frequency analysis of strongly nonlinear generalized duffing oscillators using he s frequency amplitude formulation and he s energy balance method in this paper he s frequency amplitude formulation hfaf and he s energy balance method hebm are employed to solve the generalized duffing equation in the form of x x a 3x 3 a 5x 5 a 7x 7 a nx n 0 for any arbitrary power of n a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the hfaf hebm and exact integration method numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

application of he s amplitude frequency formulation to a nonlinear oscillator with discontinuity in this paper he s amplitude frequency formulation is used to obtain a periodic solution of a nonlinear oscillator with discontinuity the accuracy arrives at 0 73 even when the amplitude tends to infinity

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

topological properties of banach spaces weakly compact sets in banach spaces are fragmentable we introduce the weaker notion of the a fragmentability of a subset of a banach space a notion that has a far wider range of applicability we prove that all weakly cech analytic subsets of a banach space are a fragmented this implies that all banach spaces having kadec norms are a fragmented however l is not fragmented

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

fractional wave equation with a frictional memory kernel of mittag leffler type abstract in this paper we give an analytical treatment of a fractional wave equation with caputo time fractional derivative and frictional memory kernel of mittag leffler type this problem generalizes a recently solved problem 62 of a wave equation for a vibrating string in presence of a fractional friction with power law memory kernel such equations can be used in the context of modeling processes in complex and viscoelastic media

he s frequency formulation for nonlinear oscillators using a golden mean location he s frequency formulation derived on the basis of an ancient chinese mathematical method is an effective method for treating nonlinear oscillators there are different approaches for choosing location points in the formulation this paper suggests another location point an example is given and the accuracy is improved

the general time fractional wave equation for a vibrating string the solution of a general time fractional wave equation for a vibrating string is obtained in terms of the mittag leffler type functions and complete set of eigenfunctions of the sturm liouville problem the time fractional derivative used is taken in the caputo sense and the method of separation of variables and the laplace transform method are used to solve the equation some results for special cases of the initial and boundary conditions are obtained and it is shown that the corresponding solutions of the integer order equations are special cases of those of time fractional equations the proposed general equation may be used for modeling different processes in complex or viscoelastic media disordered materials etc

application of the modified frequency formulation to a nonlinear oscillator he s frequency formulation is used to obtain the relationship between the frequency and amplitude of a nonlinear oscillator the general approach is to choose two linear oscillators in this paper however one linear oscillator and the duffing oscillator are chosen as trial equations the solution procedure is of utter simplicity while the result is of high accuracy

alur dual renormings of banach spaces we give a covering type characterization for the class of dual banach spaces with an equivalent alur dual norm

fractional calculus with an integral operator containing a generalized mittag leffler function in the kernel in this paper we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized mittag leffler functions e a b c k z email protected n 0 c k email protected email protected z nn z b email protected c r a max 0 r k 1 r k 0 in its kernel l n being the familiar pochhammer symbol a number of corollaries and consequences of the main results presented here are also considered

fractional fokker planck dynamics numerical algorithm and simulations anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional fokker planck dynamics via the underlying continuous time random walk an efficient numerical algorithm is developed which is applicable for an arbitrary potential this algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials normal and fractional diffusion are compared through their time evolution of the probability density in state space moreover we discuss the stationary probability density of the fractional current values

an ancient chinese mathematical algorithm and its application to nonlinear oscillators an ancient chinese mathematical method is briefly introduced and its application to nonlinear oscillators is elucidated where he s amplitude frequency formulation is outlined three examples are given to show the extremely simple solution procedure and remarkably accurate solutions

approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire in this paper the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented at the beginning of the study the equation of motion is derived in a detailed way he s max min approach he s frequency amplitude method and the parameter expansion method are implemented to solve the established model the numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations and a very good agreement is observed the relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results

polyhedral norms on non separable banach spaces abstract we prove the existence of equivalent polyhedral norms on a number of classes of non separable spaces the majority of which being of the form c k in particular we obtain a complete characterization of those trees t such that c 0 t admits an equivalent polyhedral norm

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

analysis of a nonlinear oscillator with discontinuity in this paper a nonlinear oscillator with discontinuity is analyzed using he s amplitude frequency formulation the natural frequency of the oscillator system can be obtained using concise and straightforward calculation steps good agreement between the results from the proposed and other classical approaches is reached

weak locally uniformly rotund norms and descriptive compact spaces we prove that a dual banach space x has an equivalent w lur norm if and only if the weak topology has a isolated network we give sufficient conditions for the existence of equivalent norms with various kinds of differentiability

effects of a fractional friction with power law memory kernel on string vibrations in this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power law memory kernel the exact solution is obtained in terms of the mittag leffler type functions and a generalized integral operator containing a four parameter mittag leffler function in the kernel the method of separation of the variables laplace transform method and sturm liouville problem are used to solve the equation analytically the asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the tauberian theorems the proposed model may be used for describing processes where the memory effects of complex media could not be neglected

