ON FREE AND PROJECTIVE S-SPACES AND FLOWS OVER A TOPOLOGICAL MONOID

2010 ◽  
Vol 03 (03) ◽  
pp. 443-456 ◽  
Author(s):  
Behnam Khosravi
Keyword(s):  
Underlying Space ◽  
Topological Monoid

In this paper, we study free and projective flows and S-spaces, and we characterize free and projective flows over a compact topological monoid S. Similarly, we characterize the same objects in the category of S-spaces for an arbitrary topological monoid S. In fact, we show that any projective S-space is topologically isomorphic to ⊕iϵISei where ei are idempotents in S and ⊕ iϵISei denotes the discrete topological sum of the underlying space of Sei. This is the same result as the category S-Act that the projective objects in this category are the coproducts of Sei's, where ei are idempotents in S.

Anatomization of the systems of dimension relaxation for facial recognition

2020 ◽  
pp. 1-11
Author(s):  
Mayamin Hamid Raha ◽  
Tonmoay Deb ◽  
Mahieyin Rahmun ◽  
Tim Chen
Keyword(s):  
Face Recognition ◽  
Main Idea ◽  
Principal Component ◽  
Feature Space ◽  
Projection Methods ◽  
Data Representation ◽  
Linear Discriminant ◽  
Class Differences ◽  
Underlying Space ◽  
Maximum Variance

Face recognition is the most efficient image analysis application, and the reduction of dimensionality is an essential requirement. The curse of dimensionality occurs with the increase in dimensionality, the sample density decreases exponentially. Dimensionality Reduction is the process of taking into account the dimensionality of the feature space by obtaining a set of principal features. The purpose of this manuscript is to demonstrate a comparative study of Principal Component Analysis and Linear Discriminant Analysis methods which are two of the highly popular appearance-based face recognition projection methods. PCA creates a flat dimensional data representation that describes as much data variance as possible, while LDA finds the vectors that best discriminate between classes in the underlying space. The main idea of PCA is to transform high dimensional input space into the function space that displays the maximum variance. Traditional LDA feature selection is obtained by maximizing class differences and minimizing class distance.

scholarly journals Ideal Convergence and Completeness of a Normed Space

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 897 ◽  
Author(s):  
Fernando León-Saavedra ◽  
Francisco Javier Pérez-Fernández ◽  
María del Pilar Romero de la Rosa ◽  
Antonio Sala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Convergence Space ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Underlying Space ◽  
Phd Thesis

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

scholarly journals Geometry and topology of the space of Kähler metrics on singular varieties

2018 ◽  
Vol 154 (8) ◽  
pp. 1593-1632 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Vincent Guedj
Keyword(s):  
Einstein Metrics ◽  
Normal Space ◽  
Fano Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Metric Properties ◽  
Singular Varieties ◽  
Underlying Space ◽  
And Topology ◽  
Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

A Solution Method for an Optimal Controlled Vibrating Circle Shell by Measure and Classical Trajectory

2011 ◽  
Vol 52-54 ◽  
pp. 1855-1860
Author(s):  
J.A. Fakharzadeh ◽  
F.N. Jafarpoor
Keyword(s):  
Wave Equations ◽  
Classical Trajectory ◽  
Polar Coordinates ◽  
Approximation Schemes ◽  
Solution Technique ◽  
Underlying Space ◽  
The Mean ◽  
Positive Coefficients ◽  
And Control ◽  
Finite Linear

The mean idea of this paper is to present a new combinatorial solution technique for the controlled vibrating circle shell systems. Based on the classical results of the wave equations on circle domains, the trajectory is considered as a finite trigonometric series with unknown coefficients in polar coordinates. Then, the problem is transferred to one in which its unknowns are a positive Radon measure and some positive coefficients. Extending the underlying space helps us to prove the existence of the solution. By using the density properties and some approximation schemes, the problem is deformed into a finite linear programming and the nearly optimal trajectory and control are identified simultaneously. A numerical example is also given.

scholarly journals A TORSIONAL TOPOLOGICAL INVARIANT

2007 ◽  
Vol 22 (29) ◽  
pp. 5237-5244 ◽  
Author(s):  
H. T. NIEH
Keyword(s):  
Affine Connection ◽  
Euler Class ◽  
Christoffel Symbol ◽  
Space Time ◽  
Topological Invariant ◽  
Point Of View ◽  
Underlying Space ◽  
Recent Developments ◽  
Theory Point ◽  
Curvature And Torsion

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

scholarly journals Some properties of shift operators on algebras generated by $*$-polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 206-212
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Countable System ◽  
Shift Operators ◽  
Underlying Space ◽  
Algebra Of Functions ◽  
Algebras Of Holomorphic Functions ◽  
Proper Subalgebra

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

scholarly journals Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1153
Author(s):  
Najla Altwaijry ◽  
Tahani Aldhaban ◽  
Souhail Chebbi ◽  
Hong-Kun Xu
Keyword(s):  
Approximation Method ◽  
Geometric Property ◽  
Nonexpansive Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mann Iteration ◽  
Underlying Space ◽  
Uniformly Smooth ◽  
Regularization Parameters ◽  
Uniformly Smooth Banach Spaces

We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.

scholarly journals HOMOTOPY OF POSETS, NET-COHOMOLOGY AND SUPERSELECTION SECTORS IN GLOBALLY HYPERBOLIC SPACE-TIMES

2005 ◽  
Vol 17 (09) ◽  
pp. 1021-1070 ◽  
Author(s):  
GIUSEPPE RUZZI
Keyword(s):  
Hyperbolic Space ◽  
Space Time ◽  
Fundamental Groups ◽  
Mathematical Framework ◽  
Permutation Symmetry ◽  
Index Set ◽  
Globally Hyperbolic ◽  
Underlying Space ◽  
Local Observables ◽  
Suitable Change

We study sharply localized sectors, known as sectors of DHR-type, of a net of local observables, in arbitrary globally hyperbolic space-times with dimension ≥ 3. We show that these sectors define, as it happens in Minkowski space, a C*-category in which the charge structure manifests itself by the existence of a tensor product, a permutation symmetry and a conjugation. The mathematical framework is that of the net-cohomology of posets according to J. E. Roberts. The net of local observables is indexed by a poset formed by a basis for the topology of the space-time ordered under inclusion. The category of sectors, is equivalent to the category of 1-cocycles of the poset with values in the net. We succeed in analyzing the structure of this category because we show how topological properties of the space-time are encoded in the poset used as index set: the first homotopy group of a poset is introduced and it is shown that the fundamental group of the poset and one of the underlying space-time are isomorphic; any 1-cocycle defines a unitary representation of these fundamental groups. Another important result is the invariance of the net-cohomology under a suitable change of index set of the net.

scholarly journals Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Senlin Wu ◽  
Xinjian Dong ◽  
Dan Wang
Keyword(s):  
Unit Sphere ◽  
Linear Spaces ◽  
Strictly Convex ◽  
Normed Linear Spaces ◽  
Underlying Space ◽  
Pythagorean Orthogonality

We introduce the circle-uniqueness of Pythagorean orthogonality in normed linear spaces and show that Pythagorean orthogonality is circle-unique if and only if the underlying space is strictly convex. Further related results providing more detailed relations between circle-uniqueness of Pythagorean orthogonality and the shape of the unit sphere are also presented.

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