the gauss bonnet chern theorem a probabilistic perspective we prove that the euler form of a metric connection on a real oriented vector bundle e over a compact oriented manifold m can be identified as a current with the expectation of the random current defined by the zero locus of a certain random section of the bundle we also explain how to reconstruct probabilistically the metric and the connection on e from the statistics of random sections of e contents

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

generalized bergman kernels on symplectic manifolds we study the near diagonal asymptotic expansion of the generalized bergman kernel of the renormalized bochner laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold we show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them as a consequence we calculate the density of states function of the bochner laplacian and establish a symplectic version of the convergence of the induced fubini study metric we also discuss generalizations of the asymptotic expansion for non compact or singular manifolds as well as their applications our approach is inspired by the analytic localization techniques of bismut and lebeau

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

the voronoi tessellation generated from eigenvalues of complex random matrices the voronoi froth generated from eigenvalues of asymmetric complex random matrices is studied by numerical simulation it is more regular than the random voronoi froth irvf generated from a poisson process the existence of a unique tessellation called here random matrix voronoi froth rmvf follows from the universality of the distribution of eigenvalues which is also briefly commented on the geometrical and topological properties of the rmvf have been characterised an empirical and accurate distribution function is also proposed for the cell side length of a rvf deviations from the aboav weaire law are discussed their magnitude may be interpreted as a measure ofthe departure from an equilibrium structure in the frame of the statistical crystallography theory of rivier

general quantum surface of section method a new method for exact quantization of general bound hamiltonian systems is presented it is the quantum analogue of the classical poincare surface of section sos reduction of classical dynamics the quantum poincare mapping is shown to be the product of the two generalized non unitary but compact on shell scattering operators of the two scattering hamiltonians which are obtained from the original bound one by cutting the f dimensional configuration space cs the along the f 1 dimensional configurational sos and attaching the flat quasi one dimensional waveguides instead the quantum poincare mapping has fixed points at the eigenenergies of the original bound hamiltonian the energy dependent quantum propagator e h 1 can be decomposed in terms of the four energy dependent propagators which propagate from and or to cs to and or from configurational sos which may generally be composed of many disconnected parts i show that in the semiclassical limit h cross to 0 the quantum poincare mapping converges to the bogomolny propagator and explain how the higher order semiclassical corrections can be obtained systematically

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

bridge functions near the liquid vapor coexistence curve in binary lennard jones mixtures we have carried out extensive molecular dynamics simulation studies of binary lennard jones mixtures to calculate directly the bridge function at state points lying in a very narrow single fluid phase region between the vapor liquid and solid liquid coexistence lines lamm and hall fluid phase equilib 182 37 2001 194 197 197 2002 by varying the density close to the liquid vapor coexistence line significant deviations are observed at intermediate distances between the simulated bridge function and two widely used approximate closures in the integral equation theory of liquids viz the hybrid mean spherical approximation and the duh henderson closures the overall qualitative agreement remains the same with small variation in temperature that brings the system closer to either the liquid vapor or liquid solid coexistence curve we also report a comparison of the direct and indirect correlation functions obtained from our simulation studies as well as from the integral equation theory of liquids our results emphasize the need for developing new closures applicable to binary fluid mixtures over a wide range of thermodynamic parameters

critical end point behavior in a binary fluid mixture we consider the liquid gas phase boundary in a binary fluid mixture near its critical end point using general scaling arguments we show that the diameter of the liquid gas coexistence curve exhibits singular behaviour as the critical end point is approached this prediction is tested by means of extensive monte carlo simulations of a symmetrical lennard jones binary mixture within the grand canonical ensemble the simulation results show clear evidence for the proposed singularity as well as confirming a previously predicted singularity in the coexistence chemical potential fisher and upton phys rev lett 65 2402 1990 the results suggest that the observed singularities particularly that in the coexistence diameter should also be detectable experimentally

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

equidistribution and convergence speed for zeros of holomorphic sections of singular hermitian line bundles abstract we establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular hermitian line bundle with an estimate of the convergence speed

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

sorting permutations and binary strings by length weighted rearrangements abstract comparative genomics is a line of study which aims at determining dis similarities between genomes one way of doing this is to infer the large scale mutations also called genome rearrangements that are responsible for transforming one genome into another under the parsimony principle numerous efforts have been made over the past years regarding the traditional approach in which all genome rearrangements have the same unit cost however since long rearrangements are more likely to disturb the organism some works considered the length weighted approach in which the cost of each rearrangement is a function of its length the problem is thus modeled as follows given two genomes g 1 and g 2 and a set of allowed genome rearrangements each with a given cost determine their distance i e the total cost of a minimum cost sequence of rearrangements from that transforms g 1 into g 2 in this paper we study length weighted reversals and transpositions mainly considering their prefix variants in which the leftmost part of the genome must always be included we present approximation algorithms and bounds on the distances and diameters i e maximum possible distance over any two genomes of same length for eight problems when genomes are represented as permutations and another eight problems when genomes are represented as binary strings in each case considering two different cost functions

field theoretical analysis of singularities at critical end points abstract continuum models with critical end points are considered whose hamiltonian h depends on two densities and field theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical end point singularities like the thermal singularity t 2 of the spectator phase boundary and the coexistence singularities t 1 or t of the secondary density the appearance of a discontinuity eigenexponent associated with the critical end point is confirmed and the mechanism by which it arises in field theory is clarified

