Cartesian Product Research Articles

Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma

AbstractIn this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$ . We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

An investigation of continuous-time quantum walk on hypercube in view of Cartesian product structure

In this paper, continuous-time quantum walk on hypercube is discussed in view of Cartesian product structure. We find that the [Formula: see text]-fold Cartesian power of the complete graph [Formula: see text] is the [Formula: see text]-dimensional hypercube, which give us new ideas for the study of quantum walk on hypercube. Combining the product structure, the spectral distribution of the graph and the quantum decomposition of the adjacency matrix, the probability amplitudes of the continuous-time quantum walker’s position at time [Formula: see text] are given, and it is discussed that the probability distribution for the continuous-time case is uniform when [Formula: see text]. The application of this product structure greatly improves the study of quantum walk on complex graphs, which has far-reaching influence and great significance.

Theoretical framework for a decision support system for micro-enterprise supermarket investment risk assessment using novel picture fuzzy hypersoft graph.

Risk evaluation has always been of great interest for individuals wanting to invest in various businesses, especially in the marketing and product sale centres. A finely detailed evaluation of the risk factor can lead to better returns in terms of investment in a particular business. Considering this idea, this paper aims to evaluate the risk factor of investing in different nature of products in a supermarket for a better proportioning of investment based on the product's sales. This is achieved using novel Picture fuzzy Hypersoft Graphs. Picture Fuzzy Hypersoft set (PFHSs) is employed in this technique, a hybrid structure of Picture Fuzzy set and Hypersoft Set. These structures work best for evaluating uncertainty using membership, non-membership, neutral, and multi-argument functions, making them ideal for Risk Evaluation studies. Also, the concept of the PFHS graph with the help of the PFHS set is introduced with some operations like the cartesian product, composition, union, direct product, and lexicographic product. This method presented in the paper provides new insight into product sale risk analysis with a pictorial representation of its associated factors.

On maximum Wiener index of directed grids

This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid Gm, n is the Cartesian product Pm□Pn of paths on m and n vertices, and in a particular case when m = 2, it is a called the ladder graph Ln. Kraner Šumenjak et al. in 2021 proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of Ln, is obtained when all layers isomorphic to one factor are directed paths directed in the same way except one (corresponding to an endvertex of the other factor) which is a directed path directed in the opposite way. Then they conjectured that the natural generalization of this orientation to Gm, n will attain the maximum Wiener index among all orientations of Gm, n. In this paper we disprove the conjecture by showing that a comb-like orientation of Gm, n has significiantly bigger Wiener index.

An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

Cartesian Product of Quadratic Residue Graphs

Cartesian Product of Quadratic Residue Graphs

An algebraic approach to the variants of convexity for soft expert approximate function with intuitionistic fuzzy setting

A new area of research called intuitionistic fuzzy soft expert set is expected to overcome the drawbacks of an intuitionistic fuzzy soft set in terms of eligibility for soft expert-argument approximate function. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert-argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, ( α , v ) -convexity(concave) sets, strongly ( α , v ) -convexity (concave) sets, strictly ( α , v ) -convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and results on the basis of these specified notions. Additionally, by using a theoretic cum analytical approach, various elements of computational geometry, such as convex hull and convex cone, are theorized and some pertinent results are generalized.

On the power graph of a monogenic semigroup

Let [Formula: see text] be a monogenic semigroup with zero. Here, we consider the power graph [Formula: see text] over [Formula: see text] with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] divides [Formula: see text] or [Formula: see text] divides [Formula: see text], where [Formula: see text]. We compute various graph parameters of [Formula: see text] and topological indices based on distance of vertices. Finally, we compute some graph parameters of the cartesian product [Formula: see text] of graphs [Formula: see text] and [Formula: see text].

An inexact multiblock alternating direction method for grasping-force optimization of multifingered robotic hands

In this paper, we present an inexact multiblock alternating direction method for the point-contact friction model of the force-optimization problem (FOP). The friction-cone constraints of the FOP are reformulated as the Cartesian product of circular cones. We focus on the convex quadratic circular-cone programming model of the FOP, which is an exact cone-programming model. Coupled with the separable convex quadratic objective function, we recast the circular-cone-programming model as a multiblock separable cone model. A parallel inexact multiblock alternating direction method is used to solve the FOP. We prove the global convergence of the proposed method. Simulation results of the three-fingered FOP are reported, which verified the efficiency of the proposed method.

NEUTROSOPHIC FUZZY BI-IDEAL OF BS-ALGEBRAS

The main aim of this paper is to introduce the new concept of neutrosophic fuzzy bi-ideal of BS-algebras. Some algebraic nature are investigated. Neutrosophic fuzzy bi-ideal of BS-algebras is also applied in Cartesian product. Finally, we also provide the homomorphic behaviour of neutrosophic fuzzy bi-ideal of BS-algebras.

Mathematical analysis of generative adversarial networks based on complex picture fuzzy soft information

Generative Adversarial Networks (GANs) are the models that generate data samples from the statistical distribution of the data. It is one of the most well-known branches of machine learning and deep learning. Different techniques are involved in the processing and production of visual data, which sometimes gives rise to misperception uncertainties. Bearing this issue in mind, we define some solid mathematical concepts to model and resolve the stated problem named complex picture fuzzy soft relations (CPFSRs) which is defined by the Cartesian product (CP) of two complex picture fuzzy soft sets (CPFSSs). The major objective of this study is to develop some innovative and useful notions that may be used to handle difficult and inconsistent information in practical situations. The proposed notion is foremost and superior to the prevailing ideas, where the presented idea is the improved technique of two different theories, named picture fuzzy set (PFS) and soft set (SS). Additionally, it presents the picture fuzzy soft set (PFSS) in professional decision-making by reducing complexions. The evaluated CPFSRs are the improved versions of soft relations, fuzzy relations, complex soft relations, and complex fuzzy relations. Therefore, this paper provides modeling methodologies based on CPFSRs which are used for the analysis of electing the best GAN for effective working. In the process, the score functions are also formulated and analyzed. Finally, a comparative study of existing techniques has been done to show the validity of the proposed work.

Prediction Model of a Generative Adversarial Network Using the Concept of Complex Picture Fuzzy Soft Information

A computer vision model known as a generative adversarial network (GAN) creates all the visuals, including images, movies, and sounds. One of the most well-known subfields of deep learning and machine learning is generative adversarial networks. It is employed for text-to-image translations, as well as image-to-image and conceptual image-to-image translations. Different techniques are used in the processing and generation of visual data, which can lead to confusion and uncertainty. With this in mind, we define some solid mathematical concepts to model and solve the aforementioned problem. Complex picture fuzzy soft relations are defined in this study by taking the Cartesian product of two complex picture fuzzy soft sets. Furthermore, the types of complex picture fuzzy soft relations are explained, and their results are also discussed. The complex picture fuzzy soft relation has an extensive structure comprising membership, abstinence, and non-membership degrees with multidimensional variables. Therefore, this paper provides modeling methodologies based on complex picture fuzzy soft relations, which are used for the analysis of generative adversarial networks. In the process, the score functions are also formulated. Finally, a comparative analysis of existing techniques was performed to show the validity of the proposed work.

Three-Dimensional Graph Products with Unbounded Stack-Number

We prove that the stack-number of the strong product of three n-vertex paths is $$\Theta (n^{1/3})$$ . The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $$\Delta _0$$ such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree $$\Delta _0$$ . We show that $$\Delta _0\in \{6,7\}$$ .

INVERSE DOMINATION IN SOME OPRATIONS ON BIPOLAR FUZZY GRAPHS

In this paper the concept of inverse domintion in some operations on bipolar fuzzy graphs was introduced and investigated the bound of γ8 of some operations on bipolar fuzzy graphs are obtained like union, join,Cartesian product, strong product and composition.

Interactive Effect of Learning Rate and Batch Size to Implement Transfer Learning for Brain Tumor Classification

For classifying brain tumors with small datasets, the knowledge-based transfer learning (KBTL) approach has performed very well in attaining an optimized classification model. However, its successful implementation is typically affected by different hyperparameters, specifically the learning rate (LR), batch size (BS), and their joint influence. In general, most of the existing research could not achieve the desired performance because the work addressed only one hyperparameter tuning. This study adopted a Cartesian product matrix-based approach, to interpret the effect of both hyperparameters and their interaction on the performance of models. To evaluate their impact, 56 two-tuple hyperparameters from the Cartesian product matrix were used as inputs to perform an extensive exercise, comprising 504 simulations for three cutting-edge architecture-based pre-trained Deep Learning (DL) models, ResNet18, ResNet50, and ResNet101. Additionally, the impact was also assessed by using three well-known optimizers (solvers): SGDM, Adam, and RMSProp. The performance assessment showed that the framework is an efficient framework to attain optimal values of two important hyperparameters (LR and BS) and consequently an optimized model with an accuracy of 99.56%. Further, our results showed that both hyperparameters have a significant impact individually as well as interactively, with a trade-off in between. Further, the evaluation space was extended by using the statistical ANOVA analysis to validate the main findings. F-test returned with p < 0.05, confirming that both hyperparameters not only have a significant impact on the model performance independently, but that there exists an interaction between the hyperparameters for a combination of their levels.

