Variational Formula Research Articles

A generalization of multiple zeta values. Part 1: Recurrent sums

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

Rare event asymptotics for exploration processes for random graphs

Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps. Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

Information theoretic limits of learning a sparse rule

We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per sample when the system size grows to infinity. This result allows us to derive an expression for the minimum mean-square error (MMSE) of the Bayesian estimator when the signal entries have a discrete distribution with finite support. We find that, for such signals and suitable vanishing scalings of the sparsity and sampling rate, the MMSE is nonincreasing piecewise constant. In specific instances the MMSE even displays an all-or-nothing phase transition, that is, the MMSE sharply jumps from its maximum value to zero at a critical sampling rate. The all-or-nothing phenomenon has previously been shown to occur in high-dimensional linear regression. Our analysis goes beyond the linear case and applies to learning the weights of a perceptron with general activation function in a teacher-student scenario. In particular, we discuss an all-or-nothing phenomenon for the generalization error with a sublinear set of training examples.

On the L p Brunn-Minkowski theory and the L p Minkowski problem for C-coconvex sets

Abstract Let $C$ be a pointed closed convex cone in ${{\mathbb {R}}^n}$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet }=C\setminus A$ is a closed convex set. For $0<p<1$, the $p$-co-sum of $C$-coconvex sets is introduced and the corresponding $L_p$ Brunn–Minkowski inequality for $C$-coconvex sets is established. We also define the $L_p$ surface area measures, for $0\neq p\in {\mathbb {R}}$, of certain $C$-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associated with a family of functions obtained from the $p$-co-sum. This motivates the $L_p$ Minkowski problem aiming to characterize the $L_p$ surface area measures of $C$-coconvex sets. The existence of solutions to the $L_p$ Minkowski problem for all $0\neq p\in {\mathbb {R}}$ is established. The $L_p$ Minkowski inequality for $0<p<1$ is proved and is used to obtain the uniqueness of the solutions to the $L_p$ Minkowski problem for $0<p<1$. For $p=0$, we introduce $(1-\tau )\diamond A_1\oplus _0\tau \diamond A_2$, the log-co-sum of two $C$-coconvex sets $A_{1}$ and $A_{2}$ with respect to $\tau \in (0, 1)$, and prove the log-Brunn–Minkowski inequality of $C$-coconvex sets. The log-Minkowski inequality is also obtained and is applied to prove the uniqueness of the solutions to the log-Minkowski problem that characterizes the cone-volume measures of $C$-coconvex sets. Our result solves an open problem raised by Schneider [41].

Variational formulas for the exit time of Hunt processes generated by semi-Dirichlet forms

Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincaré inequalities.

Exponential stability and synchronisation of fuzzy Mittag–Leffler discrete-time Cohen–Grossberg neural networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper.

An Estimate for the Steklov Zeta Function of a Planar Domain Derived from a First Variation Formula

We consider the Steklov zeta function \(\zeta _\varOmega \) of a smooth bounded simply connected planar domain \(\varOmega \subset {\mathbb {R}}^2\) of perimeter \(2\pi \). We provide a first variation formula for \(\zeta _\varOmega \) under a smooth deformation of the domain. On the base of the formula, we prove that, for every \(s\in (-1,0)\cup (0,1)\), the difference \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\) is non-negative and is equal to zero if and only if \(\varOmega \) is a round disk (\(\zeta _\mathrm{R}\) is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\ge 0\) for \(s\in (-\infty ,-1]\cup (1,\infty )\); the latter fact was proved in our previous paper (2018) in a different way. We also provide an alternative proof of the equality \(\zeta '_\varOmega (0)=2\zeta '_\mathrm{R}(0)\) obtained by Edward and Wu (Determinant of the Neumann operator on smooth Jordan curves. Proc Am Math Soc 111(2):357–363, 1991).

Lieb type convexity for positive operator monotone decreasing functions

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions, together with strictly positive linear maps of matrices. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions, by allowing positive operator monotone decreasing functions instead of power maps.Our proof is based on variational formula for trace functionals based on the Legendre transform, and a strengthened convexity of positive operator monotone decreasing functions in a previous work of Kirihata and the second named author. We also provide the generalization to the framework of unital tracial C -algebras based on Petz's work.

Propagation Phenomena in Periodic Patchy Landscapes with Interface Conditions

This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, \(c^*\), exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for \(c^*\) by using the principal eigenvalues of certain linear periodic eigenvalue problems.

Geometry of log-concave functions: the $$L_p$$ Asplund sum and the $$L_{p}$$ Minkowski problem

The aim of this paper is to develop a basic framework of the $$L_p$$ theory for the geometry of log-concave functions, which can be viewed as a functional “lifting” of the $$L_p$$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $$L_p$$ Asplund sum of log-concave functions for all $$p>1$$ and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $$L_p$$ setting. Based on these, we further establish an $$L_p$$ Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our $$L_p$$ surface area measure for log-concave functions. Consequently, the $$L_p$$ Minkowski problem for log-concave functions, which aims to characterize the $$L_p$$ surface area measure for log-concave functions, is introduced. The existence of solutions to the $$L_p$$ Minkowski problem for log-concave functions is obtained for $$p>1$$ under some mild conditions on the pre-given Borel measures.

A note on eigenvalues estimates for one-dimensional diffusion operators

Dealing with one-dimensional diffusion operators, we obtain upper and lower variational formulae on the eigenvalues given by the max–min principle, generalizing the celebrated result of Chen and Wang on the spectral gap. Our inequalities reveal to be sharp at least when the eigenvalues considered belong to the discrete spectrum of the operator, since in this case both lower and upper bounds coincide and involve the associated eigenfunctions. Based on the intertwinings between diffusion operators and some convenient gradients with weights, our approach also allows to estimate the gap between the two first positive eigenvalues when the spectral gap belongs to the discrete spectrum.

