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.     

Lipschitz functions,Schatten ideals,and unbounded derivations

It is proved that, for any Lipschitz function f(t ₁, ..., t _n ) of n variables, the corresponding map f _op: (A ₁, ...,A _n ) → f(A ₁, ..., A _n ) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S ^p , p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S ^p . It is also proved that the map f _op is Fréchet differentiable in the norm of S ^p if f is continuously differentiable.     

Classi localmente filtrali

Riassunto Si dà la definizione di classe ?localmente filtrale?. Si diceche K è una classe localmente filtrale se per ogni n∈ω, per ogni A ₀,...,A _n−1, εK e per ogni famiglia di sotto-insiemi V_i di A_i (i∈n) con V_i finiti, la classe {B ₀,...,B _n−1 delle algebre generate da V₀, ..., V_n−1 è costituita da algebre finite ed è filtrale. Si dimostra che seK è localmente filtrale alloraV _L(K)=IR _L DS(K) e si dà un teorema di caratterizzazione per queste classi.

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Summary We define a ?classe localmente filtrale? as follows: LetK be a class of similar algebras;K is a ?classe localmente filtrale? if for andn ∈ ω and for anyA ₀,...,A _n−1 ink and for any family of finite subsetsV _i ofA _i(i∈n), the class {B ₀,...,B _n−1 of algebras generated byV _o, ...,V _n−1 consists of finite algebras and is ?filtrale?. We show that ifK is ?localmente filtrale? thenV _L (K)=IR _L DS(K) and we give a characterization theorem for these classes.

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Lavoro eseguito nell'ambito dei gruppi di ricerca matematici del C.N.R. per l'anno 1970–71.     

A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type with bipartite seed. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the A _n case while providing a novel interpretation for the B _n , C _n , and D _n cases.     

Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

Aggregation in a one-dimensional stochastic gas model with finite polynomial moments of particle speeds

We consider a one-dimensional system of auto-gravitating sticky particles with random initial speeds and describe the process of aggregation in terms of the largest cluster size L_n at any fixed time prior to the critical time. We study the asymptotic behavior of L_n for the warm gas, i.e., for a system of particles with nonzero initial speeds v_i(0) = u_i, where (u_i) is a family of i.i.d. random variables with mean zero, unit variance, and finite pth moment E(|u_i|^p) < ∞, p ≥ 2, and obtain sharp lower and upper bounds for L_n(t). Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 158–179.     

The Hadwiger Number of Jordan Regions Is Unbounded

We show that for every n > 0 there is a planar topological disk A₀ and n translates A₁, A₂, ..., A_n of A₀ such that the interiors of A₀, ..., A_n are pairwise disjoint, but with each A_i touching A₀ for 1 ≤ i ≤ n.     

F-Polynomials in Quantum Cluster Algebras

F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this paper, we define and prove the existence of analogous quantum F-polynomials for quantum cluster algebras. We prove some properties of quantum F-polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A_n\mbox{A}_{n} quantum cluster algebras.     

Preprojective Cluster Variables of Acyclic Cluster Algebras

It is proved that any cluster-tilted algebra defined in the cluster category 𝒞(H) has the same representation type as the initial hereditary algebra H. For any valued quiver (Γ, Ω), an injection from the subset 𝒫?(Ω) of the cluster category 𝒞(Ω) consisting of indecomposable preprojective objects, preinjective objects, and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra 𝒜_Ω is given. The images are called “preprojective cluster variables”. It is proved that all preprojective cluster variables other than u_i have denominators u ^dim M in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ, Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x₁,…, x_n) be a cluster of the cluster algebra 𝒜_Ω, and V the cluster tilting object in 𝒞(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than x_i is x ^dim M, where M is the indecomposable Λ-module corresponding to y. This result is a generalization of the corresponding result of Caldero–Chapoton–Schiffler to the non-simply-laced case.     

Gargano,Lewinter and Malerba问题的解

§1　IntroductionLet G be a graph with vertex-set V(G) ={ v1 ,v2 ,...,vn} .A labeling of G is a bijectionL:V(G)→{ 1,2 ,...,n} ,where L (vi) is the label of a vertex vi.A labeled graph is anordered pair (G,L) consisting of a graph G and its labeling L.Definition1.An increasing nonconsecutive path in a labeled graph(G,L) is a path(u1 ,u2 ,...,uk) in G such thatL(ui) + 1相似文献   

Estimation of the extreme value and the extreme points

Summary Letf be a continuous function defined on some domainA andX ₁,X ₂, ... be iid random variables. We estimate the extreme value off onA by studying the limiting distribution of min {f(X ₁), ...,f(X _n )} or max {f(X ₁), ...,f(X _n )} properly normalized. Sufficient conditions for the existence of the limiting distribution as well as a characterization of the limiting distribution relative to the extreme points off will be provided. A discussion of the multidimensional case is also carried out. Partially supported by CNPq-No. 301508/84.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.     

Bar-invariant bases of the quantum cluster algebra of type <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>2</Subscript><Superscript>(2)</Superscript></Stack>

We construct bar-invariant ℤ[q ^±1/2]-bases of the quantum cluster algebra of the valued quiver A ₂⁽²⁾, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

Large-deviation probability and the local dimension of sets

Let X ₁ , X ₂ , ..., X_n be n independent identically distributed real random variables and S_n = Σ _n=1 ⁿ X_i. We obtain precise asymptotics forP (S_n ∈ nA) for rather arbitrary Borel sets A₁ in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (S_n ∈ γ_nA), with γ_n/n → ∞. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On the Proximal Point Algorithm

Let A be a maximal monotone operator in a real Hilbert space H and let {u _n } be the sequence in H given by the proximal point algorithm, defined by u _n =(I+c _n A)⁻¹(u _n−1−f _n ), ∀ n≥1, with u ₀=z, where c _n >0 and f _n ∈H. We show, among other things, that under suitable conditions, u _n converges weakly or strongly to a zero of A if and only if lim inf  _n→+∞|w _n |<+∞, where w _n =(∑ _k=1 ⁿ c _k )⁻¹∑ _k=1 ⁿ c _k u _k . Our results extend previous results by several authors who obtained similar results by assuming A ⁻¹(0)≠φ.