Related papers: When Are There Continuous Choices for the Mean Val…
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous…
Our goal in this work is to present some mean value type theorems that are not studied in classic calculus and analysis courses. They are simple theorems yet with large applicability in mathematical analysis (for example, in the study of…
Modern statistics provides an ever-expanding toolkit for estimating unknown parameters. Consequently, applied statisticians frequently face a difficult decision: retain a parameter estimate from a familiar method or replace it with an…
The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems,…
Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the…
An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among…
This paper studies the differentiability of the value function of switched linear systems under arbitrary switching and controlled switching, referred to as worst-case and optimal value functions respectively. First, we show that the value…
We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set $\mathcal{O}$. The stopping horizon is either random, equal…
We address the decision problem for a fragment of real analysis involving differentiable functions with continuous first derivatives. The proposed theory, besides the operators of Tarski's theory of reals, includes predicates for…
Within the gossamer numbers which extend the real numbers to include infinitesimals and infinities we prove the Fundamental Theorem of Calculus (FTC). Riemann sums are also considered in the gossamer number system, and their non-uniqueness…
In this work a mean value theorem of Pompeiu's type for functions of two variables is presented. Other related results are given as well.
We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that…
A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points…
We estimate the average of any arithmetic function $k$ over the values of any smooth polynomial in many variables provided only that $k$ has a distribution in arithmetic progressions of fixed modulus. We give several applications of this…
Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the…
This paper presents the Functional Machine Calculus (FMC) as a simple model of higher-order computation with "reader/writer" effects: higher-order mutable store, input/output, and probabilistic and non-deterministic computation. The FMC…
Let $\mu$ be a measure on $[-1,1]$. Then for every continuous function $f:\mathbb{R}\to\mathbb{R}$ and $\alpha>0$ one can define its averaging $f_{\alpha}:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_{\alpha}(x) = \int_{-1}^{1}…
The concept of sequential choice functions is introduced and studied. This concept applies to the reduction of the problem of stable matchings with sequential workers to a situation where the workers are linear.
If uncertainty is modelled by a probability measure, decisions are typically made by choosing the option with the highest expected utility. If an imprecise probability model is used instead, this decision rule can be generalised in several…
We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and…