on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

boundary trace embedding theorems for variable exponent sobolev spaces we study boundary trace embedding theorems for variable exponent sobolev space w1 p let be an open bounded or unbounded domain in rn satisfying strong local lipschitz condition under the hypotheses that p l 1 infp x supp x n we prove that there is a continuous boundary trace embedding w1 p lq provided q a measurable function on satisfies condition p x q x n 1 p x n p x for x

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

existence of infinitely many solutions for a neumann problem involving the p x laplacian abstract in this paper we consider the neumann problem involving the p x laplacian of the type div u p x 2 u x u p x 2 u f x u g x u in u 0 on we prove the existence of infinitely many solutions of the problem under weaker hypotheses by applying a variational principle due to b ricceri and the theory of the variable exponent sobolev spaces our results are an improvement and generalization of the relative results obtained by b ricceri for the p laplacian case

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

a constrained maximum likelihood approach to estimating switching regressions abstract it is widely known that the likelihood function for the switching regression model is unbounded if the error variances are unconstrained this paper shows that a constrained maximum likelihood formulation makes the likelihood function bounded relatively mild constraints are imposed on the parameters and if the true parameters satisfy the constraints there is a global maximizer of the likelihood function on the constrained parameter space which is consistent asymptotically normal and efficient a well known em algorithm is modified in order to compute constrained maximizers of the likelihood function

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

singular solutions of the p laplace equation soit un sous ensemble ouvert de r n n 1 et p un nombre reel plus grand que 1 on suppose que 0 o et u est p harmonique dans on etudie le comportement de la solution faible u pres de o de l equation apu div u p 2 u 0

multiplicity results for a differential inclusion problem with non standard growth abstract in this paper we examine the multiplicity of solutions of a differential inclusion problem involving p x laplacian of the type p p x u v x u p x 2 u j 1 x u x j 2 x u x in u n 0 on by using the nonsmooth version of ricceri variational principle we get three critical points of the corresponding energy motreanu panagiotopoulos type functional which are the solutions of p

extending the akaike information criterion to mixture regression models we examine the problem of jointly selecting the number of components and variables in finite mixture regression models we find that the akaike information criterion is unsatisfactory for this purpose because it overestimates the number of components which in turn results in incorrect variables being retained in the model therefore we derive a new information criterion the mixture regression criterion mrc that yields marked improvement in model selection due to what we call the clustering penalty function moreover we prove the asymptotic efficiency of the mrc we show that it performs well in monte carlo studies for the same or different covariates across components with equal or unequal sample sizes we also present an empirical example on sales territory management to illustrate the application and efficacy of the mrc finally we generalize the mrc to mixture quasi likelihood and mixture autoregressive models thus extending its applicability to non gaussian models discrete responses and d

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

a method for computation of velocity profiles by inversion of large offset records this paper describes a new method for recovering velocity profiles utilizing both phase and amplitude information including wide angle arrivals post and precritical reflections this method is based on a double spatial transformation with a minimization procedure the first transformation is slant stacking of the observed wave field seismogram the second is projecting the slant stacked wave field into the domain of horizontal slowness p and depth z in this domain the inverse problem is reduced to finding the critical path p v 1 z where v z is the true velocity of the compressional waves a numerical algorithm based on a minimization technique is used to find the critical path which is equivalent to the set of turning points of the critically reflected rays when this path is found then the following criteria are satisfied 1 most of the energy is concentrated away from the precritical region 2 the computed reflection coefficients reach their maximum on this path and 3 for horizontally st

a constrained formulation of maximum likelihood estimation for normal mixture distributions in this context the normal densities are sometimes referred to as component densities the log likelihood function l zy corresponding to a random sample xl xnj is defined by l ty ek 1 10g pm xk y the first problem associated with maximum likelihood estimation arises from the unboundedness of l qy on r day 1969 a global maximum likelihood estimate always fails to exist in addition the unboundedness of l y causes failures of optimization algorithms of both the em redner and walker 1984 and quasi newton fowlkes 1979 types in spite of the unboundedness of l 4y statistical theory kiefer 1978 guarantees that a particular local maximizer of l y is strongly consistent and asymptotically efficient several local maximizers can exist for a given sample and the other major maximum likelihood difficulty is in determining when the correct one has been found day 1969 noted that spurious maximizers corresponding to parameter points having some component standard deviations very small relative to others are generated by any small number of sample points grouped sufficiently close together the spurious maximizers like the unboundedness of l zy can create difflculties when using the em or quasi newton algorithms the constrained maximum likelihood formulation presented in the next section avoids the unconstrained problems of singularities and spurious maximizers of the type noted by day section 3 contains the statement and proof of a consistency result for the corresponding constrained estimator

inversion of refraction data by wave field continuation the process of wave equation continuation migration is adapted for refraction data in order to produce velocity depth models directly from the recorded data the procedure consists of two linear transformations a slant stack of the data produces a wave field in the p plane which is then downward continued using o as the imaging condition the result is that the data wave field is linearly transformed from the time distance domain into the slowness depth domain where the velocity profile can be picked directly no travel time picking is involved and all the data are present throughout the inversion the method is iterative because it is necessary to specify a velocity function for the continuation the solution produced by a given iteration is used as the continuation velocity function for the next step convergence is determined when the output wave field images the same velocity depth function as was input to the continuation the method obviates the problems associated with determining the envelope of solutions that are consistent with the observations since the time resolution in the data is transformed into a depth resolution in the slowness depth domain the method is illustrated with several synthetic examples and with a refraction line recorded in the imperial valley california

inversion of reflection traveltimes and amplitudes a bayesian nonlinear inversion scheme is used to invert traveltime and amplitude data from supercritical reflections reflection amplitudes are determined using a spherical wave model where the amplitude of a reflected spherical wave is given by a sommerfeld integral this integral is evaluated numerically by integrating over a contour in the complex plane the spherical wave model differs from plane wave models in the regions around the critical points for p and s refraction in these regions the spherical wave amplitudes are sensitive to frequency with a dependance on the ratio h between the reflector depth h and the seismic wave length the accuracies of the inversion scheme and the spherical reflection model are tested using control data from the university of houston physical model tank the solution obtained by inversion agrees with the known solution to within 3 percent for all parameters and the amplitudes predicted by the spherical wave model agree remarkably well with the observed data

remarks on eigenvalue problems involving the p x laplacian abstract this paper deals with the eigenvalue problem involving the p x laplacian of the form div u p x 2 u u q x 2 u in u 0 on where is a bounded domain in r n p c 0 inf x p x 1 q l 1 q x q x e p x for x e is a positive constant p x is the sobolev critical exponent it is shown that for every t 0 the problem has at least one sequence of solutions u n t n t such that 1 p x u n t p x t and n t as n the principal eigenvalues for the problem in several important cases are discussed especially the similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed some known results on the eigenvalue problem are extended

on the p x laplacian robin eigenvalue problem abstract consider robin eigenvalue problem involving the p x laplacian on a smooth bounded domain as follows p x u u p x 2 u in u p x 2 u x u p x 2 u 0 on we prove the existence of infinitely many eigenvalue sequences if p x constant and also present some sufficient conditions for which there is no principal eigenvalue and the set of all eigenvalues is not closed

direct mapping of seismic data to the domain of intercept time and ray parameter a plane wave decomposition marine seismic data recorded as a function of source receiver offset and traveltime are mapped directly to the domain of intercept or vertical delay time and horizontal ray parameter this is a plane wave decomposition based on beam forming of wide aperture seismic array data to determine automatically the loci of coherent seismic reflection and refraction events in this computation semblance in addition to the required slowness or horizontal ray parameter stack is found for linear x t trajectories across subarrays subsequently semblance is used to derive a windowing filter that is applied to the slowness stack to determine the points of stationary phase and eliminate aliasing the resulting filtered slowness stacks for multiple subarrays can then be linearly transformed and combined according to ray parameter range and time the resulting function of intercept time and horizontal ray parameter offers significant computational and interpretational advantages for the case of horizontal homogeneou

