Operational total space theory of principal 2-bundles I: Operational geometric framework

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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A geometric framework for rigid body attitude estimation

Automatica ◽

10.1016/j.automatica.2021.109494 ◽

2021 ◽

Vol 128 ◽

pp. 109494

Author(s):

Yujendra Bharathi Mitikiri ◽

Kamran Mohseni

Keyword(s):

Rigid Body ◽

Attitude Estimation ◽

Geometric Framework ◽

Body Attitude

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Finite cell method for functionally graded materials based on V-models and homogenized microstructures

Advanced Modeling and Simulation in Engineering Sciences ◽

10.1186/s40323-020-00182-1 ◽

2020 ◽

Vol 7 (1) ◽

Author(s):

Benjamin Wassermann ◽

Nina Korshunova ◽

Stefan Kollmannsberger ◽

Ernst Rank ◽

Gershon Elber

Keyword(s):

Functionally Graded Materials ◽

Large Scale ◽

Functionally Graded ◽

Material Behavior ◽

Modeling Framework ◽

Finite Cell Method ◽

Geometric Framework ◽

Graded Materials ◽

Cell Method ◽

Finite Cell

AbstractThis paper proposes an extension of the finite cell method (FCM) to V-rep models, a novel geometric framework for volumetric representations. This combination of an embedded domain approach (FCM) and a new modeling framework (V-rep) forms the basis for an efficient and accurate simulation of mechanical artifacts, which are not only characterized by complex shapes but also by their non-standard interior structure. These types of objects gain more and more interest in the context of the new design opportunities opened by additive manufacturing, in particular when graded or micro-structured material is applied. Two different types of functionally graded materials (FGM) are considered: The first one, multi-material FGM is described using the inherent property of V-rep models to assign different properties throughout the interior of a domain. The second, single-material FGM—which is heterogeneously micro-structured—characterizes the effective material behavior of representative volume elements by homogenization and performs large-scale simulations using the embedded domain approach.

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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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