definition of the riesz derivative and its application to space fractional quantum mechanics we investigate and compare different representations of the riesz derivative which plays an important role in anomalous diffusion and space fractional quantum mechanics in particular we show that a certain representation of the riesz derivative r x that is generally given as also valid for 1 behaves no differently than the other definition given in terms of its fourier transform in the light of this we discuss the 1 limit of the space fractional quantum mechanics and its consistency

global bifurcation of steady gravity water waves with critical layers we construct families of two dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed in particular establishing the existence of waves of large amplitude a riemann hilbert problem approach is used to recast the governing equations as a one dimensional elliptic pseudodifferential equation with a scalar constraint the structural properties of this formulation which arises as the euler lagrange equation of an energy functional enable us to develop a theory of analytic global bifurcation

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

steady periodic water waves with constant vorticity regularity and local bifurcation this paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed using conformal mappings the free boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable the new formulation leads to a regularity result and by use of bifurcation theory to the existence of waves of small amplitude even in the presence of stagnation points in the flow

transmission through locally periodic potentials in space fractional quantum mechanics abstract we study transmission through locally periodic potentials in the framework of space fractional quantum mechanics sfqm in particular we calculate the transmission probabilities t n for a particle in a chain of n dirac barriers and for the other case a chain of n square barriers we use a transfer matrix m matrix approach in the context of sfqm to obtain equations for t n for both systems bandlike structures emerge even for a small number of barriers of as low as n 4 and are more apparent for lower values of levy index considered

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

steady periodic waves and formal stability for fixed depth rotational equatorial flows abstract in this paper we apply the bifurcation theory to study the steady two dimensional periodic equatorial water waves for the f plane approximation we consider the waves with vorticity which propagate with a specified fixed depth between the thermocline and the upper boundary of the centre layer moreover we present a functional j and its first variation corresponds to the exact equations in the transformed variables using the second variation of j we prove a formal stability result for the bifurcation inducing the laminar solution

the fractional d alembert s formulas abstract in this paper we develop generalized d alembert s formulas for abstract fractional integro differential equations and fractional differential equations on banach spaces some examples are given to illustrate our abstract results and the probability interpretations of these fractional d alembert s formulas are also given moreover we also provide d alembert s formulas for abstract fractional telegraph equations

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

comment on on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 recently we have reanalyzed the consistency of the solutions of the space fractional schrodinger equation found in a piecewise manner and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent in this comment for clarity we present additional information about the critical integrals and describe how their analytic continuation is accomplished

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

regularly open sets in fuzzy topological spaces abstract this paper is devoted to the study of the role of fuzzy regularly open sets we prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings we prove that under a fuzzy almost continuous and fuzzy almost open map the inverse image of a fuzzy regularly open set is fuzzy regularly open further we define a new type of fuzzy separation axioms fuzzy almost separation axioms it is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology for example in a fuzzy almost t1 space not every fuzzy singleton is closed also a fuzzy space which is fuzzy almost t 2 1 2 as well as fuzzy almost t0 is fuzzy almost regular while in general topology we have to take an almost t2 space in place of almost t0 space

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

characteristic matrix of covering and its application to boolean matrix decomposition covering based rough sets provide an efficient means of dealing with covering data which occur widely in practical applications boolean matrix decomposition has frequently been applied to data mining and machine learning in this paper three types of existing covering approximation operators are represented by boolean matrices and then used in boolean matrix decomposition first we define two characteristic matrices of a covering through these boolean characteristic matrices three types of existing covering approximation operator are concisely and equivalently represented second these operator representations are applied to boolean matrix decomposition which has a close relationship with nonnegative matrix factorization a popular and efficient technique for machine learning we provide a sufficient and necessary condition for a square boolean matrix to decompose into the boolean product of another matrix and its transpose we then develop an algorithm for this boolean matrix decomposition finally these three covering approximation operators are axiomatized using boolean matrices this work presents an interesting viewpoint from which to investigate covering based rough set theory and its applications

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 the solution of a generalized higgs boson equation in the de sitter space time through an efficient and hamiltonian scheme abstract the present work is the first paper in the literature to report on a hamiltonian discretization of the fractional higgs boson equation in the de sitter space time and its theoretical analysis more precisely we design herein a numerically efficient finite difference hamiltonian technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with generalized potential and riesz space fractional derivatives of orders in 1 2 an energy integral for the model is readily available and we propose a nonlinear implicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties in the discrete scenario a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the de sitter space time and find results qualitatively similar to those available in the literature for the sake of convenience we provide the matlab code of an alternative linear discretization of the method presented in this work this linear implicit approach is thoroughly analyzed also

