holomorphic quantization on the torus and finite quantum mechanics we explicitly construct the quantization of classical linear maps of on toroidal phase space of arbitrary modulus using the holomorphic chiral version of the metaplectic representation we show that finite quantum mechanics fqm on tori of arbitrary integer discretization is a consistent restriction of the holomorphic quantization of to the subgroup being the principal congruent subgroup on a finite dimensional hilbert space the generators of the rotation group for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates

periodic orbit maslov indices for systems with time reversal symmetry we show that in time reversal invariant systems a pair of periodic orbits related by time reversal symmetry have the same maslov index previously this result had been implicitly assumed in the semiclassical derivation of the gaussian orthogonal ensemble spectral form factor

on bargmann representations of the wigner function by using the localized character of canonical coherent states we give a straightforward derivation of the bargmann integral representation of the wigner function w a non integral representation is presented in terms of a quadratic form w v fv where f is a self adjoint matrix whose entries are tabulated functions and v is a vector depending in a simple recursive way on the derivatives of the bargmann function such a representation may be of use in numerical computations we discuss a relation involving the geometry of the wigner function and the spatial uncertainty of the coherent state basis we use to represent it

first order transitions in spherical models finite size scaling finite size behavior near the first order phase boundary of ferromagnetic spherical models is investigated for block and cylinder shaped systems ind dimensions the bulk thermodynamic singularities are rounded and asymptotically for large size obey appropriate scaling laws both short range interactions and long range couplings decaying like 1 rd with 0 are analyzed the short range results agree precisely with a recently developed scaling theory foro n symmetric systems in the limitn more generally the scaling functions are universal depending only on explicit aspects of the shape and interactions enter only in the spin wave or goldstone mode contributions which appear technically as corrections to scaling an appendix analyzes the truncation error in the approximation by many fold sums of multivariate integrals with integrands diverging like jaj j2 as 0

spaces of type w growth of hermite coefficients wigner distribution and bargmann transform abstract in their famous monograph on generalized functions gelfand and shilov introduce the function spaces w m w and w m these spaces consist of c functions and holomorphic functions respectively with growth behaviour specified by suitable convex functions m and in this paper we study the spaces wmmx where mx denotes young s dual function corresponding to m we characterize the hermite expansion coefficients of the functions in wmmx their fourier transforms wigner distributions and bargmann transforms in particular we prove that wmmx wm wmx

finite size effects in a field theoretic model with long range exchange interaction we present a systematic approach to the calculation of finite size fs effects for ano n field theoretic model with both short range sr and long range lr exchange interactions the lr exchange interaction decays at large distances as 1 rd 2 2 0 0 renormalization group calculations ind du e are performed for a system with a fully finite block geometry under periodic boundary conditions we calculate the fs shift of the critical temperature and the fs renormalized coupling constant of the model to one loop order the universal scaling variable is obtained and the fs scaling hypothesis is verified

the weight hierarchies of some product codes bounds on the weight hierarchies of the product of two simplex codes two first order reed muller codes and the product of a simplex code and a first order reed muller code are determined the weight hierarchies of the product of two hamming codes and the product of a hamming code and an even weight code are also discussed

information theoretic characterization of quantum chaos hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a hamiltonian system from increasing we demonstrate numerically that hypersensitivity to perturbation is present in the following quantum maps the quantum kicked top the quantum baker s map the quantum lazy baker s map and the quantum sawtooth and cat maps

commensurability effects on the spectra of integrable systems the eigenvalue spectra of two simple integrable systems a three dimensional anisotropic harmonic oscillator aho and a particle in a cuboid are studied as the aspect ratios are changed from rational to irrational values for rational aspect ratios commensurable case there are large number theoretic degeneracies which grow with energy these degeneracies disappear when the aspect ratios are irrational incommensurable case however for a given set of rational aspect ratios commensurable behavior ensues only at large energies the number of distinct eigenvalues per unit energy interval which is indicative of the degree of degeneracies is seen to approach a constant asymptotically for rational aspect ratios the asymptotic constant is determined for all sets of aspect ratios for the aho and is estimated semi analytically for several sets of aspect ratios for the cuboid the crossover from incommensurable to commensurable behaviour is studied by following a sequence of rational aspect ratios which approaches an irrational limit a scaling form the crossover is suggested and explored numerically a clear indication of scaling is seen for the aho while the evidence is suggestive but not definitive for the cuboid the distribution of energy level spacings in the commensurable regime is determined for the aho and a sequence of cuboids

mobius mellin and mathematical physics abstract we examine some results and techniques of analytic number theory which have application or potential application in mathematical physics we consider inversion formulae for lattice sums various transformations of infinite series and products functional equations and scaling relations with selected applications in electrostatics and statistical mechanics in the analysis the mellin transform and the riemann zeta function play a key role

