a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

state space formulas for the computation of the gap we consider the computation of the gap between the graphs of rational transfer functions the gap is related to the norm of a parallel projection which can be computed as a regular indefinite linear quadratic optimal control problem with partially specified initial conditions the use of coprime factorizations is avoided in this way and riccati equations are derived directly in terms of the state space realizations the computation needed to check whether the gap is smaller than some fixed number is shown to be an eigenvector eigenvalue problem for a hamiltonian matrix in which the state space parameters of minimal realizations of the transfer functions are stacked

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

hamiltonian formalism for nonlocal lagrangians a hamiltonian formalism is set up for nonlocal lagrangian systems the method is based on obtaining an equivalent singular first order lagrangian which is processed according to the standard legendre transformation and then the resulting hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints finally the standard results for local lagrangians of any order are obtained as a particular case

technical communique the pbh test for multidimensional ltid systems in this article the classical popov belevitch hautus controllability test is generalized to higher dimensions

the bethe permanent of a nonnegative matrix it has recently been observed that the permanent of a nonnegative square matrix i e of a square matrix containing only nonnegative real entries can very well be approximated by solving a certain bethe free energy function minimization problem with the help of the sum product algorithm we call the resulting approximation of the permanent the bethe permanent in this paper we give reasons why this approach to approximating the permanent works well namely we show that the bethe free energy function is convex and that the sum product algorithm finds its minimum efficiently we then discuss the fact that the permanent is lower bounded by the bethe permanent and we comment on potential upper bounds on the permanent based on the bethe permanent we also present a combinatorial characterization of the bethe permanent in terms of permanents of so called lifted versions of the matrix under consideration moreover we comment on possibilities to modify the bethe permanent so that it approximates the permanent even better and we conclude the paper with some observations and conjectures about permanent based pseudocodewords and permanent based kernels

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

loop calculus in statistical physics and information science considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series the leading term in the series is the bethe peierls belief propagation bp contribution the rest are expressed as loop contributions on the factor graph and calculated directly using the bp solution the series unveils a small parameter that often makes the bp approximation so successful applications of the loop calculus in statistical physics and information science are discussed

belief propagation and loop calculus for the permanent of a non negative matrix we consider computation of the permanent of a positive n n non negative matrix p pji i j 1 n or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph kn n the problem is known to be of likely exponential complexity stated as the partition function z of a graphical model the problem allows for exact loop calculus representation chertkov m and chernyak v 2006 phys rev e 72 065102 in terms of an interior minimum of the bethe free energy functional over non integer doubly stochastic matrix of marginal beliefs ji i j 1 n also correspondent to a fixed point of the iterative message passing algorithm of the belief propagation bp type our main result is an explicit expression of the exact partition function permanent in terms of the matrix of bp marginals as z perm p zbp perm ji 1 ji i j 1 ji where zbp is the bp expression for the permanent stated explicitly in terms of we give two derivations of the formula a direct one based on the bethe free energy and an alternative one combining the ihara graph function and the loop calculus approaches assuming that the matrix of the bp marginals is calculated we provide two lower bounds and one upper bound to estimate the multiplicative term two complementary lower bounds are based on the gurvits van der waerden theorem and on a relation between the modified permanent and determinant respectively

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

proper representations of multivariate linear differential systems abstract a proper representation of a linear differential system is a representation with no singularity at infinity it is shown that such a representation always exists it turns out that for proper representations having minimal number of rows is equivalent to having minimal total row degree one is led therefore to a natural definition of the notion of minimality what is remarkable is that a minimal proper representation is uniquely determined up to premultiplication by a unimodular polynomial matrix of special form this uniqueness result allows in particular to introduce important integer invariants

on the rest state stability of an objective fractional derivative viscoelastic fluid model the one dimensional fractional derivative maxwell model has been found very useful in modeling the linear viscoelastic response in the glass transition and relaxation regions that motivated further work on generalizations to nonlinear viscoelastic fractional constitutive equations in the work of palade et al int j eng sci 37 315 1999 a fully objective constitutive equation for an incompressible fluid reducible to the linear fractional derivative maxwell model under the small deformations hypothesis is given together with a state of rest stability analysis in the stability section of the aforementioned work for some physically aceptable perturbation inputs it has been shown that no continuously differentiable solutions may be found to the corresponding equations of motion in this work we extend the stability analysis framework to l2 hilbert functional space and prove the existence of solutions to the corresponding equations of motions

