analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

gaps in the differential forms spectrum on cyclic coverings we are interested in the spectrum of the hodge de rham operator on a mathbb z covering x over a compact manifold m of dimension n 1 let be a hypersurface in m which does not disconnect m and such that m is a fundamental domain of the covering if the cohomology group hn 2 is trivial we can construct for each n in mathbb n a metric g gn on m such that the hodge de rham operator on the covering x g has at least n gaps in its essential spectrum if h n 2 sigma ne 0 the same statement holds true for the hodge de rham operators on p forms provided p notin n 2 n 2 1

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities abstract this paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set n d the optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable is lower semicontinuous for the topology of convergence in distribution the multiparameter version of prophet inequality for the one parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality

geometric quantisation of the mic kepler problem the geometric quantisation scheme is applied to the compact kaehler orbit manifolds of the modified kepler problem thus the authors obtain the quantisation of the magnetic charge and energy spectrum of the corresponding quantum problem a new regularisation of the standard kepler problem is presented

error control for the approximation of allen cahn and cahn hilliard equations with a logarithmic potential a fully computable upper bound for the finite element approximation error of allen cahn and cahn hilliard equations with logarithmic potentials is derived numerical experiments show that for the sharp interface limit this bound is robust past topological changes modifications of the abstract results to derive quasi optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold

an energetic variational approach for the cahn hilliard equation with dynamic boundary condition model derivation and mathematical analysis the cahn hilliard equation is a fundamental model that describes phase separation processes of binary mixtures in recent years several types of dynamic boundary conditions have been proposed in order to account for possible short range interactions of the material with the solid wall our first aim in this paper is to propose a new class of dynamic boundary conditions for the cahn hilliard equation in a rather general setting the derivation is based on an energetic variational approach that combines the least action principle and onsager s principle of maximum energy dissipation one feature of our model is that it naturally fulfills three important physical constraints conservation of mass dissipation of energy and force balance relations next we provide a comprehensive analysis of the resulting system of partial differential equations under suitable assumptions we prove the existence and uniqueness of global weak strong solutions to the initial boundary value problem with or without surface diffusion furthermore we establish the uniqueness of the asymptotic limit as t to infty and characterize the stability of local energy minimizers for the system

micz kepler problems in all dimensions the kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance the micz kepler problems are its natural cousins and have been previously generalized from dimension 3 to dimension 5 in this paper we construct and analyze the quantum micz kepler problems in all dimensions higher than 2

bounded imaginary powers of cone differential operators on higher order mellin sobolev spaces and applications to the cahn hilliard equation abstract extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted l p spaces h p 0 b over a manifold with conical singularities we show how the same assumptions also yield the existence of bounded imaginary powers on higher order mellin sobolev spaces h p s b s 0 as an application we consider the cahn hilliard equation on a manifold with possibly warped conical singularities relying on our work for the case of straight cones we first establish r sectoriality and thus maximal regularity for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by clement and li we also obtain the short time asymptotics of the solution near the conical point

asymptotic stability of solutions for 1 d compressible navier stokes cahn hilliard system abstract this paper is concerned with the evolution of the periodic boundary value problem and the mixed boundary value problem for a compressible mixture of binary fluids modeled by the navier stokes cahn hilliard system in one dimensional space the global existence and the large time behavior of the strong solutions for these two systems are studied the solutions are proved to be asymptotically stable even for the large initial disturbance of the density and the large velocity data we show that the average concentration difference for the two components of the initial state determines the long time behavior of the diffusive interface for the two phase flow

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

an l bound for solutions of the cahn hilliard equation in this paper we consider the equation partial t u varepsilon delta u varepsilon sum limits i j 1 n partial ij f ij u varepsilon x varepsilon 2 delta 2 u varepsilon with fij constant when u e is large and we obtain l bounds independent of 0

the classical dynamic symmetry for the sp 1 kepler problems a poisson realization of the simple real lie algebra so 4n on the phase space of each sp 1 kepler problem is exhibited as a consequence one obtains the laplace runge lenz vector for each classical sp 1 kepler problem the verification of these poisson realizations is greatly simplified via an idea of weinstein the totality of these poisson realizations is shown to be equivalent to the canonical poisson realization of so 4n on the poisson manifold t h n sp 1 here h n hn 0 and the hamiltonian action of sp 1 on t h n is induced from the natural right action of sp 1 on h n

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

an index formula for the intersection euler characteristic of an infinite cone the aim of this article is to establish an index formula for the intersection euler characteristic of a cone the main actor is the model witten laplacian on the infinite cone first we study its spectral properties and establish a mckean singer type formula we also give an explicit formula for the zeta function of the model witten laplacian in a second step we apply local index techniques to the model witten laplacian by combining these two steps we express the absolute and relative intersection euler characteristic of the cone as a sum of two terms a term which is local and a second term which is the cheeger invariant

the advantage of using non measurable stop rules comparisons are made between the expected returns using measurable and non measurable stop rules in discrete time stopping problems in the independent case a natural sufficient condition preservation of independence is found for the expected return of every bounded non measurable stopping function to be equal to that of a measurable one and for that of every unbounded non measurable stopping function to be arbitrarily close to that of a measurable one for non negative and for uniformly bounded independent random variables universal sharp bounds are found for the advantage of using non measurable stopping functions over using measurable ones partial results for the dependent case are obtained

analysis of the cahn hilliard equation with a relaxation boundary condition modeling the contact angle dynamics we analyze the cahn hilliard equation with a relaxation boundary condition modeling the evolution of an interface in contact with the solid boundary an l estimate is established which enables us to prove the global existence of the solution we also study the sharp interface limit of the system the dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations

stability of cahn hilliard fronts we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity c 1999 john wiley sons inc

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

convergence of solutions for the fractional cahn hilliard system abstract this paper deals with the cauchy dirichlet problem for the fractional cahn hilliard equation the main results consist of global in time existence of weak solutions characterization of parabolic smoothing effects implying under proper condition eventual boundedness of trajectories and convergence of each solution to a single equilibrium in particular to prove the convergence result a variant of the so called ojasiewicz simon inequality is provided for the fractional dirichlet laplacian and possibly non analytic but c 1 nonlinearities

global existence and asymptotics of solutions of the cahn hilliard equation abstract this paper is concerned with the cauchy problem of the cahn hilliard equation u t u 2 u 0 x r n t 0 u t 0 u 0 x x r n first we construct a local smooth solution u t x to the above cauchy problem then by combining some a priori estimates sobolev s embedding theorem and the continuity argument the local smooth solution u t x is extended step by step to all t 0 provided that the smooth nonlinear function u satisfies a certain local growth condition at some fixed point u r and that u 0 x u l 1 r n is suitably small secondly we show that the global smooth solution u t x satisfies the following temporal decay estimates d k u t x u l p r n c 1 t k 4 n 4 1 1 p t 0 k 0 1 here p 1 c 0 is a constant depending on and 0 is any positive constant which can be chosen sufficiently small at last we show that under a strong assumption on the growth of the nonlinear function u at u u the asymptotics of solutions of the above cauchy problem is described by u 0 t n 4 g x t 4 here 0 r n u 0 x u d x g x r n exp 4 i x d

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

bounded imaginary powers of differential operators on manifolds with conical singularities we study the minimal and maximal closed extension of a differential operator a on a manifold b with conical singularities when a acts as an unbounded operator on weighted l p spaces over b 1 p under suitable ellipticity assumptions we can define a family of complex powers a z z we also obtain sufficient information on the resolvent of a to show the boundedness of the purely imaginary powers examples concern unique solvability and maximal regularity for the solution of the cauchy problem for the laplacian on conical manifolds as well as certain quasilinear diffusion equations

resolvents of cone pseudodifferential operators asymptotic expansions and applications we study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted sobolev spaces over a compact manifold with boundary we obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

analysis and approximation of a fractional cahn hilliard equation we derive a fractional cahn hilliard equation fche by considering a gradient flow in the negative order sobolev space h alpha alpha in 0 1 where the choice alpha 1 corresponds to the classical cahn hilliard equation while the choice alpha 0 recovers the allen cahn equation the existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order alpha and that it indeed reduces the free energy we then turn to the delicate question of the l infty boundedness of the solution and establish an l infty bound for the fche in the case where the nonlinearity is a quartic polynomial as a consequence of the estimates we are able to show that the fourier galerkin method delivers a spectral rate of convergence for the fche in the case of a semidiscrete approximation scheme finally we present results obtained using computational simulation of the fche for a variety of choices of fractional order alpha it is

