Finite Subset Research Articles

Simultaneous perception–action design via invariant finite belief sets

Although perception is an increasingly dominant portion of the overall computational cost for autonomous systems, only a fraction of the information perceived is likely to be relevant to the current task. To alleviate these perception costs, we develop a novel simultaneous perception–action design framework wherein an agent senses only the task-relevant information. This formulation differs from that of a partially observable Markov decision process, since the agent is free to synthesize not only its policy for action selection but also its belief-dependent observation function. The method enables the agent to balance its perception costs with those incurred by operating in its environment. To obtain a computationally tractable solution, we approximate the value function using a novel method of invariant finite belief sets, wherein the agent acts exclusively on a finite subset of the continuous belief space. We solve the approximate problem through value iteration in which a linear program is solved individually for each belief state in the set, in each iteration. Finally, we prove that the value functions, under an assumption on their structure, converge to their continuous state-space values as the sample density increases.

Robust MILP formulations for the two-stage weighted vertex [formula omitted]-center problem

The weighted vertex p-center problem (PCP) consists of locating p facilities among a set of available sites such that the maximum weighted distance (or travel time) from any demand node to its closest located facility is minimized. This paper studies the exact solution of the two-stage robust weighted vertex p-center problem (RPCP2). In this problem, the location of the facilities is fixed in the first stage while the demand node allocations are recourse decisions fixed once the uncertainty is revealed. The problem is modeled by box uncertainty sets on both the demands and the distances. We introduce five different robust reformulations based on MILP formulations of (PCP) from the literature. We prove that considering a finite subset of scenarios is sufficient to obtain an optimal solution of (RPCP2). We leverage this result to introduce a column-and-constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem optimally. We highlight how these algorithms can also be adapted to solve the single-stage problem (RPCP1) which is obtained when no recourse is considered. We present a numerical study to compare the performances of these formulations on randomly generated instances and on a case study from the literature.

Diversity on \(\wp_{fin}\) (X)

In tight-span theory, a diversity is a generalization of metric space where the metric is defined over the set \(\wp_{fin}\) (X) which is composed of finite subsets of X. In this paper we are going to generalize the results of D. Silvestru and C. Gosa to derive some sharp inequalities for the diameter diversity. This sharp inequality can be used to study models with diversity in a collective manner.

Schreier families and -(almost) greedy bases

AbstractLet $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$ . In this paper, we introduce and characterize $\mathcal {F}$ -(almost) greedy bases. Given such a family $\mathcal {F}$ , a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$ -greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$ , $m \in \mathbb {N}$ , and $G_m(x)$ , we have $$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$ Here, $G_m(x)$ is a greedy sum of x of order m, and $\mathbb {K}$ is the scalar field. From the definition, any $\mathcal {F}$ -greedy basis is quasi-greedy, and so the notion of being $\mathcal {F}$ -greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal {F}$ -greedy bases as being $\mathcal {F}$ -unconditional, $\mathcal {F}$ -disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal {F}$ -almost greedy bases.Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$ -greedy. For a countable ordinal $\alpha $ , we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$ , where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $ . We show that for each $\alpha $ , there is a basis that is $\mathcal {S}_{\alpha }$ -greedy but is not $\mathcal {S}_{\alpha +1}$ -greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $ , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

Sums of transcendental dilates

AbstractWe show that there is an absolute constant such that for any finite subset of and any transcendental number . By a construction of Konyagin and Łaba, this is best possible up to the constant .

Product-free sets in approximate subgroups of distal groups

Recall that a subset X of a group G is ‘product-free’ if X2∩X=∅, i.e. if xy∉X for all x,y∈X. Let G be a group definable in a distal structure. We prove there are constants c>0 and δ∈(0,1) such that every finite subset X⊆G distinct from {1} contains a product-free subset of size at least δ|X|c+1/|X2|c. In particular, every finite k-approximate subgroup of G distinct from {1} contains a product-free subset of density at least δ/kc.The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma.

