ frequencies of order 0.1 c/yr.However, viscosity will not let single big outbursts every 10 yr create photometric high states lasting for several years.In the VDD paradigm, a photometric high state of a disc could result from outbursts with a frequency and effectiveness sufficient to sustain a massive disc.The lowest level would be formed by mini-outbursts with single Δ frequencies.A second clock could come into play for outbursts perhaps associated with the brightening in 28 Cyg around mJD 630.At least a third level would be required to regulate the first two processes and cause cycles of a few years length.The example of the brightening around mJD 630 illustrates that the amplitude amplification can be a highly nonlinear process, either in true NRP amplitudes or their effect on mass loss or both.One way of achieving slow clocking could be 'hierarchical' Δ frequencies, i.e., combinations of Δ frequencies.In 28 Cyg, they could be f_Δ32 and, if real, 0.00995 c/d (Sect. <ref>).In their most basic form, they might explain seemingly particularly simple cases as depicted in Fig. 5 of <cit.> in which long-lasting bright states cyclically repeat on two similar timescales of order 500-1000 d.Irrespective of how several NRP wave patterns combine to Δ frequencies, it seems plausible that for best coupling efficiency their angular structures must be the same, i.e., they should agree in their ℓ and m indices.In μ Cen <cit.> and 28 Cyg <cit.> the available spectroscopic evidence supports this expectation.The work of <cit.> has furthermore demonstrated that ℓ = -m = 2 modes dominate the spectral variability of many Be stars.If equality of the mode indices is a requirement for mass-loss-modulating difference frequencies, this could be an important filter meaning that not any two arbitrary NRP modes can combine to cause mass loss.Indications of mass lift-ups occurring over some range in stellar latitude may place first constraints on the pulsational velocity field (ℓ and m).However, spectroscopic searches for Δ frequencies could be more effective in determining the geometry of the variability.From models, it would be attractive to learn whether energy leakage from pulsations to mass loss can establish another effective filter of the NRP mode spectrum.A robust distinction between stellar and circumstellar phenomena is critical for the identification of frequencies for asteroseismology. High-cadence long-term space photometry has the potential of achieving this.For the understanding of the Be phenomenon at large it is comforting that with the VDD model a tool is available that permits observed circumstellar variabilities to be cast into coherent sequences of dynamical states.The authors thank the BRITE operations staff for their untiring efforts to deliver data of the quality that enabled this investigation.They are also very grateful to the many amateur astronomers who, in response to a specific request, took numerous Hα spectra and shared them through BeSS.This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.This research has made use of NASA's Astrophysics Data System.This work has made use of the BeSS database, operated at LESIA, Observatoire de Meudon, France: http://basebe.obspm.fr.ACC acknowledges the support from CNPq (grant 307594/2015-7) and FAPESP (grant 2015/17967-7).DP acknowledges financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-MCTIC Brazil; grant ). M.R.G. acknowledges the support from CAPES PROEX Programa Astronomia.GH thanks the Polish NCN for support (grant 2015/18/A/ST9/00578).AFJM is grateful for financial aid from NSERC (Canada) and FRQNT (Quebec).APi acknowledges support from the Polish NCN grant no. 2016/21/B/ST9/01126.APo acknowledges support through NCN grant No. 2013/11/N/ST6/03051. GAW acknowledges Discovery Grant support from the Natural Sciences and Engineering Research Council (NSERC) of Canada.The Polish contribution to the BRITE project is supported by Polish Ministry of Science and Higher Education, and the Polish National Science Center (NCN, grant 2011/01/M/ST9/05914). | http://arxiv.org/abs/1708.07360v1 | {
"authors": [
"D. Baade",
"A. Pigulski",
"Th. Rivinius",
"A. C. Carciofi",
"D. Panoglou",
"M. Ghoreyshi",
"G. Handler",
"R. Kuschnig",
"A. F. J. Moffat",
"H. Pablo",
"A. Popowicz",
"G. A. Wade",
"W. W. Weiss",
"K. Zwintz"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170824114939",
"title": "Short-term variability and mass loss in Be stars III. BRITE and SMEI satellite photometry of 28 Cygni"
} |
^1N. L. Dukhov All-Russia Research Institute of Automatics, 127055 Moscow, Russia ^2National Research Nuclear University (MEPhI), 115409 Moscow, Russia ^3V. A. Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 125009 Moscow, Russia ^4Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, 125412 Moscow, Russia ^5Institute of Spectroscopy, Russian Academy of Sciences, 142190 Moscow region, Troitsk, Russia ^6Moscow Institute of Physics and Technology, 141700 Moscow region, Dolgoprudny, Russia We consider dynamics of a disordered ensemble of qubits interacting with single mode photon field, which is described by exactly solvable inhomogeneous Dicke model. In particular, we concentrate on the crossover from few-qubit systems to the system of many qubits and analyze how collective behavior of coupled qubits-cavity system emerges despite of the broadening. We show that quantum interference effects survive in the mesoscopic regime – dynamics of an entangled Bell state encoded into the qubit subsystem remains highly sensitive to the symmetry of the total wave function. Moreover, relaxation of these states is slowed down due to the formation of collective dark states weakly coupled to light. Dark states also significantly influence dynamics of excitations of photon subsystem by absorbing them into the qubit subsystem and releasing quasiperiodically in time. We argue that predicted phenomena can be useful in quantum technologies based on superconducting qubits. For instance, they provide tools to deeply probe both collective and quantum properties of such artificial macroscopic systems. 02.30Ik, 42.50.Ct, 03.65.Fd Dynamics of mesoscopic qubit ensemble coupled to cavity: role of collective dark statesDecember 30, 2023 =======================================================================================§ INTRODUCTION Controllable manipulation by quantum states of spin-photon coupled systems attracts a considerable current interest, since ensembles of spins/atoms interacting with the quantized electromagnetic field are promising candidates for implementation of quantum information and computation devices <cit.>. One of the most perspective applications of such systems is storage of quantum information <cit.