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

normality and modulability indices part ii convex cones in hilbert spaces this paper proposes and compares several ways of measuring the degree of normality of a convex cone contained in a normed space the dual concept of modulability is also considered other notions like solidity and sharpness are also analyzed from a quantitative point of view

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

generalized distributed order diffusion equations with composite time fractional derivative abstract in this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the fourier laplace transform method we represent solutions in terms of infinite series in fox h functions the fractional and second moments are derived by using mittag leffler functions we observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives generalized uniformly distributed order diffusion equation as a model for strong anomalous behavior is analyzed by using tauberian theorem some previously obtained results are special cases of those presented in this paper

on banach spaces whose norm open sets are fσ sets in the weak topology we investigate non separable banach spaces whose norm open sets are countable unions of sets closed in the weak topology and a narrower class of banach spaces with a network for the norm topology which is discrete in the weak topology in particular we answer a question of arhangel skii exhibiting various examples of non separable function spaces c k with a discrete network for the pointwise topology and consistently we answer some questions of edgar and oncina concerning borel structures and kadec renormings in banach spaces

continuous functions on compact totally ordered spaces abstract the banach space c k of continuous functions on k with its pointwise topology is shown to be fragmented by pointwise closed sets when k is a compact totally ordered set when k is a dendron and when k is a compact hausdorff space with a closed crossfree subbase for its topology in some special instances c k will have an equivalent locally uniformly convex norm

average locally uniform rotundity and a class of nonlinear maps abstract we consider some topological characterizations of dual banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of slicely continuous maps

convex sets which are intersections of closed balls in this paper we aim to investigate different questions concerning the stability of the set m of all intersections of closed balls in a normed space we are mainly concerned with i the stability of m under the closure of the vector sums ii the stability under the addition of balls we prove that i and ii are different properties which have strong connections with the geometry of the space they have interest both in finite and infinite dimension in the former case there is a link with linear programming theory we also study two more stability properties related to the well known binary intersection property mazur sets and mazur spaces are introduced as a natural family satisfying i we prove that every two dimensional normed space is a mazur space a result which distinguishes dimension d 2 from dimension d 3 we also discuss the connections between mazur spaces and porosity

generalized langevin equation with tempered memory kernel abstract we study a generalized langevin equation for a free particle in presence of a truncated power law and mittag leffler memory kernel it is shown that in presence of truncation the particle from subdiffusive behavior in the short time limit turns to normal diffusion in the long time limit the case of harmonic oscillator is considered as well and the relaxation functions and the normalized displacement correlation function are represented in an exact form by considering external time dependent periodic force we obtain resonant behavior even in case of a free particle due to the influence of the environment on the particle movement additionally the double peak phenomenon in the imaginary part of the complex susceptibility is observed it is obtained that the truncation parameter has a huge influence on the behavior of these quantities and it is shown how the truncation parameter changes the critical frequencies the normalized displacement correlation function for a fractional generalized langevin equation is investigated as well all the results are exact and given in terms of the three parameter mittag leffler function and the prabhakar generalized integral operator which in the kernel contains a three parameter mittag leffler function such kind of truncated langevin equation motion can be of high relevance for the description of lateral diffusion of lipids and proteins in cell membranes

uniformly gâteaux smooth approximations on c0 γ abstract every lipschitz mapping from c 0 into a banach space y can be uniformly approximated by lipschitz mappings that are simultaneously uniformly g teaux smooth and c frechet smooth

hilbert generated spaces we classify several classes of the subspaces of banach spaces x for which there is a bounded linear operator from a hilbert space onto a dense subset in x dually we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in hilbert spaces or in c0 spaces in their weak topology the existence of such embeddings is characterized by the existence of certain uniformly g teaux smooth norms

economic fluctuations and anomalous diffusion we quantify the relation between trading activity measured by the number of transactions n deltat and the price change g deltat for a given stock over a time interval t t deltat to this end we analyze a database documenting every transaction for 1000 u s stocks for the two year period 1994 1995 we find that price movements are equivalent to a complex variant of classic diffusion where the diffusion constant fluctuates drastically in time we relate the analog for stock price fluctuations of the diffusion constant known in economics as the volatility to two microscopic quantities i the number of transactions n deltat in deltat which is the analog of the number of collisions and ii the variance w 2 deltat of the price changes for all transactions in deltat which is the analog of the local mean square displacement between collisions our results are consistent with the interpretation that the power law tails of p g deltat are due to p w deltat and the long range correlations in g deltat are due to n deltat

gruenhage compacta and strictly convex dual norms abstract we prove that if k is a gruenhage compact space then c k admits an equivalent strictly convex dual norm as a corollary we show that if x is a banach space and x span k where k is a gruenhage compact in the w topology and is equivalent to a coarser w lower semicontinuous norm on x then x admits an equivalent strictly convex dual norm we give a partial converse to the first result by showing that if is a tree then c 0 admits an equivalent strictly convex dual norm if and only if is a gruenhage space finally we present some stability properties satisfied by gruenhage spaces in particular gruenhage spaces are stable under perfect images