Nonincreasing Sequence Research Articles

On a conjecture that strengthens Kundu's k $k$‐factor theorem

AbstractLet be a nonincreasing degree sequence with even . In 1974, Kundu showed that if is graphic, then some realization of has a ‐factor. For , Busch et al. and later Seacrest for showed that if and is graphic, then there is a realization with a ‐factor whose edges can be partitioned into a ‐factor and edge‐disjoint 1‐factors. We improve this to any . In 1978, Brualdi and then Busch et al. in 2012, conjectured that . The conjecture is still open for . However, Busch et al. showed the conjecture is true when or . We explore this conjecture by first developing new tools that generalize edge‐exchanges. With these new tools, we can drop the assumption is graphic and show that if then has a realization with edge‐disjoint 1‐factors. From this we confirm the conjecture when or when is graphic and .

Some properties of discrete classes of weights satisfying reverse Hölder inequality and reverse inequality for nonincreasing sequences

Abstract In this paper, we prove some new discrete inequalities with weights of Hardy’s type and employ them to establish some properties of discrete weighted Gehring class of sequences satisfying reverse Hölder inequality. We also consider the weighted class of sequences satisfying a reverse inequality for nonincreasing sequences and prove some higher summability theorems.

Distributed online constrained convex optimization with event-triggered communication

This paper focuses on the distributed online convex optimization problem with time-varying inequality constraints over a network of agents, where each agent collaborates with its neighboring agents to minimize the cumulative network-wide loss over time. To reduce communication overhead between the agents, we propose a distributed event-triggered online primal–dual algorithm over a time-varying directed graph. With several classes of appropriately chose decreasing parameter sequences and non-increasing event-triggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. Finally, a numerical simulation example is provided to verify the theoretical results.

Sequences of Operators, Monotone in the Sense of Contractive Domination

A sequence of operators Tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_n$$\\end{document} from a Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {H}}}$$\\end{document} to Hilbert spaces Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {K}}}_n$$\\end{document} which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {H}}}$$\\end{document} to a Hilbert space K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {K}}}$$\\end{document}. Moreover, the closability or closedness of Tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_n$$\\end{document} is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.

ON SEQUENCES OF ELEMENTARY TRANSFORMATIONS IN THE INTEGER PARTITIONS LATTICE

An integer partition, or simply, a partition is a nonincreasing sequence \(\lambda = (\lambda_1, \lambda_2, \dots)\) of nonnegative integers that contains only a finite number of nonzero components. The length \(\ell(\lambda)\) of a partition \(\lambda\) is the number of its nonzero components. For convenience, a partition \(\lambda\) will often be written in the form \(\lambda=(\lambda_1, \dots, \lambda_t)\), where \(t\geq\ell(\lambda)\); i.e., we will omit the zeros, starting from some zero component, not forgetting that the sequence is infinite. Let there be natural numbers \(i,j\in\{1,\dots,\ell(\lambda)+1\}\) such that (1) \(\lambda_i-1\geq \lambda_{i+1}\); (2) \(\lambda_{j-1}\geq \lambda_j+1\); (3) \(\lambda_i=\lambda_j+\delta\), where \(\delta\geq2\). We will say that the partition \(\eta={(\lambda_1, \dots, \lambda_i-1, \dots, \lambda_j+1, \dots, \lambda_n)}\) is obtained from a partition \(\lambda=(\lambda_1, \dots, \lambda_i, \dots, \lambda_j, \dots, \lambda_n)\) by an elementary transformation of the first type. Let \(\lambda_i-1\geq \lambda_{i+1}\), where \(i\leq \ell(\lambda)\). A transformation that replaces \(\lambda\) by \(\eta=(\lambda_1, \dots, \lambda_{i-1}, \lambda_i-1, \lambda_{i+1}, \dots)\) will be called an elementary transformation of the second type. The authors showed earlier that a partition \(\mu\) dominates a partition \(\lambda\) if and only if \(\lambda\) can be obtained from \(\mu\) by a finite number (possibly a zero one) of elementary transformations of the pointed types. Let \(\lambda\) and \(\mu\) be two arbitrary partitions such that \(\mu\) dominates \(\lambda\). This work aims to study the shortest sequences of elementary transformations from \(\mu\) to \(\lambda\). As a result, we have built an algorithm that finds all the shortest sequences of this type.

A methodology using a non-increasing sequence applied in the solution of heat transfer problems

This research aims to propose a numerical methodology that employs a non-increasing sequence to approximate the solution for a large class of ordinary differential equations like the ones described in complex non-linear heat transfer through porous fins. It also aims to compare this proposed methodology with an earlier one using a non-decreasing sequence of elements and to provide an upper-bound estimation for the error. The comparison between the non-increasing and non-decreasing sequences showed excellent agreement when applied to an example of convection and radiation in porous fins.

