analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

local stability of dynamical processes in random media abstract a particular type of random dynamical process is considered in which the stochasticity is introduced through randomly fluctuating parameters a method of local multipliers is developed for treating the local stability of such dynamical processes corresponding to infinite dimensional dynamical systems the method is illustrated by several examples by the random diffusion equation random wave equation and random schrodinger equation the evolution equation for the density matrix of a quasiopen statistical system subject to the action of random surrounding is considered the stationary solutions to this equation are found to be unstable against arbitrary small finite random perturbations the notion of random structural stability is introduced

self similar variational perturbation theory for critical exponents we extend field theoretic variational perturbation theory by self similar approximation theory which greatly accelerates convergence this is illustrated by recalculating the critical exponents of o n symmetric ensuremath phi 4 theory from only three loop perturbation expansions in 4 ensuremath dimensions we obtain analytic results for the exponents which are close to those derived recently from ordinary field theoretic variational perturbational theory to seventh order in particular the specific heat exponent is found to be in good agreement with best measured exponent ensuremath alpha ensuremath approx ensuremath 0 0127 of the specific heat peak in superfluid helium found in a satellite experiment in addition our analytic expressions reproduce also the exactly known large n behavior of the exponents

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

global existence and blow up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions abstract in this paper we consider nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions we establish respectively the conditions on nonlinearities to guarantee that u x t exists globally or blows up at some finite time if blow up occurs we obtain upper and lower bounds of the blow up time

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

the blow up behavior of the heat equation with neumann boundary conditions abstract we establish the blow up rate for the solution of the heat equation u t u xx 0 x t 0 subject to neumann boundary conditions u x 0 t 0 u x 1 t e u 1 t with convex initial data

k3 surfaces with involution equivariant analytic torsion and automorphic forms on the moduli space in this paper we introduce an invariant of a k3 surface with 2 action equipped with a 2 invariant kahler metric which we obtain using the equivariant analytic torsion of the trivial line bundle this invariant is shown to be independent of the choice of the kahler metric it can be viewed as a function on the moduli space of k3 surfaces with involution the main result of this paper is that this function can be identified with an automorphic form which characterizes the discriminant locus in particular we show that the ray singer analytic torsion of the trivial line bundle on an enriques surface with ricci flat kahler metric is given by the value of the norm of the borcherds function at its period point

global and blow up solutions for nonlinear parabolic equations with robin boundary conditions abstract in this paper we discuss the blow up for classical solutions to the following class of parabolic equations with robin boundary condition b u t g u u f u in 0 t u n u 0 on 0 t u x 0 h x 0 in where is a bounded domain of r n n 2 with smooth boundary by constructing some appropriate auxiliary functions and using a first order differential inequality technique we derive conditions on the data which guarantee the blow up or the global existence of the solution for the blow up solution a lower bound on blow up time is also obtained moreover some examples are presented to illustrate the applications

enriques surfaces analytic discriminants and borcherds s φ function in bor 96 borcherds constructed a non vanishing weight 4 modular form on the moduli space of marked polarized enriques surface of degree 2 by considering the twisted denominator function of the fake monster lie algebra associated to an automorphism of order 2 of the leech lattice fixing an 8 dimensional subspace in jt 94 and jt 96 we defined and studied a meromorphic multi valued modular form of weight 2 which we call the k3 analytic discriminant on the moduli space of marked polarized k3 surfaces of degree 2d in certain cases including when where p k are distinct primes our meromorphic form is actually a holomorphic form our construction involves a determinant of the laplacian on a polarized k3 surface with respect to the calabi yau metric together with the l 2 norm of the image of the period map with respect to a properly scaled holomorphic two form since the universal cover of any enriques surface is a k3 surface we can restrict the k3 analytic discriminant to the moduli space of degree 2 enriques surfaces the main result of this paper is the observation that the square of our degree 2 analytic discriminant viewed as a function on the moduli space of degree 2 enriques surfaces is equal to the borcherd s function up to a universal multiplicative constant this result generalizes known results in the study of generalized kac moody algebras and elliptic curves and suggests further connections with higher dimensional calabi yau varieties specifically those which can be realized as complete intersections in some possibly weighted projective space

blow up solutions and global solutions for nonlinear parabolic problems abstract this paper deals with the blow up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in a finite time by a new approach moreover upper estimates of the blow up time blow up rate and global solutions are obtained also

