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In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization — the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) — belong to this class, and introduce a new algorithm — the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the problem that was posed by I. Raşa in [10] and was proved based on a probabilistic approach by Mrowiec, Rajba and Wąsowicz in [8] . We end this paper by proposing two open problems related to Raşa's proposed inequality. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of structured fractional minimization problems where the numerator part of the objective is the sum of a convex function and a Lipschitz differentiable (possibly) nonconvex function, while the denominator part is a convex function. By exploiting the structure of the problem, we propose a first-order algorithm, namely, a proximal-gradient-subgradient algorithm with backtracked extrapolation (PGSA_BE) for solving this type of optimization problem. It is worth pointing out that there are a few differences between our backtracked extrapolation and other popular extrapolations used in convex and nonconvex optimization. One of such differences is as follows: if the new iterate obtained from the extrapolated iteration satisfies a backtracking condition, then this new iterate will be replaced by the one generated from the non-extrapolated iteration. We show that any accumulation point of the sequence generated by PGSA_BE is a critical point of the problem regarded. In addition, by assuming that some auxiliary functions satisfy the Kurdyka-Łojasiewicz property, we are able to establish global convergence of the entire sequence, in the case where the denominator is locally Lipschitz differentiable, or its conjugate satisfies the calmness condition. Finally, we present some preliminary numerical results to illustrate the efficiency of PGSA_BE. [ABSTRACT FROM AUTHOR]

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by e i t : up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. [ABSTRACT FROM AUTHOR]

This paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses. This problem is motivated by a variety of applications, such as ultrasound and radar. We show that for univariate and bivariate stream of pulses, one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem, provided that a pulse-dependent separation condition is satisfied. The main result of this paper states that the recovery is robust to additive noise or model mismatch. [ABSTRACT FROM AUTHOR]

In this paper, we establish an existence and uniqueness result for the system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in R K by analytical approach. The method is based on the approximation of the penalized systems of SPDEs. [ABSTRACT FROM AUTHOR]

The C 1 , 1 estimate of the Dirichlet problem for degenerate k -Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function f is only assumed to satisfy f 1 / (k − 1) ∈ C 1 , 1. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain. [ABSTRACT FROM AUTHOR]

Consider the semilinear elliptic equation − Δ u = λ f (u) in the unit ball B 1 ⊂ R N , with Dirichlet data u | ∂ B 1 = 0 , where λ ≥ 0 is a real parameter and f is a C 1 positive, nondecreasing and convex function in [ 0 , ∞) such that f (s) / s → ∞ as s → ∞. In this paper we study the behavior of f ′ (u ⁎) near the origin when u ⁎ , the extremal solution of the previous problem associated to λ = λ ⁎ , is singular. This answers to an open problems posed by Brezis and Vázquez [2, Open problem 5]. [ABSTRACT FROM AUTHOR]

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm. [ABSTRACT FROM AUTHOR]

Classical stochastic demography predicts that environmental stochasticity reduces population growth rates and, thereby, can increase extinction risk. In contrast, in a 1978 Theoretical Population Biology paper, Gillespie demonstrated with his stochastic additive scale and concave fitness function (SAS-CFF) model that environmental stochasticity can promote genetic diversity. Extending the SAS-CFF to account for demography, I examine the simultaneous effects of environmental stochasticity on genetic diversity and population persistence. Explicit expressions for the per-capita growth rates of rare alleles and the population at low-density are derived. Consistent with Gillespie's analysis, if the log-fitness function is concave and allelic responses to the environment are not perfectly correlated, then per-capita growth rates of rare alleles are positive and genetic diversity is maintained in the sense of stochastic persistence i.e. allelic frequencies tend to stay away from zero almost-surely and in probability. Alternatively, if the log-fitness function is convex, then per-capita growth rates of rare alleles are negative and an allele asymptotically fixates with probability one. If the population's low-density, per-capita growth rate is positive, then the population persists in the sense of stochastic persistence, else it goes asymptotically extinct with probability one. In contrast to per-capita growth rates of rare alleles, the population's per-capita growth rate is a decreasing function of the concavity of the log-fitness function. Moreover, when the log-fitness function is concave, allelic diversity increases the population's per-capita growth rate while decreasing the per-capita growth rate of rare alleles; when the log-fitness function is convex, environmental stochasticity decreases the per-capita growth rate of rare alleles, but increases the population's per-capita growth rate. Collectively, these results (i) highlight how mechanisms promoting population persistence may be at odds with mechanisms promoting genetic diversity, and (ii) provide conditions under which population persistence relies on existing standing genetic variation. [ABSTRACT FROM AUTHOR]