adjoints of elliptic cone operators we study the adjointness problem for the closed extensions of a general b elliptic operator a x diff m b m e 0 initially defined as an unbounded operator a c c m e x l 2 b m e x l 2 b m e inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the case where a is a symmetric semibounded operator is of particular interest and we give a complete description of the domain of the friedrichs extension of such an operator

a simple proof of stability of fronts for the cahn hilliard equation we apply a method developed in our earlier work on a non local phase kinetics equation to give a simple proof of the non linear stability of fronts for the cahn hilliard equation

regular singular sturm liouville operators and their zeta determinants abstract we consider sturm liouville operators on the line segment 0 1 with general regular singular potentials and separated boundary conditions we establish existence and a formula for the associated zeta determinant in terms of the wronski determinant of a fundamental system of solutions adapted to the boundary conditions this generalizes the earlier work of the first author treating general regular singular potentials but only the dirichlet boundary conditions at the singular end and the recent results by kirsten loya park for general separated boundary conditions but only special regular singular potentials

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

stop rule inequalities for uniformly bounded sequences of random variables if x0 xi is an arbitrarily dependent sequence of random variables taking values in 0 i and if v xa0 x is the supremum over stop rules t of ex then the set of ordered pairs x v x v x0 xa x and v e maxj a 1 for some x0 x is precisely the set c x v x v x i 1 1 ix 0 x and the set of ordered pairs x v x v x0 xi and v e sup x for some x0 xi is precisely the set

resolvents of elliptic cone operators we prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions this is achieved by the construction of a suitable parametrix and reduction to the boundary special attention is devoted to the clarification of the analytic structure of the resolvent

convergence of the cahn hilliard equation to the mullins sekerka problem in spherical symmetry abstract we show that as e 0 the solution of the cahn hilliard equation formula converges to a solution of the mullins sekerka problem formula where denotes a normal v the normal velocity and k the sum of principal curvatures of the interface provided the solutions are radially symmetric we use energy type estimates to show that the solution of the cahn hilliard equation can be approximated by the well known stationary wave solution that corresponds to the potential w

approximate solutions of the cahn hilliard equation via corrections to the mullins sekerka motion we develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the cahn hilliard equation suitable for the study of its sharp interface limit the method is based on the hilbert expansion used in kinetic theory besides its relative simplicity it leads to calculable higher order corrections to the interface motion

prophet inequalities for averages of independent non negative random variables on demontre un theoreme qui affine les resultats de krengel et sucheston 1977 1978 dans lesquels la constante faible 2 1 3 etait obtenue

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

an efficient time adaptivity based on chemical potential for surface cahn hilliard equation using finite element approximation abstract we present numerical simulations for the surface cahn hilliard equation which describes phase separation phenomenon occurred on general surfaces the spatial discretization is based on surface finite element method while the temporal discretization methods are first and second order stabilized semi implicit schemes which guarantee the free energy decay an efficient and parameter free adaptive time stepping strategy is proposed based on the numerical energy stability generated by stabilized semi implicit scheme the main idea is to use discrete chemical potential the byproduct of stabilized semi implicit scheme to estimate the variation of numerical energy that is used as a indicator to update the time step the operation avoid calculating numerical energy with gauss integral in each time step and reduce the calculated cost in addition optimal error estimate of first order stabilized semi implicit scheme in the case of curved surface are provided finally numerical experiments are presented to demonstrate the stability accuracy and efficiency of the proposed algorithms

an l bound for solutions of a fractional cahn hilliard equation abstract recently some fractional cahn hilliard equations are proposed for phase transition and image process etc which have attracted a lot of attention in this paper we concern the l bound of the solutions to a class of fractional cahn hilliard equations which extends the results of integer order by an invariant derivative technique the crucial uniform estimates related to the kernel function of fractional cahn hilliard are established the technique presented here can be also applied to the other related fractional models

a note on the optimal temporal decay estimates of solutions to the cahn hilliard equation abstract this paper is concerned with the optimal temporal decay estimates on the solutions of the cauchy problem of the cahn hilliard equation it is shown in liu wang and zhao 2007 11 that such a cauchy problem admits a unique global smooth solution u t x provided that the smooth nonlinear function u satisfies a local growth condition furthermore if u satisfies a somewhat stronger local growth condition the optimal temporal decay estimates on u t x are also obtained in liu wang and zhao 2007 11 thus a natural question is how to deduce the optimal temporal decay estimates on u t x only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding cauchy problem and the main purpose of this paper is devoted to this problem our analysis is motivated by the technique developed recently in ukai yang and zhao 2006 15 with a slight modification

