Finite Subset Research Articles

The finer selection properties

For a function f from ω to ω, a topological space X satisfies ⋃f(Γ,Γ) if for each sequence (Un:n∈ω,Unhas no finite subcover) of elements of Γ, select for each n a finite subset Fn⊆Un such that |Fn|≤f(n) for all n and {⋃Fn:n∈ω} is an element of Γ, where Γ denotes the family of all open γ-covers of X. In this paper, we prove the following results.(1)Assume the Continuum Hypothesis. There is a set of real numbers that satisfies ⋃fin(Γ,Γ) and S1(Γ,O) but not ⋃id(Γ,Γ), where id is the identity function from ω to ω.(2)Assume b=c. There is a set of real numbers that satisfies ⋃id(Γ,Γ) but not ⋃k(Γ,Γ) for all natural numbers k≥1.(3)Assume the Continuum Hypothesis. For each natural number k≥2, there is a set of real numbers that satisfies ⋃k+1(Γ,Γ) but not ⋃k(Γ,Γ). These results answer an open problem proposed by Zdomskyy and a conjecture proposed by Tsaban.

Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

We consider continuous-time random walks on a random locally finite subset of mathbb {R}^d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group mathbb {G}=mathbb {R}^d or mathbb {G}=mathbb {Z}^d. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on mathbb {Z}^d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

Residue pattern for infinitely many primes

Let [Formula: see text] be a nonempty finite subset of [Formula: see text] and [Formula: see text] be an arbitrary map (choice of signs for [Formula: see text]). We will say that [Formula: see text] has residue pattern [Formula: see text] modulo [Formula: see text] if [Formula: see text], where [Formula: see text] is the Legendre symbol mod [Formula: see text]. For a given nonempty finite subset [Formula: see text] of [Formula: see text] with a choice of signs [Formula: see text] and a real number [Formula: see text], we obtain an asymptotic formula for the number of primes [Formula: see text] of the form [Formula: see text] such that [Formula: see text] has residue pattern [Formula: see text] modulo [Formula: see text] which also satisfies [Formula: see text] where [Formula: see text] are integers with [Formula: see text] and [Formula: see text]. For an irrational [Formula: see text] and a real [Formula: see text], we also obtain an asymptotic formula for the same but primes [Formula: see text] of the form [Formula: see text] under certain assumption on [Formula: see text].

Projectively-compact spinor vertices and space-time spin-locality in higher-spin theory

The concepts of compact and projectively-compact spin-local spinor vertices are introduced. Vertices of this type are shown to be space-time spin-local, i.e., their restriction to any finite subset of fields is space-time local. The known spinor spin-local cubic vertices with the minimal number of space-time derivatives are verified to be projectively-compact. This has the important consequence that spinor spin-locality of the respective quartic vertices would imply their space-time spin-locality. More generally, it is argued that the proper class of solutions of the non-linear higher-spin equations that leads to the minimally non-local (presumably space-time spin-local) vertices is represented by the projectively-compact vertices. The related aspects of the higher-spin holographic correspondence are briefly discussed.

Subalgebras in K[x] of small codimension

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in mathbb {K}[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(alpha )=f(beta ) for alpha ,beta in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

Medical Diagnosis and Pattern Recognition Based on Generalized Dice Similarity Measures for Managing Intuitionistic Hesitant Fuzzy Information

Pattern recognition is the computerized identification of shapes, designs, and reliabilities in information. It has applications in information compression, machine learning, statistical information analysis, signal processing, image analysis, information retrieval, bioinformatics, and computer graphics. Similarly, a medical diagnosis is a procedure to illustrate or identify diseases or disorders, which would account for a person’s symptoms and signs. Moreover, to illustrate the relationship between any two pieces of intuitionistic hesitant fuzzy (IHF) information, the theory of generalized dice similarity (GDS) measures played an important and valuable role in the field of genuine life dilemmas. The main influence of GDS measures is that we can easily obtain a lot of measures by using different values of parameters, which is the main part of every measure, called DGS measures. The major influence of this theory is to utilize the well-known and valuable theory of dice similarity measures (DSMs) (four different types of DSMs) under the assumption of the IHF set (IHFS), because the IHFS covers the membership grade (MG) and non-membership grade (NMG) in the form of a finite subset of [0, 1], with the rule that the sum of the supremum of the duplet is limited to [0, 1]. Furthermore, we pioneered the main theory of generalized DSMs (GDSMs) computed based on IHFS, called the IHF dice similarity measure, IHF weighted dice similarity measure, IHF GDS measure, and IHF weighted GDS measure, and computed their special cases with the help of parameters. Additionally, to evaluate the proficiency and capability of pioneered measures, we analyzed two different types of applications based on constructed measures, called medical diagnosis and pattern recognition problems, to determine the supremacy and consistency of the presented approaches. Finally, based on practical application, we enhanced the worth of the evaluated measures with the help of a comparative analysis of proposed and existing measures.

