Related papers: Inequalities in calculus: methods of prooving resu…
This is a handbook of simple proofs of the convergence of gradient and stochastic gradient descent type methods. We consider functions that are Lipschitz, smooth, convex, strongly convex, and/or Polyak-{\L}ojasiewicz functions. Our focus is…
In the seminal book M\'echanique analitique, Lagrange, 1788, the notion of a Lagrange multiplier was first introduced in order to study a smooth minimization problem subject to equality constraints. The idea is that, under some regularity…
We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a…
In this paper we develop a general theoretical tool for the establishment of the boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
This book deals with functions allowing to express the dissimilarity (discrepancy) between two data fields or ''divergence functions'' with the aim of applications to linear inverse problems. Most of the divergences found in the litterature…
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity…
Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well…
We give some new refinements and reverses Young inequalities in both additive-type and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some…
We introduce some symmetric homogeneous means, and then show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give the tighter bounds of the logarithmic mean than the…
In this article we prove sharp Landau--Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $\mathbb{R}^d$, $d\geq 1$, that are absolutely continuous with respect to the Lebesgue measure. In…
Various inequalities (Boole inequality, Chung-Erd\"os inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra `quantum…
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov…
We prove inequalities on symmetric tensor sums of positive definite operators. In particular, we prove multivariable operator inequalities inspired by generalizations to the well-known Hlawka and Popoviciu inequalities. As corollaries, we…
A non-linear Black-Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the non-linear reaction function allows for an equivalent transformation of the semi-linear…
The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Certain excess versions of the Minkowski and H\"older inequalities are given. These new results generalize and improve the Minkowski and H\"older inequalities.
By use of a modified Nunokawa's lemma, we obtain some new conditions for univalence. Also, some sharp inequalities concerning univalent functions are presented.
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…