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

levy path integrals of particle on circle and some applications fractional quantum mechanics in the non relativistic case is described by the space fractional schrodinger equation containing the fractional riesz operator which is a non local operator the non locality makes it difficult to solve the fractional schrodinger equation in the local potentials in this paper we study the solution of the fractional schrodinger equation of a free particle moving on a circle by use of the levy path integrals approach we present the levy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case in addition the laplace transform energy time transform and momentum representation of the free particle kernel are also obtained the results of this paper contain the ones in standard quantum mechanics as special cases the problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems and the method of levy path integrals can be generalized to study more complex local potentials

comment on on the consistency of solutions of the space fractional schrodinger equation j math phys 53 042105 2012 in bayin s paper j math phys 53 042105 2012 10 1063 1 4705268 he claims to prove the consistency of the purported piece wise solutions to the fractional schrodinger equation for an infinite square well however his calculation uses standard contour integral techniques despite the absence of an analytic integrand the correct calculation is presented and supports our earlier work proving that the purported piece wise solutions do not solve the fractional schrodinger equation for an infinite square well m jeng s l y xu e hawkins and j m schwarz j math phys 51 062102 2010 10 1063 1 3430552

fractional schrodinger equation for a particle moving in a potential well in this paper the fractional schrodinger equation that contains the quantum riesz fractional derivative instead of the laplace operator is revisited for the case of a particle moving in the infinite potential well in the recent papers m jeng s l y xu e hawkins and j m schwarz on the nonlocality of the fractional schrodinger equation j math phys 51 062102 2010 10 1063 1 3430552 and s s bayin on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 10 1063 1 4705268 published in this journal controversial opinions regarding solutions to the fractional schrodinger equation for a particle moving in the infinite potential well that were derived by laskin fractals and quantum mechanics chaos 10 780 790 2000 10 1063 1 1050284 have been given in this paper a thorough mathematical treatment of these matters is provided the problem under consideration is reformulated in terms of three integral equations with the power

bound states for multiple dirac δ wells in space fractional quantum mechanics using the momentum space approach we obtain bound states for multiple dirac wells in the framework of space fractional quantum mechanics introducing first an attractive dirac comb potential i e dirac comb with strength g g 0 in the space fractional schrodinger equation we show that the problem of obtaining eigenenergies of a system with n dirac wells can be reduced to a problem of obtaining the eigenvalues of an n n matrix as an illustration we use the present matrix formulation to derive expressions satisfied by the bound state energies of n 1 2 3 delta wells we also obtain the corresponding wave functions and express them in terms of fox s h function

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

regularity and normality of l m fuzzy topological spaces abstract in this paper regularity and normality axioms are introduced in l m fuzzy topological spaces where l and m are completely distributive lattices with order reversing involution each l m fuzzy topological space can be regarded to be regular and normal to some degree the relations among the four separation axioms are investigated they are equivalent to each other in an l m fuzzy metric space

on the consistency of the solutions of the space fractional schrodinger equation recently it was pointed out that the solutions found in the literature for the space fractional schrodinger equation in a piecewise manner are wrong except the case with the delta potential we re analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise we also discuss effective potential approach and present a free particle solution for the space and time fractional schrodinger equation in general coordinates in terms of fox s h functions

consistency problem of the solutions of the space fractional schrodinger equation recently consistency of the infinite square well solution of the space fractional schrodinger equation has been the subject of some controversy hawkins and schwarz j math phys 54 014101 2013 10 1063 1 4772533 objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional schrodinger equation s s bayin j math phys 53 042105 2012 10 1063 1 4705268 s s bayin j math phys 53 084101 2012 10 1063 1 4739758 here we show for general n that as far as the integral representation of the solution in the momentum space is concerned there is no inconsistency to pinpoint the source of a possible inconsistency we also scrutinize the different representations of the riesz derivative that plays a central role in this controversy and show that they all have the same fourier transform when evaluated with consistent assumptions

stability of standing waves for the fractional schrodinger hartree equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger hartree equation by using the profile decomposition theory and variational methods we firstly obtain the best constant of a generalized gagliardo nirenberg inequality and then prove that the standing waves are orbitally stable

on lower separation axioms abstract in 3 mashhour et al introduced the concept of fuzzy disjointness two fuzzy sets a b in x are said to be fuzzy disjoint if a co b where co b is the complement of b using this concept they introduced fti separation axioms as a generalization for the basic separation axioms ti i 1 4 in 5 singal and rajvanshi used the concepts of regular open fuzzy sets 1 and fuzzy disjointness to introduce some fuzzy almost separation axioms which are stronger than those of mashhour unfortunately some results in 5 are incorrect in this paper we use the concept of the quasi coincident relation 4 to restate lower fuzzy separation axioms and give a corrected version for some definitions and results in 5

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

a new approach to l t2 l urysohn and l completely hausdorff axioms in this paper new l t 2 l urysohn and l completely hausdorff axioms are introduced in l topological spaces relations between these axioms are established the l extended real line and its subspaces as well as pointwise metric spaces satisfy these axioms