Nondecreasing Sequence Research Articles

An Iterative Procedure for Constructing Subsolutions of Discrete-Time Optimal Control Problems

An iterative procedure for constructing subsolutions of deterministic or stochastic optimal control problems in discrete time with continuous state space is introduced. The procedure generates a nondecreasing sequence of subsolutions, giving true lower bounds on the minimal costs. Convergence of the values at any fixed initial state is shown.

Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let $P(x)$ be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of $P(x+1)$, which leads to the log-concavity of $P(x+c)$ for any $c\geq 1$ due to Llamas and Martínez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.

Subdegree growth rates of infinite primitive permutation groups

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$ on $\Omega$ are called the $\alpha$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $\alpha$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $\Omega$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture there might exist a number $c$ which perhaps depends upon $G$, perhaps only on $m$, such that $m_r \leq c(m-2)^{r-1}$. In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs. A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.

Approximation and Interpolation by Entire Functions of Several Variables

AbstractLet f : ℝn → ℝ be C∞ and let h: ℝn → ℝ be positive and continuous. For any unbounded nondecreasing sequence ﹛ck﹜ of nonnegative real numbers and for any sequence without accumulation points ﹛xm﹜ in ℝn, there exists an entire function g : ℂn → ℂ taking real values on ℝn such thatThis is a version for functions of several variables of the case n = 1 due to L. Hoischen.

A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel

Let {φk} be an orthonormal system on a quasi-metric measure space \({\mathbb{X}}\), {lk} be a nondecreasing sequence of numbers with lim k→∞lk=∞. A diffusion polynomial of degree L is an element of the span of {φk:lk≤L}. The heat kernel is defined formally by \(K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}\). If T is a (differential) operator, and both Kt and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖p≤c1Lc‖P‖p. In particular, we are interested in the case when \({\mathbb{X}}\) is a Riemannian manifold, T is a derivative operator, and \(p\not=2\). In the case when \({\mathbb{X}}\) is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion polynomial valued operators.

Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

Let $\mathbf{d}=d_1\leq d_2\leq\dots\leq d_n$ be a nondecreasing sequence of $n$ positive integers whose sum is even. Let $\mathcal{G}_{n,\mathbf{d}}$ denote the set of graphs with vertex set $[n]=\{1,2,\dots,n\}$ in which the degree of vertex $i$ is $d_i$. Let $G_{n,\mathbf{d}}$ be chosen uniformly at random from $\mathcal{G}_{n,\mathbf{d}}$. It will be apparent from section 4.3 that all of the sequences we are considering will be graphic. We give a condition on $\mathbf{d}$ under which we can show that whp $\mathcal{G}_{n,\mathbf{d}}$ is Hamiltonian. This condition is satisfied by graphs with exponential tails as well those with power law tails.

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form ∑ n = 0 ∞ n v n n ! where ( v n ) n = 0 ∞ is either a non-negative integer sequence with υ n = o(log n/log log n) or a non-decreasing integer sequence with v n = o ( 2 n / 3 ) .

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.

Realizable growth vectors of affine control systems

In this paper, the following problem is considered: for a given nondecreasing finite sequence of positive integers, determine if there exists an affine control system whose growth vector coincides with this sequence. The answer is "yes" if and only if the Taylor series of a certain function constructed via elements of the sequence has nonnegative coefficients. It turns out that this problem is closely connected with properties of "core Lie subalgebras" of affine control systems which are responsible for a homogeneous approximation. We give a representation of core Lie subalgebras as free Lie algebras and describe sets of all possible core Lie subalgebras for systems with a fixed growth vector.

Degree sequences of k-multi-hypertournaments

Let n and k (n ≥ k > 1) be two non-negative integers. A k-multi-hypertournament on n vertices is a pair (V, A), where V is a set of vertices with |V| = n, and A is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V, A contains at least one (at most k!) of the k! k-tuples whose entries belong to S. The necessary and sufficient conditions for a non-decreasing sequence of non-negative integers to be the out-degree sequence (in-degree sequence) of some k-multi-hypertournament are given.

A central limit theorem for integer partitions

Recently, Hwang proved a central limit theorem for restricted Λ-partitions, where Λ can be any nondecreasing sequence of integers tending to infinity that satisfies certain technical conditions. In particular, one of these conditions is that the associated Dirichlet series has only a single pole on the abscissa of convergence. In the present paper, we show that this condition can be relaxed, and provide some natural examples that arise from the study of integers with restrictions on their digital (base-b) expansion.

Counting Nondecreasing Integer Sequences that Lie Below a Barrier

Given a barrier $0 \leq b_0 \leq b_1 \leq \cdots$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq \cdots \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formulæ for $f(n)$ include an $n \times n$ determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to $n$ and $a_n$, leads to a bivariate generating function, then a univariate generating function, then a linear recursion for $\{ f(n) \}$. Moreover, the coefficients of the bivariate generating function have a probabilistic interpretation, leading to an analytic inequality which is an identity for certain values of its argument.