asymmetric fluid criticality ii finite size scaling for simulations the vapor liquid critical behavior of intrinsically asymmetric fluids is studied in finite systems of linear dimensions l focusing on periodic boundary conditions as appropriate for simulations the recently propounded complete thermodynamic stackrel ensuremath rightarrow l ensuremath infty scaling theory incorporating pressure mixing in the scaling fields as well as corrections to scaling phys rev e 67 061506 2003 is extended to finite l initially in a grand canonical representation the theory allows for a yang yang anomaly in which when stackrel ensuremath rightarrow l ensuremath infty the second temperature derivative d 2 ensuremath mu ensuremath sigma dt 2 of the chemical potential along the phase boundary ensuremath mu ensuremath sigma t diverges when stackrel ensuremath rightarrow t t mathrm c ensuremath the finite size behavior of various special critical loci in the temperature density or t ensuremath rho plane in particular the k inflection susceptibility loci and the q maximal loci derived from q l t ensuremath rho l ensuremath equiv m 2 l 2 m 4 l where m ensuremath equiv ensuremath rho ensuremath ensuremath rho l is carefully elucidated and shown to be of value in estimating t mathrm c and ensuremath rho mathrm c concrete illustrations are presented for the hard core square well fluid and for the restricted primitive model electrolyte including an estimate of the correlation exponent ensuremath nu that confirms ising type character the treatment is extended to the canonical representation where further complications appear

on the quantization of the three particle toda lattice the authors compare the einstein brillouin keller quantization procedure and the canonical quantization of a three particle toda chain with periodic boundary conditions in particular the transition from very low energies at which the system may be approximated by harmonic oscillators to intermediate energies is investigated this is the regime of a general integrable nonlinear system for which they find a poissonian statistics for the energy levels in the limit of very high energies they exploit the fact that the system may be described essentially by a triangular billiard and thus can derive some exact results

beyond the heisenberg time semiclassical treatment of spectral correlations in chaotic systems with spin 1 2 the two point correlation function of chaotic systems with spin 1 2 is evaluated using periodic orbits the spectral form factor for all times thus becomes accessible equivalence with the predictions of random matrix theory for the gaussian symplectic ensemble is demonstrated a duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

critical behavior in colloid polymer mixtures theory and simulation we extensively investigated the critical behavior of mixtures of colloids and polymers via the two component asakura oosawa model and its reduction to a one component colloidal fluid using accurate theoretical and simulation techniques in particular the theoretical approach hierarchical reference theory a parola and l reatto adv phys 44 211 1995 incorporates realistically the effects of long range fluctuations on phase separation giving exponents which differ strongly from their mean field values and are in good agreement with those of the three dimensional ising model computer simulations combined with finite size scaling analysis confirm the ising universality and the accuracy of the theory although some discrepancy in the location of the critical point between one component and full mixture description remains to assess the limit of the pair interaction description we compare one component and two component results

the weight hierarchy of some reducible cyclic codes the generalized hamming weights ghws of linear codes are fundamental parameters the knowledge of which is of great interest in many applications however to determine the ghws of linear codes is difficult in general in this paper we study the ghws for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases this is achieved by extending the idea of yang et al into higher dimension and by employing some interesting combinatorial arguments it shall be noted that these cyclic codes may have arbitrary number of nonzeros

experimental versus numerical eigenvalues of a bunimovich stadium billiard a comparison we compare the statistical properties of eigenvalue sequences for a gamma 1 bunimovich stadium billiard the eigenvalues have been obtained in two ways one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator real system and the other set is calculated numerically ideal system we show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system we also discuss the influence of a family of marginally stable orbits the bouncing ball orbits in two microwave stadium billiards with different geometrical dimensions

random matrices generalized zeta functions and self similarity of zero distributions there is growing evidence for a connection between random matrix theories used in physics and the theory of the riemann zeta function and l functions the theory underlying the location of the zeros of these generalized zeta functions is one of the key unsolved problems physicists are interested because of the hilbert polya conjecture that the non trivial zeros of the zeta function correspond to the eigenvalues of some positive operator to complement the continuing theoretical work it would be useful to study empirically the locations of the zeros by different methods in this paper we use the rescaled range analysis to study the spacings between successive zeros of these functions over large ranges of the zeros the spacings have a hurst exponent of about 0 095 using sample sizes of 10 000 zeros this implies that the distribution has a high fractal dimension 1 9 and shows a lot of detailed structure the distribution is of the anti persistent fractional brownian motion type with a significant degree of anti persistence thus the high order zeros of these functions show a remarkable self similarity in their distribution over fifteen orders of magnitude for the riemann zeta function we find that the hurst exponents for the random matrix theories show a different behaviour a heuristic study of the effect of low order primes seems to show that this effect is a promising candidate to explain the results that we observe in this study we study the distribution of zeros for l functions of conductors 3 and 4 and find that the distribution is similar to that of the riemann zeta functions

