hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

approximate travelling waves for generalized kpp equations and classical mechanics we consider the existence of approximate travelling waves of generalized kpp equations in which the initial distribution can depend on a small parameter which in the limit 0 is the sum of some functions or a step function using the method of elworthy truman 1982 we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears by the feynman kac formula and the maruyama girsanov cameron martin transformation we obtain an identity from which with a late caustic assumption we see the propagation of the global wave front and the shape of the trough our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave finally we prove a huygens principle for kpp equations on complete riemannian manifolds without cut locus with some bounds on their volume element in particular cartan hadamard manifolds

duality in interacting particle systems and boson representation in the context of markov processes we show a new scheme to derive dual processes and a duality function based on a boson representation this scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators for some stochastic processes duality relations have been known which connect continuous time markov processes with discrete state space and those with continuous state space we clarify that using a generating function approach and the doi peliti method a birth death process or discrete random walk model is naturally connected to a differential equation with continuous variables which would be interpreted as a dual markov process the key point in the derivation is to use bosonic coherent states as a bra state instead of a conventional projection state as examples we apply the scheme to a simple birth coagulation process and a brownian momentum process the generator of the brownian momentum process is written by elements of the su 1 1 algebra and using a boson realization of su 1 1 we show that the same scheme is available

ricci measure for some singular riemannian metrics we define the ricci curvature as a measure for certain singular torsion free connections on the tangent bundle of a manifold the definition uses an integral formula and vector valued half densities we give relevant examples in which the ricci measure can be computed in the time dependent setting we give a weak notion of a ricci flow solution on a manifold

remarks on non compact gradient ricci solitons in this paper we show how techniques coming from stochastic analysis such as stochastic completeness in the form of the weak maximum principle at infinity parabolicity and lp liouville type results for the weighted laplacian associated to the potential may be used to obtain triviality rigidity results and scalar curvature estimates for gradient ricci solitons under lp conditions on the relevant quantities

stable and unstable einstein warped products in this article we systematically investigate the stability properties of certain warped product einstein manifolds we characterize stability of these metrics in terms of an eigenvalue condition of the einstein operator on the base manifold in particular we prove that all complete manifolds carrying imaginary killing spinors are strictly stable moreover we show that ricci flat and hyperbolic cones over kahler einstein fano manifolds and over nonnegatively curved einstein manifolds are stable if the cone has dimension n 10 on the other hand we construct unstable einstein metrics of negative scalar curvature

uniform convergence of martingales in the branching random walk in a discrete time supercritical branching random walk let z n be the point process formed by the n th generation let m lambda be the laplace transform of the intensity measure of z 1 then w n lambda int e lambda x z n dx m lambda n which is the laplace transform of z n normalized by its expected value forms a martingale for any lambda with m lambda finite but nonzero the convergence of these martingales uniformly in lambda for lambda lying in a suitable set is the first main result of this paper this will imply that on that set the martingale limit w lambda is actually an analytic function of lambda the uniform convergence results are used to obtain extensions of known results on the growth of z n nc d with n for bounded intervals d and fixed c this forms the second part of the paper where local large deviation results for z n which are uniform in c are considered finally similar results both on martingale convergence and uniform local large deviations are also obtained for continuous time models including branching brownian motion

rotational symmetry of conical kahler ricci solitons we show that expanding kahler ricci solitons which have positive holomorphic bisectional curvature and are c 2 asymptotic to a conical kahler manifold at infinity must be the u n rotationally symmetric expanding solitons constructed by cao

extended duality relations between birth death processes and partial differential equations duality relations between continuous state and discrete state stochastic processes with continuous time have already been studied and used in various research fields we propose extended duality relations which enable us to derive discrete state stochastic processes from arbitrary diffusion type partial differential equations the derivation is based on the doi peliti formalism and it will be clarified that additional states for the discrete state stochastic processes must be considered in some cases

the existence of bounded solutions of a semilinear elliptic equation on etudie l existence de la solution non constante bornee de l equation differentielle w y w 2 w p 1 w w p 1 0 dans r n n2 avec pp c ou p c n 2 n 2 est l exposant de l espace de sobolev critique pour r n

the angle criterion we complete the proof of the angle conjecture on when a pair of orientedm planes is area minimizing the nonzero sum oriented union p1 p2 is area minimizing if and only if the characterizing angles between them satisfy the inequality beta m leqq beta 1 beta m 1

finite and infinite systems of interacting diffusions we study the problem of relating the long time behavior of finite and infinite systems of locally interacting components we consider in detail a class of lincarly interacting diffusionsx t x i t i d in the regime where there is a one parameter family of nontrivial invariant measures for these systems there are naturally defined corresponding finite systems x n t left x i n t i in lambda n right with lambda n n n d cap mathbb z d our main result gives a comparison between the laws ofx t n andx n t n for timest n asn the comparison involves certain mixtures of the invariant measures for the infinite system

