open 3 manifolds whose fundamental groups have infinite center and a torus theorem for 3 orbifolds our main result is a characterization of open seifert fibered 3 manifolds in terms of the fundamental group and large scale geometric properties of a triangulation as an application we extend the seifert fiber space theorem and the torus theorem to a class of 3 orbifolds

fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

tameness persists in weakly type preserving strong limits we show that if a weakly type preserving sequence of tame hyperbolic 3 manifolds converges strongly then the limit is tame as a first corollary we observe that we can replace the assumption of strong convergence with algebraic convergence in most cases as a second corollary we observe that given a finitely generated geometrically finite kleinian group tame groups are dense in the boundary of its quasiconformal deformation space in most cases

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

numerical analysis of a two parameter fractional telegraph equation abstract in this paper we consider the two parameter fractional telegraph equation of the form c d t 0 1 u t x c d x 0 1 u t x c d t 0 u t x u t x 0 here c d t 0 c d t 0 1 c d x 0 1 are operators of the caputo type fractional derivative where 0 1 and 0 1 a numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained numerical examples are given in the final section of the paper

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

bayesian prediction for flowgraph models with covariates an application to bladder carcinoma statistical flowgraph models are an efficient tool to model multi state stochastic processes they support both frequentist and bayesian approaches inclusion of covariates is also available in this paper we propose an easy way to perform a bayesian approach with covariates results are presented with an application to bladder carcinoma data

survival and hazard functions for progressive diseases using saddlepoint approximations saddlepoint approximations for the computation of survival and hazard functions are introduced in the context of parametric survival analysis although these approximations are computationally fast accurate and relatively straightforward to implement their use in survival analysis has been lacking we approximate survival functions using the lugannani and rice saddlepoint approximation to the distribution function or by numerically integrating the saddlepoint density approximation the hazard function is approximated using the saddlepoint density and distribution functions the approximations are especially useful for consideration of survival and hazard functions for waiting times in complicated models examples include total or partial waiting times for a disease that progresses through various stages convolutions of distributions

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

stochastic network models for survival analysis abstract we present methodology giving highly accurate approximations for bayesian predictive densities and distribution functions of first passage times between states of a semi markov process with a finite number of states when the states describe a degenerative disorder with an absorbing end state such predictive distributions are the survival distributions of a patient we illustrate these methods with a variety of examples including data from the san francisco aids study we achieve our approximations using a three step sequence first we introduce advanced concepts of flowgraph theory which allow us to compute the moment generating function of the first passage time given the model parameters next we use saddlepoint approximations to convert this into a density or distribution function conditional on the model parameter finally we use monte carlo methods to remove dependence on the model parameter these methods apply quite generally to all finite state semi markov models in discrete or con

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

scott s rigidity theorem for seifert fibered spaces revisited we will present a new proof of the rigidity theorem for seifert fibered spaces of infinite 1 by scott 1983 in the case when the base of the fibration is a hyperbolic triangle 2 orbifold our proof is based on arguments in the rigidity theorem for hyperbolic 3 manifolds by gabai 1997

galerkin finite element method for two dimensional riesz space fractional diffusion equations abstract in this article a class of two dimensional riesz space fractional diffusion equations is considered some fractional spaces are established and some equivalences between fractional derivative spaces and fractional sobolev space are presented by the galerkin finite element method and backward difference method a fully discrete scheme is obtained according to lax milgram theorem the existence and uniqueness of the solution to the fully discrete scheme are investigated the stability and convergence of the scheme are also derived finally some numerical examples are given for verification of our theoretical analysis

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

surgery on double knots and symmetries w whitten conjectured pacific j math 97 1981 no 1 209 216 that no 3 manifold obtained by a nontrivial surgery on a double of a noninvertible knot is a 2 fold branched covering of s3 the authors give counterexamples to this conjecture and determine the exact range of validity of the conjecture more generally they consider closed orientable 3 manifolds obtained by nontrivial dehn surgery on a double of a non strongly invertible knot and study the symmetries of such manifolds i e the homeomorphisms of finite order on these manifolds they show that except for a finite number of surgeries these manifolds admit no nontrivial symmetry

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

the hilbert smith conjecture for three manifolds we show that every locally compact group which acts faithfully on a connected three manifold is a lie group by known reductions it suffices to show that there is no faithful action of zp the p adic integers on a connected three manifold if zp acts faithfully on m3 we find an interesting zp invariant open set u m with h2 u z and analyze the incompressible surfaces in u representing a generator of h2 u it turns out that there must be one such incompressible surface say f whose isotopy class is fixed by zp an analysis of the resulting homomorphism zp mcg f gives the desired contradiction the approach is local on m msc 2010 primary 57s10 57m60 20f34 57s05 57n10 msc 2010 secondary 54h15 55m35 57s17

