On the Ranked Excursion Heights of a Kiefer Process

Let (K(s,t), 0s1, t1) be a Kiefer process, i.e., a continuous two-parameter centered Gaussian process indexed by [0,1]×₊ whose covariance function is given by (K(s₁,t₁) K(s₂,t₂))=(s₁s₂-s₁s₂)t₁t₂, 0s₁, s₂1, t₁, t₂ 0. For each t>0, the process K(·,t) is a Brownian bridge on the scale of . Let M ₁ ^* (t) M ₂ ^* (t) M _j ^* (t) 0 be the ranked excursion heights of K(,t). In this paper, we study the path properties of the process tM _j ^* (t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M _j ^* (t) as t goes to infinity.     

Large deviations for a simple closed queueing model

We consider a simple queueing model with one service station. The arrival and service processes have intensitiesa(N–Q _t) andNf(N ^–1 Q _t), where Q_t is the queue length,N is a large integer,a>0 andf(x) is a positive continuous function. We establish the large deviation principle for the sequence of the normalized queue length processq _N ^t =N ^–1Q_t,N1 for both light (a<f(0)) and heavy (af(0)) traffic and use this result for an investigation of ergodic properties ofq _N ^t ,N 1.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Degree-Continuous Graphs

A graph G is degree-continuous if the degrees of every two adjacent vertices of G differ by at most 1. A finite nonempty set S of integers is convex if k S for every integer k with min(S)kmax(S). It is shown that for all integers r > 0 and s 0 and a convex set S with min(S) = r and max(S) = r+s, there exists a connected degree-continuous graph G with the degree set S and diameter 2s+2. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph G and convex set S of positive integers containing the integer 2, there exists a connected degree-continuous graph H with the degree set S and containing G as an induced subgraph if and only if max(S)(G) and G contains no r-regular component where r = max(S).     

A geometric condition for an l.c.K. manifold to be Kähler

We prove that a necessary and sufficient condition for a locally conformal Kähler manifold(M ⁿ , g), n 2, to be Kähler is thatM admits through every one of its points a minimal Kahler submanifold of complex dimensionp2.     

On the postulation of a general projection of a curve inP 3

Summary We say that a curve C in P ³ has maximal rank if for every integer k the restriction map r_c(k):H ⁰(P ³, O_P3(k)) H⁰ (C, O_C(k))has maximal rank. Here we prove the following results. Theorem 1Fix integers g, d with 0g3,dg+3.Fix a curve X of genus g and L Pic^d (X).If g=3and X is hyperelliptic, assume d8. Let _L(X)be the image of X by the complete linear system H ⁰(X, L). Then a general projection of _L(X)into P ³ has maximal rank. Theorem 2For every integer g0,there exists an integer d(g, 3)such that for every dd(g, 3),for every smooth curve X of genus g and every LPic^d (X) the general projection of _L(X)into P ³ has maximal rank.     

Two theorems concerning variational systems of smooth dynamical systems

A dynamical system given by a vector field of class C² in an n-dimensional, smooth, closed manifold Vⁿ let us call differentially homogeneous if for every v, w Vⁿ there exists a diffeomorphism of Vⁿ into itself such that it takes v into w and commutes with respect to motion along a trajectory for any time t. It can be shown that all of the variational systems of such a system are almost reducible.Furthermore, the dynamical systems given by the vector fieldsf(v) are considered to be ergodic in that they have the same integral invariant (nearly all of the variational systems of such a system have the same indices ₁(f)₂(f)... _n(f). It is proven that is an upper semicontinuous function off(v) when k=1, 2, ..., n.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 49–54, January, 1969.     

On the postulation of a general projection of a curve inP N,N⩾4

Summary Fix a curve X of genus g and L Pic ^d (X). Let _L(X) be the image of X through the complete linear system H⁰(X, L). Here we prove that a general projection of _L(X) intoP ^N has maximal rank if either (a) N4, 0gN–1, dg+N, or (b) dd (g, N) for suitable d(g, N).     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

On the homogeneous ideal of projectively normal curves

Summary Fix integers k, d, g with g0, dg+3, k>0, 2k<(d–g), d(g(k+1)/k) + k+1. Here we prove that for a general curve X of genus g and a general L Pic^d(X), L is normally presented.     

The M/G/1 processor-sharing model: transient behavior

This paper deals with the M/G/1 model with processor-sharing service discipline. LetL ^* (t, x) denote the number of jobs present at timet whose attained service time is not greater thanx,x0, andV ₀(t,z) the sojourn time of a tagged job placed in the system at timet and requiringz units of service. Explicit analytical expressions are obtained for the joint distribution ofL ^*(t, ·) andV ⁰(t, ·) under various initial conditions in terms of the Laplace transform with respect tot. It is shown that for initial conditions of special kind (there is one job or none) the results can be expressed in a closed form.     

