scholarly journals Solve Special Case of Some Guran Problems

2016 ◽  
Vol 8 (3) ◽  
pp. 33
Author(s):  
Ahmad Alghoussein ◽  
Ziad Kanaya ◽  
Salwa Yacoub
Keyword(s):  
Dimensional Space ◽  
Topological Spaces ◽  
Topological Groups ◽  
Open Problems ◽  
Hereditary Properties ◽  
Special Case

<p>Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by <em>I(X)</em> and <em>I(Y)</em> to be a free topological groups over a topological spaces <em>X</em> and <em>Y</em> respectively where <em>X</em> and <em>Y</em> are assumed to be a non- empty sub manifolds Which are also a closed sub sets, and <em>P</em> is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zero- dimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.</p>

scholarly journals Solve Special Case of Some Guran Problems

2016 ◽  
Vol 8 (3) ◽  
pp. 33
Author(s):  
Ahmad Alghoussein ◽  
Ziad Kanaya ◽  
Salwa Yacoub
Keyword(s):  
Dimensional Space ◽  
Topological Spaces ◽  
Topological Groups ◽  
Open Problems ◽  
Hereditary Properties ◽  
Special Case

<p>Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by <em>I(X)</em> and <em>I(Y)</em> to be a free topological groups over a topological spaces <em>X</em> and <em>Y</em> respectively where <em>X</em> and <em>Y</em> are assumed to be a non- empty sub manifolds Which are also a closed sub sets, and <em>P</em> is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zero- dimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.</p>

scholarly journals Free Abelian topological groups and the Pontryagin-Van Kampen duality

1995 ◽  
Vol 52 (2) ◽  
pp. 297-311 ◽  
Author(s):  
Vladimir Pestov
Keyword(s):  
Topological Group ◽  
Dimensional Space ◽  
Topological Spaces ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

2003 ◽  
Vol 10 (2) ◽  
pp. 209-222
Author(s):  
I. Bakhia
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

2021 ◽  
Vol 7 (1) ◽  
pp. 1224-1240
Author(s):  
Sang-Eon Han ◽  
◽  
Saeid Jafari ◽  
Jeong Min Kang ◽  
Sik Lee ◽  
...  
Keyword(s):  
Crucial Role ◽  
Dimensional Space ◽  
Digital Topology ◽  
Topological Spaces ◽  
Khalimsky Topology

<abstract><p>The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.</p></abstract>

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

Lie subgroups of the symmetry group of the equations describing a nonstationary and isentropic flow: Invariant and partially invariant solutions

1994 ◽  
Vol 72 (7-8) ◽  
pp. 362-374 ◽  
Author(s):  
A. M. Grundland ◽  
L. Lalague
Keyword(s):  
Symmetry Group ◽  
Dimensional Space ◽  
Projective Group ◽  
Invariant Solutions ◽  
Partially Invariant Solutions ◽  
Isentropic Flow ◽  
Dimensional Space Time ◽  
Group A ◽  
Special Case

We study the symmetries of the equations describing a nonstationary and isentropic flow for an ideal and compressible fluid in four-dimensional space-time. We prove that this system of equations is invariant under the Galilean-similitude group. In the special case of the adiabatic exponent γ = 5/3, corresponding to a diatomic gas, the symmetry group of this system is larger. It is invariant under the Galilean-projective group. A representatives list of subalgebras of Galilean similitude and Galilean-projective Lie algebras, obtained by the method of classification by conjugacy classes under the action of their respective Lie groups, is presented. The results are given in a normalized list and summarized in tables. Examples of invariant and nonreducible partially invariant solutions, obtained from this classification, is constructed. The final part of this work contains an analysis of this classification in connection with a further classification of the symmetry algebras for the Euler and magnetohydrodynamics equations.

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