Numeration of non-decreasing and non-increasing serial sequences

Finite sets of n-valued serial sequences are examined. Their structure is determined not only by restrictions on the number of series and series lengths, but also by restrictions on the series heights, which define the order number of series and their lengths, but also is limited to the series heights, by whose limitations the order of series of different heights is given. Solutions to numeration and generation problems are obtained for the following sets of sequences: non-decreasing and non-increasing sequences where the difference in heights of the neighboring series is either not smaller than a certain value or not greater than a certain value. Algorithms that assign smaller numbers to lexicographically lower sequences and smaller numbers to lexicographically higher sequences are developed.     

Solutions to enumeration problems for single-transition serial sequences with an adjacent series height increment bounded above

Sets of n-valued finite serial sequences are investigated. Such a sequence consists of two serial subsequences, beginning with an increasing subsequence and ending in a decreasing one (and vice versa). The structure of these sequences is determined by constraints imposed on the number of series, on series lengths, and on series heights. For sets of sequences the difference between adjacent series heights in which does not exceed a certain given value 1 ≤ |h _j+1 ? h _j | ≤ δ, two algorithms are constructed of which one assigns smaller numbers to lexicographically lower sequences and the other assigns smaller numbers to lexicographically higher sequences.     

Enumeration problems of sets of increasing and decreasing n-valued serial sequences with double-ended constraints on series heights

Enumeration problems for n-valued serial sequences are considered. Sets of increasing and decreasing sequences whose structure is specified by constraints on lengths of series and on the difference in heights of the neighboring series in the case when this difference lies between δ ₁ and δ ₂ are examined. Formulas for powers of these sets and algorithms for the direct and reverse numerations (assigning smaller numbers to lexicographically lower order sequences or smaller numbers to lexicographically higher order sequences) are obtained.     

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.     

Enumeration problems for oriented serial sequences

Sets of n-valued m-sequences of a serial structure are considered. In addition to the conventional concepts of the length of a series and of the number of series in a sequence, the concepts of a series height and of a sequence of series heights are introduced. The structure of sequences, which will be referred to as oriented, is determined from restrictions on the number and length of series and on the order of sequencing of series of various heights. A general approach to solving enumeration problems for sets of such sequences is proposed. The approach is based on formulas for the number of arrangements of elements in cells and the power of a set of height sequences. Exact solutions are derived for some restrictions, which are important for applications.     

Localization for multiple Fourier series with “J k -lacunary sequence of partial sums” in Orlicz classes

We obtain structural and geometric characteristics of sets on which weak generalized localization almost everywhere is valid for multiple trigonometric Fourier series of functions in the classes $L(\log ^ + L)^{3k + 2} (\mathbb{T}^N ),1 \leqslant k \leqslant N - 2,N \geqslant 3$ , in the case where the rectangular partial sums of these series have an “index” in which exactly k components are elements of lacunary sequences.     

Representations of finite partially ordered sets over commutative artinian uniserial rings

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.     

Odd and even hamming spheres also have minimum boundary

Combinatorial problems with a geometric flavor arise if the set of all binary sequences of a fixed length n, is provided with the Hamming distance. The Hamming distance of any two binary sequences is the number of positions in which they differ. The (outer) boundary of a set A of binary sequences is the set of all sequences outside A that are at distance 1 from some sequence in A. Harper [6] proved that among all the sets of a prescribed volume, the ‘sphere’ has minimum boundary.We show that among all the sets in which no pair of sequences have distance 1, the set of all the sequences with an even (odd) number of 1's in a Hamming ‘sphere’ has the same minimizing property. Some related results are obtained. Sets with more general extremal properties of this kind yield good error-correcting codes for multi-terminal channels.     

Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J₂ of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.     

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, SV, and the multi-index graded algebra defined by V, grVR. We prove that SV is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V(k) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V(k), allow us to obtain the (finite) minimal generating sequences for C as well as for V.We also analyze the structure of the homogeneous components of grVR. The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V(k), k?0.     

