maximal exponents of polyhedral cones iii article history received 31 may 2009 available online 14 november 2009 submitted by j bastero

on a conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture 7 that every integer n 2 2n 2 2 1 is an exponent of some n n primitive matrix in this paper we prove three results related to lewin and vitek s conjecture 1 every integer n2 2n 2 4 1 is an exponent of some n n primitive matrix 2 the conjecture is true when n is sufficiently large 3 we give a counterexample to show that the conjecture is not true in the case when n 11

holderian variational problems subject to integral constraints abstract we prove a necessary stationary condition for non differentiable isoperimetric variational problems with scale derivatives defined on the class of holder continuous functions

fractional variational problems depending on indefinite integrals abstract we obtain necessary optimality conditions for variational problems with a lagrangian depending on a caputo fractional derivative a fractional and an indefinite integral main results give fractional euler lagrange type equations and natural boundary conditions which provide a generalization of the previous results found in the literature isoperimetric problems problems with holonomic constraints and depending on higher order caputo derivatives as well as fractional lagrange problems are considered

bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial abstract suppose a is an n n nonnegative primitive matrix whose minimal polynomial has degree m we conjecture that the well known bound on the exponent of primitivity n 1 2 1 due to wielandt can be replaced by m 1 2 1 the only case for which we cannot prove the conjecture is when m 5 the number of distinct eigenvalues of a is m 1 or m and the directed graph of a has no circuits of length shorter than m 1 but at least one of its vertices lies on a circuit of length not shorter than m we show that m m 1 is always a bound on the exponent this being an improvement on wielandt s bound when m n for the case in which a is also symmetric the bound which we obtain is 2 m 1 to obtain our results we prove a lemma which shows that for a general nonnegative matrix the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph

on the exponent of a primitive digraph abstract we characterize the equality case of the upper bound d n s n 2 for the exponent of a primitive digraph in the case s 2 where n is the number of the vertices of the digraph d and s is the length of the shortest elementary circuit of d we also answer a question about the equality case when s 1 the exponent of an n n primitive nonnegative matrix a is the same as the exponent of the associated digraph d a of a so every theorem in this paper can be translated into a theorem about nonnegative matrices

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

generalizing the variational theory on time scales to include the delta indefinite integral we prove necessary optimality conditions of euler lagrange type for generalized problems of the calculus of variations on time scales with a lagrangian depending not only on the independent variable an unknown function and its delta derivative but also on a delta indefinite integral that depends on the unknown function such kinds of variational problems were considered by euler himself and have been recently investigated in j gregory generalizing variational theory to include the indefinite integral higher derivatives and a variety of means as cost variables methods appl anal 15 4 2008 427 435 our results not only provide a generalization to previous results but also give some other interesting optimality conditions as special cases

leitmann s direct method for fractional optimization problems abstract based on a method introduced by leitmann g leitmann a note on absolute extrema of certain integrals int j non linear mech 2 1967 55 59 we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control

fractional variations for dynamical systems hamilton and lagrange approaches fractional generalization of an exterior derivative for calculus of variations is defined the hamilton and lagrange approaches are considered fractional hamilton and euler lagrange equations are derived fractional equations are obtained by fractional variation of lagrangian and hamiltonian that have only integer derivatives

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

convergence analysis of a numerical scheme for two classes of non linear fractional differential equations we use the expansion formula for the fractional derivatives to reduce the problem of solving non linear fractional order differential equations arising in mechanics to the problem of solving a system of integer order differential equations we prove the convergence of the solutions to the reduced integer order systems to the solutions of the original problem the procedure is illustrated by two specific examples

calculus of variations on time scales with nabla derivatives abstract we prove a necessary optimality condition of the euler lagrange type for variational problems on time scales involving nabla derivatives of higher order the proof is done using a new and more general fundamental lemma of the calculus of variations on time scales

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

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

fractional oscillator equation transformation into integral equation and numerical solution in this paper we propose a numerical solution of a fractional oscillator equation being a class of the fractional euler lagrange equation at first we convert the fractional differential equation of order 0 to an equivalent integral equation including boundary conditions next we present a numerical solution of the integral form of the considered equation for two cases 0 1 and 1 2 we show illustrative examples of solutions for checking the correctness of the proposed solution method of the equation also we determine the convergence order of numerical schemas

hamilton jacobi fractional mechanics abstract as a continuation of rabei et al work eqab m rabei khaled i nawafleh raed s hijjawi sami i muslih dumitru baleanu the hamilton formalism with fractional derivatives j math anal appl 327 2007 891 897 the hamilton jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives the hamilton jacobi function in configuration space is obtained in a similar manner to the usual mechanics two problems are considered to demonstrate the application of the formalism the result is found to be in exact agreement with agrawal s formalism

