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

a new framework for multivariate general conformable fractional calculus and potential applications abstract some applications of fractional partial differential equations fpdes based on conformable fractional derivative cfd are discussed recently as an extension of cfd power kernel is generalized to other normalized probability distribution kernels general conformable fractional derivative gcfd provides a framework to explain the physical meaning of cfd and describe which phenomena or processes gcfd cfd are suitable to model we will establish the multivariate theory of gcfd in this paper compared with traditional hereditary fractional calculus like riemann liouville type or caputo type the unified theory of gcfd vector calculus has well forms with simplicity and elegance as applications fractional maxwell s equations of gcfd are given to describe electromagnetic fields of media that demonstrate inhomogeneous complicated and local properties and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations

on conformable fractional calculus recently the authors khalil et al 2014 introduced a new simple well behaved definition of the fractional derivative called conformable fractional derivative in this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus the fractional versions of chain rule exponential functions gronwall s inequality integration by parts taylor power series expansions laplace transforms and linear differential systems are proposed and discussed

conformable derivative approach to anomalous diffusion abstract by using a new derivative with fractional order referred to conformable derivative an alternative representation of the diffusion equation is proposed to improve the modeling of anomalous diffusion the analytical solutions of the conformable derivative model in terms of gauss kernel and error function are presented the power law of the mean square displacement for the conformable diffusion model is studied invoking the time dependent gauss kernel the parameters related to the conformable derivative model are determined by levenberg marquardt method on the basis of the experimental data of chloride ions transportation in reinforced concrete the data fitting results showed that the conformable derivative model agrees better with the experimental data than the normal diffusion equation furthermore the potential application of the proposed conformable derivative model of water flow in low permeability media is discussed

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

auxiliary equation method for time fractional differential equations with conformable derivative abstract in this paper the auxiliary equation method is applied to obtain analytical solutions of 2 1 dimensional time fractional zoomeron equation and the time fractional third order modified kdv equation in the sense of the conformable fractional derivative given equations are converted to the nonlinear ordinary differential equations of integer order and then the resulting equations are solved using a novel analytical method called the auxiliary equation method as a result some exact solutions for them are successfully established the exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective

fractional time quantum dynamics application of the fractional calculus to quantum processes is presented in particular the quantum dynamics is considered in the framework of the fractional time schr odinger equation se which differs from the standard se by the fractional time derivative ensuremath partial ensuremath partial t ensuremath rightarrow ensuremath partial ensuremath alpha ensuremath partial t ensuremath alpha it is shown that for ensuremath alpha 1 2 the fractional se is isospectral to a comb model an analytical expression for the green s functions of the systems are obtained the semiclassical limit is discussed

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

controlled 2 blocks of quasisimple groups abstract controlled p blocks introduced by alperin and broue in 1979 were classified for quasisimple groups g for odd primes in 10 we classify the controlled 2 blocks of quasisimple groups g the results imply that every nilpotent 2 block of g has abelian defect groups this agrees with one of the main results proved in 15 we also give an explicit characterization of non controlled 2 blocks of all quasisimple groups this implies that the p 2 version of the block theoretic analogue of glauberman s zj theorem proved by kessar linckelmann and robinson in 27 holds for quasisimple groups g

2 weights for unitary groups this paper gives a description of the local structures of 2 radical subgroups in a finite unitary group and proves alperin s weight conjecture for finite unitary groups when the characteristic of modular representation is even

existence of solution to a local fractional nonlinear differential equation abstract we prove existence of solution to a local fractional nonlinear differential equation with initial condition for that we introduce the notion of tube solution

corrections to stochastic barbalat s lemma and its applications jun 12 1537 1543 the proof of theorem 1 stochastic barbalat s lemma in the paper by wu is incorrect this note provides a new statement of stochastic barbalat s lemma in addition some definitions propositions and proofs are given to replace those in the paper by wu

