On free semilattice-ordered semigroups satisfying x n =x

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity x ⁿ =1. Let B _slo (k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity x ⁿ =x. We prove a generalization of the Green-Rees theorem: B _slo (k,1,n) is finite for all k≥1 if and only if B(k,0,n−1) is finite for all k≥1. We find a formula for card(B _slo (1,1,n)). We construct B _slo (k,1,n) for some concrete values of k and n.     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Smooth and path connected Banach submanifold Σ r of B(E,F) and a dimension formula in B(ℝ n ,ℝ m )

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ _r the set of all operators of finite rank r in B(E,F), and Σ _r ^# the number of path connected components of Σ _r . It is known that Σ _r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ _r . In this paper,the equality Σ _r ^# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ _r is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T _A Σ _r = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ _r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of Σ _r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ ⁿ and F = ℝ ^m , then Σ _r is a smooth and path connected submanifold of B(ℝ ⁿ , ℝ ^m ) and its dimension is dimΣ _r = (m+n)r−r ² for each r, 0 <- r < min {n,m}. Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).     

A braidlike presentation of Sp(n,p)

Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB _n+1. Ifs ₁, …,s _n are standard generators ofB _n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s ₁ ^p ,(s ₁ s ₂)⁶,s ₁ ^(p+1)/2 s ₂ ⁴ s ₁ ^(p−1)/2 s ₂ ⁻² s ₁ ⁻¹ s ₂ ² and (s ₁ s ₂ s ₃)⁴ A ⁻¹ s ₁ ⁻² A, whereA=s ₂ s ₃ ⁻¹ s ₂ ^(p−1)/2 s ₄ s ₃ ² s ₄, all of them lying in the subgroupB ₅. Sp(n, p) acts on a vector space and the image of the subgroupB _n ofB _n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B _k+1·B _k … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem. Partially supported by the Technion VPR-Fund.     

Bounded projectors in spaces of functions holomorphic in the unit ball

The paper studies some bounded operators in the Banach spaces L ^∞(B) and L ¹(B) over the unit ball B of ℂ ⁿ , the range of which are the corresponding holomorphic subspaces A _∞(φ) and A ¹(ϕ) depending on a normal pair of weight-functions {φ, ϕ}.     

Counting partitions on the abacus

In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case ℓ=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss. Dedicated to the memory of Dr. Manfred Schocker (1970–2006)     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

On Λ(p)-subsets of squares

This paper is a follow up of [B₁]. It is shown that the sequence of squares {n ²|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic method developed in [B₁] and a precise estimate of the Λ(p)-constant for the sequence of squares itself. Analogues of this phenomenon are obtained for other arithmetic sets, such as the sequence ofkth powers {n ^k |n=1, 2, ...} or the sequence of prime numbers. Sections 2 and 3 of the paper are of independent interest to orthogonal system theory.     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

Sharp concentration of the chromatic number on random graphsG n,p

The distribution of the chromatic number on random graphsG _{n, p} is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity arbitrarily slowly. If the average degreepn is less thann ^1/6, it concentrates almost surely in five consecutive integers. Large deviation estimates for martingales are used in the proof.     

Factorization for non-nevanlinna classes of analytic functions

A generalization of the Blaschke product is constructed. This product enables one to factor out the zeros of the members of certain non-Nevanlinna classes of functions analytic in the unit disc, so that the remaining (non-vanishing) functions still belong to the same class. This is done for the classesA ⁻ⁿ (0<n<∞) andB ⁻ⁿ (0<n<2) defined as follows:f ∈A ⁻ⁿ iff |f(z)|≦C _f (1−|z|)⁻ⁿ ,f ∈B ⁻ⁿ iff |f(z)|≦exp {C _f (1−|z|)⁻ⁿ }, whereC _f depends onf.     

Affine Near-Semirings Over Brandt Semigroups

In order to study the structure of A ⁺(B _n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A ⁺(B _n ) and reports the size of A ⁺(B _n ). Further, idempotents and regular elements of the semigroup reducts of A ⁺(B _n ) have also been characterized and studied some relevant semigroups in A ⁺(B _n ).     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.      

Weighted composition operators on growth spaces

Denote by Hol(B _n ) the space of all holomorphic functions in the unit ball B _n of ℂ ⁿ , n ≥ 1. Given g ∈ Hol(B _m ) and a holomorphic mapping φ: B _m → B _n , put C _φ ^g f = g · (f ∘ φ) for f ∈ Hol(B _n ). We characterize those g and φ for which C _φ ^g is a bounded (or compact) operator from the growth space A ^−log(B _n ) or A ^−β (B _n ), β > 0, to the weighted Bergman space A _α ^p (B _m ), 0 < p < ∞, α > −1. We obtain some generalizations of these results and study related integral operators.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

Uniformly cross intersecting families

Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff |A∩B|=ℓ for all A∈ A and B∈B. Denote by P _e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P _e (n) is Θ(2 ⁿ ), by Frankl and R?dl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with |A||B| = $ \left( {{*{20}c} {2\ell } \\ \ell \\ } \right) $ \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) 2 ^n−2ℓ = Θ(2 ⁿ /$ \sqrt \ell $ \sqrt \ell ), and conjectured that this is best possible. Consequently, Sgall asked whether or not P _e (n) decreases with ℓ.     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.