Self-dual sequences

A study is made on self-dual sequences. Some enumeration problems on the number of these sequences and the number of cycles of length k which can be produced by an n-stage shift register are investigated. Also, some full cycles with special properties are constructed from those sequences.     

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

     

Logarithmic behavior of some combinatorial sequences

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schröder numbers, sequences of values of some classic orthogonal polynomials and many others. The calculus method extends also to numbers indexed by two or more parameters.     

Extremal functions for sequences

Davenport-Schinzel sequences DS(s) are finite sequences of some symbols with no immediate repetition and with no alternating subsequence (i.e. of the type ababab …) of the length s. This concept based on a geometrical motivation is due to Davenport and Schinzel in the middle of 1960s. In the late 1980s strong lower and upper (superlinear) bounds on the maximum length of the DS(s) sequences on n symbols were found. DS(s) sequences are well known to computer geometrists because of their application to the estimates of the complexity of the lower envelopes.

Here we summarize some properties of the generalization of this concept and prove that the extremal functions of aa… abb… baa… abb… b grow linearly.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

The signs of three-term recurrence sequences

In this note, we present two sufficient conditions for determining the signs of three-term recurrence sequences. In order to determine the signs of some sequences, by our method, it suffices to compute a constant number of terms at the beginning. For example, in order to prove the positivity of the central Delannoy number D(n), by our method, it just needs to know the recurrence relation of D(n) and the values of D(k) for 0≤k≤2. As applications, we determine the signs of some famous sequences.     

Algorithms for enumeration of single-transition serial sequences

Sets of n-valued single-transition serial sequences consisting of two serial subsequences (an increasing one and a decreasing one) determined by constraints on the number of the series and on their lengths and heights are considered. Enumeration problems for sets of finite sequences in which the difference in height between the neighboring series is not less than some given value are solved. Algorithms that assign smaller numbers to lexicographically lower-order sequences and smaller numbers to lexicographically higher-order sequences are obtained.     

Finite unions of interpolation sequences

A unified and relatively simple proof is given for some well-known results involving finite unions of uniformly separated sequences.

     

Lyapunov sequences for exponential dichotomies

For a linear nonautonomous dynamics with discrete time, we study the relation between nonuniform exponential dichotomies and strict Lyapunov sequences. Given such a sequence, we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence. The main difficulty is to extract some information about the angles between the stable and unstable subspaces (or some appropriate notion in the case of Banach spaces) from the Lyapunov sequence. In particular, for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic Lyapunov sequences, that is, strict Lyapunov sequences defined by quadratic forms. We also construct explicitly families of strict Lyapunov sequences for each nonuniform exponential dichotomy, in terms of Lyapunov norms.     

Some new convergent sequences and inequalities of Euler's constant

In this paper, some new convergent sequences and inequalities of Euler's constant are provided. To demonstrate the superiority of our new convergent sequence over DeTemple's sequence, Vernescu's sequence and Mortici's sequences, some numerical computations are also given.     

New results on nonexistence of perfect p-ary sequences and almost p-ary sequences

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.     

NA列与两两NQD列的L^p收敛性

设{Xn,n≥1）是NA列或两两NQD列,{ank;1≤k≤n,n∈N）是实数阵列．利用矩不等式和截尾方法,研究了∑k=1^n ankXk的L^p收敛性,所获结论推广和改进了前人的相应结果．     

Schur-Nevanlinna-Potapov sequences of rational matrix functions

We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur-Nevanlinna-Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur-Nevanlinna-Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna-Pick type for matricial Schur functions.     

Uniform densities of regular sequences in the unit disk

The upper and lower uniform densities of some regular sequences are computed. These densities are used to determine sequences of sampling and interpolation for Bergman spaces of the unit disk.

     

Graphs of difference operators for p-ary sequences

For a given prime p and positive integer n, we consider the graph G _n of the difference operator acting on p-ary sequences of length n. We suggest new proofs of some results of V.I. Arnold on the graph G _n and the complexity of sequences and obtain new results for the length of a maximal cycle in the general case of p-ary sequences. We also provide estimates for the number of complicated sequences.      

On some exact sequences in the theory of cyclotomic fields

The purpose of this article is to show the existence of some exact sequences which relate the ideal class groups of cyclotomic fields to Gauss sums. These exact sequences imply the results of Hachimori, Ichimura and Beliaeva. Furthermore, we study the relations between the étale cohomology groups and some conjectures on cyclotomic fields.     

Polynomial discrepancy of sequences

Generalizing E. Hlawka's concept of polynomial discrepancy we introduce a similar concept for sequences in the unit cube and on the sphere. We investigate the relation of this polynomial discrepancy to the usual discrepancy and obtain lower and upper bounds. In a final section some computational results are established.     

Linear recurring sequences over Galois rings

Linear recurrences of maximal period over a Galois ring and over a residue class ring modulo p are studied. For any such recurrence, the coordinate sequences (in p-adic and some other expansions) are considered as linear recurring sequences over a finite field. Upper and lower bounds for the ranks (linear complexities) of these coordinate sequences are obtained. The results are based on using the properties of Galois rings and the trace-function on such rings.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 169–189, March-April, 1995.     

Hausdorff means and moment sequences

We prove Schur’s theorem on the complete symmetric functions by using Hausdorff means. This approach leads to a more general result and to some new properties of moment sequences.     

Some properties of Lagrangian sequences

Here we discuss a sequence of Lagrangians and corresponding Euler–Lagrange equations and point out some interesting properties that this particular sequence holds. This is an extension on recent results obtained on sequences of differential equations to Lagrangians as it leads to differential equations on the application of the Variational Principle.