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy–Littlewood maximal operator and the Laplace transform. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound , the error and a modulus of uniform convexity. [Copyright &y& Elsevier]

Abstract: We study the best possible constants in the Brezis–Marcus inequalities for in balls . The quantity is known by our paper [F.G. Avkhadiev, K.-J. Wirths, Unified Poincaré and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech. 87 (8–9) 26 (2007) 632–642]. In the present paper we prove the estimate and the assertion which gives that the known lower estimates in [G. Barbatis, S. Filippas, and A. Tertikas in Comm. Cont. Math. 5 (2003), no. 6, 869–881] for , are asymptotically sharp as . Also, for the 3-dimensional ball we obtain a new Brezis–Marcus type inequality which contains two parameters , and has the following form where is the first zero of the Bessel function of order and the constants are sharp for all admissible values of parameters and . [Copyright &y& Elsevier]

• A new metric learning method is presented for histogram data classification. • A convex metric learning cost function is proposed based on modified chi distance. • Local Metric Learning is proposed for facial expression recognition in the wild. • Dropout based regularizer is employed to avoid over-fitting on training data. Facial expression recognition is an interesting and challenging problem in computer vision. So far, much research has been performed in this area; however, facial expression recognition in uncontrolled conditions has remained an unresolved problem. The widely-used feature descriptors in computer vision are often histogram data. In this paper, a new metric learning method is presented for histogram data classification. In this method, chi-squared distance is appropriately modified for metric learning. Then, a convex cost function is proposed to use in metric learning optimization. Moreover, the proposed algorithm is redefined as Local Metric Learning for facial expression recognition problem. In this definition, the proposed metric learning method is applied locally on facial sub-regions. Experimental results on four histogram datasets (dslr, webcam, amazon, and caltech) as well as controlled and uncontrolled facial expression recognition datasets (CK+, SFEW, and RAF-DB) show that the proposed method has superior performance compared to the state-of-art methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions. [ABSTRACT FROM AUTHOR]

Abstract The MoF (Moment of Fluid) method is an accurate approach for interface reconstruction in numerical simulation of multi-material fluid flow. So far, most works focus on improving its accuracy and efficiency, such as developing analytic reconstruction method and deducing the iteration schemes based on high order derivatives of the objective function. In this paper, we mainly concern on improving its robustness, especially for severely deformed polygonal meshes, in which case the objective function has multiple minimum value points. By using an efficient method for solving multiple roots of the nonlinear equation in large scope, a new algorithm is developed to enhance robustness of the MoF method. The main idea of this algorithm is as follows. The first derivative of the objective function is continuous, so the minimum value points of the objective function must be the zero points of the first derivative. Instead of finding the zero points of the first derivative directly, we turn to calculating the minimum value points (also zero points) of the square of the first derivative, which is a convex function on a neighborhood of each zero point. Applying the properties of convex function, the neighbor of each extreme minimum point of it can be obtained efficiently. Then each zero point of the square of the first derivative can be obtained using the iterative formula in its neighbor. Finally, by comparing the values of the objective function at these zero points of the first derivative, the global minimum value point of the objective function can be found and is the desired solution. The new algorithm only uses the first derivative of the objective function. It doesn't need an initial guess for the solution, which has to be carefully chosen in previous works. Numerical results are presented to demonstrate the accuracy and robustness of this new algorithm. The results show that it is applicable to severely deformed polygonal mesh, even with concave cells. Highlights • A new variant of the MOF method is presented to enhance the robustness of the method. • On severely deformed meshes, the objective function may have multiple minimum values. • By solving multiple roots of a nonlinear equation, these values can be calculated. • Then by comparing these minimum values, the global minimum value can be found. • The method is very robust on highly deformed meshes, including the three-material case. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we prove some representation theorems for t-Wright convex functions, as a consequence of a support theorem, which was proved by the author in earlier paper. [Copyright &y& Elsevier]