An approximate analysis for a class of assembly-like queues

Assembly-like queues model assembly operations where separate input processes deliver different types of component (customer) and the service station assembles (serves) these input requests only when the correct mix of components (customers) is present at the input. In this work, we develop an effective approximate analytical solution for an assembly-like queueing system withN(N 2) classes of customers formingN independent Poisson arrival streams with rates {_i=1,...,N} The arrival of a class of customers is turned off whenever the number of customers of that class in the system exceeds the number for any of the other classes by a certain amount. The approximation is based on the decomposition of the originalN input stream stage into a cascade ofN-1 two-input stream stages. This allows one to refer to the theory of paired customer systems as a foundation of the analysis, and makes the problem computationally tractable. Performance measures such as server utilization, throughput, average delays, etc., can then be easily computed. For illustrative purposes, the theory and techniques presented are applied to the approximate analysis of a system withN = 3. Numerical examples show that the approximation is very accurate over a wide range of parameters of interest.     

Simplex pivots on the set packing polytope

This paper considers the set packing problem max{wx: Ax b, x 0 and integral}, whereA is anm × n 0–1 matrix,w is a 1 ×n weight vector of real numbers andb is anm × 1 vector of ones. In equality form, its linear programming relaxation is max{wx: (x, y) P(A)} whereP(A) = {(x, y):Ax +I _m y =b, x0,y0}. Letx ¹ be any feasible solution to the set packing problem that is not optimal and lety ¹ =b – Ax ¹; then (x ¹,y ¹) is an integral extreme point ofP(A). We show that there exists a sequence of simplex pivots from (x ¹,y ¹) to (x*,y*), wherex* is an optimal solution to the set packing problem andy* =b – Ax*, that satisfies the following properties. Each pivot column has positive reduced weight and each pivot element equals plus one. The number of pivots equals the number of components ofx* that are nonbasic in (x ¹,y ¹).This research was supported by NSF Grants ECS-8005360 and ECS-8307473 to Cornell University.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Remarks on some existence theorems for optimal control

First, a remark is made that a growth condition contained in previous papers by Cesari concerning existence theorems for optimal controls can be replaced by a slightly more general condition. In this more general condition, a constantM 0 is replaced by any functionM (t)0 which is assumed to beL-integrable in every finite interval.Then, the remark is made that the same condition, which is usually required to be satisfied by the functionsf ₀(t, x, u),f(t, x, u) characterizing the control, can be required to be satisfied only by the admissible pairsx(t),u(t) of the class in which the optimum is being sought. This generalization requires a subtle argument. The new condition parallels now the usual conditions of the type |x| ^p dtM, which are required to be satisfied by the admissible pairs of the class .The work of L. Cesari was partially supported by the Office of Scientific Research, United States Air Force, Grant No. 942-65. The work of T. Nishiura was partially supported by the National Science Foundation, Grant No. GP-7676.     

An inequality linking packing and covering densities of plane convex bodies

For every convex body K in R ², let (K) denote the packing density of K, i.e. the density of the tightest packing of congruent copies of K in R ², and let (K) denote the covering density of K, i.e. the density of the thinnest covering of R ² with congruent copies of K. It is shown here that 4(K)3(K) for every convex body K in R ². This inequality is the strongest possible, since if E is an ellipse, then the equality 4(E)=3(E) holds. Two corollaries are presented, and a summary of known bounds for packing and covering densities is given.     

On the gi/m/∞ service system with batch arrivals and different types of service distributions

We study the infinite-server system with batch arrivals ands different types of customers. With probabilityp _i an arriving customer is of typei (i=1,..., s) and requires an exponentially distributed service time with parameter _i (G ^GI /M ₁ ...M _s /). For theGI ^GI /M ₁...M _s / system it is shown that the binomial moments of thes-variate distribution of the number of type-i customers in the system at batch arrival epochs are determined by a recurrence relation and, in steady state, can be computed recursively. Furthermore, forG ^GI /M ₁...M _s /, relations between the distributions (and their binomial moments) of the system size vector at batch arrival and random epochs are given. Thus, earlier results by Takács [14], Gastwirth [9], Holman et al. [11], Brandt et al. [3] and Franken [6] are generalized.     

Vacation policies in an M/G/1 type queueing system with finite capacity

This paper deals with a queueing system with finite capacity in which the server passes from the active state to the inactive state each time a service terminates withv customers left in the system. During the active (inactive) phases, the arrival process is Poisson with parameter (₀). Denoting byu _n the duration of thenth inactive phase and byx _n the number of customers present at the end of thenth inactive phase, we assume that the bivariate random vectors {(v _n ,x _n ),n 1} are i.i.d. withx _n v+l a.s. The stationary queue length distributions immediately after a departure and at an arbitrary instant are related to the corresponding distributions in the classical model.