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

local monotonicity and mean value formulas for evolving riemannian manifolds we derive identities for general flows of riemannian metrics that may be regarded as local mean value monotonicity or lyapunov formulae these generalize pre vious work of the first author for mean curvature flow and other nonlinear diusions our results apply in particular to ricci flow where they yield a local monotone quantity di rectly analogous to perelman s reduced volume v and a local identity related to perelman s average energy f

non singular solutions to the normalized ricci flow equation in this paper we study non singular solutions to ricci flow on a closed manifold of dimension at least 4 amongst other things we prove that if m is a closed 4 manifold on which the normalized ricci flow exists for all time t 0 with uniformly bounded sectional curvature then the euler characteristic chi m ge 0 moreover the 4 manifold satisfies one of the followings i m is a shrinking ricci soliton ii m admits a positive rank f structure iii the hitchin thorpe type inequality holds 2 chi m ge 3 tau m where chi m resp tau m is the euler characteristic resp signature of m

on friedrichs poincare type inequalities friedrichs and poincare type inequalities are important and widely used in the area of partial differential equations and numerical analysis most of their proofs appearing in references are the argument of reduction to absurdity in this paper we give direct proofs of friedrichs type inequalities in h 1 and poincare type inequalities in some subspaces of w 1 p the dependencies of the inequality coefficients on the domain and some sub domains are illustrated explicitly

rigidity of asymptotically conical shrinking gradient ricci solitons we show that if two gradient shrinking ricci solitons are asymptotic along some end of each to the same regular cone 0 dr rg then the soliton metrics must be isometric on some neighborhoods of infinity of these ends our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness as an application we prove that the only complete connected gradient shrinking ricci soliton asymptotic to a rotationally symmetric cone is the gaussian soliton on r

a remark on compact ricci solitons it is shown that a complete shrinking soliton is compact if and only if it is bounded and moreover in such a case it has finite fundamental group

hamiltonian stationary shrinkers and expanders for lagrangian mean curvature flows we construct examples of shrinkers and expanders for lagrangian mean curvature flows these examples are hamiltonian stationary and asymptotic to the union of two hamiltonian stationary cones found by schoen and wolfson in sw the schoen wolfson cones cp q are obstructions to the existence problems of special lagrangians or lagrangian minimal surfaces in the variational approach it is known that these cone singularities cannot be resolved by any smooth oriented lagrangian submanifolds the shrinkers and expanders that we found can be glued together to yield solutions of the brakke motiona weak formulation of the mean curvature flow for any coprime pair p q other than 2 1 we construct such a solution that resolves any single schoen wolfson cone cp q this thus provides an evidence to schoen wolfson s conjecture that the 2 1 cone is the only area minimizing cone higher dimensional generalizations are also obtained

the kahler ricci flow on fano bundles we study the behavior of the kahler ricci flow on some fano bundles which is a trivial bundle on one zariski open set we show that if the fiber is mathbb p m blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kahler class then the fibers collapse in finite time and the metrics converge sub sequentially in gromov hausdorff sense to a metric on the base

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

branching brownian motion seen from its tip it has been conjectured since the work of lalley and sellke ann probab 15 1052 1061 1987 that branching brownian motion seen from its tip e g from its rightmost particle converges to an invariant point process very recently it emerged that this can be proved in several different ways see e g brunet and derrida a branching random walk seen from the tip 2010 poissonian statistics in the extremal process of branching brownian motion 2010 arguin et al the extremal process of branching brownian motion 2011 the structure of this extremal point process turns out to be a poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process the main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process another proof and description has been obtained independently by arguin et al the extremal process of branching brownian motion 2011

on the blow up of four dimensional ricci flow singularities in this paper we prove a conjecture by feldman ilmanen knopf in 14 that the gradient shrinking soliton metric they constructed on the tautological line bundle over cp 1 is the uniform limit of blow ups of a type i ricci ow singularity on a closed manifold we use this result to show that limits of blow ups of ricci ow singularities on closed four dimensional manifolds do not necessarily have non negative ricci curvature

