(S − 1,S) Perishable systems with stochastic leadtimes

This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent. The Markov renewal techniques are employed to obtain the various operating characteristics for the case of Poisson demand and exponential lifetimes. The problem of minimizing the steady state expected cost rate is also discussed, and in the special case of exponential processing times, the optimal stock level is derived explicitly.     

The Contraction of S2p−1 to Hp−1

 A contraction of the sphere , considered as the homogeneous space , to the Heisenberg group is defined. The infinite dimensional irreducible unitary representations of Heisenberg group are then shown to be the limits of the irreducible representations of which are class-1 with respect to . Our results generalise the earlier results of Fulvio Ricci. (Received 1 July 1998; in revised form 3 November 1998)     

Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Sets of Minimality of (1 − 1)-Rational Functions

We describe dynamical systems associated to (1 ? 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems.     

(2s−1) Designs withs intersection numbers

It follows from Ray-Chaudhuri and Wilson (1975) that a (2s–1)-(v, k, ) design has at leasts block intersection numbers. The extremal case of such designs having exactlys intersection numbers is studied. Some necessary conditions on the parameters and intersection numbers are obtained. The following characterization results are proved: (i) A (2s–1)-design with exactlys intersection numbers is the Witt 5-(24, 8, 1) design if and only ifs3 and the sum of the intersection numbers is less than or equal tos(s–1); (ii) A tight 2s-design, that is a 2s-design with exactlys intersection numbers, is the Witt 4-(23, 7, 1) design if and only ifs2 and the sum of the intersection numbers is less than or equal tos ².M. S. Shrikhande acknowledges support from a Central Michigan University Summer Fellowship Award No. 42137.     

Orthogonal polynomials for Jacobi-exponential weights (1−x 2) ρ e −Q(x) on (−1,1)

This paper deals with orthogonal polynomials for Jacobi-exponential weights (1?x ²) ^ρ e ^?Q(x) on (?1,1) and gives bounds on orthogonal polynomials, zeros, and Christofel functions. In addition, restricted range inequalities are also obtained.     

How often is 84(g−1) achieved?

For any finitely generated group Γ, the asymptotics of the set of orders of finite quotient groups of Γ are determined by the minimum dimension of a complex linear group containing an infinite quotient of Γ. We give a proof and an application to the asymptotic behavior of the set of integersg for which the Hurwitz bound is sharp. Partially supported by NSF Grant DMS-97-27553.     

Invariant Sp(1)-instantons on S 2 × S 2

In this paper all (anti)self-dual invariant connections on homogeneous quaternionic line bundles over S ² × S ² are calculated and described in terms of the isotropy homomorphism of the bundle using Wang's theorem. These are the canonical connections on bundles with an (anti)symmetric twist and an S ¹-parametrized family of flat structures on bundles with a simple twist.     

A note on the diophantine equation (a n − 1)(b n − 1) = x 2

Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a ⁿ ? 1)(b ⁿ ? 1) = x ² are determined.     

Embedding (p,p−1) graphs in their complements

A graphG is embeddable in its complement ifG is isomorphic with a subgraph of . A complete characterization is given of those (p,p−1) graphs which are embeddable in their complements. In particular, letG be a (p,p−1) graph wherep≧6 ifp is even andp≧9 ifp is odd; thenG is embeddable in if and only ifG is neither the starK _1,p−1 norK _1,n ∪C ₃ withn≧4.     

SU(m+n)SU(m)×SU(n) ISOSCALAR FACTORS AND S(f_1+f_2)S(f_1)×S(f_2) OUTER-PRODUCT ISOSCALAR FACTORS

It is proved that the SU(m+n)SU(m)×SU(n) isoscalar factors (ISF) are equal to the S(f_1+f_2) outer-product ISF of the permutation group. Since the latter only depend on the partition labels, the values of the SU(m+n)SU(m)×SU(n) ISF do not depend on m and n explicitely. Consequently for a f(=f_1+f_2)-particle system, by evaluating the S(f) S(f_1)×S(f_2) outer-product ISF we can obtain all (an infinite number) of the SU (m+n) SU(m)×SU(n) ISF (or the f_2-particle coefficients of fractional parentage) for arbitrary m and n at a single stroke, in stead of one m and one n at a time. A simple method, the eigenfunction method, is given for evaluating the SU(m+n) SU(m)×SU(n) single particle ISF, while the many-particle ISF can be calculated in terms of the outer-product reduction coefficients and the transformation coefficients from the Yamanouchi basis to the S(f_1+f_2) S(f_1)×S(f_2) basis.     

Affine Hypersurfaces Admitting a Pointwise SO(n − 1) Symmetry

In this article we obtain a classification of strictly locally convex affine hypersurfaces in ${\mathbb{A}^{n+1}}$ for which the geometrical structure is pointwise invariant under the group SO(n ? 1) represented by rotations around a fixed axis in the tangent space. This generalises the results obtained by Lu and Scharlach (Results Math 48:275–300, 2005) for n =  3.     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.