the congruence class of the solutions of some matrix equations the congruence class of a least square solution for the following matrix equations ax b a xa d axb dand axxb ef is presented also we derive necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the singular value decomposition svd and generalized singular value decomposition gsvd

an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

the fast hartley transform algorithm the fast hartley transform fht is similar to the cooley tukey fast fourier transform fft but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the fft through use of the fht discrete cosine transforms dct and discrete fourier transforms dft can be obtained the recursive nature of the fht algorithm derived in this paper enables us to generate the next higher order fht from two identical lower order fht s in practice this recursive relationship offers flexibility in programming different sizes of transforms while the orderly structure of its signal flow graphs indicates an ease of implementation in vlsi

actions of inverse semigroups arising from partial actions of groups abstract in this work we present a definition of crossed product for actions of inverse semigroups on c algebras without resorting to covariant representations as done by sieben in related work we also show the existence of an isomorphism between the crossed product by a partial action of a group g and the crossed product by a related action of s g an inverse semigroup associated to g introduced by the first named author

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

iterated collocation methods for volterra integral equations with delay arguments in this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for volterra integral equations with constant delay this analysis includes continuous collocation based volterra runge kutta methods as well as iterated collocation methods and their discretizations

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

implicit runge kutta methods of optimal order for volterra integro differential equations implicit runge kutta methods with m stages and optimal order p 2m for the approximate solution of volterra integro differential equations can be viewed as fully dis cretized collocation methods in certain polynomial spline spaces the choice of the quadrature formulas needed for the full discretization of the collocations is investigated and it is shown that in contrast to ordinary differential equations there exist for fixed m several optimal methods 1 introduction in this paper we shall be concerned with the approximate solution of the initial value problem for a nonlinear first order volterra integro differential equation

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

the interplay between steinberg algebras and skew rings abstract we study the interplay between steinberg algebras and skew rings for a partial action of a group in a hausdorff locally compact totally disconnected topological space we realize the associated partial skew group ring as a steinberg algebra over the transformation groupoid attached to the partial action we then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings over commutative algebras in terms of continuous orbit equivalence of the associated partial actions finally we show that any steinberg algebra associated to a hausdorff ample groupoid can be seen as a partial skew inverse semigroup ring

generalized sliding fft and its application to implementation of block lms adaptive filters this paper has two contributions first the concept of the generalized sliding fast fourier transform gsfft as an efficient implementation of the hopping fft is introduced application of the gsfft is broad and not limited to what has been considered in this paper the frequency domain block lms fblms adaptive filters are then revised and their implementations for block lengths less than the length of the adaptive filter are studied the gsfft and the available pruned ffts are used to give an efficient implementation of these filters in the particular case of the block length equal to one where the fblms algorithm reduces to the frequency domain lms flms algorithm it is shown that the latter can be implemented with the order of m complexity where m is the length of the adaptive filter

preconditioners based on the alternating direction implicit algorithm for the 2d steady state diffusion equation with orthotropic heterogeneous coefficients in this paper we combine the alternating direction implicit adi algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2d steady state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions specifically we adopt the compound iteration idea and use adi iterations as the preconditioner for the outside krylov subspace method that is used to solve the preconditioned linear system an efficient algorithm to perform each adi iteration is crucial to the efficiency of the overall iterative scheme we exploit the kronecker product structure in the matrices inherited from the tensor product basis functions to achieve high efficiency in each adi iteration meanwhile in order to reduce the number of krylov subspace iterations we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions numerical results demonstrated the efficiency and quality of the proposed preconditioner

