Triangulating Topological Spaces

1997 ◽  
Vol 07 (04) ◽  
pp. 365-378 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Nimish R. Shah
Keyword(s):  
Homotopy Equivalent ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Voronoi Cells ◽  
Delaunay Complex ◽  
Common Intersection ◽  
Finite Set ◽  
The Common ◽  
Nerve Theorem

Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text] are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for [Formula: see text] and [Formula: see text] be homeomorphic.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

scholarly journals On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates

1991 ◽  
Vol 43 (1) ◽  
pp. 19-33 ◽  
Author(s):  
Charles K. Chui ◽  
Amos Ron
Keyword(s):  
Laplace Transform ◽  
Critical Points ◽  
Linear Independence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compactly Supported ◽  
Integer Translates ◽  
Box Spline ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractThe problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices

1952 ◽  
Vol 4 ◽  
pp. 31-42 ◽  
Author(s):  
J. K. Goldhaber
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Characteristic Roots ◽  
Finite Set ◽  
Homomorphic Mapping ◽  
Necessary And Sufficient

The problem of determining the conditions under which a finite set of matrices A1A2, … , Ak has the property that their characteristic roots λ1j, λ2j, … , λki (j = 1, 2, …, n) may be so ordered that every polynomial f(A1A2 … , Ak) in these matrices has characteristic roots f(λ1j, λ2j …,λki) (j = 1, 2, … , n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (Ai〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.

scholarly journals Metrisation of Moore spaces and abstract topological manifolds

1997 ◽  
Vol 56 (3) ◽  
pp. 395-401 ◽  
Author(s):  
David L. Fearnley
Keyword(s):  
Recent Work ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Global Coherence ◽  
Topological Manifolds ◽  
Necessary And Sufficient ◽  
Moore Spaces ◽  
Significant Class

The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

Property (G), Regularity, and Semi-Equicontinuity

1973 ◽  
Vol 16 (4) ◽  
pp. 587-594
Author(s):  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Function Spaces ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Semitopological Group ◽  
Function Of Two Variables ◽  
The Way

This note, motivated by [2], [3], and [4], is devoted to an investigation of properties related to equicontinuity in function spaces of topological spaces. In §2, we study the property (G) defined in [3], and the regularity defined in [4]. A sufficient condition for the simultaneous continuity of a function of two variables, which is analogous to a well known result in equicontinuity, is given at the end of the section. In §3, we relate the regularity with the semi-equicontinuity defined in [2], by localizing the semi-equicontinuity in an obvious way which leads us to weaken some of the hypotheses used in [2]. By the way of constructing an example, we also obtained a sufficient condition for a regular semitopological group to be a topological group.

scholarly journals EMBEDDINGS IN COSET MONOIDS

2008 ◽  
Vol 85 (1) ◽  
pp. 75-80
Author(s):  
JAMES EAST
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Simple Proof ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Simple Description ◽  
Group Of Units ◽  
Finite Monoids ◽  
Finite Set ◽  
Symmetric Inverse Semigroup

AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

scholarly journals Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

2019 ◽  
Vol 8 (3) ◽  
Author(s):  
Vika Yugi Kurniawan
Keyword(s):  
Vector Space ◽  
Directed Graph ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Paper Study ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Indecomposable Representations ◽  
Finite Set ◽  
Necessary And Sufficient

A directed graph is also called as a quiver  where  is a finite set of vertices,  is a set of arrows, and  are two maps from  to . A representation  of a quiver  is an assignment of a vector space  to each vertex  of  and a linear mapping  to each arrow.  We denote by  the direct sum of representasions  and  of a quiver  . A representation  is called indecomposable if  is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

Estimation of the Common Probability of Success for Several Binomials —A Simulation Study

1998 ◽  
Vol 48 (1-2) ◽  
pp. 61-72
Author(s):  
Joydeep Bhanja
Keyword(s):  
Lower Bound ◽  
Simulation Study ◽  
Regularity Conditions ◽  
Moment Estimate ◽  
Probability Of Success ◽  
Finite Set ◽  
The Common ◽  
Asymptotically Efficient ◽  
Common Probability

In this paper we consider an example where for each i, i = 1,2, ... , n, the observations Xij , j = 1, 2, ... , k are i.i.d . Binomial ( ni, θ). Based on a theory developed by us earlier, we propose estimates of θ which are asymptotically efficient under the assumption that k ≥ 2, the ni 's come from a finite set { 1, 2, ... , q} and some mild regularity conditions on the sequence { ni} and θ hold. We present the results of a simulation whlch indicate, among other thlngs, the asymptotic lower bound to variance is lower than or approximately equal to simulated Variances and a simple moment estimate of θ does as well as the asymptotically efficient estimates.

Strongly set-colorable graphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950007
Author(s):  
Kumar Abhishek
Keyword(s):  
Necessary Conditions ◽  
Sufficient Condition ◽  
Finite Set

In [S. M. Hegde, Set colorings of graphs, European J. Combin. 30 (2009) 986–995.] Hegde introduced the notion of set colorings of a graph [Formula: see text] as an assignment of distinct subsets of a finite set [Formula: see text] of [Formula: see text] colors to the vertices of [Formula: see text] such that all the colors of the edges which are obtained as the symmetric differences of the subsets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of [Formula: see text] are the set of all nonempty subsets of [Formula: see text] then the coloring is said to be a strong set-coloring and [Formula: see text] is said to be strongly set-colorable. In this paper, we report some new necessary conditions and propose a conjuncture for the sufficient condition for a graph to admit strong set-coloring. We also identify and characterize some new classes of graphs admitting strong set-coloring. In addition to these, we also propose strategies to construct infinite families graphs admitting strong set-coloring.

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