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In this paper, for any two elements y; u of a BCK-algebra X, we assign a subset of X, denoted by Sy(u), and investigate some related properties. We show that Sy(u) is a subalgebra of X for all y; u 2 X. Using these subalgebras, we characterize the involutive BCK-algebras, and give a necessary and sufficient condition for a bounded BCK-algebra to be a commutative BCK-chain. Finally, we show that the set of all subalgebras Sy(u) forms a bounded distributive lattice. [ABSTRACT FROM AUTHOR]

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to a quasirelativistic Hamiltonian matrix, that has the same electronic eigenstates as the original Dirac matrix. This transformation involves a matrix X, for which an exact identity is derived, and which can be constructed either in a noniterative way or by various iteration schemes, without requiring an expansion parameter. The convergence behavior of five different iteration schemes is studied numerically, with very promising results. [ABSTRACT FROM AUTHOR]

The effective vibronic Hamiltonian for a linear tetra-atomic molecule in a Π state has been investigated. In addition to the usual vibrational and Renner–Teller coupling terms, the bending mode anharmonicity, spin-orbit coupling, and Fermi resonance interactions have been added to the model. Terms in the Hamiltonian up to the fourth order are given explicitly for molecules of C∞υ symmetry and simplifications for symmetric D∞h molecules are discussed. The matrix elements for the HCCS free radical have been obtained and are used to analyze the observed ground-state levels of HCCS and DCCS in a companion paper. The Sears resonance vibronic interaction that couples levels with the selection rules ΔK=±1, ΔΣ=∓1, and ΔP=0 has also been studied and the matrix elements derived. The determinable combinations of signs for the major parameters in the model are discussed. [ABSTRACT FROM AUTHOR]

Abstract The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the symmetry algebra of the harmonic oscillator on the 2-sphere. These two algebras are recognized as the commutant of a o (2) ⊕ o (2) subalgebra of o (4) in the oscillator representation of the universal algebra U (u (4)). This connection is further related to the embedding of the (discrete) Hahn algebra in U (su (1 , 1)) ⊗ U (su (1 , 1)) in light of the dual action of the pair (o (4) , su (1 , 1)) on the state vectors of four harmonic oscillators. The two-dimensional singular oscillator is naturally seen by dimensional reduction to have the Higgs algebra as its symmetry algebra. Highlights • Hahn algebra encodes the bispectrality of the Clebsch-Gordan coefficients of su (1 , 1). • Higgs algebra describes the symmetries of various superintegrable models. • The isomorphic Higgs/Hahn algebras are commutants in the universal algebra of u (4). • Howe duality connects commutant picture and embedding in su (1 , 1) ⊗ su (1 , 1). • Higgs symmetry of 2D singular oscillator recognized from dimensional reduction. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) be the extended centroid of R, H and G two generalized derivations of R, L a noncentral Lie ideal of R, I a nonzero ideal of R. The left annihilator of S ⊆ R is denoted by lR(S) and defined by lR(S) = {x ∊ Rǀ xS = 0}. Suppose that S = {H(un)un +unG(un) ǀ u ∊ L} and T = {H(xn)xn +xnG(xn) ǀ x ∊ I}, where n ≥ 1 is a fixed integer. In the paper, we investigate the cases when the sets lR(S) and lR(T ) are nonzero. [ABSTRACT FROM AUTHOR]

This is a short account of some of the work of L. G. (Laci) Kovács on varieties of groups. [ABSTRACT FROM PUBLISHER]

In a residuated lattice, divisible fuzzy filters and strong fuzzy filters are introduced, and their properties are investigated. Characterizations of a divisible and strong fuzzy filter are discussed. Conditions for a fuzzy filter to be divisible are established. Relations between a divisible fuzzy filter and a strong fuzzy filter are considered. [ABSTRACT FROM AUTHOR]