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

bounding prefix transposition distance for strings and permutations a transposition is an operation that exchanges two adjacent substrings transpositions over permutations the sequences with no repeated symbols are related to genome rearrangements if one of the substrings is restricted to a prefix then it is called a prefix transposition the prefix transposition distance between a pair of strings permutations is the minimum number of prefix transpositions required to transform a given string permutation into another given string permutation for a permutation of length n we improve the current prefix transposition distance upper bound of n log 8n to n log 9 2n for arbitrary strings we prove new prefix transposition distance upper and lower bounds for binary strings of length n we show that n 2 is an upper bound we show that the prefix transposition distance problem is np complete for binary strings

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

tighter upper bound for sorting permutations with prefix transpositions permutations are sequences where each symbol in the given alphabet appears exactly once a transposition is an operation that exchanges two adjacent sublists in a permutation if one of these sublists is restricted to be a prefix then one obtains a prefix transposition transpositions over permutations have applications in genome rearrangements and computer interconnection networks the set of permutations with n symbols derived from the alphabet 0 1 n 1 form a symmetric group that we denote by p n the prefix transposition distance between two permutations p n and p n is the minimum number of prefix transpositions also called moves needed to transform into a prefix transposition can be modeled by a permutation operation a permutation is type of sequence and it is also an operation the generators for prefix transposition operation are a subset of permutation operation as a result transforming an arbitrary p n to an arbitrary p n is equivalent to sorting some p n thus upper bound for transforming any p n into any p n with prefix transpositions is simply the upper bound to sort any permutation p n with moves in the article that establishes the best known upper bound for prefix transposition distance over p n as n log 9 2 n explicit proofs were not given for some cases in this article we provide the missing proofs validating the stated upper bound furthermore we establish an upper bound of n log 7 2 n for prefix transposition distance over p n

an 18 11 n upper bound for sorting by prefix reversals the pancake problem asks for the minimum number of prefix reversals sufficient for sorting any permutation of length n we improve the upper bound for the pancake problem to 18 11 n o 1 1 6363 n

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

universality and scaling of correlations between zeros on complex manifolds we study the limit as n of the correlations between simultaneous zeros of random sections of the powers l n of a positive holomorphic line bundle l over a compact complex manifold m when distances are rescaled so that the average density of zeros is independent of n we show that the limit correlation is independent of the line bundle and depends only on the dimension of m and the codimension of the zero sets we also provide some explicit formulas for pair correlations in particular we prove that hannay s limit pair correlation function for su 2 polynomials holds for all compact riemann surfaces

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

critical endpoint behavior in an asymmetric ising model application of wang landau sampling to calculate the density of states using the wang landau sampling method with a two dimensional random walk we determine the density of states for an asymmetric ising model with two and three body interactions on a triangular lattice in the presence of an external field with an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite size analyses at and away from the critical endpoint we observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint and the exponents for both divergences agree well with previous theoretical predictions

a semiclassical quantization of area preserving maps abstract a semiclassical path integral formalism is developed for a class of area preserving maps the quasi energy spectrum of these systems is represented as a sum over closed orbits in the manner of gutzwiller the method is illustrated with some numerical tests of the standard map

universality results for zeros of random holomorphic sections in this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular hermitian holomorphic line bundles on a compact kahler complex space x namely under mild moment assumptions we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections in the case when x is a compact kahler manifold we also prove an off diagonal exponential decay estimate for the bergman kernels of a sequence of positive line bundles on x

super potentials of positive closed currents intersection theory and dynamics we introduce a notion of super potential for positive closed currents of bidegree p p on projective spaces this gives a calculus on positive closed currents of arbitrary bidegree we define in particular the intersection of such currents and the pull back operator by meromorphic maps one of the main tools is the introduction of structural discs in the space of positive closed currents which gives a geometry on that space we apply the theory of super potentials to construct green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms

sorting permutations with transpositions in o n3 amortized time abstract the problem of sorting a permutation over alphabet 1 2 3 n with the minimum number of transpositions is known to be intractable that is the computation of transposition distance i e d t is hard such sorting is equivalent to determining the distance between any of the symmetric group sn over under the transposition operation it has applications in computer interconnection networks where a node is denoted by a permutation and genetics where a genome is modeled by a permutation the maximum distance between any pair of permutations in sn is the diameter i e diam s n of the corresponding cayley graph s n it corresponds to the maximum latency in the computer interconnection network modeled by s n and the maximum genetic dissimilarity between any pair in sn we call transpositions prefix transpositions suffix transpositions and prefix suffix transpositions as block moves we design an algorithm to compute d t in sn and compute diam s n with transpositions in o n n 3 time and o n n 2 space at an amortized time of o n 3 the same can be computed for the remaining block moves in o n 2 log n amortized time and o n n space

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions