A remark on finiteness of the lower spectrum of uniformly elliptic operators with singular potentials

In this paper, we extend the result of Donig[Do] and show the finiteness of the lower spectrum of the uniformly elliptic operators–·(A(x))+q with singular potentialsq which belong to the Kato class. Even in the case of the Schrödinger operator–+q, our proof simplifies the one in [Do].     

Edge-isoperimetric inequalities in the grid

The grid graph is the graph on [k] ⁿ ={0,...,k–1} ⁿ in whichx=(x _i ) ₁ ⁿ is joined toy=(y _i ) ₁ ⁿ if for somei we have |x _i –y _i |=1 andx _j =y _j for allji. In this paper we give a lower bound for the number of edges between a subset of [k] ⁿ of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA[k] ⁿ satisfiesk ⁿ /4|A|3k ⁿ /4 then there are at leastk ^n–1 edges betweenA and its complement.Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.We also give a best possible upper bound for the number of edges spanned by a subset of [k] ⁿ of given cardinality. In particular, forr=1,...,k we show that ifA[k] ⁿ satisfies |A|r ⁿ then the subgraph of [k] ⁿ induced byA has average degree at most 2n(1–1/r).Research partially supported by NSF Grant DMS-8806097     

Two-dimensional variational vibroequilibria and Faraday’s drops

When contacting with acoustically-vibrated structures a fluid volume can take a [time-averaged] geometric shape differing from capillary equilibrium. In accordance with theorems by Beyer et al. (2001) this shape (vibroequilibrium) furnishes a local minimum of a [quasi-potential energy] functional. The variational problem contains five dimensionless parameters evaluating the fluid volume, the wave number of acoustic field in the fluid domain, the contact angle and two newly-introduced numbers (1, 2) giving relationships between (surface tension, gravitation) and Kapitsas vibrational forces/energy. The paper focuses on negligible small wave numbers (incompressible fluid) and two-dimensional flows. Although the variational problem may in some isolated cases have analytical solutions, it requires in general numerical approaches. Numerical examples simulate experiments by Wolf (1969) and Ganiyev et al. (1977) on vibroequilibria in horizontally vibrating tanks. These show that there appear at least two types of stable vibroequilibria associated with symmetric (possible non-connected) and asymmetric surface shapes. The paper represents also numerical results on flattening and vibrostabilisation of a drop hanging beneath a vibrating plate (experiments by Faraday (1831)).Received: October 17, 2002; revised: June 30, 2003     

Optimally scaled matrices,necessary and sufficient conditions

Summary We shall in this paper consider the problem of determination a row or column scaling of a matrixA, which minimizes the condition number ofA. This problem was studied by several authors. For the cases of the maximum norm and of the sum norm the scale problem was completely solved by Bauer [1] and Sluis [5]. The condition ofA subordinate to the pair of euclidean norms is the ratio /, where and are the maximal and minimal eigenvalue of (A ^H A)^1/2 respectively. The euclidean case was considered by Forsythe and Strauss [3]. Shapiro [6] proposed some approaches to a numerical solution in this case. The main result of this paper is the presentation of necessary and sufficient conditions for optimal scaling in terms of maximizing and minimizing vectors. A uniqueness proof for the solution is offered provided some normality assumption is satisfied.     

Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L^m-norm, m [1,], of non-negative solutions to the Cauchy problem _t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established.     

Graph products and monochromatic multiplicities

Arcane two-edge-colourings of complete graphs were described in [13], in which there are significantly fewer monochromaticK _r 's than in a random colouring (so disproving a conjecture of Erds [2]). Jagger, ovíek and Thomason [7] showed that the same colourings have fewer monochromaticG's than do random colourings for any graphG containingK ₄.The purpose of this note is to point out that these colourings are not as obscure as might appear. There is in fact a large, natural and easily described class of colourings of the above kind; the specific examples used in [13] and [7] fall into this class.     

Superoptimal singular values and indices of matrix functions

LetG be a matrix-valued function on the unit circle which is the sum of a continuous function and anH function. We establish an inequality between corresponding terms of the sequence of singular values of the Hankel operator {s _j (H _G )} and a sequence formed from the superoptimal singular values ofG with repetitions. The number of times each superoptimal singular value is repeated is a positive integer index which is the winding number of a related scalar function on the circle and gives information about the superoptimal error functionG-Q withQH . In the second part of the paper we establish a property of invariance of the sum of the indices corresponding to a particular superoptimal singular value. This establishes the truth of two conjectures made in [PeY].The research was supported by an NSF grant in modern analysis.     

A Remark on the Linearized Stability of Positive Solutions for Systems Involving the p-Laplacian

The aim of this paper is to prove some stability result for nonlinear elliptic systems of the form where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(|∇ z|^p-2∇ z); p > 2, Ω is a bounded domain in R^N (N > 1) with smooth boundary where with h = 1 when α = 1, λ is a positive parameter and f,g are C² functin on [0,∞) × [0,∞). We prove stability and instability results of positive stationary solutions under various choices of f and g.     

Covering the hypercube with a bounded number of disjoint snakes

We present a construction of an induced cycle in then-dimensional hypercubeI[n] (n2), and a subgroup _n ofI[n] considered as the group ₂ ⁿ , such that | _n |16 and the induced cycle uses exactly one element of every coset of _n . This proves that for anyn2 the vertices ofI[n] can be covered using at most 16 vertex-disjoint induced cycles.     

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.     

On Negative Inertia of Pick Matrices Associated with Generalized Schur Functions

It is known [6] that for every function f in the generalized Schur class and every nonempty open subset Ω of the unit disk , there exist points z₁,...,z_n ∈Ω such that the n × nPick matrix has κ negative eigenvalues. In this paper we discuss existence of an integer n₀ such that any Pick matrix based on z₁,...,z_n ∈Ω with n ≥ n₀ has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if , then such a number n₀ does not exist unless f is a ratio of two finite Blaschke products; in the latter case the minimal value of n₀ can be found. We show also that if the closure of Ω is contained in then such a number n₀ exists for every function f in .     

Functional calculus and its application to operator algebras

In this paper we will prove that any unital isometric representation ofH () with finitely connected domain has property (A ₁(r)) for somer1, which generalizes the same conclusion in [2] and [8].     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Extremal solutions for nonlinear parabolic problems with discontinuities

This paper examines nonlinear parabolic initial-boundary value problems with a discontinuous forcing term, which is locally of bounded variation. Assuming that there exist an upper solution and a lower solution , we prove the existence of a maximal and of a minimal solution within the order interval [,] L ^P (P xZ). Our approach is based on a Jordan-type decomposition for the discontinuous forcing term and on a fixed point theorem for nondecreasing maps in ordered Banach spaces.     

On the Chromatic Number of Triangle-Free Graphs of Large Minimum Degree

We prove that, for each xed real number c > 1/3, the triangle-free graphs of minimum degree at least cn (where n is the number of vertices) have bounded chromatic number. This problem was raised by Erds and Simonovits in 1973 who pointed out that there is no such result for c < 1/3.     

Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation where and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions which blows-up and concentrates as at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is      

On Non-Linear Rayleigh quotients

The stability with respect top of the non-linear eigenvalue problem div(|u| ^p–2u)+|u| ^p–2 u=0 is studied.     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

Finite clusters in high-density continuous percolation: Compression and sphericality

Summary A percolation process inR ^d is considered in which the sites are a Poisson process with intensity and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other. The distribution of the number of sites in the clusterC of the origin is examined, and related to the geometry ofC. It is shown that when andk are large, there is a characteristic radius such that conditionally on |C|=k, the convex hull ofC closely approximates a ball of radius , with high probability. When the normal volumek/ thatk points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this -ball is much greater than the ambient density . For larger normal volumes there is less compression. This can be compared to Bernoulli bond percolation on the square lattice in two dimensions, where an analog of this compression is known not to occur.Research supported by NSF grant number DMS-9006395     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.