Related papers: The Fundamental Theorems of Interval Analysis
We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the…
Recently, it was shown that most popular IR measures are not interval-scaled, implying that decades of experimental IR research used potentially improper methods, which may have produced questionable results. However, it was unclear if and…
A monotone function interval is the set of monotone functions that lie pointwise between two fixed monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings.…
This note addresses the input and output of intervals in the sense of interval arithmetic and interval constraints. The most obvious, and so far most widely used notation, for intervals has drawbacks that we remedy with a new notation that…
The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
This study addresses the often-overlooked issue of measurability at intermediate points when applying Taylor's theorems to random functions and random vectors (e.g., likelihood functions with respect to estimators) in statistics. Classical…
Two articles published by Information Science discuss the derivatives of interval functions, in the sense of Svetoslav Markov. The authors of these articles tried to characterize for which functions and points such derivatives exist.…
Extensions of previous linear regression models for interval data are presented. A more flexible simple linear model is formalized. The new model may express cross-relationships between mid-points and spreads of the interval data in a…
The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple…
Moore and Yang defined an integral notion, based on an extension of Riemann sums, for inclusion monotonic continuous interval functions, where the integrations limits are real numbers. This integral notion extend the usual integration of…
The paper "A General Theory of IR Evaluation Measures" develops a formal framework to determine whether IR evaluation measures are interval scales. This comment shows some limitations about its conclusions.
The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…
Based on the total integrability we first define an integral of a real valued function f as an interval function associated to its antiderivative F. By introducing the concept of the residue of a function into the real analysis, the…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when…
A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…
In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…