scholarly journals K-loop Structures Raised by the Direct Limits of Pseudo Unitary $U(p, a_{n})$ and Pseudo Orthogonal $O(p, a_{n})$ Groups

2017 ◽  
Vol 9 (5) ◽  
pp. 37
Author(s):  
ALPER BULUT
Keyword(s):  
Relativistic Theory ◽  
Direct Limit ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Inverse Property ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Bol Loop

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.

scholarly journals On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

2018 ◽  
Vol 15 (12) ◽  
pp. 1850206
Author(s):  
Fernand Pelletier
Keyword(s):  
Direct Limit ◽  
Symplectic Forms ◽  
Banach Manifolds ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Darboux Theorem

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.

scholarly journals Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

2019 ◽  
Vol 71 (1) ◽  
pp. 131-152 ◽  
Author(s):  
Helge Glöckner
Keyword(s):  
Lie Groups ◽  
Direct Limit ◽  
Topological Groups ◽  
Test Function ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Function Group ◽  
The Given ◽  
Locally Convex ◽  
Infinite Dimensional Lie Groups

AbstractWe prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

scholarly journals Real Dimension Groups

2013 ◽  
Vol 56 (3) ◽  
pp. 551-563 ◽  
Author(s):  
David Handelman
Keyword(s):  
Direct Limit ◽  
Polynomial Rings ◽  
Vector Spaces ◽  
Real Vector ◽  
Product Decomposition ◽  
Finite Dimensional ◽  
Dimension Groups ◽  
Ordered Vector Spaces ◽  
Direct Limits ◽  
Directed Set

AbstractDimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

scholarly journals On involutive Lie algebras having a Cartan decomposition

2004 ◽  
Vol 69 (2) ◽  
pp. 191-202 ◽  
Author(s):  
A. J. Calderón Martín
Keyword(s):  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Direct Limit ◽  
Arbitrary Dimension ◽  
Cartan Decomposition ◽  
Infinite Dimensional ◽  
Finite Dimensional

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H in the sense of Y. Billig and A. Pianzola for involutive complex Lie algebras L of arbitrary dimension. If L has such a decomposition and is infinite dimensional and simple, we show it is *-isomorphic to a direct limit of classical finite dimensional simple involutive Lie algebras of the same type A, B, C, or D.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals The Farkas lemma of Shimizu, Aiyoshi and Katayama

1985 ◽  
Vol 31 (3) ◽  
pp. 445-450 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Convex Programming ◽  
Infinite Dimensional ◽  
Farkas Lemma ◽  
Finite Dimensional

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

scholarly journals Free non-associative principal train algebras

1984 ◽  
Vol 27 (3) ◽  
pp. 313-319 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Finite Number ◽  
Ground Field ◽  
Linear Forms ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Senses ◽  
Special Train ◽  
Complex Coefficients ◽  
Infinite Linear

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

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