discrete symmetries in the weyl expansion for quantum billiards we consider two and three dimensional quantum billiards with discrete symmetries the boundary condition is either dirichlet or neumann we derive the first terms of the weyl expansion for the level density projected onto the irreducible representations of the symmetry group the formulae require only knowledge of the character table of the group and the geometrical properties such as surface perimeter etc of sub parts of the billiard invariant under a group transformation as an illustration the method is applied to the icosahedral billiard

families of minimal involutive surfaces in projective space abstract we present an algorithm that can be used to determine if a given surface in complex projective 3 space is minimal involutive and we apply it to the construction of explicit examples of families of minimal involutive surfaces of degrees 3 and 4

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

g twisted braces and orbifold landau ginzburg models given an algebra with a group g action we construct brace structures for its g twisted hochschild cochains an an application we construct g frobenius algebras for orbifold landau ginzburg b models and present explicit orbifold cup product formula for all invertible polynomials

singular foliations with trivial canonical class this paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds

on the spectrum of the dirichlet laplacian for horn shaped regions in rn with infinite volume abstract an inequality for trace et d is proven where d is the dirichlet laplacian for horn shaped regions d in rn the results of rozenbljum and simon for the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 0 are easily recovered an example of a horn shaped region in r2 where that asymptotics is exponential is given

trace properties of the dirichlet laplacian on obtient des bornes superieures et inferieures pour tr exp ht en termes de la fonction r x

on the density of algebraic foliations without algebraic invariant sets let x be a smooth complex projective variety of dimension greater than or equal to 2 l an ample line bundle and k 0 an integer we prove that a generic global section of the twisted tangent sheaf x l k gives rise to a foliation of x without any proper algebraic invariant subvarieties of nonzero dimension as a corollary we obtain a dynamical characterization of ampleness for line bundles over smooth projective surfaces

foliations of multiprojective spaces and a conjecture of bernstein and lunts we use foliations of multiprojective spaces defined by hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of bernstein and lunts according to which the symbol of a generic first order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle

antithetic variate splitting for steady sate simulations abstract obtaining precise estimates of parameters of infinite horizon or steady state simulations can be expensive because of the need to discard initial outputs to mitigate the effects of initial conditions we consider splitting independent replications at the point of output truncation into dependent replications to reduce point estimator variance and or simulation cost

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

symmetry decomposition of potentials with channels we discuss the symmetry decomposition of the average density of states for the two dimensional potential and for its three dimensional generalization in both problems the energetically accessible phase space is non compact due to the existence of infinite channels along the axes it is known that in two dimensions the phase space volume is infinite in these channels thus yielding non standard forms for the average density of states here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels leading to terms which are essentially leading order we verify these results numerically and also observe a peculiar numerical effect which we associate with the channels we additionally show that the next order corrections are anomalously weak being at least two powers of smaller than one would expect in three dimensions the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials however there are still non generic effects related to some of the group elements

influence of intermittency on quantum spectra bound chaotic systems typically exhibit intermittency some semiclassical properties of intermittency are studied the hyperbola billiard and the x2y2 model are used as strongly intermittent model systems the almost integrable motion in the arms of the potential may be treated in the adiabatic approximation the corresponding adiabatic hamiltonian may be semiclassically quantized yielding surprisingly good agreement for the hyperbola billiard it is then demonstrated by means of the semiclassical trace formula how this result may be related to families of periodic orbit exclusively exploring the potential arms it is discussed how this result implies an integrable component in the spectrum possible implications for the resummation problem of the trace formula as well as for the statistical properties of energy levels are discussed

an analytical evaluation of alternative strategies in steady state simulation we consider the simple technique of making independent and probabilistically identical simulation replications to estimate the steady state mean of a stochastic process for a fixed total sample size using a first order autoregressive model with high autocorrelation and time dependent distributions we analytically quantify and investigate the effects of the number of replications and of initial deletions of output on several different measures of point estimator and confidence interval quality this analysis leads to general recommendations on the choice of replications and deletion in particular we conclude that deletion of some amount of the initial output in a replication can be an effective and efficient method of dealing with initialization bias this conclusion differs from several previous studies on the efficacy of deletion

accelerated convergence in the simulation of countably infinite state markov chains this paper describes a method of obtaining results from the simulation of a countably infinite state positive recurrent aperiodic markov chain at a cost considerably below that required to achieve the same accuracy with pure random sampling reorganizing k independent epochs or tours simulated serially into k replications simulated in parallel can induce selected joint distributions across replications that produce the cost savings the joint distributions follow from the use of rotation sampling a special case of the antithetic variate method the chains considered are of the band type so that for some integer i a transition from any state i 0 1 2 can move no further than to states i i and i i the paper shows that an estimator of interest has variance bounded above by oi 2ln k4 k2 when using rotation sampling as compared to a variance o1 k for independent sampling moreover the mean cost of simulation based on rotation sampling has an upper bound oi ln k2 as compared to a cost of at least ok for independent sampling the paper also describes how one can exploit special structure in a model together with rotation sampling to improve the bound on variance for essentially the same mean cost

adventures of the coupled yang mills oscillators i semiclassical expansion we study the quantum mechanical motion in the x2y2 potentials with n 2 3 which arise in the spatially homogeneous limit of the yang mills ym equations these systems show strong stochasticity in the classical limit 0 and exhibit a quantum mechanical confinement feature we calculate the partition function z t going beyond the thomas fermi tf approximation by means of the semiclassical expansion using the wigner kirkwood wk method we derive a novel compact form of the differential equation for the wk function after separating the motion in the channels of the equipotential surface from the motion in the central region we show that the leading higher order corrections to the tf term vanish up to eighth order in if we treat the quantum motion in the hyperbolic channels correctly by adiabatic separation of the degrees of freedom finally we obtain an asymptotic expansion of the partition function in terms of the parameter g24t3