expanding solitons with non negative curvature operator coming out of cones we consider ricci flow of complete riemannian manifolds which have bounded non negative curvature operator non zero asymptotic volume ratio and no boundary we prove scale invariant estimates for these solutions using these estimates we show that there is a limit solution obtained by scaling down this solution at a fixed point in space this limit solution is an expanding soliton coming out of the asymptotic cone at infinity

metric contraction of the cone divisor by the conical kahler ricci flow we use the momentum construction of calabi to study the conical kahler ricci flow on hirzebruch surfaces with cone angle along the exceptional curve and show that either the flow gromov hausdorff converges to the riemann sphere or a single point in finite time or the flow contracts only the cone divisor to a single point and gromov hausdorff converges to a two dimensional projective orbifold the limiting behaviour depends only on the cone angle numerical properties of the initial kahler class and the degree of the hirzebruch surface this gives the first example of the conical kahler ricci flow contracting the cone divisor to a single point and shows that the conical flow may contract curves of self intersection less than 1 as opposed to the smooth kahler ricci flow at the end we introduce a conjectural picture of the geometry of finite time non collapsing singularities of the flow on kahler surfaces in general

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

the ergodicity of a class of reversible reaction diffusion processes we build on recent results of durrett ding and liggett to establish ergodicity in a class of reversible reaction diffusion processes

fractional derivative estimates in gevrey spaces global regularity and decay for solutions to semilinear equations in rn abstract we propose a unified functional analytic approach to study the uniform analytic gevrey regularity and the decay of solutions to semilinear elliptic equations on r n first we develop a fractional calculus for nonlinear maps in banach spaces of l p based gevrey functions 1 h p s r n regularity with s scr 0 depending on the nonlinearity next we investigate the type of decay polynomial or exponential of the derivatives of solutions to semilinear elliptic equations provided they decay a priori slowly as o x x for some small 0 the restrictions involved in our results are optimal in particular given a hyperplane l we construct 2d 2 strongly singular solutions locally in h p s r n for s

genealogy of extremal particles of branching brownian motion branching brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism by virtue of the groundbreaking work by m bramson on the convergence of solutions of the fisher kpp equation to traveling waves the law of the rightmost particle in the limit of large times is rather well understood in this work we address the full statistics of the extremal particles first second third largest etc in particular we prove that in the large t limit such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0 or within a distance of order 1 from time t the approach relies on characterizing up to a certain level of precision the paths of the extremal particles as a byproduct a heuristic picture of branching brownian motion at the edge emerges which sheds light on the still unknown limiting extremal process 2011 wiley periodicals inc

on travelling waves for the stochastic fisher kolmogorov petrovsky piscunov equation this paper is concerned with properties of the wave speed for the stochastically perturbed fisher kolmogorov petrovsky piscunov fkpp equation it was shown in the classical 1937 paper by kolmogorov petrovsky and piscunov that the large time behavior of the solution to the fkpp equation with heaviside initial data is a travelling wave in a seminal 1995 paper mueller and sowers proved that this also holds for a stochastically perturbed fkpp equation the wave speed depends on the strength of the noise in this paper bounds on the asymptotic behavior of the wave speed c as 0 and are obtained

some cheeger gromov taylor type compactness theorems via m bakry emery and m modified ricci curvatures abstract we establish some new compactness criteria for complete riemannian manifolds via m bakry emery and m modified ricci curvatures in the spirit of cheeger et al 1982 our results improve and generalize previous compactness theorems for complete riemannian manifolds via m bakry emery and m modified ricci curvatures obtained by fernandez lopez and garcia rio 2008 limoncu 2010 2012 qian 1997 wei and wylie 2009 and wu 2018

on dualities for ssep and asep with open boundary conditions duality relations for simple exclusion processes with general open boundaries are discussed it is shown that a combination of spin operators and bosonic operators enables us to have a unified discussion about duality relations with open boundaries as for the symmetric simple exclusion process ssep more general results than those from previous studies are obtained it is clarified that not only the absorbing sites but also additional sites called copying sites are needed for the boundaries in the dual process for the ssep the role of the copying sites is to conserve information about the particle states on the boundary sites similar discussions are applied to the asymmetric simple exclusion process asep in which the q analogues are employed and it is clarified that the asep with open boundaries has a complicated dual process on the boundaries

interacting particles the stochastic fisher kolmogorov petrovsky piscounov equation and duality abstract the stochastic fisher kolmogorov petrovsky piscunov equation is t u x t d xx u u 1 u e u 1 u x t for 0 u 1 where x t is a gaussian white noise process in space and time here d and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of ito equations solutions of this stochastic partial differential equation have an exact connection to the a a a reaction diffusion system at appropriate values of the rate coefficients and particles diffusion constant this relationship is called duality by the probabilists it is not via some hydrodynamic description of the interacting particle system in this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic fisher kolmogorov petrovsky piscunov equation

