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In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXAH=B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature. [ABSTRACT FROM AUTHOR]

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

We study high order numerical quadratures to one dimensional delta function integrals in this paper. This is motivated by the fact that traditional numerical quadratures give only first order accuracy in general. We provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures. Discrete delta functions and support size formulas satisfying these criteria are designed. Numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures. [ABSTRACT FROM AUTHOR]

These remarks are based on a talk the writer gave at the 11th International Conference in Fixed Point Theory and Applications, held at Galatasaray University in Istanbul, Turkey, July 20-24, 2015. They represent selected thoughts on a career in research, largely devoted to metric fixed point theory, that has spanned over 50 years. [ABSTRACT FROM AUTHOR]

This paper discusses the eigenvalue problem for the real symmetric matrix, especially the determination of the largest eigenvalue. The largest eigenvalue is the maximum extremum of the objective function called the Rayleigh quotient and can be determined by the steepest descent method. It is known, however, that the steepest descent method suffers from slow convergence because it converges linearly. Especially, when the largest and the nest largest eigenvalues have very close values, the convergence is particularly slow. This paper analyzes this situation and shows that the convergence can be accelerated by combining the steepest descent method with a technique called shaking. Finally, it is demonstrated by a numerical example that the convergence is accelerated drastically by the proposed technique. [ABSTRACT FROM AUTHOR]

The article is a comment on an article "Matrix Theory and Cost Allocation," by researchers T.H. Williams and C.H. Griffin, published in the July 1964 issue of the journal "The Accounting Review." According to the author, Williams and Griffin were the first to select this topic as an illustration of the application of matrix algebra to accounting problems. The model presented by Williams and Griffin is a matrix formulation of the popular simultaneous equations method of past years. It was pointed out that the aggregate cost of service departments after allocation in the Williams and Griffin model is more than the combined direct cost before allocation. The scope of this paper has two dimensions: first, to attempt to clarify the matrix algebra approaches that have already been posed and place them in perspective; second, to present a new matrix model of reciprocal service cost allocation and relate it to those previously presented. It would be assumed that each service department will have some of its cost allocated to some of the producing departments.

Let R be any ring with identity. An element a ∊ R is called nil-clean, if a = e + n where e is an idempotent element and n is a nil-potent element. In this paper we give necessary and sufficient conditions for a 2 × 2 matrix over an integral domain R to be nil-clean. [ABSTRACT FROM AUTHOR]

The paper investigates classes of words equivalent under different ways of counting subwords. We present a general method of constructing sequences of equivalent words. Apart from obtaining numerical characterizations of words, we also construct sequences of words generating equal arithmetical power sums. Our notions of equivalence include the Parikh equivalence resulting from Parikh matrices, recently widely studied. Consequently, we are able to settle some open problems in this area. [ABSTRACT FROM AUTHOR]

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n. [ABSTRACT FROM AUTHOR]

The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense "irrelevant" from those which are "relevant" or "nonredundant" in the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid in RMQ, whereas the former are only materially valid. RMQ is a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper how RMQ by its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics. [ABSTRACT FROM AUTHOR]

In this paper, we apply the dual approach developed by A. Chambolle for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches. [ABSTRACT FROM AUTHOR]

In this paper. the design problem of the dynamic output feedback controller for the asymptotic stabilization of a class of linear delay differential systems of the neutral type is considered. A criterion for the existence of such controller is derived based on the matrix inequality approach combined with the Lyapunov method. A parametrized characterization of the controller is given in terms of the feasible solutions to certain matrix inequalities, which can be solved by various convex optimization algorithms. A numerical example is given to illustrate the proposed design method. [ABSTRACT FROM AUTHOR]

