Bipartite Distance-regular Graphs, Part II

This is a continuation of “Bipartite Distance-regular Graphs, Part I”. We continue our study of the Terwilliger algebra T of a bipartite distance-regular graph. In this part we focus on the thin irreducible T-modules of endpoint 2 and on those distance-regular graphs for which every irreducible T-module of endpoint 2 is thin. Revised: June 2, 1997     

Spin Leonard pairs

Let ${\mathbb K}$ denote a field, and let V denote a vector space over ${\mathbb K}$ of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U ^?1 = U*^?1 AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.     

The Transit Route Arc-Node Service Maximization problem

This article presents a new method for determining optimal transit routes. The Transit Route Arc-Node Service Maximization model is a mathematical model that maximizes the service value of a route, rather than minimizing cost. Cost (distance) is considered as a budget constraint on the extent of the route. The mathematical formulation modifies and exploits the structure of linear programming problems designed for the traveling salesman problem. An innovative divide-and-conquer solution procedure is presented that not only makes the transit routing problem tractable, but also provides a range of high-quality alternate routes for consideration, some of which have substantially varying geometries. Variant formulations are provided for several common transit route types. The model is tested through its application to an existing street network in Richardson, TX. Optimal numeric results are obtained for several problem instances, and these results demonstrate that increased route cost is not correlated with increased service provision.     

Copper exchanged beta zeolites for the catalytic oxidation of ammonia

Copper exchanged on beta zeolites are extremely active and selective for the catalytic oxidation of ammonia to nitrogen and water and this activity correlates to the ease of reduction of the copper species.     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .