scholarly journals Intersecting and Cross-Intersecting Families of Labeled Sets

10.37236/884 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Peter Borg
Keyword(s):  
Sharp Bound ◽  
General Inequality ◽  
The Family ◽  
Intersecting Families ◽  
Special Case

A family ${\cal A}$ of sets is said to be intersecting if any two sets in ${\cal A}$ intersect. Families ${\cal A}_1, ..., {\cal A}_p$ are said to be cross-intersecting if, for any $i, j \in \{1, ..., p\}$ such that $i \neq j$, any set in ${\cal A}_i$ intersects any set in ${\cal A}_j$. For ${\bf k} = (k_1, ..., k_n) \in {\Bbb N}^n$, $2 \leq k_1 \leq ... \leq k_n$, let ${\cal L}_{\bf{k}}$ be the family of labeled $n$-sets given by ${\cal L}_{\bf{k}} := \{\{(1,l_1), ..., (n,l_n)\} \colon l_i \in \{1, ..., k_i\}, i = 1, ..., n\}$. We point out a relationship between intersecting families and cross-intersecting families of labeled sets, and we show that, if ${\cal A}_1, ..., {\cal A}_p$ are cross-intersecting sub-families of ${\cal L}_{\bf{k}}$, then $$ \sum_{j = 1}^p |{\cal A}_j| \leq \left\{ \matrix{ k_1k_2...k_n & \hbox{if $p \leq k_1$};\cr pk_2...k_n & \hbox{if $p \geq k_1$}.\cr } \right. $$ We also determine the cases of equality. We then obtain a more general inequality, a special case of which is a sharp bound for cross-intersecting families of permutations.

scholarly journals A Sharp Bound for the Product of Weights of Cross-Intersecting Families

10.37236/5782 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Peter Borg
Keyword(s):  
General Setting ◽  
Sharp Bound ◽  
Integer Sequences ◽  
Sharp Bounds ◽  
The Family ◽  
Intersecting Families

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $\{1, \dots, n\}$ of size at most $k$, and let $\mathcal{S}_{n,k}$ denote the family of sets in ${[n] \choose \leq k}$ that contain $1$. The author recently showed that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $|\mathcal{A}||\mathcal{B}| \leq \mathcal{S}_{m,r}||\mathcal{S}_{n,s}|$. We prove a version of this result for the more general setting of \emph{weighted} sets. We show that if $g : {[m] \choose \leq r} \rightarrow \mathbb{R}^+$ and $h : {[n] \choose \leq s} \rightarrow \mathbb{R}^+$ are functions that obey certain conditions, $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $$\sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{S}_{m,r}} g(C) \sum_{D \in \mathcal{S}_{n,s}} h(D).$$The bound is attained by taking $\mathcal{A} = \mathcal{S}_{m,r}$ and $\mathcal{B} = \mathcal{S}_{n,s}$. We also show that this result yields new sharp bounds for the product of sizes of cross-intersecting families of integer sequences and of cross-intersecting families of multisets.

scholarly journals On the union of intersecting families

2019 ◽  
Vol 28 (06) ◽  
pp. 826-839
Author(s):  
David Ellis ◽  
Noam Lifshitz
Keyword(s):  
Fixed Integer ◽  
Empty Intersection ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
Element Set

AbstractA family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

A family of densities derived from the three-parameter Dirichlet process

2002 ◽  
Vol 39 (4) ◽  
pp. 764-774 ◽  
Author(s):  
Matthew A. Carlton
Keyword(s):  
State Space ◽  
Density Function ◽  
Dirichlet Process ◽  
Dirichlet Distribution ◽  
The State ◽  
Measurable Partition ◽  
Multivariate Distributions ◽  
The Family ◽  
Multivariate Density ◽  
Special Case

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

scholarly journals Exact inbreeding coefficient and effective size of finite populations under partial sib mating.

Genetics ◽  
1995 ◽  
Vol 140 (1) ◽  
pp. 357-363
Author(s):  
J Wang
Keyword(s):  
Poisson Distribution ◽  
Family Size ◽  
Inbreeding Coefficient ◽  
Stochastic Simulations ◽  
Effective Size ◽  
Finite Populations ◽  
A Value ◽  
The Family ◽  
Sib Mating ◽  
Special Case

Abstract An exact recurrence equation for inbreeding coefficient is derived for a partially sib-mated population of N individuals mated in N/2 pairs. From the equation, a formula for effective size (Ne) taking second order terms of 1/N into consideration is derived. When the family sizes are Poisson or equally distributed, the formula reduces to Ne = [(4 - 3 beta) N/(4 - 2 beta)] + 1 or Ne = [(4 - 3 beta) N/(2 - 2 beta)] - 8/(4 - 3 beta), approximately. For the special case of sib-mating exclusion and Poisson distribution of family size, the formula simplifies to Ne = N + 1, which differs from the previous results derived by many authors by a value of one. Stochastic simulations are run to check our results where disagreements with others are involved.

