on additive transformations preserving a multiplicative matrix function abstract let m n k be the ring of all n n matrices over a division ring k and f be a multiplicative matrix function from m n k to a multiplicative abelian group with zero g 0 f ab f a f b a b m n k we call an additive transformation on m n k preserves a multiplicative matrix function f if f a f a a m n k in this paper we characterize all additive surjective transformations on m n k over any division ring k chk 2 that leave a non trivial multiplicative matrix function invariant applications to several related preservers are considered

a generalization of the craig sakamoto theorem to euclidean jordan algebras abstract letac and massam 10 extended the craig sakamoto theorem to euclidean jordan algebras in this paper we give another proof of this generalization by reformulating the result in terms of rank and determinant equalities and by proving the result in each of the simple euclidean jordan algebras

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

spectra of sparse non hermitian random matrices an analytical solution we present the exact analytical expression for the spectrum of a sparse non hermitian random matrix ensemble generalizing two standard results in random matrix theory this analytical expression constitutes a non hermitian version of the kesten mckay measure as well as a sparse realization of girko s elliptic law our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs which are locally treelike in this context we show analytically that the convergence rate of a transport process on a very sparse graph depends in a nonmonotonic way upon the degree of symmetry of the graph edges

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

on exponential of split quaternionic matrices abstract the exponential of a matrix plays an important role in the theory of lie groups the main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their lie algebras into the groups since the set of split quaternions is a noncommutative algebra the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers therefore we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix

color sparse representations for image processing review models and prospects sparse representations have been extended to deal with color images composed of three channels a review of dictionary learning based sparse representations for color images is made here detailing the differences between the models and comparing their results on the real and simulated data these models are considered in a unifying framework that is based on the degrees of freedom of the linear filtering transformation of the color channels moreover this allows it to be shown that the scalar quaternionic linear model is equivalent to constrained matrix based color filtering which highlights the filtering implicitly applied through this model based on this reformulation the new color filtering model is introduced using unconstrained filters in this model spatial morphologies of color images are encoded by atoms and colors are encoded by color filters color variability is no longer captured in increasing the dictionary size but with color filters this gives an efficient color representation

commuting graphs of some subsets in simple rings abstract let d be a division ring with center f and n 1 a natural number for s m n d the commuting graph of s denoted by s is the graph with vertex set s z s such that distinct vertices a and b are adjacent if and only if ab ba in this paper we prove that if n 2 and a n i t are the sets of all non invertible nilpotent idempotent matrices and involutions respectively then for any division ring d a n i and t are connected graphs we show that if n 2 and u is the set of all upper triangular matrices then for every algebraic division ring d u is a connected graph also it is shown that if r is the set of all reducible matrices and m n d is algebraic over f then for n 2 r is a connected graph finally we prove that for n 2 m n h is a connected graph where h is the ring of real quaternions

sliding mode control design for nonlinear systems using probability density function shaping in this paper we propose a sliding mode based stochastic distribution control algorithm for nonlinear systems where the sliding mode controller is designed to stabilize the stochastic system and stochastic distribution control tries to shape the sliding surface as close as possible to the desired probability density function kullback leibler divergence is introduced to the stochastic distribution control and the parameter of the stochastic distribution controller is updated at each sample interval rather than using a batch mode it is shown that the estimated weight vector will converge to its ideal value and the system will be asymptotically stable under the rank condition which is much weaker than the persistent excitation condition the effectiveness of the proposed algorithm is illustrated by simulation

