The Absolute Ε-Entropy of a Compact Set of Infinitely Differentiable Aperiodic Functions

We calculate asymptotics for the Kolmogorov ε-entropy of the compact set of infinitely differentiable aperiodic functions embedded continuously into the space of continuous functions on a closed finite interval.     

On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl's theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl's theorem for f(T).     

On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant such that for we have .     

On a Constructive Proof of Kolmogorov’s Superposition Theorem

Kolmogorov (Dokl. Akad. Nauk USSR, 14(5):953–956, 1957) showed that any multivariate continuous function can be represented as a superposition of one-dimensional functions, i.e., $$f(x_{1},\ldots,x_{n})=\sum_{q=0}^{2n}\varPhi _{q}\Biggl(\sum_{p=1}^{n}\psi_{q,p}(x_{p})\Biggr).$$ The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Φ _q and ψ _q,p , respectively. Sprecher (Neural Netw. 9(5):765–772, 1996; Neural Netw. 10(3):447–457, 1997) gave a constructive proof of Kolmogorov’s superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ by ψ _p,q :=λ _p ψ(x _p +qa) with appropriate values λ _p ,a∈?. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Köppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474–479, 2002) suggested a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen’s ψ, and we present a correct constructive proof of Kolmogorov’s superposition theorem for continuous inner functions ψ similar to Sprecher’s approach.     

Newton’s Method for Minimizing a Convex Twice Differentiable Function on a Preconvex Set

The problem of minimizing a convex twice differentiable function on the set-theoretic difference between a convex set and the union of several convex sets is considered. A generalization of Newton’s method for solving problems with convex constraints is proposed. The convergence of the algorithm is analyzed.     

Nonsaturable quadrature formulas on an interval (on Babenko’s Problem)

An interpolation quadrature formula with a weight function from L_p[–1, 1] (1 < p < ∞) whose error is estimated in terms of Chebyshev smoothness characteristics is shown to be nonsaturable.     

On a Problem of Cameron’s on Inexhaustible Graphs

A graph G is inexhaustible if whenever a vertex of G is deleted the remaining graph is isomorphic to G. We address a question of Cameron [6], who asked which countable graphs are inexhaustible. In particular, we prove that there are continuum many countable inexhaustible graphs with properties in common with the infinite random graph, including adjacency properties and universality. Locally finite inexhaustible graphs and forests are investigated, as is a semigroup structure on the class of inexhaustible graphs. We extend a result of [7] on homogeneous inexhaustible graphs to pseudo-homogeneous inexhaustible graphs.The authors gratefully acknowledge support from the Natural Science and Engineering Research Council of Canada (NSERC).     

Kolmogorov＇s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov＇s ε-entropy for the bounded sets with small ＂tail＂ in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov＇s ε-entropy of the global attractor for lattice systems.     

Solution of Ponomarev’s Problem of Condensation onto Compact Sets

Siberian Mathematical Journal - Assuming the Continuum Hypothesis (CH), we prove that there exists a&nbsp;perfectly normal compact topological space&nbsp; $ Z $ and a&nbsp;countable set...     

Resonance Problem for a Class of Duffing’s Equations

Consider the Duffing's equation+g(x)=f(t),(1)where g∈C(R,R)and f∈P≡{f∈C(R,R);f is ω-periodic for some ω>0}.The functiong is said to be resonant if there exists f∈P such that eq.(1) has no bounded solutions on[0,∞).Using a generalized version of the Poincare-Birkhoff fixed point theorem,theauthors establish conditions on g which guarantee the following result holds:for any f∈Pwith period ω,there exists K≥0 such that eq.(1) has infinitely many kω-periodic solutionsfor every integer k≥K.In such a case,g is clearly non-resonant.     

Corrigendum: “On the Exceptional Set for Goldbach’s Problem in Short Intervals”

We correct the proof of the log-free hybrid zero-density estimate given in On the Exceptional Set for Goldbachs Problem in Short Intervals, Monatsh Math 132: 49–65.     

Sharpened versions of a Kolmogorov’s inequality

Sharpened versions of a Kolmogorov’s inequality for sums of independent Bernoulli random variables are proved.     

On Rubel’s Problem in the Class of Linear Operators on the Space of Analytic Functions

In this paper, we describe all derivation pairs of linear operators that act in spaces of functions analytic in domains.     

On Hardy’s Uncertainty Principle for Solvable Locally Compact Groups

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ? for locally compact Abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.     

On the Exceptional Set for Goldbach’s Problemin Short Intervals

 Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References (Received 15 May 2000; in revised form 11 October 2000)     

On the Exceptional Set for Goldbach’s Problemin Short Intervals

 Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References