Open Convex Research Articles

Strip deformations of decorated hyperbolic polygons

AbstractIn this paper, we study the hyperbolic and parabolic strip deformations of ideal (possibly once‐punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger–Guéritaud–Kassel. We also give a version of this result for the undecorated ideal polygons and once‐punctured ideal polygons.

Around strongly operator convex functions

We establish the subadditivity of strongly operator convex functions on (0,∞) and (−∞,0). By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on (−∞,0). We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on (0,∞). Moreover, we study strongly operator convex functions on (a,∞) with −∞<a and on the left half-line (−∞,b) with b<∞. We demonstrate that any nonconstant strongly operator convex function on (a,∞) is strictly operator decreasing, and any nonconstant strongly operator convex function on (−∞,b) is strictly operator monotone. Consequently, for a strongly operator convex function g on (a,∞) or (−∞,b), we provide lower bounds for |g(A)−g(B)| whenever A−B>0.

Generalized Choi–Davis–Jensen’s Operator Inequalities and Their Applications

The original Choi–Davis–Jensen’s inequality, known for its extensive applications in various scientific and engineering fields, has inspired researchers to pursue its generalizations. In this study, we extend the Choi–Davis–Jensen’s inequality by introducing a nonlinear map instead of a normalized linear map and generalize the concept of operator convex functions to include any continuous function defined within a compact region. Notably, operators can be matrices with structural symmetry, enhancing the scope and applicability of our results. The Stone–Weierstrass theorem and the Kantorovich function play crucial roles in the formulation and proof of these generalized Choi–Davis–Jensen’s inequalities. Furthermore, we demonstrate an application of this generalized inequality in the context of statistical physics.

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications

By using optimal mass transport theory, we provide a direct proof to the sharp L p L^p -log-Sobolev inequality ( p ≥ 1 ) (p\geq 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n E\subseteq \mathbb R^n . The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the L p L^p -log-Sobolev inequality. The characterization of the equality cases is new for p ≥ n p\geq n even in the unweighted setting and E = R n E=\mathbb R^n . As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.

Accurate TDOA estimation using matching cumulative estimator for LFM signal with small samples

This paper introduces two matching cumulative estimators designed for achieving gridless time difference of arrival (TDOA) estimation for linear frequency modulated (LFM) signals with limited samples. A designed matching basis function, customized to the source signal waveform, is employed to calculate with the received data. Time delay matching is then utilized to formulate a monotonic objective function related to the estimated TDOA, enabling gridless estimation. To enhance robustness, especially in scenarios characterized by small samples and low signal to noise ratios (SNRs), a nonlinear cumulative transformation, comprised of a series of strongly convex operations, is incorporated to mitigate the impact of noise on the peak calculation of the objective function. Simulation results validate the efficacy of the two proposed TDOA estimators, particularly under challenging conditions involving small samples and low SNRs.

On the entropy and information of Gaussian mixtures

AbstractWe establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if are independent scalar Gaussian mixtures, then the entropy of is concave in , thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix‐valued function acting on densities of mixtures in . As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

Stochastic modelling of symmetric positive definite material tensors

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly ‘higher’ spatial invariance class. We formulate a modelling framework which not only respects these two requirements—positive definiteness and invariance—but also allows a fine control over orientation on one hand, and strength / size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties—scaling, orientation, and prescribed material symmetry—on the desired quantities of interest.

Operator upper bounds for Davis-Choi-Jensen's difference in Hilbert spaces

In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference Ph (f (A)) - f (Ph (A)) for any convex function f : I → R, any selfadjoint operator A in H with the spectrum Sp (A) ⊂ I and any linear, positive and normalized map Ph : B (H) → B (K), where H and K are Hilbert spaces. Some examples of convex and operator convex functions are also provided.

