realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

an accelerated cyclic reduction based solvent method for solving quadratic eigenvalue problem of gyroscopic systems abstract the quadratic eigenvalue problem qep 2 m g k x 0 with m t m being positive definite k t k being negative definite and g t g is associated with gyroscopic systems in guo 2004 a cyclic reduction based solvent crs method was proposed to compute all eigenvalues of the above mentioned qep firstly the problem is converted to find a suitable solvent of the quadratic matrix equation qme m x 2 g x k 0 then using a cayley transformation and a proper substitution the qme is transformed into the nonlinear matrix equation nme z a t z 1 a q with a m k g and q 2 m k the problem finally can be solved by applying the cr method to obtain the maximal symmetric positive definite solution of the nme as long as the qep has no eigenvalues on the imaginary axis or for some cases where the qep has eigenvalues on the imaginary axis however when all eigenvalues of the qep are far away from or near the origin the cayley transformation seems not to be the best one and the convergence rate of the crs method proposed in guo 2004 might be further improved in this paper inspired by using a doubling algorithm to solve the qme we use a mobius transformation instead of the cayley transformation to present an accelerated crs acrs method for solving the qep of gyroscopic systems in addition we discuss the selection strategies of optimal parameter for the acrs method numerical results demonstrate the efficiency of our method

state for 2 d systems abstract a new definition of state for n d systems is given in a noncausal context this definition is based on a deterministic markovian like property it is shown that for the particular case of ar 2 d systems it yields systems that can be described by a special kind of first order equations the solutions of these equations can be simulated by means of a local line by line computational scheme

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

existence and multiplicity results for nonlinear periodic boundary value problems abstract this paper studies the existence of positive solutions for periodic boundary value problems the criteria for the existence nonexistence and multiplicity of positive solutions are established by using the global continuation theorem fixed point index theory and approximate method the results obtained herein generalize and complement some previous findings of j r graef l kong h wang existence multiplicity and dependence on a parameter for a periodic boundary value problem j differential equations 245 2008 1185 1197 and some other known results

a numerical method for quadratic eigenvalue problems of gyroscopic systems abstract we consider the quadratic eigenvalues problem qep of gyroscopic systems 2 m g k x 0 where m m g g and k k r n n with m being positive definite guo numerical solution of a quadratic eigenvalue problem linear algebra and its applications 385 2004 391 406 showed that all eigenvalues of the qep can be found by solving the maximal solution of a nonlinear matrix equation z a z 1 a q with quadratic convergence when the qep has no eigenvalues on the imaginary axis the convergence becomes linear or more slower guo 2004 when the qep allows purely imaginary eigenvalues having even partial multiplicities in this paper we consider the general case when the qep has eigenvalues on the imaginary axis we propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts while keeps other eigenpairs unchanged this transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically numerical examples illustrate the efficiency of our method

on the hermitian positive definite solutions of nonlinear matrix equation x s σ m i 1 a i x t i a i q in this paper the hermitian positive definite solutions of nonlinear matrix equation x email protected i 1 ma i x t ia i q are considered where q is a hermitian positive definite matrix a i are nonsingular complex matrices s m are positive numbers and 0

converting high order linear pdes to first order abstract a straightforward way requiring no calculations is offered to convert a system of high order constant coefficient linear pdes into a system of first order pdes

existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem abstract in this paper we study the existence of nontrivial periodic solutions to the following nonlinear differential equation u t a t u t f t u t t r u 0 u u 0 u where a r r is an periodic continuous function with a t 0 f r r r is continuous may take negative values and can be sign changing without making any nonnegative assumption on nonlinearity by using the first eigenvalue corresponding to the relevant linear operator and the topological degree the existence of nontrivial periodic solutions to the above periodic boundary value problem is established finally three examples are given to demonstrate the validity of our main results