Plastic pairs of metric spaces

We address pairs (X,Y) of metric spaces with the following property: for every mapping f:X→Y the existence of points x,y∈X with d(f(x),f(y))>d(x,y) implies the existence of x˜,y˜∈X for which d(f(x˜),f(y˜))<d(x˜,y˜). We give sufficient conditions for this property and for its uniform version in terms of finite ε-nets and finite ε-separated subsets.

The Fermat–Steiner Problem in the Space of Compact Subsets of the Euclidean Plane

The Fermat–Steiner problem is the problem of finding all points of a metric space Y such that the sum of the distances from them to points of a certain fixed finite subset A of the space Y is minimal. In this paper, we examine the Fermat–Steiner problem in the case where Y is the space of compact subsets of the Euclidean plane endowed with the Hausdorff metric, and points of A are finite pairwise disjoint compact sets.

Algebras from finite group actions and a question of Eilenberg and Schützenberger

In 1976 S. Eilenberg and M.-P. Schützenberger posed the following diabolical question: if A is a finite algebraic structure, Σ is the set of all identities true in A, and there exists a finite subset F of Σ such that F and Σ have exactly the same finite models, must there also exist a finite subset F′ of Σ such that F′ and Σ have exactly the same finite and infinite models? (That is, must the identities of A be “finitely based”?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the “inherently nonfinitely based” algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Schützenberger. As a corollary, we give the first known examples of inherently nonfinitely based “automatic algebras” constructed from group actions.

An elementary result on infinite and finite direct sums of modules

Let R be a ring, and consider an R-module M given with two (generally infinite) direct sum decompositions, A⊕(⨁i∈ICi)=M=B⊕(⨁j∈JDj), such that the submodules A and B, and the Dj, are all finitely generated. We show that there then exist finite subsets I0⊆I, J0⊆J, and a direct summand Y⊆⨁i∈I0Ci, such that A⊕Y=B⊕(⨁j∈J0Dj).We then note some ways that this result can and cannot be generalized, and pose some related questions.

Triangulations of non-archimedean curves, semi-stable reduction, and ramification

Let K be a complete discretely valued field with algebraically closed residue field and let ℭ be a smooth projective and geometrically connected algebraic K-curve of genus g. Assume that g⩾2, so that there exists a minimal finite Galois extension L of K such that ℭ L admits a semi-stable model. In this paper, we study the extension L∣K in terms of the minimal triangulation of C, a distinguished finite subset of the Berkovich analytification C of ℭ. We prove that the least common multiple d of the multiplicities of the points of the minimal triangulation always divides the degree [L:K]. Moreover, in the special case when d is prime to the residue characteristic of K, then we show that d=[L:K], obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito’s theorem to the wildly ramified case.

Total Orderization Invariant Maps on Distributive Lattices

Given any finite subset A of order n of a distributive lattice and kin {1,ldots ,n}, there is a natural extension of the median operation to n variables which generalizes the notion of the kth smallest element of A. By applying each of these operations to A, a totally ordered set to(A) is obtained. We refer to to(A) as the total orderization of A. After developing a brief theory of total orderization invariant maps on distributive lattices, it is shown in this paper how these functions generalize and provide new characterizations for symmetric continuous positively homogeneous functions, bounded orthosymmetric multilinear maps, and certain power sum polynomials on vector lattices. These theorems generalize several results by Bernau, Huijsmans, Kusraev, Azouzi, Boulabiar, Buskes, Boyd, Ryan, and Snigireva and in turn reveal novel properties of the various maps studied in this paper.

Разрешимость теории конечных подмножеств безатомных булевых алгебр

В работе рассматриваются алгебраические системы, где в качестве носителя выступают конечные подмножества некоторой безатомной булевой алгебры. Для полученной системы мы вводим новое отношение для конечных подмножеств: считаем, что одно подмножество состоит в отношении с другим подмножеством в том и только том случае, когда все элементы одного подмножества меньше всех элементов другого. Мы демонстрируем, что теория построенной таким образом новой системы сводится к теории безатомных булевых алгебр. Следовательно, также как и теория исходной системы, теория новой системы оказывается разрешимой. We consider atomless boolean algebras, and study algebraic structures where the universe consists of finite subsets of such an algebra. On these structures, we define the new relation between finite subsets: we say that one set is less than another one iff all elements of the first set are less than all elements of the second one. We show that the theory of constructed structure is reducible to the theory of original boolean algebra. Hence, the theory of constructed structure is decidable.