Abstract: A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. The main result in this paper establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. The paper characterizes the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided. [ABSTRACT FROM AUTHOR]

Abstract: In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93–99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193–201]. In these papers, in the particular case of a differentiable objective function f taking values in and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions. [Copyright &y& Elsevier]

Abstract: In 1979, in an interesting paper, R.J. Morris introduced the notion of convex set function defined on an atomless finite measure space. After a short period this notion, as well as generalizations of it, began to be studied in several papers. The aim was to obtain results similar to those known for usual convex (or generalized convex) functions. Unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct. In this way there were stated even false results. The aim of this paper is to point out that using some simple ideas it is possible, on one hand, to deduce the correct results by means of convex analysis and, on the other hand, to emphasize the reasons for which there are problems with other results. [Copyright &y& Elsevier]

This paper is concerned with various fine properties of the functional D (A) ≐ ∫ T n det 1 n − 1 ⁡ (A (x)) d x introduced in [34]. This functional is defined on X p , which is the cone of matrix fields A ∈ L p (T n ; Sym + (n)) with div (A) a bounded measure. We start by correcting a mistake we noted in our [13, Corollary 7] , which concerns the upper semicontinuity of D (A) in X p. We give a proof of a refined correct statement, and we will use it to study the behaviour of D (A) when A ∈ X n n − 1 , which is the critical integrability for D (A). One of our main results gives an explicit bound of the measure generated by D (A k) for a sequence of such matrix fields { A k } k. In particular it allows us to characterize the upper semicontinuity of D (A) in the case A ∈ X n n − 1 in terms of the measure generated by the variation of { div A k } k. We show by explicit example that this characterization fails in X p if p < n n − 1. As a by-product of our characterization we also recover and generalize a result of P.-L. Lions [26,27] on the lack of compactness in the study of Sobolev embeddings. Furthermore, in analogy with Monge-Ampère theory, we give sufficient conditions under which det 1 n − 1 ⁡ (A) ∈ H 1 (T n) when A ∈ X n n − 1 , generalising the celebrated result of S. Müller [30] when A = cof D 2 φ , for a convex function φ. [ABSTRACT FROM AUTHOR]

The lighting situation in which a picture was taken has an impact on its color. Illuminant estimation is crucial in computer vision because the colors of objects vary as illumination changes. For this reason, numerous methods for estimating the illuminant have been suggested. In this paper, we suggest a novel statistic-based method for estimating single and multiple illuminants using convex functions. In this respect, convex functions are used in the two subsequent steps of normalization and weight creation. After using weighted K-means to segment the picture, each segment's associated illuminations are determined. The illumination map for the input image is estimated as a final stage. In this study, we also analyze the effect of convexity on color constancy algorithms and present proofs for the convexity of some statistic-based algorithms. Four different single and multi-illuminant datasets have been used to evaluate the proposed algorithm in terms of two evaluation metrics; recovery and reproduction angular error. We believe that the proposed method could be considered one of the statistical state-of-the-art algorithms. In addition, it has competitive results when compared to most learning-based and deep-learning methods. Further advantages of the proposed algorithm include its simplicity of implementation and low execution time. [Display omitted] • A new statistic-based method with easy implementation and no learning required. • A general statistic-based method for estimating single and multiple illuminations. • Presenting an efficient method that is one of the statistical state-of-the-art methods. • Estimating the local and global illuminations using convex functions. • Algorithm stability against changing the number of light sources and databases. [ABSTRACT FROM AUTHOR]