Ranks with Respect to a Projective Variety and a Cost-Function

Let X⊂Pr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=[1,+∞)∪+∞ such that w(a)=1 for a non-empty open subset of X. For any q∈Pr, the rank rX,w(q) of q with respect to (X,w) is the minimum of all ∑a∈Sw(a), where S is a finite subset of X spanning q. We have rX,w(q)<+∞ for all q. We discuss this definition and classify extremal cases of pairs (X,q). We give upper bounds for all rX,w(q) (twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors.

Domination ratio of a family of integer distance digraphs with arbitrary degree

An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $\Gamma(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$. Our result generalizes a previous result on the domination ratio of the graph $\Gamma(\mathbb{Z},\{1,s\})$ with $s\in\mathbb{Z}\setminus\{0,1\}$ and also implies the domination number of certain circulant graphs $\Gamma(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the finite cyclic group of integers modulo $n$ and $S$ is a subset of $\mathbb{Z}_n$.

A Galois Connection Approach to Wei-Type Duality Theorems

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}$ </tex-math></inline-formula> , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-matroids defined over finite sets and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

On product-one sequences over subsets of groups

Let G be a group and G_0 subseteq G be a subset. A sequence over G_0 means a finite sequence of terms from G_0, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of G and over the whole group G, with a special emphasis on the infinite dihedral group.

Borda-optimal taxation of labour income

I numerically compute Borda-optimal, i.e., optimal based on the Borda count as the normative criterion, labour-income tax schedules for the United States. I do so in the context of a Mirrlees-style model with quasilinear preferences and a constant elasticity of labour supply. Because the Borda count is defined for finitely many alternatives, the computations restrict attention to a finite subset of the set of continuous, piecewise linear tax schedules with (in the baseline analysis) four or fewer pieces.

POWERS OF SUBSETS IN FREE PERIODIC GROUPS

It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|&gt;4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

A note on the moduli spaces of holomorphic and logarithmic connections over a compact Riemann surface

Let X be a compact Riemann surface of genus $$g \ge 3$$ . We consider the moduli space of holomorphic connections over X and the moduli space of logarithmic connections singular over a finite subset of X with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces and show that they are constant under certain conditions. We show the Torelli-type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces.

Light Spanners for High Dimensional Norms via Stochastic Decompositions

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har–Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with \(\tilde{O}(n^{1+1/t^2})\) edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of \(\ell _p\) with \(1<p\le 2\). Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of \(\ell _p\) for \(1<p\le 2\) has an O(t)-spanner with \(n^{1+\tilde{O}(1/t^p)}\) edges and lightness \(n^{\tilde{O}(1/t^p)}\). In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness \(O(n^{1/t})\). We exhibit the following tradeoff: metrics with decomposability parameter \(\nu =\nu (t)\) admit an O(t)-spanner with lightness \(\tilde{O}(\nu ^{1/t})\). For example, metrics with doubling constant \(\lambda \), graphs of genus g, and graphs of treewidth k, all have spanners with stretch O(t) and lightness \(\tilde{O}(\lambda ^{1/t})\), \(\tilde{O}(g^{1/t})\), \(\tilde{O}(k^{1/t})\) respectively. While these families do admit a (\(1+\epsilon \))-spanner, its lightness depend exponentially on the dimension (resp. \(\log g\), k). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

The cardinality of the partitions of a set in the absence of the Axiom of Choice

Abstract In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|&amp;lt;2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, $2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|&amp;lt;|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.

Construction of the fitness function depending on a set of competing strategies based on the analysis of population dynamics

The purpose of this work is to construct a fitness function that depends on the set of coexisting competing hereditary elements based on population dynamics in the “predator– prey” model with the logistic growth of prey. Materials and methods. The work uses the generalized Volterra model. The planktivorous fish plays the role of a predator. Many different species of zooplankton are considered as prey, which differ from each other in the hereditary strategies of daily vertical migrations. The model takes into account the intraspecific competition of prey. The peculiarity of the model consists of the presence of pairs of hereditary strategies in which the carriers of the first can displace the carriers of the second and vice versa — the carriers of the second can displace the carriers of the first, depending on the set of competing strategies in which they coexist. To restore the fitness function, the ranking method is used, which is reduced to the classification of ordered pairs of hereditary strategies into two classes “the first strategy displaces the second” and “the second displaces the first”. Results. The article presents a new methodology for constructing the fitness function. The technique involves two stages. First, the fitness function is reconstructed for a certain finite subset of elements on the basis of processing data on the long-term dynamics and comparing their competitive advantages. At the second stage, the form of the fitness function is derived for an arbitrary set of elements. It uses the features of interspecies interaction reflected in the model. With the help of the constructed fitness function, an evolutionarily stable regime of daily vertical migrations of zooplankton is modeled by numerically solving the minimax problem. Conclusion. The proposed method for constructing a fitness function that depends on a set of competing strategies is quite general and can be applied to a wide range of models of population dynamics. The strategy of diel vertical migrations of zooplankton constructed as a result of modeling is in good agreement with empirical data.

