The Existence of （v, 4, 1） Disjoint Difference Families with v a Prime Power

A （v, k, λ） difference family （（v, k, λ）-DF in short） over an abelian group G of order v, is a collection F=（Bi｜i ∈ I} of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely A ways as a difference of two elements lying in some base blocks in F. A （v, k, λ）-DDF is a difference family with disjoint blocks. In this paper, by using Weil＇s theorem on character sum estimates, it is proved that there exists a （p^n, 4, 1）-DDF, where p = 1 （rood 12） is a prime number and n ≥1.     

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by: . Based on two independent samples it is proposed to estimate λ by , whereF _n (x) andG _n (x) are the empirical distribution functions ofF(x) andG(x) respectively and and are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of is presented and a two sample goodness-of-fit test is presented based on . Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f ₀ (x), withf ₀ (x) completely known and for testing symmetry, i.e., testingH ₀:f(x)=f(−x).     

Existence of Positive Solutions for Operator Equations and Applications to Semipositone Problems

In this paper we study the existence of positive solutions of the following operator equation in a Banach space E: where G(x, λ) = λKFx+e₀, K: E↦ E is a linear completely continuous operator, F: P↦ E is a nonlinear continuous , bounded operator, e₀∈ E, λ is a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e₀ may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently small, and that under certain sub-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in the literature.     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

Dynamics of rational maps: Lyapunov exponents,bifurcations, and capacity

 Let L(f)=∫log∥Df∥dμ _f denote the Lyapunov exponent of a rational map, f:P ¹→P ¹ . In this paper, we show that for any holomorphic family of rational maps {f _λ :λX} of degree d>1, T(f)=dd ^c L(f _λ ) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent:

Some results on distance two labelling of outerplanar graphs

Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Numeration systems,fractals and stochastic processes

A numeration system Ω is a compactification of the set of real numbers keeping the actions of addition and positive multiplication in a natural way. That is, Ω is a compact metrizable space with #Ω≥2 to which ℝ acts additively andG acts multiplicatively satisfying the distributive law, whereG is a nontrivial closed multiplicative subgroup of ℝ₊. Moreover, the additive action is minimal and uniquely ergodic with 0-topological entropy, while the multiplication by λ has |log λ|-topological entropy attained uniquely by the unique invariant probability measure under the additive action. We construct Ω as above as a colored tiling space corresponding to a weighted substitution. This framework contains especially the substitution dynamical systems and β-transformation systems with periodic expansion of 1, both of which have discreteG. It also contains systems withG=ℝ₊. We study α-homogeneous cocycles on it with respect to the addition. They are interesting from the point of view of fractal functions or sets as well as self-similar processes. We obtain the zeta-functions of Ω with respect to the multiplication.     

On some examples of duality groups

In [1, 3] it was shown: Theorem A. If G is the fundamental group of a finite graph of λ-dimensional duality groups with |G _o(e) : G _e | < ∞ and |G _τ(e) : G _e | < ∞ for every edge e of the corresponding G-tree, then G is an (λ + 1)-dimensional duality group. Here we use the methods of Brown and Geoghegan in [3] to obtain examples of duality groups under weaker conditions than those of Theorem A. Received: 5 June 2007     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Free products of finitary skew linear groups

We give two very different proofs of the following result. Let {G _γ:λɛΛ : λ ∈ Λ} be a family of finitary skew linear groups of the same characteristic p ≧ 0. Then the free product of the G _γ is isomorphic to some finitary skew linear group of characteristic p. This extends recent work of R. J. H. Minty on the skew linear case and of O. Puglisi on the finitary linear case.     

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

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 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.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ ϕ(λ,μ)dE_λTdF_μ, where E_λ, F_μ are the spectral functions of tow self-adjoint operators A, B on a Hilbert space and T is a bounded operator. In principal, the case where A has finite spectrum is studied. Nonlinear estimates of ‖f(A)T-T f(B)‖ in terms of the norm of ‖AT-TB‖ for f∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 148–173. Translated by S. V. Kislyakov.     

On precover classes

Sunto LetG andH be abstract classes of modules. The classH is said to have theG-property if to each infinite cardinal λ there exists a cardinal κ>λ such that for everyF∈H with |F|≥κ and every its submoduleK with |F/K|≤λ there exists a submoduleL ofK such thatF/L/teG and |F/L|<κ. This condition is stronger than the condition (P) requiringL≠0 instead of |F/L|<κ, which was introduced and investigated in [8]. In this note we are going to study the relations of this more general condition to the existence of precovers with respect to some classes of modules. As an application we obtain some sufficient conditions for the existence of σ-torsionfree precovers related to a given hereditary torsion theory σ for the categoryR-mod. This result is closely related to and in some sense extends that of [5]. The research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/03/0937 and also by the institutional grant MSM 113 200 007.     

Bases of admissible inference rules for table modal logics of depth 2

In the present article, we prove the theorem which states that every table modal logic λ of depth 2 over S4 has a finite basis of admissible inference rules. In addition, it is established that a finite algebra ℒ belongs to F_ω(λ)^Q iff there exist numbers n₁…, n_k such that (Lemma 5). Let F be a λ-frame of depth 2 and b a cluster of the second layer in F. We show that for any n₁,…,n_k, there exist no p-morphisms from (F_n1⊔…⊔F_nk)⁺ a local component K (b) such that, for any n, there is no p-morphism from any local component of F_n onto K (b) (Lemma 6). Translated fromAlgebra i Logika, Vol. 35, pp. 612–622, September–October, 1996.     

(<Emphasis Type="Italic">p, λ</Emphasis>)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK _λ ^p (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK _λ ^p (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK _λ ^p (A) we prove that ⊕ _i≥0 Ext _A ⁱ (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK _λ ^p (A) under certain conditions.