Effective design of parallel matrix multiplication algorithms relies on the consideration of many interdependent issues based on the underlying parallel machine or network upon which such algorithms will be implemented, as well as, the type of methodology utilized by an algorithm. In this paper, we determine the parallel complexity of multiplying two (not necessarily square) matrices on parallel distributed-memory machines and/or networks. In other words, we provided an achievable parallel run-time that can not be beaten by any algorithm (known or unknown) for solving this problem. In addition, any algorithm that claims to be optimal must attain this run-time. In order to obtain results that are general and useful throughout a span of machines, we base our results on the well-known LogP model. Furthermore, three important criteria must be considered in order to determine the running time of a parallel algorithm; namely, (i) local computational tasks, (ii) the initial data layout, and (iii) the communication schedule. We provide optimality results by first proving general lower bounds on parallel run-time. These lower bounds lead to significant insights on (i)–(iii) above. In particular, we present what types of data layouts and communication schedules are needed in order to obtain optimal run-times. We prove that no one data layout can achieve optimal running times for all cases. Instead, optimal layouts depend on the dimensions of each matrix, and on the number of processors. Lastly, optimal algorithms are provided. [ABSTRACT FROM AUTHOR]

On modern computers, the performance of programs is often limited by memory latency rather than by processor cycle time. To reduce the impact of memory latency, the restructuring compiler community has developed locality-enhancing program transformations such as loop permutation and tiling. These transformations work well for perfectly nested loops (loops in which all assignment statements are contained in the innermost loop), but their performance on codes such as matrix factorizations that contain imperfectly nested loops leaves much to be desired. In this paper, we propose an alternative approach called data-centric transformation. Instead of reasoning directly about the control structure of the program, a compiler using the data-centric approach chooses an order for the arrival of data elements in the cache, determines what computations should be performed when that data arrives, and generates the appropriate code. At runtime, program execution will automatically pull data into the cache in an order that corresponds approximately to the order chosen by the compiler; since statements that touch a data structure element are scheduled close together, locality is improved. The idea of data-centric transformation is very general, and in this paper, we discuss a particular transformation called data-shackling. We have implemented shackling in the SGI MIPSPro compiler which already has a sophisticated implementation of control-centric transformations for locality enhancement. We present experimental results on the SGI Octane comparing the performance of the two approaches, and show that for dense numerical linear algebra codes, data-shackling does better by factors of two to five. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the mathematical strategies permitting the treatment of second order unilateral systems involving singular mass, damping, and stiffness matrices. Reduction methods are used here to transform second order differential inclusions in first order ones and classical results on differential inclusions are considered in order to obtain solutions. Friction and impact problems arising in Unilateral Mechanics are studied so as to illustrate the theoretical approach. [ABSTRACT FROM AUTHOR]

The paper reviews the role of sensors as a variable depicting information in a stochastic program. Through examples, and with mathematical analysis of random measures, it is shown how policy variables affect the sensors in a given problem. Sensors are amenable to mathematical derivations. It is shown how the trade-off between the gain from introducing new information into the problem, and the cost of that action, can be addressed. Suggestions for further investigation are offered. [ABSTRACT FROM AUTHOR]

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29–43] and Galin [Uspekhi Mat. Nauk, 27 (1972),pp. 241–242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found. [ABSTRACT FROM AUTHOR]

This paper gives a quantitative result that if A is a substochastic matrix and has r eigenvalues which are sufficiently close to 1, then A has r disjoint principal submatrices which are nearly stochastic. [ABSTRACT FROM AUTHOR]

The decomplexification technique proposed for coneigenvalue problems in a recent paper by T. Jiang et al. is discussed and compared with the classical decomplexification technique for eigenvalue problems. Simple explanations for both techniques are presented, and their properties concerning special matrix classes are indicated. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

A matrixover a field is minimal if for every matrixthat commutes with, the centralizer ofis a subset of the centralizer of. In this paper, we study the minimal matrices over the field with two elements. We characterize the minimal matrices with their minimal polynomial of the form, whereis an irreducible polynomial and. We also characterize all minimal matrices with spectrum in. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply a complete pivoting strategy to compute the left-looking version of AINV preconditioner for linear systems. As the preprocessing, the MultiLevel Nested Dissection reordering has also been applied. We have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the University of Florida Sparse Matrix Collection. Numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left-looking version of AINV preconditioner. [Copyright &y& Elsevier]

Abstract: In this paper, we give some necessary and sufficient conditions for the solvability to the system of matrix equations and provide an expression of the general solution to (0.1). Furthermore, we obtain the maximal and minimal ranks of and in (0.1). The findings of this paper extend the known results in the literatures. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]

The goal of this paper is to construct explicitly a p-adic family of representations (which are dihedral representations), to construct their attached (φ, Γ)-modules by writing down explicit matrices for φ and for the action of Γ, and finally to determine which of these are trianguline. [ABSTRACT FROM AUTHOR]