a fractional dirac equation and its solution this paper presents a fractional dirac equation and its solution the fractional dirac equation may be obtained using a fractional variational principle and a fractional klein gordon equation both methods are considered here we extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of caputo derivatives by applying the variational principle to a fractional action s we obtain the fractional euler lagrange equations of motion we present a lagrangian and a hamiltonian for the fractional dirac equation of order we also use a fractional klein gordon equation to obtain the fractional dirac equation which is the same as that obtained using the fractional variational principle eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard dirac equation we also provide expressions for the path integral quantization for the fractional dirac field which in the limit 1 approaches to the path integral for the regular dirac field it is hoped that the fractional dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics

stability criteria for first order dynamical systems in a behavioral framework abstract this paper addresses the stability of dynamical systems described by first order differential equations in the behavioral framework some algebraic stability criteria are derived under the assumption that the behavior does not include distributions the derivation of our criteria is based on quadratic differential forms and two variable polynomial matrices cf willems and trentelman siam j optim control 36 1998 1703 1749 willems proceedings of the 30th ieee conference on decision and control 1991 pp 900 904 which are appropriate tools for system synthesis and analysis in the behavioral approach thus our result and approach will contribute to the developement of the behavioral system theory furthermore our criteria are represented in terms of linear matrix inequalities and algebraic matrix equations which leads to easy examination of the stability with the help of appropriate software packages

reduced polynomial matrices in several variables we define a higher dimensional analogue of a column reduced polynomial matrix called a reduced polynomial resolution as an exact sequence of polynomial matrices having the predictable degree property a reduced polynomial resolution is associated to any linear time invariant differential system it is uniquely determined up to isomorphism and provides a number of important integer invariants of the system

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

supervisory control of hybrid systems within a behavioural framework abstract this contribution addresses the synthesis of supervisory control for hybrid systems with discrete external signals such systems are in general neither l complete nor can they be represented by finite state machines we find an l complete approximation abstraction l for represent it by a finite state machine and investigate the control problem for the approximation if a solution exists we synthesize the maximally permissive supervisor for l we show that it also solves the control problem for the hybrid system if no solution exists approximation accuracy can be increased by computing a k complete abstraction k k l this paper is entirely set within the framework on willems behavioural systems theory

well posedness of a linearized fractional derivative fluid model the one dimensional fractional derivative maxwell model e g palade et al rheol acta 35 1996 265 of importance in modeling the linear viscoelastic response in the glass transition region has been generalized in palade et al internat j engrg sci 37 1999 315 to objective three dimensional constitutive equations ces for both fluids and solids regarding the rest state stability of the fluid ce in heibig and palade j math phys 49 2008 043101 we gave a proof for the existence of weak solutions to the corresponding boundary value problem the aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

from time series to linear system part ii exact modelling abstract in the second part of this paper the problem of finding an exact model for a q dimensional infinite time series is considered first a mathematical vocabulary for discussing exact modelling is developed it is then shown how the results of part i guarantee the existence of a most powerful ar model for an observed time series two algorithms for obtaining such an ar model are subsequently derived one of these algorithms gives a shortest lag input output model the problem of obtaining a minimal state space realization of the observed time series is also considered in order to do that realization theory based on the truncated behaviour is developed as an extensive example the classical situation with impulse response measurements is discussed

the fractional boltzmann transport equation the fractional boltzmann transport equation is derived making use of the fractional hamilton s equations based on the fractional actionlike variational approach by simply defining a distribution function and inspecting its time derivative many important results in statistical physics can be derived

transcendental bernstein series for solving nonlinear variable order fractional optimal control problems abstract this paper deals with finding an approximate solution of variable order fractional optimal control problems v focps based on lagrange multiplier optimization technique in combination with the new operational matrix of variable order vo fractional derivatives to carry out the proposed approach we firstly develop the well known bernstein polynomials to the new series of functions namely transcendental bernstein series tbs then we implement these basis functions to approximate the solutions of v focps in fact the series expansion in tbs with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation the convergence analysis of our proposed method will be guaranteed by proving a new theorem for the tbs to investigate the practical computational efficiency and accuracy of the presented method some numerical examples are provided the experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions

on a riemann liouville generalized taylor s formula in this paper a generalized taylor s formula of the kindfx j 0najx a j 1 1 tnx whereaj r x a 0 1 is established such expression is precisely the classical taylor s formula in the particular case 1 in addition detailed expressions fortn x andaj involving the riemann liouville fractional derivative and some applications are also given

fractional hamiltonian analysis of higher order derivatives systems the fractional hamiltonian analysis of 1 1 dimensional field theory is investigated and the fractional ostrogradski s formulation is obtained the fractional path integral of both simple harmonic oscillator with an acceleration squares part and a damped oscillator are analyzed the classical results are obtained when fractional derivatives are replaced with the integer order derivatives