>. There are various physical realizations of spin-photon coupled systems, which range from superconducting artificial atoms (qubits) coupled to microwave resonators to nitrogen-vacancy (NV) centers in diamond and include even hybrid circuits combining two or more physical systems <cit.>. Effective parameters of state-of-art spin-photon systems can be very different as well as numbers of coherently interacting spins (qubits). For instance, a typical number of NV centers in the ensemble is macroscopically large, while the coupling of a single center to the cavity mode is very weak. In contrast, state-of-art superconducting quantum circuits are limited by tens of qubits, while interaction between a single qubit and the microwave radiation can be relatively strong <cit.>. Nevertheless, essentially any solid-state physical realization is characterized by the inhomogeneous broadening of the density of states, i.e., by the splitting between excitation frequencies of individual spins. Broadening is caused by fundamental mechanisms and therefore is poorly controlled: for example, an excitation frequency of superconducting flux qubits depends exponentially on Josephson energies of contacts embedded into their structure <cit.>, which makes it highly sensitive to characteristics of nanometer-scale Josephson junctions. For NV-centers, inhomogeneous broadening is induced by background disorder <cit.>.Disorder in spin excitation energies is usually considered as a negative phenomenon, which prevents quantum information processing by introducing a decoherence <cit.>. In order to overcome this problem, various ideas for the spectral engineering of spin density profile <cit.> or spectral hole burning <cit.> have been proposed (mostly in the context of NV centers in diamond). However, inhomogeneous broadening can also play a positive role since it may be utilized for the construction of multi-mode quantum memories <cit.>. Theoretical description of such systems is usually provided within Tavis-Cummings (Dicke) model, while interpretations are developed in terms of so-called radiant and subradiant modes, which turn out to be coupled if broadening is present in the system, since in this case they do not match exact eigenstates of the Dicke Hamiltonian <cit.>. In other words, the excitation stored initially in the radiant mode is finally absorbed by a bath of subradiant states. A smart idea aimed to circumvent this problem is based on the hole burning technique: spins within certain 'dangerous' energy intervals, which are predominantly responsible for the interaction with subradiant modes, are neutralized for some time period by an external pulse <cit.>. As a result,special light-matter quantum states can be engineered, these states being mostly localized within the spin subsystem. If energy dissipation in a cavity is much larger than in the spin subsystem, as usually valid for realizations based on natural quantum systems, by using this approach it is possible to efficiently protect quantum state from the cavity decay, which might lead to realizations of quantum memory prototypes.The aim of the present paper is a general study of dynamics of inhomogeneously broadened spin ensembles of mesoscopic rather than macroscopic sizes. This is especially actual for possible realizations of such ensembles coupled to microwave resonators within superconducting platform (sometimes referred to as quantum metamaterials, see, e.g., Refs. <cit.>) and perhaps some other future solid-state circuits. In particular, we focus on the crossover from the system of just few qubits to the macroscopic system and study how collective dynamical behavior emerges along this crossover. This is done using an exact solution of Dicke Hamiltonian via Bethe ansatz <cit.>. We restrict ourselves to the regime of weak excitation, i.e., when there is no more that one excitation in the system. We also consider different initial conditions and show that they can result in qualitatively different dynamics. Our approach provides a simple and pictorial understanding for main features of system's evolution. It allows to obtain a direct access to Hamiltonian eigenstates, which can be classified as dark and bright, and their properties, as well as to study explicitly role of such states in the system's dynamics. In contrast to earlier studies <cit.>, we mostly concentrate on the dynamics starting from excitations within spin subsystem being motivated by recent experimental advances in hybrid systems <cit.> and consider mesoscopic qubit ensembles.We show that in the limit of just few spins there appear Rabi-like oscillations between spin excitations and photon mode irrespective of the initial condition (excitation either inspin or photon subsystem), as expected. However, as spin number increases, dynamics becomes highly sensitive to initial conditions. The most counterintuitive behavior is found for the initial condition of excitation in the spin subsystem – spin excited state becomes frozen through what we call Zeno-like effect <cit.>: its relaxation time grows with the number of spins, so that in the limit of infinite spin number the excited state does not decay at all. For finite systems, there appear periodic partial revivals of an excited state, while the period of revivals grows with the spin number. We also demonstrate that certain collective excitations encoded into the spin subsystem and characterized by a finite entanglement, such as an antisymmetric Bell state, become even more robust with respect to the influence of environment of remaining spins despite of the fact that entanglement is, in general, a very fragile entity. This collective state being constructed from a couple of spins, essentially does not decay at all even in presence of a bath of remaining spins, provided these two spins have excitation energies neighboring in the energy space. Nevertheless, for larger separation between spin excitation frequencies, the evolution remains highly sensitive to the symmetry of the wave function, i.e., to the minus or plus sign in the Bell state. This result highlights a nontrivial role of quantum interference effects for disordered ensembles in a mesoscopic regime. Since entanglement is a key resource for quantum technologies, while quantum interference effects are essential for the experimental demonstration of "quantumness" of artificial macroscopic systems, our conclusions might be important for applications. The Zeno-like effect we found is directly linked to the formation of a quasi-continuum of Hamiltonian eigenstates poorly coupled to light which we refer to as collective dark states. They have similarities with subradiant states of homogeneous model the latter states being totally uncoupled from light <cit.