On Tverberg partitions

A theorem of Tverberg from 1966 asserts that every set X ? ? ^d of n = T(d, r) = (d + 1)(r ? 1) + 1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a ₁,..., a _r satisfying n = a ₁ + ··· + a _r ), in which the parts a _i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a _i ≤ d + 1 for all i = 1,..., r, there exists a set X ? ? of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a ₁,..., a _r .     

Multiple recurrence and convergence for hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $

converge in mean and we determine their limit. For example, this is the case if a(t) = t ^3/2, t log t, or t ² + (log t)². Furthermore, if {a ₁(t), …, a _? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a _i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a ₁(n)], …,m + [a _? (n)]}.

     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.     

On a question of Sárközy on gaps of product sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b ₁ < b ₂ < …} of all products a _i a _j with a _i , a _j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b _n+1 ? b _n ? α ^?3 and that for t ≥ 2 there are infinitely many t-gaps b _n+t ? b _n ? t ² α ^?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a _i /a _j , a _i , a _j ∈ A} when A ? {1, …, N} and |A| = αN.     

A realization theorem in the context of the Schur-Szegő composition

Every real polynomial of degree n in one variable with root ?1 can be represented as the Schur-Szeg? composition of n ? 1 polynomials of the form (x + 1) ^n?1(x + a _i ), where the numbers a _i are uniquely determined up to permutation. Some a _i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ? ρ, r ? [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a _i .     

Tauber-Konstanten für Verschiedene Tauberbedingungen bei den Kreisverfahren der Limitierungstheorie

Till now, we know Tauberian constants for the ‘Kreisverfahren’ with the conditions lim sup |n ^1/2 a _n|<∞ and lim sup |n ¹ a _n|<∞. Now, we obtain constants for the more general condition lim sup |n ^pa_n|<∞ with anyp(=∞<p<+∞). These constants are not always 0 or ∞, even if 1/2<p<1; therefore the Tauberian condition lim sup |n ^pa_n|<∞ is ‘appropriate’ for 1/2≦p≦1.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Almost everywhere convergence of sequences of two-dimensional Walsh–Fejér means of integrable functions

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let \(a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}\) such that a _j (n+1)≧δsup _k≦n a _j (n) (j=1,2, n∈?) for some δ>0 and a ₁(+∞)=a ₂(+∞)=+∞. Then for each integrable function f∈L ¹(I ²) we have the a.e. relation \(\lim_{n\to\infty}\sigma_{a_{1}(n), a_{2}(n)}f = f\). It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].     

On Singular Points of Meromorphic Functions Determined by Continued Fractions

It is shown that Leighton’s conjecture about singular points of meromorphic functions represented by C-fractions K^∞_n=1(a _n z ^αn /1) with exponents α₁, α₂,... tending to infinity, which was proved by Gonchar for a nondecreasing sequence of exponents, holds also for meromorphic functions represented by continued fractions K^∞_n=1(a _n A _n (z)/1), where A₁,A₂,... is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

Small prime solutions to cubic equations

Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|~(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).     

On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

Rearranged series by Haar system

For the orthonormal Haar system {X _n} the paper proves that for each 0 < ? < 1 there exist a measurable set E ? [0, 1] with measure | E | > 1 ? ? and a series of the form Σ _n=1 ^∞ a _n X _n with a _i ↘ 0, such that for every function f ∈ L ¹(0, 1) one can find a function \(\tilde f\) ∈ L ¹(0, 1) coinciding with f on E, and a series of the form

$\sum\limits_{i = 1}^\infty {\delta _i a_i \chi _i } where \delta _i = 0 or 1$

, that would converge to \(\tilde f\) in L ¹(0, 1).

     

Peixoto graph of Morse-Smale diffeomorphisms on manifolds of dimension greater than three

Let M ⁿ be a closed orientable manifold of dimension greater than three and G ₁(M ⁿ ) be the class of orientation-preserving Morse-Smale diffeomorphisms on M ⁿ such that the set of unstable separatrices of every f ∈ G ₁(M ⁿ ) is one-dimensional and does not contain heteroclinic orbits. We show that the Peixoto graph is a complete invariant of topological conjugacy in G ₁(M ⁿ ).     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

Properties of quasi-Boolean function on quasi-Boolean algebra

In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x ₁, …, x _n ) = (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ), the term (a ∧ C) can be deleted from Ψ(x ₁, …, x _n )? i.e., (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ) = (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p )? When a = 1: we divide our discussion into two cases. (1) ?₁(Ψ,C) = ø, C can not be deleted; ?₁(Ψ,C) ≠ ø, if S _i ⁰ ≠ ø (1 ≤ i ≤ q), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (m∧C)∨(a ₁∧C ₁)∨…∨(a _p ∧C _p ) = (a ₁∧C ₁)∨…∨(a _p ∧C _p ) ? (m ∧ C) ∨ C ₁ ∨ … ∨C _p = C ₁ ∨ … ∨C _p . Two possible cases are listed as follows, (1) ?₂(Ψ,C) = ø, the term (m∧C) can not be deleted; (2) ?₂(Ψ,C) ≠ ø, if (?i ₀) such that \(S'_{i_0 } \) = ø, then (m∧C) can be deleted, otherwise ((m∧C)∨C ₁∨…∨C _q )(v ₁, …, v _n ) = (C ₁ ∨ … ∨ C _q )(v ₁, …, v _n )(?(v ₁, …, v _n ) ∈ L ₃ ⁿ ) ? (C ₁ ^′ ∨ … ∨ C _q ^′ )(u ₁, …, u _q ) = 1(?(u ₁, …, u _q ) ∈ B ₂ ⁿ ).     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).