Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

On ohtsuki’s invariants of integral homology 3-spheres

We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ _n of integral homology 3-spheres extracted from Reshetikhin-TuraevSU(2) quantum invariants. Several interesting consequences will follow from our computation of λ₂. One of them says that λ₂ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ₁ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial. The first author is supported in part by NSF and the second author is supported by an NSF Postdoctoral Fellowship.     

图和线图的谱性质

Let G be a simple connected graph with n vertices and m edges,Lo be the line graph of G and λ1(LG)≥λ2 (LG)≥...≥λm(LG) be the eigenvalues of the graph LG,.. In this paper, the range of eigenvalues of a line graph is considered. Some sharp upper bounds and sharp lower bounds of the eigenvalues of Lc. are obtained. In oarticular,it is oroved that-2cos(π/n)≤λn-1(LG)≤n-4 and λn(LG)=-2 if and only if G is bipartite.     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s _λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 14–46, April, 2006.     

On the replications of certain multiplicative designs

Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA ^tA=diag (k ₁-λ₁,…,k _n-λ_n),k _j>λ_j>0,λ₁=…=λ_e,λ_e+1=…=λ_n are obtained which extend previous results for such matrices.     

On eigenvalues of symmetric (+1, −1) matrices

To every symmetric matrixA with entries ±1, we associate a graph G(A), and ask (for two different definitions of distance) for the distance ofG(A) to the nearest complete bipartite graph (cbg). Letλ ₁(A),λ ¹ (A) be respectively the algebraically largest and least eigenvalues ofA. The Frobenius distance (see Section 4) to the nearest cbg is bounded above and below by functions ofn −λ ₁ (A), wheren=ord A. The ordinary distance (see Section 1) to the nearest cbg is shown to be bounded above and below by functions ofλ ¹ (A). A curious corollary is: there exists a functionf (independent ofn, and given by (1.1)), such that |λ _i (A) | ≦f(λ ¹(A), whereλ _i (A) is any eigenvalue ofA other thanλ _i (A). This work was supported (in part) by the U.S. Army under contract #DAHC04-C-0023.     

Embedding nearly-spanning bounded degree trees

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above. Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of New Jersey. Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

A New Gradient Method with an Optimal Stepsize Property

The gradient method for the symmetric positive definite linear system is as follows

On the Iwasawa invariants of totally real number fields

We shall show two sufficient conditions under which the Iwasawa invariants λ _k and μ _k of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK _n be the composite ofk and thel ⁿ-th cyclotomic extension of the fieldQ of rational numbers. LetC _n be the factor group of thel-class group ofK _n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ₁ be an idempotent of the group ringZ _l[Gal(K ₁/k)] defined in the below. We shall prove λ _k = μ _k =0 if there is a natural numbern such that ε₁ C _n vanishes, under additional conditions concerning ramifications inK _n/k.     

Examples of λ-invariants

In this paper, we will examine some of the implications of the results in [C1] for the Iwasawa invariants, λ _p , of the cyclotomic fields Q(ζ _n ), wherep + n. In particular, a number of examples, for various primesp, are given.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

On the number of real eigenvalues of a certain boundary-value problem for a second-order equation with fractional derivative

The asymptotics as α → 0+ of the number of real eigenvalues λ _n (α) of the problem y″(x)+λD ₀ ^α (x) = 0, 0 < x < 1, y(0) = y(1) = 0, is obtained. The minimization of real eigenvalues is carried out. It is proved that . __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 137–155, 2006.     

Proof of some conjectures on the mean-value of Titchmarsh series — III

With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σ _n ^∞ =1a_n λ _n ^−s be a generalized Dirichlet series with 1 =λ ₁ <λ ₂ < …,λ _n≤Dn, andλ _n+1 -λ _n ≥D ^{− 1} λ _n+1 ^{− a} where α>0 andD(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for . These results will be applied in other papers to appear later.     

Spectral stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Ω₁ and Ω₂ of ℝ ^N with Lipschitz boundaries and such that Ω₂ ⊂ Ω₁, and we obtain two-sided estimates for the eigenvalues λ _n,2 of the Robin Laplacian in Ω₂ via the eigenvalues λ _{n, 1} of the Robin Laplacian in Ω₁. Our estimates depend on the measure of the set difference Ω\Ω₂ and on suitably defined characteristics of vicinity of the boundaries ∂Ω₁ and ∂Ω₂, and of the functions defined on ∂Ω₁ and on ∂Ω₂ that enter the Robin boundary conditions.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ℋ_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ℋ ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋ_q,1(B _n )- module D^λ remains irreducible on restriction to ℋ_q(D _n ).     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.