solving the schrodinger equation with use of 1 n perturbation theory the large n expansion provides a powerful new tool for solving the schrodinger equation in this paper we present simple recursion formulas which facilitate the calculation we do some numerical calculations which illustrate the speed and accuracy of the technique

fine tuning some resistant rules for outlier labeling abstract a previous study examined the performance of a standard rule from exploratory data analysis which uses the sample fourths fl and fu and labels as outside any observations below fl k fu fl or above fu k fu fl customarily with k 1 5 in terms of the order statistics x 1 x 2 x n the standard definition of the fourths is fl x f and fu x n 1 f where f n 3 2 and denotes the greatest integer function the results of that study suggest that finer interpolation for the fourths might yield smoother behavior in the face of varying sample size in this article we show that using f i n 4 5 12 to define the fourths produces the desired smoothness corresponding to a common definition of quartiles fq n 4 1 4 leads to similar results instead of allowing the some outside rate per sample the probability that a sample contains one or more outside observations analogous to the experimentwise error rate in simultaneous inference to vary some us

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

comparing classical and resistant outlier rules abstract in this article we compare rosner s 1983 extension of the outlier t test the generalized esd procedure to the boxplot rules tukey s 1977 resistant outlier labeling approach the underlying principles are contrasted and the behavior of the procedures is compared in a variety of sampling situations to facilitate power type comparisons of procedures that are based on different paradigms the conditional outside rate a performance criterion for assessing outlier labeling procedures is introduced this quantity represents the probability that an observation yi is labeled an outlier conditional on its distance from the center e yi e some exact expressions as well as simple approximations and a monte carlo swindle are developed the conditional outside rate together with the outside rate per observation and the some outside rate per sample are applied to compare performance of the classical and boxplot rules based on monte carlo simulation of samples of sizes n 10 30 and 50

long time behavior of solutions of fisher kpp equation with advection and free boundaries we consider fisher kpp equation with advection ut uxx ux f u for x g t h t where g t and h t are two free boundaries satisfying stefan conditions this equation is used to describe the population dynamics in advective environments we study the influence of the advection coefficient on the long time behavior of the solutions we find two parameters c0 and with c0 0 which play key roles in the dynamics here c0 is the minimal speed of the traveling waves of fisher kpp equation more precisely by studying a family of the initial data 0 where is some compactly supported positive function we show that 1 in case 0 c0 there exists 0 such that spreading happens when i e u t 1 locally uniformly in r and vanishing happens when 0 i e g t h t remains bounded and u t 0 uniformly in g t h t 2 in case c0 there exists 0 such that virtual spreading happens when i e u t 0 locally uniformly in g t and u t ct 1 locally uniformly in r for some c c0 vanishing happens when 0 and in the transition case u t o t v c0 t uniformly the latter is a traveling wave with a big head near the free boundary x c0 t and with an infinite long tail on the left 3 in case c0 there exists 0 such that virtual spreading happens when and u t 0 uniformly in g t h t when 0 4 in case vanishing happens for any solution

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

regularity and asymptotic behavior of nonlinear stefan problems we study the following nonlinear stefan problem left begin aligned u t d delta u g u quad rm for x in omega t t 0 u 0 rm and u t mu nabla x u 2 quad rm for x in gamma t t 0 u 0 x u 0 x quad rm for x in omega 0 end aligned right where omega t subset mathbb r n n geqq 2 is bounded by the free boundary gamma t with omega 0 omega 0 and d are given positive constants the initial function u0 is positive in omega 0 and vanishes on partial omega 0 the class of nonlinear functions g u includes the standard monostable bistable and combustion type nonlinearities we show that the free boundary gamma t is smooth outside the closed convex hull of omega 0 and as t to infty either omega t expands to the entire mathbb r n or it stays bounded moreover in the former case gamma t converges to the unit sphere when normalized and in the latter case u to 0 uniformly when g u au bu 2 we further prove that in the case omega t expands to mathbb r n u to a b as t to infty and the spreading speed of the free boundary converges to a positive constant moreover there exists mu geqq 0 such that omega t expands to mathbb r n exactly when mu mu

resistant and test based outlier rejection effects on gaussian one and two sample inference resistant and sequential test based procedures for the detection of multiple outliers are compared in gaussian one and two sample problems subsequent to statistical rejection of putative outliers operating characteristics of hypothesis tests applied to samples that are outlier free may differ from their nominal values outlier rejection rules may be calibrated however to control the rate of erroneous outlier labeling in pure gaussian samples the small sample behavior of resistant location and scale estimators used in outlier detection is examined through simulation yielding new formulas for calibration of boxplot based and shorth based outlier rejection rules effects of different outlier criteria and calibration rubrics on postrejection hypothesis tests are compared in a variety of contamination settings and real world examples tests conducted subsequent to calibrated rejection have approximately nominal operating characteristics whether or not outliers are present the selection of an outlier cr

global bifurcation of periodic solutions to ordinary differential equations abstract general homotopy continuation and bifurcation results are discussed and proved for a class of semiflows these results and further conley homotopy index methods are applied to obtain a type of global bifurcation for periodic solutions of second order ordinary differential equations

on a free boundary problem for the curvature flow with driving force we study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles our first main result concerns the classification of solutions every solution falls into one of the three categories namely area expanding area bounded and area shrinking types we then study in detail the asymptotic behavior of solutions in each category among other things we show that solutions are asymptotically self similar both in the area expanding and the area shrinking cases while solutions converge to either a stationary solution or a traveling wave in the area bounded case we also prove results on the concavity properties of solutions one of the main tools of this paper is the intersection number principle however in order to deal with solutions with free boundaries we introduce what we call the extended intersection number principle which turns out to be exceedingly useful in handling curves with moving endpoints

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

shifted large n expansion for the power law potentials in the klein gordon equation with applications analytic expressions for the spectrum and the wavefunction resulting from the relativistic generalisation of the shifted large n expansion method as applied to the general power law potentials v r b arv are presented these expressions obtained in the context of the klein gordon equation are then used to compute the mass spectra and leptonic decay widths of mesonic states composed of heavy quarks the relativistic corrections are found to be in good agreement with those obtained by elaborate analytic wkb and numerical methods compact analytic results of the shifted large n expansion for relativistic systems are seen to be applicable to a much wider class of problems than are most other approximation methods

on a free boundary problem for a reaction diffusion advection logistic model in heterogeneous environment abstract in this paper we investigate a reaction diffusion advection equation with a free boundary which models the spreading of an invasive species in one dimensional heterogeneous environments we assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal and the spreading mechanism of species is determined by a stefan type condition investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters also we establish the long time behavior of the solution and the asymptotic spreading speed finally some biological implications are discussed

a reaction diffusion advection free boundary problem for a two species competition system abstract in this paper we investigate a reaction diffusion advection competition system with one of the two species is a superior competitor and the other is an inferior competitor in this system the double free boundaries represent the expanding front in a one dimensional habitat the main goal is to understand the effect of small advection term on the dynamics of the two species through double free boundaries first we provide the long time behavior of the species besides we get a spreading vanishing dichotomy and criteria governing spreading and vanishing furthermore when spreading occurs the asymptotic spreading speeds of double free boundaries are estimated

performance of some resistant rules for outlier labeling abstract the techniques of exploratory data analysis include a resistant rule for identifying possible outliers in univariate data using the lower and upper fourths fl and fu approximate quartiles it labels as outside any observations below fl 1 5 fu fl or above fu 1 5 fu fl for example in the ordered sample 5 2 0 1 8 fl 2 and fu 1 so any observation below 6 5 or above 5 5 is outside thus the rule labels 8 as outside some related rules also use cutoffs of the form fl k fu fl and fu k fu fl this approach avoids the need to specify the number of possible outliers in advance as long as they are not too numerous any outliers do not affect the location of the cutoffs to describe the performance of these rules we define the some outside rate per sample as the probability that a sample will contain one or more outside observations its complement is the all inside rate per sample we also define the outside rate per observation as the average fraction of outs

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary abstract we will study a free boundary problem of the nonlinear diffusion equations of the form u t u x x f u t 0 c t x h t where f is c 1 function satisfying f 0 0 c 0 is a given constant and h t is a free boundary which is determined by a stefan like condition this model may be used to describe the spreading of a new or invasive species with population density u t x over a one dimensional habitat the free boundary x h t represents the spreading front in this model we impose zero dirichlet boundary condition at left moving boundary x c t this means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c in this paper we will extend the results of a trichotomy result obtained in 23 to general monostable bistable and combustion types of nonlinearities we show that the long time dynamical behavior of solutions can be expressed by unified fashion that is for any initial data the unique solution exhibits exactly one of the behaviors spreading vanishing and transition we also give the asymptotic profile of the solution over the whole domain when spreading happens the approach here is quite different from that used in 23

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

percentage points for a generalized esd many outlier procedure a generalized extreme studentized deviate esd many outlier procedure is given for detecting from 1 to k outliers in a data set this procedure has an advantage over the original esd many outlier procedure rosner 1975 in that it controls the type i error both under the hypothesis of no outliers and under the alternative hypotheses of 1 2 k l outliers a method is given for approximating percentiles for this procedure based on the t distribution this method is shown to be adequately accurate using monte carlo simulation for detecting up to 10 outliers in samples as small as 25 tables are given for implementing this method for n 25 1 50 10 100 50 500 k 10 05 ol 005

