trace expansions for elliptic cone operators with stationary domains we analyze the behavior of the trace of the resolvent of an ellip tic cone differential operator as the spectral parameter tends to infinity the resolvent splits into two components one associated with the minimal exten sion of the operator and another of finite rank depending on the particular choice of domain we give a full asymptotic expansion of the first compo nent and expand the component of finite rank in the case where the domain is stationary the results make use and develop further our previous investi gations on the analytic and geometric structure of the resolvent the analysis of nonstationary domains considerably more intricate is pursued elsewhere

uniqueness and short time regularity of the weak kahler ricci flow let x be a compact kahler manifold we prove that the kahler ricci flow starting from arbitrary closed positive 1 1 currents is smooth outside some analytic subset this regularity result is optimal meaning that the flow has positive lelong numbers for short time if the initial current has we also prove that the flow is unique when starting from currents with zero lelong numbers

eta invariants with spectral boundary conditions we study the asymptotics of the heat trace where p is an operator of dirac type where f is an auxiliary smooth smearing function which is used to localize the problem and where we impose spectral boundary conditions using functorial techniques and special case calculations the boundary part of the leading coefficients in the asymptotic expansion is found

ellipticity conditions for the lax operator of the kp equations the lax pseudo differential operator plays a key role in studying the general set of kadomtsev petviashvili equations although it is normally treated in a formal way without worrying about a complete characterization of its mathematical properties the aim of this paper is therefore to investigate the ellipticity condition for this purpose after a careful evaluation of the kernel with the associated symbol the majorization ensuring ellipticity is studied in detail this leads to non trivial restrictions on the admissible set of potentials in the lax operator when their time evolution is also considered the ellipticity conditions turn out to involve derivatives of the logarithm of the function

boundary value problems for first order elliptic wedge operators we develop an elliptic theory based in l 2 of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum in particular we do not require the indicial roots to be constant along the base of the boundary fibration our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the shapiro lopatinskii condition and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators wedge operators arise in the global analysis on manifolds with incomplete edge singularities our theory settles in the first order case the long standing open problem to develop a robust elliptic theory of boundary value problems for such operators

an algebra of boundary value problems not requiring shapiro lopatinskij conditions abstract we construct an algebra of pseudo differential boundary value problems that contains the classical shapiro lopatinskij elliptic problems as well as all differential elliptic problems of dirac type with aps boundary conditions together with their parametrices global pseudo differential projections on the boundary are used to define ellipticity and to show the fredholm property in suitable scales of spaces

analytic torsion for twisted de rham complexes we dene analytic torsion x e h 2 det h x e h for the twisted de rham complex consisting of the spaces of dieren tial forms on a compact oriented riemannian man ifold x valued in a at vector bundle e with a dieren tial given by r e h where r e is a at connection on e h is an odd degree closed dieren tial form on x and h x e h denotes the cohomology of this z2 graded complex the denition uses pseudo dieren tial operators and residue traces we show that when dim x is odd x e h is independent of the choice of metrics on x and e and of the representative h in the cohomology class h we explain the relation to generalized geometry when h is a 3 form we demonstrate some basic func torial properties when h is a top degree form we compute the torsion dene its simplicial counterpart and prove an analogue of the cheeger mtheorem we also establish the rela tionship of an invariant form of the analytic torsion for t dual circle bundles with 3 form uxes

adjoint ideals and gorenstein blowups in two dimensional regular local rings in this article we investigate when a complete ideal in a two dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter in particular we show that this question is closely connected to the gorenstein property of the blowup along the ideal

analytic approximations of plurisubharmonic singularities we study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual monge ampere masses they are characterized in terms of green functions for demailly s approximations relative types and valuations furthermore the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals

the degrees of plane curves with prescribed log canonical threshold abstract we obtain a bound on the degree of curves with fixed log canonical threshold at very general points on p 2

an acceleration procedure for discrete velocity approximation of the boltzmann collision operator abstract in this paper we investigate a method of realization of the discrete velocity approximation of the boltzmann collision operator studied by palczewski schneider and bobylev for this realization we propose an acceleration procedure which reduces the computational complexity of the method the efficiency of the acceleration procedure is demonstrated through a set of numerical tests which include space homogeneous relaxation problems and space nonhomogeneous problem of shock wave formation