amenability for fell bundles given a fell bundle b over a discrete group we construct its reduced cross sectional algebra c r b in analogy with the reduced crossed products defined for c dynamical systems when the reduced and full cross sectional algebras of b are isomorphic we say that the bundle is amenable we then formulate an approximation property which we prove to be a sufficient condition for amenability a theory of graded c algebras possessing a conditional expectation is developed with an eye on the fell bundle that one naturally associates to the grading we show for instance that all such algebras are isomorphic to c r b when the bundle is amenable we also study induced ideals in graded c algebras and obtain a generalization of results of str til and voiculescu on af algebras and of nica on quasi lattice ordered groups a brief comment is made on the relevance to our theory of a certain open problem in the theory of exact c algebras an application is given to the case of an fn grading of the cuntz krieger algebras oa recently discovered by quigg and raeburn specifically we show that the cuntz krieger bundle satisfies the approximation property and hence is amenable for all matrices a with entries in 0 1 even if a does not satisfy the well known property i studied by cuntz and krieger in their paper

preconditioned hss iteration method and its non alternating variant for continuous sylvester equations abstract bai 2010 proposed an efficient hermitian and skew hermitian splitting hss iteration method for solving a broad class of large sparse continuous sylvester equations to further improve the efficiency of the hss method in this paper we present a preconditioned hss phss iteration method and its non alternating variant nphss for this matrix equation the convergence properties of the phss and nphss methods are studied in depth and the quasi optimal values of the iteration parameters for the two methods are also derived moreover to reduce the computational cost we establish the inexact variants of the two iteration methods numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

an iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation axb c an iteration method is constructed to solve the linear matrix equation axb c over symmetric x by this iteration method the solvability of the equation axb c over symmetric x can be determined automatically when the equation axb c is consistent over symmetric x its solution can be obtained within finite iteration steps and its least norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new matrix equation axb c finally numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation axb c

collocation methods for nonlinear volterra integro differential equations with infinite delay in this paper we study the numerical solution of nonlinear volterra integrodifferential equations with infinite delay by spline collocation and related runge kutta type methods the kernel function in these equations is of the form k t 8 y t y s with a representative example given by volterra s population equation where we have k t a y t y 8 a t s g y t y 8

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

spline collocation methods for nonlinear volterra integral equations with unknown delay in this paper we introduce and study polynomial spline collocation methods for systems of volterra integral equations with unknown lower integral limit arising in mathematical economics their discretization leads to implicit runge kutta type methods the global convergence and local superconvergence properties of these methods are proved and the theory is illustrated by a numerical example arising in the application of such equations in certain mathematical models of liquidation

m adic polyadic invariance and its applications the concept of m adic invariance allows approximation of a linear time invariant lti system by a linear m adic invariant lmi system or equivalently approximation of a circulant matrix by a super circulant matrix the approximation reduces the number of multiplies required for computing n point cyclic convolution to 2 log m n 1 n where n mn the error introduced by the approximation can be removed if desired by subsequent processing in one concrete case determination of a small number of noncontiguous frequencies this approach approximation and subsequent correction can effect substantial savings in a number of multiplies compared to both fast fourier transform fft algorithm and direct discrete fourier transform ddft these applications are preceded by a tutorial presentation of concepts which are basic to m adic invariant systems

preconditioned hss methods for the solution of non hermitian positive definite linear systems and applications to the discrete convection diffusion equation we study the role of preconditioning strategies recently developed for coercive problems in connection with a two step iterative method based on the hermitian skew hermitian splitting hss of the coefficient matrix proposed by bai golub and ng for the solution of nonsymmetric linear systems whose real part is coercive as a model problem we consider finite differences fd matrix sequences an a p n discretizing the elliptic convection diffusion problem with being a plurirectangle of rd with a x being a uniformly positive function and p x denoting the reynolds function here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes more precisely in connection with preconditioned hss gmres like methods we consider the preconditioning sequence pn a n pn a dn1 2 a an 1 0 dn1 2 a where dn a is the suitably scaled main diagonal of an a 0 if a x is positive and regular enough then the preconditioned sequence shows a strong clustering at unity so that the sequence pn a n turns out to be a superlinear preconditioning sequence for an a 0 n where an a 0 represents a good approximation of re an a p namely the real part of an a p the computational interest is due to the fact that the preconditioned hss method has a convergence behavior depending on the spectral properties of pn 1 a re an a p n pn 1 a an a 0 n therefore the solution of a linear system with coefficient matrix an a p is reduced to computations involving diagonals and to the use of fast poisson solvers for an 1 0 n some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques

gramian based model reduction for data sparse systems model reduction is a common task within the simulation control and optimization of complex dynamical systems for instance in control and optimization problems for instationary partial differential equations pdes the associated large scale linear systems resulting from a semi discretization have to be solved repeatedly model reduction is a more and more accepted technique to attack these problems in reasonable time here we investigate model reduction for parabolic pdes based on balanced truncation bt the system theoretic properties of bt and the related method of singular perturbation approximation spa provide desirable features for the reduced order system the major computational task in bt is the solution of large scale lyapunov or stein equations depending on whether the semidiscretized or fully discretized system is considered therefore the method is often considered to be of limited use for really large scale applications i e problems that require gg1 gbyte of storage so that memory access becomes an issue here we develop effective implementations of bt and spa by exploiting the structure of the underlying pde problem this is achieved by a data sparse approximation of the semi discretized differential operator using the hierarchical matrix format furthermore we review how to integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the lyapunov equations and suggest an iterative scheme for solving stein equations employing formatted arithmetic these approaches are well suited for a class of practically relevant problems and allow the application of bt and spa to systems coming from two dimensional 2d and three dimensional 3d finite element method fem and boundary element method bem discretizations the resulting methods are most beneficial in the context of bem discretizations for 3d problems where large scale dense matrices define the semi discretized control system we also analyze the additional approximation error in the reduced order model introduced by the hierarchical matrix approximation the efficiency of the methods is demonstrated for linear scalar convection diffusion reaction equations the numerical results confirm that the model reduction bt error and the hierarchical matrix approximation error can be balanced by choosing the tolerances and parameters involved appropriately

strongly graded groupoids and strongly graded steinberg algebras abstract we study strongly graded groupoids which are topological groupoids g equipped with a continuous surjective functor g to a discrete group such that 1 1 1 for all we introduce the category of graded g sheaves and prove an analogue of dade s theorem g is strongly graded if and only if every graded g sheaf is induced by a g e sheaf the steinberg algebra of a graded ample groupoid is graded and we prove that the algebra is strongly graded if and only if the groupoid is applying this result we obtain a complete graphical characterisation of strongly graded leavitt path and kumjian pask algebras

new matrix iterative methods for constraint solutions of the matrix equation axb c in this paper two new matrix iterative methods are presented to solve the matrix equation axb c the minimum residual problem min x email protected email protected and the matrix nearness problem min x s email protected x x where s is the set of constraint matrices such as symmetric symmetric r symmetric and r s symmetric and s e is the solution set of above matrix equation or minimum residual problem these matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in deng et al 2006 13 huang et al 2008 15 peng 2005 16 and lei and liao 2007 17 paige s algorithms are used as the frame method for deriving these matrix iterative methods numerical examples are used to illustrate the efficiency of these new methods

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

the ideal structure of algebraic partial crossed products given a partial action of a discrete group g on a hausdorff locally compact totally disconnected topological space x we consider the correponding partial action of g on the algebra lc x consisting of all locally constant compactly supported functions on x taking values in a given field k we then study the ideal structure of the algebraic partial crossed product lc x g after developing a theory of induced ideals we show that every ideal in lc x g may be obtained as the intersection of ideals induced from isotropy groups thus proving an algebraic version of the effros hahn conjecture