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

a weak competition system with advection and free boundaries abstract in order to study the influence of advection speed on the competitive outcomes of two invading species a reaction diffusion advection weak competition system with four free boundaries in one dimensional space is proposed and investigated in the case of small advection speed the explicit classification of the competitive outcomes the estimation of the spreading speed the long time behavior of the solutions and the minimal habitat size which determines whether the species can always spread or not are obtained the results are similar as the case with no advection in the case of large advection speed both two species cannot spread successfully but may virtual spread downstream in the case of medium sized advection speed some competitive outcomes and the long time behaviors of the solutions are also obtained mathematical results suggest that the competitive outcomes which depend on the advection speed and moving parameters are very complicated some criteria for spreading vanishing and virtual spreading are also established in all cases

spreading in advective environment modeled by a reaction diffusion equation with free boundaries abstract we consider a reaction diffusion advection equation of the form u t u x x u x f u for x g t h t where g t and h t are two free boundaries satisfying stefan conditions f is a bistable type of nonlinearity this equation is used to describe the population dynamics in advective environment we study the influence of the advection coefficient on the dynamics of the solutions we find a parameter c 0 where c 0 0 is the speed of the traveling wave of the equation u t u x x f u such that when 0 c 0 resp when c 0 there is a vanishing transition spreading resp vanishing transition virtual spreading trichotomy result for the long time behavior of the solutions when all the solutions vanish i e h t g t is bounded and u 0 uniformly

outlier labeling with boxplot procedures in this article we focus on the detection of possible outliers based on the widely used boxplot procedures the outliers in a set of data are defined to be a subset of observations that appear to be inconsistent with the remaining observations we identify the outliers by constructing a boxplot with its lower fence lf and upper fence uf either a satisfying the requirement that if the given sample is outlier free then the probability that one or more of the sample data would fall outside the region lf uf is equal to a prescribed small value or b taken to be the tolerance limits derived from an outlier free random sample within which a specified large proportion of the sampled population would be asserted to fall with a given large probability exact expressions that can be routinely used to evaluate the constants needed in the construction of the boxplot s outlier region for samples taken from the family of location scale distributions are obtained for both procedures this article sh

1 n expansion for the yukawa potential revisited using the form prescribed by moreno and zepeda for the n dimensional yukawa potential and employing the method of 1 n expansion we have obtained in a recent paper the energies of the ground state and the first excited state of a three dimensional yukawa potential in the present addendum we introduce a modified prescription for the n dimensional yukawa potential which enables us also to calculate the energies of the higher angular momentum states our results are in very good agreement with the accurate numerical values

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

blow up rates of solutions of initial boundary value problems for a quasi linear parabolic equation abstract we consider initial boundary value problems for a quasi linear parabolic equation k t k 2 k k with zero dirichlet boundary conditions and positive initial data it has known that each of solutions blows up at a finite time with the rate faster than t t 1 in this paper it is proved that sup k t t t 1 log log t t 1 as t t under some assumptions our strategy is based on analysis for curve shortening flows that with self crossing brought by s b angenent and j j l velazquez in addition we prove some of numerical conjectures by watterson which are keys to provide the blow up rate

on a reaction diffusion equation with robin and free boundary conditions this paper studies the following problem ut uxx f u 0 0 u 0 t bux 0 t t 0 u h t t 0 h t ux h t t t 0 h 0 h0 u x 0 x 0 x h0 where f is an unbalanced bistable nonlinearity b 0 0 and is a compactly supported c2 function we prove that there exists 0 such that vanishing happens when 0 and u t converges as t to 0 uniformly in 0 h t spreading happens when i e h t c t tends to a constant for some c 0 u t converges to a positive stationary solution locally uniformly in 0 and to a traveling semi wave with speed c near x h t in the transition case when u t v t h2 0 h t tends to 0 as t where t is a maximum point of u t and v is the unique even positive solution of v f v 0 subject to v 0 moreover with respect to b and f t pln t q o 1 for some p 0 and q r or t z for some root z of v z bv z

transition operators of diffusions reduce zero crossing if u t x is a solution of a one dimensional parabolic secondorder linear partial differential equation pde then it is known that under suitable conditions the number of zero crossings of the function u t decreases that is does not increase as time t increases such theorems have applications to the study of blow up of solutions of semilinear pde time dependent sturm liouville theory curve shrinking problems and control theory we generalise the pde results by showing that the transition operator of a possibly time inhomogenous one dimensional diffusion reduces the number of zero crossings of a function or even suitably interpreted a signed measure our proof is completely probabilistic and depends in a transparent manner on little more than the sample path continuity of diffusion processes

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

1 n expansion for gaussian potential the method of large n expansion has been applied to an attractive radial gaussian potential to obtain its bound state energy levels it has been shown that the shifted 1 n expansion technique provides a better approximation than the method of unshifted 1 n expansion and yields energy values which are in good agreement with the accurate numerical and analytic results

