on the class of reduced order models obtainable by projection the paper investigates the properties of general reduced order models obtained by projection of a higher order system it answers questions such as are any two models of different orders related by a projection is it possible to obtain the same reduced order model using different projections how to find if it exists a projection that relates the two models etc it is shown that answers to those questions can be obtained by investigating the properties of a certain matrix pencil the key tool is the kronecker canonical form and in some cases of square systems the problem becomes that of generalized eigenvalues

analytical methods for solving the time fractional swift hohenberg s h equation in this paper the most effective methods the homotopy perturbation method hpm and the differential transform method dtm are applied to obtain the approximate solutions of the nonlinear time fractional swift hohenberg s h equation the basic philosophy of these methods does not involve linearization weak nonlinearity assumptions or perturbation theory numerical solutions for various combinations of the parameters a eigenvalue parameter l length and a fractional index are obtained the solutions of the s h equation are useful for studies of shear thinning effects in non newtonian fluid flows at the end the solutions obtained are also presented graphically

linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

complements to cost equivalent realizations abstract skelton 1981 and skelton and yousuff 1983 use component cost analysis to obtain a reduced order model that is coat equivalent to a full order stochastic system this paper presents a more direct and numerically efficient method to obtain the same reduced order model which is shown to be a first order autoregressive model

homotopy perturbation method for quadratic riccati differential equation and comparison with adomian s decomposition method abstract in this paper a homotopy perturbation method is proposed to solve quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the proposed method is very effective and simple

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

homotopy methods for solving the optimal projection equations for the h2 reduced order model problem the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints due to the resemblance of these equations to standard matrix lyapunov equations they are called modified lyapunov equations the algorithms proposed herein utilize probability one homotopy theory as the main tool it is shown that there is a family of systems the homotopy that make a continuous transformation from some initial system to the final system with a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable controllable and observable one method which solves the equations in their original form requires a decomposition of the projection matrix using the drazin inverse of a matrix it is shown that the appropriate inverse is a differentiable function an effective algorithm for computing the derivative of the projection matrix that involves solving a set of sylvester equations is given another class of methods considers the equations in a modified form using a decomposition of the pseudogramians based on a contragredient transformation some freedom is left in making an exact match between the number of equations and the number of unknowns thus effectively generating a family of methods

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

inexact solves in interpolatory model reduction we investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem we give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods we show that when a petrov galerkin framework is employed for the inexact solves the associated reduced order model is an exact interpolatory model for a nearby full order system thus demonstrating backward stability we also give evidence that for h2 optimal interpolation points interpolatory model reduction is robust with respect to perturbations due to inexact solves finally we demonstrate the effectiveness of direct use of inexact solves in optimal h2 approximation the result is an effective model reduction strategy that is applicable in realistically large scale settings

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

a comparison of homotopies for alternative formulations of the l 2 optimal model order reduction problem abstract a fundamental problem in control systems theory is finding a reduced order model that is optimal in the l 2 sense to a given full order system model the numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous a number of homotopy based approaches have been developed the primary numerical issues are the number of degrees of freedom in the homotopy parameter vector the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm this paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms the results show that the numerical well conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent

analysis of the rational krylov subspace and adi methods for solving the lyapunov equation for large scale problems an effective approach for solving the algebraic lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation although krylov subspaces have been used for a long time only more recent developments have shown that rational krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit adi recurrence in this paper we develop a convergence analysis of the rational krylov subspace method rksm based on the kronecker product formulation and on potential theory moreover we propose new enlightening relations between this approach and the adi method our results provide solid theoretical ground for recent numerical evidence of the superiority of rksm over adi when the involved parameters cannot be computed optimally as is the case in many practical application problems

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

model reduction via truncation an interpolation point of view in this paper we focus our attention on linear time invariant continuous time linear systems with one input and one output siso lti systems we consider the problem of constructing a reduced order system via truncation of the original system given a siso strictly proper transfer function t s of mcmillan degree n and a strictly proper siso transfer function t over cap s of mcmillan degree n n we prove that t over cap s can always be constructed via truncation of the system t s the proof is mainly based on interpolation theory and more precisely on multipoint pade interpolation moreover new results about krylov subspaces are developed c 2003 elsevier inc all rights reserved

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

component cost analysis of large scale systems abstract a brief summary of component cost analysis cca is given cca provides a means to determine the contribution of each dynamical element of the system in the overall performance metric using cca a theory of cost equivalent realizations is developed cost equivalent realizations are realizations that have the same value of the performance metric as the original system and they can be of lower order than kalman s minimal realization

frequency weighted model reduction via optimal projection the problem of order reduction with frequency weighting is considered necessary conditions characterizing the reduced order model which minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting are given the problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input when the weighting matrix is singular at infinity the reduced order model has a direct transmission term even if the original system is strictly proper this additional degree of freedom can be used to obtain better approximation over a prespecified frequency range