Scheduling production of multiple part-types in a system with pre-known demands and deterministic inactive time intervals

We introduce and formalise a scheduling problem that consists in determining an optimum policy (i.e. one that minimises total inventory holding costs) to produce n part-types using a system that is able to share its capacity at all times among these part-types and that switches between an active and an inactive state for pre-known periods of time. Consequently, when active the system must produce enough reserves to meet the demand during the inactive interval. We show that there is always a simple optimum policy in which the production of the part-types is prioritised and, provided the units are properly defined, the optimum priority ordering corresponds to a non-decreasing sequence of the unit holding costs of the part-types.

Score lists in [h-k]-bipartite hypertournaments

Let m, n, h and k be integers such that m ≥ h > 1 and n ≥ k > 1. An [h-k]-bipartite hypertournament on m + n vertices is a triple (U, V, E), with two vertex sets U and V, |U| = m, |V| = n, together with an arc set E, a set of (h + k)-tuples of vertices, with exactly h vertices from U and exactly k vertices from V, called arcs, such that for any h-subset U1 of U and k-subset V1 of V, E contains exactly one of the (h + k)! (h + k)-tuples whose h entries belong to U1 and k entries belong to V1. We obtain necessary and sufficient conditions for a pair of nondecreasing sequences of nonnegative integers to be the losing score lists or score lists of some [h-k]-bipartite hypertournament.

Complexity of the packing coloring problem for trees

Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i -th class have pairwise distance greater than i . The main result of this paper is a solution of an open problem of Goddard et al. showing that the decision whether a tree allows a packing coloring with at most k classes is NP-complete. We further discuss specific cases when this problem allows an efficient algorithm. Namely, we show that it is decideable in polynomial time for graphs of bounded treewidth and diameter, and fixed parameter tractable for chordal graphs. We accompany these results by several observations on a closely related variant of the packing coloring problem, where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers.

Distribution of alternation points in best rational approximations

We study the convergence of counting measures of alternation point sets in best rational approximations to the equilibrium measure. It is shown that, for any prescribed nondecreasing sequence of denominator degrees, there exists a function analytic on [0, 1] and a sequence of numerator degrees such that the corresponding sequence of measures does not converge to the equilibrium measure of the interval.

A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem

This paper presents a preprocessing procedure for the 0–1 multidimensional knapsack problem. First, a non-increasing sequence of upper bounds is generated by solving LP-relaxations. Then, a non-decreasing sequence of lower bounds is built using dynamic programming. The comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal, or to fix a subset of variables to their optimal values. In addition, a heuristic solution is obtained. Computational experiments with a set of large-scale instances show the efficiency of our reduction scheme. Particularly, it is shown that our approach allows to reduce the CPU time of a leading commercial software.

Degree sequence of oriented k-hypergraphs

In this paper, we define the degree of a vertex in oriented hypergraphs and give a necessary and sufficient condition for a nondecreasing sequence of integers to be a degree sequence of an oriented 3-hypergraphs.

Stabilization of max-plus-linear systems using model predictive control: The unconstrained case

Max-plus-linear (MPL) systems are a class of event-driven nonlinear dynamic systems that can be described by models that are “linear” in the max-plus algebra. In this paper we derive a solution to a finite-horizon model predictive control (MPC) problem for MPL systems where the cost is designed to provide a trade-off between minimizing the due date error and a just-in-time production. In general, MPC can deal with complex input and states constraints. However, in this paper we assume that these are not present and it is only required that the input should be a nondecreasing sequence, i.e. we consider the “unconstrained” case. Despite the fact that the controlled system is nonlinear, by employing recent results in max-plus theory we are able to provide sufficient conditions such that the MPC controller is determined analytically and moreover the stability in terms of Lyapunov and in terms of boundedness of the closed-loop system is guaranteed a priori.

Optimality of the Holm procedure among general step-down multiple testing procedures

We study the class of general step-down multiple testing procedures, which contains the usually considered procedures determined by a nondecreasing sequence of thresholds (we call them threshold step-down, or TSD, procedures) as a parametric subclass. We show that all procedures in this class satisfying the natural condition of monotonicity and controlling the family-wise error rate (FWER) at a prescribed level are dominated by one of them — the classical Holm procedure. This generalizes an earlier result pertaining to the subclass of TSD procedures [Lehmann, E.L., Romano, J.P., 2005. Testing Statistical Hypotheses, 3rd ed. Springer, New York]. We also derive a relation between the levels at which a monotone step-down procedure controls the FWER and the generalized FWER (the probability of k or more false rejections).