Distributed inertial online game algorithm for tracking generalized Nash equilibria.

This paper is concerned with the distributed generalized Nash equilibrium (GNE) tracking problem of noncooperative games in dynamic environments, where the cost function and/or the coupled constraint function are time-varying and revealed to each agent after it makes a decision. We first consider the case without coupled constraints and propose a distributed inertial online game (D-IOG) algorithm based on the mirror descent method. The proposed algorithm is capable of tracking Nash equilibrium (NE) through a time-varying communication graph and has the potential of achieving a low average regret. With an appropriate non-increasing stepsize sequence and an inertial parameter, the regrets can grow sublinearly if the deviation of the NE sequence grows sublinearly. Second, the time-varying coupled constraints are further investigated, and a modified D-IOG algorithm for tracking GNE is proposed based on the primal-dual and mirror descent methods. Then, the upper bounds of regrets and constraint violation are derived. Moreover, inertia and two information transmission modes are discussed. Finally, two simulation examples are provided to illustrate the effectiveness of the D-IOG algorithms.

Maximally edge‐connected realizations and Kundu's k $k$‐factor theorem

AbstractA simple graph with edge‐connectivity and minimum degree is maximally edge‐connected if . In 1964, given a nonincreasing degree sequence , Jack Edmonds showed that there is a realization of that is ‐edge‐connected if and only if with when . We strengthen Edmonds's result by showing that given a realization of if is a spanning subgraph of with such that when , then there is a maximally edge‐connected realization of with as a subgraph. Our theorem tells us that there is a maximally edge‐connected realization of that differs from by at most edges. For , if has a spanning forest with components, then our theorem says there is a maximally edge‐connected realization that differs from by at most edges. As an application we combine our work with Kundu's ‐factor theorem to show there is a maximally edge‐connected realization with a ‐factor for and present a partial result to a conjecture that strengthens the regular case of Kundu's ‐factor theorem.

AROUND THE ERDÖS–GALLAI CRITERION

By an (integer) partition we mean a non-increasing sequence \(\lambda=(\lambda_1, \lambda_2, \dots)\) of non-negative integers that contains a finite number of non-zero components. A partition \(\lambda\) is said to be graphic if there exists a graph \(G\) such that \(\lambda = \mathrm{dpt}\,G\), where we denote by \(\mathrm{dpt}\,G\) the degree partition of \(G\) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.

The potential-Ramsey numbers [formula omitted] and [formula omitted

A nonincreasing sequence ρ=(ρ1,…,ρn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ρ. Given two graphs G1 and G2, Busch et al. introduced the potential-Ramsey number of G1 and G2, denoted rpot(G1,G2), is the smallest nonnegative integer m such that for every m-term graphic sequence ρ, there is a realization G of ρ with G1⊆G or with G2⊆G¯, where G¯ is the complement of G. For t≥2 and 0≤k≤⌊t2⌋, let Kt−k be the graph obtained from Kt by deleting k independent edges. Busch et al. determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=0. Du and Yin determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=1. In this paper, we further determine rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for 2≤k≤⌊t2⌋.

(H_{p}-L_{p})-Type inequalities for subsequences of Nörlund means of Walsh–Fourier series

We investigate the subsequence {t_{2^{n}}f } of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences. In particular, we prove that a large class of such summability methods are not bounded from the martingale Hardy spaces H_{p} to the space mathit{weak-}L_{p} for 0< p<1/(1+alpha ) , where 0<alpha <1. Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.

On the maximal solution of the conjugate discrete-time algebraic Riccati equation

In this paper we consider a class of conjugate discrete-time Riccati equations, arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Under some mild assumptions and the framework of the fixed-point iteration, a constructive proof is given for the existence of the maximal solution to the conjugate discrete-time Riccati equation, in which the control weighting matrix is nonsingular and its constant term is Hermitian. Moreover, starting with a suitable initial matrix, we also show that the nonincreasing sequence generated by the fixed-point iteration converges at least linearly to the maximal solution of the Riccati equation. An example is given to demonstrate the correctness of our main theorem and provide considerable insights into the study of another meaningful solutions.

Center of Distances and Central Cantor Sets

We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space (X,,d) is defined by S(A) :={ alpha in [0,,+infty ): forall xin A exists yin A d(x,,y)=alpha } . Given a nonincreasing sequence (a_{n}) of positive numbers converging to 0, the set E(a_{n}) := left{ xin {mathbb {R}}: exists Asubset {mathbb {N}} x=,sum _{nin A}a_{n}right} is called the achievement set of the sequence (a_{n}). This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series sum a_{n}, and consequently S(C)supset left{ 0right} cup left{ a_{n}:nin {mathbb {N}}right} . We try to check which central Cantor sets have the minimal possible center of distances and which have not.