Simultaneous visibility in the integer lattice

Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let S be a finite subset of Zk. The asymptotic density of the set of lattice points, visible from all points of S, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.

Diversities and the Generalized Circumradius

The generalized circumradius of a set of points Asubseteq mathbb {R}^d with respect to a convex body K equals the minimum value of lambda ge 0 such that a translate of lambda K contains A. Each choice of K gives a different function on the set of bounded subsets of mathbb {R}^d; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalized circumradius to a finite subset of mathbb {R}^d. We obtain elegant characterizations in the case that K is a simplex or parallelotope.

Almost sure central limit theorems for the maxima of randomly chosen random variables

In this paper, we give an almost sure central limit theorem (ASCLT) version of a maximum limit theorem (MLT) with an arbitrary sequence {dn, n ≥ 1} of weighted means of max{Xk, k ∈ An}, where {Xn, n ≥ 1} is a sequence of independent random variables, and {An, n ≥ 1} is a sequence of almost surely finite random subsets of positive integers independent of {Xn, n ≥ 1}. Thus we generalize the cases considered in the literature: (i) the nonrandom version of ASCLT for the MLT; (ii) the version of ASCLT for randomly indexed MLT; and (iii) the version of maximum schema of observed and unobserved random variables. We complete the paper with illustrative examples.

On Locally Finite Orthomodular Lattices

Abstract Let us denote by ℒ ℱ $[\mathcal{L}\mathcal{F}$ the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ provided each finite subset of L generates in L a finite subOML). In this note, we first show how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We find ℒ ℱ $[\mathcal{L}\mathcal{F}$ considerably large though, obviously, not all OMLs belong to ℒ ℱ $[\mathcal{L}\mathcal{F}$ . Then we study states on the OMLs of ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We show that local finiteness may to a certain extent make up for distributivity. For instance, we show that if L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ and if for any finite subOML K there is a state s: K → [0, 1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of ℒ ℱ $[\mathcal{L}\mathcal{F}$ relevant to the quantum logic theory.

On generalisations of the Aharoni–Pouzet base exchange theorem

AbstractThe Greene–Magnanti theorem states that if is a finite matroid, and are bases and is a partition, then there is a partition such that is a base for every . The special case where each is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite‐dimensional vector spaces. Later, he and Aharoni answered this conjecture affirmatively not just for vector spaces but also for infinite matroids. We prove two generalisations of their result. On the one hand, we show that ‘being a singleton’ can be relaxed to ‘being finite’ and this is sharp in the sense that the exclusion of infinite sets is really necessary. In addition, we prove that if and are bases, then there is a bijection between their finite subsets such that is a base for every . In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.

On Borsuk–Ulam theorems and convex sets

AbstractThe Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if is a continuous function on the unit circle S1 in such that for all , then there is a finite subset X of S1 of diameter at most (in the standard metric in which the circle has circumference of length 2π) such the convex hull of contains .

A Kneser-Type Theorem for Restricted Sumsets

Let be a commutative group and and be nonempty finite subsets of and . Kneser’s theorem is a fundamental result in additive number theory, and it establishes that . For any subset of , write . For any , set . An important problem in additive number theory is to find a Kneser-type theorem for the restricted sumsets . In particular, more than 20 years ago V. Lev proved that if , for all (resp., ) there is at most one (resp., ) such that (resp., ), and ; then In the same paper, Lev proposed as a problem to improve to something of the form with whenever . Lev’s problem has been solved for some particular groups and some specific subsets of . However, it remains open for arbitrary groups and arbitrary large subsets of . Here, as a consequence of the main result of this paper, it is shown that if we take instead of in the lower bound of , then indeed we can take as the coefficient of something of the form with whenever .