Disjoint Nonempty Subsets Research Articles

Geometry of the space of triangulations of a compact manifold

In this paper we study the spaceT M of triangulations of an arbitrary compact manifoldM of dimension greater than or equal to four. This space can be endowed with the metric defined as the minimal number of bistellar operations required to transform one of two considered triangulations into the other. Recently, this space became and object of study in Quantum Gravity because it can be regarded as a “toy” discrete model of the space of Riemannian structures onM. Our main result can be informally explained as follows: LetM be either any compact manifold of dimension greater than four or any compact four-dimensional manifold from a certain class described in the paper. We prove that for a certain constantC>1 depending only on the dimension ofM and for all sufficiently largeN the subsetT M(N) ofT M formed by all triangulations ofM with ≦N simplices can be represented as the union of at least [C N] disjoint non-empty subsets such that any two of these subsets are “very far” from each other in the metric ofT M. As a corollary, we show that for any functional from a very wide class of functionals onT M the number of its “deep” local minima inT M(N) grows at least exponentially withN, whenN→∞.

Sets of Prime Numbers Satisfying a Divisibility Condition

We study setsPofkprimes that satisfy the condition gcd(∏A−∏B, ∏P)=1 wheneverAandBare disjoint non-empty subsets ofP. It is known that such sets of primes exist for all positive integers k.It is of interest to know the asymptotic behavior ofnk, the smallest natural number that is the product of k such primes. In this paper we derive asymptotic bounds fornk.

High-Energy Behavior of Total Scattering Cross Sections for 3-body Quantum Systems

Let H be the Schrodinger operator obtained by separating the kinetic energy of the center of mass from H. H acts in M :~L(R), and its explicit form depends on the coordinates of R. We adopt the Jacobi coordinates. A partition of the set {1, 2, 3} into nonempty disjoint subsets is called a cluster decomposition. We call {(1), (2), (3)} (resp. {(i, /), k} , i<j) a 3-cluster decomposition (resp. a 2-cluster decomposition). We denote by Az the set of all 2-cluster decompositions. For a^A2 with a={(i, j), k] , we define the Jacobi coordinates (*a, yJ by

On the equal-subset-sum problem

Given a set S of n integers, the problem is to decide whether there exist two disjoint nonempty subsets S 1, S 2⊆ S whose elements sum up to the same value. We show this problem to be NP-complete and we present an approximation algorithm with worst-case ratio 1.324 for the optimization version of this problem.

COMPUTATION OF DIMENSION POLYNOMIALS

The consideration of differential versions of Hilbert dimension polynomials is due to A. Einstein [1] and E. Kolchin [2] (one can find the coverage of the theory of differential dimension polynomials in [6]). In this paper we introduce the notion of a dimension polynomial of a subset of ℕm associated with arbitrary partition of the set {1,…, m} into disjoint nonempty subsets (m∈ℕ, ℕ denoting the set of all nonnegative integers). The theory of such polynomials is developed. The importance of our considerations is connected with the fact that the computation of differential and difference dimen sion polynomials may be reduced to the computation of some dimension polynomials of subsets of ℕm where m∈ℕ (see [3, p. 115], [5]). We also give some methods and algorithms for computation of dimension polynomials.

Multiterminal xcut problems

Ani-j xcut of a setV={1, ...,n} is defined to be a partition ofV into two disjoint nonempty subsets such that bothi andj are contained in the same subset. When partitions are associated with costs, we define thei-j xcut problem to be the problem of computing ani-j xcut of minimum cost. This paper contains a proof that the\((\begin{array}{*{20}c} n \\ 2 \\ \end{array} )\) minimum xcut problems have at mostn distinct optimal solution values. These solutions can be compactly represented by a set ofn partitions in such a way that the optimal solution to any of the problems can be found inO(n) time. For a special additive cost function that naturally arises in connection to graphs, some interesting properties of the set of optimal solutions that lead to a very simple algorithm are presented.

Certain ideals for which I1· Ig= I1∩·∩ Ig

Let I 1,·, I g be proper ideals of a Noetherian ring R. Then it is shown that I 1·I g=I 1∩·∩I g=∩ {I j:(I 1·I j−1I j>+1 ·I g) n; j=1,·,g} for all n≥1 if and only if each P ∈ Ass( R/ I 1· I g ) contains exactly one of the ideals I j . As an application it is shown that if I 1,·, I g are generated by disjoint nonempty subsets of a permutable R-sequence, then I m 1 1·I m g g=I m 1 1∩·∩I m g g=∩ }I m j j (I 1·I j−1I j+1 ·I g) n; j=1,·, g} and Ass( R/ I m 1· I m g )=∪ g j =1 Ass ( R/ I j ) for all positive integers m 1, ·, m g , n. Then asymptotic and essential theory versions of these results are proved.

