Nonempty Disjoint Research Articles

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Equidistant sets on Alexandrov surfaces

We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.

Spectral conditions for edge connectivity and spanning tree packing number in (multi-)graphs

A multigraph is a graph with possible multiple edges, but no loops. Let t be a positive integer. Let Gt be the set of simple graphs (or multigraphs) such that for each G∈Gt there exist at least t+1 non-empty disjoint proper subsets V1,V2,…,Vt+1⊆V(G) satisfying V(G)∖(V1∪V2∪⋯∪Vt+1)≠ϕ and edge connectivity κ′(G)=e(Vi,V(G)∖Vi) for i=1,2,…,t+1. Let D(G) and A(G) denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) G, and let μi(G) be the ith largest eigenvalue of the Laplacian matrix L(G)=D(G)+A(G). In this paper, we investigate the relationship between μn−2(G) and edge connectivity or spanning tree packing number of a graph G∈G1, respectively. We also give the relationship between μn−3(G) and edge connectivity or spanning tree packing number of a graph G∈G2, respectively. Moreover, we generalize all the results about L(G) to a more general matrix aD(G)+A(G), where a is a real number with a≥−1.

Degenerate $ r $-truncated Stirling numbers

<abstract><p>For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.</p></abstract>

Path partition of planar graphs with girth at least six

A graph [Formula: see text] has a [Formula: see text]-partition if [Formula: see text] can be partitioned into two nonempty disjoint subsets [Formula: see text] and [Formula: see text] so that [Formula: see text] and [Formula: see text] are graphs whose components are paths of order at most [Formula: see text] and [Formula: see text], respectively. In this paper, we proved that every planar graph with girth at least six giving that [Formula: see text]-cycle is not intersecting with [Formula: see text]-cycle admits a [Formula: see text]-partition, where [Formula: see text] and [Formula: see text].

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number χt(G) is the least integer k for which G admits a coloring with k colors such that each color class induces a (t−1)-degenerate subgraph of G. So χ1 is the chromatic number and χ2 is the point arboricity. The point partition number χt with t≥1 was introduced by Lick and White. A graph G is called χt-critical if every proper subgraph H of G satisfies χt(H)<χt(G). In this paper we prove that if G is a χt-critical graph whose order satisfies |G|≤2χt(G)−2, then G can be obtained from two non-empty disjoint subgraphs G1 and G2 by adding t edges between any pair u,v of vertices with u∈V(G1) and v∈V(G2). Based on this result we establish the minimum number of edges possible in a χt-critical graph G of order n and with χt(G)=k, provided that n≤2k−1 and t is even. For t=1 the corresponding two results were obtained in 1963 by Tibor Gallai.

Three Representations for Set Partitions

The Set Partitioning Problem (SPP) aims to obtain non-empty disjoint subsets of objects such that their union equals the whole set of objects, and the partition meets some prespecified criteria. The ubiquity of SPP is impressive, given that it has a lot of theoretical and practical motivations. In the theoretical side, the study of the SPP is closely related to Bell numbers, Stirling numbers of the second kind, integer partitions, Eulerian numbers, Restricted Growth Strings (RGS), factoradic number system, power calculations, etc. In the practical side, SPP is intimately related to classification problems, clustering problems, reduction of dimensionality problems, and so on. In this work, three representations for instances of SPP are presented, these representations use: Restricted Growth Strings (RGS), factoradic number system, and a number system with a fixed base. Two cases for these representations will be presented: where the number of subsets is unbounded (i.e. the number of subsets can be the number of objects); and where the number of subsets is less than the number of objects. Bidirectional mappings between these three representations will be introduced, also the mapping among these three representations and the power of a base is defined. Given, that these three representations can be used to solve instances of SPP using exact, greedy, and metaheuristic algorithms, that require to do small changes to one possible solution and/or recombination of two possible solutions, definitions of mutation and recombination operators for the three representations will be shown. In order to motivate the use of the three representations for the solution of particular instances of SPP, it was decided to present their application to solve an instance of a set partition of integers problem (SPIP) using a simple genetic algorithm.