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

computational methods for modifying seemingly unrelated regressions models computational efficient methods for updating seemingly unrelated regressions models with new observations are proposed a recursive algorithm to solve a series of updating problems is developed the algorithm is based on orthogonal transformations and has as main computational tool the updated generalized qr decomposition ugqrd strategies to compute the orthogonal factorizations by exploiting the block sparse structure of the matrices are designed the problems of adding and deleting exogenous variables from the seemingly unrelated regressions model have also been investigated the solution of these problems utilize the strategies for computing the ugqrd

associativity of crossed products by partial actions enveloping actions and partial representations given a partial action a of a group g on an associative algebra a we consider the crossed product a g using the algebras of multipliers we generalize a result of exel 1997 on the associativity of a g obtained in the context of c algebras in particular we prove that a g is associative provided that a is semiprime we also give a criterion for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations as an application we endow partial group algebras with a crossed product structure

diagonal preserving isomorphisms of etale groupoid algebras abstract work of jean renault shows that for topologically principal etale groupoids a diagonal preserving isomorphism of reduced c algebras yields an isomorphism of groupoids several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses in this paper we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids our approach has the additional feature that we only need to impose conditions on one of the two groupoids applications are given to leavitt path algebras

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

the ideal structure of steinberg algebras abstract given an ample action of an inverse semigroup on a locally compact and hausdorff topological space we study the ideal structure of the crossed product algebra associated with it by developing a theory of induced ideals we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras this can be interpreted as an algebraic version of the effros hahn conjecture finally as an application of our result we study the ideal structure of a steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra

circle actions on c algebras partial automorphisms and a generalized pimsner voiculescu exact sequence abstract we introduce a method to study c algebras possessing an action of the circle group from the point of view of their internal structure and their k theory under relatively mild conditions our structure theorem shows that any c algebra where an action of the circle is given arises as the result of a construction that generalizes crossed products by the group of integers such a generalized crossed product construction is carried out for any partial automorphism of a c algebra where by a partial automorphism we mean an isomorphism between two ideals of the given algebra our second main result is an extension to crossed products by partial automorphisms of the celebrated pimsner voiculescu exact sequence for k groups the representation theory of the algebra arising from our construction is shown to parallel the representation theory for c dynamical systems in particular we generalize several of the main results relating to regular and covariant representations of crossed products

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

parameterized preconditioned hermitian and skew hermitian splitting iteration method for a class of linear matrix equations recently wang et al 2013 proposed an efficient hermitian and skew hermitian splitting hss iteration method to approximate the solution of linear matrix equation to further improve the efficiency of the hss method we in this paper establish a parameterized preconditioned hss pphss iteration method by introducing two hermitian positive definite preconditioners and two more different iteration parameters for the hss method the convergence properties of the pphss method used for solving linear matrix equation a x b c are strictly analyzed and the quasi optimal values of the iteration parameters for the pphss method are also derived numerical results demonstrate the feasibility and the efficiency of the pphss iteration method

general 2 2 system of nonlinear integral equations and its approximate solution abstract in this note we consider a general 2 2 system of nonlinear volterra type integral equations the modified newton method modified nm is used to reduce the nonlinear problems into 2 2 linear system of algebraic integral equations of volterra type the latter equation is solved by discretization method nystrom method with gauss legendre quadrature is applied for the kernel integrals and newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained finally numerical examples are provided to show the validity and efficiency of the method presented numerical results reveal that the proposed methods is efficient and accurate comparisons with other methods for the same problem are also presented

stationary splitting iterative methods for the matrix equation axb c abstract stationary splitting iterative methods for solving a x b c are considered in this paper the main tool to derive our new method is the induced splitting of a given nonsingular matrix a m n by a matrix h such that i h 1 exists convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants in particular for certain surface fitting applications our method is much more efficient than the progressive iterative approximation pia a conventional iterative method often used in computer aided geometric design cagd

the jacobi and gauss seidel type iteration methods for the matrix equation a x b c in this paper the jacobi and gauss seidel type iteration methods are proposed for solving the matrix equation a x b c which are based on the splitting schemes of the matrices a and b the convergence and computational cost of these iteration methods are discussed furthermore we give the preconditioned jacobi and gauss seidel type iteration methods numerical examples are given to demonstrate the efficiency of these methods proposed in this paper

