Archimedean L -factors on GL ( n ) × GL ( n ) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, π _v × π _v ^′ ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π _v and _v ^′ . In this paper we show that, if ν is archimedean, and π _v and _v ^′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.     

The Chebyshev theory of a variation of L approximation

A new method of approximation is proposed which maintains almost all of the essentials of the Chebyshev theory of best uniform approximation, while also using an L-type measure of approximation.     

Best approximation in L(μ, X), II

The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L¹(μ, Y) is proximinal in L¹(μ, X) if and only if L^p(μ, Y) is proximinal in L^p(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L¹(μ, Y) is proximinal in L¹(μ, X).     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

L2 approximation with trigonometric n-nomials

We compare the degree of approximation to L²(?π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, ? ∞ < k < ∞.     

（L^p,L^q） Estimates for Multilinear Oscillatory Singular Integralswith Smooth Phases

In this paper,the authors establish the weighted (L^p,L^q) estimates for a class of multilinear oscillatory singular integrals with smooth phases.Certain endpoint estimates are also considered.     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

Weighted Uncertainty Principles in L

We determine sufficient conditions on positive weights W and V such that there exists continuous, strictly increasing functions Φ and Ψ on [0, ∞) such that Φ(0)=0=Ψ(0) and whenever f: R→R is a continuous integrable function. We also give an example that shows the optimality of our conditions.     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

An L \infty/ L 1-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L_\infty norm and in the L₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m + n log n) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(m log log n + n log n), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the log log nfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(m log log n + n log n) algorithm for finding a maximum-weight clique—which matches the best known algorithm.     

Constrained Lp-Approximation by Generalized n-Convex Functions Induced by ECT-Systems

The problem of finding a best L_p-approximation (1 ≤ p < ∞) to a function in L_p from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L₁ are investigated. Some special results for generalized monotone and convex cases are obtained.     

No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.