Class dimension of association schemes in singular linear spaces

2017 ◽  
Vol 09 (01) ◽  
pp. 1750005
Author(s):  
Haixia Guo ◽  
You Gao
Keyword(s):  
Association Scheme ◽  
Upper Bounds ◽  
Association Schemes ◽  
Resolving Set ◽  
Linear Spaces ◽  
Set Of Points ◽  
Special Case ◽  
Singular Linear Spaces

A resolving set for an association scheme [Formula: see text] is a set of points [Formula: see text] such that, for all [Formula: see text], the ordered list of relations [Formula: see text] uniquely determines [Formula: see text], where [Formula: see text] denotes the relation [Formula: see text] containing the pair [Formula: see text] in [Formula: see text]. In this paper, we determine upper bounds on class dimension for a family of association schemes in singular linear spaces, and construct their resolving sets for a special case.

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

scholarly journals A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes

10.37236/4423 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Hiroshi Nozaki ◽  
Hirotake Kurihara
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Equivalent Condition ◽  
Polynomial Property ◽  
Equivalent Conditions ◽  
Symmetric Association Scheme

We give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

On Some Issues in Shakedown Analysis

2001 ◽  
Vol 68 (5) ◽  
pp. 799-808 ◽  
Author(s):  
G. Maier
Keyword(s):  
Heterogeneous Media ◽  
Direct Methods ◽  
Cost Effective ◽  
Upper Bounds ◽  
Saturated Porous Media ◽  
Safety Factors ◽  
Shakedown Analysis ◽  
Time Stepping ◽  
Kinematic Theorem ◽  
Special Case

Shakedown analysis, and its more classical special case of limit analysis, basically consists of “direct” (as distinct from time-stepping) methods apt to assess safety factors for variable repeated external actions and procedures which provide upper bounds on history-dependent quantities. The issues reviewed and briefly discussed herein are: some recent engineering-oriented and cost-effective methods resting on Koiter’s kinematic theorem and applied to periodic heterogeneous media; recent extensions (after the earlier ones to dynamics and creep) to another area characterized by time derivatives, namely poroplasticity of fluid-saturated porous media. Links with some classical or more consolidated direct methods are pointed out.

Polarization of Separating Invariants

2008 ◽  
Vol 60 (3) ◽  
pp. 556-571 ◽  
Author(s):  
Jan Draisma ◽  
Gregor Kemper ◽  
David Wehlau
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Group Actions ◽  
Upper Bounds ◽  
Weyl’S Theorem ◽  
Finite Group Actions ◽  
Separating Invariants ◽  
The Difference ◽  
Special Case

AbstractWe prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue ofWeyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

scholarly journals Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Roots Of Unity ◽  
Galois Rings ◽  
Type Complex ◽  
Complex Hadamard Matrix

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

scholarly journals On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Allen Herman ◽  
Gurmail Singh
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Group Rings ◽  
The 1960S ◽  
Torsion Units

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

2019 ◽  
Vol 29 (02) ◽  
pp. 95-120 ◽  
Author(s):  
Prosenjit Bose ◽  
André van Renssen
Keyword(s):  
Line Segment ◽  
Large Family ◽  
Upper Bounds ◽  
The Other ◽  
Additional Property ◽  
Graph Partitions ◽  
Set Of Points ◽  
The Given

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets

1999 ◽  
Vol 51 (2) ◽  
pp. 326-346 ◽  
Author(s):  
W. J. Martin ◽  
D. R. Stinson
Keyword(s):  
Linear Programming ◽  
Association Scheme ◽  
Type Theorem ◽  
Orthogonal Arrays ◽  
Association Schemes ◽  
Theoretic Approach ◽  
Linear Programming Bound ◽  
Ordered Orthogonal Arrays

AbstractIn an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

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