On multi-transitivity with respect to a vector

A topological dynamical system(X,f)is said to be multi-transitive if for every n∈N the system(Xn,f×f2××fn)is transitive.We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets,answering a question proposed by Kwietniak and Oprocha(2012).We also show that multi-transitive systems are Li-Yorke chaotic.     

Hyperbolicity for Conservative Diffeomorphisms

This paper introduces the notion of robust hyperbolicity along periodic orbits homoclinically related to p (RNUHP) for conservative diffeomorphisms. It is proved that if f ∈ Diff¹⁺ _m (M) is RNUHP, then f is Anosov. It is also shown that f admits a dominated splitting, provided that f is expansive conservative stable.     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

Approximations by convolutions and antiderivatives

Let g be a given function in L ¹ = L ¹(0, 1), and let B be one of the spaces L ^p (0, 1), 1 ≤ p < ∞, or C ₀[0, 1]. We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on g, we prove the equivalence in B of the systems f _n * g and I f _n , where f _n ∈ L ¹, n ∈ ?, and I f = f * 1 is the antiderivative of f. As a consequence, we obtain criteria for the completeness and basis property in B of subsystems of antiderivatives of g.     

Linear transformations of monotone functions on the discrete cube

For a function f:{0,1}ⁿ→R and an invertible linear transformation L∈GL_n(2), we consider the function Lf:{0,1}ⁿ→R defined by Lf(x)=f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf)≥I(f), where I(f) is the total influence of f. Second, we conjecture that if both f and L(f) are monotone, then f=L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular.     

Schatten class membership of Hankel operators on the unit sphere

Let H²(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H²(S), where the symbol function f is allowed to be arbitrary in L²(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0.     

Local n-connectivity of decomposition spaces

A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LC^{n + 1}-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LC^{n + 1}provided X is LC^{n + 1}.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCⁿ.     

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.     

Extremal convex sets

Letf be an extended real valued function on the classK ⁿ of closed convex subsets of euclideann-dimensional space. A setK∈K ⁿ is said to bef-maximal if the conditionsK′∈K ⁿ ,K?K′,K≠K′ implyf(K)<f(K′), andf-minimal ifK′∈K ⁿ,K′∈K,K′≠K impliesf(K′)<f(K). In the cases whenf is the circumradius or inradius allf-maximal andf-minimal sets are determined. Under a certain regularity assumption a corresponding result is obtained for the minimal width. Moreover, a general existence theorem is established and a result concerning the existence of extremal sets with respect to packing and covering densities is proved.     

Cuntz-Krieger algebras representations from orbits of interval maps

Let f be an expansive Markov interval map with finite transition matrix Af. Then for every point, we yield an irreducible representation of the Cuntz-Krieger algebra OAf and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit. The restriction of each representation to the gauge part of OAf is decomposed into irreducible representations, according to the decomposition of the orbit.     

A Trotter-Kato type result for a second order difference inclusion in a Hilbert space

A Trotter-Kato type result is proved for a class of second order difference inclusions in a real Hilbert space. The equation contains a nonhomogeneous term f and is governed by a nonlinear operator A, which is supposed to be maximal monotone and strongly monotone. The associated boundary conditions are also of monotone type. One shows that, if An is a sequence of operators which converges to A in the sense of resolvent and fn converges to f in a weighted l²-space, then under additional hypotheses, the sequence of the solutions of the difference inclusion associated to An and fn is uniformly convergent to the solution of the original problem.     

Quantitative relation between noise sensitivity and influences

A Boolean function f: {0,1} ⁿ → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ _p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand’s lemma, which easily generalizes in various directions, including non-monotone functions.     

On a problem of Oppenheim concerning “factorisatio numerorum”

Let f(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is “highly factorable” if f(m)<f(n) for all m < n. We prove that f(n)=n·L(n)^?1+0(1) for n highly factorable, where $\text{L(n)=}\text{exp}\text{{}\text{log}\text{n}\text{logloglog}\text{n}\text{loglog}\text{n}\text{}}$. This result corrects the 1926 paper of Oppenheim where it is asserted that f(n)=n·L(n)^?2+0(1). Some results on the multiplicative structure of highly factorable numbers are proved and a table of them up to 10⁹ is provided. Of independent interest, a new lower bound is established for the function Ψ(x, y), the number of n≤x free of prime factors exceeding y.     

On the boundedness below of trigonometric polynomials of best approximation

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α _n } _n=0 ^∞ , are uniformly bounded below if the conditions a _n = O(n ^?1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent ?1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L _2π ¹ .     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

A subclass of parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. A function f in A satisfying the normalization is said to be in the class SP_n if Dⁿf is a parabolic starlike function, where Dⁿ is a notation of the Salagean operator. In this paper, several basic properties and characteristics of the class SP_n are investigated. These include subordination, convolution properties, class-preserving integral operators, and Fekete-Szegö problems.     

Normal families of meromorphic functions concerning shared values

In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f^′−afn and g^′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3.