nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

spectral asymptotics for laplacians on self similar sets abstract given a self similar dirichlet form on a self similar set we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a geometric counting function defined through a family of coverings of the self similar set naturally associated with the dirichlet space secondly under sub gaussian heat kernel upper bound we prove a detailed short time asymptotic behavior of the partition function which is the laplace stieltjes transform of the eigenvalue counting function associated with the dirichlet form this result can be applicable to a class of infinitely ramified self similar sets including generalized sierpinski carpets and is an extension of the result given recently by b m hambly for the brownian motion on generalized sierpinski carpets moreover we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function

map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

pullback attractors in v for non autonomous 2d navier stokes equations and their tempered behaviour abstract in this paper the asymptotic behaviour of the solutions to a non autonomous 2d navier stokes model is analyzed when the initial datum belongs to v for two frameworks the universe of fixed bounded sets and also for another universe given by a tempered condition the existence of pullback attractors in these different universes is established and thanks to regularity properties the relation between these several families of attractors and the corresponding in h is successfully studied finally two results about the tempered behaviour in v and h 2 2 of the pullback attractors when time goes to are obtained

global attractors for p laplacian equation abstract the existence of a l 2 w 0 1 p l q global attractor is proved for the p laplacian equation u t div u p 2 u f u g on a bounded domain r n n 3 with dirichlet boundary condition where p 2 the nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 u q k f u u c 2 u q k and f u l where q 2 is arbitrary there is no other restriction on p and q the asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method called asymptotic a priori estimate

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

the generalized fractional trace variational identity and fractional integrable couplings of kaup newell hierarchy starting from fractional isospectral problems and general bilinear forms the generalized fractional trace identity is presented then a new explicit lie algebra is introduced for which the fractional integrable couplings of kaup newell hierarchy are derived furthermore we obtained the fractional hamiltonian structures of the fractional integrable couplings of kaup newell hierarchy the methods derived by us can be generalized to other fractional integrable couplings

a general form of the generalized taylor s formula with some applications in this article a new general form of fractional power series is introduced some results of classical power series are circulated and proved to this new form a new general form of the generalized taylor s formula is also obtained applications including fractional power series solutions for fdes are analyzed results reveal that the new expansion is effective for formulating the solutions in this article a new general form of fractional power series is introduced in the sense of the caputo fractional derivative using this approach some results of the classical power series are circulated and proved to this fractional power series whilst a new general form of the generalized taylor s formula is also obtained some applications including fractional power series solutions for higher order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results the results reveal that the new fractional expansion is very effective straightforward and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components

existence of a global attractor for a p laplacian equation in rn abstract in this paper we prove the existence of the l 2 r n w 1 p r n l q r n global attractor for the p laplacian equation u t div u p 2 u u p 2 u f u g with a more general nonlinear term f f 1 a x f 2 in r n where a l 1 r n l r n and f 1 f 2 satisfy the arbitrary q order polynomial growth condition without any restriction on q p q 2 p 2

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal

random walks with self similar clusters abstract we construct a random walk on a lattice having a hierarchy of self similar clusters built into the distribution function of allowed jumps the random walk is a discrete analog of a levy flight and coincides with the levy flight in the continum limit the fourier transform of the jump distribution function is the continuous nondifferentiable function of weierstrass we show that for cluster formation it is necessary that the mean squared displacement per jump be infinite and that the random walk be transient we interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker the difference in dimensions is related to the fractal hausdorff besicovitch dimension of the self similar clusters

spatial log periodic oscillations of first passage observables in fractals for transport processes in geometrically restricted domains the mean first passage time mfpt admits a general scaling dependence on space parameters for diffusion anomalous diffusion and diffusion in disordered or fractal media for transport in self similar fractal structures we obtain an expression for the source target distance dependence of the mfpt that exhibits both the leading power law behavior depending on the hausdorff and spectral dimension of the fractal as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure we also present refined numerical results for the sierpinski gasket that confirm this oscillatory behavior

existence and upper semicontinuity of pullback attractors for non autonomous p laplacian parabolic problems abstract we study the asymptotic behavior of parabolic p laplacian problems of the form u t t div d t u t p 2 u t u t p 2 u t b t u t in a bounded smooth domain in r n where n 1 p 2 d l t with 0 d t x m a e in t 0 and for each 0 we have d s x d t x c s t for all x s t t for some positive constants and c moreover d d 1 in l t as 1 we prove that for each 0 the evolution process of this problem has a pullback attractor and we show that the family of pullback attractors behaves upper semicontinuously at 1