It is conjectured that the eigenvalues of random Schr¨odinger operators at the localization transition in dimensions d ≥ 2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n × m boxes with 1 ⪡ m ⪡ n so that the eigenvalues in low disorder converge to Sine1, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner' famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices. [ABSTRACT FROM AUTHOR]

Letdenote the rank-numerical range of an-by-complex matrix. We give a characterization for, where, via the compressions and the principal submatrices of. As an application, the matrixsatisfying, whereis the classical numerical range ofand, is under consideration. We show that iffor some, thenis unitarily similar to, whereis a 2-by-2 matrix,is a-by-matrix and. [ABSTRACT FROM PUBLISHER]

Ann-by-n() weighted shift matrixAis one of the formwhere the’s, called the weights ofA, are complex numbers. Letdenote the-by-principal submatrix ofAobtained by deleting itsjth row andjth column. We show that the boundary of numerical rangeW(A)has a line segment if and only if the’s are nonzero andfor some. This refines previous results of Tsai and Wu on numerical ranges of weighted shift matrices. In addition, we give an example showing that there is a weighted shift matrix with line segments on the boundary of its numerical range such that the moduli of its weights are not periodic. [ABSTRACT FROM PUBLISHER]

We describe improved versions of indeX-calculus algorithms for solving discrete logarithm problems in Jacobians of high-genus hyperelliptic curves defined over even characteristic fields. Our first improvement is to incorporate several ideas for the low-genus case by Gaudry and Theriault, including the large prime variant and using a smaller factor base, into the large-genus algorithm of Enge and Gaudry. We eXtend the analysis in a 2001 paper by Jacobson, Menzes, and Stein to our new algorithm, allowing us to predict accurately the number of random walk steps required to find all relations, and to select optimal degree bounds for the factor base. Our second improvement is the adaptation of sieving techniques from Flassenberg and Paulus, and Jacobson to our setting. The new algorithms are applied to concrete problem instances arising from the Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating significant improvements in practice. [ABSTRACT FROM AUTHOR]

This paper begins a systematic study of weakly cartesian properties of monads that determine familiar varieties of universal algebras. While these properties clearly fail to hold for groups, rings, and many other related classical algebraic structures, their analysis becomes non-trivial in the case of semimodules over semirings, to which our main results are devoted. In particular necessary and sufficient conditions on a semiring S, under which the free semimodule monad has: (a) its underlying functor weakly cartesian, (b) its unit a weakly cartesian natural transformation, (c) its multiplication a weakly cartesian natural transformation, are obtained. [ABSTRACT FROM AUTHOR]

We prove that every matrix in SL 2(ℤ[1/p]) can be written as a product of at most five elementary matrices. This statement can also be interpreted in terms of short division chains in ℤ[1/p]. Similar bounds for the number of factors were known previously only under the generalized Riemann hypothesis. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a Tau method for solving the singular Lane-Emden equation-a nonlinear ordinary differential equation on a semi-infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane-Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well-known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

An appropriate distance is an essential ingredient in various real-world learning tasks. Distance metric learning proposes to study a metric, which is capable of reflecting the data configuration much better in comparison with the commonly used methods. We offer an algorithm for simultaneous learning the Mahalanobis like distance and K-means clustering aiming to incorporate data rescaling and clustering so that the data separability grows iteratively in the rescaled space with its sequential clustering. At each step of the algorithm execution, a global optimization problem is resolved in order to minimize the cluster distortions resting upon the current cluster configuration. The obtained weight matrix can also be used as a cluster validation characteristic. Namely, closeness of such matrices learned during a sample process can indicate the clusters readiness; i.e. estimates the true number of clusters. Numerical experiments performed on synthetic and on real datasets verify the high reliability of the proposed method. [ABSTRACT FROM AUTHOR]

The Golub-Kahan-Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of $$X \in \mathbb{R }^{m \times n}, m \ge n$$, given by where $$U \in \mathbb{R }^{m \times n}$$ is left orthogonal, $$V \in \mathbb{R }^{n \times n}$$ is orthogonal, and $$B \in \mathbb{R }^{n \times n}$$ is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of $$U$$ and $$V$$ tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. The use of an approach started by Simon and Zha (SIAM J Sci Stat Comput, 21:2257-2274, ) that reorthogonalizes only one of the two left orthogonal matrices $$U$$ and $$V$$ is shown to be very effective by the results presented here. Supposing that $$V$$ is the matrix reorthogonalized, the reorthogonalized GKL algorithm proposed here is modeled as the Householder Q-R factorization of $$\left( \begin{array}{c} 0_{n \times k} \\ X V_k \end{array}\right) $$ where $$V_k = V(:,1:k)$$. That model is used to show that if $$\varepsilon _M $$ is the machine unit and where $$\mathbf{tril }(\cdot )$$ is the strictly lower triangular part of the contents, then: (1) the GKL recurrence produces Krylov spaces generated by a nearby matrix $$X + \delta X$$, $$\Vert \delta X\Vert _F = \mathcal O (\varepsilon _M + \bar{\eta }) \Vert X\Vert _F$$; (2) singular values converge in the Lanczos process at the rate expected from the GKL algorithm in exact arithmetic on a nearby matrix; (3) a new proposed algorithm for recovering leading left singular vectors produces better bounds on loss of orthogonality and residual errors. [ABSTRACT FROM AUTHOR]

Unified Parallel C (UPC) is a Partitioned Global Address Space (PGAS) language whose popularity has increased during the last years owing to its high programmability and reasonable performance through an efficient exploitation of data locality, especially on hierarchical architectures like multicore clusters. However, the performance issues that arise in this language due to the irregular structure of sparse matrix operations have not yet been studied. Among them, the selection of an adequate storage format for the sparse matrices can significantly improve the efficiency of the parallel codes. This paper presents an evaluation, using UPC, of the most common sparse storage formats with different implementations of the matrix-vector and matrix-matrix products, which are key kernels in many scientific applications. [ABSTRACT FROM AUTHOR]

In this work, we investigate the derivations of n-dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank n − 1 are described. [ABSTRACT FROM PUBLISHER]

A real m × n matrix A and a vector y ∈ ℝ determine the discrete l-regularization (DLR) problem where | · | denotes the l-norm of a vector and ρ ≥ 0 is a nonnegative parameter. In this paper, we provide a detailed analysis of this problem which include a characterization of all solutions to (0.1), remarks about the geometry of the solution set and an effective iterative algorithm for numerical solution of (0.1). We are specially interested in the behavior of the solution of (0.1) as a function of ρ and in this regard, we prove in general the existence of critical values of ρ between which the l-norm of any solution remains constant. These general remarks are significantly refined when A is a strictly totally positive ( STP) matrix. The importance of STP matrices is well-established [, ]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in ℝ. Throughout our analysis of the DLR problem, we emphasize the importance of the dual maximum problem by demonstrating that any solution of it leads to a solution of the DLR problem, and vice versa. [ABSTRACT FROM AUTHOR]

We investigate decompositions of positive matrices as weighted sums of orthogonal projections, and apply them to the construction of fusion frames when fusion frame operators are prescribed. Examples are provided to demonstrate the simplicity and flexibility in this practical construction of fusion frames. As an application, we provide an method constructing Parseval fusion frames that are optimal for the one packet erasure problem. [ABSTRACT FROM AUTHOR]

