Distributions defined as limits I. Distributions as derivatives; continuity

1957 ◽  
Vol 53 (1) ◽  
pp. 76-92 ◽  
Author(s):  
J. R. Ravetz
Keyword(s):  
Regular Sequence ◽  
Continuous Functions ◽  
General Level ◽  
Natural Manner ◽  
Basic Theorem ◽  
Regular Sequences ◽  
Alternative Approaches ◽  
Definition Of ◽  
Generalized Limits ◽  
Products Of Distributions

The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals Alternative approaches in the conceptualization of social capital

2007 ◽  
Vol 52 (174-175) ◽  
pp. 152-167
Author(s):  
Natasa Golubovic ◽  
Srdjan Golubovic
Keyword(s):  
Social Capital ◽  
Social Processes ◽  
Social Capital Theory ◽  
Consistent Theory ◽  
Capital Theory ◽  
Theoretical Position ◽  
The Social ◽  
Alternative Approaches ◽  
The Subject ◽  
Definition Of

Despite the great interest for the concept and a considerable number of papers that deal with the subject of social capital, yet there is no unique and consistent definition of social capital. Forming a consistent theory of social capital is hindered by the presence of several different approaches in the analysis of this phenomenon. Depending on the author?s theoretical position in the definition of social capital or the analysis of its sources, components and outcomes, the emphasis rests on different social processes and relationships. The aim of this paper is to analyze alternative approaches in the conceptualization of social capital, their advantages and shortfalls, and their implications for the development of the social capital theory.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

Dynamical black holes

2020 ◽  
pp. 312-336
Author(s):  
Piotr T. Chruściel
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Cauchy Data ◽  
Einstein Equations ◽  
Black Hole Spacetimes ◽  
Alternative Approaches ◽  
Definition Of ◽  
Meaningful Definition ◽  
Stability Results ◽  
Black Hole Solutions

In this chapter we review what is known about dynamical black hole-solutions of Einstein equations. We discuss the Robinson–Trautman black holes, with or without a cosmological constant. We review the Cauchy-data approach to the construction of black-hole spacetimes. We propose some alternative approaches to a meaningful definition of black hole in a dynamical spacetime, and we review the nonlinear stability results for black-hole solutions of vacuum Einstein equations.

scholarly journals On Regular Sequences

1966 ◽  
Vol 27 (1) ◽  
pp. 355-356 ◽  
Author(s):  
J. Dieudonné
Keyword(s):  
Algebraic Geometry ◽  
Noetherian Ring ◽  
Regular Sequence ◽  
Local Rings ◽  
Regular Sequences

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

scholarly journals Boundaries in digital planes

1990 ◽  
Vol 3 (1) ◽  
pp. 27-55 ◽  
Author(s):  
Efim Khalimsky ◽  
Ralph Kopperman ◽  
Paul R. Meyer
Keyword(s):  
Jordan Curve ◽  
Ordered Set ◽  
Continuous Functions ◽  
Jordan Curve Theorem ◽  
Connected Sets ◽  
Digital Plane ◽  
Parameter Interval ◽  
Totally Ordered Set ◽  
Definition Of ◽  
Natural Way

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

Phylogenetic pattern and the quantification of organismal biodiversity

1994 ◽  
Vol 345 (1311) ◽  
pp. 45-58 ◽  
Keyword(s):  
General Framework ◽  
Phylogenetic Diversity ◽  
Species Level ◽  
Option Value ◽  
Alternative Definition ◽  
Phylogenetic Pattern ◽  
Alternative Approaches ◽  
Definition Of ◽  
Different Levels ◽  
Proper Consideration