the extremal process of branching brownian motion we prove that the extremal process of branching brownian motion in the limit of large times converges weakly to a cluster point process the limiting process is a randomly shifted poisson cluster process where the positions of the clusters is a poisson process with intensity measure with exponential density the law of the individual clusters is characterized as branching brownian motions conditioned to perform unusually large displacements and its existence is proved the proof combines three main ingredients first the results of bramson on the convergence of solutions of the kolmogorov petrovsky piscounov equation with general initial conditions to standing waves second the integral representations of such waves as first obtained by lalley and sellke in the case of heaviside initial conditions third a proper identification of the tail of the extremal process with an auxiliary process based on the work of chauvin and rouault which fully captures the large time asymptotics of the extremal process the analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses

upper bounds on the first eigenvalue for a diffusion operator via bakry emery ricci curvature let l delta nabla varphi cdot nabla be a symmetric diffusion operator with an invariant measure d mu e varphi dx on a complete riemannian manifold in this paper we prove li yau gradient estimates for weighted elliptic equations on the complete manifold with nabla varphi leq theta and dimensional bakry emery ricci curvature bounded below by some negative constant based on this we give an upper bound on the first eigenvalue of the diffusion operator l on this kind manifold and thereby generalize a cheng s result on the laplacian case math z 143 289 297 1975

martingale convergence and the stopped branching random walk we discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines applying a method of lyons 1997 and lyons pemantle and peres 1995 concerning size biased branching trees we establish a relationship between stopping lines and certain stopping times consequently we develop conditions under which these supermartingales are also martingales further we prove a generalization of biggins martingale convergence theorem biggins 1977a within this context

integral pinched shrinking ricci solitons abstract we prove that a n dimensional 4 n 6 compact gradient shrinking ricci soliton satisfying a l n 2 pinching condition is isometric to a quotient of the round s n the proof relies mainly on sharp algebraic curvature estimates the yamabe sobolev inequality and an improved rigidity result for integral pinched einstein metrics

exact asymptotics of the freezing transition of a logarithmically correlated random energy model we consider a logarithmically correlated random energy model namely a model for directed polymers on a cayley tree which was introduced by derrida and spohn we prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the kpp equation thus translating bramson s work on the kpp equation into a discrete time case we also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

collapsing of the chern ricci flow on elliptic surfaces we investigate the chern ricci flow an evolution equation of hermitian metrics generalizing the kahler ricci flow on elliptic bundles over a riemann surface of genus greater than one we show that starting at any gauduchon metric the flow collapses the elliptic fibers and the metrics converge to the pullback of a kahler einstein metric from the base some of our estimates are new even for the kahler ricci flow a consequence of our result is that on every minimal non kahler surface of kodaira dimension one the chern ricci flow converges in the sense of gromov hausdorff to an orbifold kahler einstein metric on a riemann surface

travelling waves for the fkpp equation via probabilistic arguments we outline a completely probabilistic study of travelling wave solutions of the fkpp reaction diffusion equation that are monotone and connect 0 to 1 the necessary asymptotics of such travelling waves are proved using martingale and brownian motion techniques recalling the connection between the fkpp equation and branching brownian motion through the work of mckean and neveu we show how the necessary asymptotics and results about branching brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed

rate of expansion of an inhomogeneous branching process of brownian particles let x be the b0 qn x branching diffusion where b0is the exp left int limits o t k b s ds right subprocess of bm r1 and qn x is the probability that a particle dying at x produces n offspring q0 q1 0 put m x nqn x we assume qn n 2 m and k are all continuous but m is not necessarily bounded if k x m x 0 as x then we prove that rt t 2 2 1 2 as t a s and in mean of any order where rtis the position of the rightmost particle at time t and 0 is the largest eigenvalue of 1 2 d2 dx2 q q x k x m x 1

the bakry emery ricci tensor and its applications to some compactness theorems let m g be a complete and connected riemannian manifold of dimension n by using the bakry emery ricci curvature tensor on m we prove two theorems which correspond to the myers compactness theorem