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

fractional bosonic strings the aim of this paper is to present a simple generalization of bosonic string theory in the framework of the theory of fractional variational problems specifically we present a fractional extension of the polyakov action for which we compute the general form of the equations of motion and discuss the connection between the new fractional action and a generalization the nambu goto action consequently we analyze the symmetries of the modified polyakov action and try to fix the gauge following the classical procedures then we solve the equations of motion in a simplified setting finally we present a hamiltonian description of the classical fractional bosonic string and introduce the fractional light cone gauge it is important to remark that throughout the whole paper we thoroughly discuss how to recover the known results as an integer limit of the presented model published by aip publishing

new quadrature approach based on operational matrix for solving a class of fractional variational problems this manuscript presents a new numerical approach to approximate the solution of a class of fractional variational problems the presented approach is consisting of using the shifted legendre orthonormal polynomials as basis functions of the operational matrix of fractional derivatives described in the caputo sense and that of fractional integrals described in the sense of riemann liouville with the help of the legendre gauss quadrature formula together with the lagrange multipliers method for converting such fractional variational problems into easier problems that consist of solving an algebraic system in the unknown coefficients the convergence of the proposed method is analyzed finally in order to demonstrate the accuracy of the present method some test problems are introduced with their approximate solutions and comparisons with other numerical approaches

discrete direct methods in the fractional calculus of variations finite differences as a subclass of direct methods in the calculus of variations consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem this article generalizes the same idea for fractional variational problems we consider a minimization problem with a lagrangian that depends on the left riemann liouville fractional derivative using the grunwald letnikov definition we approximate the objective functional in an equispaced grid as a multi variable function of the values of the unknown function on mesh points the problem is then transformed to an ordinary static optimization problem the solution to the latter problem gives an approximation to the original fractional problem on mesh points

fractional variational calculus for nondifferentiable functions we prove necessary optimality conditions in the class of continuous functions for variational problems defined with jumarie s modified riemann liouville derivative the fractional basic problem of the calculus of variations with free boundary conditions is considered as well as problems with isoperimetric and holonomic constraints

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

approximation of fractional integrals by means of derivatives we obtain a new decomposition of the riemann liouville operators of fractional integration as a series involving derivatives of integer order the new formulas are valid for functions of class c n n n and allow us to develop suitable numerical approximations with known estimations for the error the usefulness of the obtained results in solving fractional integral equations and fractional problems of the calculus of variations is illustrated

a bound on the exponent of primitivity in terms of diameter abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers two conjectures due to hartwig neumann and lin in 1984 or so are that a m 1 2 1 and a d2 1 where m is the degree of the minimal polynomial of a and d is the diameter of the directed graph of a it is well known that the latter is stronger than the former because d m 1 in a recent paper we have proved a m 1 2 1 in this paper we prove the conjecture a d2 1 on that basis

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

leitmann s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales abstract the fundamental problem of the calculus of variations on time scales concerns the minimization of a delta integral over all trajectories satisfying given boundary conditions this includes the discrete time the quantum and the continuous classical calculus of variations as particular cases in this note we follow leitmann s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale

multiobjective fractional variational calculus in terms of a combined caputo derivative abstract the study of fractional variational problems in terms of a combined fractional caputo derivative is introduced necessary optimality conditions of euler lagrange type for the basic isoperimetric and lagrange variational problems are proved as well as transversality and sufficient optimality conditions this allows to obtain necessary and sufficient pareto optimality conditions for multiobjective fractional variational problems

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

extending bauer s corollary to fractional derivatives we comment on the method of dreisigmeyer and young j phys a math gen 368297 to model nonconservative systems with fractional derivatives it was previously hoped that using fractional derivatives in an action would allow us to derive a single retarded equation of motion using a variational principle it is proven that under certain reasonable assumptions the method of dreisigmeyer and young fails

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

generalized euler lagrange equations for fractional variational problems with free boundary conditions in this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems fvps with completely free boundary conditions have an extremum the fractional derivatives are defined in the caputo sense first we present the necessary conditions for the problem with only one dependent variable and then we generalize them to problems with multiple dependent variables we also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables it is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives the new results reduce to the known necessary conditions some examples are presented to demonstrate the applicability of the new formulations

spectral theory for tensor products of hermitian holomorphic line bundles let l be a quasipositive line bundle over a compact complex hermitian manifold it has been conjectured that the smallest positive eigenvalue of the laplacian acting on the k th tensor power of l is uniformly bounded independent of k various special cases of the conjecture are proved but a counterexample is given to the general conjecture

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

a diameter bound on the exponent of a primitive directed graph abstract a directed graph g is primitive if there exists a positive integer k such that for every pair of vertices u v g there is a walk from u to v of length k the least such k is called the exponent of g we define g k to be the directed graph having the same vertex set as g and arcs u v if and only if there is a walk in g of length k from vertex u to vertex v a well known upper bound for the exponent of a primitive directed graph g of order n is n 1 2 1 due to h wielandt in 1950 our main result is the following refinement of the wielandt bound if g is a primitive directed graph with diameter d then the exponent of g is at most d 2 1 we construct the primitive graphs for which equality is attained and we generalize the bound to the class of irreducible matrices in the course of proving the main result we find the following which is interesting in its own right if g is a primitive directed graph with diameter d then the diameter of g k is at most d for all positive integers k

proof of a conjecture about the exponent of primitive matrices abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in the matrix ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers the upper bound on a due to wielandt is a n 1 2 1 and a better bound for a due to hartwig and neumann is a m m 1 where m is the degree of the minimal polynomial of a also hartwig and neumann conjecture that a m 1 2 1 which had been suggested in 1984 in this paper we prove this conjecture

on lewin and vitek s conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture that every integer w 1 i n2 2n 2 1 is the exponent of some n x n primitive matrix in this paper we prove that this conjecture is true except for n 11 the problem of determining the exponent set e is completely solved

higher order hahn s quantum variational calculus abstract we prove a necessary optimality condition of euler lagrange type for quantum variational problems involving hahn s derivatives of higher order

fractional variational calculus with classical and combined caputo derivatives abstract we give a proper fractional extension of the classical calculus of variations by considering variational functionals with a lagrangian depending on a combined caputo fractional derivative and the classical derivative euler lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved

an approximate method for numerically solving fractional order optimal control problems of general form in this article we discuss fractional order optimal control problems focps and their solutions by means of rational approximation the methodology developed here allows us to solve a very large class of focps linear nonlinear time invariant time variant siso mimo state input constrained free terminal conditions etc by converting them into a general rational form of optimal control problem ocp the fractional differentiation operator used in the focp is approximated using oustaloup s approximation into a state space realization form the original problem is then reformulated to fit the definition used in general purpose optimal control problem ocp solvers such as riots 95 a solver created as a matlab toolbox illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time ocp is also solved

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

generalized fractional calculus with applications to the calculus of variations we study operators that are generalizations of the classical riemann liouville fractional integral and of the riemann liouville and caputo fractional derivatives a useful formula relating the generalized fractional derivatives is proved as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual such results are explored in the context of dynamic optimization by considering problems of the calculus of variations with general fractional operators necessary optimality conditions of euler lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of riemann liouville or caputo as an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem illustrative optimization problems are considered

direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method this paper presents an adaptive legendre gauss pseudospectral method for solving a type of constrained fractional variational problems fvps the fractional derivative is defined in the caputo sense in the presented method by dividing the domain of the problem into a uniform mesh the given fvp reduces to a nonlinear mathematical programming problem and there is no need to solve the complicated fractional euler lagrange equations the method developed in this paper adjusts both the mesh spacing and the number of collocation points on each subinterval in order to improve the accuracy the method is easy to implement and yields very accurate results some error estimates and convergence properties of the method are discussed numerical examples are included to confirm the efficiency and convergence of the proposed method

a problem on the exponent of primitive digraphs abstract a digraph d a is called primitive if and only if a the 0 1 connection matrix of d a is primitive the exponent of primitivity of d a is defined to be d a min k z a k 0 where z denotes the set of positive integers in a recent paper we have proved the conjecture d a m 1 2 1 due to robert e hartwig and michael neumann where m is the degree of the minimal polynomial of a in this paper we characterize the equality case of the upper bound d a m 1 2 1