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

a generalized spectral collocation method with tunable accuracy for fractional differential equations with end point singularities we develop spectral collocation methods for fractional differential equations with variable order with two end point singularities specifically we derive three term recurrence relations for both integrals and derivatives of the weighted jacobi polynomials of the form 1 x mu 1 1 x mu 2 p j a b x a b mu 1 mu 2 1 which leads to the desired differentiation matrices we apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end point singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters mu 1 and mu 2 even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

a tunable finite difference method for fractional differential equations with non smooth solutions in this work a finite difference method of tunable accuracy for fractional differential equations fdes with end point singularities is developed modified weighted shifted grunwald letnikov wsgl formulas are proposed to approximate the left and right riemann liouville fractional operators which show better accuracy than the original wsgl formulas due to the use of the correction terms finite difference schemes are constructed to solve two fractional boundary value problems and a space fractional allen cahn equation even if the singularity of the considered fdes is unknown satisfactory numerical solutions can still be obtained by suitably tuning the correction terms various numerical examples are presented to verify the effectiveness of the present method and comparisons with other known methods are also made that demonstrate higher accuracy of the current method

boundary slopes of immersed surfaces in 3 manifolds author s hass joel rubinstein j hyam wang shicheng abstract this paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3 manifold with torus boundary in the case of hyperbolic 3 manifolds we obtain uniform quadratic bounds in g for the number of possible slopes independent of the 3 manifold we also look at some related quantities such as how many times the slopes of two such surfaces of specified genus can intersect

bootstrap confidence bands for sojourn distributions in multistate semi markov models with right censoring transient semi markov processes have traditionally been used to describe the transitions of a patient through the various states of a multistate survival model a survival distribution in this context is a sojourn through the states until passage to a fatal absorbing state or certain endpoint states using complete sojourn data this paper shows how such survival distributions and associated hazard functions can be estimated nonparametrically and also how nonparametric bootstrap pointwise confidence bands can be constructed for them when patients are subject to independent right censoring from each state during the sojourn limitations to the estimability of such survival distributions that result from random censoring with bounded support are clarified the methods are applicable to any sort of sojourn through any finite state process of arbitrary complexity involving feedback into previously occupied states copyright 2012 oxford university press

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

the cannon thurston map for punctured surface groups let be the fundamental group of a compact surface group with non empty boundary we suppose that admits a properly discontinuous strictly type preserving action on hyperbolic 3 space such that there is a positive lower bound on the translation lengths of loxodromic elements we describe the cannon thurston map in this case in particular we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3 space where the action on the circle is obtained by taking any finite area complete hyperbolic structure on the surface and lifting to the boundary of hyperbolic 2 space we deduce that the limit set is locally connected hence a dentrite in the singly degenerate case moreover we show that the cannon thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3 manifold for closed surface bundles over the circle this was obtained by cannon and thurston some generalisations and variations have been obtained by minsky mitra alperin dicks porti mcmullen and cannon we deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected

critical heegaard surfaces in this paper we introduce critical surfaces which are described via a 1 complex whose definition is reminiscent of the curve complex our main result is that if the minimal genus common stabilization of a pair of strongly irreducible heegaard splittings of a 3 manifold is not critical then the manifold contains an incompressible surface conversely we also show that if a non haken 3 manifold admits at most one heegaard splitting of each genus then it does not contain a critical heegaard surface in the final section we discuss how this work leads to a natural metric on the space of strongly irreducible heegaard splittings as well as many new and interesting open questions

min max minimal hypersurfaces in non compact manifolds in this work we prove the existence of embedded closed minimal hypersurfaces in non compact manifolds containing a bounded open subset with smooth and strictly mean concave boundary and a natural behavior on the geometry at infinity for doing this we develop a modified min max theory for the area functional following almgren and pitts setting to produce minimal hypersurfaces with intersecting properties in particular we prove that any strictly mean concave region of a compact riemannian manifold without boundary intersects a closed minimal hypersurface

on generalized fractional kinetic equations abstract in a recent paper saxena et al astro phys space sci 282 2002 281 developed solutions of generalized fractional kinetic equations in terms of mittag leffler functions the object of the present paper is to derive the solution of further generalized fractional kinetic equations their relation to fundamental laws of physics is briefly discussed results are obtained in a compact form in terms of generalized mittag leffler functions and a number of representations of these functions which are widely distributed in the literature are compiled for the first time