trace expansions and the noncommutative residue for manifolds with boundary for a pseudodifferential boundary operator a of order 2 z and class 0 in the boutet de monvel calculus on a compact n dimensional manifold with boundary we consider the function tr ab s where b is an auxiliary system formed of the dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary we prove that tr ab s has a meromorphic extension to c with poles at the half integers s n j 2 j 2 n possibly double for s 0 and we prove that its residue at 0 equals the noncommutative residue of a as defined by fedosov golse leichtnam and schrohe by a different method to achieve this we establish a full asymptotic expansion of tr a b k in powersl 2 and log powersl 2 log where the noncommutative residue equals the coefficient of the highest order log power there is a related expansion of tr ae tb the paper will appear in journal reine angew math crelle s journal then is a holomorphic function of s for large re s we show that it extends to a mero morphic function on the whole complex plane with at most double poles moreover we prove that the noncommutative residue res a of a can be recovered as a residue in this

the ζ determinant and the additivity of the η invariant on the smooth self adjoint grassmannian in this paper we discuss the existence of the determinant of a dirac operator on a compact manifold with boundary we show that the determinant is well defined in the case of the operator with a domain determined by a boundary condition from the smooth self adjoint grassmannian discussed in the papers 5 13 29 we prove a generalization of a pasting formula for the invariant see 34 the results of the paper are used in the recent proof of the projective equality of the determinant and quillen determinant on see 30 31

weyl asymptotics of bisingular operators and dirichlet divisor problem we consider a class of pseudodifferential operators with crossed vector valued symbols defined on the product of two closed manifolds we study the asymptotic expansion of the counting function of positive selfadjoint operators in this class using a general theorem of aramaki we can determine the first term of the asymptotic expansion of the counting function and in a special case we are able to find the second term we give also some examples emphasizing connections with problems of analytic number theory in particular with dirichlet divisor function

residues of the eta function for an operator of dirac type abstract we compute the asymptotics of tr l 2 pe tp 2 where p is a first order operator of dirac type this is equivalent to evaluating the residues of the eta function

a sharp effectiveness result of demailly s strong openness conjecture abstract in this article we establish a sharp effectiveness result of demailly s strong openness conjecture we also establish a sharp effectiveness result related to a conjecture posed by demailly and kollar

smeared heat kernel coefficients on the ball and generalized cone we consider smeared zeta functions and heat kernel coefficients on the bounded generalized cone in arbitrary dimensions the specific case of a ball is analyzed in detail and used to restrict the form of the heat kernel coefficients an on smooth manifolds with boundary supplemented by conformal transformation techniques it is used to provide an effective scheme for the calculation of the an as an application the complete a5 2 coefficient is given

high energy limits of laplace type and dirac type eigenfunctions and frame flows we relate high energy limits of laplace type and dirac type operators to frame flows on the corresponding manifolds and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators observables for the corresponding quantum systems are matrix valued pseudodifferential operators and therefore the system remains non commutative in the high energy limit we discuss to what extent the space of stationary high energy states behaves classically

measure valued solutions to conservation laws motivated by traffic modelling motivated by a traffic model we study a conservation law whose solutions naturally need to be bounded radon measures we investigate the relations between this equation and finite systems of conservation laws that converge to it

adiabatic limits of the η invariants the odd dimensional atiyah patodi singer problem we study the invariant of boundary value problems of atiyah patodi singer type we prove the formula for the spectral flow of the families overs1 assuming a product structure in a collar neighbourhood of the boundary we show that the invariant behaves the same way as on a closed manifold we also study the adiabatic limit of the spectral invariants in nice cases we are able to split them into a contribution from the interior one from the cylinder and an error term then we show that the error term disappears with the increasing length of the cylinder

multiplier ideal sheaves associated with weights of log canonical threshold one abstract in this article we will characterize the multiplier ideal sheaves associated with weights of log canonical threshold one by restricting the weights to complex regular surface

spectral functions for the schrodinger operator on r with a singular potential in this article we analyze the spectral zeta function the heat kernel and the resolvent of the operator d2 dr2 r2 r2 over the interval 0 for 1 4 depending on the self adjoint extension chosen nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed in particular for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s k k n0 do occur this implies that the small t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk ln t l 1 for k l n0 the corresponding statements for the resolvent are also shown furthermore we evaluate the zeta determinant of the operator for all values of and any self adjoint extension

fast numerical method for the boltzmann equation on non uniform grids we introduce a new fast numerical method for computing discontinuous solutions to the boltzmann equation and illustrate it by numerical examples a combination of adaptive grids for approximation of the distribution function and an approximate fast fourier transform on non uniform grids for computing smooth terms in the boltzmann collision integral is used

