Related papers: The Fundamental Theorems of Interval Analysis
Interval arithmetic is hardly feasible without directed rounding as provided, for example, by the IEEE floating-point standard. Equally essential for interval methods is directed rounding for conversion between the external decimal and…
Extended regular expressions with counting and interleaving are widely used in practice. However the related theoretical studies for this kind of expressions currently cannot meet the need of practical work. This paper develops syntax…
This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with…
Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…
We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified.…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
This paper provides a rigorous and gap-free proof of the index theorem used in the theory of regular economy. In the index theorem that is the subject of this paper, the assumptions for the excess demand function are only several usual…
The paper gives a brief account of the spaces of interval functions defined through the concepts of H-continuity, D-continuity and S-continuity. All three continuity concepts generalize the usual concept of continuity for real (point…
The behaviors of various confidence/credible interval constructions are explored, particularly in the region of low statistics where methods diverge most. We highlight a number of challenges, such as the treatment of nuisance parameters,…
This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property,…
It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably,…
We explain the concept of p-values presupposing only rudimentary probability theory. We also use the occasion to introduce the notion of p-function, so that p-values are values of a p-function. The explanation is restricted to the discrete…
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and…
"Theorems for Free!" (Wadler, FPCA 1989) is a slogan for a technique that allows to derive statements about functions just from their types. So far, the statements considered have always had a purely extensional flavor: statements relating…
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…
Complex interval arithmetic is a powerful tool for the analysis of computational errors. The naturally arising rectangular, polar, and circular (together called primitive) interval types are not closed under simple arithmetic operations,…
When knowledge is obtained from a database, it is only possible to deduce confidence intervals for probability values. With confidence intervals replacing point values, the results in the set covering model include interval constraints for…
The expressiveness of Metric Temporal Logic (MTL) has been extensively studied throughout the last two decades. In particular, it has been shown that the \emph{interval-based} semantics of MTL is strictly more expressive than the…