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

prym varieties and the geodesic flow onso n on etudie la relation entre l espace ou le flot apparait et la variete abelienne sur laquelle le flot se linearise

superconnection currents and complex immersions let i m m be an immersion of complex manifolds and let be a complex of holomorphic hermitian vector bundles on m which provides a projective resolution of the sheaf of sections of a holomorphic vector bundle on m we study the convergence as u of a class of superconnection currents u introduced by quillen and we calculate the limit microlocal estimates of the speed of convergence are also given

a necessary and sufficient condition for global existence for a degenerate parabolic boundary value problem consider the degenerate parabolic boundary value problemut u f u on 0 in which is a bounded domain inrnand thec 0 functionsfand are nonnegative and nondecreasing with s f s 0 ifs 0 and 0 0 assume homogeneous neumann boundary conditions and an initial condition that is nonnegative nontrivial and continuous on because the function is not sufficiently nice to allow this problem to have a classical solution we consider generalized solutions in a manner similar to that of benilan crandall and sacks appl math optim 17 1988 203 224 we show that this initial boundary value problem has such a nonnegative generalized solution if and only ifi 0 ds 1 f s

blow up and global existence for nonlinear parabolic equations with neumann boundary conditions in this paper we study the global and blow up solutions of the following problem h u t a u t b x u g t f u in dx 0 t u n 0on dx 0 t u x 0 u 0 x 0in d where d r n is a bounded domain with smooth boundary d by constructing auxiliary functions and using maximum principles and comparison principles the sufficient conditions for the existence of global solution an upper estimate of the global solution the sufficient conditions for the existence of the blow up solution an upper bound for the blow up time and an upper estimate of the blow up rate are specified under some appropriate assumptions on the functions a b f g and h

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

probabilistic approach to pattern selection abstract the problem of pattern selection arises when the evolution equations have many solutions whereas observed patterns constitute a much more restricted set an approach is advanced for treating the problem of pattern selection by defining the probability distribution of patterns then the most probable pattern naturally corresponds to the largest probability weight this approach provides the ordering principle for the multiplicity of solutions explaining why some of them are more preferable than others the approach is applied to solving the problem of turbulent photon filamentation in resonant media

blow up and global solutions for nonlinear reaction diffusion equations with neumann boundary conditions abstract the type of problem under consideration is 1 u ln 1 u t ln 1 u u 1 u ln 1 u in d 0 t u n 0 on d 0 t u x 0 u 0 x 0 in d where d r n is a bounded domain with smooth boundary d n 2 it is proved that if 1 0 the positive solution u x t blows up globally in d whereas if 0 1 the positive solution u x t is global solution furthermore an upper bound of the blow up time an upper estimate of the blow up rate and an upper estimate of the global solutions are given

expansion exponents for nonequilibrium systems abstract local expansion exponents for nonequilibrium dynamical systems described by partial differential equations are introduced these exponents show whether the system phase volume expands contracts or is conserved in time the ways of calculating the exponents are discussed the principle of minimal expansion provides the basis for treating the problem of pattern selection the exponents are also defined for stochastic dynamical systems the analysis of the expansion exponent behavior for quasi isolated systems results in the formulation of two other principles the principle of asymptotic expansion tells that the phase volumes of quasi isolated systems expand at asymptotically large times the principle of time irreversibility follows from the asymptotic phase expansion since the direction of time arrow can be defined by the asymptotic expansion of phase volume

duality for relative prymians associated to k3 double covers of del pezzo surfaces of degree 2 markushevich and tikhomirov provided a construction of an irreducible symplectic v manifold of dimension 4 the relative compactified prym variety of a family of curves with involution which is a lagrangian fibration with polarization of type 1 2 we give a characterization of the dual lagrangian fibration we also identify the moduli space of lagrangian fibrations of this type and show that the duality defines a rational involution on it