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

biased movement and the ideal free distribution in some free boundary problems abstract some invasive species can persist in their habitat but eventually spread very slow in a nonlinear fashion to expand their habitat range in order to capture this phenomenon we consider reaction diffusion advection models with a free boundary modeling the spreading and the biased movement of species in one dimensional spatially heterogeneous environments under a condition of low resource quality we find that large advection can lead to the spreading of the species but the spreading speed goes asymptotically to zero moreover we investigate the effect of the resource on the dynamics of the current problem finally we bring the notion of an ideal free distribution ifd into free boundary problems to understand the mechanism such that the species can eventually match the environmental quality perfectly under the current problem setting the ifd may not hold even if the population plays an ideal free strategy we then provide a sufficient condition for the ifd to be reached when using an ideal free strategy

a new technique for deriving the large n solution of the klein gordon equation a new procedure for deriving the energy values of the klein gordon equation with a potential having both scalar and vector components is described using the h cross expansion and the quantization condition recursion formulae were obtained which can readily be applied to any 1 n expansion scheme these formulae have simple structures both for background and excited states and provide in principle the calculation of 1 n corrections up to an arbitrary order contrary to the methods elaborated formerly our technique does not involve converting the initial equation to the shortened schrodinger like form for the coulomb potential the solutions of the klein gordon and shortened equations are compared

a bayesian approach to outlier detection and residual analysis summary an approach to detecting outliers in a linear model is developed an outlier is defined to be an observation with a large random error generated by the linear model under consideration outliers are detected by examining the posterior distribution of the random errors an augmented residual plot is also suggested as a graphical aid in finding outliers we propose a precise definition of an outlier in a linear model which appears to lead to simple ways of exploring data for the possibility of outliers the definition is such that if the parameters of the model are known then it is also known which observations are outliers alternatively if the parameters are unknown the posterior distribution can be used to calculate the posterior probability that any observation is an outlier in a linear model with normally distributed random errors ei with mean zero and variance a 2we declare the ith observation to be an outlier if iei i ko for some choice of k the value of k can be chosen so that the prior probability of an outlier is small and thus outliers are observations which are more extreme than is usually expected realizations of normally distributed errors of more than about three standard deviations from the mean are certainly surprising and worth further investigation such outlying observations can occur under the assumed model however and this should be taken into account when deciding what to do with outliers and in choosing k note that ei is the actual realization of the random error not the usual estimated residual i the problem of outliers is studied and thoroughly reviewed by barnett lewis 1984 hawkins 1980 beckman cook 1983 and pettit smith 1985 the usual bayesian approach to outlier detection uses the definition given by freeman 1980 freeman defines an outlier to be any observation that has not been generated by the mechanism that generated the majority of observations in the data set freeman s definition therefore requires that a model for the generation of outliers be specified and is implemented by for example box tiao 1968 guttman dutter freeman 1978 and abraham box 1978 our method differs in that we define outliers as arising from the model under consideration rather than arising from a separate expanded model our approach is similar to that described by zellner moulton 1985 and is an extension of the philosophy

spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection diffusion equations abstract in this study we consider free boundary problems for nonlinear advection diffusion equations of the form u t u x x u x f u for t 0 g t x h t where x g t and x h t are free boundaries this problem may be used to describe the spreading of a biological or chemical species where the free boundaries represent the expanding fronts when f is a logistic nonlinearity it has been shown that the asymptotic spreading speeds of the two fronts h t and g t are different due to the advection term in this study for monostable bistable and combustion types nonlinearities we give much sharper estimates of the different spreading speeds of the fronts and we also prove that the solution converges to a semi wave in c 2 norm as t when spreading occurs we develop new approaches and extend a previous result to the problem with the advection term

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

the stefan problem for the fisher kpp equation abstract we study the fisher kpp equation with a free boundary governed by a one phase stefan condition such a problem arises in the modeling of the propagation of a new or invasive species with the free boundary representing the propagation front in one space dimension this problem was investigated in du and lin 2010 11 and the radially symmetric case in higher space dimensions was studied in du and guo 2011 10 in both cases a spreading vanishing dichotomy was established namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state or fails to establish and dies out in the long run moreover in the case of spreading the asymptotic spreading speed was determined in this paper we consider the non radially symmetric case in such a situation similar to the classical stefan problem smooth solutions need not exist even if the initial data are smooth we thus introduce and study the weak solution for a class of free boundary problems that include the fisher kpp as a special case we establish the existence and uniqueness of the weak solution and through suitable comparison arguments we extend some of the results obtained earlier in du and lin 2010 11 and du and guo 2011 10 to this general case we also show that the classical aronson weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3