Biodiversity can be explored at a number of different levels and in principle may be separately quantified at each. Phylogenetic pattern has the potential to quantify and estimate biodiversity at the finest scale, that is, variation among species in features or attributes. This scale is an important one for conservation, as it should form the basis for prioritizing conservation efforts at the species level. Further, recent published objections to differentially weighting species are answered by defining option value at this feature-level. Unfortunately, there has been no consensus on exactly how phylogeny can be used to value species, possibly because proper consideration of the link between pattern and underlying features generally has been unresolved. ‘Phylogenetic diversity’ (PD) represents just one of several approaches that do consider diversity at the feature-level explicitly. These alternative approaches are discussed in the context of a general framework for using pattern to quantify diversity at a level below that of the original objects. The pattern framework highlights that estimation of biodiversity at a lower level using pattern will require decisions about the nature of the units of diversity, the kind of pattern to be used, the model relating unit items to pattern, and finally how this implies a pattern-based measure reflecting biodiversity. An alternative published model for relating features to a particular form of phylogenetic pattern is considered, and shown to make unwarranted assumptions. A possible alternative definition of the underlying units of diversity is examined, which may represent a different form of option value, also quantifiable using phylogeny. A possible alternative pattern to a phylogenetic tree for the prediction of feature diversity is also discussed. The appeal of these alternative approaches depends on the goals of conservation; in addition, justification for prioritizing or weighting requires that any practical approach avoid arbitrary, unwarranted, assumptions.

Levallois

2020 ◽  
Author(s):  
Michael Chazan
Keyword(s):  
Early Modern ◽  
Stone Tools ◽  
Stone Tool ◽  
General Level ◽  
Significant Component ◽  
Tool Manufacture ◽  
Core Technology ◽  
Artifact Analysis ◽  
Definition Of ◽  
The 19Th Century

Levallois refers to a way of making stone tools that is a significant component of the technological adaptations of both Neanderthals and early modern humans. Although distinctive Levallois artifacts were identified already in the 19th century, a consensus on the definition of the Levallois and clear criteria for distinguishing Levallois from non-Levallois artifacts remain elusive. At a general level, Levallois is one variant on prepared core technology. In a prepared core approach to stone tool manufacture, the worked material (the core) is configured and maintained to allow for the production of detached pieces (flakes) whose morphology is constrained by the production process. The difficulty for archaeologists is that Levallois refers to a particular process of manufacture rather than a discrete finality. The study of Levallois exposes limitations of typological approaches to artifact analysis and forces a consideration of the challenges in creating a solid empirical basis for characterizing technological processes.

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

An interactive, visual approach to developing and applying parametric three-dimensional spatial grammars

2011 ◽  
Vol 25 (4) ◽  
pp. 333-356 ◽  
Author(s):  
Frank Hoisl ◽  
Kristina Shea
Keyword(s):  
Computer Aided Design ◽  
Three Dimensional ◽  
Visual Development ◽  
General Level ◽  
Design Environment ◽  
Prototype Implementation ◽  
Left Hand ◽  
Spatial Grammar ◽  
The Creation ◽  
Definition Of

AbstractSpatial grammars are rule based, generative systems for the specification of formal languages. Set and shape grammar formulations of spatial grammars enable the definition of spatial design languages and the creation of alternative designs. Since the introduction of the underlying formalism, they have been successfully applied to different domains including visual arts, architecture, and engineering. Although many spatial grammars exist on paper, only a few, limited spatial grammar systems have been computationally implemented to date; this is especially true for three-dimensional (3-D) systems. Most spatial grammars are hard-coded, that is, once implemented, the vocabulary and rules cannot be changed without reprogramming. This article presents a new approach and prototype implementation for a 3-D spatial grammar interpreter that enables interactive, visual development and application of grammar rules. The method is based on a set grammar that uses a set of parameterized primitives and includes the definition of nonparametric and parametric rules, as well as their automatic application. A method for the automatic matching of the left hand side of a rule in a current working shape, including defining parametric relations, is outlined. A prototype implementation is presented and used to illustrate the approach through three examples: the “kindergarten grammar,” vehicle wheel rims, and cylinder cooling fins. This approach puts the creation and use of 3-D spatial grammars on a more general level and supports designers with facilitated definition and application of their own rules in a familiar computer-aided design environment without requiring programming.

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