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

elliptic boundary problems on manifolds with polycylindrical ends abstract we investigate general shapiro lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends this is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension and we then deal with boundary value problems for cusp differential operators we introduce an adapted boutet de monvel s calculus of pseudodifferential boundary value problems and construct parametrices for elliptic cusp operators within this calculus fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

convergence of discrete velocity schemes for the boltzmann equation in this paper we prove the convergence of two discrete velocity deterministic schemes for the boltzmann equation namely buet s scheme and a new finite volume scheme that we introduce here we write the discretized equation in the form of a boltzmann continuous equation in order to be in the framework of the diperna lions theory of renormalized solutions in order to prove convergence we have to overcome two difficulties the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero

on the gluing problem for the spectral invariants of dirac operators abstract in this paper we solve the gluing problem for the determinant of a dirac laplacian to do so we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems the analysis of the resolvent of the dirac operator and the introduction of an auxiliary model problem moreover as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis

on the index of geometrical operators for riemannian manifolds with boundary abstract we use heat equation methods and invariance theory to compute the index of the classical elliptic complexes with pseudo differential boundary conditions of atiyah patodi singer type where the structure are not product near the boundary

a weakly polyhomogeneous calculus for pseudodifferential boundary problems abstract in a joint work with r seeley a calculus of weakly parametric pseudodifferential operators on closed manifolds was introduced and used to obtain complete asymptotic expansions of traces of resolvents and heat operators associated with the atiyah patodi singer problem the present paper establishes a generalization to pseudodifferential boundary operators defining weakly polyhomogeneous singular green operators poisson operators and trace operators associated with a manifold with boundary as well as a suitable transmission condition for pseudodifferential operators full composition formulas are established for the calculus which contains the resolvents of aps type problems the operators in the calculus have complete asymptotic trace expansions in the parameter when of trace class with polynomial and logarithmic terms

thresholds valuations and k stability abstract let x be a normal complex projective variety with at worst klt singularities and l a big line bundle on x we use valuations to study the log canonical threshold of l as well as another invariant the stability threshold the latter generalizes a notion by fujita and odaka and can be used to characterize when a q fano variety is k semistable or uniformly k stable it can also be used to generalize volume bounds due to fujita and liu the two thresholds can be written as infima of certain functionals on the space of valuations on x when l is ample we prove that these infima are attained in the toric case toric valuations achieve these infima and we obtain simple expressions for the two thresholds in terms of the moment polytope of l

zeta forms and the local family index theorem for a family f of elliptic pseudodifferential operators we show there is a natural zeta form f s and zeta determinant form dot f in the ring of smooth differential forms on the parameterizing manifold generalizing the classical single operator zeta function and zeta determinant we show that the zeta forms extend the atiyah bott seeley formula for the index of an elliptic operator to a family of elliptic operators while the zeta determinant form leads to a graded chern class form for the index bundle globally the zeta form and zeta determinant form exist only at the level of k theory as maps to cohomology

aps boundary conditions eta invariants and adiabatic limits we prove an adiabatic limit formula for the eta invariant of a manifold with boundary the eta invariant is defined using the atiyah patodisinger boundary condition and the underlying manifold is fibered over a manifold with boundary our result extends the work of bismut cheeger to manifolds with boundary

local syzygies of multiplier ideals the purpose of this note is to prove an elementary but somewhat surprising result concerning the syzygies of multiplier ideals it follows in particular that in dimensions three or higher multiplier ideals are very special among all integrally closed ideals let x be a smooth complex algebraic variety of dimension d and let b ox be an ideal sheaf given a rational or real number c 0 one can construct the multiplier ideal

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

valuations and multiplier ideals this article is the third of a series of work on a new approach to the study of singularities of various objects in a local two dimensional setting our focus in the present paper is on multiplier ideals and singularity exponents in the discussion below we fix an equicharacteristic zero two dimensional regular local ring r m with algebraically closed residue field an important example is the ring r do o holomorphic germs at the origin in c2 in fj1 we introduced the space v consisting of all r u 00 valued valuations on r centered at m and normalized by m 1 this space is naturally a tree it is a union of uncountably many real segments patched together in such a way that v remains homotopic to a point it is also an r tree in the classical sense for a natural metric we hence call v the valuative tree it encodes in a natural way all possible blowups of r centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects the points in v that are not ends form the subtree vqm of quasimonomial valuations these valuations which can alternatively be characterized as abhyankar valuations of rank 1 or as

valuative analysis of planar plurisubharmonic functions we show that valuations on the ring r of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions giving rise to generalized lelong numbers in the sense of demailly any plurisubharmonic function thus defines a real valued function on the set mathcal v of valuations on r and by way of a natural laplace operator defined in terms of the tree structure on mathcal v a positive measure on mathcal v this measure contains a great deal of information on the singularity at the origin under mild regularity assumptions it yields an exact formula for the mixed monge ampere mass of two plurisubharmonic functions as a consequence any generalized lelong number can be interpreted as an average of valuations using our machinery we also show that the singularity of any positive closed 1 1 current t can be attenuated in the following sense there exists a finite composition of blowups such that the pull back of t decomposes into two parts the first associated to a divisor with normal crossing support the second having small lelong numbers