Two by two squares in set partitions

AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 &lt; min B2 &lt; ⋯ &lt; min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i &lt; j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

Edge connectivity, packing spanning trees, and eigenvalues of graphs

ABSTRACT Let be the set of simple graphs (or multigraphs) G such that for each there exists at least two non-empty disjoint proper subsets satisfying and edge connectivity for . A multigraph is a graph with possible multiple edges, but no loops. Let be the maximum number of edge-disjoint spanning trees of a graph G. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of , we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of and of a simple graph or a multigraph , respectively.

Bratteli–Vershik models for partial actions of ℤ

Let [Formula: see text] and [Formula: see text] be open subsets of the Cantor set with nonempty disjoint complements, and let [Formula: see text] be a homeomorphism with dense orbits. Building on the ideas of Herman, Putnam and Skau, we show that the partial action induced by [Formula: see text] can be realized as the Vershik map on an ordered Bratteli diagram, and that any two such diagrams are equivalent.

Strongly Connectedness in Closure Space

A Čech closure space (X, u) is a set X with Čech closure operator u: P(X) → P(X) where P(X) is a power set of X, which satisfies u𝝓=𝝓, A ⊆uA for every A⊆X, u (A⋃B) = uA⋃uB, for all A, B ⊆ X. Many properties which hold in topological space hold in closure space as well. A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei‟s are nonempty disjoint closed subsets of X, then X≠ E1∪E2∪. We further extend the concept of strongly connectedness in closure space. The aim of this paper is to introduce and study the concept of strongly connectedness in closure space.

Set partition asymptotics and a conjecture of Gould and Quaintance

The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B1,B2,…,Bk of non-empty disjoint subsets of S such that ⋃i=1kBi=S, as well as to certain classes of partitions of [n].

The existence of a linear horseshoe of continuous maps of dendrites

Assume that a continuous map f defined on a dendrite X has a horseshoe (A, B), where A and B are nonempty disjoint subcontinua in X. In this paper we obtain conditions for the structure of sets A and B under which some iteration of f has a linear horseshoe.

Jourdain, Russell and the Axiom of Choice: A New Document

June 25, 2012 (9:21 pm) E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1 060 red.wpd 1 Jourdain’s principal historical writings are gathered together in Selected Essays on the History of Set Theory and Logics (1906–1918), ed. I. Grattan-Guinness (Bologna: clueb, 1991) and his introduction to G. Cantor, Contributions to the Founding of the Theory of TransWnite Numbers, trans. and ed. Jourdain (La Salle, Ill.: Open Court, 1915; repr. New York: Dover, 1955). 2 I. Grattan-Guinness, Dear Russellz—zDear Jourdain: a Commentary on Russelly’s Logic, Based on His Correspondence with Philip Jourdain (London: Duckworth; New York: Columbia U. P., 1977); hereafter “RJy”. russell: the Journal of Bertrand Russell Studies n.s. 32 (summer 2012): 69–74 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 ocuments JOURDAIN, RUSSELL AND THE AXIOM OF CHOICE: A NEW DOCUMENT I. Grattan-Guinness Middlesex U. Business School Hendon, London nw4 4bt, uk Centre for Philosophy of Natural and Social Science, l.s.e. London wc2a 2ae, uk ivor2@mdx.ac.uk i.wcareer P hilip Edward Bertrand Jourdain (1879–1919) went up to Trinity College Cambridge in 1899 with a scholarship in mathematics. Two years later he attended a special College course in mathematical logic oTered by Bertrand Russell, the Wrst of its kind ever taught in a British university. This contact with Russell was of major consequence for his intellectual career, for he focused upon set theory and its history, and also on the histories of mathematical analysis and of logic, and aspects of the history and philosophy of mechanics and of science.1 He corresponded at length with Russell until his death in 1919; the exchange forms the core of my book on their relationship.2 From his youth Jourdain suTered from a creeping paralysis called “Friedreich ’s ataxia”. It prevented him from taking the Part 2 Tripos; thus he did not graduate as a Wrangler and so could not compete for a fellowship at Trinity. So he made his living from a few scholarships and from freelance writing as an June 25, 2012 (9:21 pm) E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1 060 red.wpd 70 i. grattan-guinness 3 E. Zermelo, “Beweis, dass jede Menge wohlgeordnet werden kann”, Mathematische Annalen 59 (1904): 514–16. 4 See especially H. Rubin and J.yE Rubin, Equivalents of the Axiom of Choice, 2nd ed. (Amsterdam: North-Holland, 1985). 5 The history of this/these axioms has been charted in much detail. See especially J. Cassinet and M. Guillemot, “L’axiome du choix dans les mathématiques de Cauchy (1821) à Gödel (1940)”, 2 vols. (U. of Toulouse double docteur d’état des sciences, 1983); F.yA. MedvedeT, Rannyaya istoriya aksiomi vibora (Moscow: Nauka, 1982); and G.yH. Moore, Zermelo’s Axiom of Choice … (New York: Springer, 1982). RJ discusses the axioms in the contexts of the correspondence. independent scholar. In particular, the American house The Open Court Publishing Company employed him, especially from 1912 on their principal journal The Monist, commissioning papers (and also some books) from colleagues, in particular Russell. He seems to have served as its general editor soon after the death in February 1919 of its founder editor, Paul Carus, but he died himself early in October. Those last months form the time-frame of this paper. 2.wobsession Jourdain’s principal research interest in the foundations of mathematics came to focus upon the denumerable “axiom of choice” in set theory and mathematics , as Ernst Zermelo named it a few years after introducing it in 1904.3 Take an inWnite ensemble of non-empty and pairwise disjoint sets and choose a member from each set in independent actions; then the axiom asserts that the collection of chosen members can always be regarded as a genuine set. But this manner of forming a set roused considerable doubts and opposition among many (though not all) set-theorists and logicians. Some of them rejected the axiom altogether; others tried to reprove theorems that used the axiom by proofs that avoided it; several sought hopefully more congenial assumptions that were logically equivalent to the axiom, and eventually many came to light.4 Logicists such...

