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Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable…
In this paper, we explore two fundamental theorems of differential calculus: Rolle's Theorem and the Mean Value Theorem (MVT). These theorems play a crucial role in the development of theoretical and practical results in mathematics,…
Various forms of Mean Value Theorems are available in the literature. If we use Flett's Mean Value Theorem in Extended Generalized Mean Value Theorem then what would the new theorem look like. A sincere effort is done to develop this…
Given a continuous function from Euclidean space to the real line, we analyze (under some natural assumption on the function), the set of values it takes on translates of lattices. Our results are of the flavor: For almost any translate,…
Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous…
Assuming that the two integrals in the Change of Variable Formula for the Riemann integral on the real line are finite, one can rightfully ask if they have equal value. We give a positive answer to this question. The proof is very easy to…
The paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's Lemma for restricted power series is the main tool of the proof.
Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two…
The mean value theorem of calculus states that, given a differentiable function $f$ on an interval $[a, b]$, there exists at least one mean value abscissa $c$ such that the slope of the tangent line at $c$ is equal to the slope of the…
In this paper a special type of difference equations is investigated. The impulses start abruptly at some points and their action continue on given finite intervals. This type of equations is used to model a real process. An algorithm,…
We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
We prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices. We apply these results to study dependence of the solution of the initial…
Matrix theory, foundational in diverse fields such as mathematics, physics, and computational sciences, typically categorizes matrices based strictly on their invertibility-determined by a sharply defined singular or nonsingular…
Some monotone increasing sequences of the lower bounds for the minimum eigenvalue of $M$-matrices are given. It is proved that these sequences are convergent and improve some existing results. Numerical examples show that these sequences…
Multivariable, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that the best notion…
The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
It is widely claimed that the quantile function is equivariant under increasing transformations. We show by a counterexample that this is not true (even for strictly increasing transformations). However, we show that the quantile function…