Solutions to problems about potentially K s,t -bigraphic pair

Abstract Let S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) , where a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y , E ) G=\left(X\cup Y,E) such that a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are the degrees of the vertices in X X and Y Y , respectively. In this case, G G is referred to as a realization of S S . Given a bigraphic pair S S , and a complete bipartite graph K s , t {K}_{s,t} , we say that S S is a potentially K s , t {K}_{s,t} -bigraphic pair if some realization of S S contains K s , t {K}_{s,t} as a subgraph (with s s vertices in the part of size m m and t t in the part of size n n ). Ferrara et al. (Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) to be the minimum integer k k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n}) with σ ( S ) = a 1 + ⋯ + a m ≥ k \sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge k is a potentially K s , t {K}_{s,t} -bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ 9 s 4 t 4 n\ge m\ge 9{s}^{4}{t}^{4} . In this paper, we further determine σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ s n\ge m\ge s and n + m ≥ 2 t 2 + t + s n+m\ge 2{t}^{2}+t+s . As two corollaries, if n ≥ m ≥ t 2 + t + s 2 n\ge m\ge {t}^{2}+\frac{t+s}{2} or if n ≥ m ≥ s n\ge m\ge s and n ≥ 2 t 2 + t n\ge 2{t}^{2}+t , the values σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

On a new application of almost non-increasing sequence to Ultraspherical Series associated with (N,p,q)k means

On a new application of almost non-increasing sequence to Ultraspherical Series associated with (N,p,q)k means

A-statistical convergence with a rate and applications to approximation

A = (ank) be a regular summability matrix. In the present paper we deal with subspaces of the space of A?statistically convergent sequences obtained by the rate at which the A?statistical limit tends to zero. We prove that a sequence is the A?strongly convergent if and only if it is the A?statistically convergent and the A?uniformly integrable with the rate of o (an) where a = (an) is a positive nonincreasing sequence. We also make a link between the A?strong convergence and the A?distributional convergence with the rate of o (an). Finally, as an application we present an approximation theorem of Korovkin type.

A Note on Derivative of Sine Series with Square Root

Chaundy and Jolliffe proved that ifanis a nonnegative, nonincreasing real sequence, then series∑ansinnxconverges uniformly if and only ifnan⟶0. The purpose of this paper is to show that ifnanis nonincreasing andnan⟶0, then the seriesfx=∑ansinnxcan be differentiated term-by-term onc,dforc,d&gt;0. However,f′0may not exist.

Trigonometric series with noninteger harmonics

Let {ck} be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and α>0 be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series ∑kcksin⁡kαx and ∑kckcos⁡kαx on the real line and its bounded subsets.

Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence (a(n)) the sequence (na(n)) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset {mathcal {AOS}} of ell _{1} consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in {mathcal {AOS}} and its subsets; this kind of study is known as lineability. Finally we show that {mathcal {AOS}} is a residual mathcal {G}_{delta sigma } but not an {mathcal {F}}_{sigma delta } text {-set} .

Game-theoretic upper expectations for discrete-time finite-state uncertain processes

Game-theoretic upper expectations are joint (global) probability models that mathematically describe the behaviour of uncertain processes in terms of supermartingales; capital processes corresponding to available betting strategies. Compared to (the more common) measure-theoretic expectation functionals, they are not bounded to restrictive assumptions such as measurability or precision, yet succeed in preserving, or even generalising many of their fundamental properties. We focus on a discrete-time setting where local state spaces are finite and, in this specific context, build on the existing work of Shafer and Vovk, the main developers of the framework of game-theoretic upper expectations. In a first part, we study Shafer and Vovk's characterisation of a local upper expectation and show how it is related to Walley's behavioural notion of coherence. The second part consists in a study of game-theoretic upper expectations on a more global level, where several alternative definitions, as well as a broad range of properties are derived, e.g. the law of iterated upper expectations, compatibility with local models, coherence properties, … Our main contribution, however, concerns the continuity behaviour of these operators. We prove continuity with respect to non-increasing sequences of so-called lower cuts and continuity with respect to non-increasing sequences of finitary functions. We moreover show that the game-theoretic upper expectation is uniquely determined by its values on the domain of bounded below limits of finitary functions, and show in addition that, for any such limit, the limiting sequence can be constructed in such a way that the game-theoretic upper expectation is continuous with respect to this particular sequence.