Finite Subset Research Articles

Time complexity of the analyst's traveling salesman algorithm

The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of $\R^N$ is contained on a curve of finite length. We show that for finite sets, the algorithm constructed in \cite{Schul-Hilbert,BNV} that solves the Analyst's Traveling Salesman Problem has polynomial time complexity and we determine the sharp exponent.

Finite matchability under the matroidal Hall's condition

Aharoni and Ziv conjectured that if M and N are finitary matroids on E, then a certain “Hall-like” condition is sufficient to guarantee the existence of an M-independent spanning set of N. We show that their condition ensures that every finite subset of E is N-spanned by an M-independent set.

Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

Multi-Criteria Decision Making Model For Hotel Selection Problem Under Complex Dual Hesitant Fuzzy Information

ABSTRACT The notion of the “complex dual hesitant fuzzy set (CDHFS)” is the combination of the “dual hesitant fuzzy set (DHFS)” and the “complex fuzzy set (CFS).” It is characterized by two degrees, namely the membership and nonmembership, in the form of a finite subset on a unit disc in the complex plane. CDHFS is useful for dealing with real-world problems involving uncertain or hard-to-predict information. Also, to approximate smoothly, the Einstein operators are well-known aggregation operators, while prioritized operators are effective tools for prioritization among criteria. Therefore, the goal of this study is to develop some prioritized aggregation operators under the CDHFS environment; namely the complex dual hesitant fuzzy prioritized averaging (CDHFPA) operator, the complex dual hesitant fuzzy prioritized geometric (CDHFPG) operator, the complex dual hesitant fuzzy Einstein prioritized averaging (CDHFEPA) operator, and complex dual hesitant fuzzy Einstein prioritized geometric (CDHFEPG) operator. Some properties of the proposed operators are investigated in detail. In addition, a multi-criteria decision-making (MCDM) method based on the proposed operators with the complex dual hesitant fuzzy setting is developed. Moreover, a numerical example is given for the application and effectiveness of the developed MCDM approach. A comparison study is also done with existing methods to show that the proposed MCDM method is better and more reliable. The study finds that if the expert’s preference is used to choose the right aggregation operators, the decision maker will have access to a wide range of compromise solutions.

Fredholm separating maps on continuous function spaces

For locally compact Hausdorff spaces X and Y, we initiate a study of Fredholm separating maps T from C(X) into C(Y). Our key aim is to completely recognize the structures of the kernel space and the range space of T in terms of merging points to show that X and Y are homeomorphic after removing related finite subsets; as the main result, we describe the interaction between continuity of T and closedness of its range; in particular, under certain condition, we conclude that T can be represented as a weighted composition operator.

ON LARGE EXTERNALLY DEFINABLE SETS IN NIP

Abstract We study cofinal systems of finite subsets of $\omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

Projective joint spectra and characters of representations of [formula omitted

For a tuple of square complex-valued N×N matrices A1,…,An the determinant of their linear combination x1A1+⋯+xnAn, which is called a pencil, is a homogeneous polynomial of degree N in C[x1,...xn]. Zero-set of this polynomial is an algebraic set in the projective space CPn−1. This set is called the determinantal hypersurface or determinantal manifold of the tuple (A1,...,An). It was shown in [7] that determinantal hypersurfaces contain substantial information about representations of finite Coxeter groups. Namely, if G is a non-special Coxeter group of type A,B, or D, ρ1 and ρ2 are two linear representations of G, and the determinantal hypersurfaces of images of the Coxeter generators of G under ρ1 and ρ2 coincide as divisors in the projective space, the characters of ρ1 and ρ2 are equal, and, therefore, ρ1 and ρ2 are equivalent. In [30] this result was extended in the characters part to affine Coxeter groups of types B,C, and D. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of A type are not covered by this result, as their combinatorics is quite different. In this paper we explicitly construct a finite set in A˜n having the same property. We also show that every group which is a semidirect product of a finite group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensional representation of the group, the determinantal hypersurface of images of the set determines the representation character.

Commuting maps on idempotents and functional identities on finite subsets of fields

Let F be a field, n ≥ 3 and let M n be the ring of all n × n matrices with entries in F . For a given subset H of M n , consider H ¯ = H ∪ V with V = { α E ii ∣ α ∈ F and E ii is a matrix unit for each i = 1 , … , n } . In this paper, under a mild technical assumption on F , we describe additive maps G : M n → M n satisfying [ G ( X ) , X ] = G ( X ) X − XG ( X ) = 0 for all X ∈ H ¯ in the following settings: H = E = { A ∈ M n ∣ A is idempotent } ; H = S = { A ∈ M n ∣ A is algebraic of degree 2 } ; H = P = { X ∈ M n ( F ) ∣ tr ( X ) ∈ L } where L is the prime field of F . These maps are so-called commuting on H ¯ . Our findings will allow us to conclude that for H = S the map G has the so-called standard form. Moreover, we will show that G is commuting on E ¯ if and only if G is commuting on P ¯ . Our investigation will lead us to consider functional identities on F of the form H 1 ( r ) s + H 2 ( s ) r = 0 with r, s belonging to a 2-point subset of F . At the end, we will discuss about the existence of non-standard additive commuting maps on E ¯ which are not commuting on the set of rank 1 matrices and vice versa.

