weighted norm estimates for sobolev spaces we give sufficient conditions for estimates of the form t g u x g d u x c u lls p u e hszp to holds where u x is a measure and llull5 p is the norm of the sobolev space hs p if d u dxs this reduces to the usual sobolev inequality the general form has much wider applications in both linear and nonlinear partial differential equations an application is given in the last section introduction if v x is a function in lt lt rn then it follows easily from holder s inequality and the sobolev imbedding theorem that 1 11 vu liq c u lls p u e hs p provided 2 1 t l q l p 1 t s n l q the last inequality in 2 must be strict if either p 1 or n sp here llwllq denotes the norm in lq and llull5 p denotes the norm in the sobolev space hs p hs p rn for precise definitions see 1 several authors have shown that inequality 1 can hold even when v x is not in some lt s 7ace stummel 8 proved 1 for p q s 2 and v x satisfying l q 3 sup iv x iqlx yl ndx x y lx yl 1 for some a 4 balslev 9 proved it for 1 p q x s a positive integer and v x satisfying 3 for some a sq this was extended to s any positive real number by schechter 10 for p q 2 it was shown in 11 that one can take a sq in 3 berger and schechter 12 proved 1 under the conditions 4 1 p q x l p s n l q they showed that 1 holds provided there exists an a such that 5 0 ol nq s n l q l p received by the editors november 25 1985 and in revised form january 19 1987 1980 mclthemcltics subject classification 1985 revision primary 26a84 secondary 35j10 31987 american mathematical society 0002 9947 87 1 00 25 per page

evolution of a modulated kp soliton the author investigates the propagation of an initial profile consisting of a planar kp soliton with some small modulations using the solution of the cauchy problem for the linearized kp equation find that for large times the modulations move away from the peak of the profile leaving behind a stable soliton a generalization of this method is formulated for the study of the stability of solutions of other integrable equations

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

a survey on fuzzy fractional variational problems abstract this paper focuses on introducing and studying generalized fuzzy euler lagrange equation and fuzzy isoperimetric problem according to the concept of caputo type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional euler lagrange equation for both constrained and unconstrained fuzzy fractional variational problems the fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy euler lagrange finally two examples are given to describe the proposed approach

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

directed polymers in a random medium 1 d expansion and the n tree approximation the authors develop a method which allows one to obtain 1 d expansions for the problem of directed polymers in a random medium using this method they expand thermal properties such as the free energy the specific heat the overlaps and the fluctuations of the transverse displacement their results are consistent with the existence of a finite upper critical dimension above which the low temperature phase is mean field like i e characterised by a broad distribution of overlaps broken replica symmetry and transverse fluctuations which scale as the length of the polymer

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

gradient method for thermal phase transitions the authors present a new numerical method of obtaining critical temperatures and exponents for ferromagnetic spin models and dynamical transitions this method is based on the spreading of damage in a temperature gradient they illustrate the method on the two dimensional ising model and present new results on the or ising spin glass in two dimensions they also obtain new critical exponents describing the behaviour of the width and the length of the damage front

roughness exponent in the domany kinzel cellular automaton we propose a method to detect phase transitions in discrete lattice models based on the roughness exponent this approach is applied to the one dimensional domany kinzel cellular automaton ca our results obtained by numerical simulations show that the roughness exponent method detects the frozen active phase transition directly from the ca spatio temporal configurations without any reference to thermodynamical potentials order parameters or response functions

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

the domany kinzel cellular automaton phase diagram the boundaries between the three phases of the domany kinzel probabilistic cellular automaton are determined with high accuracy via the gradient method the difficulties the extrapolation to the thermodynamic limit are circumvented and the critical exponents are also presented

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

nearly weakly orthonormal sequences singular value estimates and calderon zygmund operators abstract the direct way to estimate the singular values of a compact operator is to decompose it as a sum of orthogonal rank one pieces however such decompositions can generally not be found in practice we give a variation of the decomposition using nearly weakly orthonormal nwo sequences the nwo decomposition is easier to do in examples but is still strong enough to give good singular value estimates we illustrate this by giving sharp trace ideal estimates for the double layer potential and for the first and higher commutators of multiplication operators and calderon zygmund operators in particular nwo sequences seem well suited for dealing with operators which depend nonlinearly on their symbol we also show that the class of operators which admit nwo expansions is well behaved under composition under conjugation by weights and under conjugation by changes of variable

on damage spreading transitions we study the damage spreading transition in a generic one dimensional stochastic cellular automaton with two inputs domany kinzel model using an original formalism for the description of the microscopic dynamics of the model we are able to show analytically that the evolution of the damage between two systems driven by the same noise has the same structure as a directed percolation problem by means of a mean field approximation we map the density phase transition into the damage phase transition obtaining a reliable phase diagram we extend this analysis to all symmetric cellular automata with two inputs including the ising model with heat bath dynamics

