Beurling’s analyticity theorem for quantum differences

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ _n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.     

Generically finite morphisms and formal neighborhoods of arcs

Let f : X → Y be a morphism of pure-dimensional schemes of the same dimension, with X smooth. We prove that if is an arc on X having finite order e along the ramification subscheme R _f of X, and if its image δ = f _∞(γ) on Y does not lie in J _∞(Y _sing), then the induced map T _γ J _∞(X) → T _δ J _∞(Y) is injective, with a cokernel of dimension e. In particular, if Y is smooth too, and if we denote by and the formal neighborhoods of and , then the induced morphism is a closed embedding of codimension e.      

Supremizers of inner <Emphasis Type="Italic">γ</Emphasis>-convex functions

A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a such that holds for all satisfying ||x ₀ − x ₁|| = νγ and . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such satisfying lim . For instance, if an upper bounded and inner γ-convex function, which is defined on a convex and bounded subset D of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of D relative to aff D or at a γ-extreme point of D, and if D is open relative to aff D or if dim D ≤ 2 then there is certainly a supremizer at a γ-extreme point of D. Another example is: if D is an affine set and is inner γ-convex and bounded above, then for all , and if 2 ≤ dim D < ∞ then each is a supremizer of f.      

On the action of Lipschitz functions on vector-valued random sums

Let X be a Banach space and let (ξ_j)_j ≧ 1 be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent:

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form

The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity

In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx ： X→L（E） be a continuous map, and each R（Tx） be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^＋： X→L（E） is continuous if and only if ｜｜Tx^＋｜｜ is locally bounded, where Tx^＋ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement： For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that （0, λ^2）lohtatn in ∩x∈∪0C／σ（Tx^＋Tx）, where a（T） denotes the spectrum of operator T.     

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.     

On the solutions of a boundary value problem arising in free convection with prescribed heat flux

For given , c < 0, we are concerned with the solution f _b of the differential equation f ′′′ + ff ′′ + g(f ′) = 0 satisfying the initial conditions f(0) = a, f ′ (0) = b, f ′′ (0) = c, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists b _* > 0 such that f _b exists on [0, + ∞) and is such that as t → + ∞, if and only if b ≥ b _*. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.      

Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like

A property of the β-Cauchy-type integral with continuous density

  A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator ). It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

Affine restriction for radial surfaces

Suppose dμ is affine surface measure on a convex radial surface Γ(x) = (x, γ(|x|)), a ≤ |x| < b, in . Under appropriate smoothness and growth conditions on γ, we prove and Fourier restriction estimates for Γ.     

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T _φ , we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by T _φ . For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λ^α with 0 < α < ∞, and if f is an H ^p -function (p ≥ 1) with T _φ f = 0, then f ∈ Λ^α and

The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.