Fuzzifying bornological linear spaces

2021 ◽  
pp. 1-12
Author(s):  
Zhen-Yu Jin ◽  
Cong-Hua Yan
Keyword(s):  
Linear Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Probabilistic Normed Spaces ◽  
Necessary And Sufficient ◽  
Topological Linear

In this paper, a notion of fuzzifying bornological linear spaces is introduced and the necessary and sufficient condition for fuzzifying bornologies to be compatible with linear structure is discussed. The characterizations of convergence and separation in fuzzifying bornological linear spaces are showed. In particular, some examples with respect to linear fuzzifying bornologies induced by probabilistic normed spaces and fuzzifying topological linear spaces are also provided.

scholarly journals NORMS ON CARTESIAN PRODUCT OF LINEAR SPACES

1990 ◽  
Vol 21 (1) ◽  
pp. 35-39
Author(s):  
Chi-Kwong Li ◽  
Nam-Kiu Tsing
Keyword(s):  
Product Space ◽  
Cartesian Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Spaces ◽  
Complex Linear ◽  
Necessary And Sufficient

Let $X_i, (i=1, \cdots, n)$ be real or complex linear spaces, each equipped with a norm $||\cdot||_i$. Standard ways of constructing norms $||\cdot||$ on the Cartestian product $X =X_1 \times \cdots \times X_n$ are to define \[ ||(x_1, \cdots, x_n)||=\phi(||x_1||_1, \cdots, ||x_n||_n)\] via some functions $\phi$ on $\mathbb{R}^n$. Common examples of $\phi$ in standard texbooks are norms on $\mathbb{R}^n$. This may mislead peoples to think that any norm $\phi$ on $\mathbb{R}^n$ can induce a norm on the product space $X$ in the above way. In this note we show that this is actually false and characterize the functions $\phi$ that can give rise to norms on $X$ in the above manner. It turns out that a necessary and sufficient condition on $\phi$ is : for any $a_1, \cdots, a_n, b_1, \cdots, b_n\ge 0$, (I) $\phi(a_1, \cdots, a_n)>0$ if $(a_1, \cdots, a_n)\neq (0, \cdots,0)$; (II) $\phi(\alpha(a_1, \cdots, a_n))= \alpha \phi(a_1, \cdots, a_n)$ if $\alpha\ge 0$; (III) $\phi(c_1, \cdots, c_n)\le \phi(a_1, \cdots, a_n)+ \phi(b_1, \cdots, b_n)$ if $(c_1, \cdots, c_n)= (a_1, \cdots, a_n)+ (b_1, \cdots, b_n)$;(IV) $\phi(a_1, \cdots, a_n) \le \phi(b_1, \cdots, b_n)$ if $a_i \le b_i$ for all $i$. Several interesting consequences of the result are discussed.

scholarly journals Weakly externally hyperconvex subsets and hyperconvex gluings

2016 ◽  
Vol 09 (03) ◽  
pp. 379-407
Author(s):  
Benjamin Miesch ◽  
Maël Pavón
Keyword(s):  
Metric Spaces ◽  
Maximum Norm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Spaces ◽  
Full Characterization ◽  
Finite Dimensional ◽  
Hyperconvex Space ◽  
Necessary And Sufficient

We give a necessary and sufficient condition under which gluings of hyperconvex metric spaces along weakly externally hyperconvex subsets are hyperconvex. This leads to a full characterization of hyperconvex gluings of two isometric copies of the same hyperconvex space. Furthermore, we investigate the case of gluings of finite dimensional hyperconvex linear spaces along linear subspaces. For this purpose, we characterize the weakly externally hyperconvex subsets of [Formula: see text] endowed with the maximum norm.

scholarly journals The moment spaces of normed linear spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 277-283
Author(s):  
Fowzi Ahmad Sejeeni ◽  
Matooq Ahmad Badri
Keyword(s):  
Normed Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Independent Sequence ◽  
Linear Spaces ◽  
Linearly Independent ◽  
The Moment ◽  
Moment Spaces ◽  
Necessary And Sufficient

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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