Spatial mixing and the random‐cluster dynamics on lattices

AbstractAn important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .

Improved bound for the Gerver-Ramsey collinearity problem

Let S be a finite subset of Zn. A vector sequence (zi) is an S-walk if and only if zi+1−zi is an element of S for all i. Gerver and Ramsey showed in 1979 that for S⊂Z3 there exists an infinite S-walk in which no 511+1=48,828,126 points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to 189.

Uniformly expanding random walks on manifolds

In this paper we construct uniformly expanding random walks on smooth manifolds. Potrie showed that given any open set U of , there exists an uniformly expanding random walk µ supported on a finite subset of U. In this paper we extend those results to closed manifolds of any dimension, building on the work of Potrie and Chung to build a robust class of examples. Adapting to higher dimensions, we work with a new definition of uniform expansion that measures volume growth in subspaces rather than norm growth of single vectors.

Hereditary classes of ordered sets of width at most two

This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs.Several open problems about hereditary classes of relational structures which have been considered over the years have a positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ruškuc and Vatter.In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fraïssé.We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver’s theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo.Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized.We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of N ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs.

Rota’s basis conjecture holds for random bases of vector spaces

In 1989, Rota conjectured that, given n bases B1,…,Bn of the vector space Fn over some field F, one can always decompose the multi-set B1∪⋯∪Bn into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases B1,…,Bn. More specifically, our first result shows that Rota’s basis conjecture holds with probability 1−o(1) as n→∞ if the bases B1,…,Bn are chosen independently uniformly at random among all bases of Fqn for some finite field Fq (the analogous result is trivially true for an infinite field F). In other words, the conjecture is true for almost all choices of bases B1,…,Bn⊆Fqn. Our second, more general, result concerns random bases B1,…,Bn⊆Sn for some given finite subset S⊆F (in other words, bases B1,…,Bn where all vectors have entries in S). We show that when choosing bases B1,…,Bn⊆Sn independently uniformly at random among all bases that are subsets of Sn, then again Rota’s basis conjecture holds with probability 1−o(1) as n→∞.

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

Can we recover an integral quadratic form by representing all its subforms?

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.

Liminary C⁎-algebras with boolean spectrum

Liminary C⁎-algebras with boolean spectrum are the main topic of Cignoli's joint paper with G.A. Elliott and the present author (Cignoli et al., 1993 [9]). Solving the analogue of Kaplansky's problem for these algebras, in that paper it is proved that the Murray von Neumann order of projections is sufficient to uniquely recover the C⁎-algebraic structure. In this paper we continue the study of these algebras. Among others, we prove that the Elliott partial semigroup of any such algebra A is canonically extendible to a locally finite MV-algebraE(A), in the sense that every finite subset of E(A) generates a finite subalgebra of E(A). Further, every extremal state s of K0(A) has the property that s(K0(A)) is a cyclic subgroup of R, and K0(A) has general comparability. We characterize central projections in these C*-algebras as fixpoints for the “eccentricity” partial order ⊑ on E(A) introduced in the present author's paper (Mundici, 2023 [25]).

Tree approximation in quasi-trees

n this paper, we investigate the geometric properties of quasi-trees and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is (1,C) -quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces (1987) can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree.

Partitioned difference families and harmonious linear spaces

A linear space is harmonious if it is resolvable and admits an automorphism group acting sharply transitively on the points and transitively on the parallel classes. Generalizing old results by the first author et al. [11] we present some difference methods to construct harmonious linear spaces. In particular, it is shown that, for any finite non-singleton subset K of Z+, there are infinitely many values of v for which there exists a partitioned difference family that is the base parallel class of a harmonious linear space with v points whose block sizes are precisely the elements of K.

Error Correction for Discrete Tomography

Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number d of directions. In this paper we consider functions f : A → R where A is a finite subset of ℤ2 and R an integral domain. Several reconstruction methods have been introduced in the literature. Recently Ceko, Pagani and Tijdeman developed a fast method to reconstruct a function with the same line sums as f. Up to here we assumed that the line sums are exact. Some authors have developed methods to recover the function f under suitable conditions by using the redundancy of data. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible. Moreover, we prove that if it is known that the line sums in k given directions are correct, then the line sums in every other direction can be corrected provided that the number of wrong line sums in that direction is less than k/2.

On property-(R 1) and relative Chebyshev centers in Banach spaces-II

We continue to study (strong) property-(R 1) in Banach spaces. As discussed by Pai & Nowroji in [On restricted centers of sets, J. Approx. Theory, 66(2), 170–189 (1991)], this study corresponds to a triplet , where X is a Banach space, V is a closed convex set, and is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then has strong property-(R 1), where represents the compact subsets of X. It is established that for any . This extends the well-known fact that a compact subset of a Lindenstrauss space X admits a nonempty Chebyshev center in X. We extend our observation that is Lipschitz continuous in if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and represents the set of all finite subsets of BX then we observe that BY exhibits the condition for simultaneously strongly proximinal (viz. property-(P 1)) in X for if satisfies strong property-(R 1), where represents the set of all finite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ ∞, then exhibits the strong property-(R 1), where represents the set of all compact subsets of ℓ ∞. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces.