We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions sa...
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We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.
In the dual L phi* of a Delta 2-Orlicz space L phi, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology tau(L phi*,...
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In the dual L phi* of a Delta 2-Orlicz space L phi, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology tau(L phi*,L phi) if and only if on each order interval [-zeta,zeta]={xi:-zeta <=xi <=zeta}(zeta is an element of L phi*), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlos type result: every norm bounded sequence (xi n)n in L phi* admits a sequence of forward convex combinations xi over bar n is an element of conv(xi n,xi n+1, horizontal ellipsis ) such that supn|xi over bar n|is an element of L phi* and xi over bar n converges a.s.
Given 0 ≤ R1 ≤ R2 ≤ ∞, CVG(R1, R2) denotes the class of normalized convex functions f in the unit disc U, for which df(U) satisfies a Blaschke Rolling Circles Criterion with radii R1 and R2. Necessary and sufficie...
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We introduce and study the notion of sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgree...
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We introduce and study the notion of sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-subdifferential of a proper function f which contains the Clarke-Rockafellar subdifferential of f under some mild assumptions on f and sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We show that some well known properties of the convex function, namely Lipschitz property on the interior of its domain, remain valid for the large class of sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-convex functions.
Let X-n := {x(i)}(i=0)(n) be a given set of (n + 1) pairwise distinct points in R-d (called nodes or sample points), let P = conv(X-n), let f be a convex function with Lipschitz continuous gradient on P and lambda := ...
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Let X-n := {x(i)}(i=0)(n) be a given set of (n + 1) pairwise distinct points in R-d (called nodes or sample points), let P = conv(X-n), let f be a convex function with Lipschitz continuous gradient on P and lambda := {lambda(i)}(i=0)(n) be a set of barycentric coordinates with respect to the point set X-n. We analyze the error estimate between land its barycentric approximation: B-n[f](x) = Sigma(i=0)lambda(i)(x)f(x(i)), (x is an element of P) and present the best possible pointwise error estimates off. Additionally, we describe the optimal barycentric coordinates that provide the best operator B-n for approximating f by B-n[f]. We show that the set of (linear finite element) barycentric coordinates generated by the Delaunay triangulation gives access to efficient algorithms for computing optimal approximations. Finally, numerical examples are used to show the success of the method. (C) 2014 Elsevier Inc. All rights reserved.
Some Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrat...
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Some Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Holder-McCarthy's inequality for positive operators and Ky Fan's inequality for real numbers are given as well. (C) 2011 Elsevier Inc. All rights reserved.
The main purpose of this paper is to introduce a new class U H(q, s, lambda, beta, k) of functions which are analytic in the open unit disk Delta = {z : z epsilon C and vertical bar z vertical bar < 1}. We obtain v...
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The main purpose of this paper is to introduce a new class U H(q, s, lambda, beta, k) of functions which are analytic in the open unit disk Delta = {z : z epsilon C and vertical bar z vertical bar < 1}. We obtain various results including characterization, coefficient estimates, and distortion and covering theorems, for functions belonging to the class U H(q, s, lambda, beta, k). (C) 2007 Elsevier Inc. All rights reserved.
Recently there has been renewed interest in the problem of finding under and over estimations on the set of convex functions to a given nonnegative linear functional;that is, approximations which estimate always below...
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Recently there has been renewed interest in the problem of finding under and over estimations on the set of convex functions to a given nonnegative linear functional;that is, approximations which estimate always below (or above) the functional over a family of convex functions. The most important example of such an approximation problem is given by the multidimensional versions of the midpoint (rectangle) rule and the trapezoidal rule, which provide under and over estimations to the true value of the integral on the set of convex functions (also known as the HermiteHadamard inequality). In this paper, we introduce a general method of constructing new families of under/overestimators on the set of convex functions for a general class of linear functionals. In particular, under the regularity condition, namely the functions belonging to C-2(Omega) (not necessarily convex), we will show that the error estimations based on such estimators are always controlled by the error associated with using the quadratic function. The result is also extended to the class of Lipschitz functions. We also propose a modified approximation technique to derive a general class of under/over estimators with better error estimates. (C) 2014 Elsevier Inc. All rights reserved.
A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the e...
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A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.
The electrical capacitance tomography technology has potential benefits for the process industry by providing visualization of material distributions. One of the main technical gaps and impediments that must be overco...
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The electrical capacitance tomography technology has potential benefits for the process industry by providing visualization of material distributions. One of the main technical gaps and impediments that must be overcome is the low-quality tomogram. To address this problem, this study introduces the data-guided prior and combines it with the electrical measurement mechanism and the sparsity prior to produce a new difference of convex functions programming problem that turns the image reconstruction problem into an optimization problem. The data-guided prior is learned from a provided dataset and captures the details of imaging targets since it is a specific image. A new numerical scheme that allows a complex optimization problem to be split into a few less difficult subproblems is developed to solve the challenging difference of convex functions programming problem. A new dimensionality reduction method is developed and combined with the relevance vector machine to generate a new learning engine for the forecast of the data-guided prior. The new imaging method fuses multisource information and unifies data-guided and measurement physics modeling paradigms. Performance evaluation results have validated that the new method successfully works on a series of test tasks with higher reconstruction quality and lower noise sensitivity than the popular imaging methods.
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