A characterization of Q-polynomial association schemes

We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.     

The Subconstituent Algebra of an Association Scheme, (Part I)

We introduce a method for studying commutative association schemes with many vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple -algebra T = T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y.In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter.We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then sufficiently large dimension means dimension at least four.We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur.We close with some conjectures and open problems.     

The Subconstituent Algebra of an Association Scheme (Part III)

This is the continuation of an article from the previous issue. In this part, we focus on the thin P- and Q-polynomial association schemes. We provide some combinatorial characterizations of these objects and exhibit the known examples with diameter at least 6. For each example, we give the irreducible modules for the subconstituent algebra. We close with some conjectures and open problems.     

Linking systems in nonelementary abelian groups

Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

The Subconstituent Algebra of an Association Scheme (Part II)

This is a continuation of an article from the previous issue. In this section, we determine the structure of a thin, irreducible module for the subconstituent algebra of a P- and Q- polynomial association scheme. Such a module is naturally associated with a Leonard system. The isomorphism class of the module is determined by this Leonard system, which in turn is determined by four parameters: the endpoint, the dual endpoint, the diameter, and an additional parameter f. If the module has sufficiently large dimension, the parameter f takes one of a certain set of values indexed by a bounded integer parameter e.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

Similarity mass and approximate reasoning

It can reflect the nature of approximate reasoning and meet more application expectations to design the approximate reasoning matching schemes and the corresponding algorithms with similarity relation Q instead of equivalence relation R. In this paper, based on similarity relation Q, we introduce type V matching scheme and corresponding approximate reasoning type V Q-algorithm with the given input A* and knowledge A → B. Besides, we present completeness of type V and its perfection on knowledge base K in Q-logic ℂ _Q in this paper.     

Constructions for large sets of <Emphasis Type="Italic">L</Emphasis>-intersecting Steiner triple systems

Recently, Franek et al. introduced large sets of v − 1 L-intersecting Steiner triple systems of order v (STS(v)) and gave four constructions for them (Des., Codes and Cryptogr., 26 (2002), 243–256). In this paper, we mainly focus on large sets of v − 1{0, 1}-intersecting STS(v) and large sets of v + 1{1}-intersecting STS(v). For this purpose, we introduce a concept of L-intersecting partitionable candelabra system (L-PCS) of order v with q(v) subsystems and establish a relationship between L-PCS and large set of q(v)L-intersecting STS(v). Some constructions for L-PCSs are also presented by 3-wise balanced designs. These facilitate the production of some new infinite classes of these large sets. Research supported by Tianyuan Mathematics Foundation of NSFC Grant 10526032 and Natural Science Foundation of Universities of Jiangsu Province Grant 05KJB110111.     

The split hypercubic association scheme

This paper presents a new series of association schemes which are derived from the hypercubic association scheme. Some properties of PBIB designs having these split hypercubic association schemes are given and the incidence matrixN is studied. This research was supported by National Research Grant Number A8483.     

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

Transition function for the Toda chain

We use the method of Λ-operators developed by Derkachov, Korchemsky, and Manashov to derive eigenfunctions for the open Toda chain. Using the diagram technique developed for these Λ-operators, we reproduce the Sklyanin measure and study the properties of the Λ-operators. This approach to the open Toda chain eigenfunctions reproduces the Gauss-Givental representation for these eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 371–390, March, 2007.     

Variational approach to various nonlinear problems in geometry and physics

In this survey, we will summarize the existence results of nonlinear partial differential equations which arises from geometry or physics by using variational method. We use the method to study Kazdan-Warner problem, Chern-Simons-Higgs model, Toda systems, and the prescribed Q-curvature problem in 4-dimension.      

Local c- and E-optimal Designs for Exponential Regression Models

In this paper we investigate local E- and c-optimal designs for exponential regression models of the form . We establish a numerical method for the construction of efficient and local optimal designs, which is based on two results. First, we consider for fixed k the limit μ _i → γ (i = 1, ... , k) and show that the optimal designs converge weakly to the optimal designs in a heteroscedastic polynomial regression model. It is then demonstrated that in this model the optimal designs can be easily determined by standard numerical software. Secondly, it is proved that the support points and weights of the local optimal designs in the exponential regression model are analytic functions of the nonlinear parameters μ ₁, ... , μ _k . This result is used for the numerical calculation of the local E-optimal designs by means of a Taylor expansion for any vector (μ ₁, ... , μ _k ). It is also demonstrated that in the models under consideration E-optimal designs are usually more efficient for estimating individual parameters than D-optimal designs.