>. In the case of a single photon present in the system in the initial moment, dark eigenstates also significantly affect dynamics of a mesoscopic ensemble – they absorb the photon into the spin subsystem and then periodically release it giving rise to sharp revivals.Our results are potentially useful for quantum states protection, storage and engineering. We believe that they can be also used to deeply probe quantum mechanical nature as well as collective properties of mesoscopic ensembles of artificial macroscopic spins, such as superconducting qubits, coupled to cavities. § HAMILTONIAN AND PRELIMINARIES We consider an ensemble of two-level systems coupled to a single mode photon field. This coupled system is described by Dicke Hamiltonian of the formH=∑_j=1^L ϵ_j σ_j^+ σ_j^- + ω a^† a + g ∑_j=1^L (a^†σ_j^- + a σ_j^+),where a^† and a correspond to the boson degree of freedom:[a, a^†]=1,while σ_j^±, σ_j^z correspond to the paulion degrees of freedom and describe a set of L two-level systems:[σ_j^+, σ_j^-]=2σ_j^z,[σ_j^z, σ_j^±] = ±σ_j^±. The Hamiltonian (<ref>) commutes with the operator of the total pseudo-particle number, i.e., the number of bosons plus the number of excited two-level systems. Let us denote this number as M. Pseudo-particles of Dicke model are often referred to as excitations (of noninteracting system), but they should not be confused with excited states within a sector of given M (of interacting system). Namely, for any fixed M, there are different eigenstates of the Hamiltonian. At given M, the lowest energy state is the ground state, while others represent excited states. Note that ground state energies for different values of M can be also quite different. For example, if interaction constant g is large enough, a global ground state can be attained at some nonzero M. This behavior is closely related to the so called superradiant transition <cit.>.Note that the Hamiltonian (<ref>) is based on rotating wave approximation, which neglects counterrotating terms of the form g(a σ_j^- + a^†σ_j^+). These terms do not conserve excitation number and it is known that they can be omitted provided the detuning between the cavity and spin is not too large, |ϵ_j - ω| ≪ω, see, e.g., Ref. <cit.>. However, counterotating processes have to be taken into account even in the resonance, but only in certain specific situations, such as parametric and periodic excitation of a coupled qubit-cavity system <cit.>. In the situation we consider in this article, counterrotating terms can indeed be safely neglected, since we are mostly interested in the interaction between the spin ensemble and cavity, which are close to the resonance, and do not treat such parametric excitations.We also introduce an operator S^†(λ) defined asS^†(λ) = a^†+∑_j=1^Lg/λ-ϵ_jσ_j^+,which is parametrized by the energy-like quantity (rapidity) λ. The state of the form∏ _n=1^MS^†(λ_n) |↓↓…↓, 0 ⟩is an eigenstate of the Hamiltonian, provided rapidities satisfy a set of Bethe equations <cit.>λ_n-ω/g+∑_m≠ n^M 2g/λ_n - λ_m - ∑_j=1^L g/λ_n - ϵ_j=0,while the eigenenergies E are expressed through the roots λ_n asE = ∑_n=1^M λ_n.There are in general multiple solutions of Eq. (<ref>), i.e., many sets {λ_n }, which form an energy spectrum within a sector of a given M according to Eq. (<ref>).We restrict ourselves to the regime of a single pseudo-particle, M=1. In this case, there is only single rapidity λ, which satisfies a single Bethe equationλ-ω/g - ∑_j=1^L g/λ - ϵ_j=0.This is a polynomial equation of order L+1 and has the same number of solutions, which we refer to as λ^(α). It can be readily extracted from Eq. (<ref>) that all solutions are real. They correspond to different eigenstates of the Hamiltonian in the same sector M=1. The unnormalized eigenfunctions can be represented as|Φ_α⟩= S^†(λ^(α)) |↓↓…↓, 0 ⟩.It is easy to find a norm as⟨Φ_α|Φ_α⟩= 1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2.The normalized wave function thus reads as|φ_α⟩ = 1/√(⟨Φ_α|Φ_α⟩) |Φ_α⟩. In Appendix A, we show how these results can be used to analyze system's evolution starting from different initial conditions, but corresponding to the same sector M=1. This number is of course conserved during the free evolution.Note that, in absence of inhomogeneous broadening, Bethe states of Dicke model do not form a complete set (see, e.g., Ref. <cit.>), since degenerate nonradiating states, decoupled from light, cannot be obtained through Eq. (<ref>).§ FLAT DISTRIBUTION WITH NEARLY CONSTANT DENSITY OF STATES§.§ Hamiltonian eigenstates In the limit of a single spin L=1, the Hamiltonian (<ref>) reduces to the well known Jaynes-Cummings Hamiltonian. In this case, there exist only two solutions of Bethe equation (<ref>). These are shown schematically in Fig. 1, where a full resonance between spin excitation energy and photon frequency is assumed. In the case of many spins L, the set of Bethe roots becomes drastically different due to the splitting between spin energies. This situation is also illustrated schematically in Fig. 1 for distribution of spin energies having abrupt terminations. In this case, in addition to the two separated roots relevant for Jaynes-Cummings model, new roots do appear, which are confined between neighboring spin energies. Note that in the limit of strong interaction g, when splitting between spin energies is irrelevant, two separated roots become responsible for Rabi oscillations of the whole ensemble of spins with frequency g√(L). Among physically meaningful distributions of the density of states induced by disorder are the Gaussian and Lorentziandistributions <cit.> or q-Gaussian distribution relevant for NV-centers <cit.>. Furthermore, simplified equally-spaced distribution of ϵ between the two cut-offs is of fundamental importance, since it allows to grasp main features of system's dynamics produced by splitting of excitation frequencies. Physically, it might correspond to the broad distribution, for which a central part, most strongly interacting with the photon mode, has nearly constant density of states. In the next Section, we will briefly describe the effects due to distributions with smooth tails.Now let us concentrate on such an equally-spaced distribution of spin energies ϵ_j with the difference between neighboring ϵ_j denoted as d. We assume that the width of the distribution Ω, i.e., the difference between maximum and minimum excitation energies, is independent on L, while the number of spins L is large. We then consider the limit L →∞ and find explicitly a leading order in L behavior as well as dominant corrections in 1/L essential for mesoscopic systems. In this limit, d → 0. In order to construct 1/L expansion, we assume that g√(L) is L-independent. Thus, d/g ∼ 1/√(L)≪ 1 in this limit, so that there are many spin excitation frequencies within the energy scale g. We also focus on the most interesting regime, when Ω and g√(L) are of the same order that results in a very rich phase diagram already for a static systems. While the former quantity is a natural scale to characterize broadening, the latter provides coupling energy between photon and spin subsystem in absence of broadening. Thus, at Ω∼ g√(L), there exists a pronounced competition between collective action of the whole ensemble of spins and their individual behavior<cit.