Идентификация областей возможных оценок параметров моделей полносвязной линейной регрессии

&lt;p&gt;This article is devoted to the study of fully connected linear regression models, in which the observed variables contain errors, and the pairs of true variables are interconnected by linear functional dependencies. When estimating fully connected regressions, the main problem is the correct choice of the error variances ratios of the variables. If the choice is made incorrectly, then the fully connected regression estimates will be biased. The purpose of this article is to find the dependence of main parameters possible estimates areas on the possible error variances ratios of the variables in fully connected regressions. For the first time, with the help of matrix algebra elements, the inverse problem is solved - analytical dependences of the error variances ratios of variables on the main parameters are obtained. These dependences make it possible to identify the parameters possible estimates areas in which the necessary condition for the extremum of the objective function is satisfied. It is proved that, under certain conditions, for any error variances ratios of the variables, the parameters estimates always lie inside an open convex polygon located only in one of the orthants of the multidimensional space. In this case, the signs of the estimates always agree with the signs of the corresponding correlation coefficients. A numerical experiment was carried out, confirming the correctness of the results obtained.&lt;/p&gt;

Equality cases in monotonicity of quasi-entropies, Lieb’s concavity and Ando’s convexity

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

Quasi-Arithmetic Means Ad Libitum

Let alpha _1, ldots , alpha _m be two or more positive reals with sum 1, let Csubseteq {mathbb {R}}^k be an open convex set, and f: Crightarrow {mathbb {R}}^k be a continuous injection with convex image. For each nonempty set Ssubseteq C, let {mathscr {M}}(S) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights alpha _1,ldots ,alpha _m, that is, the family M(S)=f-1α1f(x1)+⋯+αmf(xm):x1,…,xm∈S.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathscr {M}}(S)= \\left\\{ f^{-1}\\left( \\alpha _1f(x_1)+\\cdots +\\alpha _mf(x_m)\\right) : x_1,\\ldots ,x_m \\in S \\right\\} . \\end{aligned}$$\\end{document}We provide a simple necessary and sufficient condition on S for which the infinite iteration bigcup _{n}{mathscr {M}}^n(S) is relatively dense in the convex hull of S.

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.

Planar Convex Codes are Decidable

We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.

Operator Convexity of an Integral Transform with Applications

For a continuous and positive function w(λ), λ &gt; 0 and μ a positive measure on (0, ∞) we consider the following integral transformwhere the integral is assumed to exist for t &gt; 0.We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) -f(t)] t-1 is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s differenceunder some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

On the convergence of numerical integration as a finite matrix approximation to multiplication operator

We study the convergence of a family of numerical integration methods where the numerical integration is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for a wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen’s operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

Extendability of continuous quasiconvex functions from subspaces

Let Y be a subspace of a topological vector space X, and A⊂X an open convex set that intersects Y. We say that the property (QE) [property (CE)] holds if every continuous quasiconvex [continuous convex] function on A∩Y admits a continuous quasiconvex [continuous convex] extension defined on A. We study relations between (QE) and (CE) properties, proving that (QE) always implies (CE) and that, under suitable hypotheses (satisfied for example if X is a normed space and Y is a closed subspace of X), the two properties are equivalent. By combining the previous implications between (QE) and (CE) properties with known results about the property (CE), we obtain some new positive results about the extension of quasiconvex continuous functions. In particular, we generalize the results contained in [9] to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which (QE) does not hold.

$\alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations

In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha&gt; 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.

Linear topological invariants for kernels of convolution and differential operators

We establish the condition (Ω) for smooth kernels of various types of convolution and differential operators. By the (DN)-(Ω) splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel (DF)-spaces whose strong duals satisfy the condition (DN), e.g., the space D′(Y) of distributions over an open set Y⊆Rn or the space S′(Rn) of tempered distributions. Most notably, we show that:(i)EP(X)={f∈E(X)|P(D)f=0} satisfies (Ω) for any differential operator P(D) and any open convex set X⊆Rd.(ii)Let P∈C[ξ1,ξ2] and X⊆R2 open be such that P(D):E(X)→E(X) is surjective. Then, EP(X) satisfies (Ω).(iii)Let μ∈E′(Rd) be such that E(Rd)→E(Rd),f↦μ⁎f is surjective. Then, {f∈E(Rd)|μ⁎f=0} satisfies (Ω). The central result in this paper is that the space of smooth zero solutions of a general convolution equation satisfies the condition (Ω) if and only if the space of distributional zero solutions of the equation satisfies the condition (PΩ). The above and related statements then follow from known results concerning (PΩ) for distributional kernels of convolution and differential operators [3,15,16].

Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.