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

an energy stable linear diffusive crank nicolson scheme for the cahn hilliard gradient flow abstract we propose and analyze a linearly stabilized semi implicit diffusive crank nicolson scheme for the cahn hilliard gradient flow in this scheme the nonlinear bulk force is treated explicitly with two second order stabilization terms this treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation rigorous error analysis is carried out for the fully discrete scheme when the time step size and the space step size are small enough second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of 1 e here e is the thickness of the interface numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

zeta determinants for regular singular laplace type operators we discuss a specific class of regular singular laplace type operators with matrix coefficients their zeta determinants were studied by kirsten loya and park manuscr math 125 95 2008 on the basis of the contour integral method with general boundary conditions at the singularity and dirichlet boundary conditions at the regular boundary we complete the arguments of kirsten loya and park by explicitly verifying that the contour integral method indeed applies in the regular singular setup further we extend the zeta determinant computations to generalized neumann boundary conditions at the regular boundary and apply our results to compute zeta determinants of laplacians on a bounded generalized cone with relative ideal boundary conditions

pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions abstract in this paper we study the pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions by the green function method furthermore we obtain the l p convergence rate of the solution

a statistical version of prophet inequalities all classical prophet inequalities for independent random variables hold also in the case where only a noise corrupted version of those variables is observable that is if the pairs x 1 z 1 x n z n are independent with arbitrary known joint distributions and only the sequence z 1 z n is observable then all prophet inequalities which would hold if the x s were directly observable still hold even though the expected x values i e the payoffs for both the prophet and statistician will be different our model includes for example the case when z i x i y i where the y s are any sequence of independent random variables

multiphase allen cahn and cahn hilliard models and their discretizations with the effect of pairwise surface tensions in this paper the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an n component mixture are studied for the general choice of phase variables the unisolvent property of the coefficient matrix involved in the n phase models based on the pairwise surface tensions is established moreover the symmetric positive definite property of the coefficient matrix on an n 1 n 1 dimensional hyperplane which is of fundamental importance to the well posedness of the models can be proved equivalent to some physical condition for pairwise surface tensions the n phase allen cahn and n phase cahn hilliard equations can then be derived from the free energy functional a natural property is that the resulting dynamics of concentrations are independent of phase variables chosen finite element discretizations for n phase models can be obtained as a natural extension of the existing discretizations for the two phase model the discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

prophet inequalities for finite stage multiparameter optimal stopping problems abstract this paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set n d in the classical optimal stopping problems the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules known as prophet inequalities have been studied by many authors prophet inequalities in the case of finite stage two parameter optimal stopping problems are extended to the case of finite stage general multiparameter optimal stopping problems

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

additive comparisons of stopping values and supremum values for finite stage multiparameter stochastic processes abstract this paper concerns the optimal stopping problem for discrete time multiparameter stochastic processes with the index set n d in the classical optimal stopping problems the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules known as prophet inequalities have been studied by many authors ratio comparisons between these values in the case of multiparameter optimal stopping problems are studied by krengel and sucheston 1981 9 and tanaka 2007 2006 14 15 in this paper an additive comparison in the case of finite stage multiparameter optimal stopping problems is given

asymptotic behavior of solutions to conservation laws with diffusion type terms of regularity gain and regularity loss abstract in this paper we study asymptotic behaviors of solutions to the cauchy problem of nonlinear conservation laws with a diffusion type source term related to an index s r for s 1 and s 1 the diffusion type term takes on a characteristic of regularity gain and regularity loss on the high frequency domain respectively by combining the green function method with the energy method we overcome the weakly dissipative structure of the equation for the case of s 1 and obtain the global existence and optimal l p norm time decay rates of solutions in the case of regularity gain pointwise estimates of solutions are shown by using the refined analysis on the green function

analysis of the viscous cahn hilliard equation in r n abstract solvability of cauchy s problem in r n for an extended viscous cahn hilliard equation is studied the problem is considered first in a standard sobolev space h 1 r n next a notion of the h solution is introduced that is well adapted to the structure of the viscous cahn hilliard equation several properties of the solutions are reported in particular those connected with their asymptotic behavior we collect here also known properties of the unbounded operator 1 in r n that are needed in our considerations