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

reduced order models with delay the paper considers the problem of h 2 optimal approximation of a high order system by a low order system with delay an algorithm to solve this problem is presented it consists of optimization with respect to the delay only where the rational part of the model and the quadratic cost are given by a solution of an equivalent order reduction problem without delay it is shown by means of an example that introduction of delay into the model leads in certain cases to considerable improvement in the approximation the results are extended to multiple input multiple output systems and also to frequency weighted order reduction

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

the use of fake algebraic riccati equations for co channel demodulation this paper describes a method for nonlinear filtering based on an adaptive observer which guarantees the local stability of the linearized error system a fake algebraic riccati equation is employed in the calculation of the filter gain the design procedure attempts to produce a stable filter at the expense of optimality this contrasts with the extended kalman filter ekf which attempts to preserve optimality via its linearization procedure at the expense of stability a passivity approach is applied to deduce stability conditions for the filter error system the performance is compared with an ekf for a co channel frequency demodulation application

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

fractional market dynamics a new extension of a fractality concept in nancial mathematics has been developed we have introduced a new fractional langevin type stochastic dierential equation that diers from the standard langevin equation i by replacing the rst order derivative with respect to time by the fractional derivative of order and ii by replacing white noise gaussian stochastic force by the generalized shot noise each pulse of which has a random amplitude with the stable levy distribution as an application of the developed fractional non gaussian dynamical approach the expression for the probability distribution function pdf of the returns has been established it is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of s p 500 returns c 2000 elsevier science b v all rights reserved

solution of bvps for fourth order integro differential equations by using homotopy perturbation method in this study the homotopy perturbation method proposed by ji huan he is applied to solve both linear and nonlinear boundary value problems for fourth order integro differential equations the analysis is accompanied by numerical examples the results show that the homotopy perturbation method is of high accuracy more convenient and efficient for solving integro differential equations

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

on the solution of the riccati equation by the taylor matrix method in this paper we suggest a matrix method to solve the riccati differential equation in terms of taylor polynomials this method is based on first taking the truncated taylor series of the function in equations and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

iterated he s homotopy perturbation method for quadratic riccati differential equation abstract in this paper iterated he s homotopy perturbation method is proposed to solving quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the method is very effective and simple

model reduction and clusterization of large scale bidirectional networks this paper proposes two model reduction methods for large scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization first we present a krylov based model reduction method that guarantees a specified error precision in terms of the h norm positive tridiagonalization allows us to derive an approximation error bound for the input to state model reduction without computationally expensive operations such as matrix factorization second we propose a novel model reduction method that preserves network topology among clusters i e node sets in this approach we introduce the notion of cluster uncontrollability based on positive tridiagonalization and then derive its theoretical relation to the approximation error this error analysis enables us to construct clusters that can be aggregated with a small approximation error the efficiency of both methods is verified through numerical examples including a large scale complex network

intersection numbers of polygon spaces we study the intersection ring of the space m 1 m of polygons in r 3 we find homology cycles dual to generators of this ring and prove a recursion relation in m the number of edges for their intersection numbers this result is an analog of the recursion relation appearing in the work of witten and kontsevich on moduli spaces of punctured curves and in the work of weitsman on moduli spaces of flat connections on two manifolds of genus g with m marked points based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples among others we study the special case of equilateral polygon spaces where all i s are the same and compare our results with the expressions for these particular spaces that have been determined by kamiyama and tezuka finally we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of yoshida as well as with equivalent expressions for polygon spaces obtained by takakura and konno through different techniques

l2 and l2 l model reduction via linear matrix inequalities necessary and sufficient conditions are derived for the existance of a solution to the continuous time and discrete time suboptimal l2 and l2 l model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint in addition explicit parametrizations of all reduced order models that correspond to a feasible solution are presented in terms of contractive matrices

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

projection properties of the l 2 optimal reduced order model the paper investigates the optimal l 2 reduced order model in the context of reduced order models obtained by projection it deals first with the general case of such systems and shows that the existence and uniqueness of a projection that relates two models of different orders can be inferred from the kronecker canonical form of a certain linear pencil combining the l 2 optimality conditions into that framework gives a different point of view on the optimal reduced order model it is shown that the zeros of the optimal approximation error transfer function are the mirror image of the reduced model poles with the same multiplicity in non square systems and doubled multiplicity in square systems the latter case enhances an existing result for siso systems that the multiplicity of the error zeros exceeds that of the optimal reduced model poles by one it is also shown that the optimal projection which is obtained in some solutions of the problem is discontinuous at the optimal solution if the reduced

model order reduction by balanced proper orthogonal decomposition and by rational interpolation we show that model order reduction by rational interpolation also known as moment matching or rational krylov can be seen as the special case of balanced proper orthogonal decomposition where all the snapshots are retained and particular numerical procedures are used to obtain the snapshots

