scholarly journals The special atom space and Haar wavelets in higher dimensions

2020 ◽  
Vol 53 (1) ◽  
pp. 131-151
Author(s):  
Eddy Kwessi ◽  
Geraldo de Souza ◽  
Ngalla Djitte ◽  
Mariama Ndiaye
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Higher Dimensions ◽  
Haar Wavelets ◽  
Lipschitz Spaces ◽  
The Real ◽  
Haar Functions ◽  
Wide Range ◽  
Definition Of ◽  
Almost All

AbstractIn this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

scholarly journals Some remarks on incomplete gamma type function γ*(α, x_)

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Emin Özcağ ◽  
İnci Egeb
Keyword(s):  
Recurrence Relation ◽  
Real Line ◽  
Summable Function ◽  
Type Function ◽  
The Real ◽  
Definition Of ◽  
Derivatives Of

The incomplete gamma type function ?*(?, x_) is defined as locally summable function on the real line for ?>0 by ?*(?,x_) = {?x0 |u|?-1 e-u du, x?0; 0, x > 0 = ?-x_0 |u|?-1 e-u du the integral divergining ? ? 0 and by using the recurrence relation ?*(? + 1,x_) = -??*(?,x_) - x?_ e-x the definition of ?*(?, x_) can be extended to the negative non-integer values of ?. Recently the authors [8] defined ?*(-m, x_) for m = 0, 1, 2,... . In this paper we define the derivatives of the incomplete gamma type function ?*(?, x_) as a distribution for all ? < 0.

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.

Alfonso Fontanelli’s Cadences and the Seconda Pratica

2013 ◽  
Vol 30 (1) ◽  
pp. 49-102
Author(s):  
Stefano La Via
Keyword(s):  
Large Scale ◽  
Specific Aspect ◽  
Compositional Techniques ◽  
The Real ◽  
Wide Range ◽  
Definition Of ◽  
Alternative Solutions ◽  
Musical Analyses

In his brilliant studies and accurate editions Anthony Newcomb has shown Alfonso Fontanelli’s contributions to the definition of “the new Ferrarese style of the 1590s” and, therefore, to the birth of the seconda pratica. My article focuses on a specific aspect of Fontanelli’s polyphonic writing: the handling of cadences for not only syntactical and tonally structural but also expressive purposes. The literary-musical analyses of some of the most representative settings published in Fontanelli’s two books of madrigals (1595 and 1604)—including masterpieces such as “Tu miri, o vago ed amoroso fiore” (Anonymous), “Io piango, ed ella il volto” (Petrarca), “Lasso, non odo più Filli mia cara” (Anonymous), and “Dovrò dunque morire” (Rinuccini)—shows, above all, the unusually wide range of Fontanelli’s cadential palette. He used not only traditional models (such as the perfect, authentic, Phrygian, and half cadences) but also a great variety of alternative solutions (including what Newcomb has named “evaporated” and “oblique” cadences) that are often so experimental and bold as to escape rigid classification. In the context of a basically chromatic, dissonant, harmonically restless, and tonally unfocused polyphonic flow such cadential variety seems to reflect Fontanelli’s intention not only to underscore the conceptual and emotional meanings represented in the verbal text but also to sharpen their large-scale affective contrasts. In these and other experimental traits of his “cadential style” Fontanelli further developed (possibly through the mediation of Jacques de Wert, and also under the influence of composers such as Luzzaschi and Gesualdo) those basic compositional techniques and exegetic principles that Cipriano de Rore, the real father of the seconda pratica, had already established in his later madrigals, and that Vincenzo Galilei, in turn, had neatly codified in his treatise on counterpoint (ca. 1588–1591).

scholarly journals Characterizing the continuous functionals

1983 ◽  
Vol 48 (4) ◽  
pp. 965-969 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Rational Numbers ◽  
The Real ◽  
Finite Systems ◽  
Several Variables ◽  
Finite Structures ◽  
Starting Point ◽  
Topological Completion ◽  
Continuous Functionals

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

scholarly journals Compactness in Metric Spaces

2016 ◽  
Vol 24 (3) ◽  
pp. 167-172
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Real Line ◽  
Metric Spaces ◽  
Topological Properties ◽  
The Real ◽  
The Third ◽  
Norm Space ◽  
Number Sequences ◽  
Sequential Compactness ◽  
Countable Compactness ◽  
Definition Of

Summary In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].