phase diagram of a stochastic cellular automaton with long range interactions abstract a stochastic one dimensional cellular automaton with long range spatial interactions is introduced in this model the state probability of a given site at time t depends on the state of all the other sites at time t 1 through a power law of the type 1 r r being the distance between sites for this model reduces to the domany kinzel cellular automaton the dynamical phase diagram is analyzed using monte carlo simulations for 0 we found the existence of two different regimes one for 0 1 and the other for 1 it is shown that in the first regime the phase diagram becomes independent of regarding the frozen active phase transition in this regime a strong evidence is found that the mean field prediction for this model becomes exact a result already encountered in magnetic systems it is also shown that for replicas evolving under the same noise the long range interactions fully suppress the spreading of damage for 0 1

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

the spectrum of the schrodinger operator on decrit le spectre negatif de l operateur de schrodinger avec un potentiel singulier on determine la valeur exacte du fond du spectre et on etablit une double estimation

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

scaling of overhangs in 1 1 dimensional directed processes in a gradient when a gradient is imposed in the control parameter of a directed process in a 1 1 dimensional directed process a front may occur the position of this front evolves in steps of size one in the direction opposite to the gradient and in steps of size equal to or larger than one in the other direction the larger steps are due to overhangs there is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular i e it takes on a finite value in the thermodynamic limit if there is a critical point this step size distribution is a power law at this point with an exponent larger than two and which the authors conjecture to be equal to 3 beta nu perpendicular to beta is the order parameter exponent and nu perpendicular to is the spatial correlation length exponent in a finite gradient there is a second exponent governing the upper cut off in the step size distribution this exponent is equal to nu perpendicular to 1 nu perpendicular to these two exponents govern the convergence of the position of the front towards the critical value of the control parameter the authors exemplify their discussion with directed site percolation and with two versions of a model originally designed to study catalytic reactions

on the ulam stability of first order linear fuzzy differential equations under generalized differentiability in this paper we investigate under some suitable conditions and generalized differentiability the ulam stability problems of three variants of first order linear fuzzy differential equations respectively

fuzzy versions of hyers ulam rassias theorem we introduce three reasonable versions of fuzzy approximately additive functions in fuzzy normed spaces more precisely we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in a fuzzy sense

static critical exponents from the dynamics of damage spreading and overhangs in the ising model with temperature gradient by studying damage spreading in spin models in a temperature gradient it is possible to determine not only the critical temperature and the correlation length exponent v but also a second critical exponent beta with high precision this is done by studying the dynamics of damage spreading in particular the overhangs which develop in the damage front the authors demonstrate the method for the two dimensional ising model using heat bath dynamics and discuss differences with metropolis dynamics

hyers ulam stability of fuzzy fractional volterra integral equations with the kernel ψ function via successive approximation method abstract in this paper we investigative the hyers ulam stability and the hyers ulam rassias stability of fuzzy fractional volterra integral equations ffvies involving the kernel function we propose conditions for the existence uniqueness and stability of ffvie by employing the method of successive approximation to visualize the theoretical results some examples and numerical simulations are given

a percolation approach to the kauffman model a study is made of the spreading of damage in the random but deterministic kauffman model on the square lattice with the spreading from one edge of the lattice the critical value of the parameterp c above which the system becomes chaotic is found to bep c 0 298 the possibility of suppression of the chaotic phase by noise is also studied it is found that forp p c an extremely large noise levelg 0 99 is required if possible at all

some properties of the space of fuzzy valued continuous functions on a compact set in this paper a generalized form of the bolzano theorem in classical analysis to fuzzy number space and a characterization of compact subsets in fuzzy number space are given some properties of the fuzzy valued continuous functions defined on a compact set k are studied completeness of the space c k e 1 of fuzzy valued continuous functions on k endowed with the supremum metric d is proved a characterization of compact subsets in the space c k e 1 d is presented which is a generalization of the arzela ascoli theorem in classical analysis

a decomposition theorem for bmo and applications abstract every function having bounded mean oscillation bmo on the line is written as a sum of coefficients times normalized rational functions with the coefficients satisfying a carleson measure packing condition this decomposition theorem and related techniques are used to obtain operator norm estimates for first and second order commutators and direct and converse singular value estimates for hankel operators and products of hankel operators

fuzzy stability of the jensen functional equation we establish a generalized hyers ulam rassias stability theorem in the fuzzy sense in particular we introduce the notion of fuzzy approximate jensen mapping and prove that if a fuzzy approximate jensen mapping is continuous at a point then we can approximate it by an everywhere continuous jensen mapping as a fuzzy version of a theorem of schwaiger we also show that if every fuzzy approximate jensen type mapping from the natural numbers into a fuzzy normed space can be approximated by an additive mapping then the fuzzy norm is complete

