A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is λ _i when an external Markov process J(?) is in state i. It is assumed that molecules decay after an exponential time with mean μ ^?1. The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped up by a factor N ^α , for some α>0, whereas the arrival rates become N λ _i , for N large. The main result of this paper is a functional central limit theorem (F-CLT) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if α > 1 the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the F-CLT is the usual \(\sqrt {N}\), whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the F-CLT is N ^1?α/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(?).     

Analytic and Numerical Solutions of an <Emphasis Type="Italic">N</Emphasis> Duplicate-Server System

We consider an N duplicate-server system, where each server consists of two reconfigurable duplicated units which are subject to breakdowns. This system is studied analytically using generating-functions, and also numerically using the matrix-geometric procedure. Using the generating-function approach we obtain a recursive expression of the queue-length probability distribution for N = 1. This expression is difficult to generalize to N ? 2. The numerical method is applicable for any value of N. For any N, we also obtain the condition for stability and the availability of the system.     

Three-dimensional <Emphasis Type="Italic">N</Emphasis>=4 supersymmetry in harmonic <Emphasis Type="Italic">N</Emphasis>=3 superspace

We consider the map of three-dimensional N=4 superfields to the N=3 harmonic superspace. The left and right representations of the N=4 superconformal group are constructed on N=3 analytic superfields. These representations are convenient for describing N=4 superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the N=4 invariance in the non-Abelian N=3 Yang-Mills theory.     

Stability and absence of binding for multi-polaron systems

We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich’s 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling \(\sqrt{\alpha}\), and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.     

Generalized Wigner-Inönü contractions and Maxwell (super)algebras

We consider a class of generalized Wigner-Inönü contractions for the semidirect product of two particularly related semisimple Lie (super)algebras. A special class of such contractions provides the D = 4 Maxwell algebra and the recently introduced simple D = 4 Maxwell superalgebra. Further we present two types of D = 4 N-extended Maxwell superalgebras, the nonstandard one for any N with ½N(N?1) central charges and the standard one, for even N = 2k, with k(2k ? 1) internal symmetry generators.     

Hearing the weights of weighted projective planes

Which properties of an orbifold can we “hear,” i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes \(M := {\mathbb{C}}P^2(N_1, N_2, N_3)\) with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on M determine the weights N ₁, N ₂, and N ₃. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of N ₁, N ₂, and N ₃, we can hear whether M is endowed with an extremal Kähler metric.     

A note on a mixed routing and scheduling problem on a grid graph

We consider a particular case of the Fleet Quickest Routing Problem (FQRP) on a grid graph of m × n nodes that are placed in m levels and n columns. Starting nodes are placed at the first (bottom) level, and nodes of arrival are placed at the mth level. A feasible solution of FQRP consists in n Manhattan paths, one for each vehicle, such that capacity constraints are respected. We establish m*, i.e. the number of levels that ensures the existence of a solution to FQRP in any possible permutation of n destinations. In particular, m* is the minimum number of levels sufficient to solve any instance of FQRP involving n vehicles, when they move in the ways that the literature has until now assumed. Existing algorithms give solutions that require, for some values of n, more levels than m*. For this reason, we provide algorithm CaR, which gives a solution in a graph m* × n, as a minor contribution.     

Online and Quasi-online Colorings of Wedges and Intervals

We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors N intervals of the real line using \({\Theta }(\log N/k)\) colors such that for every point p, contained in at least k intervals, not all the intervals containing p have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors are enough to color wedges (independent of N and k). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms.     

Adjoining batch Markov arrival processes of a Markov chain

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D _k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

A priori estimates and blow-up behavior for solutions of −QNu = Veu in bounded domain in ℝN

Let Q_N be an N-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, N-Laplacian operator and the anisotropic Laplacian operator. We firstly obtain the properties of Q_N, which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of Q_Nu in bounded domain in ?^N, N ≥ 2 are established. Finally, the blow-up behavior of the only singular point is also considered.     

The smallest singular value of random rectangular matrices with no moment assumptions on entries

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N₀ depending only on β and δ with the following property: for any N,n such that N ≥ max(N₀, δ_n), any N × n random matrix A = (a_ij) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value s_n of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

The extrinsic holonomy Lie algebra of a parallel submanifold

We investigate parallel submanifolds of a Riemannian symmetric space N. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full, complete, parallel submanifolds of N, usually called “1-fullness” of M. Furthermore, for every parallel submanifold \({M\subset N}\) we consider the pullback bundle T N| _M with the linear connection induced by \({\nabla^N}\) . Then there exists a distinguished parallel subbundle \({\mathcal {O}M}\) , usually called the “second osculating bundle” of M. Given a parallel isometric immersion from a symmetric space M into N, we can describe the “extrinsic” holonomy Lie algebra of \({\mathcal {O} M}\) by means of the second fundamental form and the curvature tensor of N at some fixed point. If moreover N is simply connected and M is even a full symmetric submanifold of N, then we will calculate the “extrinsic” holonomy Lie algebra of T N| _M in an explicit form.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

Additive properties of product sets in the field $$ \mathbb{F}_{p^2 } $$

For given positive integer n and ε > 0 we consider an arbitrary nonempty subset A of a field consisting of p ² elements such that its cardinality exceeds p ^2/n?ε . We study the possibility to represent an arbitrary element of the field as a sum of at most N(n, ε) elements from the nth degree of the set A. An upper estimate for the number N(n, ε) is obtained when it is possible.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

On the failure of concentration for the ℓ∞-ball

Let (X, d) be a compact metric space and µ a Borel probability on X. For each N ≥ 1 let d_∞^N be the ?_∞-product on X^N of copies of d, and consider 1-Lipschitz functions X^N → ? for d_∞^N.     

The Fundamental Function of Spaces Generated by Interpolation Methods Associated to Polygons

We consider the K- and J-spaces generated from an N-tuple of rearrangement-invariant function spaces by using the interpolation methods associated to polygons. We compute their fundamental functions and we characterize their associate spaces. Furthermore, an application is given to interpolation of N-tuples of Marcinkiewicz spaces.