height pairings on shimura curves and p adic uniformization in a recent paper 16 of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0 cycles on shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic eisenstein series of genus 2 on the one hand the height pairing can be written as a sum of local height pairings for example if the 0 cycles have disjoint support on the generic fiber then their height pairing is a sum of an archimedean contribution and a contribution from each of the finitely many finite primes p for which the cycles meet in the fiber at p on the other hand it turns out that the non singular part of the fourier expansion of the central derivative of the metaplectic eisenstein series also has a decomposition into a sum of contributions indexed by the places of q then one would like to compare the height pairing and the fourier coefficients by proving an identity of local contributions place by place in loc cit the identity for the archimedean place was proved and it was shown that the identity at a non archimedean place of good reduction is a consequence of results of gross and keating 10 for the algebraicgeometric side and of kitaoka 12 for the analytic side it then remains to consider the finite primes p where the shimura curve has bad reduction these are of three sorts i the primes p at which the quaternion algebra defining the shimura curve is split but which divide the level ii the primes p at which the quaternion algebra remains division but no level structure is imposed and iii the primes p at which the quaternion algebra remains division and where a higher level structure is imposed at p so far little is known concerning cases i and iii

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

strong enhancement of noise induced escape by nonadiabatic periodic driving due to transient chaos we have found a mechanism by which a moderately weak nonadiabatic periodic driving may significantly facilitate noise induced interwell transitions in an underdamped multiwell system the mechanism is associated with the onset of a homoclinic tangle in the noise free system if the ratio of the driving amplitude a to the damping ensuremath gamma exceeds a critical value ensuremath sim 1 then the basins of attraction of the linear responses related to different wells are mixed in a complex manner in some layer associated with the separatrix of the undriven nondissipative system and the minimal energy in such layer is lower than the top of the barrier thus the energy to which the system needs to be activated by the noise to be able to make a transition is lower than the top of the barrier

an explicit koppelman formula for ddc and green currents on mathbb p n we compute a quite explicit koppelman formula for dd c on projective space and obtain green currents for closed p p currents

a fixed point formula of lefschetz type in arakelov geometry iv the modular height of c m abelian varieties we give a new proof of a slightly weaker form of a theorem of p colmez c2 par 2 this theorem corollary 5 8 gives a formula for the faltings height of abelian varieties with complex multiplication by a c m eld whose galois group over q is abelian it reduces to the formula of chowla and selberg in the case of elliptic curves we show that the formula can be deduced from the arithmetic

intersecting subvarieties of g_m n with algebraic subgroups let x subset g m n be an irreducible closed subvariety defined over bar q we bound the height of algebraic points on x that are in a certain sense close to the union of all algebraic subgroup of g m n of dimension m n dim x the bound we obtain is effective and will be expressed as a function of the height of x the degree of x and n we then apply this bound to derive certain finiteness results if m is also strictly less than n dim x

green currents for holomorphic automorphisms of compact kahler manifolds and some additional regularity properties very likely the currents t will describe the distribution of invariant manifolds of codimension s corresponding to the small est lyapounov exponents let dp denote the dynamical degree of order p of it describes the growth under iteration of the volume of p dimensional manifolds when d 1 it is natural to introduce first a positive closed 1 l current as ifnyoj 7 lim

a fixed point formula of lefschetz type in arakelov geometry i statement and proof we consider arithmetic varieties endowed with an action of the group scheme of n th roots of unity and we define equivariant arithmetic k 0 theory for these varieties we use the equivariant analytic torsion to define direct image maps in this context and we prove a riemann roch theorem for the natural transformation of equivariant arithmetic k 0 theory induced by the restriction to the fixed point scheme this theorem can be viewed as an analog in the context of arakelov geometry of the regular case of the theorem proved by p baum w fulton and g quart in bafq we show that it implies an equivariant refinement of the arithmetic riemann roch theorem in a form conjectured by j m bismut cf b2 par l p 353 and also ch soule s question in sabk 1 5 p 162