Finite Approximations and Similarity of Languages

A new framework to measure distances (similarity) between formal languages and between grammars based on distances between words is introduced. It is based on approximating languages by their finite subsets and using monotone sequences of such finite approximations to define an infinite language in the limit. Distances between finite languages are defined and extended to distances between monotone sequences of finite languages leading to distances between infinite languages. The framework captures several distances studied in the literature. Context-free grammars with energy are introduced to enable finite approximations emphasizing “syntactically important” parts of words. Grammars with energy are also used to extend distances between monotone sequences of finite languages to distances between context-free grammars. A basic toolkit for monotone sequences of finite languages and distances between languages resp. grammars is provided. As part of this toolkit a non-symmetric version of distances is defined, providing additional characterisation of distances in general. Additional properties of distances between grammars are derived by restricting the“energy use” of grammars with energy. Some methods of estimating the distances are presented to be used in cases where the distance is not computable or difficult to compute.

The Jacobson radical for an inconsistency predicate

As a form of the Axiom of Choice about relatively simple structures (posets), Hausdorff’s Maximal Chain Principle appears to be little amenable to computational interpretation. This received view, however, requires revision: maximal chains are more reminiscent of maximal ideals than it seems at first glance. The latter live in richer algebraic structures (rings), and thus are readier to be put under computational scrutiny. Exploiting this, and of course the analogy between maximal chains and maximal ideals, the concept of Jacobson radical carries over from a ring to an arbitrary set with an abstract inconsistency predicate: that is, a distinguished monotone family of finite subsets. All this makes possible not only to generalise Hausdorff’s principle, but also to express it as a syntactical conservation theorem. The latter, which encompasses the desired computational core of Hausdorff’s principle, is obtained by a generalised inductive definition. The over-all setting is constructive set theory.

Weak convergence and spectrality of infinite convolutions

Let {Ak}k=1∞ be a sequence of finite subsets of Rd satisfying that #Ak≥2 for all integers k≥1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolutionν=δA1⁎δA2⁎⋯⁎δAn⁎⋯, where all sets Ak⊆R+d and δA=1#A∑a∈Aδa. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in R and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.

Multiplier norm of finite subsets of discrete groups

Let G G be a discrete group, V N ( G ) VN(G) the von Neumann algebra generated by the left translation operators and A ( G ) A(G) the Fourier algebra of G G . Then V N ( G ) VN(G) can be considered as a subspace of ℓ 2 ( G ) \ell ^2(G) and A ( G ) = ℓ 2 ( G ) ∗ ℓ 2 ( G ) A(G) = \ell ^2(G)*\ell ^2(G) . The Banach space dual of A ( G ) A(G) is V N ( G ) VN(G) . The duality between φ ∈ V N ( G ) \varphi \in VN(G) and u ∈ A ( G ) u \in A(G) is ⟨ φ , u ⟩ = ∑ x ∈ G φ ( x ) u ( x ) \langle \varphi ,u \rangle = \sum _{x \in G}{\varphi (x)u(x)} , if φ ∈ ℓ 1 ( G ) \varphi \in \ell ^{1}(G) or u ∈ ℓ 2 ( G ) u \in \ell ^2(G) ; in these two cases, φ u ∈ ℓ 1 ( G ) \varphi u \in \ell ^{1}(G) . We will show that if G G is infinite then there exist φ ∈ V N ( G ) \varphi \in VN(G) and u ∈ A ( G ) u \in A(G) such that φ u ∉ ℓ 1 ( G ) \varphi u \notin \ell ^{1}(G) . When G G is countably infinite, this implies that there exist φ ∈ V N ( G ) \varphi \in VN(G) , u ∈ A ( G ) u \in A(G) and an increasing sequence of finite subsets F n F_n , G = ⋃ F n G = \bigcup F_n such that the sequence ∑ x ∈ F n φ ( x ) u ( x ) \sum _{x \in F_n}{\varphi (x)u(x)} is divergent and therefore ‖ χ F n ‖ M A ( G ) \|{\chi _{F}}_n \|_{MA(G)} is unbounded. This leads us to give a preliminary study of the growth of ‖ χ F ‖ M A ( G ) \|\chi _{F}\|_{MA(G)} as | F | |F| , the size of the finite subset F F of G G , is increasing. Recall that when G = Z G = \mathbb {Z} , the additive group of integers, ‖ χ F ‖ M A ( Z ) = 1 2 π ∫ 0 2 π | ∑ k ∈ F e i k x | d x \|\chi _{F}\|_{MA(\mathbb {Z})} = \frac {1}{2\pi }\int _0^{2\pi } \big |\sum _{k \in F} {e^{ikx}}\big | \, d{x} .