Capabilities for uncertainty quantification (UQ) with respect to nuclear data have been developed at PSI in the recent years and applied to the UAM benchmark. The guiding principle for the PSIUQ development has been to implement nonintrusive "black box" UQ techniques in state-of-the-art, production-quality codes used already for routine analyses. Two complimentary UQ techniques have been developed thus far: (i) direct perturbation (DP) and (ii) stochastic sampling (SS).The DP technique is, first and foremost, a robust and versatile sensitivity coefficient calculation, applicable to all types of input and output. Using standard uncertainty propagation, the sensitivity coefficients are folded with variance/covariance matrices (VCMs) leading to a local first-order UQ method. The complementary SS technique samples uncertain inputs according to their joint probability distributions and provides a global, all-order UQ method. This paper describes both DP and SS implemented in the lattice physics code CASMO-5MX (a special PSI-modified version of CASMO-5M) and a preliminary SS technique implemented in MCNPX, routinely used in criticality safety and fluence analyses. Results are presented for the UAM benchmark exercises I-1 (cell) and I-2 (assembly). [ABSTRACT FROM AUTHOR]

For a column-stochastic matrix, consideration was given to determination of the eigenvector which corresponds to the unit eigenvalue. Such problems are encountered in many applications,-in particular, at ranking the web pages (PageRank). Since the PageRank problem is of special interest for larger matrices, the emphasis was made on the power method for direct iterative calculation of the eigenvector. Several variants of regularization of the power methods were compared, and their relations were considered. The distinctions of their realizations were given. [ABSTRACT FROM AUTHOR]

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrainedregression/ generalized-inverses work in this situation to provide a solution in the face of rank deficiency. [ABSTRACT FROM AUTHOR]

Nowadays, the unprecedented computing power of desktop personal computers and efficient computational methodologies such as the global gradient algorithm (GGA) make large water-distribution-system modeling feasible. However, many network analysis applications, such as optimization models, require running numerous hydraulic simulations with modified input parameters. Therefore, a methodology that can reduce the computational burden of network analysis and still provide the required model accuracy is needed. This paper presents a matrix transformation approach to convert the classic GGA, which is implemented within the widely available freeware EPANET 2, into a more computationally efficient enhanced global gradient algorithm (EGGA). The latter achieves improved efficiency by reducing the size of the mathematical problem through the transformed topological representation of the original network model. By removing serial nodes and serial pipe sections from the original topological representation while preserving those elements in both energy and mass balance equations, EGGA significantly improves the model's computational efficiency without forfeiting its hydraulic accuracy. The computational efficiency and effectiveness of the EGGA approach are demonstrated on four real-life networks. Results show that the computational burden of the EGGA model is significantly lower than that of its GGA counterpart, particularly as the size of the network and/or number of service connections increases. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider certain matrices A of size with exactly k ones in each row and each column where . We say that A is -isolated if there is a single zero entry in some submatrix of A. The submatrix need not be contiguous, i.e. formed from consecutive rows and consecutive columns of A. A is said to be -stable if A contains no -isolated zeros. Several examples are presented. We show that if A is -stable where then and we characterize the case of equality, when the matrix is not equivalent to a direct sum of matrices. [Copyright &y& Elsevier]

Background: Many complex systems can be represented as networks, and how a network breaks up into subnetworks or communities is of wide interest. However, the development of a method to detect nodes important to communities that is both fast and accurate is a very challenging and open problem. Methodology/Principal Findings: In this manuscript, we introduce a new approach to characterize the node importance to communities. First, a centrality metric is proposed to measure the importance of network nodes to community structure using the spectrum of the adjacency matrix. We define the node importance to communities as the relative change in the eigenvalues of the network adjacency matrix upon their removal. Second, we also propose an index to distinguish two kinds of important nodes in communities, i.e., ''community core'' and ''bridge''. Conclusions/Significance: Our indices are only relied on the spectrum of the graph matrix. They are applied in many artificial networks as well as many real-world networks. This new methodology gives us a basic approach to solve this challenging problem and provides a realistic result. [ABSTRACT FROM AUTHOR]

Abstract: This paper gives an account of the contributions of Melvin Henriksen and John Isbell to the abstract theory of f-rings and formally real f-rings, with particular attention to the manner in which their work was framed by universal algebra. I describe the origins of the Pierce–Birkhoff Conjecture and present some other unsolved problems suggested by their work. [Copyright &y& Elsevier]