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

the robin eigenvalue problem for the p x laplacian as p abstract we study the asymptotic behavior as p of the first eigenvalues and the corresponding eigenfunctions for the p x laplacian with robin boundary conditions in an open bounded domain r n with smooth boundary we obtain uniform bounds for the sequence of first eigenvalues suitably rescaled and we prove that the positive first eigenfunctions converge uniformly in to a viscosity solution of a problem involving the laplacian subject to appropriate boundary conditions

imaging method utilizing attenuation and speed parameters in inverse scattering techniques methods for imaging the internal structures of an object using an acoustic wave field are provided in one aspect for example a method of imaging internals of a physical object using acoustic waves may include transmitting an acoustic wave field toward the object receiving a resultant acoustic wave field with a receiver where the resultant acoustic wave field is in response to the transmitted acoustic wave field reflected from or transmitted through the object and determining a predicted resultant acoustic wave field derived from a model of the object the method may also include determining a residual between the predicted resultant acoustic wave field and the resultant acoustic wave field and back propagating the residual to determine corrections to the model of the object in another aspect the above recited steps may be further iterated to successively refine the model of the object over a number of iterations until a predefined condition is reached

pde and ode limit problems for p x laplacian parabolic equations in this work we prove continuity of solutions with respect to initial conditions and parameters and we prove upper semicontinuity of a family of global attractors for problems of the form ut div d u p x 2 u b u in a bounded smooth domain in rn

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

a note on phillips 1991 a constrained maximum likelihood approach to estimating switching regressions abstract phillips phillips r f 1991 a constrained maximum likelihood approach to estimating switching regressions journal of econometrics 48 241 262 proposed a constrained maximum likelihood approach to estimating the parameters in a switching regression model in this note we propose a new approach which leads to a proof of a more general result than phillips s specifically we prove that the constrained mle cmle is still strongly consistent when the constant c decreases to 0 at the rate of exp n 1 2 log n as n increases to with 1 we also suggest a suitable hence c n for practice based on simulation results

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

p x laplacian equations satisfying cerami condition the goal of this paper is to prove the existence and multiplicity of solutions for the p x laplacian equations without the well known ambrosetti rabinowitz type growth conditions the critical point theorems with cerami condition are used

markov switching model selection using kullback leibler divergence abstract in markov switching regression models we use kullback leibler kl divergence between the true and candidate models to select the number of states and variables simultaneously specifically we derive a new information criterion markov switching criterion msc which is an estimate of kl divergence msc imposes an appropriate penalty to mitigate the over retention of states in the markov chain and it performs well in monte carlo studies with single and multiple states small and large samples and low and high noise we illustrate the usefulness of msc via applications to the u s business cycle and to media advertising

on the eigenvalues of weighted p x laplacian on rn abstract this paper following the theory of partial differential equations on variable exponent sobolev spaces is mainly concerned with the p x laplacian eigenvalue problem with a weight function on r n the results show that the spectrum of such problems contains a continuous family of eigenvalues

eigenvalues of the p x laplacian neumann problems abstract we study the eigenvalues of the p x laplacian operator with zero neumann boundary condition on a bounded domain where p x is a continuous function defined on the domain with p x 1 we show that similarly to the p laplacian case the smallest eigenvalue of the problem is 0 and it is simple and the supremum of all the eigenvalues is infinity however unlike the p laplacian case for very general variable exponent p x the first eigenvalue is not isolated that is the infimum of all positive eigenvalues of the problem is 0 we also study some properties of the set of functions having p x average value zero