universal bounds for global solutions of a diffusion equation with a nonlocal reaction term we consider the nonlinear heat equation with nonlocal reaction term in space ut u up in smoothly bounded domains we prove the existence of a universal bound for all nonnegative global solutions of this equation moreover in contrast with similar recent results for equations with local reaction terms this is shown to hold for all p 1 as an interesting by product of our proof we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

blowing up of solution for a nonlocal reaction diffusion problem in combustion theory abstract in a nonlocal reaction diffusion model in combustion theory the reaction function involves a physical parameter which is a measure of the strength of the reaction mechanism the purpose of this paper is to show the existence of a critical value such that for a unique global time dependent solution exists and converges to a steady state solution as t and for the solution blows up in finite time a characterization as well as upper and lower bounds of are given

the computation of the principal square roots of centrosymmetric h matrices in this paper two algorithms for computing the principal square root of a centrosymmetric h matrix with positive diagonal entries are proposed it is showed that our algorithms ensure significant savings in computational costs as compared to the case of an arbitrary h matrix a with positive diagonal entries

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

on global attractor for m laplacian parabolic equation with local and nonlocal nonlinearity abstract in this paper we study the long time behavior of solutions for m laplacian parabolic equation u t m u a x u u f 0 u k y u y t d y g x in 0 with the initial data u x 0 u 0 x l q q 1 and zero boundary condition in two cases for a x a 0 0 and a x 0 are considered we obtain the existence and l p estimate of global attractor a in l p for any p max 1 q the attractor a is in fact a bounded set in w 0 1 m l if a x a 0 0 in and a is bounded in w 0 1 m l p if a x 0 in

order conditions for numerical methods for partitioned ordinary differential equations motivated by the consideration of runge kutta formulas for partitioned systems the theory of p series is studied this theory yields the general structure of the order conditions for numerical methods for partitioned systems and in addition for nystrom methods fory f y y for rosenbrock type methods with inexact jacobian w methods it is a direct generalization of the theory of butcher series 7 8 in a later publication the theory ofp series will be used for the derivation of order conditions for runge kutta type methods for volterra integral equations 1

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

a note on unconditional maximum norm contractivity of diagonally split runge kutta methods in this paper we consider diagonally split runge kutta methods for the numerical solution of initial value problems for ordinary differential equations this class of numerical methods was recently introduced by bellen jackiewicz and zennaro siam j numer anal 31 1994 pp 499 523 and comprises the well known class of runge kutta methods their results strongly indicate that diagonally split runge kutta methods break the order barrier p leq 1 for unconditional contractivity in the maximum norm in this paper we investigate the effect of the requirement of unconditional contractivity in the maximum norm on the accuracy of a diagonally split runge kutta method besides the classical order p we deal with an order of accuracy r which is relevant to the case where the method is applied to dissipative initial value problems that are arbitrarily stiff we show that if a diagonally split runge kutta method is unconditionally contractive in the maximum norm then it has orders p r which satisfy p

the general solution to a system of real quaternion matrix equations in this paper we consider the system of matrix equations a 1x c 1 a 2x c 2 a 3xb 3 c 3 and a 4xb 4 c 4 over the real quaternion algebra a necessary and sufficient condition for the existence and the expression of the general solution to the system are given as particular cases the corresponding results on other systems over are also obtained

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

contractivity of waveform relaxation runge kutta iterations and related limit methods for dissipative systems in the maximum norm contractivity properties of runge kutta methods are analyzed with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm in general this type of implementation which is quite appropriate in a parallel computing environment improves the stability properties of runge kutta methods as a result of this analysis a new class of methods is determined which is different from runge kutta methods but closely related to them and which combines its high order of accuracy and unconditional contractivity in the maximum norm this is not possible for classical runge kutta methods

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

global and blow up solutions to a p laplacian equation with nonlocal source this paper deals with a p laplacian equation ut div u p 2 u uq x t dx with null dirichlet boundary conditions in a bounded domain rn where p 2 q 1 under appropriate hypotheses we establish local theory of the solution and obtain that the solution either exists globally or blows up in finite time

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

backward errors for eigenproblem of two kinds of structured matrices in this paper we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew centrosymmetric matrices by making use of the properties of symmetric centrosymmetric and symmetric skew centrosymmetric matrices we derive explicit formulae for the backward errors of approximate eigenpairs

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

unconditional contractivity in the maximum norm of diagonally split runge kutta methods we consider the new class of diagonally split runge kutta dsrk methods 1 5 for the standard initial value problem 1 1 by using the theory of p series we derive the order conditions on the method s coefficients stability analysis is also carried out by showing that the method is unconditionally contractive in the maximum norm if and only if the method is based on particular pairs of discrete and continuous runge kutta methods for an effective implementation of the method an iteration scheme is proposed which preserves the contractivity of any iterate

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

structured least squares problems and inverse eigenvalue problems for p q reflexive matrices abstract a new meaningful structured matrix p q reflexive matrix is defined without the common assumption that p or q is unitary a general solution is derived for its structured least squares problem as a necessary and sufficient condition being presented for the solvability of its structured inverse eigenvalue problem structured constrains are firstly given to guarantee the existence of the solution the optimal approximation problem is also considered under spectral constrains