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

krein like extensions and the lower boundedness problem for elliptic operators abstract for selfadjoint extensions a of a symmetric densely defined positive operator a min the lower boundedness problem is the question of whether a is lower bounded if and only if an associated operator t in abstract boundary spaces is lower bounded it holds when the friedrichs extension a has compact inverse grubb 1974 also gorbachuk and mikhailets 1976 this applies to elliptic operators a on bounded domains for exterior domains a 1 is not compact and whereas the lower bounds satisfy m t m a the implication of lower boundedness from t to a has only been known when m t m a we now show it for general t the operator a a corresponding to t a i generalizing the krein von neumann extension a 0 appears here its possible lower boundedness for all real a is decisive we study this krein like extension showing for bounded domains that the discrete eigenvalues satisfy n t a a c a t n 2 m o t n 1 e 2 m for t

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

effectiveness of demailly s strong openness conjecture and related problems in this article stimulated by the effectiveness in berndtsson s solution of the openness conjecture and continuing our solution of demailly s strong openness conjecture we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result we explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier which is implicitly contained in guan and zhou 2014 arxiv 1401 7158 we also obtain optimal effectiveness of the conjectures of demailly kollar and jonsson mustat respectively

refined analytic torsion given an acyclic representation alpha of the fundamental group of a compact oriented odd dimensional manifold which is close enough to an acyclic unitary representation we define a refinement t alpha of the ray singer torsion associated to alpha which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by turaev t alpha is equal to the graded determinant of the odd signature operator up to a correction term the metric anomaly needed to make it independent of the choice of the riemannian metric t alpha is a holomorphic function on the space of such representations of the fundamental group when alpha is a unitary representation the absolute value of t alpha is equal to the ray singer torsion and the phase of t alpha is proportional to the eta invariant of the odd signature operator the fact that the ray singer torsion and the eta invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants in particular using these methods we compute the quotient of the refined analytic torsion and turaev s refinement of the combinatorial torsion generalizing in this way the classical cheeger m uller theorem as an application we extend and improve a result of farber about the relationship between the farber turaev absolute torsion and the eta invariant as part of our construction of t alpha we prove several new results about determinants and eta invariants of non self adjoint elliptic operators

the hybrid spectral problem and robin boundary conditions the hybrid spectral problem where the field satisfies dirichlet conditions d on part of the boundary of the relevant domain and neumann n on the remainder is discussed in simple terms a conjecture for the c1 coefficient is presented and the conformal determinant on a 2 disc where the d and n regions are semi circles is derived comments on higher coefficients are made a separable second order hemisphere hybrid problem is introduced that involves robin boundary conditions and leads to logarithmic terms in the heat kernel expansion which are evaluated explicitly

parameter dependence of the bergman kernels abstract let t 1 t 1 be a family of bounded pseudoconvex domains and t p s h t let k t z w denote the bergman kernel with weight t on t we study the continuity and holder continuity of k t z w in t several applications to singularity theory of psh functions are given including a new proof of the openness 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

depths of multiplier ideals with lengths of constancy zero abstract in this note we study the behavior of depths of multiplier ideals under restriction when the lengths of constancy are zero

η forms and determinant lines abstract we construct a canonical zeta function quillen connection on the determinant line bundle for a family of aps elliptic boundary value problems and show that its curvature is the 2 form component of a relative eta form

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

the poincare series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface abstract for a simple complete ideal of a local ring at a closed point on a smooth complex algebraic surface we introduce an algebraic object named poincare series p that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to this paper is devoted to prove that p is a rational function giving an explicit expression for it

when does the subadditivity theorem for multiplier ideals hold demailly ein and lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals we prove that in the two dimensional case the subadditivity theorem holds on log terminal singularities however in the higher dimensional case we have several counterexamples we consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three dimensional toric ring

