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.     

Solving enumeration problems for serial sequences with a constant difference in adjacent series heights

In this paper, the sets of n-valued serial sequences are considered. The structure of such series is defined by constraints on the number of series, the length of series, and the height of series. The problem of recalculation, numeration, and generation is solved for the sets of ascending, descending, and one-transitive sequences with constant differences in the adjacent series heights.     

Unions of increasing and intersections of decreasing sequences of convex sets

We prove that a convex setC is a polytope if and only ifC is not the union of any strictly increasing sequence of convex sets. In addition, we attempt (with partial success) to characterize, in intrinsic geometric terms, those convex subsetsC of a convex setX such thatC is not the intersection of any strictly decreasing sequence of convex subsets ofX.     

Variational problems with several volume constraints on the level sets

We consider functionals of the kind on , and we study the problem , where consists of those functions u whose level sets satisfy certain volume constraints , where denotes Lebesgue measure, and , are given numbers. Examples show that this problem may have no solution, even for simple smooth F. As a consequence, we relax the constraint to , i.e. , and we show that the minimizers over exist and are H?lder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function F, every minimizer over actually belongs to . Received: 31 January 2001 / Accepted: 23 February 2001 / Published online: 23 July 2001     

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.     

Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces

We consider nested sequences of linear or convex closed sets of the form arising in estimation and other inverse problems. We show that such sequences may fail to converge in any of the recently studied set convergences other than Mosco convergence. We also provide a positive result concerning the epislice convergence of related sequences of functions.Research partially supported by NSERC operating grants.     

Generalized limits and representations of attraction sets in problems with constraints of asymptotic character

Doklady Mathematics -     

An application of q-concavity to increasing sequences of Stein open sets

We prove that in a normal Stein space a relatively compact open set is a domain of holomorphy provided that it is an increasing union of Stein open subspaces. Received: 13 December 2004     

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.     

Enumeration of permutations with prescribed up-down and inversion sequences

A method is described by which the enumeration of permutations of 1, 2, … n with a prescribed sequence A of rises and falls, or a prescribed sequence B of inversions of order, or with both A and B, is effected in terms of numbers derived from the representation theory of the symmetric group. A connexion with Schensted pairs of standard Young tableaux is also discussed.     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

Point sets and sequences with small discrepancy

A systematic theory of a class of point sets called (t, m, s)-nets and of a class of sequences called (t, s)-sequences is developed. On the basis of this theory, point sets and sequences in thes-dimensional unit cube with the smallest discrepancy that is currently known are constructed. Various connections with other areas arise in this work, e.g. with orthogonal latin squares, finite projective planes, finite fields, and algebraic coding theory. Applications of the theory of (t, m, s)-nets to two methods for pseudorandom number generation, namely the digital multistep method and the GFSR method, are presented. Several open problems, mostly of a combinatorial nature, are stated.The author wants to thank the Universität Hannover (West Germany) for its hospitality during the period when most of this research was carried out. The results of this paper were presented at the Colloquium on Irregularities of Partitions at Fertöd (Hungary), July 7–11, 1986.     

Pontryagin's principle for problems with isoperimetric constraints and problems with inequality terminal constraints: Reply

A reply is made to the comment of Forster and Long.     

Pontryagin's principle for problems with isoperimetric constraints and problems with inequality terminal constraints: Comment

Schmitendorf's formulation of the terminal conditions is shown to be a special case of Hestenes' formulation, despite his claim to the contrary. Imaginative use of Hestenes' control parameters enables the application of Hestenes' theorem to a wide variety of problems. Schmitendorf's rank condition on the terminal constraints can be dispensed with.The authors acknowledge with thanks comments by N. Vousden and W. Schmitendorf.     

Distribution functions and trigonometric series with monotonically decreasing coefficients

The order of the distribution function of the sum of a cosine series with monotonically decreasing coefficients is determined. Theorems concerning integrability and convergence are proved for certain integral classes.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 3–10, July, 1971.The author wishes to thank O. D. Tsereteli for directing this work and P. L. Ul'yanov for the interest he showed in it.     

Hausdorff dimension of Moran sets with increasing spacing

For a family of homogeneous Moran sets, where at each level, subintervals are arranged with in-creasing spaces between neighboring subintervals from left to right, we obtained a formula of the Hausdorff dimensions.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.