scholarly journals Universal simulation of Hamiltonians using a finite set of control operations

2002 ◽  
Vol 2 (2) ◽  
pp. 133-150
Author(s):  
P. Wocjan ◽  
M. Rotteler ◽  
D. Janzing ◽  
T. Beth
Keyword(s):  
Quantum System ◽  
Finite Groups ◽  
Sufficient Condition ◽  
Large Group ◽  
Group Characters ◽  
Finite Set ◽  
Dimension 2 ◽  
Hamiltonian Evolution

Any quantum system with a non-trivial Hamiltonian is able to simulate any other Hamiltonian evolution provided that a sufficiently large group of unitary control operations is available. We show that there exist finite groups with this property and present a sufficient condition in terms of group characters. We give examples of such groups in dimension 2 and 3. Furthermore, we show that it is possible to simulate an arbitrary bipartite interaction by a given one using such groups acting locally on the subsystems.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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.

On Pólya's Theorem

1967 ◽  
Vol 19 ◽  
pp. 792-799 ◽  
Author(s):  
J. Sheehan
Keyword(s):  
Finite Groups ◽  
Scalar Product ◽  
Combinatorial Analysis ◽  
Simple Extension ◽  
Group Characters ◽  
Polya's Theorem ◽  
Representations Of Finite Groups ◽  
Pólya’S Theorem ◽  
Special Case ◽  
Superposition Theorem

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

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.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Determining Subgroups of a Given Finite Index in a Finitely Presented Group

1974 ◽  
Vol 26 (4) ◽  
pp. 769-782 ◽  
Author(s):  
Anke Dietze ◽  
Mary Schaps
Keyword(s):  
Finite Index ◽  
Finite Groups ◽  
Infinite Groups ◽  
Matrix Groups ◽  
Finitely Presented Group ◽  
Basic Object ◽  
Finite Set ◽  
Object Of Study ◽  
Finitely Presented

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

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 CONNECTIVITY PROPERTIES OF MCKAY QUIVERS

2020 ◽  
pp. 1-13
Author(s):  
HAZEL BROWNE
Keyword(s):  
Finite Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Self Duality ◽  
Finite Dimensional ◽  
Representations Of Finite Groups ◽  
Necessary And Sufficient ◽  
Mckay Quiver

Abstract We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

Some Characterizations about the Structure of Finite Groups

2012 ◽  
Vol 19 (03) ◽  
pp. 563-568
Author(s):  
Lili Wang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Condition ◽  
A Finite Group

A subgroup H of a finite group G is called an ℋ-subgroup of G if NG(H) ∩ Hg ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G).

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