Asymptotic stability and other properties of trajectories and transfer sequences leading to the bargaining sets

The foundation of a dynamic theory for the bargaining sets started withStearns, when he constructed transfer sequences which always converge to appropriate bargaining sets. A continuous analogue was developed byBillera, where sequences where replaced by solutions of systems of differential equations. In this paper we show that the nucleolus is locally asymptotically stable both with respect toStearns' sequences andBillera's solutions if and only if it is an isolated point of the appropriate bargaining set. No other point of the bargaining set can be locally asymptotically stable. Furthermore, it is always stable in these processes. As by-products of the study we derive the results ofBillera andStearns in a different fashion. We also show that along the non-trivial trajectories and sequences, the vector of the excesses of the payoffs, arranged in a non-increasing order, always decreases lexicographically, thus each bargaining set can be viewed as resulting from a certain monotone process operating on the payoff vectors.     

Isosingular Sets and Deflation

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.     

On the enumeration of certain sets of planted plane trees

Using the definition of planted plane trees given by D. A. Klarner (“A correspondence between sets of trees,” Indag. Math.31 (1969), 292–296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of (r + 1)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.     

‘Duality’ of the Nikodym property and the Hahn property: Densities defined by sequences of matrices

The authors investigated in Boos and Leiger (2008) [5] the ‘duality’ of the Nikodym property (NP) of the set of all null sets of the density defined by any nonnegative matrix and the Hahn property (HP) of the strong null domain of it. In this paper, the investigation of the intimated duality is continued by considering densities defined by sequences of nonnegative matrices. These considerations are motivated by the known result that the ideal of the null sets of the uniform density has NP. In this context the general notion of S-convergence of double sequences (cf. Drewnowski, 2002 [8]) containing Pringsheim convergence, Hardy convergence and uniform convergence of double sequences is used.     

A generalization of the notion of microscopic sets

We investigate families of subsets of the real line defined by nonincreasing sequences of positive real numbers. One of these families coincides with the σ-ideal of microscopic sets. We prove that the union of our families is equal to the σ-ideal of Lebesgue measure zero sets and the intersection of all such families is the σ-ideal of sets of strong measure zero. We also study other properties concerning homeomorphisms between sets of the first category and sets from our families.     

On uniqueness sets for multiple Walsh series

We study uniqueness sets for multiple Walsh series under ρ-regular (or bounded) convergence in rectangles. We prove that a countable set is a uniqueness set for such a series under this convergence. We construct a class of perfect uniqueness sets for multiple Walsh series under this convergence. We show that the notion of index of a perfect set does not solve the problem of whether this set belongs to the class of uniqueness sets. We note that the results of this paper remain valid for several rearranged multiple Walsh series.     

Universality properties of Taylor series inside the domain of holomorphy

It is proven that the Taylor series of functions holomorphic in C_∞?{1} generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown that such sets necessarily are polar sets.     

A trace representation of binary Jacobi sequences

We determine the trace function representation, or equivalently, the Fourier spectral sequences of binary Jacobi sequences of period pq, where p and q are two distinct odd primes. This includes the twin-prime sequences of period p(p+2) whenever both p and p+2 are primes, corresponding to cyclic Hadamard difference sets.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.     

Constructions of optimal low-hit-zone frequency hopping sequence sets

In recent years, the study relating to low-hit-zone frequency hopping sequence sets, including the bounds on the Hamming correlations within the low hit zone and the optimal constructions, has become a new research area attracting the attention of many related researchers. In this paper, two constructions of optimal frequency hopping sequence sets with low hit zone have been employed, one of which is based on m-sequence while the other is based on the decimated sequences of m-sequence. Moreover, in the special case of \(k=n-1\), the construction based on the decimated sequences of m-sequence also yields low-hit-zone frequency hopping sequence sets with optimal periodic partial Hamming correlation property.