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

multilevel matrices with involutory symmetries and skew symmetries let n n1 n2 nk where k 1 and n1 nk are integers 1 for 1 i k let pi j 1i 1nj and qi j i 1knj and suppose that ui cni ni is a nontrivial involution i e ui ui 1 ini let ri ipi ui iqi 1 i k and denote r r1 rk if 0 1 l 2k 1 let i 1kli 2i 1 be its binary expansion we say that a cn n is r symmetric if riari 1 li a 1 i k thus we are considering matrices with k levels of block structure and an involutory symmetry or skew symmetry at each level we characterize the class of all r symmetric matrices and study their properties the theory divides into two parts corresponding to 0 and 0 problems involving an r 0 symmetric matrix split into the corresponding problems for 2k 1 matrices with orders summing to n while problems involving an r symmetric matrix with 0 split into the corresponding problems for 2k 1 1 matrices with orders summing to n the latter is also true of a b c where b is r 0 symmetric and c is r symmetric with 0

positivity and conservation properties of some integration schemes for mass action kinetics the numerical schemes approximating chemical reactions according to the mass action law should reproduce at least two properties of the corresponding physical system mass conservation and nonnegativity of the concentrations this paper analyzes the equations of mass action kinetics providing a proof of the existence uniqueness and positivity of the solution under mild hypotheses on the reaction rate and the stoichiometric coefficients we then consider some classic integration schemes in terms of conservation positivity and accuracy compared to schemes tailored for production destruction systems and propose an original scheme which guarantees conservation and nonnegativity of the solution and has order of convergence between 2 and 3

global existence and gradient estimates for the quasilinear parabolic equations of m laplacian type with a nonlinear convection term abstract in this paper we derive precise estimates for u t including smoothing effects near t 0 and decay as t as well as global existence of the solutions u t to the initial boundary value problem in a bounded domain in r n for the quasilinear parabolic equation of the m laplacian type with a nonlinear convection term b u u for the initial data u 0 we only assume u 0 l q 1 q

a linear transformation and its properties with special applications in time series filtering abstract the main purpose of this paper is to introduce a linear transformation called t and to derive its algebraic properties by means of permutation matrices that represent it to demonstrate the importance of the t transformation for the estimation of latent variables in time series decomposition we obtain a general expression for smoothing matrices characterized by symmetric and asymmetric weighting systems we show that the submatrix of the symmetric weights to be applied to central observations is t invariant whereas the submatrices of the asymmetric weights to be applied to initial and final observations are the t transform of each other by virtue of this relation the properties of the t transformation provide useful information on the smoothing of time series data finally we illustrate the role of the t transformation on the weighting systems of several smoothers often applied for trend cycle estimation such as the locally weighted regression smoother loess the cubic smoothing spline the gaussian kernel and the 13 term trend cycle henderson filter

a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices to reduce the costs of computing matrix vector product ax related to a centrosymmetric matrix a as compared to the case of an arbitrary matrix a two algorithms were proposed recently one was designed by melman a melman symmetric centrosymmetric matrix vector multiplication linear algebra appl 320 2000 193 198 for symmetric centrosymmetric matrices another was presented by fassbender and ikramov h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 for general centrosymmetric matrices in this note we further discuss this topic of computing ax where a is a generalized centrosymmetric matrix we firstly investigate the reducibility of a generalized centrosymmetric matrix and then provide an algorithm which can be viewed as a generalization of one h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 we show that our algorithm is suitable for computing many matrix vector products with the same matrix furthermore we show that similar results can be obtained for certain subclasses of generalized centrosymmetric matrices as compared to the case of a centrosymmetric matrix and analogue gains are available for generalized skew centrosymmetric or generalized centrohermitian matrices

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

asymptotic behaviour of nonlocal reaction diffusion equations abstract the existence of a global attractor in l 2 is established for a reaction diffusion equation on a bounded domain in r d with dirichlet boundary conditions where the reaction term contains an operator f l 2 l 2 which is nonlocal and possibly nonlinear existence of weak solutions is established but uniqueness is not required compactness of the multivalued flow is obtained via estimates obtained from limits of galerkin approximations in contrast with the usual situation these limits apply for all and not just for almost all time instants

procrustes problems for p q η reflexive matrices this paper characterizes p q h reflexive matrices showing that a p q h reflexive matrix can be represented in terms of k matrices a a b c m a x n b where a b h modk m a and n b are dimensions of the t a and t b eigenspaces of p and q respectively a general solution of procrustes problems of p q h reflexive matrices is presented in terms of lower order matrices a 1 h 1 a h 1 1 a h k a h 1 k 1 a k h under the condition that p and q are unitary

hermitian hermitian r symmetric and hermitian r skew symmetric procrustes problems let r cn n be a nontrivial unitary involution i e r r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i hermitian matrices ii hermitian r symmetric matrices iii hermitian r skew symmetric matrices given z w cn m we characterize the matrices a in s that minimize az w frobenius norm and given an arbitrary e cn n we find the unique matrix among the minimizers of az w in s that minimizes a e we also obtain necessary and sufficient conditions for existence of a s such that az w and assuming that the conditions are satisfied characterize the set of all such a