Discrete Toda lattices and the Laplace method

We apply the Laplace cascade method to systems of discrete equations of the form u _i+1,j+1 = f(u _i+1,j , u _i,j+1, u _i,j , u _i,j?1), where u _ij , i, j ∈ ?, is an element of a sequence of unknown vectors. We introduce the concept of a generalized Laplace invariant and the related property that the systems is “of the Liouville type.” We prove a series of statements about the correctness of the definition of the generalized invariant and its applicability for seeking solutions and integrals of the system. We give some examples of systems of the Liouville type.     

Three-term recurrence relations with matrix coefficients. The completely indefinite case

In the space {ol _p ² of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj ^uj+1+Aj ^uj+ _B ^j−1/* uj−1, whereu−1 = 0,u ₀,u ₁, … ∈ ℂ ^p ,A _j andB _j arep × p matrices with entries from ℂ,A _j ^* =A_j, and the inversesB _j ⁻¹ (j = 0, 1, …) exist. We state a necessary and sufficient condition for the deficiency numbers of the operatorL to be maximal; this corresponds to the completely indefinite case for the expressionl. Tests for incomplete indefiniteness and complete indefiniteness forl in terms of the coefficientsA _j andB _j are derived. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 709–716, May, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00333.     

Solvability of second-order left-invariant differential operators on the Heisenberg group satisfying a cone condition

We discuss local solvability of operators of the form where theV _j are left-invariant vector fields on the Heisenberg group such that [V _j ,V _j+n ]=U for 1≤j≤n andA=(a _jk )=A ₁+iA ₂ is a complex symmetric matrix satisfying the “cone condition” |A ₂|≤CA ₁. The authors acknowledge the support for this work by the European Commission through the European HCM-program “Fourier Analysis” and the TMR network “Harmonic Analysis”.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.     

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L ²(ℝ) function such that {ψ jk(x)} = {2^j/2 ψ(2 ^j x −k), j, k ∈ ℤ},is a tight frame for L ² (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Evolution linear equations of the Sobolev type on a graph

We analyze the stability and solvability of the Cauchy problem for the equations λu _jt − u _jtxx = βu _jxx − αu _jxxxx +γu _j + f _j , which appear in filtration theory and are defined on a finite connected directed graph with continuity and flow balance conditions at its vertices.     

Sets of Double and Triple Weights of Trees

Let T be a weighted tree with n leaves numbered by the set {1, . . . , n}. Let D _{i, j} (T) be the distance between the leaves i and j. Let D_i,j,k(T) = \frac12(D_i,j(T)+D_j,k(T)+D_i,k(T)){{D_{i,j,k}(T) = \frac{1}{2}(D_{i,j}(T)+D_{j,k}(T)+D_{i,k}(T))}} . We will call such numbers “triple weights” of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a n-set to be double weights of a tree. By using the same ideas, we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves. Besides we propose a slight modification of Saitou-Nei’s Neighbour-Joining algorithm to reconstruct trees from the data D _{i, j} .     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

A class of problems for which cyclic relaxation converges linearly

The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed. We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form for a _i,j ,b _i ,c _i ∈ℝ_≥0 with max {min {b ₁,b ₂,…,b _n },min {c ₁,c ₂,…,c _n }}>0 over the n-dimensional interval [l ₁,u ₁]×[l ₂,u ₂]×⋅⋅⋅×[l _n ,u _n ] with 0<l _i <u _i for 1≤i≤n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.     

Singularly perturbed elliptic boundary value problems

Summary The paper treats elliptic operators of the form L(ɛ∂₁, ..., ɛ∂_n), where L is a polynomial in a variables of order 2m₁, and ɛ is a small parameter. Solutionsu _ɛ of Lu=0 in a half space satisfyng conditions B_j(ɛ∂₁, ɛ∂₂, ..., ɛ∂_n)u=ɛγ_jϕ_j(x)(j=1, ..., m₁) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties of the interior limit u₀=u _ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables coefficients. Supported in part by N. S. F. Grant GP-11660. Entrata in Redazione il 9 gennaio 1971.     

On the Hamilton-Waterloo Problem

 The Hamilton-Waterloo problem asks for a 2-factorisation of K _v in which r of the 2-factors consist of cycles of lengths a ₁,a ₂,…,a _t and the remaining s 2-factors consist of cycles of lengths b ₁,b ₂,…,b _u (where necessarily ∑ _i=1 ^t a _i =∑ _j=1 ^u b _j =v). In this paper we consider the Hamilton-Waterloo problem in the case a _i =m, 1≤i≤t and b _j =n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}. Received: July 5, 2000     

Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

Iterated forcing and normal ideals onω 1

We prove that suitable iteration does not collapse ℵ₁ [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω ₁ is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω ₁, {ie345-1}(ω ₁)/(D ω ₁ +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ₀,<ℵ₂) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration). This research was partially supported by the NSF. This paper was largely written during the author’s visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality.     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

Weak Cartan-type second main theorem for holomorphic curves

In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, there exists a positive integer M such that ｜｜（q - （n ＋ 1） ε）Tf（r）≤∑j^q=1 1/dj Nf^M（r,Dj）＋o（Tf（r））, where ＂｜｜＂ means the estimate holds for all large r outside a set of finite Lebesgue measure.