Linear relations of refined enumerations of alternating sign matrices

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of n×n alternating sign matrices where the top row as well as the left and the right column is fixed and the number of n×n alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.     

Combining stir-bar sorptive extraction and large volume injection-gas chromatography-mass spectrometry for the determination of benzotriazole UV stabilizers in wastewater matrices

Stir‐bar sorptive extraction and liquid desorption followed by large volume injection‐gas chromatography coupled to mass spectrometry under selected ion monitoring mode acquisition (SBSE‐LD/LVI‐GC‐MS(SIM)) was applied for the determination of six benzotriazole UV stabilizers (Tinuvin P, Allyl‐bzt, Tinuvin 320, Tinuvin 326, Tinuvin 327 and Tinuvin 328) in wastewater matrices. Parameters affecting the performance of extraction and desorption steps were thoroughly evaluated using uni‐ and multivariate optimization strategies, based on the use of experimental factorial designs. Assays performed with stir bars, coated with 24 μL of polydimethylsiloxane, on 25 mL of ultra‐pure water samples spiked at the 0.5 ng/mL level, yielded recoveries ranging from 47.9±1.4% (Tinuvin P) to 103.1±3.6% (Tinuvin 326), under optimized experimental conditions. When applied to complex matrices (e.g. wastewater), the methodology showed also excellent linear dynamic ranges (0.02–10.00 ng/mL) for the six benzotriazole UV stabilizers studied with correlation coefficients higher than 0.9970, limits of quantification in between 0.004 and 0.015 ng/mL, suitable repeatability (RSD<12.7%) and reproducibility (RSD<4.5%). The application of the proposed methodology to urban sewage waters from Spain and Portugal wastewater plants revealed the presence of low contents of some benzotriazole UV stabilizers.     

Linear matrix differential dynamical systems with fuzzy matrices

In this paper, we investigate linear first-order fuzzy matrix differential dynamical systems where the coefficients matrix is described by a fuzzy matrix. We show some properties of the matrix differential dynamical systems, and their phase portraits are described by means of examples.     

Some properties of the vertical structure operators for hydrostatic atmospheric models

Hydrostatic atmospheric models in a generalized vertical coordinate are considered. The governing equations are linearized with respect to a general basic state including arbitrary vertical variation of the reference temperature profile. Vertical structure equations are derived and studied for both continuous and discretized models. Some properties of vertical normal modes are defined and a comparison between continuous and discrete cases is made.     

Four Mutually Orthogonal Latin Squares of Order 14

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012     

Intersection Matrices Revisited

Several intersection matrices of s‐subsets versus k‐subsets of a v‐set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, use of the derivative operator and some related operators reveals some connections between intersection matrices and the “combinatorics of creation‐annihilation.” As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose–Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 383–397, 2012     

Localization of generalized eigenvalues by Cartesian ovals

In this paper, we consider the localization of generalized eigenvalues, and we discuss ways in which the Gersgorin set for generalized eigenvalues can be approximated. Earlier, Stewart proposed an approximation using a chordal metric. We will obtain here an improved approximation, and using the concept of generalized diagonal dominance, we prove that the new approximation has some of the basic properties of the original Ger?gorin set, which makes it a handy tool for generalized eigenvalue localization. In addition, an isolation property is proved for both the generalized Ger?gorin set and its approximation. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

On iterative techniques for computing flow in large two-dimensional discrete fracture networks

Computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain. For large systems, this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively. Distributions of fracture lengths spanning over several orders of magnitude, and the randomness of fracture orientations and locations, lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern. In addition to the rapid increase in computational effort with increase in the size of the fracture network, the spread in the distribution of fracture parameters, such as length and transmissivity, dramatically influences the convergence behavior of the system of linear equations. An overview of the discrete fracture network (DFN) methodology for computation of flow is presented along with a comparative study of various Krylov subspace iterative methods for the resulting class of sparse matrices. The rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks. Salient features of the observed trends in the convergence pattern are discussed and guidelines for design of DFN algorithms are provided.