Row Sums Research Articles

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.

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.

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.

Multi-alternating sign matrices

We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.

Designing an event-triggered [formula omitted] filter with possibly inconsistent modes for Markov jump systems

The design of event-triggered H∞ filters for Markov jump systems is studied. The transition rates of the Markov chain allow for being uncertain and even totally unknown. Meanwhile, the plant considered and its filter are allowed to have possibly inconsistent modes, resulting in a hidden Markov model (HMM). A dual-threshold periodic event-triggered scheme is introduced to save network channel resources. Under the event-triggered scheme, a sufficient condition of mean-square exponential stability and H∞ disturbance-attenuation performance is proposed for the HMM-based filtering error system. Then, a design approach for the desired event-triggered H∞ filter is developed by eliminating potential nonlinearities, without limiting the row sum of uncertain items in the transition rate matrix to zero. Finally, two examples, including a tunnel diode circuit, are provided to verify the effectiveness of the design approach.

Bounding the row sum arithmetic mean by Perron roots of row-permuted matrices

R+n×n denotes the set of n×n non-negative matrices. For A∈R+n×n let Ω(A) be the set of all matrices that can be formed by permuting the elements within each row of A. Formally:Ω(A)={B∈R+n×n:∀i∃ a permutation ϕis.t.bi,j=ai,ϕi(j)∀j}. For B∈Ω(A) let ρ(B) denote the spectral radius or largest non-negative eigenvalue of B. We show that the arithmetic mean of the row sums of A is bounded by the maximum and minimum spectral radius of the matrices in Ω(A). Formally, we show thatminB∈Ω(A)⁡ρ(B)≤1n∑i=1n∑j=1nai,j≤maxB∈Ω(A)⁡ρ(B). For positive A we obtain necessary and sufficient conditions for these inequalities to become an equality. We also give criteria which an irreducible matrix C should satisfy so that ρ(C)=minB∈Ω(A)⁡ρ(B) or ρ(C)=maxB∈Ω(A)⁡ρ(B). These criteria are used to derive algorithms for finding such C when all the entries of A are positive.

Multigrid Incomplete Factorization Methods in Krylov Subspaces

The paper studies multigrid methods for solving systems of linear algebraic equations resulting from the seven-point discretization of the three-dimensional Dirichlet problem for an elliptic differential equation of the second order in a parallepipedal domain on a regular grid. The algorithms suggested are presented as special iteration processes of incomplete factorization in Krylov subspaces with a hierarchical recursive vector structure that corresponds to a sequence of embedded grids and gives rise to a block tridiagonal recursive representation of the coefficient matrix of the original linear algebraic system. The convergence of iterations is enhanced by using the principle of compensation of the row sums and also the symmetric successive block overrelaxation. An arbitrary m-grid method is defined recursively, based on the two-grid method. For simplicity, the algorithms are considered for linear systems with Stieltjes coefficient matrices. Issues related to generalization of the algorithms to larger classes of problems and, in particular, those with unsymmetric matrices are discussed.

Centered PSD matrices with thin spectrum are M-matrices

We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order $n$ and rank $n-1$ with eigenvalue ratio $\lambda_{\max}/\lambda_{\min}\leq n/(n-2)$ between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices.

Sequences of lower and upper bounds for the spectral radius of a nonnegative matrix

In this article, we expand a classical Frobenius' result upon consecutive k-th powers of nonnegative matrices to establish sequences of new lower and upper bounds for the spectral radius with respect to the positive integer k, each term of which is formulated by the average (k+1)-row sums of the nonnegative matrix. With the aid of the average (k+1)-row sums and taking the extreme entries of the matrix, we study new bounds generalizing existing formulae and we produce sequences of new tighter approximations for the spectral radius. The monotonicity and convergence properties of the constructed sequences are explored and certain conditions are stated under which the new bounds are sharper than Frobenius' bounds and other existing formulae. We further characterize the cases of equality in the aforesaid bounds, when the matrix is irreducible. Throughout, we perform illustrative numerical examples to showcase the efficiency of our proposed bounds and make comparisons among them.

Anti-Sum-Asymmetry Models and Orthogonal Decomposition of Anti-Sum-Symmetry Model for Ordinal Square Contingency Tables

For the analysis of C × C square contingency tables, we usually estimate using a statistical model an unknown probability distribution with high confidence from obtained observations. The statistical model that fits the data well and is easy to interpret is preferred. The anti-sum-symmetry (ASS) and anti-conditional sum-symmetry (ACSS) models have a structure that the ratio of the probability with which the sum of row and column levels is t, for t = 2, . . . , C, and the probability with which the sum of row and column levels is 2(C + 1) − t is always one and constant, respectively. This study proposes two kinds of models that the ratio of those changes exponentially depending on the sum of row and column levels. This study also gives the decomposition theorems of the ASS model using the proposed models. Moreover, we show that the value of the likelihood ratio chi-squared statistics for the ASS model is asymptotically equivalent to the sum of those for the decomposed models. We evaluate the advantage of the proposed models by applying they to a single data set of real-world grip strength data.