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

nonlinear kalman filter based on duality relations between continuous and discrete state stochastic processes an alternative application of duality relations of stochastic processes is demonstrated although conventional usages of the duality relations need analytical solutions for the dual processes here i employ numerical solutions of the dual processes and investigate the usefulness as a demonstration estimation problems of hidden variables in stochastic differential equations are discussed employing algebraic probability theory a little complicated birth death process is derived from the stochastic differential equations and an estimation method based on the ensemble kalman filter is proposed as a result the possibility for making faster computational algorithms based on the duality concepts is shown

exponential and double exponential tails for maximum of two dimensional discrete gaussian free field we study the tail behavior for the maximum of discrete gaussian free field on a 2d box with dirichlet boundary condition after centering by its expectation we show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail in particular our result implies that the variance of the maximum is of order 1 improving an o log n bound by chatterjee chaos concentration and multiple valleys 2008 and confirming a folklore conjecture an important ingredient for our proof is a result of bramson and zeitouni commun pure appl math 2010 who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

stationary states and their stability of the stepping stone model involving mutation and selection on considere un modele de diffusion qui est un processus de diffusion de dimension infinie on etudie la regularite des distributions marginales de dimension finie des etats stationnaires

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

the genealogy of branching brownian motion with absorption we consider a system of particles which perform branching brownian motion with negative drift and are killed upon reaching zero in the near critical regime where the total population stays roughly constant with approximately n particles we show that the characteristic time scale for the evolution of this population is of order logn 3 in the sense that when time is measured in these units the scaled number of particles converges to a variant of neveu s continuous state branching process furthermore the genealogy of the particles is then governed by a coalescent process known as the bolthausen sznitman coalescent this validates the non rigorous predictions by brunet derrida muller and munier for a closely related model

branching coalescing particle systems we study the ergodic behavior of systems of particles performing independent random walks binary splitting coalescence and deaths such particle systems are dual to systems of linearly interacting wright fisher diffusions used to model a population with resampling selection and mutations we use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law

the adaptive finite element method based on multi scale discretizations for eigenvalue problems in this paper adaptive finite element methods for differential operator eigenvalue problems are discussed for multi scale discretization schemes based on rayleigh quotient iteration see scheme 3 in y yang h bi a two grid discretization scheme based on shifted inverse power method siam j numer anal 49 2011 1602 1624 a reliable and efficient a posteriori error indicator is given in addition a new adaptive algorithm based on the multi scale discretizations is proposed and we apply the algorithm to the schrodinger equation for hydrogen atoms the algorithm is performed under the package of chen and satisfactory numerical results are obtained

rigidity of shrinking ricci solitons we show that a compact ricci soliton is rigid if and only if the weyl conformal tensor is harmonic in the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the ricci tensor is bounded from below

application of the virial theorem for improving eigenvalue calculations of multiparticle systems abstract the virial theorem is a fundamental property for multiparticle systems in quantum mechanics and offers an elegant relationship between kinetic and potential energies of quantum states in this paper we study applications of the virial theorem for improving eigenvalue calculations of multiparticle systems we propose the virial steepest descent scheme and the two point linear extrapolation to improve the accuracy of such eigenvalue calculations through post processing eigenvalue data obtained from standard methods mathematical analysis of both methods is presented examples are included to illustrate the efficiency of both methods

scalar v soliton equation and kahler ricci flow on symplectic quotients abstract in this paper we consider the v soliton equation which is a degenerate fully nonlinear equation introduced by la nave and tian in their work on kahler ricci flow on symplectic quotients one can apply the interpretation to study finite time singularities of the kahler ricci flow as in the case of kahler einstein metrics we can also reduce the v soliton equation to a scalar equation on kahler potentials which is of monge ampere type we formulate some preliminary estimates for such a scalar equation on a compact kahler manifold m

finite element approximations to the discrete spectrum of the schrodinger operator with the coulomb potential in the present paper the authors consider the schrodinger operator h with the coulomb potential defined in r3m where m is a positive integer both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed the spectrum of h becomes completely discrete when confined to bounded domains the error estimate of the bounded domain approximation to the discrete spectrum of h is obtained since numerical solution is difficult for a higher dimensional problem of dimension more than three the finite element analyses in this paper are restricted to the s state of the helium atom the authors transform the six dimensional schrodinger equation of the helium s state into a three dimensional form optimal error estimates for the finite element approximation to the three dimensional equation for all eigenvalues and eigenfunctions of the three dimensional equation are obtained by means of local regularization numerical results are shown in the last section

an upper diameter bound for compact ricci solitons with application to the hitchin thorpe inequality in this article stimulated by fernandez lopez and garcia rio we shall give an upper diameter bound for compact shrinking ricci solitons in terms of the range of the scalar curvature as well as in terms of the range of the potential function as an application we shall provide some new sufficient conditions for compact four dimensional shrinking ricci solitons to satisfy the hitchin thorpe inequality