Finite Subset Research Articles

Realization of finite groups as isometry groups and problems of minimality

AbstractA finite group is said to be realized by a finite subset of a Euclidean space if the isometry group of is isomorphic to . We prove that every finite group can be realized by a finite subset consisting of points, where is a generating system for . We define as the minimum number of points required to realize in for some . We establish that provides a sharp upper bound for when considering minimal generating sets. Finally, we explore the relationship between and the isometry dimension of , that is, defined as the least dimension of the Euclidean space in which can be realized.

On algebraic properties of power monoids of numerical monoids

AbstractLet S ⊂ ℕ0 be a numerical monoid and let $$\cal{P}_{\text{fin}}(S)$$ P fin ( S ) , resp. $$\cal{P}_{\text{fin},0}(S)$$ P fin , 0 ( S ) , denote the power monoid, resp. the restricted power monoid, of S, that is the set of all finite nonempty subsets of S, resp. the set of all finite nonempty subsets of S containing 0, with set addition as operation. The arithmetic of power monoids received some attention in recent literature. We complement these investigations by studying algebraic properties of power monoids, such as their prime spectrum. Moreover, we show that almost all elements of $$\cal{P}_{\text{fin},0}(S)$$ P fin , 0 ( S ) are irreducible (i.e., they are not proper sumsets).

Arbitrary Random Variables and Wiman’s Inequality

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

A study on various generalizations of generalized centers (GC) in Banach spaces

In [Generalized centers of finite sets in Banach spaces. Acta Math Univ Comenian (N.S.). 1997;66(1):83–115], Veselý developed the idea of generalized centers for finite sets in Banach spaces. In this work, we explore the concept of restricted F -center property for a triplet ( X , Y , F ( X ) ) , where Y is a subspace of a Banach space X and F ( X ) is the family of finite subsets of X. In addition, we generalize the analysis to include all closed, bounded subsets of X. Similar to how Lindenstrauss characterized n .2 . I . P . , we characterize n . X . I . P . . So, it is possible to figure out that Y has n . X . I . P . in X for all natural numbers n if and only if rad Y ( F ) = rad X ( F ) for all finite subsets F of Y. It then turns out that, for all continuous, monotone functions f, the f-radii viz. rad Y f ( F ) , rad X f ( F ) are same whenever the generalized radii viz. rad Y ( F ) , rad X ( F ) are also same, for all finite subsets F of Y. We establish a variety of characterizations of central subspaces of Banach spaces. With reference to an appropriate subfamily of closed and bounded subsets, it appears that a number of function spaces and subspaces exhibit the restricted weighted Chebyshev center property.

Acting on Totalities of infinite Processes: constructing facets of an Object conception

AbstractWe offer a new angle to explain the comprehension of mathematical infinity. We use Action—Process—Object—Schema (APOS) theory to explore the construction of different aspects of the cognitive Object related to this notion. We focus our attention on different ways of interacting with infinity, and how to act on infinite entities. Our aim is to shed light on the nature of mental constructions that take place when individuals deal with mathematical infinity. An infinite union of finite subsets of $${\mathbb{N}}$$ N provides the context for our theoretical analysis and the design of interviews conducted with graduate students and instructors. In this study we identify three types of Actions with different complexity levels. When these Actions are performed, they lead to the construction of different facets of the associated Object. We question the construction of an Object conception in infinity-related situations and offer an explanation that might change the way we think about the learning of certain mathematical concepts. We discuss the implications of our research for the advancement of APOS theory. We also offer pedagogical suggestions related to the comprehension of mathematical infinity.

Quantifying orbit detection: φ-order and φ-spectrum

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism φ, an element x∈G and a subset K⊆G, we say that the φ-order of g relative to K, φ-ordK(g), is the smallest nonnegative integer k such that gφk∈K. We prove that the set of orders, which we call φ-spectrum, is computable in two extreme cases: when K is a finite subset and when K is a recognizable subset. The finite case is proved for virtually free groups and the recognizable case for finitely presented groups. The case of finitely generated virtually abelian groups and some variations of the problem are also discussed.

New classes of modules for which the finite exchange property implies the full exchange property

Let M be a right R-module. We call a summand A of M a special summand if there exists a summand B of M with A ≅ B and A ∩ B = 0 . An internal direct sum ⊕ i ∈ I A i of submodules Ai of M is called a special local summand if each Ai is a special summand and ⊕ i ∈ F A i is a summand of M for any finite subset F ⊆ I . In this paper we give two new classes of modules for which the finite exchange property implies the full exchange property, namely, modules whose special local summands are summands, and C4-modules for which the union of any chain of special summands is a summand. More precisely, we show that if either M is a module whose special local summands are summands or M is a C4-module such that the union of any chain of special summands of itself is a summand, then M has the finite exchange property if and only if it has the full exchange property if and only if it is a clean module. As immediate applications of the aforementioned results, we generalize and extend many of the fundamental results on the subject of exchange modules.

Sumsets and entropy revisited

AbstractThe entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over .

A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws

E. Breuillard showed that finite subsets F\hspace{-.75pt} of matrices in \mathrm{GL}_{d\hspace{-.75pt}}(\hspace{-.75pt}\overline{\mathbb{Q}}\hspace{-.75pt}) generating non-virtually solvable groups have normalized height \hat{h}(F)\ge\varepsilon_d , for some positive \varepsilon_{d}>0 . The normalized height \hat{h}(F) is a measure of the arithmetic size of F and this result can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.