harmonic maps and 2 cycles realizing the thurston norm let m 3 be an oriented 3 manifold we investigate when one of the fibers or a combination of fiber components f best of a harmonic map f m 3 s 1 with morse type singularities delivers the thurston norm x f best of its homology class f best h 2 m 3 z in particular for a map f with connected fibers and any well positioned oriented surface e c m in the homology class of a fiber we show that the thurston number x satisfies an inequality x x f best f var x f here the variation var x f is can be expressed in terms of the x invariants of the fiber components and the twist f measures the complexity of the intersection of with a particular set f r of bad fiber components this complexity is tightly linked with the optimal f height of e being lifted to the f induced cyclic coverm 3 m 3 based on these invariants for any morse map f we introduce the notion of its twist x f we prove that for a harmonic f x f best x f best if and only if x f 0

a characterization of virtually embedded subsurfaces in 3 manifolds this paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3 manifold which generalizes an invariant due to rubinstein and wang the subsurface is virtually embedded if and only if the almost fiber part is aspiral and in this case the subsurface is virtually a leaf of a taut foliation besides other consequences examples are exhibited that nongeometric 3 manifolds with no seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

norms on the cohomology of hyperbolic 3 manifolds we study the relationship between two norms on the first cohomology of a hyperbolic 3 manifold the purely topological thurston norm and the more geometric harmonic norm refining recent results of bergeron engun and venkatesh as well as older work of kronheimer and mrowka we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3 manifold itself moreover we give families of examples showing that some but not all qualitative aspects of our estimates are sharp finally we exhibit closed hyperbolic 3 manifolds where the thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

an unknotting result for complete minimal surfaces r3 in this note we prove theorem if s is a complete properly embedded minimal surface in r 3 with the euclidean metric and if a is a closed connected complementary domain of z then either a is metrically r 2 0 hi or up to diffeomorphism a is made from a countable or finite collection of building blocks by attaching a proper collection of closed 1 handles each building blocks is a closed 3 ball or an open annulus x 0 1 the first theorem of this kind was b lawson s i l unknottedness theorem for minimal surfaces in s 3 that result was obtained by considering the second variation of arc length because the flat euclidean metric of r 3 does not contribute a favorable curvature term as found in s 3 a dual approach employing stable minimal surfaces dual to lawson s spanning arcs is required the key technical result proved by an analysis of the second variation operator on surfaces is that geometric planes are the only stable minimal surfaces in r 3 c p and f c s from this starting point a more topological theory has been developed by frohman hoffman meeks yau and others our argument is based to a large degree on the methods used by c frohman and w meeks f m to classify 1 ended proper complete minimal imbeddings in r 3 we present four succinct topological assertions about any 3 manifold with boundary a s which is a connected closed complementary region of a complete proper minimal surface in r 3 these statements should be thought of as axioms for this emerging theory deduced from geometry they help to mark the boundary with topology the theorem follows quickly from the axioms axiom 1 occurs explicitly as corollary 3 2 i f m axiom 2 is an immediate consequence of the methods of f m and axiom 3 is proved by an extension of these methods axiom 4 is related to an idea on the ordering of ends m y due to w meeks and s t yau these are the axioms their verification actually only requires that the mean curvature vector of 2 be everywhere zero or pointing towards a

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

optimal convergence rates for semidiscrete finite element approximations of linear space fractional partial differential equations under minimal regularity assumptions abstract we consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by jin et al 2015 and ervin et al 2018 the error estimates are proved by using two approaches one approach is to apply the duality argument in johnson 1987 for the heat equation to consider the error estimates for the linear space fractional partial differential equations this argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution another approach is to use the approximate solution operators of the corresponding fractional elliptic problems this argument can be extended to consider more general linear space fractional partial differential equations numerical examples are given to show that the numerical results are consistent with the theoretical results

stable minimal surfaces in m ℝ in 4 we developed the theory of properly embedded minimal surfaces in m r where m is a compact riemannian surface one of the first results in 4 is that a properly embedded noncompact minimal surface in m r of bounded gaussian curvature is quasiperiodic in the following sense given any sequence of vertical translates n of a subsequence of the n converges on compact subsets of m r to another properly embedded minimal surface by the curvature estimates of schoen 8 every properly embedded stable minimal surface in m r has bounded curvature therefore every properly embedded noncompact stable minimal surface in m r is quasiperiodic this quasiperiodicity property will be essential in proving the next theorem throughout this paper m denotes a compact riemannian surface