power dissipation in fractal feynman sierpinski ac circuits this paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non dissipative elements in particular we analyze the so called feynman sierpinski ladder a fractal ac circuit motivated by feynman s infinite ladder that exhibits power dissipation and wave propagation for some frequencies power dissipation in this circuit is obtained as a limit of quadratic forms and the corresponding power dissipation measure associated with harmonic potentials is constructed the latter measure is proved to be continuous and singular with respect to an appropriate hausdorff measure defined on the fractal dust of nodes of the network

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

fractal snowflake domain diffusion with boundary and interior drifts abstract we study a parabolic ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain we prove that under standard conditions a related cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial ventsell problem as a second result we prove that functions that are intrinsically lipschitz on the snowflake boundary admit euclidean lipschitz extensions to the closure of the entire domain our methods combine the fractal membrane analysis the vector analysis for local dirichlet forms and pde on fractals coercive closed forms and the analysis of lipschitz functions

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

energy and laplacian on hanoi type fractal quantum graphs this article studies potential theory and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs these spaces can be represented as a possibly infinite union of one dimensional intervals and a totally disconnected possibly uncountable compact set which roughly speaking represents the set of junction points classical quantum graphs and fractal spaces such as the hanoi attractor are included among them we begin with proving the existence of a resistance form on the hanoi attractor and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of laplacians corresponding to weakly self similar measures on the hanoi attractor these estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal we then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph finally we extend our spectral results to a large class of weakly self similar fractal quantum graphs

resistance and spectral dimension of sierpinski carpets the authors announce results which prove the existence of the spectral dimension ds for sierpinski carpets in two dimensions their method employs the einstein relation zeta d omega df using this numerical calculations of the resistance of approximations to the sierpinski carpet yield an accurate estimate for ds

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

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

log periodic critical amplitudes a perturbative approach log periodic amplitudes appear in the critical behavior of a large class of systems in particular when a discrete scale invariance is present here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial in this case the log periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated

nonlocal systems of balance laws in several space dimensions with applications to laser technology for a class of systems of nonlinear and nonlocal balance laws in several space dimensions we prove the local in time existence of solutions and their continuous dependence on the initial datum the choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam using realistic parameters solutions to this model obtained through numerical integrations meet qualitative properties of real cuts moreover the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

on a degenerate nonlocal parabolic equation with variable source abstract we study the dirichlet problem for the degenerate nonlocal parabolic equation u t div a u l 2 2 u b u l 2 2 u q x t 2 u f in q t where q t 0 t t 0 r n n 2 is a bounded domain with a sufficiently smooth boundary a b are real valued functions defined on r and q x t is a measurable function in q t with values in an interval q q 1 we show that under appropriate conditions on a b q the problem has a global in time solution and find conditions of uniqueness sufficient conditions for extinction in finite time or strict positivity of u t l 2 for all t 0 are found in the case f 0 the decay rates for u t l 2 are derived

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

asymptotic behaviour of nonlocal p laplacian reaction diffusion problems abstract in this paper we focus on studying the existence of attractors in the phase spaces l 2 and l p among others for time dependent p laplacian equations with nonlocal diffusion and nonlinearities of reaction diffusion type firstly we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term thereupon the regularising effect of the equation is shown applying an argument of a posteriori regularity since under the assumptions made we cannot guarantee the uniqueness of weak solutions in addition this argument allows to ensure the existence of an absorbing family in w 0 1 p this leads to the existence of the minimal pullback attractors in l 2 l p and some other spaces as l p relationships between these families are also established

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

long time behavior of a non autonomous parabolic equation with nonlocal diffusion and sublinear terms abstract this paper is devoted to study the asymptotic behavior of a time dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth namely we extend some previous results from the literature obtaining existence uniqueness and continuity results analyzing the stationary problem and decay of the solutions of the evolutionary problem and finally under more general assumptions ensuring the existence of pullback attractors for the associated dynamical system in both l 2 and h 1 norms relationships among these objects are established using regularizing properties of the equation

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure