on a connection between the pascal vandermonde and stirling matrices ii this article is an extension of the work presented in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear it gives a full matrix t t for which t 1 is exactly the lower triangular stochastic matrix t given in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear a new strategy to generate the elements of the matrix t t 1 is investigated a new conjecture and three combinatorial identities are also given

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

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

two routes to chaos in the fractional lorenz system with dimension continuously varying abstract the object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives we take the fractional lorenz system as example and regard one or three of its orders as bifurcation parameters in this framework we compute the corresponding bifurcation diagrams via an optimal poincare section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3 one is the process of cascaded period doubling bifurcations and the other is a crisis boundary crisis which occurs in the evolution of chaotic transient behavior we would like to point out that our investigation is the first to find out that a fractional differential equations fdes system can evolve into chaos by the crisis furthermore we observe rich dynamical phenomena in these processes such as two stage cascaded period doubling bifurcations chaotic transients and the transition from coexistence of three attractors to mono existence of a chaotic attractor these are new and interesting findings for fdes systems which to our knowledge have not been described before

the numerical rank of krylov matrices abstract in this paper we expose interesting features of krylov matrices and vandermonde matrices let s be a large symmetric matrix of order n let x be a real n vector and let k l denote the related n l krylov matrix the question considered in this paper is how the numerical rank of k l grows as l increases the key for answering this question lies in the close link between k l and the n l vandermonde matrix which is generated by the eigenvalues of s analysis of large vandermonde matrices shows that the numerical rank is expected to remain much smaller than l the proof is based on partition theorems and clustering theorems the basic tool is a new matrix equality the vandermonde pascal toeplitz equality the actual numerical rank of a vandermonde or krylov matrix depends on the distribution of the eigenvalues but often the rank is remarkable small numerical experiments illustrate these points the observation that the numerical rank of krylov matrices stays small has important practical consequences

synchronization of chaotic systems involving fractional operators of liouville caputo type with variable order abstract in this paper we propose a state observer based approach to synchronize variable order fractional vof chaotic systems in particular this work is focused on complete synchronization with a so called unidirectional master slave topology the master is described by a dynamical system in state space representation whereas the slave is described by a state observer the slave is composed of a master copy and a correction term which in turn is constituted of an estimation error and an appropriate gain that assures the synchronization the differential equations of the vof chaotic system are described by the liouville caputo and atangana baleanu caputo derivatives numerical simulations involving the synchronization of rossler oscillators chua s systems and multi scrolls are studied the simulations show that different chaotic behaviors can be obtained if different smooths functions defined in the interval 0 1 are used as the variable order of the fractional derivatives furthermore simulations show that the vof chaotic systems can be synchronized

design of a chatter free terminal sliding mode controller for nonlinear fractional order dynamical systems this paper investigates the problem of robust control of nonlinear fractional order dynamical systems in the presence of uncertainties first a novel switching surface is proposed and its finite time stability to the origin is proved subsequently using the sliding mode theory a robust fractional control law is proposed to ensure the existence of the sliding motion in finite time we use a fractional lyapunov stability theory to prove the stability of the system in a given finite time in order to avoid the chattering which is inherent in conventional sliding mode controllers we transfer the sign function of the control input into the fractional derivative of the control signal the proposed chattering free sliding mode technique is then applied for stabilisation of a broad class of three dimensional fractional order chaotic systems via a single variable driving control input simulation results reveal that the proposed fractional sliding mode controller works well for chaos control of fractional ord

notes on particular symmetric polynomials with applications this paper presents some applications using several properties of three important symmetric polynomials elementary symmetric polynomials complete symmetric polynomials and the power sum symmetric polynomials the applications includes a simple proof of el mikkawy conjecture in m e a el mikkawy appl math comput 146 2003 759 769 and a very easy proof of the newton girard formula in addition a generalization of stirling numbers is obtained