>. The situation we consider is of particular importance in the context of superconducting qubits-resonator systems, since state-of-art superconducting circuits seem to start entering such a regime, where collective properties become significant despite of the disorder <cit.>. Thus, we start from the limit of very few spins at g ≲Ω, which is addressed mainly numerically, and analyze the whole crossover to the limit g√(L)≫Ω with the particular focus on the intermediate mesoscopic regime g√(L)∼Ω.In Appendix B, we address solutions to Bethe equation (<ref>) within our model. There are in total (L+1) solutions, (L-1) of them are confined between neighboring spin excitation frequencies, while two remaining roots are, in general, separated, as shown in Fig. 1. Physically, two additional roots and confined roots describe Hamiltonian eigenstates with quite different properties. Indeed, it follows from Eq. (<ref>) that each of these states |φ^(α)⟩ consists of a superposition of a single photon state and spin excited states with excitation energies detuned from λ^(α) by energy not too large, i.e., ≲ g. Therefore, eigenstates corresponding to confined roots are coupled essentially to each spin among a set of ∼ g/d ≫ 1 spins and to the photon mode. The coupling to the photon thus appears as quite weak. Therefore, such eigenstates can be characterized as dark states. Actually, they can be imagined as superpositions of many individual excited spins centered in energy space around a given spin, each superposition being only weakly coupled to light. In the limit of homogeneous model, these eigenstates should become fully decoupled from the light being gradually transformed to usual subradiant states. In contrast, two separated roots have smaller number of surrounding spin excitation energies, and coupling to the light for these two eigenstates is stronger, so that they can be refereed to as bright states. We would like to stress that collective dark states emerge only in the limit g/d ≫ 1, since each of them must be represented by a superposition of many individual spin states in order to be dark. Indeed in the regime of just few spins and at g ≲Ω coupling to the light for all eigenstates is significant. §.§ Dynamics of the system with single spin excited in the initial moment Now we use our general results from Appendices A and B to study dynamics of the system with single spin excited in the initial moment, the excitation energy of this spin being ϵ_A. We rewrite the time dependent wave function (<ref>) in leading order as|ψ(t) ⟩≃d^2/g^2π^2∑ _α=1^L-1 e^-i λ^(α) tg/λ^(α)-ϵ_A1/1+1/π^2(lnϵ_L-ϵ_α/ϵ_α-ϵ_1)^2(a^†+∑_j=1^Lg/λ^(α)-ϵ_jσ_j^+)|↓↓…↓, 0 ⟩.Let us stress that we omitted in Eq. (<ref>) a contribution from two separated roots (bright states), which is justified in this order. For the amplitude of the probability to find the initially excited spin still in this state we have⟨ψ(t=0) |ψ(t) ⟩≃d^2/π^2∑ _α=1^L-1 e^-i λ^(α) t1/(λ^(α)-ϵ_A)^21/1+1/π^2(lnϵ_L-ϵ_α/ϵ_α-ϵ_1)^2. Let us now focus on the situation, when all energies ϵ_j are centered around ω, while ϵ_A is also in a resonance with ω. It is easy to see that, under such conditions, δ_α≃ d/2 in a vicinity of ω. It is also clear that the logarithm in the right-hand side of Eq. (<ref>) can be omitted (at least for t not too large). We then obtain in leading order for t ≲ 2π/d⟨ψ(t=0) |ψ(t) ⟩≃d^2/π^2 e^-iϵ_At∑ _α=1^L-1e^-i (ϵ_α-ϵ_A) t1/(ϵ_α-ϵ_A+d/2)^2≃8/π^2 e^-iϵ_At∑_n=0^∞cos[(2n+1)td/2]/(2n+1)^2.This function represents a simple periodic triangle wave of a period 4π / d. The initially excited spin decays on a time scale ∼ 1/d ∼ L. There is a revival after such a decay due to the finite size of the environment of other spins, i.e., the finiteness of the number of spins having frequencies close to ω. In other words, an initial excitation becomes redistributed between ∼ g/d spins most strongly interacting with an initially excited spin via the photon field. After some time, there occurs a refocusing of such a collective state into an initial state. At long times, an ideal periodic function (<ref>) becomes somehow smeared and less regular, since various subdominant contributions in 1/L come into play, which, in particular, results in certain finite occupations of strongly detuned spins. The major conclusion deduced from Eq. (<ref>) is that the excited state decays on a time scale, which grows as the number of spins (density of states) in the system increases. Thus, the excited state becomes stabilized by the continuum of dark states giving rise to what we refer to asZeno-like effect. A similar effect of "radiation trapping" is known from literature for homogeneous Dicke model <cit.>. Thus, we show that such an effect also takes place for inhomogeneously broadened ensembles and we reveal how it emerges as the number of spins in such an ensemble grows. Physically, this phenomenon might be attributed to the fact that the energy of the initial state with one of the spins excited is closer to the eigenenergies of dark states. The dynamics is then governed mostly by these dark states, which are weakly coupled to the light. Such an understanding has to be contrasted with naive expectations that an addition of extra 'parasitic' spins should only lead to a kind of a chaotization of Rabi oscillations between the given excited spin and the photon field. A chaotization occurs only in the limit of just few spins in the sense that in this limit dynamics is highly sensitive to precise detunings between spin excitation frequencies and the photon frequency. However, at L ≫ 1 it transforms to the universal triangle wave dependence sensitive only to the mean density of states in the vicinity of ω.Let us now take into account a first order correction in 1/L. The most important contribution in this order stems from the fact that we have neglected two additional roots of Eq. (<ref>), i.e., the bright states. We can readily find that these states produce a correction to the amplitude of the probability (<ref>) given byδ⟨ψ(t=0) |ψ(t) ⟩≃1/L e^-iϵ_At∑_α=0,Le^-i t (λ^(α)-ϵ_A)1/⟨Φ_0,L|Φ_0,L⟩g^2L/(λ^(α)-ϵ_A)^2.Bright states give rise to Rabi-like oscillations contributing to the total dynamics.Figure <ref> visualizes how dynamics changes as L grows. It shows the evolution of |⟨ψ(t=0) |ψ(t) ⟩| for L=4 (a), L=6 (b), L=10 (c), L=20 (d) and at g=0.05ω, Ω = 0.1ω, ϵ_A=ω. Two