A decomposition for strongly perfect graphs

AbstractA graph G is strongly perfect if every induced subgraph H of G contains a stable set that meets all the maximal cliques of H. We present a graph decomposition that preserves strong perfection: more precisely, a stitch decomposition of a graph G = (V, E) is a partition of V into nonempty disjoint subsets V1, V2 such that in every P4 with vertices in both Viapos;s, each of the three edges has an endpoint in V1 and the other in V2.We give a good characterization of graphs that admit a stitch decomposition and establish several results concerning the stitch decomposition of strongly perfect graphs.

Partitions and filters

In [2], Carlson and Simpson proved a dualized version of Ramsey's theorem obtained by coloring partitions of ω instead of subsets of ω. It was at the suggestion of Simpson that the author undertook to study the notion dual to that of a Ramsey ultrafilter. After stating the basic terminology and notation used in the paper in §1, in §2 we establish some basic properties of the lattice of all partitions of a cardinal κ. §3 is devoted to the study of families of pairwise disjoint partitions of ω. §4 is concerned with descending sequences of partitions. In §5, we give some examples of filters of partitions. Properties of such filters are discussed in §6. Co-Ramsey filters are introduced in §7, and it is shown how they can be associated with Ramsey ultrafilters. The main result of §8 is Proposition 8.1, which asserts the existence of a co-Ramsey filter under the continuum hypothesis.We use standard set theoretic conventions and notation. Let κ be a cardinal. We set κ* = κ − {0}. For every ordinal α ≤ κ, (κ)α denotes the set of those sequences X(ν), ν &lt; α, of pairwise disjoint nonempty subsets of κ such that ⋃ν&lt;αX(ν) = κ, and ⋂X(ν) &lt; ⋂X(ν′) whenever ν &lt; ν′. We also let (κ)≤α = ⋃β≤α(κ)β and (κ)&lt;α = ⋃β&lt;α(κ)β. Given X ∈ (κ)α, we put xν = ⋂X(ν) for every ν &lt; α, and we denote by Ax the set of all xν, 0 &lt; ν &lt; α.

Random walks on groups

A finite group G is partitioned into nonempty disjoint subsets C 0, C 1,…, C m such that for every g 1∈ C i the number of ordered pairs ( g 2, g 3) for which g 2∈ C ν , g 3∈ C j , and g 1 g 2 = g 3 is independent of the particular choice of g 1. A sequence of mutually independent random elements γ 0, γ 1,…, γ n ,… is chosen in G in such a way that for n = 1,2,… the probability P{γ n = g} depends only on the class C ν which contains g. Let ξ n = j if γ 0γ 1⋯γ n ∈ C j . Then {ξ n; n ≧ 0} is a homogeneous Markov chain with state space I = {0,1,…, m}. The aim of this paper is to determine the n-step transition probabilities of the Markov chain {ξ n; n ≧ 0} . The results derived in this paper lead also to a probabilistic interpretation and a generalization of group characters.

A Norm Reducing Property for Isotonized Cauchy Mean Value Functions

We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$ has the following norm reducing property: $\max_{s\in S} |\hat{\alpha}(s) - \theta(s)| \leqq \max_{s\in S} |\alpha(s) - \theta(s)|$ for all isotone $\theta(\bullet)$. Computation algorithms for $\hat{\alpha}(\bullet)$ are discussed and the norm reducing property is shown to give consistency results in several isotonic regression problems.

Aposyndetic properties of unicoherent continua

In the first part of this paper the structure of w-aposyndetic continua is studied. In particular, those continua which are w-aposyndetic but fail to be in + l)-aposyndetic are investigated. Unicoherence is shown to be a sufficient condition for an %-aposyndetic continuum to be in + l)-aposyndetic. In the final portion of the paper a stronger form of unicoherence is defined. As a point-wise property, aposyndesis and connected im kleinen are shown to be equivalent in continua with this property. Throughout this paper a continuum is a compact connected metric space and M will denote a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N. Suppose pe M and F is a closed subset of M such that p ί F. M is aposyndetic at p with respect to F if there is a subcontinuum N of M such that p e int N c N c M — F. Let n be a positive integer. If M is aposyndetic at p with respect to each subset of M consisting of n points, then M is n-aposyndetic at p. M is n-aposyndetic if it is ^-aposyndetic at each point. By convention if M is 1-aposyndetic then M is said to be aposyndetic. For other terms not defined herein, see [3], [4] and [6]. LEMMA 1. Suppose M is n-aposyndetic, peM,F is a subset of M — {p} consisting of n + 1 points, and M is not aposyndetic at p with respect to F. If F 1 and F2 are disjoint nonempty subsets of F such that F = Fι U F21 there exist subcontinua H and K such that Fιa H — K, F2 c K- H,pemt(HΓ\K), and M= H\JK.

On partitions of a finite set

A pair of partitions π 1, π 2 of a finite set S into disjoint non-empty subsets will be called conjugate if for each s∈ S, the ordered pair ( ν 1 ( s), ν 2 ( s)) determines s, where ν i ( s) denotes the cardinality of the subset of π i to which s belongs. In this note we show that S has a pair of conjugate partitions if and only if the cardinality of S is not equal to 2, 5, or 9. Partitions of this type provide a short solution to a problem arising in circuit theory.