local control in fusion systems of p blocks of finite groups abstract if p is an odd prime b a p block of a finite group g such that sl 2 p is not involved in n g q e c g q for any b subpair q e then n g z j p controls b fusion where p is a defect group of b this is a block theoretic analogue of glauberman s zj theorem several results of general interest about fusion and blocks are also proved

a new lyapunov function for stability of time varying nonlinear perturbed systems in this paper global and local uniform asymptotic stability of perturbed dynamical systems is studied by using lyapunov techniques the restriction about the perturbed term is that the perturbation is bounded by an integrable function under the assumption that the nominal system is globally uniformly asymptotically stable we use a new lyapunov function to obtain a global uniform asymptotical stability of some perturbed systems

discussion on barbalat lemma extensions for conformable fractional integrals in this paper the barbalat type lemmas for conformable fractional order integrals which can be used to conclude the convergence of a function to zero is discussed

fractional newton mechanics with conformable fractional derivative in this paper we use the conformable fractional derivative and integral to discuss the fractional newtonian mechanics the fractional version of the calculus of variations is introduced and the fractional euler lagrange equation is constructed some mechanical problems such as fractional harmonic oscillator problem the fractional damped oscillator problem and the forced oscillator problem are discussed in the one dimensional fractional dynamics

signatures of finite classical lie algebra representations the paper deals with the real classical lie algebras of types br cr and their arbitrary irreducible representations hermitian forms invariant relative to these representations are considered signature formulas for these forms are obtained

growth conditions for exponential stability of time varying perturbed systems in this paper we derive some sufficient conditions for practical uniform exponential stability of time varying perturbed systems based on lyapunov techniques whose dynamics are in general unbounded in time in the sense that the solutions are uniform stable and converge to a small neighbourhood of the origin under quite general assumptions we first present a new converse stability theorem for a large class of time varying systems which will be used to prove certain stability properties of nonlinear systems with perturbation therefore a new lyapunov function is presented that guarantees practical uniform exponential stability of perturbed systems furthermore some illustrative examples are presented

vertices for irreducible characters of a class of blocks we observe that navarro s definition of a vertex for an irreducible character of a p solvable group may be extended to irreducible characters in p blocks with defect groups contained in a normal p solvable subgroup and show that this definition is independent of the choice of n we show that the fundamental properties of navarro s vertices generalize and as a corollary show that the vertices of the irreducible brauer characters in blocks of the above form are radical and are intersections of pairs of sylow p subgroups

a simple alternative to the barbalat lemma this paper gives a simple proof of the property that if a signal is square integrable and has a bounded derivative then the signal converges to zero asymptotically

weights for symmetric and general linear groups abstract an important feature of the theory of finite groups is the number of connections and analogies with the theory of lie groups the concept of a weight has long been useful in the modular representation theory of finite lie groups in the defining characteristic of the group the idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in alperin proc sympos pure math 41 1987 369 379 where it was conjectured that the number of weights should equal the number of modular irreducible representations moreover this equality should hold block by block the conjecture has created great interest since its truth would have important consequences a synthesis of known results and solutions of outstanding problems in this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups in the latter case r may be assumed to be different from the defining characteristic p of the group since the result is known when r is p the well known analogy between the representation theory of the symmetric and general linear groups holds here too

robust stability and stabilization of fractional order linear systems with polytopic uncertainties robust stability and stabilization of fractional order uncertain linear systems with order 0 1 and 1 2 are considered in the paper a new polytopic type uncertain state space model for fractional order linear systems is addressed which allows second order uncertain parameters the uncertainty in the fractional order model appears in terms of a polytope of matrices some sufficient criteria for the robust asymptotical stable and stabilization for such fractional order uncertain linear systems are derived all the results are obtained in terms of linear matrix inequalities lmis numerical examples are presented to demonstrate the validity and feasibility of the obtained results

boundedness and convergence on fractional order systems we establish conditions to guarantee boundedness and convergence of signals described by non integer order equations using caputo derivatives the case of linear time varying unforced equations is first studied and later results for linear time varying forced equations and time varying unforced non linear equations are presented and discussed