different results are compared – the first one is obtained numerically by solving Bethe equation and the second one is an explicit result, which contains a dominant contribution (<ref>) as well as a subdominant correction (<ref>). We indeed find quite good agreement between the numerical and explicit results starting from L≈10, while qualitative agreement exists even for smaller values of L. These plots visualize how Rabi oscillations at L=1 transform into triangle wave of very large period at L ≫ 1.Despite of the universality in the large L limit, actual time evolution for the spin occupation can be very sensitive to mesoscopic fluctuations. In order to illustrate this, we plot in Fig. <ref> the same quantity |⟨ψ(t=0) |ψ(t) ⟩| calculated numerically at L=20, g=0.05ω, and Ω = 0.1ω, but for randomly distributed spin energies confined between the same cutoffs, while ϵ_A is taken to be closest to the resonance among all spins. Fig. <ref> (a) and (b) correspond to two typical realizations of disorder. In the first case, the evolution of |⟨ψ(t=0) |ψ(t) ⟩| is quite close to the similar dependence for the idealized equally-spaced distribution, while in the second case spin occupation oscillates with nearly the same period, but it does not reach zero. Such a behavior in the latter case is a direct consequence of mesoscopic fluctuations, i.e., finiteness of spin number L.We also verify that Eq. (<ref>) is consistent with the normalization condition |⟨ψ(t=0) |ψ(t=0) ⟩|=1. Indeed, it is known that ∑_n=0^∞1/(2n+1)^2=π^2/8 this identity being connected to the famous Basel problem. We would like to stress that the way, this relation appeared in our formalism, seems to be highly nontrivial. In contrast, occupations in the photon subsystem appear to be generally very small under the initial condition of single spin excited. Namely, the probability amplitude of finding a system in the state with a single photon vanishes in the limit L →∞. This result can be derived from Eq. (<ref>) yielding⟨ 0, ↓…↓ | a |ψ(t) ⟩≃4d/g π^2e^-iϵ_At∑_n=0^∞sin[(2n+1)td/2]/2n+1,which is a periodical square wave of the amplitude proportional to d/g. It is small provided there are many spin excitation energies within g. Note that two separated roots produce an additional contribution of the order of 1/L. This result is consistent with the fact that the coupling between the dark states and the photon state is very weak, while dark states are predominantly responsible for the system's dynamics in large L limit. Thus, "radiation trapping" effect for homogeneous model <cit.> is naturally recovered. If energy dissipation is present in the system and it is much larger for the cavity compared to spin subsystem, than a coupling to dark states allows to drastically reduce total effective dissipation.The obtained results are potentially important in the context of quantum information storage and quantum state protection, since they show that dark eigenstates are able to greatly enhance the life time of the single excited spin and also they reveal how stabilization of excitations within spin subsystem induced by these states emerges as the number of spins increases. Perhaps, our findings can be also used to probe properties of mesoscopic ensembles of artificial spins coupled to a cavity. By exciting one of the spins of the ensemble via an additional waveguide and tracing its evolution, it is possible to see whether the ensemble behaves collectively according to the scenario we predict or such a behavior is suppressed due to the disorder and/or decoherence processes. The problem we study might be also considered as an exactly solvable toy model for the coupled qubit-cavity system in presence of mesoscopic environment of 'parasitic' quantum two-level systems (fluctuators) interacting with the main system via photon degree of freedom and leading to decoherence. This model, however, is unable to account for the fluctuators own environment, so that it is applicable provided fluctuators are more strongly coupled to the qubit-cavity system than to their environment. The model has to be contrasted with the spin-boson model and it corresponds to pure quantum regime when an entanglement between the qubit-cavity system and fluctuators is of importance <cit.>. The existence of the exact solution allows to perform microscopic analysis for such a situation without switching to any simplified and not fully controllable approximation. In this context, we investigated the dynamics of a single qubit-cavity coupled system in presence of a mesoscopic ensemble of fluctuators with strongly detuned excitation frequencies. We found that such an ensemble produces very significant phase drift ('dephasing') for the amplitude of the probability of qubit excitation. We would like to stress that this effect is much stronger than the influence of fluctuators on qubit excited state population – even if Rabi oscillations are almost perfectly reproduced, the phase drift can be significant. Of course, it becomes larger as the detuning between the mean fluctuator frequency and photon mode is decreased. We also expect that presence of individual environments of qubits should lead to the suppression of Zeno-like effect, so that in the regime of strong energy dissipation lifetime of the qubit excited state is limited by the decay into its own environment.§.§ Dynamics of the system with single photon state in the initial moment Let us now apply our general results for Bethe roots in the equally-spaced model to the dynamics of the system starting from a single photon state. For the amplitude of probability to have still a single photon state after some evolution, we obtain from (<ref>)⟨ 0, ↓…↓ | a |ψ(t) ⟩≃d^2/π^2 g^2 e^-i d t/2∑ _α=1^L-1 e^-i ϵ_α t1/1+1/π^2(lnϵ_L-ϵ_α/ϵ_α-ϵ_1)^2+∑ _α=0,L e^-i λ^(α) t1/1+g^2L/(λ^(α)-ϵ_1)(λ^(α)-ϵ_L), where we separated contributions from dark and bright states. In the regime of weak disorder, Ω≪ g√(L), a maximum of the absolute value of the first term as a function of t scales as ∼Ω^2/g^2L, while the second term starts to represent collective Rabi oscillations of the spin ensemble as a whole with frequency g√(L) and the amplitude approaching 1. In the intermediate regime g√(L)∼Ω, amplitudes of oscillations due to these two contributions are generally of the same order.As an example, Fig. <ref> shows the evolution of the occupation of a single photon state for L=4 (a), L=12 (b), L=20 (c) and at g=0.05ω, Ω = 0.1ω calculated numerically. This figure evidences that the dynamics for small spin number is essentially chaotic. However, it becomes more ordered as number of spins grows and the system enters an intermediate regime Ω∼ g√(L). Initially, photon is mainly absorbed by a set of dark states, which transform the excitation into the spin subsystem. The