The limit distribution for the emptying times of a single-channel queueing system with Markov arrival stream admitting consolidation of states

We study the limit behavior of the emptying times of anM _r /G/1/n single-channel queueing system under heavy load conditions. It is assumed that the arrival stream is governed by a Markov chain admitting state consolidation.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 55–61, 1986.     

The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

On Some Euler Sequence Spaces of Nonabsolute Type

In the present paper, we introduce Euler sequence spaces e ₀ ^r and e _c ^r of nonabsolute type that are BK-spaces including the spaces c ₀ and c and prove that the spaces e ₀ ^r and e _c ^r are linearly isomorphic to the spaces c ₀ and c, respectively. Furthermore, some inclusion theorems are presented. Moreover, the α-, β-, γ- and continuous duals of the spaces e ₀ ^r and e _c ^r are computed and their bases are constructed. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes and are established, and characterizations of some other classes of infinite matrices are also derived by means of a given basic lemma, where 1 ≤ p ≤ ∞.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 3–17, January, 2005.     

A system of basis invariants of the group Bn

In 1968 Flatto announced the conjecture that the polynomials _±(±x ₁±x ₂±±x _n )^2s ,s=1,n, are algebraically independent. This conjecture was confirmed by Haeuslein. A new proof of this result is given.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 54–55.     

Stable Local Bases for Multivariate Spline Spaces

We present an algorithm for constructing stable local bases for the spaces ^r_d(Δ) of multivariate polynomial splines of smoothness r1 and degree dr2ⁿ+1 on an arbitrary triangulation Δ of a bounded polyhedral domain Ω ⁿ, n2.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

Optimal Interpolating Spaces Preserving Shape

In this paper we study the existence and characterization of spaces which are images of minimal-norm projections that are required to interpolate at given functionals and satisfy additional shape-preserving requirements. We will call such spaces optimal interpolating spaces preserving shape. This investigation leads to concrete solutions in classical settings and, as examples, Π_n will be determined to be such spaces with regard to certain interpolation and shape-preserving requirements on the projections. Restated, the theory of this paper gives rist to an n-dimensional Hahn–Banach extension theorem, where the minimal-norm extension is required to keep invariant a fixed cone.     

Jung's relative constant of the space Lp

We prove that in real spaces L_p[0,1], 1p <, and _p Jung's relative constant is equal to 2^–1/r, wherer=max {p,p (p–1)^–1}. We obtain upper bounds for this quantity in finite-dimensional spaces _p ⁿ which are exact in some dimensions whenp2.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 122–125, January, 1990.     

The degree of a mapping in some problems of combinatorial geometry

The problem of dividing a unit mass that is continuously distributed inR ⁿ into various ratios by a pencil ofn+1 solid angles is connected with the question of the possibility of homotopies between certain coverings of the sphereS ^n–1. We prove that it is possible to inscribe a simplex homothetic to a given simplex in any convex solid region with a smooth boundary. In all the proofs we use the concept of the degree of special mappings onto a canonical simplex and its boundary. One figure.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 62–66.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Uniqueness of Minimal Submanifolds in Euclidean Space

Let M be a properly immersed n-dimensional complete minimal submanifold in Euclidean space R^n+p of dimension n+p. Let A be the second fundamental form of the immersion, and r the extrinsic distance from the origin. Suppose M has one end and inf_t sup_r(x)>t r²(x) |A|²(x) < C(n,p), then M is an affine n-plane, where C(n,p) are constants given by C(n,1) = n – 1 and C(n,p) = (2/3)(n – 1) when p > 1.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

On the number of edge disjoint cliques in graphs of given size

In this paper, we prove that any graph ofn vertices andt _r–1(n)+m edges, wheret _r–1(n) is the Turán number, contains (1–o(1)m edge disjointK _r'sifm=o(n ²). Furthermore, we determine the maximumm such that every graph ofn vertices andt _r–1(n)+m edges containsm edge disjointK _r's ifn is sufficiently large.Research partially supported by Hungarian National Foundation for Scientific Research Grant no. 1812.     

Permuting in place: analysis of two stopping rules

Permuting in place has been first analyzed by Knuth. It uses the cycle structure of the permutation. The elements of an array to be permuted are only moved when one sees a cycle leader (smallest element in its cycle). So the essential part of such an algorithm is to test an element i about whether it is a cycle leader.Recently, Keller [Inform. Process. Lett. 81 (2002) 119–125] introduced two stopping rules: “If the last cycle leader has been detected, all elements have been moved, and no further tests are necessary” (heuristic 1), respectively “If only r elements have not been moved, then proceeding along a cycle is only useful for r steps” (heuristic 2).We analyze the average costs of these modifications applied to the standard algorithm of Knuth; they are (n+2)H_n−5n/2−1/2nlogn and respectively ((2n+1)/4)H_(n+1)/2+(1/2)H_2(n+1)/2−(1/2)((n+1)/2−n/2)−(n+1)/2(n/2)logn, as opposed to (n+1)H_n−2nnlogn in the classical case.     

On certain random simplices in

The exact density is given for the r-content of the simplicial convex hull of r + 1 independent points in ⁿ, each having a type II β distribution. The density is given in the form of an integral of Mellin-Barnes type, which even in the most general cases can be evaluated to give a series representation for the density. Some special cases are evaluated to observe the types of series that can arise. It is also shown that the r-content is asymptotically normal for large values of n, a result analogous to a result conjectured by R. E. Miles (1971, Adv. in Appl. Probab., 3 353–382).     

q-Difference intertwining operators and q-conformal invariant equations

We first recall a canonical procedure for the construction of the invariant differential operators and equations for arbitrary complex or real noncompact semisimple Lie groups. Then we present the application of this procedure to the case of quantum groups. In detail is given the construction of representations of the quantum algebra U _q (sl(n)) labelled by n–1 complex numbers and acting in the spaces of functions of n(n–1)/2 noncommuting variables, which generate a q-deformed SL(4) flag manifold. The conditions for reducibility of these representations and the procedure for the construction of the q-difference intertwining operators are given. Using these results for the case n=4 we propose infinite hierarchies of q-difference equations which are q-conformal invariant. The lowest member of one of these hierarchies are new q-Maxwell equations. We propose also new q-Minkowski spacetime which is part of a q-deformed SU(2,2) flag manifold.     

Rational Approximation with Varying Weights III

Approximation by weighted rationals of the form wⁿr_n, where r_n=p_n/q_n, p_n and q_n are polynomials of degree at most [αn] and [βn], respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given α0, β0, with α+β>0. Conditions that characterize the largest sets on which such approximation is possible are given. We apply the general theorems to Laguerre and Freud weights.     

Triangular Dissections, Aperiodic Tilings, and Jones Algebras

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type A_n determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of π/(n + 1). There are usually several possible infinite dissections compatible with a given n but a given one makes use of n/2 triangle types if n is even. Jones algebra with index [4 cos²(π/(n + 1))]⁻¹ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case n = 4, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The "tilings" associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints), using n/2 digits (if- n is even) and generalizing the Fibonacci numbers.     

Embeddings on a boolean cube

In this paper, we characterize a class of graphs which can be embedded on a boolean cube. Some of the graphs in this class are identified with the well known graphs such asmulti-dimensional mesh of trees, tree of meshes, etc. We suggest (i) an embedding of anr-dimensional mesh of trees ofn ^r (r+1)–rn ^r–1 nodes on a boolean cube of (2n) ^r nodes, and (ii) an embedding of a tree of meshes with 2n ² logn+n ² nodes on a boolean cube withn ² exp₂ (log (2 logn+1)]) nodes.