Families Of Finite Sets Research Articles

Best approximation with geometric constraints

In this paper, we consider a finite family of sets (geometric constraints) F1,F2,…,Fr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{1}, F_{2},\\ldots , F_{r}$$\\end{document} in the Euclidean space Rn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n.$$\\end{document} We show under mild conditions on the geometric constraints that the “perturbation property” of the constrained best approximation from a nonempty closed set K∩F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K \\cap F$$\\end{document} is characterized by the “convex conical hull intersection property” (CCHIP in short) at a reference feasible point in F. In this case, F is the intersection of the geometric constraints F1,F2,…,Fr,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{1}, F_{2},\\ldots , F_{r},$$\\end{document} and K is a nonempty closed convex set in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} such that K∩F≠∅.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K \\cap F \ e \\emptyset .$$\\end{document} We do this by first proving a dual cone characterization of the contingent cone of the set F. Finally, we obtain the “Lagrange multiplier characterizations” of the constrained best approximation. Several examples are given to illustrate and clarify our results.

Families of finite sets in which no set is covered by the union of the others

Let ℱ be a finite nonempty family of finite nonempty sets. We prove the following: (1) ℱ satisfies the condition of the title if and only if for every pair of distinct subfamilies {A1,…,Ar}, {B1,…,Bs} of ℱ, ⋃i=1rAi≠⋃i=1sBi. (2) If ℱ satisfies the condition of the title, then the number of subsets of ⋃A∈ℱA containing at least one set of ℱ is odd. We give two applications of these results, one to number theory and one to commutative algebra.

Integration with filters

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.

One note on positive A-computable numberings

A family of finite sets is said to be strongly coverable if it has a subfamily with the computable set of canonical indices, such that every set of the coverable family is a subset of some set of the covering subfamily. We prove that a set [Formula: see text] is low if and only if every finite strongly coverable [Formula: see text]-computable family has a positive [Formula: see text]-computable numbering. We also show that in the previous criterion the strong coverability of a finite family cannot be skipped.

A generalization of balanced tableaux and marriage problems with unique solutions

We consider families of finite sets that we call shellable and that have been characterized by Chang and by Hirst and Hughes as being the families of sets that admit unique solutions to Hall's marriage problem. In this paper, we introduce a natural generalization of Edelman and Greene's balanced tableaux that involves families of sets that satisfy Hall's marriage Condition and certain words in $[n]^m$, then prove that shellable families can be characterized by a strong existence condition relating to this generalization. As a consequence of this characterization, we show that the average number of such generalized tableaux is given by a generalization of the hook-length formula.

On the computing powers of [formula omitted]-reductions of insertion languages

We investigate the computing power of the following language operation %: Given two languages L1 over Σ and L2 over Γ with Γ⊂Σ, we consider the language operation L1%L2={u0u1⋯un|∃u=u0v1u1⋯vnun∈L1 and ∃vi∈L2(1≤∀i≤n)}. In this case we say that L(=L1%L2) is the L2-reduction of L1. This is extended to the language families as follows: L1%L2={L1%L2|L1∈L1,L2∈L2}. Among many works concerning Dyck-reductions, for the family of recursively enumerable languages RE, it was shown that LIN%{EQ}=RE (Jantzen & Petersen, 1994) with EQ={xnx‾n|n∈N} and that min-LIN%{D2}=RE (Hirose & Okawa, 1996, and Latteux & Turakainen, 1990), where LIN and min-LIN are the families of linear and minimal linear context-free languages, respectively.In this paper, we show that each recursively enumerable language L can be represented in the form L=K%D, for some K∈INS30 and a Dyck language D, where INS⁎0 (INS30) denotes the family of insertion languages (insertion languages where the maximum length of the string to be inserted is 3). We can refine it as INS⁎0%{D2}=RE, where D2 denotes the Dyck language over binary alphabet. For context-free languages, we show that INS30%F=CF, where F is the family of finite sets. This also derives that INS⁎0%{MIR}=CF with MIR={xx‾R|x∈{0,1}⁎}. Further, for regular languages, it is shown that each regular language R can be represented in the form R=K%F, for some K∈INS20 and a finite set F={abb‾a‾|a∈V}. We also present some results which characterize the computability and properties of L in the framework of L2-reduction of L1.It is intriguing to note that, from the DNA computing point of view, the notion of L-reduction is naturally motivated by a molecular biological functioning well-known as DNA(RNA) splicing occurring in most eukaryotic genes.

Well-Graded Families and the Union-Closed Sets Conjecture

The union-closed sets conjecture states that if a finite family of sets $\mathcal{F}$ is union-closed, then there must be some element contained in at least half of the sets of $\mathcal{F}$. In this work we study the relationship between the union-closed sets conjecture and union-closed families that have the property of being well-graded. In doing so, we show how the density and other properties are affected by the extra structure contained in well-graded families, and we also give several conditions under which well-graded families satisfy the union-closed sets conjecture.

Densities for sets of natural numbers vanishing on a given family

Abstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density.At the open problem session of the Workshop “Densities and their application”, held at St. Étienne in July 2013, G. Grekos asked a question whether there is a “nice” abstract upper density, whose the family of null sets is precisely a given ideal of subsets of N, where “nice” would mean the properties of the familiar densities consider in number theory.M. Di Nasso, R. Jin (2018) [3] showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when “nice” density means translation invariant and rich density (i.e. density which is onto the unit interval).In this paper we extend their result to all ideals with the Baire property.

