An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.     

The differential equation x = f ⊥ x

A theory is presented for absolutely continuous solutions of the general scalar first order autonomous o.d.e. Necessary and sufficient conditions are given for local and global existence and uniqueness, and further topics include existence-uniqueness duality, structure of the solution set and weak solutions.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

The functional equation P(f)=Q(g) in a p-adic field

Let K be a complete ultrametric algebraically closed field of characteristic π. Let P,Q be in K[x] with P′Q′ not identically 0. Consider two different functions f,g analytic or meromorphic inside a disk |x−a|<r (resp. in all K), satisfying P(f)=Q(g). By applying the Nevanlinna's values distribution Theory in characteristic π, we give sufficient conditions on the zeros of P′,Q′ to assure that both f,g are “bounded” in the disk (resp. are constant). If π≠2 and deg(P)=4, we examine the particular case when Q=λP (λ∈K) and we derive several sets of conditions characterizing the existence of two distinct functions f,g meromorphic in K such that P(f)=λP(g).     

The multiplications in GF(q)[x]⧸(a(x)) considered as linear transformations

We study in this paper the ring V(a)=GF(q)[x?(a(x)), which also can be considered as a linear space over GF(q). In this context the multiplications are considered as linear mappings of V(a) into V(a). The ideals (invariant subspaces under multiplications) are used to formulate properties of subspaces of V(a).     

The central configurations of four masses x, −x, y, −y

The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, −x, y, −y with x≠y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995-1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

Approximation of the non-stationary M(t)/M(t)/c(t)-queue using stationary queueing models: The stationary backlog-carryover approach

This paper proposes a new approach for the time-dependent analysis of stochastic and non-stationary queueing systems. The analysis of a series of stationary queueing models leads to a new approximation of time-dependent performance measures.     

The graphs G(n, k) of the Johnson Schemes are unique for n ≥ 20

Let G(n, k) denote the graph of the Johnson Scheme J(n, k), i.e., the graph whose vertices are all k-subsets of a fixed n-set, with two vertices adjacent if and only if their intersection is of size k ? 1. It is known that G(n, k) is a distance regular graph with diameter k. Much work has been devoted to the question of whether a distance regular graph with the parameters of G(n, k) must isomorphic to G(n, k). In this paper, this question is settled affirmatively for n ≥ 20. In fact the result is proved with weaker conditions.     

The parabolic map f1/e(z)=(1/e)e

We consider the exponential maps ?_λ : ? → ? defined by the formula ?_λ (z) = λe^z, λ(0,1/e]. Let J_r(?_λ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ?. Our main result is that the function λh_λ :=HD(J_r(?_λ),)), λ(0, 1/e], is continuons at the point 1/e. As a preparation for this result we deal with the map ?₁/e itself. We prove that the h₁/e-dimensional Hausdorff measure of J_r(?₁/e) is positive and finite on each horizontal strip, and that the h₁/e-dimensional packing measure of J_r(?_λ) is locally infinite at each point of J_r(?_λ). Our main technical devices are formed by the, associated with ?_λ, maps F_λ defined on some strip P of height 2π and also associated with them tonformal measures.     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

The Cn WP-Bailey chain

The purpose of this paper is to introduce the concept of C_n WP-Bailey pairs. The C_n WP-Bailey transform is obtained by applying the Cn 6φ5 summation formula. From this result, the Cn WP-Bailey lemma is deduced by making use of the Cn q-Dougall summation formula. Some applications are investigated. Finally, the case of elliptic Cn WP-Bailey pairs is discussed.     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.