Abstract: We present the full-rank representations of {2,4} and {2,3}-inverses (with given rank as well as with prescribed range and null space) as particular cases of the full-rank representation of outer inverses. As a consequence, two applications of the successive matrix squaring (SMS) algorithm from [P.S. Stanimirović, D.S. Cvetković-Ilić, Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput. 203 (2008) 19–29] are defined using the full-rank representations of {2,4} and {2,3}-inverses. The first application is used to approximate {2,4}-inverses. The second application, after appropriate modifications of the SMS iterative procedure, computes {2,3}-inverses of a given matrix. Presented numerical examples clarify the purpose of the introduced methods. [Copyright &y& Elsevier]

This paper proposes an Indirect Matrix Converter structure and two SPWM strategies (SPWM strategy and improved SPWM (ISPWM) strategy) for the remedial operation against grid side faults. With the proposed SPWM strategies for the remedial operation, load can continue its operation and allows improved system reliability even in case of grid one or two phases loss. Detailed analysis is provided for proposed strategies and a three-phase balanced R-L load. Performance of proposed SPWM strategy and improved SPWM strategy considering total harmonic distortion (THD) and lower harmonic loss is compared with each other. Simulation results are shown to demonstrate the feasibility of the proposed fault-tolerant approach to the IMC drives. [ABSTRACT FROM AUTHOR]

Abstract: We show that every can be written as a sum of orthogonal matrices () in . Moreover, we show that every can be written as a sum of orthogonal matrices in if and only if the row sums and column sums of A have the same parities. [Copyright &y& Elsevier]

We consider problems of comparing universal algebras in respect of their conditional algebraic geometries. Such comparisons admit of a quite natural algebraic interpretation. Geometric scales for varieties of algebras constructed based on these relations are a natural tool for classifying the varieties of algebras, discriminator varieties in particular. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the ACI-matrices all of whose completions have the same rank, determine the largest number of indeterminates in such partial matrices of a given size, and determine the partial matrices that attain this largest number. [Copyright &y& Elsevier]

Let A = (an,k)n,k≥0 be a non-negative matrix. Denote by Lp,q(A) the supremum of those L satisfying the following inequality: [image omitted] The purpose of this article is to establish a Bennett-type formula for [image omitted] and a Hardy-type formula for [image omitted] and [image omitted], where [image omitted] is a generalized Hausdorff matrix and 0 < p ≤ 1. Similar results are also established for [image omitted] and [image omitted] for other ranges of p and q. Our results extend [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 (2007), pp. 208-217] and [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 (2007) pp. 563-573] from [image omitted] to [image omitted] with α ≥ 0 and completely solve the value problem of [image omitted], [image omitted], [image omitted] and [image omitted] for α ∈  ∪ {0}. [ABSTRACT FROM AUTHOR]

We give a method for constructing certain orthogonal matrices of arbitrary dimension. As application, we construct orthonormal piecewise constant wavelet bases on two-dimensional domains. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations with piecewise constant argument of generalized (DEPCAGs) type, i.e., the argument is a general step function. They are hybrid equations combining properties of continuous and discrete equations. The play of the discrete part is always very important. The explicit solutions of the homogeneous and non-homogeneous linear DEPCAGs systems are obtained. Existence, uniqueness and stability of the solutions of the quasilinear DEPCAGs are under discussion. All previous results are improved. The importance of the advanced and delayed intervals will be clear. Cauchy and Green matrices type are deduced. The integral representation and Gronwall's inequality type obtained can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of DEPCAGs. [ABSTRACT FROM AUTHOR]

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In this paper, an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets. [ABSTRACT FROM AUTHOR]

Abstract: Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) . In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp. [ABSTRACT FROM AUTHOR]