An Argument for Treating Children as a ‘Special Case’

2018 ◽  
Author(s):  
Lucinda Ferguson
Keyword(s):  
Moral Consideration ◽  
Legal Language ◽  
Social Construct ◽  
Legal Decisions ◽  
Family Unit ◽  
The Social ◽  
The Family ◽  
Family Autonomy ◽  
Children's Interests ◽  
Special Case

This chapter’s argument stems from the premise that legal language should speak for itself. The ‘paramountcy’ principle suggests the prioritisation of children’s interests, and ‘children’s rights’ suggests some aspect of distinctiveness to children’s interests. But there is academic consensus in respect of both that children’s interests cannot and should not be prioritised over those of others. This chapter examines the justification for the contrary perspective, and for treating children as a prioritised ‘special case’ in all legal decisions affecting them. Four key counter-arguments frame the discussion. First, the ‘social construct’ objection: as a social construct, childhood cannot sustain the prioritisation of children’s interests over those of others. Second, the ‘vulnerability’ objection: children’s vulnerability is either not unique or suggests dependency or interdependency, not prioritisation. Third, the ‘family autonomy’ objection: parents’ rights and the family unit justify deference of children’s interests. Fourth, the ‘equality’ objection: equal moral consideration makes prioritisation unjustifiable.

Moral Formation in the Family

2019 ◽  
pp. 392-418
Author(s):  
Daniel Lapsley
Keyword(s):  
Moral Development ◽  
Personality Development ◽  
Future Research ◽  
Moral Formation ◽  
The Family ◽  
Third Line ◽  
Event Representations ◽  
Capacity For Change ◽  
Special Case

Several lessons are drawn for future research on parenting and moral formation on the basis of an historical perspective on the moral development research program. One is that sociomoral formation is a special case of personality development that draws attention to the role of attachment, event representations, autobiographical memory, and temperament for organizing dispositional coherence around morality. A second is that research on moral development in the family will be increasingly informed by study of the moral self of infancy and on the importance of early life rearing experience, widely discussed in disparate literatures from object relations to epigenetics. A third line of research might focus on parenting characteristics “beyond parenting style” to include parents’ ideological and faith commitments, their mindsets with respect to children’s personality and capacity for change, and their own sense of generativity.

scholarly journals Remark on a criterion for common transversals

1969 ◽  
Vol 10 (1) ◽  
pp. 66-67 ◽  
Author(s):  
Hazel Perfect
Keyword(s):  
Cardinal Number ◽  
General Theorem ◽  
Common Transversal ◽  
The Family ◽  
Special Case

All sets considered will be finite, and |x| will denote the cardinal number of the set X.Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.

scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (01) ◽  
pp. 266-279
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

A family of densities derived from the three-parameter Dirichlet process

2002 ◽  
Vol 39 (04) ◽  
pp. 764-774
Author(s):  
Matthew A. Carlton
Keyword(s):  
State Space ◽  
Density Function ◽  
Dirichlet Process ◽  
Dirichlet Distribution ◽  
The State ◽  
Measurable Partition ◽  
Multivariate Distributions ◽  
The Family ◽  
Multivariate Density ◽  
Special Case

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

scholarly journals STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES

2017 ◽  
Vol 82 (2) ◽  
pp. 608-628 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
DAVID FERNÁNDEZ-DUQUE
Keyword(s):  
Initial Segment ◽  
Provability Logic ◽  
Strong Completeness ◽  
Scattered Space ◽  
The Family ◽  
Image Position ◽  
Special Case

AbstractGiven a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ, we define a topology $\tau _{ + \lambda } $ in such a way that τ+0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the resulting sequence {τ+λ}λ∈Ord coincides with the family of topologies $\left\{ {\mathcal{I}_\lambda } \right\}_{\lambda \in Ord} $ used by Icard, Joosten, and the second author to provide semantics for polymodal provability logics.We prove that given any scattered space $\mathfrak{X}$ of large-enough rank and any ordinal λ > 0, GL is strongly complete for τ+λ. The special case where $\mathfrak{X} = \omega ^\omega + 1$ and λ = 1 yields a strengthening of a theorem of Abashidze and Blass.

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