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control abstract this paper investigates the projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control the pinning control method is employed in designing controllers for guaranteeing projective cluster synchronization based on the lyapunov stability theory and well known barbalat s lemma some new sufficient conditions are derived to guarantee the projective cluster synchronization of the fractional order complex network a numerical simulation is given to verify the effectiveness of the theoretical results

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

sliding mode synchronization of an uncertain fractional order chaotic system synchronization of chaotic and uncertain duffing holmes system has been done using the sliding mode control strategy regarding the synchronization task as a control problem fractional order mathematics is used to express the system and sliding mode for synchronization it has been shown that not only the performance of the proposed method is satisfying with an acceptable level of control signal but also a rather simple stability analysis is performed the latter is usually a complicated task for uncertain nonlinear chaotic systems

on a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial this paper gives a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial the generalized stirling numbers of the first and second kind denoted s n k a and s n k a respectively are given symbolically for the case n 5 as an illustrative example by using maple programming maple v r3 programming reference manual these numbers depend on n k and n distinct values a 1 a 2 a n the ordinary stirling numbers s n k and s n k may be obtained as special cases by taking a i i 1 i 1 1 n

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

fractal system as represented by singularity function a fractional slope on the log log bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole in order to represent and study its dynamical behavior a singularity function method is presented which consists of cascaded branches of a number of pole zero negative real pairs or simple rc section the distribution spectrum of the system can also be easily calculated and its accuracy depends on a prescribed error specified in the beginning the method is then extended to a multiple fractal system which consists of a number of fractional power poles the result can be simulated by a combination of singularity functions each representing a single fractal system

nonlinear observer design to synchronize fractional order chaotic systems via a scalar transmitted signal abstract in this paper based on the idea of nonlinear observer and stability theory of fractional order systems a new systematic scheme to synchronize a class of fractional order chaotic systems via a scalar transmitted signal is developed the approach is simple global and theoretically rigorous it enables synchronization of fractional order chaotic systems to be achieved in a systematic way and does not require the computation of the conditional lyapunov exponents simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

on apolarity and generic canonical forms abstract the notion of s algebra is introduced the theory of apolarity and generic canonical forms for polynomials is generalized to s algebras over the complex field c we apply this theory to the problem of finding the essential rank of general symmetric and skew symmetric tensors upper bounds for the essential ranks are found by different combinatorial coverings

a fast algorithm for solving linear systems of the pascal type abstract in this paper we present a fast algorithm of the complexity o n log n for solving the linear systems with coefficient matrix of the pascal type which is faster than the algorithm of the complexity o n 2 given in the paper m e a el mikkawy on solving linear systems of the pascal type applied mathematics and computation 136 2003 195 202 an application in solving non homogeneous differential equation with constant coefficients is given

context free grammars differential operators and formal power series abstract in this paper we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules we call such a set of rules a context free grammar because these rules act like a context free grammar in the sense of a formal language given a context free grammar we can associate each formal function with an exponential formal power series in this way we obtain grammatical interpretations of addition multiplication and functional composition of formal power series a surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation we apply this method to obtain simple demonstrations of faa di bruno s formula and some identities concerning bell polynomials stirling numbers and symmetric functions in particular the lagrange inversion formula has a simple grammatical representation from this point of view one sees that cayley s formula on labeled trees is equivalent to the lagrange inversion formula

remarks on two symmetric polynomials and some matrices by means of complete symmetric polynomials this paper gives a new proof for the vandermonde determinant formula another alternative proof for this formula is obtained via the collocation matrices it also gives a generalized relationship between the vandermonde the pascal and the stirling matrices a new approach to obtain the explicit inverse of the vandermonde matrix is investigated closed form expressions for some vandermonde related determinants are obtained

extended symmetric pascal matrices via hypergeometric functions in this paper we give a matrix representation of the hypergeometric functions of the type 2f 1 a b c x as a result we obtain a connection between the hypergeometric functions the legendre polynomials and the delannoy numbers moreover it is shown that each entry of p n x y p n x y t can be represented by the hypergeometric functions where p n x y i j x i jy i j 2i 1j 1 is the extended generalized pascal matrix which is defined by zhang and liu linear algebra appl 271 1998 169