On the 2-rainbow independent domination numbers of some graphs

By suitably adjusting the tropical algebra technique we compute the rainbow independent domination numbers of several infinite families of graphs including Cartesian products C_n Box P_m and C_n Box C_m for all n and mle 5, and generalized Petersen graphs P(n, 2) for n ge 3.

A New Alternative to Szeged, Mostar, and PI Polynomials—The SMP Polynomials

Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x=1, we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.

The automorphisms of 2-token graphs

Let Λ=(V(Λ),E(Λ)) be a graph. The 2-token graphF2(Λ) of Λ is with vertex set all the 2-subsets of V(Λ), a pair of 2-subsets are adjacent if the symmetric difference is exactly in E(Λ). In this paper, it is proved that Z2n−1⋊Aut(Λ)≲Aut(F2(Λ)), where Λ is a Cartesian product of n prime graphs. Then it is shown that the equality may happen by proving that Aut(F2(Qn))≅Z2n−1⋊Aut(Qn), where Qn is the n-dimensional hypercube with n≥3. This also provides some partial answers to [25, Problem 1.3].

The topological structures of the spaces of copulas and subcopulas

The (sub)copulas play a central role in statistics for capturing the stochastic dependence structure of random variables. A subcopula is a map from the Cartesian product of two closed subsets of the unit closed interval containing zero and one to the unit closed interval, which is grounded, 2-increasing and has uniform margins. A copula is a subcopula whose domain is the Cartesian product of the unit closed interval and itself. The uniform metric on the set of all copulas can be extended to a metric on the set of all subcopulas. In this work, we investigate topological structures of these spaces by using tools from infinite-dimensional topology. Our main result is as follows: there exists a homeomorphism from the space of subcopulas onto the Hilbert cube which sends the space of all copulas onto an end of the Hilbert cube and sends the space of exchangeable copulas onto an end of the end of the Hilbert cube. As a consequence, we obtain that the spaces of subcopulas, copulas and exchangeable copulas are homeomorphic to the Hilbert cube and hence have the fixed point property.

Some Properties Of Cartesian Product Of Two Fuzzy Normed Spaces

In this paper , the concept of the Cartesian Product of two fuzzy
 normed spaces is presented. Some basic properties and theorems on this
 concept are proved. The main goal of this paper is to prove that the
 Cartesian product of two complete fuzzy normed spaces is a complete
 fuzzy normed space.
 Key words:Fuzzy normed space , Cartesian product , Cauchy sequence ,
 complete fuzzy normed space.
 1- Introduction
 
 The fuzzy set concepts was introduced in mathematics by K.Menger
 in 1942 and reintroduced in the system theory by L.A.Zadeh in 1965.
 In 1984, Katsaras [ 1 ] , first introduced the notation of fuzzy norm on
 linear space, in the same year Wu and Fang [ 4 ] also introduced a notion of
 fuzzy normed space . Later on many other mathematicians like Felbin [ 2 ]
 , Cheng and Mordeson [ 10] , Bag and Samanta [12], J.Xiao and X.Zhu
 [8,9] , Krishna and Sarma [11] , Balopoulos and Papadopoulos [ 13] etc,
 have given different definitions of fuzzy normed spaces .
 J.Kider introduced the definition of fuzzy normed space[ 7 ] , we use this
 definition to prove that the Cartesian product of two fuzzy normed spaces
 is also fuzzy normed space.
 
 The structure of the paper is as follow : In section 2 we
 present some fundamental concepts . In section 3, the definition of fuzzy
 normed space appeared [7] is used to prove that the cartesain product of
 two fuzzy normed spaces is also fuzzy normed space, then we prove that
 the cartesain product of two complete fuzzy normed spaces is complete
 fuzzy normed space.
 
 2. Preliminaries
 In this section, we briefly recall some definitions and preliminary
 results which are used in this paper.