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

existence and uniqueness theorem for a class of delay differential equations with left and right caputo fractional derivatives the existence and uniqueness theorems for functional right left delay and left right advanced fractional functional differential equations with bounded delay and advance respectively are proved the continuity with respect to the initial function for these equations is also proved under some lipschitz kind conditions the q operator is used to transform the delay type equation to an advanced one and vice versa an example is given to clarify the results

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

on the extension of solutions of linear constant coefficient pde abstract we solve the following over determined boundary value problem the extension problem let r be a matrix whose entries are linear partial differential operators with constant coefficients let be a non empty open bounded convex set we consider the homogeneous system r f 0 in a neighborhood of subject to the boundary condition f g in a neighborhood of for a given g we give a criterion for the unique existence of a smooth solution f to this problem there are two obvious necessary conditions g is smooth and r g 0 in a neighborhood of we characterize the class of differential operators r for which the problem is solvable for any g satisfying the necessary conditions finally in the case where the solution is non unique we consider the possibility of obtaining uniqueness by fixing several components of the desired solution

analysis and numerical approximation of tempered fractional calculus of variations problems abstract in this paper we study variational problems where the cost functional involves the tempered caputo fractional derivative several important optimization conditions are derived to find the optimal solution sufficient and necessary conditions are presented for different variational problems for example the cases of integral isoperimetric problem and holonomic constraints are considered as well as problems with high order derivatives a numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

approximating the permanent with fractional belief propagation we discuss schemes for exact and approximate computations of permanents and compare them with each other specifically we analyze the belief propagation bp approach and its fractional belief propagation fbp generalization for computing the permanent of a non negative matrix known bounds and conjectures are verified in experiments and some new theoretical relations bounds and conjectures are proposed the fractional free energy ffe function is parameterized by a scalar parameter 1 1 where 1 corresponds to the bp limit and 1 corresponds to the exclusion principle but ignoring perfect matching constraints mean field mf limit ffe shows monotonicity and continuity with respect to for every non negative matrix we define its special value 1 0 to be the for which the minimum of the parameterized ffe function is equal to the permanent of the matrix where the lower and upper bounds of the interval corresponds to respective bounds for the permanent our experimental analysis suggests that the distribution of varies for different ensembles but always lies within the 1 1 2 interval moreover for all ensembles considered the behavior of is highly distinctive offering an empirical practical guidance for estimating permanents of non negative matrices via the ffe approach

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

singular lagrangians with higher derivatives the hamiltonian formalism for systems with singular lagrangians of the second order is constructed a new method is proposed for obtaining the equations of motion in the phase space for theories with singular lagrangians and the connection of the lagrangian and hamiltonian descriptions is traced in detail as an example a generalisation of the relativistic point action is considered it involves both the length and curvature of the point trajectory in spacetime

a note on a fast breakdown free algorithm for computing the determinants and the permanents of k tridiagonal matrices k tridiagonal matrices have attracted much attention in recent years which are a generalization of tridiagonal matrices in this note a breakdown free numerical algorithm of o n is presented for computing the determinants and the permanents of k tridiagonal matrices even though the algorithm is not a symbolic algorithm it never suffers from breakdown furthermore it produces exact values when all entries of the k tridiagonal matrices are given in integer in addition the algorithm can be simplified for a general symmetric toeplitz case and it generates the kth powers of fibonacci pell and jacobsthal numbers for a certain symmetric toeplitz case

taylor approximations of multidimensional linear differential systems abstractthe taylor approximations of a multidimensional linear differential system are of importance as they contain a complete information about it it is shown that in order to construct them it is sufficient to truncate the exponential trajectories only a computation of the taylor approximations is provided using purely algebraic means without requiring explicit knowledge of the trajectories

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

a numerical scheme for two dimensional optimal control problems with memory effect a new formulation for multi dimensional fractional optimal control problems is presented in this article the fractional derivatives which are coming from the formulation of the problem are defined in the riemann liouville sense some terminal conditions are imposed on the state and control variables whose dimensions need not be the same a numerical scheme is described by using the grunwald letnikov definition to approximate the riemann liouville fractional derivatives the set of fractional differential equations which are obtained after the discretization of the time domain are solved within the grunwald letnikov approximation to obtain the state and the control variable numerically a two dimensional fractional optimal control problem is studied as an example to demonstrate the performance of the scheme

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role

fractional variational problems from extended exponentially fractional integral in this letter we study the fractional variational problems from extended exponentially fractional integral both the lagrangian and hamiltonian formulations are explored and discussed in some details some interesting consequences are revealed