Alternative Tuned Mass Damper (TMD) configurations have been proposed in the literature for performance improvement and maintenance, mainly modifying their restoring force or dissipation mechanisms. In this context, Folded-Pendulum TMDs (FPTMDs) are appealing for tuning low-frequency structures because they are compact and easy to retune while avoiding sliding mechanisms. In turn, Friction Dampers (FDs) are equally attractive due to their minimal maintenance and easy desired capacity adjustment. Nevertheless, despite recent advances in integrating FDs with PTMDs, a comprehensive literature survey reveals that Reliability-Based Design Optimization (RBDO) studies for PTMDs or Folded-PTMDs with FDs cannot be found to the best of the authors' knowledge. Hence, the present paper aims at performing an original RBDO of single and multiple Friction-Folded-PTMDs in buildings subjected to stochastic excitation. Unlike the existing optimization studies in this field, a complete nonlinear description (dry friction plus large rotations) is employed because the considered applications involve high-intensity excitations, like wind and seismic loading. Finally, scenarios with multiple absorbers are included due to their associated performance and constructional advantages. Optimal design of multiple Folded-PTMDs with FDs has not yet been addressed, as they increase the optimization problem complexity leading to multimodal and convex objective functions. Accordingly, an active-learning Kriging-based Efficient Global Optimization (EGO) procedure is used, which allows for finding the optimum solutions with only a few objective function evaluations. The application cases to two distinct buildings subject to wind and seismic loadings show that using Friction-FPTMDs allows for obtaining a control strategy with an appropriate performance while ensuring practical advantages over classical TMDs. [ABSTRACT FROM AUTHOR]

This paper concerns infinitely repeated and undiscounted two-person non-zero-sum games of incomplete information on one side. Following an improvement on the proof of equilibrium existence, it establishes sufficient conditions for the existence of an equilibrium with payoffs superior to what the players would receive from observable deviation. To do this, we show that a no-where empty convex valued upper hemi-continuous correspondence from a compact metric space to a compact and convex subset of a Euclidean space can be approximated by Hausdorff-continuous convex valued correspondences that contain the original correspondence. This paper can be considered to be a first step toward a theory of equilibrium selection for these games. Examples are presented that show both the difficulty of and the desirability for stronger results than those presented here. [Copyright &y& Elsevier]

In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g -Laplace type growth. Finally, we prove (sharp) regularity for ω -semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C 1 , 1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006. [ABSTRACT FROM AUTHOR]

A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N ≤ 2 n , the defining functions φ ( z , z ¯ , u ) of all real-analytic hypersurfaces M = { v = φ ( z , z ¯ , u ) } ⊂ C n + 1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C N + 1 satisfy an universal algebraic partial differential equation D ( φ ) = 0 , where the algebraic-differential operator D = D ( n , N ) depends on n ≥ 1 , n < N ≤ 2 n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n , N as above there exists μ = μ ( n , N ) such that a Zariski generic real-analytic hypersurface M ⊂ C n + 1 of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C N + 1 . We also provide an explicit upper bound for μ in terms of n , N . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree. [ABSTRACT FROM AUTHOR]

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar. [ABSTRACT FROM AUTHOR]

We consider the L ∞ weak solutions to a type of compressible Euler equation with dissipation effects. Several studies [6,14,30] have obtained the L ∞ weak solutions to this type of system by using numerical schemes and the compensated compactness method. Therefore, the uniform boundedness of approximate solutions and the H l o c − 1 compactness of the corresponding entropy dissipation measures must be considered. It should be noted that the obtained L ∞ bounds typically increase over time. However, getting a time-independent uniform bound is important to consider the large time behavior of weak solutions. In this paper, by using invariant region theory, we prove that the L ∞ weak solutions derived by the Lax–Friedrichs scheme are uniformly bounded in time. [ABSTRACT FROM AUTHOR]

Based on convex analysis, a novel consensus algorithm of dynamical points in a CAT(0) space is developed in this paper, in which the associated communication graph uniformly contains a directed spanning tree. The proposed algorithm provides an efficient method of solving consensus problems in a general CAT(0) space, while having certain robustness against weak communication. The application of the new algorithm to the distributed fusion of phylogenetic trees is shown with demonstrative-case simulations, together with a study on the algorithm's robustness and efficiency. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ . The associated Cameron–Martin space is denoted by H . Consider two sufficiently regular convex functions U : X → R and G : X → R . We let ν = e − U μ and Ω = G − 1 ( − ∞ , 0 ] . In this paper we are interested in the W 2 , 2 regularity of the weak solutions of elliptic equations of the type (0.1) λ u − L ν , Ω u = f , where λ > 0 , f ∈ L 2 ( Ω , ν ) and L ν , Ω is the self-adjoint operator associated with the quadratic form ( ψ , φ ) ↦ ∫ Ω 〈 ∇ H ψ , ∇ H φ 〉 H d ν ψ , φ ∈ W 1 , 2 ( Ω , ν ) . In addition we will show that if u is a weak solution of problem (0.1) then it satisfies a Neumann type condition at the boundary, namely for ρ -a.e. x ∈ G − 1 ( 0 ) 〈 Tr ( ∇ H u ) ( x ) , Tr ( ∇ H G ) ( x ) 〉 H = 0 , where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator. [ABSTRACT FROM AUTHOR]