The logic of absolute value inequalities

1964 ◽  
Vol 57 (2) ◽  
pp. 73-74
Author(s):  
Marvin L. Chachere
Keyword(s):  
Real Line ◽  
Two Dimensions ◽  
Elementary Algebra ◽  
Absolute Value ◽  
The Real ◽  
Definition Of ◽  
Absolute Value Inequalities

Teachers motivated by the need to express the distance of any point on the real line from zero find the definition of absolute value useful and interesting. It is used in elementary algebra to define addition and multiplication of “directed” numbers and to describe the graphs of broken lines in one and two dimensions. Recent emphasis on inequalities and the inclusion in newer textbooks of absolute value inequalities raise several questions that cannot be handled in a perfunctory manner.

scholarly journals Generalized Exponential Trichotomies for Abstract Evolution Operators on the Real Line

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Nicolae Lupa ◽  
Mihail Megan
Keyword(s):  
Differential Equations ◽  
Real Line ◽  
Exponential Dichotomy ◽  
Natural Extension ◽  
Evolution Operators ◽  
The Real ◽  
Exponential Trichotomy ◽  
Whole Space

This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole spaceX(unlike the case of evolution operators generated by differential equations).

Expressive Features of Author Subjectivity in the Literary Ego-Text (Roman Senchin’s What Do You Want?)

2021 ◽  
pp. 23-39
Author(s):  
Olga Yu. Shum ◽  
Keyword(s):  
Literary Criticism ◽  
Time Frame ◽  
The Novel ◽  
The Real ◽  
Political Commentary ◽  
Wide Range ◽  
Literary Process ◽  
Family Chronicle ◽  
The One ◽  
Almost All

The modern literary process indicates the presence of a large number of fiction with documentary subcurrent: facts from the author’s biography are a very slight hyperbolization. An example of such work would be the novel What Do You Want? (2013) by a contemporary Russian writer Roman Senchin, which became the subject of consideration in this article. The synthesis of auto-documentary and literary principles in the story organizes self-narration with unsteady boundaries between the real (“factual”) and the fictional (“fictitious”). The specificity of the correlation of the factual and the fictitious is examined in this work using the method of literary criticism and contextual analysis. The immediate aim of the article is to identify the specificity of expressing the implicit method of author subjectivity in non-fiction. In the author’s opinion, the implicit way of expressing the reflective type of author subjectivity fits more harmoniously into the literary fabric of the work, enriching it with subtexts and hidden meanings. In the course of the study it has been determined that although the center of the story revolves around the everyday life of an ordinary Moscow family, Senchin’s work is not a slice-of-life novel, but a political commentary. The theme of What Do You Want?” is sociopolitical, the problematics are sociocultural. The narration of the novel undergoes an intense analyzing and coming to terms with the sociopolitical events that are highlighted in almost all of the scenes. The text implies that the writer comprehends his own political position and interprets its cause-effect relationships. In order to distance himself as much as possible from his own identity, Senchin uses the technique of “externalizing” and “assigns” the role of the narrator to a teenage girl Dasha, the prototype of which he himself cannot be. Dasha, being a narrator-observer, asks questions, including the one from the title, to herself and other characters, including the father “Roman Senchin”. The time frame for the narration is precisely established: 18 December 2011 – 26 February 2012; each part of the text is a certain day, there are six in total. However, it is not clear who marked the specific days – the real author of the story or, as he conceived, the narrator Dasha. The autofiction method of “externalizing” in combination with the factual plot allows considering What Do You Want? as an ego-text, which in its genre form is something between an excerpt from a family chronicle and a diary. Autofiction in the form of an ego-text allows the writer to implicate his reflection, organizing a space for discussion of the unquestioning “Roman Senchin” (his alter ego) and the doubting Dasha within a kind of a “mental diary” – a space of consciousness in which the author-subject and the narrator are united. “Bringing to light” this “mental” diary, the writer redirects it to a wide range of readers and thus shifts the story from the field of “literature without fiction” to the sphere of art

A non-constant invariant function for certain ergodic flows

2008 ◽  
Vol 28 (3) ◽  
pp. 1031-1035
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Differentiable Manifold ◽  
Invariant Function ◽  
Continuous Functions ◽  
The Real ◽  
Almost All ◽  
Uniformly Continuous

AbstractLet U be the vector space of uniformly continuous real-valued functions on the real line $\mathbb {R}$ and let U0 denote the subspace of U consisting of all bounded uniformly continuous functions. If X is a compact differentiable manifold and we are given a flow on X, then we associate with the flow a function F:X→H1(X,U/U0) that is invariant under the flow. We give examples for which the flow on X is ergodic but there is no λ∈H1(X,U/U0) such that F(p)=λ for almost all points p.

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