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 generalized and fractional derivatives and their applications to classical mechanics a generalized differential operator on the real line is defined by means of a limiting process when an additional fractional parameter is introduced this process leads to a locally defined fractional derivative the study of such generalized derivatives includes as a special case basic results involving the classical derivative and current research involving fractional differential operators all our operators satisfy properties such as the sum product quotient rules and the chain rule we study a sturm liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard sturm liouville theory as an application of the developments herein we find the general solution of a generalized harmonic oscillator we also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift the generic nature of phase plane orbits modulo a time shift is extended to the classical gravitational n body problem of newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law finally restricting the generalized derivatives to a special class of fractional derivatives we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter the results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

controlled blocks of the finite quasisimple groups for odd primes abstract we classify controlled blocks introduced by alperin and broue in 1979 for all quasisimple groups g for odd primes the results imply that every nilpotent block of g has abelian defect groups which in turn is one of the main results proved in an and eaton 2011 6 we also give an explicit characterization of non controlled blocks of all quasisimple groups g for odd primes this implies the block theoretic analogue of glauberman s zj theorem for g proved by kessar linckelmann and robinson 2002 18

simplest equation method for some time fractional partial differential equations with conformable derivative abstract the conformable fractional derivative was proposed by r khalil et al in 2014 which is natural and obeys the leibniz rule and chain rule based on the properties a class of time fractional partial differential equations can be reduced into odes using traveling wave transformation then the simplest equation method is applied to find exact solutions of some time fractional partial differential equations the exact solutions solitary wave solutions periodic function solutions rational function solutions of time fractional generalized burgers equation time fractional generalized kdv equation time fractional generalized sharma tasso olver fsto equation and time fractional fifth order kdv equation 3 1 dimensional time fractional kdv zakharov kuznetsov kdv zk equation are constructed this method presents a wide applicability to solve some nonlinear time fractional differential equations with conformable derivative

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

on fractional extensions of barbalat lemma a b s t r a c t this paper presents barbalat like lemmas for fractional order integrals which can be used to conclude about the convergence of a function to zero based on some conditions upon its fractional integral some examples in the context of asymptotic behaviour of solutions of fractional order differential equations indicate the potential application of these lemmas in control theory

on the lyapunov theory for fractional order systems abstract we provide the main features of lyapunov theory when it is formulated for fractional order systems we give consistent extensions of lyapunov lasalle and chetaev classical theorems to the case of fractional order systems we give examples to illustrate the applications of the concepts and propositions introduced

signatures of finite representation of real simple lie algebras the paper deals with the arbitrary irreducible representations of simple lie algebras of types g2 f4 br cr dr and the hermitian forms being invariant relative to this representation formulas and tables for calculating hermitian form signatures are obtained

fractional differential forms a generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders that is a fractional exterior derivative is defined this is found to generate new vector spaces of finite and infinite dimension fractional differential form spaces the definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case coordinate transformation rules are also computed the transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative the metric for the fractional form spaces is given based on the coordinate transformation rules all results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one

stochastic barbalat s lemma and its applications in the deterministic case a significant improvement on stability analysis of nonlinear systems is caused by introducing barbalat s lemma into control area after lyapunov s second method and lasalle s theorem were established this note considers the extension of barbalat s lemma to the stochastic case to this end the uniform continuity and the absolute integrability are firstly described in stochastic forms it is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily ito diffusions when it is applied to ito diffusion processes many classical results on stochastic stability are covered as special cases

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

signatures of finite representations of real outer lie algebras the paper deals with the real lie algebras of outer type and their arbitrary irreducible representations hermitian forms being invariant relative to these representations are considered signature formulae for these forms are obtained

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems