Row Sums Research Articles

A Generalized Method for Deriving Steady-State Behavior of Consistent Fuzzy Priority for Interdependent Criteria

Interdependent criteria play a crucial role in complex decision-making across various domains. Traditional methods often struggle to evaluate and prioritize criteria with intricate dependencies. This paper introduces a generalized method integrating the analytic network process (ANP), the decision-making trial and evaluation laboratory (DEMATEL), and the consistent fuzzy analytic hierarchy process (CFAHP) in a fuzzy environment. The Drazin inverse technique is applied to derive a fuzzy total priority matrix, and we normalize the row sum to achieve the steady-state fuzzy priorities. A numerical example in the information systems (IS) industry demonstrates the approach’s real-world applications. The proposed method derives narrower fuzzy spreads compared to the past fuzzy analytic network process (FANP) approaches, minimizing objective uncertainty. Total priority interdependent maps provide insights into complex technical and usability criteria relationships. Comparative analysis highlights innovations, including non-iterative convergence of the total priority matrix and the ability to understand interdependencies between criteria. The integration of the FANP’s network structure with the fuzzy DEMATEL’s influence analysis transcends the capabilities of either method in isolation, marking a significant methodological advancement. By addressing challenges such as parameter selection and mathematical complexity, this research offers new perspectives for future research and application in multi-attribute decision-making (MADM).

Research on the strategy for average consensus control of flywheel energy storage array system based on lifecycle

In the domain of clean energy, the flywheel energy storage array system (FESAS) is widely employed for efficient and renewable energy storage to stabilize power grids and smooth power variations. However, managing the lifecycle of FESAS and achieving coordinated control presents a considerable challenge. This study integrates a lifecycle parameter based on the average consensus algorithm, enabling flywheels with longer lifecycles to handle additional charging and discharging tasks, while those with shorter lifecycles participate less. This approach effectively balances flywheel lifecycles and extends the overall lifespan of FESAS. Experimental findings reveal that when the P matrix satisfies the condition of equal row sums or column sums, the power distribution parameters tend toward average consensus, enabling FESAS to efficiently smooth unbalanced power. In undirected graphs, the symmetry of the P matrix causes the auxiliary variables to tend toward average consensus, effectively suppressing unbalanced power. For directed unbalanced graphs, this study proposes Error Balancing Iterative Algorithm (EBTA) to construct a doubly stochastic matrix P, facilitating the convergence of auxiliary variables to average consensus and effectively suppressing unbalanced power. The experimental results validate the efficacy of our proposed approach. This research establishes a theoretical foundation for the intelligent coordinated control of FESAS, highlighting the pivotal role of lifecycle parameters, and offers valuable insights for enhancing the performance and lifespan of FESAS.

Clustering of high-dimensional observations

We present a novel clustering method for high-dimensional, low sample size (HDLSS) data. The method is distance-based, takes advantage of the distance concentration phenomenon and the limiting values of the dissimilarity indices to construct clusters. We describe an algorithm that orders each row of the dissimilarity matrix to estimate the change points, which define cluster boundaries. We construct an agreement matrix of the Rand indices of the row clusters. The minimum of the row sum of the agreement matrix provides us with the best clusters. We prove that the new method achieves perfect clustering as the number of features diverges for a fixed sample size. Several examples are presented to illustrate the proposed method. We compare the new method with four other clustering techniques, including high-dimensional k-means, minimal spanning tree and Hierarchical Scan. The clustering methods are applied to the Lymphoma data set.

Regression‐based network‐flow and inner‐matrix reconstruction

AbstractNetwork or matrix reconstruction is a general problem that occurs if the row‐ and column sums of a matrix are given, and the matrix entries need to be predicted conditional on the aggregated information. In this paper, we show that the predictions obtained from the iterative proportional fitting procedure (IPFP) or equivalently maximum entropy (ME) can be obtained by restricted maximum likelihood estimation relying on augmented Lagrangian optimization. Based on this equivalence, we extend the framework of network reconstruction, conditional on row and column sums, toward regression, which allows the inclusion of exogenous covariates and bootstrap‐based uncertainty quantification. More specifically, the mean of the regression model leads to the observed row and column margins. To exemplify the approach, we provide a simulation study and investigate interbank lending data, provided by the Bank for International Settlement. This dataset provides full knowledge of the real network and is, therefore, suitable to evaluate the predictions of our approach. It is shown that the inclusion of exogenous information leads to superior predictions in terms of and errors.

