On the convergence of finite state mean-field games through Γ-convergence

In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler–Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems.     

On the socles of fully invariant subgroups of Abelian p-groups

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult. Received: 4 September 2008     

Decay properties for the solutions of a partial differential equation with memory

Decay properties in energy norm for solutions of a class of partial differential equations with memory are studied by means of frequency domain methods. Our results are optimal for this class, as we are able to characterize polynomial as well as exponential decay rates. The results apply to models for viscoelastic materials. An extension to a semilinearly perturbed problem is also included. Received: 9 July 2008, Revised: 16 September 2008     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

Codimension one minimal cycles with coefficients inZ orZ p , and variational functionals on fibered spaces

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H_* (M, Z) (or, respectively, in H_* (M, Z_p) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem. We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε. As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired.     

Robust H∞ control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach

This paper addresses the problem of robust H_∞ control for a class of switched nonlinear cascade systems with parameter uncertainty using the multiple Lyapunov functions (MLFs) approach. Each subsystem under consideration is composed of two cascade-connected parts. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi equations, sufficient conditions for the solvability of the robust H_∞ control problem are presented. As an application, the hybrid robust H_∞ control problem for a class of uncertain non-switched nonlinear cascade systems is solved when no single continuous controller is effective. Finally, a numerical example is provided to demonstrate the feasibility of the proposed method.     

On priority queues with impatient customers

In this paper, we study three different problems where one class of customers is given priority over the other class. In the first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority queue length of the multi-server system exactly.      

AlmostP 0-matrices and the classQ

This paper demonstrates that within the class of thosen × n real matrices, each of which has a negative determinant, nonnegative proper principal minors and inverse with at least one positive entry, the class ofQ-matrices coincides with the class of regular matrices. Each of these classes of matrices plays an important role in the theory of the linear complementarity problem. Lastly, analogous results are obtained for nonsingular matrices which possess only nonpositive principal minors.     

Monotone optimal control for a class of Markov decision processes

This paper provides a unified framework to study monotone optimal control for a class of Markov decision processes through D-multimodularity. We demonstrate that each system in this class can be classified as either a substitution-type or a complement-type system according to the possible transition set, which can be used as a classification mechanism that integrates a variety of models in the literature. We develop a generic proof of the structural properties of both types of system. In particular, we show that D-multimodularity is a generally sufficient condition for monotone optimal control of different types of system in this class. With this unified theory, there is no need to pursue each problem ad hoc and the structural properties of this class of MDPs follow with ease.     

CMV matrices with asymptotically constant coefficients. Szegö-Blaschke class, scattering theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrices acting in l², whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in l²) behaves asymptotically as the basis associated with the free system. (2) From the point of view of the scattering problem, the most natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely and (b) the associated Gelfand-Levitan-Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an A₂ kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions close to the optimal ones are given directly in terms of the scattering data.     

Portfolio insurance under a risk-measure constraint

We study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor’s utility function subject to the risk-measure constraint. We give a full solution to this non-convex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists) and for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases).     

Approximation of fixed points of uniformly R-subweakly commuting mappings

We introduce a new class of uniformly R-subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically S-nonexpansive mappings in a Banach space with uniformly Gâteaux differentiable norm.     

The prescribed curvature problem in dimension four

Necessary and sufficient conditions for a g-valued differential 2-form on a 4-dimensional manifold to be, locally, a curvature form, are given. The dimension four is exceptional for the problem of prescribed curvature as, in this dimension, Bianchi's identities can be eliminated for a large class of Lie algebras, including semisimple algebras. Hence, the curvature forms are characterized as the solutions to a second-order partial differential system, which is proved to be formally integrable.     

The G Class of Functions and Its Applications

Abstract In this paper, we introduce the concept of the G class of functions of the parabolic class, and show the H?lder continuity of the G class of functions. The introduction of this concept contributes to the proof of the regularity and existence of the solution for the first boundary problem of parabolic equation in divergence form. Project supported by NNFC (79790130) and ZJPNFC(198013) and STDF of Shanghai     

Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives

We prove the uniform correctness of a Cauchy-type problem with two fractional derivatives and a bounded operator A. We propose a criterion for the uniform correctness of unbounded operator A.     

Optimization of one class of hyperbolic systems with smooth controls

In this paper we consider the optimal control problem for a system of first-order hyperbolic equations. The right-hand side of this system is determined by a control system of ordinary differential equations. Admissible controls belong to the class of smooth functions. We obtain an optimality condition and propose a general scheme of improvement methods for the mentioned problem.     

On invexity-type nonlinear programming problems

In this paper, we propose a new class of nonlinear programing, called SFJ-invex programming. The optimality characterization shows that a problem is SFJ-invex if and only if a Fritz John point together with its multiplier, is a Fritz John saddle point of the problem. Under any constraint qualification assumption, a problem is SFJ-invex if and only if a Kuhn-Tucker point together with its multiplier is a Kuhn-Tucker saddle point of the problem. Furthermore, a generalization of the SFJ-invex, class is developed; the applications to (h, )-convex programming, particularly geometric programming, and to generalized fractional programming provide a relaxation in constraint qualification for differentiable problems to get saddle-point type optimality criteria.The author wishes to thank the referee for helpful comments.     

Solving ℓ1 Regularization Problems With Piecewise Linear Losses

This article is concerned with the computational aspect of ?₁ regularization problems with a certain class of piecewise linear loss functions. The problem of computing the ?₁ regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise ?_∞ penalty. This article has supplementary material online.     

The d-precoloring problem for k-degenerate graphs

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.     

Rough blowup solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> critical NLS

We study the singularity formation for the cubic focusing L ²-critical nonlinear Schrödinger equation on \({\mathbb{R}^{2}}\) . In a series of recent works, Merle and Raphaël have completely described the so called log–log blowup regime and proven its stability in the energy space H ¹. Our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space L ². By blending the Merle, Raphaël techniques with the quantitative I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao for the study of the Cauchy problem for rough data, we obtain the stability of the log–log regime in H ^s for all s > 0.