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Remarks on strongly Wright-convex functions

Nelson Merentes, Kazimierz Nikodem, Sergio Rivas (2011)

Annales Polonici Mathematici

Some properties of strongly Wright-convex functions are presented. In particular it is shown that a function f:D → ℝ, where D is an open convex subset of an inner product space X, is strongly Wright-convex with modulus c if and only if it can be represented in the form f(x) = g(x)+a(x)+c||x||², x ∈ D, where g:D → ℝ is a convex function and a:X → ℝ is an additive function. A characterization of inner product spaces by strongly Wright-convex functions is also given.

Renewal Processes of Mittag-Leffler and Wright Type

Mainardi, Francesco, Gorenflo, Rudolf, Vivoli, Alessandro (2005)

Fractional Calculus and Applied Analysis

2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05.After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding...

Représentation par automate de fonctions continues de tore

F. Blanchard, B. Host, A. Maass (1996)

Journal de théorie des nombres de Bordeaux

Soient A p = { 0 , , p - 1 } et Z A p × A p un sous-système. Z est une représentation en base p d’une fonction f du tore si pour tout point x du tore, ses développements en base p sont liés par le couplage Z aux développements en base p de f ( x ) . On prouve que si f est représentable en base p alors f ( x ) = ( u x + m p - 1 ) mod 1 , où u et m A p . Réciproquement, toutes les fonctions de ce type sont représentables en base p par un transducteur. On montre finalement que les fonctions du tore qui peuvent être représentées par automate cellulaire sont exclusivement les multiplications...

Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, Raanan Schul (2017)

Analysis and Geometry in Metric Spaces

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...

Riemann Integral of Functions from ℝ into Real Banach Space

Keiko Narita, Noboru Endou, Yasunari Shidama (2013)

Formalized Mathematics

In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems about the...

Riemann Integral of Functions from R into n -dimensional Real Normed Space

Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].

Riemann Integral of Functions from R into R n

Keiichi Miyajima, Yasunari Shidama (2009)

Formalized Mathematics

In this article, we define the Riemann Integral of functions from R into Rn, and prove the linearity of this operator. The presented method is based on [21].

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