Purity and approximation with respect to a class of morphisms

2020 ◽  
pp. 2150170
Author(s):  
Lixin Mao
Keyword(s):  
General Setting ◽  
Projective Cover ◽  
Classical Characterization

We investigate purity and approximation with respect to a class of morphisms of modules. Let [Formula: see text] be a class of left [Formula: see text]-module morphisms. An epimorphism [Formula: see text] of left [Formula: see text]-modules is called [Formula: see text]-pure if for any morphism [Formula: see text] in [Formula: see text], there is a morphism [Formula: see text] such that [Formula: see text]. An [Formula: see text]-pure epimorphism [Formula: see text] is called [Formula: see text]-superfluous if every morphism [Formula: see text] with [Formula: see text] [Formula: see text]-pure is itself [Formula: see text]-pure. We get many properties of [Formula: see text]-pure and [Formula: see text]-superfluous morphisms. As an application, we generalize the classical characterization of a projective cover to a general setting of an [Formula: see text]-cover. It is proven that an epimorphism [Formula: see text] in [Formula: see text] is an [Formula: see text]-cover of [Formula: see text] if and only if [Formula: see text] has an [Formula: see text]-cover and [Formula: see text] is an [Formula: see text]-superfluous epimorphism if and only if [Formula: see text] is [Formula: see text]-pure and there is no proper submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-pure. In addition, some dual results are also given.

scholarly journals PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

2003 ◽  
Vol 02 (01) ◽  
pp. 21-50 ◽  
Author(s):  
M. FONTANA ◽  
P. JARA ◽  
E. SANTOS
Keyword(s):  
Characterization Theorem ◽  
Field Extensions ◽  
Star Operation ◽  
Classical Characterization ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Semistar Operation ◽  
Prüfer Domains ◽  
The Given

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

Classical characterization of biphoton correlation in waveguide lattices

2011 ◽  
Vol 83 (1) ◽  
Author(s):  
Robert Keil ◽  
Felix Dreisow ◽  
Matthias Heinrich ◽  
Andreas Tünnermann ◽  
Stefan Nolte ◽  
...  
Keyword(s):  
Classical Characterization

scholarly journals Solutions of the nonlinear paraxial equation due to laser plasma–interactions

2004 ◽  
Vol 22 (1) ◽  
pp. 69-74 ◽  
Author(s):  
F. OSMAN ◽  
R. BEECH ◽  
H. HORA
Keyword(s):  
Laser Plasma ◽  
Theoretical Study ◽  
Semiclassical Limit ◽  
General Setting ◽  
One Dimension ◽  
One Dimensional ◽  
Laser Plasma Interactions ◽  
Weak Limits ◽  
The One

This article presents a numerical and theoretical study of the generation and propagation of oscillation in the semiclassical limit ħ → 0 of the nonlinear paraxial equation. In a general setting of both dimension and nonlinearity, the essential differences between the “defocusing” and “focusing” cases are observed. Numerical comparisons of the oscillations are made between the linear (“free”) and the cubic (defocusing and focusing) cases in one dimension. The integrability of the one-dimensional cubic nonlinear paraxial equation is exploited to give a complete global characterization of the weak limits of the oscillations in the defocusing case.

scholarly journals Quantum and Classical Characterization of Single/Few Photon Detectors

2015 ◽  
Vol 4 (3) ◽  
pp. 200-212 ◽  
Author(s):  
M. G. Mingolla ◽  
F. Piacentini ◽  
A. Avella ◽  
M. Gramegna ◽  
L. Lolli ◽  
...  
Keyword(s):  
Photon Detectors ◽  
Classical Characterization

scholarly journals Characterization of ellipsoids as K-dense sets

2016 ◽  
Vol 146 (1) ◽  
pp. 213-223
Author(s):  
Rolando Magnanini ◽  
Michele Marini
Keyword(s):  
Convex Body ◽  
Asymptotic Formula ◽  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Measurable Set ◽  
Classical Characterization ◽  
Dense Sets

Let K ⊂ ℝN be any convex body containing the origin. A measurable set G ⊂ ℝN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r > 0, the measure of G ⋂ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In a previous work, we proved for the case N = 2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in ℝN: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, which builds upon results obtained in our previous work, relies on an asymptotic formula for the measure of G ⋂ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to Petty.

Characterization of the convergence of weighted averages in a more general setting

2013 ◽  
Vol 50 (1) ◽  
pp. 51-66
Author(s):  
Ferenc Móricz ◽  
Ulrich Stadtmüller
Keyword(s):  
Borel Measure ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
General Setting ◽  
Finite Limit ◽  
Positive Borel Measure ◽  
Weighted Averages ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)} $ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ) \to Last \to \infty $ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

A cohomological characterization of amenable actions

1991 ◽  
Vol 110 (3) ◽  
pp. 491-504
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Dynamical Systems ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Classical Characterization

AbstractWe give a new characterization of amenability for dynamical systems, in cohomological terms, which generalizes the classical characterization of amenable locally compact groups stated by Johnson.

scholarly journals Characterization of Modular Join-Semilattices

1970 ◽  
Vol 29 ◽  
pp. 43-49
Author(s):  
MR Talukder ◽  
MHB Arif
Keyword(s):  
Key Words ◽  
Classical Characterization ◽  
Join Semilattices ◽  
Distributive Semilattices

In this paper we have proved a classical characterization of modular join-semilattices. We have also given some characterizations of modular ideals of join-semilattices through congruences. Key words: Join-semilattices; modular semilattices; distributive semilattices; quotient semilattices GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 43-49  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8514

scholarly journals Unique range sets - a further study

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1499-1516
Author(s):  
Sanjay Mallick
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
General Polynomial ◽  
Unique Range Set ◽  
First Time

The purpose of the paper is to investigate the problems of unique range sets in the most general setting. Accordingly, we have studied sufficient conditions for a general polynomial to generate a unique range set which put all the variants of unique range sets into one structure. Most importantly, as an application of the main result we have been able to accommodate not only examples of critically injective polynomials but also examples of non-critically injective polynomials generating unique range sets which are for the first time being exemplified in the literature. Furthermore, some of these examples show that characterization of unique range sets generated by non-critically injective polynomials does not always demand gap polynomials which also complements the recent results by An and Banerjee-Lahiri in this direction. Moreover, one of the lemmas proved in this paper improves and generalizes some results due to Frank-Reinders and Lahiri respectively.

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