Optimal <Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript> regularity theory for parabolic equations in divergence form

This work treats L^p regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W^1,p regularity theory holds. This work was supported by SNU foundation in 2005.     

<Emphasis Type="Italic">m</Emphasis> Components-admissible solutions of systems of higher-order partial differential equations on <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.     

A new full-Newton step <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) infeasible interior-point algorithm for semidefinite optimization

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed a primal-dual infeasible interior-point algorithm with the currently best iteration bound for linear optimization problems. Since the algorithm uses only full Newton steps, it has the advantage that no line-searches are needed. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates either by finding an ε-solution or by detecting that the primal-dual problem pair has no optimal solution (X ^*,y ^*,S ^*) with vanishing duality gap such that the eigenvalues of X ^* and S ^* do not exceed ζ. The iteration bound coincides with the currently best iteration bound for semidefinite optimization problems.     

On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened ellipticity condition and right-hand sides of the class L ¹ depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 99–112, 2006.     

Inducing <Emphasis Type="Italic">W</Emphasis>-graphs

 Let ℋ be the Hecke algebra associated with a Coxeter group W. Many interesting ℋ-modules can be described using the concept of a W-graph, as introduced in the influential paper [4] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of ℋ has an associated W-graph. The purpose of this note is to show that if W _J is a parabolic subgroup of W and V is a module for the corresponding Hecke algebra ℋ _J , then a W _J -graph structure for V gives rise to a W-graph structure for the induced module ℋ⊗ _ℋJ V. In the case that W _J is the identity subgroup and V has dimension 1, our construction coincides with that given by Kazhdan and Lusztig for the regular representation. For arbitrary J and V of dimension 1 we recover the constructions of Couillens [1] and Deodhar [3]. Received: 14 June 2002; in final form: 13 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 20C08     

Asymptotic properties of combinatorial optimization problems in <Emphasis Type="Italic">p</Emphasis>-adic space

In this article we consider two well known combinatorial optimization problems (travel-ling salesman and minimum spanning tree), when n points are randomly distributed in a unit p-adic ball of dimension d. We investigate an asymptotic behavior of their solutions at large number of n. It was earlier found that the average lengths of the optimal solutions in both problems are of order n ^1−1/d . Here we show that standard deviations of the optimal lengths are of order n ^1/2−1/d if d > 1, and prove that large number laws are valid only for special subsequences of n.     

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R~N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

Duality for <Emphasis Type="Italic">ε</Emphasis>-Variational Inequality

In this paper, following the method in the proof of the composition duality principle due to Robinson and using some basic properties of the ε-subdifferential and the conjugate function of a convex function, we establish duality results for an ε-variational inequality problem. Then, we give Fenchel duality results for the ε-optimal solution of an unconstrained convex optimization problem. Moreover, we present an example to illustrate our Fenchel duality results for the ε-optimal solutions. The authors thank the referees for valuable suggestions and comments. This work was supported by Grant No. R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

On <Emphasis Type="Italic">N</Emphasis>-termed approximations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>-norms with respect to the Haar system

In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H ^s -norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

On meromorphic solutions of <Emphasis Type="Italic">f</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">g</Emphasis><Superscript>2</Superscript> = 1

We show that meromorphic solutions f, g of f ² + g ² = 1 in C² must be constant, if f _z2 and g _z1 have the same zeros (counting multiplicities). We also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations.     

A randomized Delaunay triangulation heuristic for the Euclidean Steiner tree problem in <Emphasis Type="Italic">ℜ</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

We present a heuristic for the Euclidean Steiner tree problem in ℜ ^d for d≥2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to identify the Steiner points to remove, and second-order cone programming to optimize the location of the remaining Steiner points. Unlike other ESTP heuristics relying upon Delaunay triangulation, we insert Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. We govern this neighbor generation procedure with a local search framework that extends effectively into higher dimensions. We present computational results on benchmark test problems in ℜ ^d for 2≤d≤5.     

3-Selmer groups for curves <Emphasis Type="Italic">y</Emphasis><Superscript>2</Superscript> = <Emphasis Type="Italic">x</Emphasis><Superscript>3</Superscript> + <Emphasis Type="Italic">a</Emphasis>

We explicitly perform some steps of a 3-descent algorithm for the curves y ² = x ³ + a, a a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize L ^p estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. Submitted: May 11, 2006. Accepted: September 19, 2006.     

Harnack estimate for the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-flow

We obtain the Harnack estimate of the solution to Hk-flow in Euclidean space Rn 1,for k ＞ 0.By using this estimate,we get some corollaries about the translation soliton.     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

An <Emphasis Type="Italic">M</Emphasis><Superscript>[<Emphasis Type="Italic">X</Emphasis>]</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M ^[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically. I. Atencia’s and Moreno’s research is supported by the MEC through the project MTM2005-01248.     

On Optimality Conditions for Some Nonsmooth Optimization Problems over <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Spaces

The paper deals with the minimization of an integral functional over an L^p space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calculus of proximal subgradients. The results are specialized to nonsmooth two-stage and multistage stochastic programs.The authors express their gratitude to Boris Mordukhovich (Detroit) for his extensive support during this research and to Marian Fabian (Prague) and Alexander Kruger (Ballarat) for valuable discussions. They are indebted also to two anonymous referees for helpful suggestions.The research of this author was partly supported by Grant 1075005 of the Czech Academy of SciencesThe research of this author was supported by the Deutsche Forschungsgemeinschaft