D3-Modules

A right R-module M is called a D3-module, if M ₁ and M ₂ are direct summands of M with M = M ₁ + M ₂, then M ₁ ∩ M ₂ is a direct summand of M. Following the work of Bass on projective covers, we introduce the notion of D3-covers and provide new characterizations of several well-known classes of rings in terms of D3-modules and D3-covers.     

Quasi-Armendariz Rings Relative to a Monoid

A ring R is called an M-quasi-Armendariz ring (a quasi-Armendariz ring relative to a monoid M) if whenever elements α = a ₁ g ₁ + a ₂ g ₂ + ··· + a _n g _n , β = b ₁ h ₁ + b ₂ h ₂ + ··· + b _m h _m  ? R[M] satisfy α R[M]β = 0, then a _i Rb _j  = 0 for each i, j. After discussing some basic properties of M-quasi-Armendariz rings, we consider the influence of transformation of the monoid M and the ring R on this property. Particularly, we give some sufficient conditions for the monoids M, N, and the ring R under which R is M × N-quasi-Armendariz if and only if R is M-quasi-Armendariz and N-quasi-Armendariz.     

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? S_E^×{f\in S_{E^{\times}}} supporting μ(x), or s(x ^*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.     

An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator

A good robust functional should, if possible, be efficient at the model, smooth, and have a high breakdown point. M-estimators can be made efficient and Fréchet differentiable by choosing appropriate ψ-functions but they have a breakdown point of at most 1/(p + 1) in p dimensions. On the other hand, the local smoothness of known high breakdown functionals has not been investigated. It is known that Rousseeuw's minimum volume ellipsoid estimator is not differentiable and that S-estimators based on smooth functions force a trade-off between efficiency and breakdown point. However, by using a two-step M-estimator based on the minimum volume ellipsoid we show that it is possible to obtain a highly efficient, Fréchet differentiable estimator whilst still retaining the breakdown point. This result is extended to smooth S-estimators.     

Exact emulation of a priority queue with a switch and delay lines

All-optical packet switched networking is hampered by the problem of realizing viable queues for optical packets. Packets can be buffered in delay lines, but delay lines do not functionally emulate queues from an input-output point of view. In this paper we consider the problem of exact emulation of a priority queue of size K using a switching system comprised of a switch of size (M + 1) × (M + 1), which has one distinguished input for external arrivals, one distinguished output for external departures, and fixed-length delay lines of lengths L₁, L₂, ..., L_M connecting the other inputs and outputs in pairs. We measure the complexity of such an emulation by M + 1. We prove that and present a construction which works with ; further, in our construction . We also sketch an idea for an all-optical packet switched communication network architecture based on approximate emulation of priority queues of finite size using switches and delay lines, with erasure control coding at the packet level. AMS 2000 subject classifications: Primary: 60K25; Secondary: 90B22 · 90B36 · 68R99 The work of A. D. Sarwate is supported by an NDSEG Graduate Research Fellowship which is sponsored by the U.S. Department of Defense. The work of V. Anantharam is supported by ONR grant No. N00014-1-0637, DARPA grant No. N66001-00-C-8062, and by NSF grant No. ECS 0123512.     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

Asymptotic properties of the algebraic moment range process

Let M _n denote the n-th moment space of the set of all probability measures on the interval [0, 1], P _n the uniform distribution on the set M _n and r _{n + 1} the maximal range of the (n + 1)-th moments corresponding to a random moment point C _n with distribution P _n on M _n . We study several asymptotic properties of the stochastic process (r _⌊nt⌋+1) _t∈[0,T] if n → ∞. In particular weak convergence to a Gaussian process and a large deviation principle are established.      

Some results on (k, μ)′-almost Kenmotsu manifolds

In this paper, we study the Weyl conformal curvature tensor 𝒲 and the concircular curvature tensor 𝒞 on a (k, μ)′-almost Kenmotsu manifold M²ⁿ⁺¹ of dimension greater than 3. We obtain that if M²ⁿ⁺¹ satisfies either R · 𝒲 = 0 or 𝒞 · 𝒞 = 0, then it is locally isometric to either the hyperbolic space ?²ⁿ⁺¹ (?1) or the Riemannian product ?ⁿ⁺¹(?4) × ?ⁿ.     

On the Conditions of Extending Mean Curvature Flow

The authors consider a family of smooth immersions F(·,t):Mn → Rn+1 of closed hypersurfaces in Rn+1 moving by the mean curvature flow F∈(pt,t)=-H(p,t)·ν(p,t) for t ∈ [0,T).They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded,then the flow can extend past time T.The result is similar to that in [6-9].     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

Testing the diameter of graphs

We propose a general model for testing graph properties, which extends and simplifies the bounded degree model of Goldreich and Ron [Property Testing in Bounded Degree Graphs, Proc. 31st Annual ACM Symposium on the Theory of Computing, 1997, pp. 406–415.] In this model, we present a family of algorithms that test whether the diameter of a graph is bounded by a given parameter D, or is ?‐far from any graph with diameter at most β(D). The function β(D) ranges between D+4 and 4D+2, depending on the algorithm. All our algorithms run in time polynomial in 1/?. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 165–183, 2002     

Heat flow on Finsler manifolds

This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : T_xM → ?₊ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold:

• as gradient flow on L²(M, m) for the energy
• as gradient flow on the reverse L²‐Wasserstein space ??₂(M) of probability measures on M for the relative entropy

Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove ??^{1, α} regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be ??². Moreover, we derive pointwise comparison results à la Cheeger‐Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc.     

Analysis of the M X /G/1 Queue Under D-Policy

Abstract

We study the queue length of the M ^X /G/1 queue under D-policy. We derive the queue length PGF at an arbitrary point of time. Then, we derive the mean queue length. As special cases, M/G/1, M ^X /M/1, and M/M/1 queue under D-policy are investigated. Finally, the effects of employing D-policy are discussed.     

Exponential bounds for queues with Markovian arrivals

Exponential bounds [queueb]e ^b are found for queues whose increments are described by Markov Additive Processes. This is done by application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor .     

A unified approach to the large deviations for small perturbations of random evolution equations

LetX ^ɛ = {X ^ɛ (t ; 0 ⩽t ⩽ 1 } (ɛ > 0) be the processes governed by the following stochastic differential equations:

Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M, J) of real dimension four. We construct a local peak J-plurisubharmonic function at every point p ∈ bD of finite D’Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D’Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Fast simulation of buffer overflows in tandem networks ofGI/GI/1 queues

Simply because of their rarity, the estimation of the statistics of buffer overflows in well-dimensioned queueing networks via direct simulation is extremely costly. One technique that can be used to reduce this cost is importance sampling, and it has been shown previously that large deviations theory can be used in conjunction with importance sampling to minimize the required simulation time. In this paper, we obtain results on the fast simulation of tandem networks of queues, and derive an analytic solution to the problem of finding an optimal simulation system for a class of tandem networks ofGI/GI/1 queues.Work supported by Australian Telecommunications and Electronics Research Board (ATERB). The authors wish to acknowledge the funding of the activities of the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centres Program.