The Linear Fuzzy Space: Theory and Applications

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

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Improving KKM Theory on General Metric Type Spaces

Topological Algebra and its Applications ◽

10.1515/taa-2020-0101 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-12

Author(s):

Sehie Park

Keyword(s):

Convex Space ◽

Metric Spaces ◽

Space Theory ◽

Type Space ◽

Generalized Metric ◽

Abstract Convex Space ◽

Type Spaces

Abstract A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.

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Modeling learning in knowledge space theory through bivariate Markov processes

Journal of Mathematical Psychology ◽

10.1016/j.jmp.2021.102549 ◽

2021 ◽

Vol 103 ◽

pp. 102549

Author(s):

Pasquale Anselmi ◽

Luca Stefanutti ◽

Debora de Chiusole ◽

Egidio Robusto

Keyword(s):

Markov Processes ◽

Space Theory ◽

Knowledge Space ◽

Knowledge Space Theory

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Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽

10.1515/anly-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Siran Li

Keyword(s):

Banach Space ◽

Discrete Geometry ◽

Thin Shell ◽

Euclidean Geometry ◽

Convex Geometry ◽

Regularity Theory ◽

Elliptic Partial Differential Equations ◽

High Dimensional ◽

Space Theory ◽

Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

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Embedded discourse spaces in narrative reports

Discourse Studies ◽

10.1177/1461445619893776 ◽

2019 ◽

Vol 22 (2) ◽

pp. 221-240

Author(s):

Anna Ewa Wieczorek

Keyword(s):

Bill Clinton ◽

Space Theory ◽

Presentation Of Self ◽

Reported Speech ◽

Image Construction ◽

Positive Image ◽

Narrative Representations ◽

Presidential Speeches

This article aims to discuss conceptual levels of narrative representations of utterances based on reported speech frames employed in presidential speeches. It adopts some assumptions from Chilton’s Deictic Space Theory and Cap’s Proximisation Theory, both primarily used to indicate exclusive reference, a clash of interests and threat-oriented conceptualisation of events. This article, however, extends their scope to include strategies for inclusion and positive image construction and makes a distinction between primary, secondary and tertiary embedding as discursive means that contribute to presentation of self and legitimisation. Data for this research comprise a corpus of 125 presidential speeches (25 per tenure) divided into three subcorpora: JKC – John Kennedy Corpus, BCC – Bill Clinton Corpus, and BOC – Barrack Obama Corpus. A total of 1251 instances of narrative reports have been analysed to investigate primary and multilevel embedding, which constitute the basis for this study.

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