Remarks on some generalization of the notion of microscopic sets

2020 ◽  
Vol 70 (6) ◽  
pp. 1349-1356
Author(s):  
Aleksandra Karasińska
Keyword(s):  
General Sense ◽  
Equivalent Definition ◽  
Definition Of ◽  
Null Set

AbstractWe consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

scholarly journals Corrective Feedback and EFL Learning Style

2021 ◽  
Vol 2 (10) ◽  
pp. 47-54
Author(s):  
Steven Samie
Keyword(s):  
Learning Style ◽  
Corrective Feedback ◽  
General Sense ◽  
Definition Of ◽  
Efl Learning

The present study is an attempt to address an important issue for an instructor in the classroom. It begins with an account of how to provide Corrective Feedback (CF) for individuals so that they have the highest rate of uptake in the classroom, then it is followed by the definition of the concept in its general sense of the term. Next, the study has provided information on some of the previous studies along with the researcher’s insights on the issue. Finally, the essay concludes by explaining how this term project will help me change the instructor’s attitude in my future teaching in the classroom. Some suggestions for fellow instructors are provided to give them some food for thought.

Characterizations for the n-strong Drazin invertibility in a ring

2020 ◽  
pp. 2150141
Author(s):  
Honglin Zou ◽  
Jianlong Chen ◽  
Huihui Zhu ◽  
Yujie Wei
Keyword(s):  
Positive Integer ◽  
Generalized Inverse ◽  
Product Formula ◽  
Drazin Inverse ◽  
Equivalent Definition ◽  
New Type ◽  
Definition Of ◽  
Existence Criteria

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

The linear canonical wavelet transform on some function spaces

2018 ◽  
Vol 16 (01) ◽  
pp. 1850010 ◽  
Author(s):  
Yong Guo ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Continuous Operator ◽  
Function Spaces ◽  
Linear Continuous Operator ◽  
Convolution Theorem ◽  
Linear Canonical Transform ◽  
Equivalent Definition ◽  
New Space ◽  
Definition Of

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

scholarly journals FRACTAL ANALYSIS OF HOPF BIFURCATION AT INFINITY

2012 ◽  
Vol 22 (12) ◽  
pp. 1230043 ◽  
Author(s):  
GORAN RADUNOVIĆ ◽  
DARKO ŽUBRINIĆ ◽  
VESNA ŽUPANOVIĆ
Keyword(s):  
Hopf Bifurcation ◽  
Fractal Analysis ◽  
Stereographic Projection ◽  
Riemann Sphere ◽  
Box Dimension ◽  
Euclidean Spaces ◽  
Equivalent Definition ◽  
Basic Properties ◽  
Definition Of

Using geometric inversion with respect to the origin, we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf–Takens bifurcation at infinity.

scholarly journals Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

2011 ◽  
Vol 109 (1) ◽  
pp. 71
Author(s):  
Alexander Pavlov ◽  
Ulrich Pennig ◽  
Thomas Schick
Keyword(s):  
Strict Topology ◽  
Locally Compact ◽  
Compact Spaces ◽  
Equivalent Definition ◽  
C Algebras ◽  
Definition Of ◽  
Locally Compact Spaces

Quasi-multipliers for a Hilbert $C^*$-bimodule $V$ were introduced by L. G. Brown, J. A. Mingo, and N.-T. Shen [3] as a certain subset of the Banach bidual module $V^{**}$. We give another (equivalent) definition of quasi-multipliers for Hilbert $C^*$-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over $C^*$-algebras, provided these $C^*$-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule $l_2(A)$ and for bimodules of sections of Hilbert $C^*$-bimodule bundles over locally compact spaces.

Collective Security

2014 ◽  
Author(s):  
Ademola Abass
Keyword(s):  
Cold War ◽  
United Nations ◽  
International Law ◽  
Collective Security ◽  
General Sense ◽  
Regional Organizations ◽  
The United Nations ◽  
Definition Of ◽  
The Cold War

The term collective security in a general sense is given many understandings both professional and nonprofessional. The phrase is sometimes used to describe the organization of security on a “collective” basis. Often, it is used to denote the “collective organization” of security. While neither of these uses is inherently wrong, neither succinctly captures what “collective security” implies when used by international lawyers. In international law, collective security is a term connoting something more dense and intricate, and much more slippery, than the above more straightforward expressions. The notion of collective security, its premise, and objectives are deeply contested by states and scholars. It is universally acknowledged that collective security is today organized under the United Nations; however, regional organizations, which used to focus primarily on economic matters, have attained greater prominence in collective security efforts especially since the end of the Cold War. This article examines the definition of collective security, its features and objectives, the actors that have the responsibility for operating it globally and regionally, its various manifestations, its limitations and, above all, its role in future.

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