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Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide…
Models based on approximation capabilities have recently been studied in the context of Optimal Recovery. These models, however, are not compatible with overparametrization, since model- and data-consistent functions could then be…
In this paper, new upper and lower bounds for the Trapezoid inequality of absolutely continuous functions are obtained. Applications to some special means are provided as well.
Convex analysis is fundamental to proving inequalities that have a wide variety of applications in economics and mathematics. In this paper we provide Jensen-type inequalities for functions that are, intuitively, "very" convex. These…
We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of…
We consider classes of objective functions of cardinality constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of $\gamma$-$\alpha$-augmentable functions and prove…
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$…
This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their…
Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…
Many psychological theories can be operationalized as linear inequality constraints on the parameters of multinomial distributions (e.g., discrete choice analysis). These constraints can be described in two equivalent ways: Either as the…
In our recent research endeavors, we have delved into the realm of tail bounds problems concerning bivariate random tensor means. In this context, tensors are treated as finite-dimensional operators. However, the longstanding challenge of…
The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of…
We describe a general method to obtain weak subconvexity bounds for many classes of $L$-functions. This has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms (see arxiv.org:math/0809.1636).
In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start by…
First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly…
In this paper, we establish some new inequalities of the Hermite-Hadamard like for class of (h-s)_{1,2}-convex functions which are ordinary, super-multiplicative or similarly ordered and nonnegative.
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The…
In this work, we introduce new approximation operators for univariate set-valued functions with general compact images. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new…
In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized…