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Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II

Manabu Naito (2021)

Archivum Mathematicum

We consider the half-linear differential equation of the form ( p ( t ) | x ' | α sgn x ' ) ' + q ( t ) | x | α sgn x = 0 , t t 0 , under the assumption that p ( t ) - 1 / α is integrable on [ t 0 , ) . It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as t .

Remarks on the spaces of differentiable multifunctions

Andrzej Kasperski (2011)

Banach Center Publications

In this paper we consider some spaces of differentiable multifunctions, in particular the generalized Orlicz-Sobolev spaces of multifunctions, we study completeness of them, and give some theorems.

Remarks on WDC sets

Dušan Pokorný, Luděk Zajíček (2021)

Commentationes Mathematicae Universitatis Carolinae

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets A 2 . We prove that, for such A , the distance function d A = dist ( · , A ) is a “DC aura” for A , which implies that each closed locally WDC set in 2 is a WDC set. Another consequence is that compact WDC subsets of 2 form a Borel subset of the space of all compact sets.

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...

Residue class rings of real-analytic and entire functions

Marek Golasiński, Melvin Henriksen (2006)

Colloquium Mathematicae

Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

Restricted continuity and a theorem of Luzin

Krzysztof Chris Ciesielski, Joseph Rosenblatt (2014)

Colloquium Mathematicae

Let P(X,ℱ) denote the property: For every function f: X × ℝ → ℝ, if f(x,h(x)) is continuous for every h: X → ℝ from ℱ, then f is continuous. We investigate the assumptions of a theorem of Luzin, which states that P(ℝ,ℱ) holds for X = ℝ and ℱ being the class C(X) of all continuous functions from X to ℝ. The question for which topological spaces P(X,C(X)) holds was investigated by Dalbec. Here, we examine P(ℝⁿ,ℱ) for different families ℱ. In particular, we notice that P(ℝⁿ,"C¹") holds, where...

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. Ferreyra, T. Godoy, M. Urciuolo (2004)

Studia Mathematica

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by σ ( A ) = B χ A ( x , φ ( x ) ) d x where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from L p ( ³ ) to L q ( Σ , d σ ) for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Restrictions of Fourier transforms to curves

S. W. Drury (1985)

Annales de l'institut Fourier

Let x ( t ) = ( t , 1 2 t 2 , 1 6 t 3 ) a certain curve in R 3 . We investigate inequalities of the type { | f ^ ( x ( t ) ) | b d t } 1 / b C f a for f 𝒮 ( R 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

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