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

random walks kleinian groups and bifurcation currents let be a holomorphic family of representations of a finitely generated group g into psl 2 parameterized by a complex manifold we define a notion of bifurcation current in this context that is a positive closed current on describing the bifurcations of this family of representations in a quantitative sense it is the analogue of the bifurcation current introduced by demarco for holomorphic families of rational mappings on 1 our definition relies on the theory of random products of matrices so it depends on the choice of a probability measure on g

super potentials densities of currents and number of periodic points for holomorphic maps abstract we prove that if a positive closed current is bounded by another one with bounded continuous or holder continuous super potentials then it inherits the same property there are two different methods to define wedge products of positive closed currents of arbitrary bi degree on compact kahler manifolds using super potentials and densities when the first method applies we show that the second method also applies and gives the same result as an application we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact kahler manifolds whose actions on cohomology are simple a similar result still holds for a large class of holomorphic correspondences

heights of kudla rapoport divisors and derivatives of l functions we study special cycles on integral models of shimura varieties associated with unitary similitude groups of signature n 1 1 we construct an arithmetic theta lift from harmonic maass forms of weight 2 n to the arithmetic chow group of the integral model of a unitary shimura variety by associating to a harmonic maass form f a linear combination of kudla rapoport divisors equipped with the green function given by the regularized theta lift of f our main result is an equality of two complex numbers 1 the height pairing of the arithmetic theta lift of f against a cm cycle and 2 the central derivative of the convolution l function of a weight n cusp form depending on f and the theta function of a positive definite hermitian lattice of rank n 1 when specialized to the case n 2 this result can be viewed as a variant of the gross zagier formula for shimura curves associated to unitary groups of signature 1 1 the proof relies on among other things a new method for computing improper arithmetic intersections

residue calculus and effective nullstellensatz multivariate residue calculus in the spirit of j lipman is developed from the computational point of view for example with several variants of the classical transformation law and used in order to make totally explicit the bezout identity and therefore the algebraic nullstellensatz in k x 1 x n where k is an infinite field of arbitrary characteristic such identities provide sharp size estimates for the denominator and the divisors in the bezout identity when k is the quotient field of a factorial regular ring a equipped with a size such as z or f p 1 q the estimates obtained by the authors in a previous work in the case a z are sharpened here while analytic techniques are replaced with an algebraic approach

a procedure for finding numerical trajectories on chaotic saddles abstract examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors if almost every trajectory eventually leaves some region but the region contains a chaotic set then typical trajectories will behave chaotically for a while and then will leave the region and so we will observe chaotic transients the main objective that will be addressed is the dynamic restraint problem given a region that contains a chaotic set but does not contain an attractor find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time systems with horseshoes have such regions as do systems with fractal basin boundaries as does the henon map for suitably chosen parameters we present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory the method is called the pim triple procedure since it is based on so called pim triples a pim proper interior maximum triple is three points a c b in a straight line segment such that the interior point c i e c is between a and b has the maximum escape time that is its escape time from the region is greater than the escape time of both a and b proper means the segment from a to b is smaller than a previously obtained segment we show rigorously that the pim triple procedure works in ideal situations we find it works well even in less than ideal cases this procedure can also be used for the computation of lyapunov exponents furthermore the accessible pim triple procedure a refined pim triple procedure for finding accessible trajectories on the chaotic saddle will also be discussed

identifying stochastic basin hopping by partitioning with graph modularity abstract it has been known that noise in a stochastically perturbed dynamical system can destroy what was the original zero noise case barriers in the phase space pseudobarrier noise can cause the basin hopping we use the frobenius perron operator and its finite rank approximation by the ulam galerkin method to study transport mechanism of a noisy map in order to identify the regions of high transport activity in the phase space and to determine flux across the pseudobarriers we adapt a new graph theoretical method which was developed to detect active pseudobarriers in the original phase space of the stochastic dynamic previous methods to identify basins and basin barriers require a priori knowledge of a mathematical model of the system and hence cannot be applied to observed time series data of which a mathematical model is not known here we describe a novel graph method based on optimization of the modularity measure of a network and introduce its application for determining pseudobarriers in the phase space of a multi stable system only known through observed data

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

enhancement of noise induced escape through the existence of a chaotic saddle we study the noise induced escape process in a prototype dissipative nonequilibrium system the ikeda map in the presence of a chaotic saddle embedded in the basin of attraction of the metastable state we find the novel phenomenon of a strong enhancement of noise induced escape this result is established by employing the theory of quasipotentials our finding is of general validity and should be experimentally observable