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L 1 ( 0 , ∞ ) relaxation function g namely g ′ ( t ) ≤ − ξ ( t ) H ( g ( t ) ) , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H ( s ) = s p and p covers the full admissible range [ 1 , 2 ) . We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature. [ABSTRACT FROM AUTHOR]

The fifth order WENO scheme with adaptive order is competent for solving hyperbolic conservation laws, its reconstruction is a convex combination of a fifth order linear reconstruction and three third order linear reconstructions. Note that, on uniform mesh, the computational cost of smoothness indicator for fifth order linear reconstruction is comparable with the sum of ones for three third order linear reconstructions, thus it is too heavy; on non-uniform mesh, the explicit form of smoothness indicator for fifth order linear reconstruction is difficult to be obtained, and its computational cost is much heavier than the one on uniform mesh. In order to overcome these problems, a simple smoothness indicator for fifth order linear reconstruction is proposed in this paper. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators L n ( f ) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions b n , k ( x ) ( 0 ≤ k ≤ n ) of order n are constructed by replacing x in the i th factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function p i ( x ) , where p i ( 0 ) = 0 and p i ( 1 ) = 1 for 1 ≤ i ≤ n . These operators L n ( f ) are positive linear operators that preserve constant functions, and a non-constant function γ n p ( x ) . If all the p i ( x ) are strictly increasing and strictly convex, then γ n p ( x ) is strictly increasing and strictly convex as well. Iterates L n M ( f ) of L n ( f ) are also considered. It is shown that L n M ( f ) converges to f ( 0 ) + ( f ( 1 ) − f ( 0 ) ) γ n p ( x ) as M → ∞ . Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every ( 1 , γ n p ( x ) ) -convex function f ∈ C [ 0 , 1 ] , we have L n ( f ; x ) ≥ f ( x ) ; and by invoking the total positivity of the system { b n , k ( x ) } 0 ≤ k ≤ n , we show that if f is ( 1 , γ n p ( x ) ) -convex, then L n ( f ; x ) is also ( 1 , γ n p ( x ) ) -convex. Finally we show that if all the p i ( x ) are monomial functions, then for every ( 1 , γ n + 1 p ( x ) ) -convex function f , L n ( f ; x ) ≥ L n + 1 ( f ; x ) if and only if p 1 ( x ) = ⋯ = p n ( x ) = x . [ABSTRACT FROM AUTHOR]

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap. Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolík–Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco. [ABSTRACT FROM AUTHOR]

In this paper we consider support points for the family S g 0 ( B 2 ) of mappings with g -parametric representation on the Euclidean unit ball B 2 in C 2 , where g is a univalent function on the unit disc U in C , which satisfies certain natural assumptions. We shall use the shearing process recently introduced by Bracci, to prove the existence of bounded support points for the family S g 0 ( B 2 ) . This result is in contrast to the one dimensional case, where all support points of the family S are unbounded. We also study the case of time- log ⁡ M reachable families R ˜ log ⁡ M ( id B 2 , M g ) generated by the Carathéodory family M g , and obtain certain results and applications, which show a basic difference between the theory in the case of one complex variable and that in higher dimensions. Of particular interest is the case where g is a convex (univalent) function on U . Finally, various consequences and certain conjectures are also considered. [ABSTRACT FROM AUTHOR]

In this paper we study the structure, weak continuity and approximability properties for the integer multiplicity rectifiable currents carried by the graphs of maximal semi-monotone set-valued maps on an n -dimensional convex domain. Especially, we give an enhanced version of approximation theorem for the subgradients of semi-convex functions. [ABSTRACT FROM AUTHOR]

In this paper, for any symplectic matrix P , the existence of subharmonic P l -solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct P l -solutions is proved. [ABSTRACT FROM AUTHOR]