a note on time domain simulation of feedback fractional order systems the study of feedback fractional order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional order models and that fractional order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer order controllers owing to the lack of effective analytic methods for the time domain analysis and simulation of linear feedback fractional order systems we suggest in this paper two reliable and accurate numerical methods for inverting fractional order laplace transforms one is based on computing bromwich s integral with a numerical integration scheme capable of accuracy control and the other is based on expanding the time response function in a b spline series in order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the grunwald letniknov approximation method and podlubny s analytic formulas which are in a form of double infinite series the time domain simulations of the feedback control of a fractional order process with a pd sup spl mu controller and a fractional order band limited lead compensator are worked out the simulation results indicate that a convergence problem indeed occurs in using podlubny s infinite series expressions and that the problem could not be overcome by a series acceleration scheme

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

limitations of frequency domain approximation for detecting chaos in fractional order systems abstract in this paper we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems the number and location of equilibria and also the stability of these points are compared between the original system and its frequency based approximated counterpart it is shown that the original system and its approximation are not necessarily equivalent according to the number location and stability of the fixed points this problem can cause erroneous results in special cases for instance to prove the existence of chaos in fractional order systems numerical simulations have been largely based on frequency domain approximations but in this paper we show that this method is not always reliable for detecting chaos this approximation can numerically demonstrate chaos in the non chaotic fractional order systems or eliminate chaotic behavior from a chaotic fractional order system

symmetric polynomials pascal matrices and stirling matrices we consider lower triangular matrices consisting of symmetric polynomials and we show how to factorize and invert them since binomial coefficients and stirling numbers can be represented in terms of symmetric polynomials these results contain factorizations and inverses of pascal and stirling matrices as special cases this work generalizes that of several other authors on pascal and stirling matrices

chaotic behavior in noninteger order cellular neural networks in this paper a simple system showing chaotic behavior is introduced it is based on the well known concept of cellular neural networks cnns which have already given good results in generating complex dynamics the peculiarity of the cnn model consists in the fact that it replaces the traditional first order cell with a noninteger order one the introduction of the fractional cell with a suitable choice of the coupling parameters leads to the onset of chaos in a simple two cell system a theoretical approach based on the harmonic balance theory has been used to investigate the existence of chaos a circuit realization of the proposed fractional two cell chaotic cnn is reported and the corresponding strange attractor is also shown

commentary on an open question abstract in this commentary we give the triangular recurrence relations for the elements of the matrix t t i j l 1 d 1 s 1 and its inverse t 1 t i j s 2 dl 1 1 respectively which have also answered the open question put forward by el mikkawy moreover with the recurrences a computational maple program is written to generate the elements of t and t 1

decompositions of a matrix by means of its dual matrices with applications abstract we introduce the notion of dual matrices of an infinite matrix a which are defined by the dual sequences of the rows of a and naturally connected to the pascal matrix p i j i j 0 1 2 we present the cholesky decomposition of the symmetric pascal matrix by means of its dual matrix decompositions of a vandermonde matrix are used to obtain variants of the lagrange interpolation polynomial of degree n that passes through the n 1 points i q i for i 0 1 n

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

chaos in fractional ordered liu system abstract present paper deals with fractional version of a dynamical system introduced by c liu l liu and t liu c liu l liu t liu a novel three dimensional autonomous chaos system chaos solitons fractals 39 4 2009 1950 1958 numerical investigations on the dynamics of this system have been carried out properties of the system have been analyzed by means of lyapunov exponents furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders it is noteworthy that the results obtained are consistent with the analytical conditions given in the literature

hybrid projective synchronization of chaotic fractional order systems with different dimensions abstract the hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper it is shown that the slave system can be synchronized with the projection of the master system generated through state transformation based on the stability theorem of linear fractional order systems a suitable controller for achieving the synchronization is given the hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order the corresponding numerical results verify the effectiveness of the proposed method