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints abstract the inverse eigenvalue problems play an important role in broad application areas such as system identification hopfield neural networks control design mass spring system and molecular spectroscopy this paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions m c k under prescribed submatrix constraints of the quadratic inverse eigenvalue problem m x 2 c x k x 0 the algorithm is developed based on the conjugate gradient normal equations residual minimizing cgnr method we discuss the convergence properties of the algorithm finally the performance of the algorithm is tested on two numerical examples and compared to the previous algorithm

numerical solution of a quadratic eigenvalue problem we consider the quadratic eigenvalue problem qep 2 m g k x 0 where m m t is positive denite k k t is negative denite and g g t the eigenvalues of the qep occur in quadruplets or in real or purely imaginary pairs we show that all eigenvalues of the qep can be found eciently and with the correct symmetry by nding a proper solvent x of the matrix equation mx 2 gx k 0 as long as the qep has no eigenvalues on the imaginary axis this solvent approach works well also for some cases where the qep has eigenvalues on the imaginary axis ams classication 15a18 15a24 65f30

an efficient algorithm based on lanczos type of bcr to solve constrained quadratic inverse eigenvalue problems abstract this paper proposes the lanczos type of biconjugate residual bcr algorithm for solving the quadratic inverse eigenvalue problem l 2 x 2 l 1 x l 0 x 0 where l 2 l 1 and l 0 should be partially bisymmetric under a prescribed submatrix constraint an analysis reveals that the algorithm obtains the solutions of the constrained quadratic inverse eigenvalue problem in finitely many steps in the absence of round off errors finally numerical results are performed to confirm the analysis and to illustrate the efficiency of the proposed algorithm

markov properties for systems described by pdes and first order representations abstract the relation between markovianity and representability by means of first order pdes is investigated we consider two versions of the markovian property weak and strong markovianity the weak version has been introduced in j c willems state and first order representations in v d blondel a megretski eds unsolved problems in mathematical systems control theory princeton university press princeton nj 2004 pp 54 57 and conjectured to correspond to first order representations we provide a counterexample to this conjecture for finite dimensional behaviors strong markovianity is proven to be indeed equivalent to the representability by means of first order pdes with a special structure

state space realization of 2 d finite dimensional behaviours this paper deals with the state space realization of autonomous autoregressive two dimensional 2 d systems in the context of the behavioural approach an arbitrary autoregressive 2 d system sig

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

levy walks with applications to turbulence and chaos abstract diffusion on fractal structures has been a popular topic of research in the last few years with much emphasis on the sublinear behavior in time of the mean square displacement of a random walker another type of diffusion is encountered in turbulent flows with the mean square displacement being superlinear in time we introduce a novel stochastic process called a levy walk which generalizes fractal brownian motion to provide a statistical theory for motion in the fractal media which exists in a turbulent flow the levy walk describes random but still correlated motion in space and time in a scaling fashion and is able to account for the motion of particles in a hierarchy of coherent structures we apply our model to the description of fluctuating fluid flow when kolmogorov s 5 3 law for homogeneous turbulence is used to determine the memory of the levy walk then richardson s 4 3 law of turbulent diffusion follows in the mandelbrot absolute curdling limit if as suggested by mandelbrot that turbulence is isotropic but fractal then intermittency corrections to the 5 3 law follow in a natural fashion the same process with a different space time scaling provides a description of chaos in a josephson junction

solving constrained quadratic inverse eigenvalue problem via conjugate direction method abstract inverse eigenvalue problems are among the most important problems in numerical linear algebra this paper is concerned with the problem of designing an iterative method for a quadratic inverse eigenvalue problem of the form m x 2 g x k x 0 where m g and k should be partially doubly symmetric under a prescribed submatrix constraint this kind of quadratic inverse eigenvalue problem includes the classical and generalized inverse eigenvalue problems as special cases we propose the conjugate direction cd method for computing the least frobenius norm solutions of this constrained quadratic inverse eigenvalue problem in a finite number of steps in the absence of round off errors the cd method can automatically determine the solvability of this problem finally the overall performance and efficiency of the cd method for computing approximate solutions of the problem are discussed

bifurcation of periodic orbits by perturbing high dimensional piecewise smooth integrable systems abstract this paper deals with the problem of periodic orbit bifurcations for high dimensional piecewise smooth systems under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane a formula for the first order melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits we especially can use the function to study the number of periodic orbits both in degenerate hopf bifurcations and in degenerate homoclinic bifurcations finally we present two examples to illustrate an application of the theoretical results