simplicity of partial skew group rings with applications to leavitt path algebras and topological dynamics let a be a commutative and associative ring not necessarily unital g a group and a partial action of g on ideals of a all of which have local units we show that a is maximal commutative in the partial skew group ring a g if and only if a has the ideal intersection property in a g from this we derive a criterion for simplicity of a g in terms of maximal commutativity and g simplicity of a we also provide two applications of our main results first we give a new proof of the simplicity criterion for leavitt path algebras as well as a new proof of the cuntz krieger uniqueness theorem secondly we study topological dynamics arising from partial actions on clopen subsets of a compact set

derivatives of fourth order kronecker power systems with applications in nonlinear elasticity abstract a natural way to describe systems with polynomial nonlinearities is using the kronecker product particularly third order kronecker power systems can express a wide range of systems from electronic engineering to nonlinear elasticity but such development e g equations of motions of elastic structures from nonlinear strain energy requires standard formulation for the derivative of the kronecker power of vectors with respect to the same vector such standard way cannot be found in literature this paper presents a method to obtain the derivatives of kronecker powers of vectors with respect to itself up to a power of four and also third order kronecker power systems containing those terms in a concise matrix form the matrix expression of these systems provides new approach for efficient numerical implementation organized analysis and linearization to demonstrate the strength of this method an example of application for a finite element nonlinear euler beam is also presented

a generalized preconditioned hss method for non hermitian positive definite linear systems based on the hss hermitian and skew hermitian splitting and preconditioned hss methods we will present a generalized preconditioned hss method for the large sparse non hermitian positive definite linear system our method is essentially a two parameter iteration which can extend the possibility to optimize the iterative process the iterative sequence produced by our generalized preconditioned hss method can be proven to be convergent to the unique solution of the linear system an exact parameter region of convergence for the method is strictly proved a minimum value for the upper bound of the iterative spectrum is derived which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting an efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method the optimality and efficiency are effectively testified by some comparisons with numerical results

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

the reflexive solutions of the matrix equation ax b c in this paper we study the existence of a reflexive with respect to the generalized reflection matrix p solution of the matrix equation ax b c for the special case when b i we get the result of peng and hu 1

noniterative map reconstruction using sparse matrix representations we present a method for noniterative maximum a posteriori map tomographic reconstruction which is based on the use of sparse matrix representations our approach is to precompute and store the inverse matrix required for map reconstruction this approach has generally not been used in the past because the inverse matrix is typically large and fully populated i e not sparse in order to overcome this problem we introduce two new ideas the first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding this theory is based on a distortion metric that reflects the distortions produced in the final matrix vector product rather than the distortions in the coded matrix itself the resulting algorithms are shown to require orthonormal transformations of both the measurement data and the matrix rows and columns before quantization and coding the second idea is a method for efficiently storing and computing the required orthonormal transformations which we call a sparse matrix transform smt the smt is a generalization of the classical fft in that it uses butterflies to compute an orthonormal transform but unlike an fft the smt uses the butterflies in an irregular pattern and is numerically designed to best approximate the desired transforms we demonstrate the potential of the noniterative map reconstruction with examples from optical tomography the method requires offline computation to encode the inverse transform however once these offline computations are completed the noniterative map algorithm is shown to reduce both storage and computation by well over two orders of magnitude as compared to a linear iterative reconstruction methods

a new construction framework for polyphase complete complementary codes with various lengths complete complementary codes ccc with equal sequence lengths have found numerous applications in wireless communications owing to their ideal correlation properties in modern multi user multi rate communication systems cccs with various sequence lengths for different complementary codes are desirable in this paper we propose a new construction of polyphase ccc with various lengths by using unimodular paraunitary pu matrices our key idea is to decompose a large square pu matrix into direct sum of matrices with smaller sizes which is equivalent to block diagonalization of matrices in order to construct ccc with various lengths we show that our proposed constructions not only generalize several previously known methods but also lead us to new cccs with both equal and various lengths