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

consimilarity of quaternion matrices and complex matrices this paper discusses consimilarity of quaternion matrices obtains the jordan canonical form of a quaternion matrix under consimilarity and some necessary and sufficient conditions for consimilarity of quaternion matrices the consimilarity of quaternion matrices is a natural extension of the complex consimilarity of complex matrices has many good properties including some that are essentially different from those of complex consimilarity moreover using the consimilarity of quaternion matrices we obtain a new way to discuss complex consimilarity of complex matrices

a system of generalized sylvester quaternion matrix equations and its applications let h m n be the set of all m n matrices over the real quaternion algebra we call that a h n n is hermitian if a a i j k where i j k are the quaternion units denote a a in this paper we derive some necessary and sufficient conditions for the solvability to the system of generalized sylvester real quaternion matrix equations a i x i y i b i c i z d i e i i 1 2 and give an expression of the general solution to the above mentioned system as applications we give some solvability conditions and general solution for the generalized sylvester real quaternion matrix equation a 1 x y b 1 c 1 z d 1 e 1 where z is required to be hermitian we also present some solvability conditions and general solution for the system of real quaternion matrix equations involving hermicity a i x i a i x i b i y b i c i i 1 2 where y is required to be hermitian our results include some well known results as special cases

an ilc based adaptive control for general stochastic systems with strictly decreasing entropy in this paper a new method for adaptive control of general nonlinear and non gaussian unknown stochastic systems has been proposed the method applies the minimum entropy control scheme to decrease the closed loop randomness of the output under an iterative learning control ilc basis both modeling and control of the plant are performed using dynamic neural networks for this purpose the whole control horizon is divided into a certain number of time domain subintervals called batches and a pseudo d type ilc law is employed to train the plant model and controller parameters so that the entropy of the closed loop tracking error is made to decrease batch by batch the method has the advantage of decreasing the output uncertainty versus the advances of batches along the time horizon the analysis on the proposed ilc convergence is made and a set of demonstrable experiment results is also provided to show the effectiveness of the obtained control algorithm where encouraging results have been obtained

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

quaternionic differential operators motivated by a quaternionic formulation of quantum mechanics we discuss quaternionic and complex linear differential equations we touch only a few aspects of the mathematical theory namely the resolution of the second order differential equations with constant coefficients we overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients the resolution of the complex linear schrodinger equation in the presence of quaternionic potentials represents an interesting application of the mathematical material discussed in this paper

optimization in quaternion dynamic systems gradient hessian and learning algorithms the optimization of real scalar functions of quaternion variables such as the mean square error or array output power underpins many practical applications solutions typically require the calculation of the gradient and hessian however real functions of quaternion variables are essentially nonanalytic which are prohibitive to the development of quaternion valued learning systems to address this issue we propose new definitions of quaternion gradient and hessian based on the novel generalized hamilton real ghr calculus thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field rather than using the isomorphism with the real domain as is current practice in addition unlike the existing quaternion gradients the ghr calculus allows for the product and chain rule and for a one to one correspondence of the novel quaternion gradient and hessian with their real counterparts properties of the quaternion gradient and hessian relevant to numerical applications are also introduced opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms the proposed ghr calculus is shown to yield the same generic algorithm forms as the corresponding real and complex valued algorithms advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks

eigenvectors of the discrete laplacian on regular graphs a statistical approach in an attempt to characterize the structure of eigenvectors of random regular graphs we investigate the correlations between the components of the eigenvectors associated with different vertices in addition we provide numerical observations suggesting that the eigenvectors follow a gaussian distribution following this assumption we reconstruct some properties of the nodal structure which were observed in numerical simulations but were not explained so far dakel et al 2007 approx random pp 436 48

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

the isospectral fruits of representation theory quantum graphs and drums we present a method which enables one to construct isospectral objects such as quantum graphs and drums one aspect of the method is based on representation theory arguments which are shown and proved the complementary part concerns techniques of assembly which are both stated generally and demonstrated for this purpose quantum graphs are grist to the mill we develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects we discuss the theoretical implications which include sunada s theorem of isospectrality sunada 1985 ann math 121 169 arising as a particular case of this method a gallery of new isospectral examples is presented and some known examples are shown to result from our theory

basic classes of matrices with respect to quaternionic indefinite inner product spaces basic classes of matrices or linear transformations in finite dimensional quaternionic vector spaces with a regular indefinite inner product are introduced and studied the classes include plus matrices selfadjoint skew adjoint and unitary matrices in particular results are proved concerning extensions of invariant semidefinite subspaces canonical form for unitary matrices is developed and subsequently applied to stability of periodic hamiltonian systems

