On Perfect Subsemigroups of Q+

It has long been known that in order for a subsemigroup of Q ₊, properly containing {0}, to be perfect, it is necessary that it be non-C-finite. We define quasi-C-finite semigroups in such a way that it is necessary that the semigroup be non-quasi-C-finite. Every C-finite subsemigroup of Q ₊ is quasi-C- finite, but there is a quasi-C-finite subsemigroup of Q ₊ which is not C-finite. This revised version was published online in June 2006 with corrections to the Cover Date.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

弱P—正则半群

在本文中,我们刻画弱P－正则半群上的最大幂等分离同余,并且证明了有超C－集的弱P－正则半群S[P]是拟P－正则当且仅当它同构于WB（p)[P^*]和某个弱正则^*-半群T的织积,其中B是S[P]的半带,最后,我们获得,在一般情况下,WB(p)[P]的弱P－正则半群但不是P－正则半群的条件。     

具有正则*-断面的正则半群的结构

设S是一个正则半群,如果存在一个S的子半群S~*及上的一元运算*满足条件：(1)(?)x∈S,x~*∈S~*∩V(x)；(2)(?)x∈S~*,(x~*)~*=x；(3)(?)x,y∈S,(x~*y)~*=y~*x~(**),(xy~*)~*=y~(xx)x~*则称S~*是S的一个正则*_-断面．本文刻画了具有正则*_-断面的正则半群的结构。     

On the Stieltjes moment problem on semigroups

We characterize finitely generated abelian semigroups such that every completely positive definite function (a function all of whose shifts are positive definite) is an integral of nonnegative miltiplicative real-valued functions (called nonnegative characters).     

具有拟理想正则*-断面的正则半群

本文提出了具有正则*-断面正则半群的概念，所给出的例子表明具有拟理想正则*-断面的正则半群类真包含了具有拟理想逆断面的正则半群类和正则*-半群类；最后刻画了具有拟理想正则*-断面的正则半群的结构．     

Natural solutions of rational Stieltjes moment problems

In the strong or two-point Stieltjes moment problem, one has to find a positive measure on [0,∞) for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions μ₀ and μ_∞ exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points.     

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.     

Tauberian Theorem for the Stieltjes Transform

Under weak constraints on the positive functions to be compared, we derive their asymptotic equivalence at infinity as a consequence of the asymptotic equivalence of their Stieltjes transforms at infinity.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

An Example of a Positive Semidefinite Double Sequence Which is Not a Moment Sequence

The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as (m, n) .     

Stieltjes polynomials and Gauss-Kronrod quadrature formulae for measures induced by Chebyshev polynomials

Given a fixedn1, and a (monic) orthogonal polynomial _n (·)= _n (·;d) relative to a positive measured on the interval [a, b], one can define the nonnegative measure , to which correspond the (monic) orthogonal polynomials . The coefficients in the three-term recurrence relation for , whend is a Chebyshev measure of any of the four kinds, were obtained analytically in closed form by Gautschi and Li. Here, we give explicit formulae for the Stieltjes polynomials whend is any of the four Chebyshev measures. In addition, we show that the corresponding Gauss-Kronrod quadrature formulae for each of these , based on the zeros of and , have all the desirable properties of the interlacing of nodes, their inclusion in [–1, 1], and the positivity of all quadrature weights. Exceptions occur only for the Chebyshev measured of the third or fourth kind andn even, in which case the inclusion property fails. The precise degree of exactness for each of these formulae is also determined.     

A note on the two-sided two-dimensional moment problem

It is shown that there is a positive semidefinite two-sided two-dimensional sequence f, which is not a moment sequence, such that log f(2m,2n) =O(m ² + n ²) as (m,n) →∞. This revised version was published online in June 2006 with corrections to the Cover Date.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)⁻¹dμ(t)<∞. In this paper we study the order of the uniform approximation of the function

Congruences on *-Regular Semigroups

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup. This revised version was published online in June 2006 with corrections to the Cover Date.     

基半群为0-J-严格单半群的零直并的代数

设Ａ是代数闭域ｋ上的一个具乘基Ｂ的有限维含幺结合代数，称半群Ｂ∪｛０｝为Ａ的基半群．本文给出了０ Ｊ 严格单半群的定义．对于基半群为０ Ｊ 严格单半群的零直并的代数，完全研究了它的代数表示型     

关于实方阵的正定性

本文研究一般实方阵的正定性 ,给出了方阵正定的一些充分必要条件