Counting Subwords in a Partition of a Set

A partition $\pi$ of the set $[n]=\{1,\ldots,n\}$ is a collection $\{B_1,\ldots ,B_k\}$ of nonempty disjoint subsets of $[n]$ (called blocks) whose union equals $[n]$. In this paper, we find explicit formulas for the generating functions for the number of partitions of $[n]$ containing exactly $k$ blocks where $k$ is fixed according to the number of occurrences of a subword pattern $\tau$ for several classes of patterns, including all words of length 3. In addition, we find simple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of $[n]$ containing $k$ blocks, providing both algebraic and combinatorial proofs.

Convex Partitions of Graphs induced by Paths of Order Three

Graphs and Algorithms A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.

Enumerating set partitions by the number of positions between adjacent occurrences of a letter

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

Smooth partitions and Chebyshev polynomials

A partition of the set [n]={1, 2, …, n} is a collection {B1, …, Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. A partition of [n] is said to be smooth if i ∈ Bs implies that i + 1 ∈ Bs−1 ∪ Bs ∪ Bs+1 for all i ∈ [n − 1] (B0 = Bk + 1 = ∅). This paper presents the generating function for the number of k-block, smooth partitions of [n], written in terms of Chebyshev polynomials of the second kind. There follows a formula for the number of k-block, smooth partitions of [n] written in terms of trigonometric sums. Also, by first establishing a bijection between the set of smooth partitions of [n] and a class of symmetric Dyck paths of semilength 2n − 1, we prove that the counting sequence for smooth partitions of [n] is Sloane's A005773.

On the complexity of variations of equal sum subsets

The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems.

Extremes of conditioned elliptical random vectors

Let { X n , n ⩾ 1 } be iid elliptical random vectors in R d , d ≥ 2 and let I , J be two non-empty disjoint index sets. Denote by X n , I , X n , J the subvectors of X n with indices in I , J , respectively. For any a ∈ R d such that a J is in the support of X 1 , J the conditional random sample X n , I | X n , J = a J , n ≥ 1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X 1 has distribution function in the max-domain of attraction of a univariate extreme value distribution.