Simpler characterizations of total orderization invariant maps

Given a finite subset [Formula: see text] of a distributive lattice, its total orderization [Formula: see text] is a natural transformation of [Formula: see text] into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

A generalisation of Läuchli's lemma

AbstractLäuchli showed in the absence of the Axiom of Choice () that and, consequently, for all infinite cardinals , where and are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality . In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, in the latter equation can be replaced by other cardinals which are equal to in but not in , for example, and , the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality .

On join-complete implication algebras

In this paper, first, we consider an algebra that has a binary operation and a join of arbitrary nonempty subset. A lattice implication algebra is a lattice with a binary operation, which has a join and a meet of finite nonempty subsets. In this work, the notion of join-complete implication algebras L is defined as a join-complete lattice with a binary operation, and some properties of this algebra L are searched. Moreover, we prove that the interval [a, 1] in L is a lattice implication algebra and show that L satisfies the completely distributive law when it has the smallest element 0. Finally, we state the concept of filter and multipliers of L and provide finite and infinite examples of them. In addition, we research some properties of these concepts in detail.

Interpolation of set-valued functions

Abstract Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators. Imperial College Press). By this theory, a ‘metric polynomial interpolant’ is a collection of polynomial interpolants to all the ‘metric chains’ of the given samples of $F$. For set-valued functions whose graphs have nonempty interior, the collection of these ‘metric chains’ can be infinite. Our algorithm computes a small finite subset of ‘significant metric chains’, which is sufficient for approximating $F$. For the class of Lipschitz continuous functions with samples at the roots of the Chebyshev polynomials of the first kind, we prove that the error incurred by our computed interpolant decays with increasing number of interpolation points in the same rate as in the case of interpolation by the metric polynomial interpolant. This is also demonstrated by our numerical examples. For the class of set-valued functions whose graphs have smooth boundaries, we extend our algorithm to achieve a high-precision detection of the points of topology change, followed by a high-order approximation of the boundaries of the graph of F. We further discuss the case of set-valued functions whose graphs have ‘holes’ with Hölder-type singularities at the points of change of topology. To treat this case we apply some special approximation ideas near the singular points of the holes. We analyze the approximation order of the algorithm, including the error in approximating the points of change of topology, and show by several numerical examples the capability of obtaining high-order approximation of the holes.

Two problems on subset sums

For a set A of positive integers, let P(A) denote the set of all finite subset sums of A. In this paper, we completely solve a problem of Chen and Wu by proving that if B={b1<b2<⋯} is a sequence of integers with b1≥11, 3b1+5≤b2≤4b1, 3b2+2≤b3≤3b2+b1 and 3bn−bn−2≤bn+1≤3bn(n≥3), then there exists a set of positive integers A for which P(A)=N∖B. We also partially answer a problem of Wu by determining the structure of B={b1<b2<⋯} with b1>10 and b2>3b1+4, for which there exists a set of positive integers A such that P(A∩[0,bk])=[0,2bk]∖{bi,2bk−bi:1≤i≤k}(k≥2).

Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several problems of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite superset that is internally-$k$-quasi-invariant? This parallels a similar investigation undergone by C. E. Praeger in the context of group actions.

Local and extensive fluctuations in sparsely interacting ecological communities.

Ecological communities with many species can be classified into dynamical phases. In systems with all-to-all interactions, a phase where species abundances always reach a fixed point and a phase where they continuously fluctuate have been found. The dynamics when interactions are sparse, with each species interacting with only a few others, has remained largely unexplored. Here we study a system of sparse interactions, first when interactions are of constant strength and completely unidirectional, and then when adding variability and bidirectionality. We show that in this case a phase unique to the sparse setting appears in the phase diagram, where for the same control parameters different communities may reach either a fixed point or a state where the abundances of only a finite subset of species fluctuate, and we calculate the probability for each outcome. These fluctuating species are organized around short cycles in the interaction graph, and their abundances undergo large nonlinear fluctuations. We characterize the approach from this phase to a phase with extensively many fluctuating species, and show that the probability of fluctuations grows continuously to one as the transition is approached, and that the number of fluctuating species diverges. This is qualitatively distinct from the transition to extensive fluctuations coming from a fixed point phase, which is marked by a loss of linear stability. The differences are traced back to the emergent binary character of the dynamics when far from short cycles.

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

The Vietoris hyperspace $\mathcal F(X)$ and certain generalized metric properties

In this paper, we study several generalized metric properties of the space $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. In particular, we consider such properties $(P)$ for which $\mathcal F(X)$ has $(P)$ if and only if $X$ has $(P)$. Also, we obtain some results related to the images of metric spaces under some kinds of continuous mappings.

Mandelbrot set for fractal n-gons and zeros of power series

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset G⊂C. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set G of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set Mn for fractal n-gons. We prove that Mn is connected and locally connected for any n.

Density questions in rings of the form 𝒪K[γ] ∩ K

We fix a number field [Formula: see text] and study statistical properties of the ring [Formula: see text] as [Formula: see text] varies over algebraic numbers of a fixed degree [Formula: see text]. Given [Formula: see text], we explicitly compute the density of [Formula: see text] for which [Formula: see text] and show that this does not depend on the number field [Formula: see text]. In particular, we show that the density of [Formula: see text] for which [Formula: see text] is [Formula: see text]. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define [Formula: see text] to be a certain finite subset of [Formula: see text] and show that [Formula: see text] determines the ring [Formula: see text]. We show that if [Formula: see text] satisfy [Formula: see text], then the events [Formula: see text] and [Formula: see text] are independent. As [Formula: see text], we study the asymptotics of the density of [Formula: see text] for which [Formula: see text].