Recently Nikodem, Rajba and Wąsowicz compared the classes of n -Wright-convex functions and n -Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n . The classes of strongly n -Wright-convex and strongly n -Jensen-convex functions are also compared (with the same assertion). [ABSTRACT FROM AUTHOR]

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a “polynomial ellipsoid” (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene–Krantz conjecture in the case of uniform non-tangential convergence, new results about continuous extensions (of biholomorphisms and complex geodesics), and a new Wolff–Denjoy theorem. [ABSTRACT FROM AUTHOR]

This is the first of two papers concerning saddle-shaped solutions to the semilinear equation L K u = f (u) in R 2 m , where L K is a linear elliptic integro-differential operator and f is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone { (x ′ , x ″) ∈ R m × R m : | x ′ | = | x ″ | } , and vanish only on this set. By the odd symmetry, L K coincides with a new operator L K O which acts on functions defined only on one side of the Simons cone, { | x ′ | > | x ″ | } , and that vanish on it. This operator L K O , which corresponds to reflect a function oddly and then apply L K , has a kernel on { | x ′ | > | x ″ | } which is different from K. In this first paper, we characterize the kernels K for which the new kernel is positive and therefore one can develop a theory on the saddle-shaped solution. The necessary and sufficient condition for this turns out to be that K is radially symmetric and τ ↦ K (τ) is a strictly convex function. Assuming this, we prove an energy estimate for doubly radial odd minimizers and the existence of saddle-shaped solution. In a subsequent article, part II, further qualitative properties of saddle-shaped solutions will be established, such as their asymptotic behavior, a maximum principle for the linearized operator, and their uniqueness. [ABSTRACT FROM AUTHOR]

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = { V n } be a nested sequence of subspaces of X such that V n ¯ ⊆ V n + 1 for any n ∈ N . Let e n be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup { dist ( x , V n ) } , we prove that there exist x ∈ X and n o ∈ N such that e n 3 ≤ dist ( x , V n ) ≤ 3 e n for any n ≥ n o . By using the above theorem, as a corollary we obtain classical Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Banach spaces. Also we prove versions of both Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Fréchet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fréchet spaces will be discussed. We also give a theorem improving Konyagin’s (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Fréchet spaces, such as Orlicz spaces generated by s -convex functions and locally bounded Fréchet spaces. [ABSTRACT FROM AUTHOR]

In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15] ), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau–Rockafellar formula (Rockafellar 1970, [23] ; Moreau 1966, [20] ), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization. [ABSTRACT FROM AUTHOR]

A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ on R d which can be expressed as the moment measures of suitable convex functions u , i.e. are of the form ( ∇ u ) # e − u for u : R d → R ∪ { + ∞ } and finds the corresponding u by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities ρ and the optimizer ρ turns out to be e − u . This requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport. The notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers. [ABSTRACT FROM AUTHOR]

In this paper we provide a general methodology of analysis and design of systems involving uncertainties. Available experimental data is enclosed by some geometric figures (triangle, rectangle, ellipse, parallelogram, super ellipse) of minimum area. Then these areas are inflated resorting to the Chebyshev inequality in order to take into account the forecasted data. Next step consists in evaluating response of system when uncertainties are confined to one of the above five suitably inflated geometric figures. This step involves a combined theoretical and computational analysis. We evaluate the maximum response of the system subjected to variation of uncertain parameters in each hypothesized region. The results of triangular, interval, ellipsoidal, parallelogram, and super ellipsoidal calculi are compared with the view of identifying the region that leads to minimum of maximum response. That response is identified as a result of the suggested predictive inference. The methodology thus synthesizes probabilistic notion with each of the five calculi. Using the term “pillar” in the title was inspired by the News Release (2013) on according Honda Prize to J. Tinsley Oden, stating, among others, that “Dr. Oden refers to computational science as the “third pillar” of scientific inquiry, standing beside theoretical and experimental science. Computational science serves as a new paradigm for acquiring knowledge and informing decisions important to humankind”. Analysis of systems with uncertainties necessitates employment of all three pillars. The analysis is based on the assumption that that the five shapes are each different conservative estimates of the true bounding region. The smallest of the maximal displacements in x and y directions (for a 2D system) therefore provides the closest estimate of the true displacements based on the above assumption. [ABSTRACT FROM AUTHOR]