Uncomplemented subspaces of Lorentz spaces

In the paper we construct a system of bounded functions which generates an uncomplemented subspace in the Lorentz space Λ(α) for all α∈(0,1). Lower bounds of the norms of the projector onto such subspaces are obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 57–65, July, 2000.     

Lorentz space estimates and Jacobian convergence for the Ginzburg-Landau energy with applied magnetic field

In this paper, we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in [15]. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field h _ex in certain interesting regimes of h _ex . This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration (u, A) is comparable to the L ^{2, ∞} Lorentz space norm of ∇ _A u. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz spaces. Supported by an NSF Graduate Research Fellowship.     

Global weak solutions to the Nordström-Vlasov system

The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a priori bounds on the solutions of this system, which together with energy estimates and the smoothing effect of “momentum averaging” yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.     

Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi

We study the asymptotic expansion of the first Dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached. In certain cases, which include isosceles and right triangles, we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero, plus the constant term and estimates on the remainder. For rhombi and other triangle families such as isosceles triangles where now the angle opening approaches π, we have the first two terms plus bounds on the remainder. These results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned. Apart from being accurate near the singular limits considered, our lower bounds for the rhombus improve upon the bound by Hooker and Protter for angles up to approximately 22° and in the range (31°,54°). These results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it.     

Energy Bounds for a Compressed Elastic Film on a Substrate

We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, \({\sigma }\), and a measure of the bonding strength between the film and substrate, \({\gamma }\). We prove upper bounds on the minimum energy of the form \({\sigma }^a {\gamma }^b\) and find that there are four different parameter regimes corresponding to different values of a and b and to different folding patterns of the film. In some cases, the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds.     

Energy of generalized line graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Line graphs play an important role in the study of graph theory. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. In this paper, we establish relations between the energy of the generalized line graph of a graph G and the Laplacian and signless Laplacian energies of G. We give upper and lower bounds for the energy of generalized line graphs. Finally, we present upper and lower bounds for some special graphs.     

Some remarks on quasilinear wave equations with null condition in 3‐D

In this paper, we give an alternative proof to the global existence result, which is originally owing to the pioneering work of Klainerman and Christodoulou, for the Cauchy problem of quasilinear wave equations with null condition in three space dimensions. The proof can display the following three features simultaneously: the Lorentz boost operator is not employed in the generalized energy estimates; the generalized energy of the solution will be always small, which was first observed by Alinhac; and the initial data are not assumed to have a compact support. Copyright © 2016 John Wiley & Sons, Ltd.     

Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

Instanton approximation, periodic ASD connections, and mean dimension

We study a moduli space of ASD connections over S³×R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the mean dimension by using a “Runge-approximation” for ASD connections, and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic ASD connections.     

Approximate receding horizon approach for Markov decision processes: average reward case

We consider an approximation scheme for solving Markov decision processes (MDPs) with countable state space, finite action space, and bounded rewards that uses an approximate solution of a fixed finite-horizon sub-MDP of a given infinite-horizon MDP to create a stationary policy, which we call “approximate receding horizon control.” We first analyze the performance of the approximate receding horizon control for infinite-horizon average reward under an ergodicity assumption, which also generalizes the result obtained by White (J. Oper. Res. Soc. 33 (1982) 253-259). We then study two examples of the approximate receding horizon control via lower bounds to the exact solution to the sub-MDP. The first control policy is based on a finite-horizon approximation of Howard's policy improvement of a single policy and the second policy is based on a generalization of the single policy improvement for multiple policies. Along the study, we also provide a simple alternative proof on the policy improvement for countable state space. We finally discuss practical implementations of these schemes via simulation.     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

Overlaps help: Improved bounds for group testing with interval queries

Given a finite ordered set of items and an unknown distinguished subset P of up to p positive elements, identify the items in P by asking the least number of queries of the type ‘‘does the subset Q intersect P?”, where Q is a subset of consecutive elements of {1,2,…,n}. This problem arises, e.g., in computational biology, in a particular method for determining splice sites in genes. We consider time-efficient algorithms where queries are arranged in a fixed number s of stages: In each stage, queries are performed in parallel. In a recent bioinformatics paper, we proved optimality (subject to lower-order terms) with respect to the number of queries, of some strategies for the special cases p=1 or s=2. Exploiting new ideas, we are now able to provide improved lower bounds for any p?2 and s?3 and improved upper bounds for larger s. Most notably, our new bounds converge as s grows. Our new query scheme uses overlapping query intervals within a stage, which is effective for large enough s. This contrasts with our previous results for s?2 where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small s?3. Anyway, the remaining gaps between the current upper and lower bounds for any fixed s?3 amount to small constant factors in the main term. The paper ends with a discussion of practical implications in the case that the positive elements are well separated.     

An alternative energy bound derivation for a generalized Hasegawa-Mima equation

We present an alternative derivation of the H¹-boundedness of solutions to a generalized Hasegawa-Mima equation, first investigated by Grauer (1998) [2]. We apply a Lyapunov function technique similar to the one used for constructing energy bounds for the Kuramoto-Sivashinsky equation. Different from Grauer (1998) [2], who uses this technique in a Fourier space approach, we employ the physical space construction of the Lyapunov function, as developed in Bronski and Gambill (2006) [11]. Our approach has the advantage that it is more transparent in what concerns the estimates and the dominant terms that are being retained. A key tool of the present work, which replaces the algebraic manipulations on the Fourier coefficients from the other approach, is a Hardy-Rellich type inequality.     

Lower bounds for the Chvátal-Gomory rank in the 0/1 cube

We revisit the method of Chvátal, Cook, and Hartmann to establish lower bounds on the Chvátal-Gomory rank, and develop a simpler method. We provide new families of polytopes in the 0/1 cube with high rank, and we describe a deterministic family achieving a rank of at least (1+1/e)n−1>n. Finally, we show how integrality gaps lead to lower bounds.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

Bounds on degrees of p-adic separating polynomials

We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Štefankovi? (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl-Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances.     

On the Lorentz Cone Complementarity Problems in Infinite-Dimensional Real Hilbert Space

In this article, we consider the Lorentz cone complementarity problems in infinite-dimensional real Hilbert space. We establish several results that are standard and important when dealing with complementarity problems. These include proving the same growth of the Fishcher–Burmeister merit function and the natural residual merit function, investigating property of bounded level sets under mild conditions via different merit functions, and providing global error bounds through the proposed merit functions. Such results are helpful for further designing solution methods for the Lorentz cone complementarity problems in Hilbert space.     

On the calculation of the James constant of Lorentz sequence spaces

In [M. Kato, L. Maligranda, On James and Jordan-von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457-465], the James constant of the 2-dimensional Lorentz sequence space d(2)(ω,q) is computed in the case where 2?q<∞. It is an open problem to compute it in the case where 1?q<2. In this paper, we completely determine the James constant of d(2)(ω,q) in the case where 1?q<2.     

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

We consider the non-homogeneous generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} -\nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.$$ Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to ${t \geq T_1}$ , where T ₁ depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for ${t \geq T_1}$ , and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.     

On convex quadratic programs with linear complementarity constraints

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y ^T Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50 %. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points; this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90 % of the gap can be closed for some QPCC problems.