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

the structure of the resolvent of elliptic pseudodifferential operators abstract we show that the resolvent kernel of an elliptic b pseudodifferential operator on a compact manifold with corners of arbitrary codimension is a polyhomogeneous or classical function on a certain manifold with corners the singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity explicit descriptions of the expansions including logarithmic terms are given in particular the asymptotics of the resolvent restricted to the diagonal follows as a corollary applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given

a proof of demailly s strong openness conjecture in this article we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function which was posed by demailly the multiplier ideal sheaf associated to a plurisubharmonic function which is an invariant of the singularities of the psh function plays an important role in several complex variables and complex geometry various properties about the multiplier ideal sheaves associated to plurisubharmonic functions have been discussed e g see 25 7 30 31 demailly s strong openness conjecture means that the strong openness property about the multiplier ideal sheaf holds in the present article we establish such a useful strong openness property on the multiplier ideal sheaf associated to any plurisubharmonic function i e demailly s strong openness conjecture is true 1 1 organization of the paper the paper is organized as follows in the present section we recall the statement of the strong openness conjecture posed by demailly and present the main result of the present paper a solution of the strong openness conjecture in section 2 we recall or give some preliminary lemmas used in the proof of the main result in section 3 we give the proof of the strong openness conjecture and present some consequences by combining the conjecture with some well known results

depths of multiplier ideals and integral closure in this note we study how the depths of multiplier ideals behave under restriction we also study possible values of the depths of multiplier ideals in the filtrations induced from maximal ideal sheaves we then use it to give a sufficient condition for the integral closedness of the product of a multiplier ideal and a power of maximal ideal sheaf in the spirit of huneke

toric plurisubharmonic functions and analytic adjoint ideal sheaves in the first part of this paper we study the properties of some particular plurisubharmonic functions namely the toric ones the main result of this part is a precise description of their multiplier ideal sheaves which generalizes the algebraic case studied by howald in the second part almost entirely independent of the first one we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting this enables us to give an analogue of howald s theorem for adjoint ideals attached to monomial ideals finally using the local ohsawa takegoshi manivel theorem we prove the existence of the so called generalized adjunction exact sequence which enables us to recover a weak version of the global extension theorem of manivel for compact kahler manifolds

filtrations and local syzygies of multiplier ideals abstract recently robert lazarsfeld and kyungyong lee proved an interesting result on local syzygies of multiplier ideals in this note we continue their investigation by showing an asymptotic behavior of local syzygies of multiplier ideals in filtrations induced by maximal ideal sheaves

poles of zeta and eta functions for perturbations of the atiyah patodi singer problem the zeta and eta functions of a differential operator of dirac type on a compact n dimensional manifold provided with a well posed pseudodifferential boundary condition have been shown in g99 to be meromorphic on with simple or double poles on the real axis extending results from g99 we show how perturbations of the boundary condition of order j affect the poles in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when j n this applies to perturbations of spectral boundary conditions also when the structure is non product and the problem is non selfadjoint

the local and global parts of the basic zeta coefficient for operators on manifolds with boundary for operators on a compact manifold x with boundary x the basic zeta coefficient c0 b p1 t is the regular value at s 0 of the zeta function tr b p 1 t s where b p g is a pseudodifferential boundary operator in the boutet de monvel calculus for example the solution operator of a classical elliptic problem and p1 t is a realization of an elliptic differential operator p1 having a ray free of eigenvalues relative formulas e g for the difference between the constants with two different choices of p1 t have been known for some time and are local we here determine c0 b p1 t itself with even order p1 showing how it is put together of local residue type integrals generalizing the noncommutative residues of wodzicki guillemin fedosov golse leichtnam schrohe and global canonical trace type integrals generalizing the canonical trace of kontsevich and vishik formed of hadamard finite parts our formula generalizes a formula shown recently by paycha and scott for manifolds without boundary it leads in particular to new definitions of noncommutative residues of expressions involving log p1 t since the complex powers of p1 t lie far outside the boutet de monvel calculus the standard consideration of holomorphic families is not really useful here instead we have developed a resolvent parametric method where results from our calculus of parameter dependent boundary operators can be used

existence stability and convergence of solutions of discrete velocity models to the boltzmann equation we prove the convergence of finite difference approximations to solutions of the boltzmann equation an essential step is the proof of convergence of discrete approximations to the collision integral this proof relies on our previous results on the consistency of this approximation for the space homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the boltzmann equation in the space dependent case we prove weak convergence to diperna lions solutions

on log canonical thresholds of reducible plane curves l etf x y be an irreducible power series j igusa found a formula for the log canonical threshold of f in terms of the first pair of puiseux exponents in this paper we give an alternate proof for the result using shokurov s adjunction theory and the classification of log terminal pairs the method also gives a generalization of igusa s result to the case of reducible curves

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results