Partial decay on simple manifolds

Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnérowicz-York equation of General Relativity); Ricci curvature in a weighted Kähler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.Current support: CNRS; partial support; CEE contract GADGET # SC1-0105-C.     

Smooth approximations without critical points

In any separable Banach space containing c ₀ which admits a C ^k-smooth bump, every continuous function can be approximated by a C ^k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

Variational approach to various nonlinear problems in geometry and physics

In this survey, we will summarize the existence results of nonlinear partial differential equations which arises from geometry or physics by using variational method. We use the method to study Kazdan-Warner problem, Chern-Simons-Higgs model, Toda systems, and the prescribed Q-curvature problem in 4-dimension.      

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

Large sets of <Emphasis Type="Italic">t</Emphasis>-designs from groups

This paper addresses questions related to the existence and construction of large sets of t-(v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa’s 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of “highlight” topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them. (Communicated by Peter Horák)     

An approximation procedure for solutions to Mayer problems with cost functional not of classC

We consider here control problems in the Mayer form, with a cost functional which is continuous, but not necessarily of classC ¹. The usual necessary conditions for such problems cannot be applied, since they require that the cost functional beC ¹. We describe a convergence procedure, based upon approximations to the cost functional, that will yield an optimal trajectory to such control systems. Criteria are given which justify that this approximation procedure will yield a trajectory and control, satisfying certain prescribed conditions, which minimize the above cost functional within a specified class of such pairs. Examples are presented.     

Models and Approximations for Call Center Design

A call center is a facility for delivering telephone service, both incoming and outgoing. This paper addresses optimal staffing of call centers, modeled as M/G/n queues whose offered traffic consists of multiple customer streams, each with an individual priority, arrival rate, service distribution and grade of service (GoS) stated in terms of equilibrium tail waiting time probabilities or mean waiting times. The paper proposes a methodology for deriving the approximate minimal number of servers that suffices to guarantee the prescribed GoS of all customer streams. The methodology is based on an analytic approximation, called the Scaling-Erlang (SE) approximation, which maps the M/G/n queue to an approximating, suitably scaled M/G/1 queue, for which waiting time statistics are available via the Pollaczek-Khintchine formula in terms of Laplace transforms. The SE approximation is then generalized to M/G/n queues with multiple types of customers and non-preemptive priorities, yielding the Priority Scaling-Erlang (PSE) approximation. A simple goal-seeking search, utilizing SE/PSE approximations, is presented for the optimal staffing level, subject to GoS constraints. The efficacy of the methodology is demonstrated by comparing the number of servers estimated via the PSE approximation to their counterparts obtained by simulation. A number of case studies confirm that the SE/PSE approximations yield optimal staffing results in excellent agreement with simulation, but at a fraction of simulation time and space.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1,<Emphasis Type="Italic">α</Emphasis></Superscript> Theory for the Prescribed Mean Curvature Equation with Dirichlet Data

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. With such a solution we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C ^1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C ^1,α .     

Sparse <Emphasis Type="Italic">H</Emphasis>-Colourable Graphs of Bounded Maximum Degree

Let F be a graph of order at most k. We prove that for any integer g there is a graph G of girth at least g and of maximum degree at most 5k¹³ such that G admits a surjective homomorphism c to F, and moreover, for any F-pointed graph H with at most k vertices, and for any homomorphism h from G to H there is a unique homomorphism f from F to H such that h=fc. As a consequence, we prove that if H is a projective graph of order k, then for any finite family of prescribed mappings from a set X to V(H) (with ||=t), there is a graph G of arbitrary large girth and of maximum degree at most 5k^26mt (where m=|X|) such that and up to an automorphism of H, there are exactly t homomorphisms from G to H, each of which is an extension of an f.Supported in part by the National Science Council under grant NSC89-2115-M-110-012Final version received: June 9, 2003     

On bounded maximum width sequential confidence ellipsoids based on generalized U-statistics

For a vector of (estimable) functionals of several independent distributions, sequential confidence ellipsoids (of bounded maximum width) based on a class of generalized U-statistics are studied. A stopping rule along with a procedure for choosing the component sample sizes at each stage is developed, so that the proposed confidence ellipsoid has a confidence coefficient asymptotically (as the prescribed maximum width shrinks to zero) equal to a preassigned 1 - α (0 < α < 1), and the expected total sample size is minimized for the procedure. Asymptotic efficiency of the procedure is also studied. The case of von Mises' functionals is treated briefly at the end.     

Nonnegative inverse eigenvalue problems with partial eigendata

In this paper we consider the inverse problem of constructing an n × n real nonnegative matrix A from the prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem. To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth Newton-type method for solving its equivalent nonsmooth equation. Under some mild assumptions, the global and quadratic convergence of our method is established. We also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.     

On an inverse eigenproblem for Jacobi matrices

Recently Xu [13] proposed a new algorithm for computing a Jacobi matrix of order 2n with a given n×n leading principal submatrix and with 2n prescribed eigenvalues that satisfy certain conditions. We compare this algorithm to a scheme proposed by Boley and Golub [2], and discuss a generalization that allows the conditions on the prescribed eigenvalues to be relaxed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pairs of codes with prescribed Hamming distances and coincidences

The main problem of coding theory is to construct codes with large Hamming-distances between the code-words. In this work we describe a fast algorithm for generating pairs of q-ary codes with prescribed pairwise Hamming-distances and coincidences (for a letter s ∈ {0,1,...,q − 1}, the number of s-coincidences between codewords a and b is the number of letters s in the same positions both in a and b). The method is a generalization of a method for constructing set-systems with prescribed intersection sizes (Grolmusz (2002) Constructing set-systems with prescribed intersection sizes. J Algorithms 44:321–337), where only the case q = 2 and s = 1 was examined. As an application, we show that the modular version of the classical Delsarte-inequality does not hold for odd, non-prime power composite moduli.      

Graph decomposition with applications to subdivisions and path systems modulo k

The existence of a function α(k) (where k is a natural number) is established such that the vertex set of any graph G of minimum degree at least α(k) has a decomposition A ∪ B ∪ C such that G(A) has minimum degree at least k, each vertex of A is joined to at least k vertices of B, and no two vertices of B are separated by fewer than k vertices in G(G ∪ C). This is applied to prove the existence of subdivisions of complete bipartite graphs (complete graphs) with prescribed path lengths modulo k in graphs of sufficiently high minimum degree (chromatic number) and path systems with prescribed ends and prescribed lengths modulo k in graphs of sufficiently high connectivity.     

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

Enclosure theorems for extremals of elliptic parametric functionals

In the following paper we study parametric functionals. First we introduce a generalized mean curvature (so called F-mean curvature). This enables us to describe extremals of parametric funcionals as surfaces of prescribed F-mean curvature. Furthermore we give a differential equation for arbitrary immersions generalizing and apply this equation to surfaces of vanishing and prescribed F-mean curvature. Especially we prove non-existence results for such surfaces generalizing Theorems by Hildebrandt and Dierkes [3], [6]. Received: 11 May 2001 / Accepted: 11 July 2001 / Published online: 12 October 2001