Design of consensus and cluster consensus controller for first-order nonlinear multi-agent systems based on subgroup structure

It is well known that nonlinear multi-agent systems (MASs) with a directed topology can reach a consensus if the global coupling coefficient is strong enough. In this paper, we propose a novel distributed protocol based on the subgroup structure. Our results demonstrate that the MAS design can follow a part-to-whole approach, starting with subgroups and gradually building up to a more complex system. We firstly consider two cases for the MASs: one with a single leader and one with a leader group, where agents within the leader group can communicate with each other, but cannot receive the information of other group. We show that followers can track the leader or group leaders if intra-coupling within the subgroup and information strength received by the root node are above a certain threshold. Next, we investigate cluster consensus and practical cluster consensus for the Laplacian matrix with unequal row sum, equal row sum, and mixing of partitioned matrices. Our results reveal that row sum and community structure of the partitioned matrix are crucial in determining the final state of each agent. Our proposed control protocol is highly flexible and relies only on the cluster structure and information received by the root node, rather than global information. Moreover, our method is not limited to tracking problems, cooperative control, or multi-objective control, and can accommodate new group agents joining the system, making it highly adaptable. We provide illustrative examples to demonstrate the effectiveness of our approach.

Mathematics Exploration Extant in The Checker Game: A Mathematics Teaching Contribution

Aversion to mathematics shown by learners belonging to different age stages, worries and challenges mathematicians and/ or mathematics teachers to develop surveys which might improve learners´ mathematics learnings. “Thawing” recondite mathematics in several games may aid to provide a different mathematics vision. The current research article aims to identify and analyze the “concealed” mathematics through checker games played in Angola, mostly in Cuando Cubango province. The survey is placed in ethnomathematics and considering its sorts, qualitative methodology has been used supported by a half-structured interview, direct observation to picture recording. The research was conducted in the Menongue municipality (Cuando Cubango capital city). The three versions of checkers games were described (12 by 12; 10 by 10 and 8 by 8), some questions were created that guided this investigation (Why does the structure of the checkers game have the same number of rows and columns? Why do players when placing checkers pieces in the middle always leave two empty rows? Why is the sum of rows and columns equal to the even number? Why is each painted square on the checkers board corresponds to an odd number?) and, finally, some mathematical concepts involved in that game were explored (such as, for example, the notion of the Cartesian plane, diagonal lines, distance between two points, median (element of Statistics), numbers even and odd).

Enhanced Technique of Constructing Multiple ODD Magic Square Matrices

An enhanced technique of constructing multiple odd magic square matrices is proposed in this work. A specific rule of establishing improved odd magic to magic squares derived from the odd algebraic Latin squares is studied and programmed here. Magic squares are practically important from the properties of their equality in the sum of rows, columns and diagonals. An n x n odd magic square is an array containing the positive integers from 1 to n2, arranged so that each of the rows, columns, and the two principal diagonals have the same sum.

SOME CONSTRUCTION OF 8N-DIMENSIONAL PERFECT MAGIC CUBE WITH ARITHMETIC SEQUENCE

A magic square whose dimensions are expanded is called a magic cube. A magic cube whose properties are expanded is called a perfect magic cube. The perfect magic cube problem is how to arrange numbers in an cube (matrix) such that the sum of rows, columns, pillars, diagonals (planes and spaces) produces a magic constant of the cube. In this paper, it will be studied how to construct a perfect magic cube of order for whose entries contain an arithmetic sequence with the difference which is set to find specific patterns, and the algorithm for constructing a perfect magic cube is then implemented into programming language to solve large orders.

Frobenius-König theorem for classes of (0,±1)-matrices

The Frobenius-König Theorem has a central role in combinatorial matrix theory; it characterizes when a (0,1)-matrix X contains a permutation matrix P (meaning P≤X entrywise). Our goal is to investigate similar questions for (0,±1)-matrices, and a main result is a Frobenius-König Theorem for the class of (0,±1)-matrices with all row and column sums being 1. Moreover, some related results are shown for alternating sign matrices.

Quantum Complexity of the Kronecker Coefficients

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue of NP. This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in QMA, complementing a recent NP-hardness result of Ikenmeyer, Mulmuley, and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error. Published by the American Physical Society 2024

Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling.

Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

A Hybrid Approach for the Supply Chain Risk Assessment of the Construction Industry During the COVID-19 Pandemic

COVID-19 is an epidemic threat to human health caused by a novel coronavirus. Different countries are still struggling with whether to extend lockdown policies considering economic recession outcomes or lift the lockdown and recover the fragile economy. This research narrows the knowledge gap by applying a decision-making approach in a fuzzy environment to consider the uncertainties in the supply chain risk assessment of the construction industry during the COVID-19 pandemic. Besides, risk factors (RFs) are identified and interrelationships between RFs are depicted. RFs are categorized using expert panel surveys and literature reviews. The interdependencies between different RFs are evaluated using a combination of interval-valued intuitionistic fuzzy numbers, decision making trial and evaluation laboratory, and the Choquet integral to allow a strong interrelationship between the sum of rows and the sum of columns of RFs in structural correlation analysis. Next, priorities of RFs are obtained by considering interdependencies between different RFs using a combination of the technique for order preference by similarity to the ideal solution (TOPSIS) method with the intuitionistic fuzzy set. An intuitionistic fuzzy weighted average operator is utilized to aggregate individual opinions of decision makers for rating the importance of RFs. Results reported that the scarcity of material and financial resources due to movement restrictions with overall weights of 0.0614 and 0.413 are the most prominent RFs in the supply chain due to the COVID-19 pandemic. The results of this research can assist decision-makers in developing strategic policies to prevail over the risk challenges and subsequently aid experts in redesigning supply chains.

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

Increasing subsequences, matrix loci and Viennot shadows

Abstract Let ${\mathbf {x}}_{n \times n}$ be an $n \times n$ matrix of variables, and let ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$ . We study the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is the generating function of permutations in ${\mathfrak {S}}_n$ by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space of k-local permutation statistics. We also calculate the structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ as a graded ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$ -module.

Матрична алгебра В як евклідовий простір

Representation of information by means of hypercomplex numerical systems is used in various problems of science and technology: in classical mechanics, solid body mechanics, electrodynamics, radio electronics, computer animation, and others [1]. Often a hypercomplex system (that is, a system whose elements are considered to be hypercomplex numbers) is understood as any finite-dimensional algebra over a field. An important place among such algebraic structures is occupied by matrix algebras. The impossibility of constructing algebras with division does not at all mean the impossibility of constructing algebras without division, but their properties are close to the first ones (use of defined division). Since each algebra of finite rank can be monomorphically immersed in some complete matrix algebra, this caused, so to speak, an inverse approach to the construction of new algebras. A certain subalgebra stands out from a complete matrix algebra, which is a matrix representation of an algebra of finite rank. It is the implementation of such an approach that makes it possible to endow elements of algebra of finite rank with matrix characteristics, in particular, a canonical representation of algebra elements is constructed through the spectral representation of a matrix, and the algebra itself is endowed with a topological structure through one of the matrix norms. At the same time, an additional condition is often imposed, that it be an algebra over the field of real or complex numbers. The article constructs a real algebra of finite rank, the elements of which are matrices of the second order with the same sum of rows and columns. We endowed it with a norm and a scalar product, demonstrating that it is a Euclidean space. This algebra is a matrix representation of the algebra of hypercomplex numbers, which we called binary in our research [4].

Three New Refined Arnold Families

The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.

Properties on the Fan Product of M-matrices

In this study, the lower bound expressions of the smallest eigenvalue q(E ★ F) are derived using the row sums of two M-matrices and the maximum of the main diagonal elements. The obtained lower bounds improve the existing related conclusions. Furthermore, the results of this paper are compared with those of related results by two specific examples to verify the accuracy of the results.

Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of , centered at c with width . At smaller c, this curve receives corrections in powers of accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.

A Numerical Study of the Peridynamic Differential Operator Discretization of Incompressible Navier–Stokes Problems

We study the incompressible Navier–Stokes equations using the Projection Method. The applications of interest are the classical channel flow problems such as Couette, shear, and Poiseuille. In addition, we consider the Taylor-Green vortex and lid-driven cavity applications. For discretization, we use the Peridynamic Differential Operator (PDDO). The main emphasis of the paper is the performance of the PDDO as a discretization method under these flow problems. We present a careful numerical study with quantifications and report convergence tables with convergence rates. We also study the approximation properties of the PDDO and prove that the N-th order PDDO approximates polynomials of degree at most N exactly. As a result, we prove that the PDDO discretization guarantees the zero row sum property of the arising system matrix.