photon state occupation in this limit is represented by a superposition of small-amplitude Rabi-like oscillations due to two bright states and quasi-periodical sharp revivals of a period ∼ 1/d due to the set of dark states, which release the excitation from the spin subsystem and then absorb it again. These revivals are only partial. The time delay for the first revival increases with the increase of the number of spins in the ensemble due to the growth of the dark state number. The amplitude of oscillations due to bright states grows as L increases, so that in the regime g√(L)≫Ω they have to be transformed to Rabi oscillations of the whole ensemble with frequency g√(L), while the role of dark states in the dynamics of photon subsystem and for the initial condition considered becomes negligible. Let us stress that, in contrary, they play a dominant role in the dynamics of spin subsystem in this limit for the initial condition considered in the preceding Subsection.As for the dynamics of spin subsystem, we obtain for the amplitude of probability to have a single spin with energy ϵ_m excited⟨ 0, ↓…↓ | σ_m |ψ(t) ⟩ = ∑ _α=0^L e^-i λ^(α) t1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2g/λ^(α)-ϵ_m ≃4 d/π^2 g e^-i ϵ_m t∑_n=0^∞sin[(2n+1)td/2]/2n+1+∑ _α=0,L e^-i λ^(α) t1/1+g^2L/(λ^(α)-ϵ_1)(λ^(α)-ϵ_L)g/λ^(α)-ϵ_m,where we again separated contributions from dark and two bright states. This probability is generally small. §.§ Dynamics of the system with Bell entangled states encoded in the initial moment into a continuum of spins Let us now consider the dynamics of the system with constant density of states, when the initial state is one of the Bell states |χ_±⟩=1/√(2)(σ_A^+ ±σ_B^+)|↓↓…↓, 0 ⟩. These states thus can be symmetric and antisymmetric. They represent more sophisticated states which are characterized by a finite quantum entanglement and include more than one spin. The evolution, in this case, is described by Eq. (<ref>). It is known that, in the case of homogeneous model, the antisymmetric Bell state does not decay at all for any L <cit.>. It is of interest to explore an influence of broadening on its stability in the case of a mesoscopic ensemble.An important quantity in this context is an overlap between the initial state of the system χ_± and its state |ψ(t) ⟩ after the beginning of the evolution⟨χ_± |ψ(t) ⟩=1/2∑ _α=0^L e^-i λ^(α) t1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2[g/λ^(α)-ϵ_A±g/λ^(α)-ϵ_B]^2.In the large-L limit, this expression reduces to⟨χ_± |ψ(t) ⟩≃8/π^2e^-i (ϵ_A+ϵ_B) t/2(cos(ϵ_B-ϵ_A/2t) ∑_n=0^∞cos[(2n+1)td/2]/(2n+1)^2±d/(ϵ_A-ϵ_B)sin(ϵ_B-ϵ_A/2t) ∑_n=0^∞sin[(2n+1)td/2]/2n+1).The correspondent evolution of fidelity defined as |⟨χ_± |ψ(t) ⟩| and calculated numerically at L=20 for three different values of |ϵ_B-ϵ_A|, ϵ_A being in a resonance with ω, is plotted in Fig. <ref>. The agreement between the numerics and explicit result (<ref>) (not plotted in Fig. <ref>) is again spectacular.We see that instead of the periodic triangle wave fidelity follows much more complicated evolution. The life time of the state χ_+, i.e., the time needed for the fidelity to drop from maximum to the first zero, is of the same order as for the single excited spin, i.e., it is enhanced due to Zeno-like effect. The situation, however, is very different for χ_-. Namely, for the minimum possible separation between energies of two spins A and B, |ϵ_B-ϵ_A|=d, the life time becomes infinite, since the fidelity oscillates periodically but always remains high. Such a behavior in the case of χ_- state is predictable for two isolated spins, but we see that it survives even in presence of an environment of other spins with excitation frequencies close to the resonance, which are expected to strongly interact with a couple of two given spins. In reality, the interaction is strongly suppressed due to a specific structure of a two-spin wave function. The fidelity also does not vanish in the case |ϵ_B-ϵ_A|=2d, but becomes suppressed in its minimum due to the influence of the intermediate spin. Starting from |ϵ_B-ϵ_A|=3d, the minimum fidelity reaches zero, but the life time of χ_- state nevertheless remains longer than the lifetime for χ_+ state. Thus, χ_- turns out to be much more robust than χ_+ especially for small separations between ϵ_A and ϵ_B. However, the effect of difference between ϵ_A and ϵ_B is quite strong and it reflects significant deviations of behavior from the case of homogeneous model. Nevertheless, the evolution of fidelity remains highly sensitive to the plus or minus sign in the Bell state, i.e., quantum interference effects in mesoscopic regime appear as quite robust with respect to broadening.These results demonstrate that it is possible to stabilize in the spin subsystem collective quantum states involving few spins and not only single spin states. Surprisingly, they can be even more stable than single spin states and are able to support finite entanglement, in principle, for an arbitrary time, which is rather unexpectable, because entanglement is usually considered as a rather fragile entity.Note that it also follows from Eq. (<ref>) that the first-order correction produced by two separated roots is suppressed in the case of χ_- state due to the minus sign in the sum. This is consistent with the results shown in Fig. <ref>.Next, let us turn to photon subsystem and evaluate the amplitude of probability of having a single photon state. We readily obtain⟨ 0, ↓…↓ | a |ψ(t) ⟩ = 1/√(2)∑ _α=0^L e^-i λ^(α) t[g/λ^(α)-ϵ_A±g/λ^(α)-ϵ_B] 1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2.In leading order in L, it reduces to⟨ 0, ↓…↓ | a |ψ(t) ⟩≃2 √(2)/π^2d/g(e^-i ϵ_A t± e^-i ϵ_B t) ∑_n=0^∞sin[(2n+1)td/2]/2n+1Due to the presence of the term (e^-i ϵ_A t± e^-i ϵ_B t) the occupation of the single photon state in the case of antisymmetric initial condition and at ϵ_A ≈ϵ_B is dramatically reduced compared to the case of a single excited spin considered above, where it is also small. Thus, such a state turns out to be "superdark". This is why χ_- is more robust compared to χ_+. In the case of homogeneous model, χ_- becomes a true Hamiltonian eigenstate totally decoupled from light (subradiant mode <cit.>).The results of this Subsection might be also used for testing quantum mechanical nature of artificial spin-cavity systems including a manifestation of quantum interference effects, which are crucial for an unambiguous demonstration of "quantumness" of such engineered macroscopic systems. The entangled state between two qubits can be created via additional tunable cavities coupled to these particular qubits. By performing a two-qubit tomography it is then possible to follow the entanglement dynamics and to deeply probe coherent properties of qubits-cavity coupled systems. Particularly, the evolution must be sensitive to the symmetry of Bell state (plus or minus sign).