Extremal Union-Closed Set Families

A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let $${{\mathcal {F}}}$$ be a union-closed family and $$\bigcup _{F\in {\mathcal {F}}} F$$ be the universe of $${\mathcal {F}}$$. Among all union-closed families isomorphic to $${{\mathcal {F}}}$$, we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.

Note on the Union-Closed Sets Conjecture

The union-closed sets conjecture states that if a finite family of sets $\mathcal{A} \neq \{\varnothing\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\mathcal{A}$, is at least $\frac{1}{2} \log_2 |\mathcal{A}|$. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show that Reimer's condition alone is not enough to imply that there is an element in at least half the sets.

Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm

In this article we consider a consistent convex feasibility problem in a real Hilbert space defined by a finite family of sets $C_i$. We are interested, in particular, in the case where for each $i$, $C_i={Fix} U_i=\{z\in \mathcal H\mid p_i(z)=0\}$, $U_i\colon\mathcal H\rightarrow \mathcal H$ is a cutter and $p_i\colon\mathcal H\rightarrow [0,\infty)$ is a proximity function. Moreover, we make the following assumption: the computation of $p_i$ is at most as difficult as the evaluation of $U_i$ and this is at most as difficult as projecting onto $C_i$. We study a double-layer fixed point algorithm which applies two types of controls in every iteration step. The first one---the outer control---is assumed to be almost cyclic. The second one---the inner control---determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions. The convergence results presented in this manuscript depend on the conditions which, first, bind together the sets, the ope...

Topological equivalences of $$\mathbf {CUT}$$ CUT and $$\mathbf {CUT(Fin)}$$ CUT ( Fin )

In this note, two well-known topological facts regarding cofinite and cocountable-like topologies over uncountable sets are shown to be equivalent either to the Countable Union Theorem or to the Countable Union Theorem for countable families of finite sets.

Systems of distant representatives in Euclidean space

Given a finite family of sets, Hall's classical marriage theorem provides a necessary and sufficient condition for the existence of a system of distinct representatives for the sets in the family. Here we extend this result to a geometric setting: given a finite family of objects in the Euclidean space (e.g., convex bodies), we provide a sufficient condition for the existence of a system of distinct representatives for the objects that are also distant from each other. For a wide variety of geometric objects, this sufficient condition is also necessary in an asymptotic sense (i.e., apart from constant factors, the inequalities are the best possible). Our methods are constructive and lead to efficient algorithms for computing such representatives.

Set Graphs. V. On representing graphs as membership digraphs

An undirected graph is commonly represented as a set of vertices and a set of doubletons of vertices; but one can also represent vertices by finite sets so as to ensure that membership mimics, over those sets, the edge relation of the graph. This alternative modeling, applied to connected claw-free graphs, recently gave crucial clues for obtaining simpler proofs of some of their properties (e.g., Hamiltonicity of the square of the graph). This paper adds a computer-checked contribution. On the one hand we discuss our development, by means of the Ref verifier, of two theorems on representing graphs by families of finite sets: a weaker theorem pertains to general graphs, and a stronger one to connected claw-free graphs. Before proving those theorems, we must show that every graph admits an acyclic, weakly extensional orientation, which becomes fully extensional when connectivity and claw-freeness are met. This preliminary work enables injective decoration, a la Mostowski, of the vertices by the sought-for finite sets. By this new scenario, we complement our earlier formalization with Ref of two classical properties of connected claw-free graphs. On the other hand, our present work provides another example of the ease with which graph-theoretic results are proved with the Ref verifier. For example, we managed to define and exploit the notion of connected graph without resorting to the notion of path.

The Boolean prime ideal theorem and products of cofinite topologies

We show: The Boolean Prime Ideal theorem is equivalent to each one of the statements: “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

Studying designs via multisets

A new method to study families of finite sets, in particular t-designs, by studying families of multisets (also called lists) and their relationships with families of sets, is developed. Notion of the tag for a subset defined earlier by one of the authors is extended to a submultiset. A new concept t-(v, k, ?) list design is defined and studied. Basic existence theory for designs is extended to a new set up of list designs. In particular tags are used to prove that signed t-list designs exist whenever necessary conditions are satisfied. The concepts of homomorphisms and block spreading are extended to this new set up.

Unique decipherability in the additive monoid of sets of numbers

Sets of integers form a monoid, where the product of two sets A and B is defined as the set containing a+b for all $a\in A$ and $b\in B$ . We give a characterization of when a family of finite sets is a code in this monoid, that is when the sets do not satisfy any nontrivial relation. We also extend this result for some infinite sets, including all infinite rational sets.

On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

Omega-limit sets for spiral maps

We investigate a class of homeomorphisms of a cylinder, with alltrajectories convergent to the cylinder base and one fixed point inthe base. Let A be a nonempty finite or countable family ofsets, each of which can be a priori an $\omega$-limit set. Then thereis a homeomorphism from our class, for which A is the family ofall $\omega$-limit sets.

Enumeration Degrees and Enumerability of Familes

We study the enumerability of families relative to the enumeration degrees. It is shown that if a family of finite sets is e-reducible to every non-zero e-degree, then the family is computably enumerable (c.e). On the another hand, we will find a non-c.e. family which is e-reducible to all non-zero e-degree. This allows to construct a model, whose (extended) degree spectrum coincides with the non-zero e-degrees.