Collection Of Finite Sets Research Articles

A filter on a collection of finite sets and Eberlein compacta

We introduce a σ-ideal on ω1×ω1 and a filter on the collection of graphs of strictly decreasing partial functions on ω1 taking values in ω1. We use them to prove that a certain space is a non-bisequential Eberlein compactum.

Convolution Estimates and Number of Disjoint Partitions

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate the number from above by $|X|^{c(n)}$ where $$c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}.$$ This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.

A Bound on Partitioning Clusters

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound$$ | \{ (A_1,A_2,A) \in X \times X \times X: A = A_1 \uplus A_2 \} | \leq |X|^{3/p} $$where $|X|$ denotes the cardinality of $X$, and $p := \log_3 \frac{27}{4} = 1.73814\dots$, so that $\frac{3}{p} = 1.72598\dots$. Furthermore, the exponent $p$ cannot be replaced by any larger quantity. This improves upon the trivial bound of $|X|^2$. The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification.In a similar vein, we show that for any subset $A$ of a discrete cube $\{0,1\}^n$, the additive energy of $A$ (the number of quadruples $(a_1,a_2,a_3,a_4)$ in $A^4$ with $a_1+a_2=a_3+a_4$) is at most $|A|^{\log_2 6}$, and that this exponent is best possible.

Some theorems on passing from local to global presence of properties of functions

When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words "a subset of"). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some $n$, the class consists of the potentially partial recursive functions of $n$ variables, and the collection consists of sets of $n$-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and the collection consists of subsets of the first of them (the condition is also necessary under a weak assumption on the second one). The third theorem is of a similar character but, instead of continuity, it concerns computability in the spirit of the one in effective topological spaces.

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowskiʼs study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X 2 has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin–Keisler order on βω. We also introduce and study the concepts of < κ-thin and superslim dense sets.

Determining Functionals for the Strongly Damped Nonlinear Wave Equation

We consider the existence of a wide collection of finite sets of functionals on the phase space that completely determines asymptotic behavior of solutions to the strongly damped nonlinear wave equation. The proof makes use of energy methods and the concept of the completeness defect. We also show that the number of determining nodes is two, that is, the asymptotic behavior of solutions is determined by the values of two sufficiently close points in the interval [0, 1].

On determining functionals for stochastic navier-stokes equations

We prove the existence of a wide collection of finite sets of functionals that completely determine the long-term behaviour of solutions to 2D Navier-Stokes equations with random initial data and excited by an additive white noise. This collection contains finite sets of determining modes, nodes and local volume averages. We also show that determining functionals can be defined on only one of the components of the velocity vector. To characterize sets of determining functionals we invoke the concept of completeness defect. Our method is general and can be applied to other dissipative evolutionary infinite-dimensional equations driven by additive stochastic perturbations

Union-closed families

A union-closed family F is a finite collection of sets not all empty, such that any union of elements of F is itself an element of F . Peter Frankl conjectured in 1979 that for any such family, there is an element in at least half of its sets. But the problem remains unsolved. We find a number of equivalent conjectures, and we prove the conjecture in special cases, including for example all families involving up to seven elements or having up to 28 sets, extending the previously known result for up to 18 sets. We also prove a general theorem stating exactly when a subfamily is enough to guarantee the existence of an element from the subfamily which is in half the sets of the whole family.

Cardinality questions concerning semilattices of finite breadth

It is shown that if L is a lattice in which every element has only finitely many predecessors and (∗) every element has no more than k immediate predecessors, for some positive integer k, then |L| ⩽ ℵk−1. An example is constructed in which k = 2 and |L| = ℵ1, but the question of whether |L| = ℵk−1 is possible for k > 2 is left unanswered. The conclusion |L|⩽ℵk−1 also holds if, instead of (∗), we substitute the weaker condition (∗)′: foranysubsetFofk + 1 elementsofL, x ⩽ sup(F⧹{x}) forsomex ∈ F. If, in addition, it is assumed that L satisfies a modularity condition, then it turns out that L must in fact be countable. For sets and set operations, the following results can be stated: if F is a collection of finite sets which is closed under finite union (resp. closed under finite intersection and directed upward) and F has the property that for any k + 1 sets in F, one of the sets is contained in the union (resp. contains the intersection) of the others, then |F| ⩽ ℵk−1. More generally, we show that if L is a join-semilattice in which every element has fewer than ℵλ predecessors and L satisfies (∗)′, then |L| ⩽ ℵλ+k−1. (This turns out to be an application of a result of Erdös and Hajnal on set-mappings.) When ℵλ is regular, an example is constructed in which k = 2 and |L| = ℵλ + 1.

Classifications of generalized index sets of open classes

The Rice-Shapiro Theorem [4] says that the index set of a class of recursively enumerable (r.e.) sets is r.e. if and only if consists of all sets which extend an element of a canonically enumerable sequence of finite sets. If an index of a difference of r.e. (d.r.e.) sets is defined to be the pair of indices of the r.e. sets of which it is the difference, then the following generalization due to Hay [3] is obtained: The index set of a class of d.r.e. sets is d.r.e. if and only if is empty or consists of all sets which extend a single fixed finite set. In that paper Hay also classifies index sets of classes consisting of d.r.e. sets which extend one of a finite collection of finite sets. These sets turn out to be finite Boolean combinations of r.e. sets. The question then arises “What about the classification of the index set of a class consisting of d.r.e. sets which extend an element of a canonically enumerable sequence of finite sets?” The results in this paper come from an attempt to answer this question.Since classes of sets which are Boolean combinations of r.e. sets form a hierarchy (the finite Ershov hierarchy, see Ershov [1]) with the r.e. and d.r.e. sets respectively levels 1 and 2 of this hierarchy, we may define index sets of classes of level n sets. If is a class of level n sets which extend some element of a canonically enumerable sequence of finite sets and if we let co-, then we extend the original classification question to the classification of the index sets of the classes and co-.Now if the sequence of finite sets enumerates only finitely many sets or if only finitely many of the finite sets are minimal under inclusion, then it is a routine computation to verify that the index sets of and co- are in the finite Ershov hierarchy. Thus we are interested in the case in which infinitely many of the sequence of finite sets are minimal under inclusion. However if the infinite sequence is fairly simple, for instance{0}, {1}, {2}, … then the r.e. index set of co- is Σ20-complete as well as the index sets of and co- for all levels n &gt; 2. Since the finite Ershov hierarchy does not exhaust ⊿20 there is a lot of “room” between these two extreme cases.