§ EFFECTS OF STATISTICS Let us briefly discuss the effects arising due to deviation from equally-spaced distribution towards Gaussian or similar types of statistics characterized by smooth tails. We found numerically that if the typical energy scale associated with the deviation from equally-spaced distribution (such as mean variance for Gaussian distribution) is much larger than g, no qualitative change of behavior for dynamics starting from excitations within spin subsystem is observed at short times ∼ 2π d_0, d_0 being an inverse maximum density of states. Namely, leading order contributions to the quantities found above are qualitatively very similar. This happens because most strong interaction between the photon and spin subsystems is due to spins with energies close to the resonance with ω, while the influence of remaining spins become larger at long times. Nevertheless, subdominant contributions do change since no well defined bright state survives under Gaussian-like distributions with tails; this, for instance, results in absence of Rabi-like oscillations, which are transformed into small amplitude chaotic dependencies similar to the ones visible in Fig. <ref>.However, rather significant changes do occur provided the scale associated with the deviation from equally-spaced distribution is lowered to ≲ g. In this case, dynamics becomes more irregular. Nevertheless, the typical relaxation time of a single spin excited remains to be long. Moreover, our general conclusion about extreme robustness of antisymmetric Bell state χ_-, which do not decay at all for neighboring spins A and B, remains valid as well. § CONCLUSIONS We studied dynamics of mesoscopic ensemble of qubits coupled to a single mode cavity and concentrated on effects of disorder in qubits excitation frequencies and the regime of a weak excitation. In particular, we analyzed how collective properties of such a coupled system are formed as the number of qubits in the ensemble grows. This is done using a Bethe ansatz solution of Dicke model, which provides a direct access to Hamiltonian exact eigenstates as well as simple and pictorial understanding based on them.We found that dark states weakly coupled to light gradually emerge and start to play a very important role upon the crossover from few-qubit systems to large ensembles. They are similar to subradiant modes of homogeneous Dicke model, which are totally decoupled from the light. The role of dark states in the free evolution dynamics is more important for initial conditions corresponding to excitations within qubit subsystem.Our main conclusions are as follows:(a) Despite of inhomogeneous broadening, single qubit excitation becomes stabilized in the infinite qubit number limit through what can be referred to as Zeno-like effect. It is induced by the gradual formation of dark states.(b) Surprisingly, antisymmetric entangled Bell state constructed from the individual states of two qubits can be even more stable than single qubit excitation, provided excitation frequencies of these two qubits are not far from each other in the energy space. However, as separation grows, inhomogeneous broadening is able to suppress this effect. Nevertheless, the evolution remainshighly sensitive to the symmetry of the total wave function, which highlights a nontrivial role of quantum interference effects in mesoscopic regime.(c) The effect of finite number of spins is generally manifested through partial revivals of the initial state, the period of revivals being proportional to the number of qubits. This scenario explains how Rabi-like oscillations in the limit of just few qubits are transformed into collective and highly cooperative behavior in the limit of many qubits.(d) The role of dark states is less essential for initial condition corresponding to the single photon. They however are still of importance in the mesoscopic regime, when they are able to absorb photon into the qubit subsystem and then to release it quasiperiodically in time. A characteristic period of such revivals grows with the number of qubits, while the amplitude lowers. They coexist with Rabi-like oscillations between the whole ensemble of qubits and photon field so that in the infinite number limit such oscillations are naturally recovered.Our results provide additional insights to physics of qubits-cavity coupled systems and might be of importance for quantum states engineering, protection, and storage. In particular, we believe that our theoretical predictions can be used to deeply probe both coherent and collective properties of mesoscopic ensembles of artificial spins coupled to cavities. For instance, Zeno-like effect is definitely linked to the formation of collective properties of the ensemble despite of the disorder in excitation frequencies. On the other hand, a sensitivity of the evolution of the entangled Bell state to the symmetry of the wave function is based on quantum interference effects, while experimental demonstration of such effects is crucial for the unambiguous evidence of true "quantumness" of engineered macroscopic artificial systems.Useful comments by A. N. Rubtsov, A. A. Elistratov, and S. V. Remizov are acknowledged. This work is supported by Advanced Research Foundation (project no. 7/076/2016-2020). W. V. P. acknowledges a support from RFBR (project no. 15-02-02128). Yu. E. L. acknowledges a support from RFBR (project no. 17-02-01134). § TIME DEPENDENCE Let us consider the dynamics of the system starting from some initial state |ψ(t=0)⟩ corresponding to M=1 this number being conserved during the evolution. We expand the time-dependent wave function over the orthonormal basis |φ_α⟩ as|ψ(t) ⟩=∑ _α=0^L C_α(t) |φ_α⟩,whereC_α(t=0)=⟨φ_α | ψ(t=0) ⟩.These coefficients at arbitrary t>0 can be readily found from the Schrödinger equationC_α(t)=C_α(t=0) exp (-i λ^(α) t). Below we consider several initial conditions and general distribution of spin energies. These general results are then used to analyze system's dynamics explicitly for an equally-spaced distribution of spin energies. §.§ Single spin excited Let us obtain the time dependent wave function corresponding to the initial state defined as|ψ(t=0)⟩= σ_A^+|↓↓…↓, 0 ⟩,which describes a single spin excited, while all the remaining spins are in their ground states and there is no photon in the system. Such states have a direct relation to states engineered in Ref. by a spectral hole burning technique. It is of interest to explore its dynamics as the number of spins in the system grows. Note that in the limit L=1 there appear Rabi oscillations between the single spin and photon field. The naive expectation is that an addition of extra spins would just lead to a sort of a chaotization of Rabi oscillations. We will show that this is not the case.We readily obtain from Eq. (<ref>)C_α(t=0)=⟨φ_α | ψ(t=0) ⟩ = 1/√(⟨Φ_α|Φ_α⟩)g/λ^(α)-ϵ_A.In the explicit form, time dependent wave function reads|ψ(t) ⟩=∑ _α=0^L e^-i λ^(α) tg/λ^(α)-ϵ_A1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2(a^†+∑_j=1^Lg/λ^(α)-ϵ_jσ_j^+)|↓↓…↓, 0 ⟩. §.§ Single photon We may also consider the initial condition of another type, which corresponds to the single photon in a cavity|ψ(t=0)⟩ = a^†|↓↓…↓, 0 ⟩.From Eq. (<ref>), we again findC_α(t=0)=⟨φ_α | ψ(t=0) ⟩ = 1.In the explicit form, time dependent wave function is|ψ(t) ⟩=∑ _α=0^L e^-i λ^(α) t1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2(a^†+∑_j=1^Lg/λ^(α)-ϵ_jσ_j^+)|↓↓…↓, 0 ⟩. §.§ Bell state encoded into the spin subsystem Let us now consider a dynamics of the system with the initial state being one of the two Bell states encoded into two spins within the spin subsystem|χ_±⟩=1/√(2)(σ_A^+ ±σ_B^+)|↓↓…↓, 0 ⟩.Using the developed approach, we arrive at the explicit form of the time dependent wave function|ψ(t) ⟩=1/√(2)∑ _α=0^L e^-i λ^(α) t[g/λ^(α)-ϵ_A±g/λ^(α)-ϵ_B] 1/1+∑_j=1^Lg^2/(λ^(α)-ϵ_j)^2(a^†+∑_j=1^Lg/λ^(α)-ϵ_jσ_j^+)|↓↓…↓, 0 ⟩.§ ROOTS FOR EQUALLY-SPACED DISTRIBUTION For the confined roots, it is convenient to represent λ^(α) as ϵ_α+δ_α, where δ_α < d. In order to find δ_α, we split the sum in Eq. (<ref>) into two contributionsλ^(α)-ω/g^2 = ∑_k=0^min(α-2,L-α)[ 1/dk+d+δ_α-1/dk-δ_α] +[∑_k=min(α-2,L-α)+1^α-21/dk+d+δ_α - ∑_k=min(α-2,L-α)+1^L-α1/dk-δ_α].The first contribution is over the energies ϵ_j extending symmetrically from ϵ_α in both sides, while the second one includes contributions of the remaining energies ϵ_j. Since we consider the regime of large L ≫ 1, we may replace upper limits of summation in the first contribution by +∞. This term then reduces to∑_k=0^+∞[ 1/dk+d+δ_α-1/dk-δ_α] = π/d(πδ_α/d).We may also replace summation by integration in the remaining terms in Eq. (<ref>) as well as λ^(α) by ϵ_α in its left-hand side. We then arrive at the expression for δ_α given byδ_α≃d/π^-1( 1/π[d/g^2(ϵ_α - ω) + lnϵ_L-ϵ_α/ϵ_α-ϵ_1] ).An important special case is a distribution centered around ω, which leads to δ_α≃ d/2 for ϵ_α approaching ω.Using a similar method, we evaluate ⟨Φ_α|Φ_α⟩ as⟨Φ_α|Φ_α⟩≃ g^2 π^2/d^21/sin^2 (πδ^(α)/d).The expression of sin^2 (πδ^(α)/d) can be readily obtained from Eq. (<ref>) yielding⟨Φ_α|Φ_α⟩≃ g^2 π^2/d^2[1+1/π^2(lnϵ_L-ϵ_α/ϵ_α-ϵ_1)^2]. Let us now consider two additional roots of Eq. (<ref>), which we denote as λ^(0) and λ^(L), while λ^(0) < ϵ_1 and λ^(L) > ϵ_L. In order to find these roots, we may replace summation by integration in this equation, which is allowed provided ϵ_1-λ^(0)≫ d and λ^(L)-ϵ_L ≫ d. The equations for λ^(0) and λ^(L) are identical and they read asλ^(0,L)-ω≃g^2L/Ωlnλ^(0,L)-ϵ_1/λ^(0,L)-ϵ_L.These are transcendental equations, which can be solved numerically. Let us stress that the dependence of λ^(0,L) on g is nonanalytic. The norms for two corresponding eigenfunctions are⟨Φ_0,L|Φ_0,L⟩≃ 1+ g^2 L/(λ^(0,L)-ϵ_1)(λ^(0,L)-ϵ_L). 1A. Reiserer and G.Rempe, Rev. Mod. Phys. 87, 1379 (2015).2B. Hacker, S. Welte, G. Rempe, and S. Ritter, Nature 536, 193 (2016).3H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, Nature 497, 86 (2013).MoiseevS. A. Moiseev, S. N. Andrianov, and F. F. Gubaidullin, Phys. Rev. A 82, 022311 (2010).mem1J. Nunn, K. Reim, K. C. Lee, V. O. Lorenz, B. J. Sussman, I. A. Walmsley, and D. Jaksch, Phys. Rev. Lett. 101, 260502 (2008).MolmerJ. H. Wesenberg, Z. Kurucz, and K. Mølmer, Phys. Rev. A 83, 023826 (2011).Molmer1B. Julsgaard and K. Mølmer, Phys. Rev. A 88, 062324 (2013).ShulgaK. V. Shulga, P. Yang, G. P. Fedorov, M. V. Fistul, M. Weides, and A. V. Ustinov, JETP Lett. 105, 47 (2017).NorihybZe-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori, Rev. Mod. Phys. 85, 623 (2013).GirvinR. J. Schoelkopf and S. Girvin, Nature 451, 664 (2008).MachaP. Macha, G. Oelsner, J.-M. Reiner, M. Marthaler, S. André, G. Schön, U. Hübner, H.-G. Meyer, E. Ilichev, and A. V. Ustinov, Nat. Commun. 5, 5146 (2014).ZagoskinA. M. Zagoskin, D. Felbacq, and E. Rousseau, EPJ Quantum Tech. 3, 2 (2016).qubitYu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).LukinP. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K. Yeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D. Lukin, and R. L. Walsworth, Phys. Rev. B 82, 201201(R) (2010).Krimer1S. Putz, A. Angerer, D. O. Krimer, R. Glattauer, W. J. Munro, S. Rotter, J. Schmiedmayer, and J. Majer, Nat. Phot. 11, 36 (2017).Krimer2D. O. Krimer, B. Hartl, and S. Rotter, Phys. Rev. Lett. 115, 033601 (2015).Krimer3D. O. Krimer, M. Zens, S. Putz, and S. Rotter, LaserPhoton. Rev. 10, 1023 (2016)eng1 G. Bensky, D. Petrosyan, J. Majer, J. Schmiedmayer, and G. Kurizki, Phys. Rev. A 86, 012310 (2012).eng2 J. Cai, F. Jelezko, N. Katz, A. Retzker, and M.B Plenio, New J. Phys. 14, 093030 (2012).GaudinM. Gaudin, J. Phys. (Paris) 37, 1087 (1976).LiebK. Hepp and E. H. Lieb, Ann. Phys. (NY) 76, 453 (1983).WiegmannA. M. Tsvelick and P. B. Wiegmann, Adv. Physics 32, 453 (1983).AndreevA. V. Andreev, V. Gurarie, and L. Radzihovsky, Phys. Rev. Lett. 93, 130402 (2004).YuzbE. A. Yuzbashyan, V. B. Kuznetsov, and B. L. Altshuler, Phys. Rev. B 72, 144524 (2005).Loss1O. Tsyplyatyev and D. Loss, Phys. Rev. A 80, 023803 (2009); ibid. 82, 024305 (2012).StronglyC. Sträter, O. Tsyplyatyev, and A. Faribault, Phys. Rev. B 86, 195101 (2012).ZenoB. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).CummingsF. W. Cummings and A. Dorri, Phys. Rev. A 28, 2282 (1983).GarrawayB. M. Garraway, Phil. Trans. R. Soc. A 369, 1137 (2011).LozovikN. B. Narozhny, A. M. Fedotov, and Yu. E. Lozovik, Phys. Rev. A 64, 053807 (2001).ShapiroD. S. Shapiro, A. A. Zhukov, W. V. Pogosov, and Yu. E. Lozovik, Phys. Rev. A 91, 063814 (2015); A. A. Zhukov, D. S. Shapiro, W. V. Pogosov, and Yu. E. Lozovik, Phys. Rev. A 93, 063845 (2016).TalalaevO. Babelon and D. Talalaev, J. Stat. Mech. 0706, 06013 (2007).DukJ. Dukelsky, S. Pittel, and G. Sierra, Rev. Mod. Phys. 76, 643 (2004).YudsonV. I. Rupasov and V. I. Yudson, JETP 60, 927 (1984).LittlewoodP. R. Eastham and P. B. Littlewood, Phys. Rev. B 64, 235101 (2001).NucPhys2017W. V. Pogosov, D. S. Shapiro, L. V. Bork, and A. I. Onishchenko, Nucl. Phys. B 919, 218 (2017).GalperinH. J. Wold, H. Brox, Y. M. Galperin, and J. Bergli, Phys. Rev. B 86, 205404 (2012). | http://arxiv.org/abs/1708.07645v1 | {
"authors": [
"A. A. Zhukov",
"D. S. Shapiro",
"W. V. Pogosov",
"Yu. E. Lozovik"
],
"categories": [
"quant-ph",
"cond-mat.supr-con",
"nlin.SI"
],
"primary_category": "quant-ph",
"published": "20170825081646",
"title": "Dynamics of mesoscopic qubit ensemble coupled to cavity: role of collective dark states"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.