Dimension Dim Research Articles

Minimal amenable subshift with full mean topological dimension

Abstract Let G be an infinite countable amenable group and P a polyhedron with the topological dimension dim ⁡ ( P ) < ∞ . We construct a minimal subshift (X, G) of ( P G , G ) such that its mean topological dimension is equal to dim ⁡ ( P ) . This result answers the question of Dou (2017 Discrete Contin. Dyn. Syst. 37 1411–24). Moreover, it extends the work of Jin and Qiao (2023 arXiv:2102.10339) for Z -action.

Local Partition Dimension of Grid Graph and Its Application to the Coordinates of Potential Disaster Areas in Jember Regency

Partition dimension was introduced as a part of interesting topic in graph theory. It was focus to observe about distance. The local partition dimension is an expansion of the partition dimension by adding certain conditions to the representation of the vertices 𝑣 to which is Π denoted by 𝑟(𝑣| Π). One of the conditions that must be met for 𝑟(𝑣| Π) is discussed. The minimum value of k so that there is a local distinguishing partition from V (G) is the local partition dimension of G or it can can be said that the distance of each neighbor is different. The local partition dimension of a graph G is denoted 〖pd〗_l (G). In this study, we used an axiomatic deductive methods and pattern recognition. In order to construct the discriminating set on the metric dimension (dim) and the discriminating partition on the partition dimension (pd), the pattern detection method looks for patterns in which the coordinate values are minimum and different. By all observations, the local partition dimensions of P_n×P_m Grid graph has two condition about the results of resolving partition. The Result of local partition dimension of a Grid graph 〖〖pd〗_l (P〗_n×P_m)=2, where n≥2 dan m≥2. In addition, it will decide how to convert the coordinates of areas in the Jember district that are prone to flooding and landslides into metric dimensions. It was about Coordinates of Flood and Landslide Disaster Locations in Jember Regency. The number of local and minimal partition sets generated for flood-prone areas in Jember Regency is 〖pd〗_l (G_Jember)=3.

Extending the reach of the point-to-set principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces Rn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two algorithmic dimensions—computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim⁡(x) and Dim(x) to individual pointsx∈X—to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces.

ESO: An enhanced snake optimizer for real-world engineering problems

Meta-heuristic algorithms are an essential way to solve realistic optimization problems. Developing effective, accurate, and stable meta-heuristic algorithms has become the goal of optimization research. Snake Optimizer (SO) is a novel algorithm with good optimization results. However, due to the limitations of natural laws, the parameters are more fixed values in the exploration and exploitation phase, so the SO algorithm quickly falls into local optimization and slowly converges. This paper proposes an Enhanced Snake Optimizer (ESO) by introducing a novel opposition-based learning strategy and new dynamic update mechanisms (parameter dynamic update strategy, sine–cosine composite perturbation factors, Tent-chaos & Cauchy mutation) to achieve better performance. The effectiveness of ESO has been tested on 23 classic benchmark functions and the CEC 2019 function set. 13 functions of 23 classic benchmark functions have variants that belong to multiple dimensions (Dim = 30, 100, 500, 1000, and 2000). In addition, ESO is also used to solve four real-world engineering design problems. Experimental results and two statistical tests show that the proposed ESO performs better than the other 13 state-of-the-art algorithms, including SO. The MATLAB code of ESO is available from: https://ww2.mathworks.cn/matlabcentral/fileexchange/120173-eso-an-enhanced-snake-optimizer.

A new 2-norm generated by bounded linear functionals on a normed space

In this work, we introduce a new 2-norm generated by bounded linear functionals on a normed space X with dimension dim(X) ? 2, and investigate its relationship with the G?hler?s 2-norm [Lineare 2-normierte R?ume, Math. Nachr.]. We also derive a norm on X to explore its relation with the usual norm on X.

Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Given a simple connected graph G, the metric dimension dim(G) (and edge metric dimension edim(G)) is defined as the cardinality of a smallest vertex subset S⊆V(G) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S. It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim(G) and edim(G) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

On dimensions of finite approximation and functors preserving ε-nets

The dimension of finite approximation dim F ξ is defined for any point ξ of the space F(X), where F is a metrizable seminormal functor and X is a metric compactum. Such points can represent closed subsets, probability measures, maximal linked systems of closed sets, etc. The analysis shows that for many functorial constructions of general topology (exponential functor, probability measures functor, superextension functor, etc.) the dimension of finite approximation dimF ξ does not exceed the box dimension dim B of the support supp(ξ) of this point. In this connection, the problem naturally arises of how to describe the class of functors for which the indicated restriction on the dimensionality of the finite approximation holds. This paper introduces the notion of a metrizable functor that preserves ε-nets. The condition that ε-nets be preserved by the functor F proved to be sufficient for the inequality dim F ξ B(supp(ξ)) for any point ξ ∈ F(X). It is proved that a number of well-known metrizable functors preserve ε-nets.

The dimension Dind of finite topological T 0-spaces

Abstract A. V. Arhangelskii introduced the dimension Dind and properties of this dimension have been studied for various classes of topological spaces. In this paper, we study this dimension for finite T 0-spaces. Especially, we prove that in the realm of finite T 0-spaces, Dind is less than or equal to the small inductive dimension ind, the large inductive dimension Ind and the covering dimension dim. We also study the “gaps” between Dind and the dimensions ind, Ind and dim, presenting various examples which shows these “gaps”. Moreover, in this field of spaces, we give characterizations of Dind, inserting the meaning of the maximal family of pairwise disjoint open sets, and study properties of this dimension.

Dimension spectra of lines1

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

Fields of dimension one algebraic over a global or local field need not be of type C1

Let (K,v) be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension E/K satisfying the following: (i) E has dimension dim(E)≤1, i.e. the Brauer group Br(E′) is trivial, for every algebraic extension E′/E; (ii) finite extensions of E are not C1-fields. This, applied to the maximal algebraic extension K of the field Q of rational numbers in the field Qp of p-adic numbers, for a given prime p, proves the existence of an algebraic extension Ep/Q, such that dim(Ep)≤1, Ep is not a C1-field, and Ep has a Henselian valuation of residual characteristic p.

Effect of Feeding Barley, Corn, and a Barley/Corn Blend on Beef Composition and End-Product Palatability.

This study evaluated the relationship among palatability attributes, volatile compounds, and fatty acid (FA) profiles in meat from barley, corn, and blended (50:50, barley and corn) grain-fed steers. Multiple correspondence analysis with three dimensions (Dim) explained 62.2% of the total variability among samples. The Dim 1 and 2 (53.3%) separated pure from blended grain-fed beef samples. Blended grain beef was linked to a number of volatiles including (E,E)-2,4-decadienal, hexanal, 1-octen-3-ol, and 2,3-octanedione. In addition, blended grain-fed beef was linked to fat-like and rancid flavors, stale-cardboard, metallic, cruciferous, and fat-like aroma descriptors, and negative categories for flavor intensity (FI), off-flavor, and tenderness. A possible combination of linoleic and linolenic acids in the blended diet, lower rumen pH, and incomplete biohydrogenation of blended grain-fed polyunsaturates could have increased (p ≤ 0.05) long-chain n-6 fatty acids (LCFA) in blended grain-fed beef, leading to more accumulation of FA oxidation products in the blended than in barley and corn grain-fed meat samples. The Dim 3 (8.9%) allowed corn separation from barley grain beef. Barley grain-fed beef was mainly linked to alkanes and beef positive FI, whereas corn grain-fed beef was associated with pyrazines, in addition to aldehydes related to n-6 LCFA oxidation.

Metric Dimension on Sparse Graphs and its Applications to Zero Forcing Sets

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. Our main result consists in proving that the increase of the metric dimension of an edge addition can be amortized in the sense that if the graph consists of a spanning tree $T$ plus $c$ edges, then the metric dimension of $G$ is at most the metric dimension of $T$ plus $6c$. We then use this result to prove a weakening of a conjecture of Eroh et al. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a subset $S$ of black vertices (whereas the other vertices are colored white) of $G$ such that all the vertices will turned black after applying finitely many times the following rule: a white vertex is turned black if it is the only white neighbor of a black vertex. Eroh et al. conjectured that, for any graph $G$, $dim(G)\leq Z(G) + c(G)$, where $c(G)$ is the number of edges that have to be removed from $G$ to get a forest. They proved the conjecture is true for trees and unicyclic graphs. We prove a weaker version of the conjecture: $dim(G)\leq Z(G)+6c(G)$ holds for any graph. We also prove that the conjecture is true for graphs with edge disjoint cycles, widely generalizing the unicyclic result of Eroh et al.

Efficient Computation of the Large Inductive Dimension Using Order- and Graph-theoretic Means

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim(X; 𝒯) of a finite topological space (X; 𝒯). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind(X; 𝒯) and the large inductive dimension Ind(X; 𝒯) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind(X; 𝒯), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind(X; 𝒯), where the specialisation pre-order of (X; 𝒯) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.

Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

Taking the covering dimension dim as notion for the dimension of a topological space, we first specify the number zdimT0(n) of zero-dimensional T0-spaces on {1,…,n} and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,…,n} by means of two mappings po and P that yield the number po(n) of partial orders on {1,…,n} and the set P(n) of partitions of {1,…,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdimT0(n) and modify one for P in such a way that it computes zdimT0(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdimT0(1) to zdimT0(n) and the Stirling numbers of the second kind S(n, 1) to S(n, n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdimT0(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

Boolean Functions with Multiplicative Complexity 3 and 4.

Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fischer and Peralta (2002), and Find et al. (2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimension dim(f) of a Boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity of f is at least [dim(f)/2]. For MC 3, this implies that there are no equivalence classes other than those 24 identified in Çalık et al. (2018). Using the techniques from Çalık et al. and the new relation between the dimension and MC, we identify all 1277 equivalence classes having MC 4. We also provide a closed formula for the number of n-variable functions with MC 3 and 4. These results allow us to construct AND-optimal circuits for Boolean functions that have MC 4 or less, independent of the number of variables they are defined on.

Metric and upper dimension of zero divisor graphs associated to commutative rings

Abstract Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 𝕑 2 n {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

On resolving domination number of friendship graph and its operation

Let G = (V, E) be a simple, finite, and connected graph of order n. A dominating set D ⊆ V(G) such every vertex not in D is adjacent to at least one member of D. A dominating set of smallest size is called a minimum dominating set and it is known as the domination number. The domination number is the minimum cardinality of a dominating set and denoted by γ(G). The other hand, for an ordered set W = {w 1, w 2, w 3, …, wk } of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k − vector r(v|W) = (d(v, w 1), d(v, w 2), d(v, w 3), …, d(v, wk )), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for G is its metric dimension dim(G). A Set of vertices of a graph G that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called resolving domination number γr (G). In this paper, we discussed the resolving domination number of friendship graphs and its operation.

Approximating pointwise products of Laplacian eigenfunctions

We consider Laplacian eigenfunctions on a d-dimensional bounded domain M (or a d-dimensional compact manifold M) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions (eℓ)ℓ∈N. We study the subspace of all pointwise productsAn=span{ei(x)ej(x):1≤i,j≤n}⊆L2(M). Clearly, that vector space has dimension dim(An)=n(n+1)/2. We prove that products eiej of eigenfunctions are simple in a certain sense: for any ε>0, there exists a low-dimensional vector space Bn that almost contains all products. More precisely, denoting the orthogonal projection ΠBn:L2(M)→Bn, we have∀1≤i,j≤n‖eiej−ΠBn(eiej)‖L2≤ε and the size of the space dim(Bn) is relatively small: for every δ>0,dim(Bn)≲M,δε−δn1+δ. We obtain the same sort of bounds for products of arbitrary length, as well for approximation in H−1 norm. Pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.

Bounds on the Bend Number of Split and Cocomparability Graphs

A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A k-bend path is a path made up of at most k + 1 line segments. A Bk-VPG representation of a graph is a collection of k-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a Bk-VPG representation are calledBk-VPG graphs. The bend number of a graph G, denoted by bend(G), is the minimum integer k for which G has a Bk-VPG representation. In this paper, we study the relationship between poset dimension and bend number of cocomparability graphs. It is known that the poset dimension dim(G) of a cocomparability graph G is greater than or equal to its bend number bend(G). Cohen et al. (order2016) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each $m, t \in \mathbb {N}$ , there exists a cocomparability graph Gt, m with t m. The techniques used to prove the above result allow us to partially address the open question posed by Chaplick et al. (wg2012) who asked whether Bk-VPG-chordal $\subsetneq ~B_{k + 1}$ -VPG-chordal for all $k \in \mathbb {N}$ . We address this by proving that there are infinitely many $m \in \mathbb {N}$ such that Bm-VPG-split $\subsetneq ~B_{m + 1}$ -VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all $t \in \mathbb {N}$ , Bt-VPG- $Forb(C_{\geq 5})~\subsetneq ~B_{4t + 29}$ -VPG-Forb(C≥ 5), where Forb(C≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all $t \in \mathbb {N}$ , PBt-VPG-split $\subsetneq PB_{36t + 80}$ -VPG-split, where PBt-VPG denotes the class of graphs with proper bend number at most t (i.e. it has a Bt-VPG representation in which two paths have only finitely many intersection points, each intersection point belongs to exactly two paths, and whenever two paths intersect they cross each other).

Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

We formulate the conditional Kolmogorov complexity of x given y at precision r , where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways; (1) We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E . We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. (2)We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim( x | y ) and Dim( x | y ) of x given y , where x and y are points in Euclidean spaces. Intuitively, these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y . We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim( x ) and Dim( x ) and the mutual dimensions mdim( x : y ) and Mdim( x : y ).