Szemerédi's Regularity Lemma is an important tool for analysing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412–1417]. As an application we give some new representations for the Drazin inverse of a block matrix. [Copyright &y& Elsevier]

The paper investigates the inclusion intervals of matrix singular values. By employing the digraph of a matrix, some new inclusion intervals of matrix singular values are presented. These intervals are based mainly on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that these results improve and generalize some known results of Li et al. (Numer. Linear Algebra Appl. 2007; 14(2):115–128) and Kolotilina (J. Math. Sci. 2006; 137(3):4794–4800). Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Bezier curves with n shape parameters and triangular Bezier surfaces with 3n(n+1)/2 shape parameters are presented in this paper. The geometric significance of the shape parameters and the geometric properties of these curves and surfaces are discussed. The shapes of the curves and the surfaces can be modified intuitively, foreseeably and precisely by changing the values of the shape parameters. [ABSTRACT FROM AUTHOR]

In the present investigation, we obtain some subordination and superordination results involving Hadamard product for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results also pointed out. [ABSTRACT FROM AUTHOR]

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsov's (2009) first-degree proof system ⊢B related to SIXTEEN3. The system GB is a standard Gentzen-type sequent calculus, FB is a four-place (horizontal) matrix sequent calculus, and QB is a quadruple (vertical) matrix sequent calculus. In contrast with GB, the calculus FB satisfies the subformula property, and the calculus QB reflects Odintsov's coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of ⊢B are proved. In addition, it is shown how the sequent systems for ⊢B can be extended to cut-free sequent calculi for Odintsov's LB, which is an extension of ⊢B by adding classical implication and negation connectives. [ABSTRACT FROM AUTHOR]

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and ∣G∣ = p4, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open. [ABSTRACT FROM AUTHOR]

We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator. [ABSTRACT FROM AUTHOR]

Abstract: A variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse are established. Some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two, three and four ordered matrices to be independent in the generalized inverse . As special cases, necessary and sufficient conditions are derived for two, three and four ordered matrices to be independent in the weighted Moore–Penrose inverse and the Drazin inverse. Some known results can be regarded as the special cases of the results in this paper. [Copyright &y& Elsevier]

In this paper we start with the Hardy's well-known Tauberian theorem for Cesàro means formulated as follows, if the sequence X = (xn)n satisfies limC1X = L and Δxn = O(1/n) then limX = L. We give here some extensions to the cases when the Cesàro operator is replaced by the operator of weighted means Nq defined by Hardy and by the operator C(λ). We show among other things that under some conditions if the sequence X = (xn)n satisfies limNqX = L1 and limQnΔqnxn = L2 then limX = L1. Similarly under other conditions it is also shown that limX = L1 if X satisfies the conditions limNqX = L1 and limqn-1QnΔxn = L2. Next we determine sequences μ for which the conditions [C(λ)X]n and μnΔxn are convergent imply together that X is convergent. This result is an extension of Hardy's Tauberian theorem and the other results extend in a certain sense some Tauberian theorems given by Hardy and Littlewood. [ABSTRACT FROM AUTHOR]

In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schröder (2007). For large sparse palindromic QEPs, we develop a generalized T-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting T-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the polynomial Krylov approximations for the computation of φ(A)v, where A is a square matrix, v represents a given vector, and φ is a suitable function which can be employed in modern integrators for differential problems. Our aim consists of proposing and analyzing innovative a posteriori error estimates which allow a good control of the approximation procedure. The effectiveness of the results we provide is tested on some numerical examples of interest. [ABSTRACT FROM AUTHOR]

The truncated SPIKE algorithm is a parallel solver for linear systems which are banded and strictly diagonally dominant by rows. There are machines for which the current implementation of the algorithm is faster and scales better than the corresponding solver in ScaLAPACK (PDDBTRF/PDDBTRS). In this paper we prove that the SPIKE matrix is strictly diagonally dominant by rows with a degree no less than the original matrix. We establish tight upper bounds on the decay rate of the spikes as well as the truncation error. We analyze the error of the method and present the results of some numerical experiments which show that the accuracy of the truncated SPIKE algorithm is comparable to LAPACK and ScaLAPACK. [ABSTRACT FROM AUTHOR]

Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated. We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type λ = αμ for PEPs of arbitrary degree. [ABSTRACT FROM AUTHOR]

We study the convergence of additive and multiplicative Schwarz iterations applied to singular M-matrices and Markov chains. We do our investigations in order to solve consistent linear systems or to calculate a probability distribution vector of a Markov chain. It turns out that for a certain set of equations we are able to prove convergence for both methods with a reliable degree of freedom concerning the overlap. These new convergence theorems are based on a graph theoretical approach and represent the main results of this paper. Other applications of the introduced theory are also discussed. [ABSTRACT FROM AUTHOR]

The question of which multivariate generalized autoregressive conditionally heteroskedastic (GARCH) models in the vec form are representable in the BEKK form is addressed. Using results from linear algebra, it is established that all vec models not representable in the simplest BEKK form contain matrices as parameters that map the vectorized positive semidefinite matrices into a strict subset of themselves. Moreover, a general result from linear algebra is presented implying that in dimension two the models are equivalent, and in dimension three a simple analytically tractable example for a vec model having no BEKK representation is given. [ABSTRACT FROM AUTHOR]

This paper presents some recent filtering methods based on the lower-rank tensor approximation approach for denoising tensor signals. In this approach, multicomponent data are represented by tensors, that is, multiway arrays, and the presented tensor filtering methods rely on multilinear algebra. First, the classical channel-by-channel SVD-based filtering method is overviewed. Then, an extension of the classical matrix filtering method is presented. It is based on the lower rank- (K1, … , KN) truncation of the higher order SVD which performs a multimode principal component analysis (PCA) and is implicitly developed for an additive white Gaussian noise. Two tensor filtering methods recently developed by the authors are also overviewed. The first method consists of an improvement of the multimode PCA-based tensor filtering in the case of an additive correlated Gaussian noise. This improvement is specially done thanks to the fourth order cumulant slice matrix. The second method consists of an extension of Wiener filtering for data tensors. The performances and comparative results between all these tensor filtering methods are presented for the cases of noise reduction in color images, multispectral images, and multicomponent seismic data. [ABSTRACT FROM AUTHOR]

We investigate the sensitivity of the problem of nonorthogonal (matrix) joint diagonalization (NOJD). First, we consider the uniqueness conditions for the problem of exact joint diagonalization (EJD), which is closely related to the issue of uniqueness in tensor decompositions. As a byproduct, we derive the well-known identifiability conditions for independent component analysis (ICA) based on an EJD formulation of ICA. We next introduce some known cost functions for NOJD and derive flows based on these cost functions for NOJD. Then we define and investigate the noise sensitivity of the stationary points of these flows. We show that the condition number of the joint diagonalizer and uniqueness of the joint diagonalizer as measured by modulus of uniqueness (as defined in this paper) affect the sensitivity. We also investigate the effect of the number of matrices on the sensitivity. Our numerical experiments confirm the theoretical results. [ABSTRACT FROM AUTHOR]

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders, and ranks, regardless of the choice of norm (or even Br`egman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank-3 tensor has an optimal rank-2 approximation. The notable exceptions to this misbehavior are rank-1 tensors and order-2 tensors (i.e., matrices). In a more positive spirit, we propose a natural way of overcoming the ill-posedness of the low-rank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete low-dimensional examples as a first step toward more general results. To this end, we present a detailed analysis of equivalence classes of 2×2×2 tensors, and we develop methods for extending results upward to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, it can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular, we make extensive use of the hyperdeterminant Δ on ℝ2×2×2. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibriu [ABSTRACT FROM AUTHOR]

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ⊞λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ⊞λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ. [ABSTRACT FROM AUTHOR]

Let γ = 2cos π/5 and let H5 be the Hecke group associated to γ. In this paper, the normalizers of the congruence subgroups H 05 (I) in PSL (2, 핫[γ]) are studied in the case where I = (2)αI′, (2, I′) = 1 and I′ is a prime ideal. [ABSTRACT FROM AUTHOR]

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 픸핋핃픸핊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He. In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 픾픸ℙ. [ABSTRACT FROM AUTHOR]

In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SOc. [ABSTRACT FROM AUTHOR]