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

numerical approximations for a phase field moving contact line model with variable densities and viscosities we consider the numerical approximations of a two phase hydrodynamics coupled phase field model that incorporates the variable densities viscosities and moving contact line boundary conditions the model is a nonlinear coupled system that consists of incompressible navierstokes equations with the generalized navier boundary condition and the cahnhilliard equations with moving contact line boundary conditions by some subtle explicitimplicit treatments to nonlinear terms we develop two efficient unconditionally energy stable numerical schemes in particular a linear decoupled energy stable scheme for the system with static contact line condition and a nonlinear energy stable scheme for the system with dynamic contact line condition an efficient spectral galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes various numerical results show that the proposed schemes are efficient and accurate

the dirac operator on spaces with conical singularities and positive scalar curvatures on obtient un theoreme d annulation de type bochner et on demontre un theoreme d indice dans le cas singulier on etudie la relation avec des varietes a bord

on the homotopy invariance of higher signatures for manifolds with boundary if m is a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or gromov hyperbolic we show that the higher signatures of m are oriented homotopy invariants we give applications to the question of when higher signatures of closed manifolds are cut and paste invariant

the sp 1 kepler problems let n 2 be a positive integer to each irreducible representation of sp 1 an sp 1 kepler problem in dimension 4n 3 is constructed and analyzed this system is superintegrable and when n 2 it is equivalent to a generalized micz kepler problem in dimension of 5 the dynamical symmetry group of this system is o 4n with the hilbert space of bound states h being the unitary highest weight representation of o 4n with highest weight 1 1 1 which occurs at the rightmost nontrivial reduction point in the enright howe wallach classification diagram for the unitary highest weight modules here is the highest weight of furthermore it is shown that the correspondence h is the theta correspondence for dual pair sp 1 o 4n sp 8n r

ratio comparisons of supremum and stop rule expectations suppose x1 x2 xn are independent non negative random variables with finite positive expectations let t n denote the stop rules for x1 xn the main result of this paper is that e max x1 x n 2 sup ex t tet n the proof given is constructive and sharpens the corresponding weak inequalities of krengel and sucheston and of garling

harmonic analysis on constant curvature surfaces with point singularities let rn2 denote the n fold covering space of the plane with the origin deleted 0 n parametrized by polar coordinates r with periodic of period 2 n and equipped with the flat riemannian metric dr2 r2 d 2 there is a natural harmonic analysis with the fourier transform given by f c 12 0 f r j v r eiv r dr d fourier inversion formula f c 0 f r j v r eiv d dv and plancherel formula f 22 2 0 f v 2 d dv in the case n and analogous expressions with v restricted to the values kn when n is finite this harmonic analysis realizes the joint spectral decompositions of the self adjoint operators i and where is the dirichlet or friedrichs extension of the laplace beltrami operator defined on d this paper gives a detailed development of this harmonic analysis including the effect of basic differentiation and multiplication operators on the fourier transform the theory of l2 sobolev spaces hk and the distinctions between h2 the domain of and the space of functions f with both f and f in l2 in the process it is necessary to study certain generalizations of the hilbert transform that are related to the theory of hankel transforms two applications of this harmonic analysis are given to the study of a natural radon transform or x ray transform on rn2 and to the solution of the heat equation on rn2 the latter also gives information about the heat equation in a sector of the plane with dirichlet or neumann boundary conditions and about brownian motion in the plane taking into account the winding number of a brownian path with respect to the origin analogous problems for spaces of non zero constant curvature are discussed and in this context some higher dimensional spaces arise naturally

local rigidity of 3 dimensional cone manifolds we investigate the local deformation space of 3 dimensional conemanifold structures of constant curvature 1 0 1 and coneangles under this assumption on the cone angles the singular locus will be a trivalent graph in the hyperbolic and the spherical case our main result is a vanishing theorem for the first l 2 cohomology group of the smooth part of the cone manifold with coefficients in the flat bundle of infinitesimal isometries weconclude local rigidity from this in the euclidean case we prove that the first l 2 cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms