Nested Log-Concavity

We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t ₀ ≥ 0 such that for any t ≥ t ₀ it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave.     

Simion猜想和对数凹性

以N(m,n;λ,u)表示在m×n的矩形格的左上角和右下角分别删掉分拆λ和μ的Ferrers图后从左下角到右上角格路的数目。Simion猜想对任意分拆λ,N(-k,k;λ,φ)关于k是对数凹的。本文证明了,如果序列x_0,x_1,…,x_n为对数凹的,则序列y_k=∑_(i=k)~n(a+i b+k)x_i亦为对数凹的,并给出其对Simion猜想的应用。本文还证明对所有分拆λ和μ,N(-k,k;λ,μ)关于k是对数凹的。     

Bell Numbers, Log-Concavity, and Log-Convexity

Let {b _k (n)} _n=0 ^∞ be the Bell numbers of order k. It is proved that the sequence {b _k (n)/n!} _n=0 ^∞ is log-concave and the sequence {b _k (n)} _n=0 ^∞ is log-convex, or equivalently, the following inequalities hold for all n?0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} _n=0 ^∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} _n=0 ^∞ is log-convex, then α(n)α(m)?α(n+m), ?n,m?0. On the other hand, if {α(n)/n!} _n=0 ^∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.     

On Infinite Cycles I

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Eulers theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.To the memory of C. St. J. A. Nash-Williams     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

On Directly Infinite Rings

A ring R with identity 1 is said to be directly finite if for any a, b R, ab = 1 implies that ba = 1; otherwise R is directly infinite. With N the set of nonnegative integers, let B be the ring of N x N matrices over the ring of integers generated by two particular matrices. Properties of directly infinite rings are explored in relation to the ring B. This is made possible by various characterizations of the ring B one of which is that it is torsion free and generated by a bicyclic subsemigroup of its multiplicative semigroup. Some ideals and all idempotents of the ring B are constructed. The concepts of directly finite and chain finite idempotents are introduced in an arbitrary ring and applied to the ring B.     

On Infinite Cycles II

We adapt the cycle space of a finite or locally finite graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S¹ in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts.To the memory of C. St. J. A. Nash-Williams     

Symmetry and Log-Concavity Results for Statistics on Fibonacci Tableaux

In this paper, we study the properties of the inversion statistic and the Fibonacci major index, Fibmaj, as defined on standard Fibonacci tableaux. We prove that these two statistics are symmetric and log-concave over all standard Fibonacci tableaux of a given shape μ and provide two combinatorial proofs of the symmetry result, one a direct bijection on the set of tableaux and the other utilizing 0, 1-fillings of a staircase shape. We conjecture that the inversion and Fibmaj statistics are log-concave over all standard Fibonacci tableaux of a given size n. In addition, we show a well-known bijection between standard Fibonacci tableaux of size n and involutions in S _n which takes the Fibmaj statistic to a new statistic called the submajor index on involutions.     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.     

Multiple and Polynomial Recurrence for Abelian Actions in Infinite Measure

The (C,F)-construction from a previous paper of the first authoris applied to produce a number of funny rank one infinite measurepreserving actions of discrete countable Abelian groups G with‘unusual’ multiple recurrence properties. In particular,the following are constructed for each p N{}:

1. a p-recurrent actionT=(T_g)_gG such that (if p) no one transformationT_g is (p+1)-recurrentfor every element g of infinite order;
2. an action T=(T_g)_gGsuch that for every finite sequence g1,...,g_rGwithout torsionthe transformation T_g1x...x T_gr is ergodic,p-recurrent but(if p) not (p+1)-recurrent;
3. a p-polynomially recurrent (C,F)-transformationwhich (if p)is not (p+1)-recurrent.

-recurrence here meansmultiple recurrence. Moreover, it is shown that there existsa (C,F)-transformation which is rigid (and hence multiply recurrent)but not polynomially recurrent. Nevertheless, the subset ofpolynomially recurrent transformations is generic in the groupof infinite measure preserving transformations endowed withthe weak topology.     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

On Commutativity and Centrality in Infinite Rings

We show that if R is an infinite ring such that XY ∩ YX ≠ ? for all infinite subsets X and Y, then R is commutative. We also prove that in an infinite ring R, an element a ∈ R is central if and only if aX ∩ Xa ≠ ? for all infinite subsets X.     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

On Infinite Disjoint Congruence Covering Systems

In this paper we show that for any α∈(0,l] there exists an infinite disjoint covering system {a1(modni)}i=1∞ such that     

On Quasi-invariance of Infinite Product Measures

Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard Borel space. A characterization of l ²-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat. 36:1107–1112, 1965) and by Hora (Math. Z. 206:169–192, 1991; J. Theor. Probab. 5:71–100, 1992) to all connected Lie groups.      

无穷级数与无穷乘积

通过构造新的级数以研究原来级数通项的极限性质,从而得到其敛散性.该方法在精细判别和无穷乘积研究有重要有用.     

无穷时滞的泛函微分方程的多重正周期解(英文)

In this paper, we investigate the existence of multiple positive periodic solutions for functional differential equations with infinite delay by applying the Krasnoselskii fixed point theorem for cone map and the Leggett-Williams fixed point theorem.     

On Infinite Galois Theory for Division Rings

本文分二部分，第一部分把作者所建立的线性变换完全环之间的有限结构定理扩展到无限的情形。第二部分应用此扩展了的结构定理研究除环上的无限Galois理论。我们的理论包含通常除环上的有限Galois理论。     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.