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

hyperpolygon spaces and moduli spaces of parabolic higgs bundles abstract given an n tuple of positive real numbers we consider the hyperpolygon space x the hyperkahler quotient analogue to the kahler moduli space of polygons in r 3 we prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable rank 2 holomorphically trivial parabolic higgs bundles over c p 1 with fixed determinant and trace free higgs field this isomorphism allows us to prove that hyperpolygon spaces x undergo an elementary transformation in the sense of mukai as crosses a wall in the space of its admissible values we describe the changes in the core of x as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles moreover we study the intersection rings of the core components of x in particular we find generators of these rings prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers using our isomorphism we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles thus determining their intersection rings as a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic higgs bundles and of the components of their nilpotent cone

approximated gramians and balanced realization of lightly damped flexible structures this paper presents a method of approximating the balanced realization for lightly damped flexible structures with nonmodal damping the damped system is treated as a perturbation from the undamped system and the controllability and observability gramians are given as a power series in the perturbation scaling factor the approximation utilizes the special structure of second order mechanical systems to obtain closed form expressions for the series coefficient matrices the series form leads also to an approximated balancing algorithm which gives the balancing transformation explicitly

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

sylvester equations and projection based model reduction in this paper we establish a connection between krylov subspace techniques for multipoint pade interpolation and the use of sylvester equations for constructing reduced order models we also briefly point out that this connection partly extends to adi type techniques and to the smith iteration for computing approximate solutions of lyapunov equations

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

controller reduction by component cost analysis component cost analysis 1 is used to develop a method for controller reduction the reduction of the controller is based upon the participation of the controller states in the value of a quadratic performance metric the controller states which have the smallest contribution to the performance metric are truncated to produce the reduced controllers an error index is defined to evaluate the reduced controller compared to the optimal lqg controller and bounds on this index are derived a numerical example is included to illustrate the procedure

a new application of he s variational iteration method for quadratic riccati differential equation by using adomian s polynomials in this paper the quadratic riccati differential equation is solved by he s variational iteration method with considering adomian s polynomials comparisons were made between adomian s decomposition method adm homotopy perturbation method hpm and the exact solution in this application we do not have secular terms and if l lagrange multiplier is equal 1 then the adomian s decomposition method is obtained the results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems

hyperpolygon spaces and their cores given an n tuple of positive real numbers 1 an konno 2000 defines the hyperpolygon space x a a hyperkahler analogue of the kahler variety m a parametrizing polygons in r 3 with edge lengths 1 an the polygon space m can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector from this point of view x a is the hyperkahler quiver variety defined by nakajima a quiver variety admits a natural c action and the union of the precompact orbits is called the core we study the components of the core of x interpreting each one as a moduli space of pairs of polygons in r 3 with certain properties konno gives a presentation of the cohomology ring of x a we extend this result by computing the c equivariant cohomology ring as well as the ordinary and equivariant cohomology rings of the core components

the second kind chebyshev wavelet method for solving fractional differential equations abstract in this paper the second kind chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations we first construct the second kind chebyshev wavelet and then derive the operational matrix of fractional order integration the operational matrix of fractional order integration is utilized to reduce the fractional differential equations to system of algebraic equations in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

an efficient approach for solving the riccati equation with fractional orders the present study introduces a novel and simple analytical method for the solution of fractional order riccati differential equation in this approach the solution considered as a taylor series expansion converges rapidly to the nonlinear problem new homotopy perturbation method nhpm depends only on two components of the homotopy series the method is illustrated by applications and the results obtained are compared with those of the exact solution moreover comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient

interpolatory h h model reduction abstract we introduce an approach to h model reduction that is founded on ideas originating in realization theory interpolatory h 2 optimal model reduction and complex chebyshev approximation within this new framework we are able to formulate a method that remains effective in large scale settings with the main cost dominated by sparse linear solves by employing loewner data driven partial realizations within each optimization cycle computationally demanding h norm calculations can be completely avoided several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better h performance than those produced by balanced truncation these models often are as good as and occasionally better than those models produced by optimal hankel norm approximation in all cases reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal hankel norm approximation

universal families of extensions if x denotes a projective variety over a field k and jr and y coherent sheaves on x the vector space exti sr y parametrizes the extensions of f by v on x and there is a universal extension of p t by p y on xx exti y 5 such that for every k rational point u of exti ir y its restriction to x x u is just the extension represented by u moreover p p extk st v parametrizes the classes of nonsplitting extensions of 9 by y on x module k and there is a universal family of extensions of pt f by pf5 p 1 on x x p over p such that for every k rational point p of p its restriction to x x p represents just the class of extensions given by p it is the purpose of this paper not only to prove these two statements but to generalize them to the relative case of a flat projective morphism f x y of noetherian schemes they turn out to be a consequence of grothendieck s universal properties of vector bundles and projective bundles cp 2 section i 91 special cases of these results appeared several times in the literature to the best of my knowledge first in 5 lemma 3 11 and second in 6 lemma 2 41 the technical tool is the base change theory for relative ext sheaves which is outlined in section 1 in section 2 some relations between relative ext sheaves and families of extensions are given section 3 contains the construction of the universal family of extensions and section 4 the construction of the universal family of classes of nonsplitting extensions concerning the notation f x y means a flat projective morphism of noetherian schemes and r and ce coherent ox modules flat over y the proofs of sections 2 to 4 are more generally correct if only there is a base change theory for relative ext sheaves for example if one uses the base change results of 1 valid for arbitrary proper flat morphisms of complex spaces one gets the corresponding results in this case too 101 oozl 8693183 3 00