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

hermitian vector bundles and extension groups on arithmetic schemes i geometry of numbers we define and investigate extension groups in the context of arakelov geometry the arithmetic extension groups ext xi f g we introduce are extensions by groups of analytic types of the usual extension groups extxi f g attached to ox modules f and g over an arithmetic scheme x in this paper we focus on the first arithmetic extension group ext x1 f g the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over x and we especially consider the case when x is an arithmetic curve namely the spectrum specok of the ring of integers in some number field k then the study of arithmetic extensions over x is related to old and new problems concerning lattices and the geometry of numbers namely for any two hermitian vector bundles f and g over x specok we attach a logarithmic sizesf g to any element of ext x1 f g and we give an upper bound on sf g in terms of slope invariants of f and g we further illustrate this notion by relating the sizes of restrictions to points in p1 z of the universal extension over pz1 to the geometry of psl2 z acting on poincare s upper half plane and by deducing some quantitative results in reduction theory from our previous upper bound on sizes finally we investigate the behaviour of size by base change i e under extension of the ground field k to a larger number field k when the base field k is q we establish that the size which cannot increase under base change is actually invariant when the field k is an abelian extension of k or when f g is a direct sum of root lattices and of lattices of voronoi s first kind the appendices contain results concerning extensions in categories of sheaves on ringed spaces and lattices of voronoi s first kind which might also be of independent interest

intersection theory on shimura surfaces ii this is the third of a series of papers relating intersections of special cycles on the integral model of a shimura surface to fourier coefficients of hilbert modular forms more precisely we embed the shimura curve over associated to a rational quaternion algebra into the shimura surface associated to the base change of the quaternion algebra to a real quadratic field after extending the associated moduli problems over we obtain an arithmetic threefold with a embedded arithmetic surface which we view as a cycle of codimension one we then construct a family indexed by totally positive algebraic integers in the real quadratic field of codimension two cycles complex multiplication points on the arithmetic threefold the intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the fourier coefficients of a very particular hilbert modular form of weight 3 2 the results are higher dimensional variants of results of kudla rapoport yang which relate intersection multiplicities of special cycles on the integral model of a shimura curve to fourier coefficients of a modular form in two variables

new constructive aspects of the lovasz local lemma the lovasz local lemma lll is a powerful tool that gives sufficient conditions for avoiding all of a given set of bad events with positive probability a series of results have provided algorithms to efficiently construct structures whose existence is non constructively guaranteed by the lll culminating in the recent breakthrough of moser and tardos 2010 for the full asymmetric lll we show that the output distribution of the moser tardos algorithm well approximates the conditional lll distribution the distribution obtained by conditioning on all bad events being avoided we show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results we also show that when a lll application provides a small amount of slack the number of resamplings of the moser tardos algorithm is nearly linear in the number of underlying independent variables not events and can thus be used to give efficient constructions in cases where the underlying proof applies the lll to super polynomially many events even in cases where finding a bad event that holds is computationally hard we show that applying the algorithm to avoid a polynomial sized core subset of bad events leads to a desired outcome with high probability this is shown via a simple union bound over the probabilities of non core events in the conditional lll distribution and automatically leads to simple and efficient monte carlo and in most cases rnc algorithms we demonstrate this idea on several applications we give the first constant factor approximation algorithm for the santa claus problem by making a lll based proof of feige constructive we provide monte carlo algorithms for acyclic edge coloring nonrepetitive graph colorings and ramsey type graphs in all these applications the algorithm falls directly out of the non constructive lll based proof our algorithms are very simple often provide better bounds than previous algorithms and are in several cases the first efficient algorithms known as a second type of application we show that the properties of the conditional lll distribution can be used in cases beyond the critical dependency threshold of the lll avoiding all bad events is impossible in these cases as the first even nonconstructive result of this kind we show that by sampling a selected smaller core from the lll distribution we can avoid a fraction of bad events that is higher than the expectation max k sat is an illustrative example of this