the quaternionic second weighted zeta function of a graph and the study determinant abstract we establish a generalization of the second weighted zeta function of a graph to the case of quaternions for an arc weighted graph whose weights are quaternions we define the second weighted zeta function by using the study determinant that is a quaternionic determinant for quaternionic matrices defined by study this definition is regarded as a quaternionic analogue of the determinant expression of hashimoto type for the ihara zeta function of a graph we derive the study determinant expression of bass type and the euler product for the quaternionic second weighted zeta function

diagonalization of bounded linear operators on separable quaternionic hilbert space by using the representation of its complex conjugate pairs we have investigated the diagonalization of a bounded linear operator on separable infinite dimensional right quaternionic hilbert space the sufficient condition for diagonalizability of quaternionic operators is derived the result is applied to anti hermitian operators which is essential for solving schrodinger equation in quaternionic quantum mechanics

trace formulae and spectral statistics for discrete laplacians on regular graphs i following the derivation of the trace formulas in the first paper in this series we establish here a connection between the spectral statistics of random regular graphs and the predictions of random matrix theory rmt this follows from the known poisson distribution of cycle counts in regular graphs in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely the result is analogous to the so called diagonal approximation in quantum chaos we also show that by assuming that the spectral correlations are given by rmt to all orders we can compute the leading deviations from the poisson distribution for cycle counts we provide numerical evidence which supports this conjecture

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

ilc based fixed structure controller design for output pdf shaping in stochastic systems using lmi techniques in this paper a generalized state space controller design for the shaping of the output probability density function pdf is presented for non gaussian dynamical stochastic systems a radial basis function rbf neural network is used to approximate the output pdf of the system such a neural network consists of a number of weights and corresponding basis functions using such an approximation the dynamics of the original stochastic system can be expressed as the dynamics between the control input and the weights of the rbf neural network the task of output pdf control can therefore be reduced to a rbf weight control together with an adaptive tuning of the basis function parameters i e the centers and widths of the basis functions to achieve this aim the control horizon is divided into certain intervals hereinafter called batches using these definitions the whole control strategy consists of three stages namely a sub space parameter identification of the dynamic nonlinear model that relates the control signal to the weights of the rbf neural network b weight tracking controller design using an lmi based convex optimization technique and c rbf basis functions shape tuning in terms of their centers and widths using an iterative learning control ilc framework among the above stages the first two are performed within each batch while stage c is carried out between any two adjacent batches such an algorithm has the advantage of the batch by batch improvement of the closed loop output pdf tracking performance moreover the controller mentioned in stage b is a general controller in a state space form stability analysis has been performed and simulation results are included to show the effectiveness of the proposed method where encouraging results have been made

exterior interior duality for discrete graphs the exterior interior duality expresses a deep connection between the laplace spectrum in bounded and connected domains in and the scattering matrices in the exterior of the domains here this link is extended to the study of the spectrum of the discrete laplacian on finite graphs for this purpose two methods are devised for associating scattering matrices to the graphs the exterior interior duality is derived for both methods

on the solutions of tridiagonal linear systems tridiagonal linear systems are of special importance being appeared in space researches and also in several well known algorithms there are many methods for solving a tridiagonal linear system most of them are suffering from the accumulation of the round off errors in this paper a new method will be given to overcome the defects of the other methods also two direct methods for solving a system of linear equations having matrices of quaternion entries as coefficients and independent terms are proposed second kind chebyshev polynomials in a matrix argument are used as a tool to do certain calculations

schur complement systems in the mixed hybrid finite element approximation of the potential fluid flow problem the mixed hybrid finite element discretization of darcy s law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components as a method of solution the reduction to three schur complement systems based on successive block elimination is considered the first and second schur complement matrices are formed eliminating the velocity and pressure variables respectively and the third schur complement matrix is obtained by elimination of a part of lagrange multipliers that come from the hybridization of a mixed method the structural properties of these consecutive schur complement matrices in terms of the discretization parameters are studied in detail based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to schur complement systems it is shown that due to special block structure the spectral properties of successive schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified theoretical results are illustrated by numerical experiments

strict double diagonal dominance in euclidean jordan algebras abstract in the first part of the paper we deal with euclidean jordan algebraic generalizations of some results of brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs as a consequence the theorems of brauer ostrowski and brauer on the location of eigenvalues are extended to the setting of euclidean jordan algebras in the second part motivated by the work of li and tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses we present some inter relations between the h property generalized strict diagonal dominance invertibility and strict double diagonal dominance in euclidean jordan algebras in addition we show that in a euclidean jordan algebra the schur complements of a strictly doubly diagonally dominant element inherit this property

a structure preserving jacobi algorithm for quaternion hermitian eigenvalue problems abstract a new real structure preserving jacobi algorithm is proposed for solving the eigenvalue problem of quaternion hermitian matrix by employing the generalized jrs symplectic jacobi rotations the new quaternion jacobi algorithm can preserve the symmetry and jrs symmetry of the real counterpart of quaternion hermitian matrix moreover the proposed algorithm only includes real operations without dimension expanding and is generally superior to the state of the art algorithm numerical experiments are reported to indicate its efficiency and accuracy

extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications abstract assume that the linear quaternion matrix expression f x 1 x 2 a a 3 x 1 b 3 a 4 x 2 b 4 where x 1 x 2 are variant quaternion matrices in this paper we derive the maximal and minimal ranks of f x 1 x 2 subject to consistent systems of quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 and a 2 x 2 c 3 x 2 b 2 c 4 moreover corresponding results on some special cases are presented as applications we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations some previous known results can be regarded as the special cases of this paper

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

an estimate for the average spectral measure of random band matrices for a class of random band matrices of band width w we prove regularity of the average spectral measure at scales w 0 99 and find its asymptotics at these scales

adaptive predictive control policy for nonlinear stochastic systems the problem of optimal adaptive predictive control for nonlinear stochastic systems is considered a classification of various suboptimal approaches is given a new approximation for the probability measure for the extended state of the system is suggested to derive a new suboptimal control law it is assumed for this approximation that the system operates in closed loop feedback mode for one part of the extended state vector and in open feedback loop mode for the other part of this vector the certainty equivalence ce assumption is used only for the first part of the extended state vector an analytical comparison for the suggested control policy shows its superiority in control quality compared with that of the open loop optimal control policy in the case of an exactly observed first part of the extended state vector the upper bound of the performance index is determined for this case the suggested control policy has a simple form for linear systems with unknown parameters a simulated example is used to demonstrate the potential of the suggested method and its superiority over the usually applied ce policy

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

conjugacy invariants of sl 2 h abstract in this paper we use the quaternionic formalism of mobius transformations on r 4 to derive conjugacy invariants on sl 2 h and hence on psl 2 h we then use these invariants to distinguish between the various conjugacy classes of psl 2 h

music algorithm for vector sensors array using biquaternions in this paper we use a biquaternion formalism to model vector sensor signals carrying polarization information this allows a concise and elegant way of handling signals with eight dimensional 8 d vector valued samples using this model we derive a biquaternionic version of the well known array processing music algorithm and we show its superiority to classically used long vector approach new results on biquaternion valued matrix spectral analysis are presented of particular interest for the biquaternion music bq music algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces we propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition evd of a hermitian biquaternionic matrix by means of a newly defined quantity the quaternion adjoint matrix the bq music estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing lv music and psa music the proposed algorithm is shown to be superior in several aspects to the existing approaches compared with lv music the bq music algorithm is more robust to modelization errors and coherent noise while it can detect less sources in comparaison with psa music our approach exhibits more accurate estimation of direction of arrival doa for a small number of sources while keeping the polarization information accessible

spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices to a unitary matrix u we associate a doubly stochastic matrix m by taking the squared modulus of each element of u to study the connection between onset of quantum chaos on graphs and ergodicity of the underlying markov chain specified by m we study the limiting distribution of the spectral gap of m when u is taken from the circular unitary ensemble and the dimension n of u is taken to infinity we prove that the limiting distribution is degenerate the gap tends to its maximal value 1 the shape of the gap distribution for finite n is also discussed

the autocorrelation function for spectral determinants of quantum graphs this paper considers the spectral determinant of quantum graph families with chaotic classical limit the secular coefficients of the spectral determinant are found to follow distributions with zero mean and variance approaching a constant in the limit of large network size for graphs without symmetries this constant is in general different from the random matrix result and depends on the classical limit a closed expression for this system dependent constant is given here explicitly in terms of the spectrum of an underlying markov process related results for graphs with time reversal symmetry are given

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

some weighted bartholdi zeta function of a digraph related to the weighted bond scattering matrix of a graph we define some weighted bartholdi zeta function of a digraph d and give a determinant expression for it we present a decomposition formula for the weighted zeta function of a group covering of d furthermore we define an l function of d and give a determinant expression for it as a corollary we express the weighted bartholdi zeta function of a group covering of d by means of its l functions

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

on solutions of the quaternion matrix equation ax b and their applications in color image restoration by using the complex representation of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the least squares solution with the least norm the least squares pure imaginary solution with the least norm and the least squares real solution with the least norm for the quaternion matrix equation ax b respectively finally we discuss their applications in color image restoration

gleason s problem and schur multipliers in the multivariable quaternionic setting abstract we define and study the counterparts of gleason s problem of the drury arveson s space and of schur multipliers when the unit ball of c n is replaced by the unit ball of h n where h denotes the algebra of real quaternions schur multipliers are characterized in terms of coisometric operator matrices in quaternionic spaces finally we define the counterpart of blaschke factors in this setting and study a one point interpolation problem

quaternionic matrices unitary similarity simultaneous triangularization and some trace identities abstract we construct six unitary trace invariants for 2 2 quaternionic matrices which separate the unitary similarity classes of such matrices and show that this set is minimal we have discovered a curious trace identity for two unit speed one parameter subgroups of sp 1 a modification gives an infinite family of trace identities for quaternions as well as for 2 2 complex matrices we were not able to locate these identities in the literature we prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field finally we consider the problem of describing the semi algebraic set of pairs x y of quaternionic n n matrices which are simultaneously triangularizable even the case n 2 which we analyze in more detail remains unsolved

a 3d ginibre point field we introduce a family of three dimensional random point fields using the concept of the quaternion determinant the kernel of each field is an n dimensional orthogonal projection on a linear space of quaternionic polynomials we find explicit formulas for the basis of the orthogonal quaternion polynomials and for the kernel of the projection for number of particles n rightarrow infty we calculate the scaling limits of the point field in the bulk and at the center of coordinates we compare our construction with the previously introduced fermi sphere point field process

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

approximate schur complement preconditioning of the lowest order nodal discretizations certain classes of nodal methods and mixed hybrid finite element methods lead to equivalent robust and accurate discretizations of second order elliptic pdes however widespread popularity of these discretizations has been hindered by the awkward linear systems which result the present work overcomes this awkwardness and develops preconditioners which yield solution algorithms for these discretizations with an efficiency comparable to that of the multigrid method for standard discretizations our approach exploits the natural partitioning of the linear system obtained by the mixed hybrid finite element method by eliminating different subsets of unknowns two schur complements are obtained with known structure replacing key matrices in this structure by lumped approximations we define three optimal preconditioners central to the optimal performance of these preconditioners is their sparsity structure which is compatible with standard finite difference discretizations and hence treated adequately with only a single multigrid cycle in this paper we restrict the discussion to the two dimensional case these techniques are readily extended to three dimensions