high orders of the weyl expansion for quantum billiards resurgence of periodic orbits and the stokes phenomenon a formalism is developed for calculating high coefficients c r of the weyl high energy expansion for the trace of the resolvent of the laplace operator in a domain b with smooth boundary b the c r are used to test the following conjectures a the sequence of c r diverges factorially controlled by the shortest accessible real or complex periodic geodesic b if this is a 2 bounce orbit it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the mobius strip which is the space of chords of b c this orbit gives an exponential contribution to the remainder when the weyl series truncated at its least term is subtracted from the resolvent the exponential switches on smoothly according to an error function where it is smallest that is across the negative energy axis stokes line these conjectures are motivated by recent results in asymptotics they survive tests for the circle billiard and for a family of curves with 2 and 3 bulges where the dominant orbit is not always the shortest and is sometimes complex for some systems which are not smooth billiards e g a particle on a ring or in a billiard where b is a polygon the weyl series terminates and so no geodesics are accessible for a particle on a compact surface of constant negative curvature only the complex geodesics are accessible from the weyl series

accelerated accuracy in the simulation of markov chains this paper describes a method of obtaining results from the simulation of an n 1 finite state positive recurrent aperiodic markov chain at a cost considerably less than that required by pure random sampling to achieve the same accuracy the method reorganizes k independent epochs or tours simulated serially into k replications simulated in parallel and therefore produces cost savings by inducing selected joint distributions across replications the joint distributions are derived by the use of rotation sampling a special case of the antithetic variate method the paper first describes the method as it applies to a finite state nearest neighbor chain in particular it shows that even for independent parallel replications the cost of achieving a specified accuracy with serial simulation relative to the cost for parallel simulation has a lower bound o k1 2 n as k when rotation sampling is used this bound is o k2 n ln k 3 a simulation of the m m 1 queueing model with finite capacity n illus

the partition function and level density for yang mills higgs quantum mechanics we calculate the partition function z t and the asymptotic integrated level density n e for yang mills higgs quantum mechanics for two and three dimensions n 2 3 due to the infinite volume of the phase space on energy shell for n 2 it is not possible to completely disentangle the coupled oscillators x2 y2 model from the higgs sector the situation is different for n 3 for which is finite the transition from order to chaos in these systems is expressed by the corresponding transitions in z t and n e analogous to the transitions in adjacent level spacing distribution from poisson distribution to wigner dyson distribution we also discuss a related system with quartic coupled oscillators and two dimensional quartic free oscillators for which contrary to ymhqm both coupling constants are dimensionless

semiclassical trace formulas for noninteracting identical particles we extend the gutzwiller trace formula to systems of noninteracting identical particles the standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families the identical nature of the particles also introduces discrete permutational symmetries we exploit the formalism of creagh and littlejohn phys rev a 44 836 1991 who have studied semiclassical dynamics in the presence of continuous symmetries to derive many body trace formulas for the full and symmetry reduced densities of states numerical studies of the three particle cardioid billiard are used to explicitly illustrate and test the results of the theory

bounds on green s functions of second order differential equations we estimate the diagonal part of the green s function for the equation 2 v x t x t 0 t 0 x b where b is a finite region of the euclidean space rd with a regular boundary in the special case v x 0 x b we also obtain bounds for the nondiagonal part of the green s function which are uniform in t

discrete symmetries and semiclassical quantization with the help of a graph and an associated adjacency matrix the problem of semiclassical quantization is discussed for physical systems with a discrete symmetry a general expression for the symmetry reduced zeta functions is derived in terms of symmetry reduced moments of the adjacency operator as an application the uniform semiclassical quantization conditions of the hecht hamiltonian are discussed within this approach

refined asymptotic expansion for the partition function of unbounded quantum billiards this paper presents a refined asymptotic expansion for the partition function t tr et of quantum billiards in the unbounded regions 0 x 0 yx 1 0 and 0 ye x 1 r2 where is the dirichlet laplacian simon ann phys 146 209 1983 j funct anal 53 84 1983 determined the leading divergence of the trace of the heat kernel for the first class of systems standard techniques are combined for the evaluation of for bounded region billiards with results by van den berg j funct anal 71 279 1987 for horn shaped regions using an optimized way of dividing the region into narrow and wide parts to determine the first three terms in the asymptotic expansion of results are also stated for bounded regions with cusps that can be obtained by the same method as an application the spectral staircase of the strongly chaotic billiard system defined in the region 0 xy 2 x 0 which has been discussed in connection with the riemann function and the search for quantum chaos is c

irreducible components of the space of holomorphic foliations of degree two in cp n n 3 in this paper we will prove that the space of holomorphic fo liations of codimension 1 and degree 2 in cp n n 3 has six irreducible components

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos

determination of resonance spectra for bound chaotic systems we consider the computation of the eigenvalues of the evolution operator the resonance spectrum by means of the zeros of a zeta function in particular we address the problems of applying this formalism to bound chaotic systems caused by e g intermittency and non completeness of the symbolic dynamics for bound intermittent systems we derive an approximation of the zeta function with the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces of the evolution operator approaching unity as a power law we also show that the dominant time scale can be much longer than the period of the shortest periodic orbit as is for example the case for the hyperbola billiard isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function semiclassical implications are briefly discussed