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

global representation of segre numbers by monge ampere products on a reduced analytic space x we introduce the concept of a generalized cycle which extends the notion of a formal sum of analytic subspaces to include also a form part we then consider a suitable equivalence relation and corresponding quotient mathcal b x that we think of as an analogue of the chow group and a refinement of de rham cohomology this group allows us to study both global and local intersection theoretic properties we provide many mathcal b analogues of classical intersection theoretic constructions for an analytic subspace v subset x we define a mathcal b segre class which is an element of mathcal b x with support in v it satisfies a global king formula and in particular its multiplicities at each point coincide with the segre numbers of v when v is cut out by a section of a vector bundle we interpret this class as a monge ampere type product for regular embeddings we construct a mathcal b analogue of the gysin morphism

the local lemma is asymptotically tight for sat the local lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science yet there are hardly any natural problems known where it provides an asymptotically tight answer the main theme of our article is to identify several of these problems among them a couple of widely studied extremal functions related to certain restricted versions of the k sat problem where the local lemma does give essentially optimal answers as our main contribution we construct unsatisfiable k cnf formulas where every clause has k distinct literals and every variable appears in at most 2 e o 1 2k k clauses the lopsided local lemma applied with an assignment of random values according to counterintuitive probabilities shows that this is asymptotically best possible the determination of this extremal function is particularly important as it represents the value where the corresponding k sat problem exhibits a complexity hardness jump from having every instance being a yes instance it becomes np hard just by allowing each variable to occur in one more clause the construction of our unsatisfiable cnf formulas is based on the binary tree approach of gebauer 2012 and thus the constructed formulas are in the class mu 1 of minimal unsatisfiable formulas having one more clause than variables the main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem this leads us to a differential equation the solution of which we are able to estimate the asymptotically optimal binary trees are then obtained through a discretization of this solution the importance of the binary trees constructed is also underlined by their appearance in many other scenarios in particular they give asymptotically precise answers for seemingly unrelated problems like the european tenure game introduced by doerr 2004 and a search problem allowing a limited number of consecutive lies as yet another consequence we slightly improve the best known bounds on the maximum degree and maximum edge degree of a k uniform maker s win hypergraph in the neighborhood conjecture of beck

schubert polynomials and arakelov theory of orthogonal flag varieties we propose a theory of combinatorially explicit schubert polynomials which represent the schubert classes in the borel presentation of the cohomology ring of the orthogonal flag variety mathfrak x rm so n b we use these polynomials to describe the arithmetic schubert calculus on mathfrak x moreover we give a method to compute the natural arithmetic chern numbers on mathfrak x and show that they are all rational numbers

twisted hilbert modular surfaces arithmetic intersections and the jacquet langlands correspondence abstract we study arithmetic intersections on quaternionic hilbert modular surfaces and shimura curves over a real quadratic field our first main result is the determination of the degree of the top arithmetic todd class of an arithmetic twisted hilbert modular surface this quantity is then related to the arithmetic volume of a shimura curve via the arithmetic grothendieck riemann roch theorem and the jacquet langlands correspondence

counting in two spin models on d regular graphs supported in part by alfred p sloan research fellowship supported in part by department of defense ndseg fellowships

heights and metrics with logarithmic singularities we prove lower bound and finiteness properties for arakelovian heights with respect to pre log log hermitian ample line bundles these heights were introduced by burgos kramer and kuhn in 2 in their extension of the arithmetic intersection theory of gillet and soule 8 aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities our results generalize the corresponding properties for the heights of cycles in bost gillet soule 1 as well as the properties established by faltings 7 for heights of points attached to hermitian ample line bundles whose metrics have logarithmic singularities we also discuss various geometric constructions where such pre log log hermitian ample line bundles naturally arise msc 14g40 primary 11j25 11j97 secondary

intersection theory in differential algebraic geometry generic intersections and the differential chow form in this paper an intersection theory for generic differential polynomials is presented the intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is shown to be an irreducible variety of dimension d 1 and order h s as a consequence the dimension conjecture for generic differential polynomials is proved based on intersection theory the chow form for an irreducible differential variety is defined and most of the properties of the chow form in the algebraic case are established for its differential counterpart furthermore the generalized differential chow form is defined and its properties are proved as an application of the generalized differential chow form the differential resultant of n 1 generic differential polynomials in n variables is defined and properties similar to that of the macaulay resultant for multivariate polynomials are proved

