Affine straight lines in family of bounded closed convex sets

We prove that affine straight lineϕ _A,B(t)=(1−t)A+tB going through bounded closed convex setsA andB is not injective if and only ifA−B is symmetric andA−A,B−B are homotetic.     

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.     

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.     

Produits infinis de resolvantes

This paper is concerned with the convergence of the sequence χ _n =(I+λ _n A)⁻¹χ _n−1 whereA is maximal monotone and λ _n >0. Various assumptions onA and λ _n are considered.      

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^μ inR ⁿ ,n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev. As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)^μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there is an interval [A _n , 1/2n−A _n ], whereA _n →1, asn→∞, with μ-values for which the property does not hold. This interval is non-empty forn≥8.     

An extremal set theoretical characterization of some steiner systems

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( _t ⁿ )( _k ^2k-t-1 )/( _t ^2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.     

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.     

A Rank Theorem of Operators between Banach Spaces

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T ₀∈B(E, F) with a generalized inverse T ₀ ⁺∈B(F, E). This paper shows that, for every T∈B(E, F) with ‖T ₀ ⁺ (T−T ₀)‖<1, B ≡ (I + T ₀ ⁺(T−T ₀))⁻¹ T ₀ ⁺ is a generalized inverse of T if and only if (I−T ₀ ⁺ T ₀)N(T) = N(T ₀), where N(·) stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T ₀ is a semi-Fredholm operator but not in general.     

Asymptotically spirallike mappings in several complex variables

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂ ⁿ . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂ ⁿ , ℂ ⁿ ), and prove that if dt < ∞ (this integral is convergent if k ₊(A) < 2m(A)), then a mapping f ∈ S(B ⁿ ) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = e ^At , {e ^−At f(·, t)} _t≥0 is a normal family on B ⁿ and f = f(·, 0). In particular, a spirallike mapping with respect to A ∈ L(ℂ ⁿ , ℂ ⁿ ) with dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2I _n , then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings. Partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. Partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from Japan Society for the Promotion of Science, 2007. Partially supported by Romanian Ministry of Education and Research, CEEX Program, Project 2-CEx06-11-10/2006.     

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.     

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z ⁿ +t−1)²+4t(witht|z ⁿ −1), which contains the well-known fields withm=4z ⁿ +1 andm=4z ²ⁿ +4 as special cases. Project supported by the National Natural Science Foundation of China.     

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

In this paper we study pseudo-Riemannian submanifolds in ℝ _n ^+k/t satisfying the condition Δx=Ax+B, whereA is an endomorphism of ℝ _n ^+k/t andB is a constant vector in ℝ _n ^+k/t . We give a characterization theorem whenA is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphismA. Supported by a FPI Grant, DGICYT, 1990. Partially supported by a DGICYT Grant No. PS87-0115-C03-03.     

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…     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

An abstract nonlinear Volterra equation

The existence, uniqueness, regularity and dependence upon data of solutions of the abstract Volterra equation u(t)+∫ ₀ ^t a(t-s)A(u(s))ds∈f(t), t≧0 are studied in a real Banach space. The nonlinear operatorA is assumed to bem-accretive and the assumptions on the kernela do not exclude the possibility that lim _t→0+ a(t)=+∞.     

On the distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).