a semi infinite horizon lq self tuning regulator for armax plants based on rls so far no extension of the celebrated self tuning property of the rls gmv adaptive controller is available for the steady state lq stochastic control the above extension is addressed in the paper namely we pose the question is it possible to adaptively get the semi infinite horizon lq stochastic control for any armax plant by using a finite number of predictors whose parameters are estimated by standard rls an adaptive control algorithm solving an underlying control problem that coincides with the semi infinite lq stochastic feedback is developed it embodies a standard rls separate identification of the parameters of t n 1 predictors of the joint i o process n being the order of the armax plant together with an appropriate control synthesis rule the proposed algorithm turns out to be a slightly modified version of musmar with the capability of turning itself to the semi infinite horizon lq stochastic regulator

quaternion var modelling and estimation quaternion vector autoregression var modeling is a natural extension of real and complex var we demonstrate how a quaternion var can be treated as a special case of structured real var we show that generalized least squares and under gaussianity maximum likelihood estimation of the parameters reduces to simple least squares estimation if the innovations are quaternion proper

tracking control for non gaussian stochastic distribution sampled data fuzzy systems abstract this paper studies the tracking control problem for the non gaussian stochastic distribution sampled data fuzzy systems first by constructing an augmented system an asynchronous proportional integral pi controller is designed such that the probability density functions pdfs of the system output can track the target one second by introducing two sets the extension reachable set and the ellipsoid set the state trajectories of the augmented system with persistent reference input signal can be restricted in a given local region namely the state constrained problem can be addressed furthermore in some existing asynchronous controller strategies it is required that the derivative of the membership function of the system not explicitly depends on the system input however in this paper by restricting the state into a specified region a new asynchronous scheme is proposed so as to the above mentioned requirement is removed finally an example is given to demonstrate the efficiency of the presented method

on the exponential generating function for non backtracking walks we derive an explicit formula for the exponential generating function associated with non backtracking walks around a graph we study both undirected and directed graphs our results allow us to derive computable expressions for non backtracking versions of network centrality measures based on the matrix exponential we find that eliminating backtracking walks in this context does not significantly increase the computational expense we show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions rigorous analysis on the star graph illustrates the effect of non backtracking and shows that unwanted localization effects can be eliminated when we restrict to non backtracking walks we also investigate the localization issue on synthetic networks

quasistationary quaternionic hamiltonians and complex stochastic maps we show that the complex projections of time dependent quasianti hermitian quaternionic hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi hermitian density matrices if and only if a quasistationarity condition is fulfilled i e if and only if is an hermitian positive time independent complex operator an example is also discussed

quaternions further contributions to a matrix oriented approach abstract by representing quaternions as four dimensional vectors and the multiplication of quaternions as matrix by vector product we employ a matrix oriented approach to the topic we investigate properties of the fundamental real matrix associated with a quaternion and examine a frequently considered quaternion equation from which the n th power of a quaternion can be determined

burnside s theorem for matrix rings over division rings abstract a version of burnside s theorem states that if f is an arbitrary field and a m n f is an irreducible or equivalently transitive subalgebra containing a rank one matrix then a m n f the present paper shows that if f is replaced by a division ring d then every transitive left subalgebra of mn d containing a rank one matrix is equal to mn d here by a left algebra we mean a ring which is also a left d module counterexamples are given in case a is irreducible but not transitive moreover it is shown that irreducible left algebras of quaternionic matrices contain rank one idempotents and their structures are classified

application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity selected groundwater flow scenarios are used in a two way comparison between the mixed hybrid finite element method and the standard finite element method also called the conforming finite element method the simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists the basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element by contrast the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential the conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity the mixed hybrid finite element method provides velocities everywhere in the field as well as potentials at the center of each element and each edge moreover the normal component of the velocity field is continuous between adjacent elements the results of the simulations are presented in the form of streamlines to avoid the problem of velocity discontinuity the method of cordes and kinzelbach 1992 is used it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method the comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy it is also superior to the conforming method in terms of computational effort the potential fields obtained by the mixed hybrid and the conforming finite element methods are the same