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

scaling of overhangs appearing in fronts in higher dimensions directed percolation in higher dimensions serves as an example of how the distribution of overhangs gives information on the critical exponents of the system overhangs appear as jumps in the position of the front formed when a gradient in the control parameter is imposed along one of the spatial directions by analyzing the overhang distribution in d 1 dimensional directed percolation the authors determine the critical exponents beta and vperpendicular to for d2 in higher dimensions the overhang distribution is insensitive to the critical region

stability of additive mappings in non archimedean fuzzy normed spaces in this paper we introduce a notion of a non archimedean fuzzy norm and study the stability of the cauchy equation in the context of non archimedean fuzzy spaces in the spirit of hyers ulam rassias gavruta as a corollary the stability of the jensen equation is established we indeed present an interdisciplinary relation between the theory of fuzzy spaces the theory of non archimedean spaces and the theory of functional equations

fractional differential systems a fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications in this paper a new formula of fuzzy caputo fractional order derivatives 0 v leq 1 in terms of shifted chebyshev polynomials is derived the proposed approach introduces a shifted chebyshev operational matrix in combination with a shifted chebyshev tau technique for the numerical solution of linear fuzzy fractional order differential equations the main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations an approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived furthermore we also discuss the convergence of the proposed method from the fuzzy perspective experimentally we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena i e viscoelasticity oscillations and resistor capacitor rc circuits comparisons are also made with solutions obtained by the laguerre polynomials and the fractional euler method

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

weighted nonlinear potential theory the potential theoretic idea of the thinness of a set at a given point is extended to the weighted nonlinear potential theoretic setting the weights representing in general singularities degeneracies and conditions on these weights are given that guarantee when two such notions are equivalent at the given point when applied to questions of boundary regularity for solutions to degenerate elliptic second order partial differential equations in bounded domains this result relates the boundary wiener criterion for one operator to that of another and in the linear case gives conditions for boundary regular points to be the same for various operators the methods also yield two weight norm inequalities for riesz potentials j llq p 1 p q oo which at least in the first order case a 1 have found some use in a number of places in analysis

evidence for a new phase in the domany kinzel cellular automaton we consider a generalized version including anisotropy of the stochastic one dimensional cellular automaton studied by domany and kinzel it recovers wolfram like deterministic cellular automata as particular cases the phase diagram presents three and not two as previously suggested phases which were detected through the numerical study of both the order parameter and the sensitivity to initial conditions the various universality classes are exhibited as well cellular automata ca are totally discrete dynamical systems discrete space discrete time and discrete number of states which find general applications in physics chemistry biology computer science etc with each site noted i is associated a variable cr which can be in k different states cr 0 1 k 1 the dynamics is defined at each time step by rules depending on the values at a previous time of o j associated with a given number of r arbitrary sites called inputs usually one considers regular lattices and the inputs refer to the sites on the local neighborhood only the local rules of a ca may be probabilistic or deterministic and the sites are simultaneously updated thus we recognize the glauber dynamics for example as a special case of a ca however in the more general context of ca the dynamics is not restricted to the usual boltzmann weight and detailed balance therefore ca do not necessarily evolve towards equilibrium thus they can model chemical reactions crystal growth models turbulence biological problems or other nonlinear processes far from thermal equilibrium

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

correlation function studies on the domany kinzel cellular automaton abstract the domany kinzel cellular automaton is a simple and yet very rich model to study phase transitions in nonequilibrium systems this model exhibits three characteristic phases frozen active and chaotic in this paper we discuss the behavior of the equal time two point correlation functions and that of the associated correlation lengths as one crosses the phase boundary both for the frozen active and active chaotic transitions we have investigated in detail how the correlation lengths diverge as one approaches the phase boundary from both sides the divergence of the correlation length coupled with the previous studies on the divergence of the susceptibility suggests that the fluctuation dissipation theorem holds true in the domany kinzel cellular automaton model time dependence of the correlation functions is also discussed

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

damage spreading in a gradient we propose a new method of analyzing the frozen chaotic transition in a cellular automaton by propagating damage in a gradient we obtain estimations forpc and for the critical exponents for the kauffman model and the mixture of or and xor rules

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

a note on ulam stability of some fuzzy number valued functional equations abstract this note investigates the ulam stability of the following fuzzy number valued functional equations in banach spaces by using the metric defined on a fuzzy number space f x y 2 f x y 2 f x and f a x b y r f x s f y