A Residual-Based Subgrid-Scale Method Combined with Spherical Harmonics Angular Approximation for Solving the Boltzmann Transport Equation

This paper presents a stabilized finite element method (FEM) and a spherical harmonics method to discretize the space and angle of the Boltzmann transport equation. The FEM is based on the subgrid-scale (SGS) model, which decomposes the unknowns into resolvable scale and SGS with an approximation for the SGS and then embeds it into a resolvable scale formulation, which yields a stabilized variational formula with only a resolvable scale. In this method, the SGS is identified as the residual of the flux, which represents the indistinguishable high-frequency component. This method is characterized by a residual equation proposed on the subgrid, thus reflecting the relationship between the residual of the flux and the residual of the source. A simple assumption is proposed that the residual of the flux is the scaling of the residual of the source. The scaling parameter is identified as a stabilization parameter, and it takes the inverse of the norm of the transport operator. This method has been verified by various benchmark problems, and the numerical results show that it has high accuracy, stability, and void applicability.

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.

Hadamard\u2019s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space

This paper concerns a variational representation formula for Wiener functionals. Let $B=\{ B_{t}\} _{t\ge 0}$ be a standard $d$-dimensional Brownian motion. Bou\'e and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$ of $B$ up to time $1$, the expectation $\mathbb{E}\!\left[ e^{F(B)}\right] $ admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also $F(B)$ to be a functional of $B$ over the whole time interval, we prove that the Bou\'e--Dupuis formula holds true provided that both $e^{F(B)}$ and $F(B)$ are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein--Uhlenbeck semigroup in $\mathbb{R}^{d}$, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the $d$-dimensional Gaussian space.

The ℓp-Gaussian-Grothendieck problem with vector spins

We study the vector spin generalization of the ℓp-Gaussian-Grothendieck problem. In other words, given integer κ≥1, we investigate the asymptotic behaviour of the ground state energy associated with the Sherrington-Kirkpatrick Hamiltonian indexed by vector spin configurations in the unit ℓp-ball. The ranges 1≤p≤2 and 2<p<∞ exhibit significantly different behaviours. When 1≤p≤2, the vector spin generalization of the ℓp-Gaussian-Grothendieck problem agrees with its scalar counterpart. In particular, its re-scaled limit is proportional to some norm of a standard Gaussian random variable. On the other hand, for 2<p<∞ the re-scaled limit of the ℓp-Gaussian-Grothendieck problem with vector spins is given by a Parisi-type variational formula.

Functional differentiation of integral operators of special form and some questions of the inverse interpolation

This article is devoted to the problem of operator interpolation and functional differentiation. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations containing the first variational derivatives of the required functional are given. For functionals defined on sets of functions and square matrices, various interpolating polynomials of the Hermitе type with nodes of the second multiplicity, which contain the first variational derivatives of the interpolated operator, are constructed. The presented solutions of the Hermitе interpolation problems are based on the algebraic Chebyshev system of functions. For analytic functions with an argument from a set of square matrices, explicit formulas for antiderivatives of functionals are obtained. The solution of some differential equations with integral operators of a special form and the first variational derivatives is found. The problem of the inverse interpolation of functions and operators is considered. Explicit schemes for constructing inverse functions and functionals, including the case of functions of a matrix variable, obtained using certain well-known results of interpolation theory, are demonstrated. Data representation is illustrated by a number of examples.

Soil Erosion Monitoring Indicators: An Approach towards Natural Resource Management in Kuresoi South, Kenya

Soil erosion is still one of the most important land problems and it is linked to land use and land cover changes. These have negative effects on land resource which ultimately affects agricultural productivity and water quality. Local monitoring systems constitute an almost compulsory component of any program or project dealing with sustainable management of natural resources. The purpose of this study was to identify soil erosion monitoring indicators in Kuresoi South, Kenya. The study was comprised of a total representative sample population of 68 respondents from Kuresoi south catchment which was achieved using Nassiuma coefficient of variation formulae. Our findings reveal a positive significant relationship between soil erosion monitoring indicators and natural resource management. Taken together, soil erosion monitoring indicators can be used in detecting change over time in soil resource.

Finite element implementation of general triangular mesh for Riesz derivative

In this work, we will study a calculation method of variation formula with Riesz fractional derivative. As far as we know, Riesz derivative is a non-local operator including 2 n directions in n − dimension space, which the difficulties for computation of variation formula rightly bother us. In this paper, we will give an accurate method to cope with element of the stiffness matrix using polynomial basis function in the general domain meshed by unstructured triangle and the proof of diagonal dominance for Riesz fractional stiffness matrix. This method can be utilized to general fractional differential equation with Riesz derivative, which especially suitable for β close to 0.5 or 1.

Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs

We consider an inhomogeneous Erdős-Rényi random graph \(G_N\) with vertex set \([N] = \{1,\dots ,N\}\) for which the pair of vertices \(i,j \in [N]\), \(i\ne j\), is connected by an edge with probability \(r(\tfrac{i}{N},\tfrac{j}{N})\), independently of other pairs of vertices. Here, \(r:\,[0,1]^2 \rightarrow (0,1)\) is a symmetric function that plays the role of a reference graphon. Let \(\lambda _N\) be the maximal eigenvalue of the adjacency matrix of \(G_N\). It is known that \(\lambda _N/N\) satisfies a large deviation principle as \(N \rightarrow \infty \). The associated rate function \(\psi _r\) is given by a variational formula that involves the rate function \(I_r\) of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of \(\psi _r\), specially when the reference graphon is of rank 1.