Asymptotics of optimal quantizers for some scalar distributions

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.     

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

     

Fixed points and stability in neutral differential equations with variable delays

In this paper we consider a linear scalar neutral delay differential equation with variable delays and give some new conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic stability theorem with a necessary and sufficient condition is proved. The results of Burton, Raffoul, and Zhang are improved and generalized.

     

The <Emphasis Type="Italic">p</Emphasis>-Harmonic Capacity of an Asymptotically Flat 3-Manifold with Non-negative Scalar Curvature

As an inclusive \({(1,3)\ni p}\)—extension of Bray–Miao’s Theorem 1 and Corollary 1 (Invent Math 172:459–475, 2008) for p = 2, this note presents a sharp isoperimetric inequality for the p-harmonic capacity of a surface in the complete, smooth, asymptotically flat 3-manifold with non-negative scalar curvature, and then an optimal Riemannian Penrose type inequality linking the ADM/total mass and the p-harmonic capacity by means of the deficit of Willmore’s energy. Even in the Euclidean 3-space, the discovered result for \({p \not =2}\) is new and non-trivial.     

Asymptotically hyperbolic metrics on the unit ball with horizons

In this paper, we construct a family of three-dimensional asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to −6. The manifolds we construct can be arbitrarily close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we use in this paper are gluing methods which are used by Miao in (Proc Am Math Soc 132(1):217–222, 2004).     

Effect of Modeling Uncertainty on Some Routing Problems

In this paper, we consider the routing problem described in Mohanty and Cassandras (Ref. 1). As in Ref. 1, we show that the optimal Bernoulli split to minimize mean time in the system is asymptotically independent of the variance of the service time. We give simple proofs of the results in that paper. We exploit the fact that the optimal split to minimize the mean queueing time is variance independent and the special structure of the Karush–Kuhn–Tucker optimality conditions to derive the optimal solution. Apart from being very straightforward, the proofs also give insight into the reason for the existence of the variance-independent asymptotically optimal routing policy.     

Multiple solutions for resonant nonlinear periodic equations

We consider a nonlinear periodic problem driven by the scalar p-Laplacian, with an asymptotically (p?1)-linear nonlinearity. We permit resonance with respect to the second positive eigenvalue of the negative periodic scalar p-Laplacian and we assume nonuniform nonresonance with respect to the first positive eigenvalue. Using a combination of variational methods, with truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions.     

On the rigidity of Riemannian–Penrose inequality for asymptotically flat 3-manifolds with corners

Mathematische Zeitschrift - In this paper we prove a rigidity result for the equality case of the Penrose inequality on 3-dimensional asymptotically flat manifolds with nonnegative scalar curvature...     

Asymptotically periodic solution and optimal harvesting policy for Gompertz system

In this paper, the single species modelled by (asymptotically) periodic Gompertz equation is investigated. It is shown that the (asymptotically) periodic system has a unique (asymptotically) periodic solution which is globally asymptotically stable for the positive solution. When the nonautonomous Gompertz equation is subject to harvesting, we study the optimal harvesting policy for the periodic system and obtain the corresponding optimal population level and the maximum sustainable yield. Further, when the functions in the exploited Gompertz system are stably bounded functions, we study the ultimately optimal harvesting policy. By choosing the average limiting maximum sustainable yield as management objective, the corresponding optimal population level is determined.     

Recursive constructions for optimal (n,4,2)‐OOCs

In [ 3 ], a general recursive construction for optical orthogonal codes is presented, that guarantees to approach the optimum asymptotically if the original families are asymptotically optimal. A challenging problem on OOCs is to obtain optimal OOCs, in particular with λ > 1. Recently we developed an algorithmic scheme based on the maximal clique problem (MCP) to search for optimal (n, 4, 2)‐OOCs for orders up to n = 44. In this paper, we concentrate on recursive constructions for optimal (n, 4, 2)‐OOCs. While “most” of the codewords can be constructed by general recursive techniques, there remains a gap in general between this and the optimal OOC. In some cases, this gap can be closed, giving recursive constructions for optimal (n, 4, 2)‐OOCs. This is predicated on reducing a series of recursive constructions for optimal (n, 4, 2)‐OOCs to a single, finite maximal clique problem. By solving these finite MCP problems, we can extend the general recursive construction for OOCs in [ 3 ] to obtain new recursive constructions that give an optimal (n · 2^x, 4, 2)‐OOC with x ≥ 3, if there exists a CSQS(n). © 2004 Wiley Periodicals, Inc.     

Maskit combinations of Poincaré-Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S³ and S²×S¹ bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.     

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi-α-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (α=1) and memory-size constrained (α=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11] and [12].     

Geometric Transient Solutions of Autonomous Scalar Maps

Smooth autonomous scalar maps with locally asymptotically stable equilibria have families of asymptotically constant solutions which decay geometrically to the equilibria. Locally, all transients converging to the equilibria have this form.     

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action. Submitted: March 16, 2007. Accepted: June 14, 2007.     

Splitting for rare event simulation: A large deviation approach to design and analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set B$B$ before another set A$A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.     

On Square metrics of scalar flag curvature

In this paper, we study the problem whether a Finsler metric of scalar flag curvature is locally projectively flat. We consider a special class of Finsler metrics — square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that in dimension n ≥ 3, any square metric of scalar flag curvature is locally projectively flat.     

Error bounds for high-resolution quantization with Rényi-α-entropy constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ? ^d under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].     

Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space ${\mathbb{H}^n}$ . The graphs are considered as unbounded hypersurfaces of ${\mathbb{H}^{n+1}}$ which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.     

On the absence of eigenvalues of Maxwell and Lamé systems in exterior domains

The arguments showing non‐existence of eigensolutions to exterior‐boundary value problems associated with systems—such as the Maxwell and Lamé system—rely on showing that such solutions would have to have compact support and therefore—by a unique continuation property—cannot be non‐trivial. Here we will focus on the first part of the argument. For a class of second order elliptic systems it will be shown that L²‐solutions in exterior domains must have compact support. Both the asymptotically isotropic Maxwell system and the Lamé system with asymptotically decaying perturbations can be reduced to this class of elliptic systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic convergence of a simulated annealing algorithm for multiobjective optimization problems

In this paper we consider a simulated annealing algorithm for multiobjective optimization problems. With a suitable choice of the acceptance probabilities, the algorithm is shown to converge asymptotically, that is, the Markov chain that describes the algorithm converges with probability one to the Pareto optimal set.