Another congruence for the Apéry numbers

In 1979 R. Apéry introduced the numbers a_n = Σ₀ⁿ(_kⁿ)²(_k^n+k)² in his irrationality proof for ζ(3). We prove some congruences for these numbers, which extend congruences previously published in J. Number Theory (Vol. 12, 14, 16, and 21).     

Numerical semigroups: Apéry sets and Hilbert series

Let a ₁,…,a _n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a ₁,…,a _n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. Dedicated to the memory of Ernst S. Selmer (1920–2006), whose calculations revealed the “Selmer group”.     

Apéry Polynomials and the Multivariate Saddle Point Method

The Apéry polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of \(\zeta (3)\) . In this paper, we present a method to study the asymptotic behavior of the sequence of Apéry polynomials \((B_{n})_{n=1}^{\infty }\) in the whole complex plane as \(n\rightarrow \infty \) . The proofs are based on a multivariate version of the complex saddle point method. Moreover, the asymptotic zero distributions for the polynomials \((B_{n})_{n=1}^{\infty }\) and for some transformed Apéry polynomials are derived by means of the theory of logarithmic potentials with external fields, establishing a characterization as the unique solution of a weighted equilibrium problem. The method applied is a general one, so that the treatment can serve as a model for the study of objects related to the Apéry polynomials.     

On the Cohen-Macaulay property for monomial curves in ℙ K 3

The arithmetical Cohen-Macaulay property for monomial curves in _K ³ with generic zero was shown in [2] to be true forn ₃>(n ₂–1)(n ₂–n ₁),n ₃>n ₂, (n ₁,n ₂,n ₃)=1. Here we establish necessary and sufficient arithmetic conditions in terms ofn ₁,n ₂,n ₃ in order for the indicated curves not to be arithmetically Cohen-Macaulay.     

On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers

Let g ≥ 2 be an integer and let (u _n ) _n≥1  be a sequence of integers which satisfies a relation u _n+1 = h(n)u _n for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u _n in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u _n+2 = h ₁(n)u _n+1 + h ₂(n)u _n with two nonconstant rational functions ${h_1(X), h_2(X) \in {\mathbb Q} [X]}Let g ≥ 2 be an integer and let (u _n ) _n≥1  be a sequence of integers which satisfies a relation u _n+1 = h(n)u _n for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u _n in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u _n+2 = h ₁(n)u _n+1 + h ₂(n)u _n with two nonconstant rational functions h₁(X), h₂(X) ? \mathbb Q [X]{h_1(X), h_2(X) \in {\mathbb Q} [X]}. This class includes the Apéry, Delannoy, Motzkin, and Schr?der numbers.     

Beukers-like supercongruences for generalized Apéry numbers

For positive integers \(f_1,f_2,m,l\), the author and Chetry defined a generalization of Apéry numbers \(A(f_1,f_2,m,l,\lambda )\) given by

$$\begin{aligned} A(f_1,f_2,m,l,\lambda ):=\sum _{j=0}^{f_2}{f_1+j\atopwithdelims ()j}^m{f_2\atopwithdelims ()j}^l\lambda ^j. \end{aligned}$$

In this article, we prove certain Beukers-like supercongruences for these generalized Apéry numbers.

     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.     

Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry

The Ramanujan Journal - There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be...     

Apéry base and polar invariants for real valuations

The main objective of this paper is to study real valuations?μ defined on the polynomial ring K[T] with coefficients in a field K such that the restriction of?μ to K is a proper valuation on K. For this, we consider the Apéry base and the iterated sequence of valuations associated with?μ and we show the main properties and the relation between both invariants. We also prove a factorization theorem for elements ${f\in K[T]}$ of the suitable degree such that?μ(f) is in the Apéry base of?μ, obtaining strong properties of their irreducible factors. In particular, some results of M. Merle, R. Ephraim and A. Granja for irreducible algebroid plane curve singularity are a special case of this theorem.     

On the index sets of Σ-subsets of the real numbers

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the Σ-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of ? ⁿ , first category subsets in ? ⁿ as well as the sets of pairs of Σ-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the Σ-formulas defining connected sets is at least Π ₁ ¹ .     

On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero

The aim of this paper is to prove that for an arbitrary set of measure zero there exists a bounded function for which the Fejér means of the Walsh-Fourier series of the function diverge. Research supported by the Georgian National Fundation for Scientific Research, Grant no. 07_225_3-100.     

On a conjecture of Hivert and Thiéry about Steenrod operators

We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.     

On the structure of invariant attracting sets of systems of finite-difference equations generating Hénon-Lozi mappings

For Hénon-Lozi mappings F, we find sufficient conditions under which on the plane there exists a domain U such that its closure is mapped by F strictly inside U. This ensures the existence of a compact invariant set in U. We prove the existence of an open set of parameter values for which this invariant set contains a zero-dimensional locally maximal topologically transitive Markov set such that the restriction of the mapping to this set is topologically conjugate to the shift automorphism in the space of sequences of two symbols. We show that if this Markov set is hyperbolic, then the above-mentioned compact invariant set coincides with the closure of the unstable manifold of F at a fixed point lying in that set and is a topologically indecomposable one-dimensional continuum. We present the parameter values for which these results hold for the Hénon mapping. We thereby prove the existence of a parameter range in which the invariant set of the Hénon mapping is a one-dimensional topologically indecomposable Brauer-Janiszewski continuum that contains a zero-dimensional locally maximal set and lies in the attraction domain of itself.     

On the cardinality of the θ-closed hull of sets

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.     

On the structure of sets of σ-monogeneity for continuous functions

The notion of the sets of -monogeneity for continuous functions is introduced which makes it possible to study pseudo-analytic properties of these functions. The theorem on the structure of these sets is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 226–232, February, 1993.