computing green currents via the heat kernel let x be a compact kahler manifold for an analytic cycle y x of complex codimension k the poincare duality assigns a class in the de rham cohomology which can be represented by a harmonic k k form y a green current gy connects both objects by the equality ddgy y y thus we can reconstruct y and its harmonic projection y from this equation because y is the locus where ddgy is not smooth while y is determined by its values on the dense open set x y furthermore the above equality tells us that y and y differ only by some exact current green currents for varieties of arbitrary dimension were systematically developed by h gillet and c soule in 3 these currents play a central role in arakelov theory see 10 and 7 on the one hand the existence of green currents follows more or less formally from the hodge decomposition for currents to do intersection theory on the other hand we need currents which behave well green forms and green currents of logarithmic type are examples of these currents see 2 and 3 here we present a method to find a canonical green current gy which depends only on the kahler manifold x and the cycle y this current gy is given by k k forms with l coefficients which are smooth on x y we start with repeating some basic results on kahler manifolds in 1 which can be found in the textbooks 5 or 11 furthermore we recall the main properties of the heat kernel before we consider the heat flow of dirac currents in section 2 obtaining a green operator for these currents in 3 we define the canonical harmonic green current and show that its harmonic projection is zero in section 4 we apply the heat kernel to intersections of properly intersecting cycles this is needed to derive a formula for the product of green currents in 5 however we have to restrict to cycles that intersect properly hence our results are weaker than those of 3 as we used a different approach we decided to present them here in section 6 we give examples which we found by a good guess 6 1 or because we know all eigen functions of the laplace operator 6 2 or from the explicit formula of the heat kernel 6 3 the last example shows how our techniques can be extended to non compact kahler manifolds for which a heat kernel exists we conclude in section 7 with some questions and remarks acknowledgment the author wishes to thank w gubler j kramer and u kuhn for many discussions which contributed to this paper the comments of the referee helped to clarify the exposition

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

dynamics of regular birational maps in p k abstract we develop the study of some spaces of currents of bidegree p p as an application we construct the equilibrium measure for a large class of birational maps of p k as intersection of green currents we show that these currents are extremal and that the corresponding measure is mixing

a dynamical pairing between two rational maps this is the publisher s final pdf the published article is copyrighted by american mathematical society and can be found at http www ams org home page

noise induced escape from attractors the author deals with several aspects of kautz 1987 1988 minimum principle for the noise induced escape from attractors he presents a slightly simpler derivation points out a connection with lyapunov exponents and invariant manifolds and generalises to chaotic maps

computations of bott chern classes on ℙ e we compute the bott chern classes of the metric eulersequence describing the relative tangent bundle of the variety p e of hyperplans of a holomorphic hermitian vector bundle e h on a complex manifold we give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle and to the computation of the height of p e with respect to the tautological quotient bundle o e 1

random walks that find perfect objects and the lovasz local lemma we give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly our work is inspired by moser s entropic method proof of the lovasz local lemma lll for satisfiability and completely bypasses the probabilistic method formulation of the lll in particular our method works when the underlying state space is entirely unstructured similarly to moser s argument the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time

a robin formula for the fekete leja transfinite diameter we generalize the classical robin formula to higher dimensions mathematics subject classification 2000 primary 32u35 32u20 14g40 secondary 31b15 31c10 the purpose of this note is to give a formula for the fekete leja transfinite diameter on cn generalizing the classical robin formula d e e v e for the usual transfinite diameter we will disengage this from a formula for the sectional capacity proved in arithmetic intersection theory 5 theorem 1 1 p 233 first recall the definition of the fekete leja transfinite diameter for a compact set e cn see 1 16 consider the set of monomials zk z1 1 zn n in the polynomial ring c z c z1 zn let n c z be the space of polynomials of total degree at most n and let k n k zn ki 0 k1 kn n be the index set for the monomial basis of n put qn k n n n n fixing n take qn independent vector variables zi zi1 zin cn i 1 qn let k1 kqn be the indices in k n the vandermonde determinant qn z1 zqn det zj i n i j 1 work supported in part by nsf grant dms 0300784 r rumely b department of mathematics university of georgia athens ga 30602 usa e mail rr math uga edu

complex multiplication cycles and kudla rapoport divisors ii we study the intersections of special cycles on a unitary shimura variety of signature n 1 1 and show that the intersection multiplicities of these cycles agree with fourier coecients of eisenstein series the results are new cases of conjectures of kudla and suggest a gross zagier theorem for unitary shimura varieties

uniform sampling through the lovasz local lemma we propose a new algorithmic framework called partial rejection sampling to draw samples exactly from a product distribution conditioned on none of a number of bad events occurring our framework builds new connections between the variable framework of the lovasz local lemma and some classical sampling algorithms such as the cycle popping algorithm for rooted spanning trees among other applications we discover new algorithms to sample satisfying assignments of k cnf formulas with bounded variable occurrences