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.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Modulus of continuity of the matrix absolute value

Lipschitz continuity of the matrix absolute value |A| = (A*A)^1/2 is studied. Let A and B be invertible, and let M ₁ = max(‖A‖, ‖B‖), M ₂ = max(‖A ⁻¹‖, ‖B ⁻¹‖). Then it is shown that

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution*

Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function Φ _N,β on ℝ ^M . In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m ^* (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 < γ _a < γ _c such that (1) If √α≦γ _a (m ^* (β )) ² , then the absolute minima of Φ are located within small balls around the points ± m ^* e ^μ , where e ^μ denotes the μ-th unit vector while (2) if √α≦γ _c (m ^* (β )) ² at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. Received: 5 August 1995 / In revised form: 22 May 1996     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

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     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

Weak Convergence of a Nonlinear Functional of a Stationary Gaussian Process. Application to the Local Time

Let {X _t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t ^β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering |X _i/N − X _(i-1)/N | ^p with p > 0, we study the properties of the Donsker line associated with p-th order variations . We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.     

Bounded pseudo-differential operators

This lecture gives an inside look into the proof of the continuity of pseudo-differential operators of orderm and typep, δ₁, δ₂ for 0≦p≦δ₁=1, 0≦p≦δ₂<1, andm/n≦p≦(δ₁+δ₂)/2. Applications are mentioned.     

Functions with slowly-growing area and harmonic majorants

Letf be a function holomorphic inU={|z|<1}, and letA(R,f) be the area off(U)∩{|w|<R}, not counting multiplicities. IfA(R,f)=O(R ^γ) asR→∞ for a γ, 0≦γ<2, then the subharmonic function exp |f| ^p has a harmonic majorant inU for eachp, 0<p<2−γ. If 0≦γ<1 further, thene ^f is of Hardy classH ^p for eachp, 0<p<∞.     

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.     

Score Lists in Tripartite Hypertournaments

Given non-negative integers l, m, n, α, β and γ with l ≥ α ≥ 1, m ≥ β ≥ 1 and n ≥ γ ≥ 1, an [α,β,γ]-tripartite hypertournament on l + m + n vertices is a four tuple (U, V, W, E), where U, V and W are three sets of vertices with |U| = l , |V| = m and |W| = n, and E is a set of (α + β + γ)-tuples of vertices, called arcs, with exactly α vertices from U, exactly β vertices from V,and exactly γ vertices from W, such that any subset U₁∪ V₁∪ W₁ of U∪ V∪ W, E contains exactly one of the (α + β + γ)! (α + β + γ) − tuples whose entries belong to U₁∪ V₁∪ W₁. We obtain necessary and sufficient conditions for three lists of non-negative integers in non-decreasing order to be the losing score lists or score lists of some [α, β, γ]-tripartite hypertournament. Supported by National Science Foundation of China (No.10501021).     

Schur multiplier characterization of a class of infinite matrices

Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.