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

the hamilton formalism with fractional derivatives recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives in this paper the passage from the lagrangian containing fractional derivatives to the hamiltonian is achieved the hamilton s equations of motion are obtained in a similar manner to the usual mechanics in addition the classical fields with fractional derivatives are investigated using hamiltonian formalism two discrete problems and one continuous are considered to demonstrate the application of the formalism the results are obtained to be in exact agreement with agrawal s formalism

semisimple commutative algebras with positive bases abstract algebras that serve as models for concurrent studying of certain aspects of both the algebra of ordinary characters and the center of the group algebra of a finite group have been considered by various authors in this article we offer another such model the main differences between our model and the known ones are 1 our model includes brauer characters and principal indecomposable characters as special cases 2 our emphasis is on the eigenvalues of the regular representation of the algebra elements an approach that gives results on values of characters ordinary central brauer principal indecomposable as well as on their products

expansion formula for fractional derivatives in variational problems abstract we modify the expansion formula introduced in t m atanackovic b stankovic an expansion formula for fractional derivatives and its applications fract calc appl anal 7 3 2004 365 378 for the left riemann liouville fractional derivative in order to apply it to various problems involving fractional derivatives as a result we obtain a new form of the fractional integration by parts formula with the benefit of a useful approximation for the right riemann liouville fractional derivative and derive a consequence of the fractional integral inequality 0 t y 0 d t y d t 0 further we use this expansion formula to transform fractional optimization minimization of a functional involving fractional derivatives to the standard constrained optimization problem it is shown that when the number of terms in the approximation tends to infinity solutions to the euler lagrange equations of the transformed problem converge in a weak sense to solutions of the original fractional euler lagrange equations an illustrative example is treated numerically

the action principle for dissipative systems in the present work we redefine and generalize the action principle for dissipative systems proposed by riewe by fixing the mathematical inconsistencies present in the original approach in order to formulate a quadratic lagrangian for non conservative systems the lagrangian functions proposed depend on mixed integer order and fractional order derivatives as examples we formulate a quadratic lagrangian for a particle under a frictional force proportional to the velocity and to the classical problem of an accelerated point charge

fractional variational principle of herglotz for a new class of problems with dependence on the boundaries and a real parameter the fractional variational problem of herglotz type for the case where the lagrangian depends on generalized fractional derivatives the free endpoints conditions and a real parameter is studied this type of problem generalizes several problems recently studied in the literature moreover it allows us to unify conservative and non conservative dynamical processes in the same model the dependence of the lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard herglotz s variational problem into a problem of a different nature

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

primitive normal matrices and covering numbers of finite groups abstract a primitive matrix is a square matrix m with nonnegative real entries such that the entries of ms are all positive for some positive integer s the smallest such s is called the primitivity index of m primitive matrices of normal type namely mmt and mtm have the same zero entries occur naturally in studying the so called conjugacy class covering number and character covering number of a finite group we show that if m is a primitive n n matrix of normal type with minimal polynomial of degree m then the primitivity index of m is at most n 2 1 m 1 this bound is then applied to improve known bounds for the various covering numbers of finite groups

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

on the existence of optimal solutions to fractional optimal control problems abstract in the paper we study control systems containing a fractional riemann liouville derivative the main result is a theorem on the existence of optimal solutions for such problems with a nonlinear integral performance index

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

nonconservative lagrangian mechanics a generalized function approach we reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle in particular a formalism is developed that allows the inclusion of fractional derivatives this is done within the lagrangian framework by treating the action as a volterra series it is then possible to derive two equations of motion one of these is an advanced equation and the other is retarded

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

formulation and integration of the standard linear viscoelastic solid with fractional order rate laws abstract a physically sound three dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented it is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of mittag leffler function type a time integration scheme based on the generalized midpoint rule together with the grunwald algorithm for numerical fractional differentiation for integration of the constitutive response is developed the predictive capability of the viscoelastic model for describing creep relaxation and damped dynamic responses is investigated both analytically and numerically the algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code abaqus the algorithm is then used together with an explicit difference scheme for integration of structural responses in numerical examples the quasi static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated

similarity solutions to nonlinear heat conduction and burgers korteweg devries fractional equations we analyze self similar solutions to a nonlinear fractional diffusion equation and fractional burgers korteweg devries equation in one spatial variable by using lie group scaling transformation we determined the similarity solutions after the introduction of the similarity variables both problems are reduced to ordinary nonlinear fractional differential equations in two special cases exact solutions to the ordinary fractional differential equation which is derived from the diffusion equation are presented in several other cases the ordinary fractional differential equations are solved numerically for several values of governing parameters in formulating the numerical procedure we use special representation of a fractional derivative that is recently obtained

generalized natural boundary conditions for fractional variational problems in terms of the caputo derivative this paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a lagrangian depending on the free end points the fractional derivatives are defined in the sense of caputo

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method