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

geometric approach to higher weights the notion of higher or generalized weights of codes is just as natural as that of the classical hamming weight the authors adopt the geometric point of view and always treat the q ary case some results and proofs being new the main goal is to present a clear picture of what is known on the subject

periodic orbits and semiclassical quantization of dispersing billiards periodic orbits in a dispersing billiard system consisting of three circular arcs are studied numerically by using a partial coding rule together with an efficient method for enumerating periodic orbits on the real billiard plane by examining several statistical measures it is shown that the length spectrum and the stability exponents are highly uncorrelated the validity of the semiclassical trace formula is also tested and a remarkable agreement of the semiclassical and quantum density of states is obtained at least for about the lower 15 levels

critical behavior of a colloid polymer mixture confined between walls we investigate the influence of confinement on phase separation in colloid polymer mixtures to describe the particle interactions the colloid polymer model of asakura and oosawa j chem phys 22 1255 1954 is used grand canonical monte carlo simulations are then applied to this model confined between two parallel hard walls separated by a distance d 5 colloid diameters we focus on the critical regime of the phase separation and look for signs of crossover from three dimensional 3d ising to two dimensional 2d ising universality to extract the critical behavior finite size scaling techniques are used including the recently proposed algorithm of kim et al phys rev lett 91 065701 2003 our results point to effective critical exponents that differ profoundly from 3d ising values and that are already very close to 2d ising values in particular we observe that the critical exponent of the order parameter in the confined system is smaller than in 3d bulk yielding a flatter binodal our results also show an increase in the critical colloid packing fraction in the confined system with respect to the bulk the latter seems consistent with theoretical expectations although subtleties due to singularities in the critical behavior of the coexistence diameter cannot be ruled out

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

resummation and the semiclassical theory of spectral statistics we address the question as to why in the semiclassical limit classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random matrix theory to do so we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits this allows us to obtain both the non oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework thus overcoming a significant problem in previous approaches in addition our results extend beyond the universal regime to describe the system specific approach to the semiclassical limit

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

orbit bifurcations and spectral statistics systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are in the semiclassical limit a combination of poisson and random matrix with relative weightings determined by the corresponding measures of regular and chaotic orbits we here identify an additional component in long range spectral statistics associated with periodic orbit bifurcations which can be semiclassically large this is illustrated for a family of perturbed cat maps it has been conjectured that in the semiclassical limit the quantum spectral statistics of classically integrable systems are generically poissonian and that those of classically chaotic systems are generically given by the average over an appropriate random matrix ensemble 1 4 in between these two extremes lie systems whose phase space is mixed that is in which regular and irregular motion coexist such systems are said to exhibit soft chaos 5 for these it has been suggested that the quantum spectral statistics are a combination of poisson and random matrix with relative weightings determined by the corresponding measures of the regular and chaotic orbits 6 our purpose here is to identify in this case an additional component in the long range statistics that is associated with periodic orbit bifurcations and which can be semiclassically large a semiclassical theory for long range spectral statistics has been developed 1 7 9 based on gutzwiller s trace formula 10 which relates the quantum density of states d e d p n e en to classical periodic orbits

energy level statistics of an integrable billiard system in a rectangle in the hyperbolic plane a separable billiard system in the hyperbolic strip a particular realization of the hyperbolic plane is investigated the author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area a 4 pi or of a hyperbolic rectangle bounded by geodesics respectively due to the simple geometry of the approximation the schrodinger equation is separable and leads to simple transcendental equations for the eigenvalues in terms of legendre functions the statistical properties of the eigenvalue spectrum are investigated and checked with respect to weyl s 1912 law level spacing number variance and spectral rigidity respectively in particular the numerical results for the spectral rigidity are in close agreement with the semi classical theory of berry 1985

test of renormalization predictions for universal finite size scaling functions we calculate universal finite size scaling functions for systems with an n component order parameter and algebraically decaying interactions just as previously found for short range interactions this leads to a singular ensuremath varepsilon expansion where ensuremath varepsilon is the distance to the upper critical dimension subsequently we check the results by numerical simulations of spin models in the same universality class our systems offer the essential advantage that ensuremath varepsilon can be varied continuously allowing an accurate examination of the region where ensuremath varepsilon is small the numerical calculations turn out to be in striking disagreement with the predicted singularity

generalized q functions the modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of q or husimi functions a parameter lambda specifies orderings in a mapping from the operator psi sigma to the corresponding phase space wavefunction where sigma is a given fiducial vector the choice lambda 0 specifies the weyl mapping and the q function so obtained is the usual one when sigma is the vacuum state more generally any choice of of lambda in the range 1 1 corresponds to orderings varying between standard and anti standard for all such orderings the generalized q functions are non negative by construction they are shown to be proportional to expectation of the system state rho with respect to a generalized displaced squeezed state which depends on lambda and position p q in phase space thus when a system has been prepared in the state rho a generalized q function is proportional to the probability of finding it in the generalized squeezed state any such q function can also be written as the smoothing of the wigner function for the system state rho by convolution with the wigner function for the generalized squeezed state

bounds on integrals of the wigner function the integral of the wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one it is shown that for systems with 1 degree of freedom the problem of determining the best possible upper and lower bounds on such an integral over an possible states reduces to the problem of finding the greatest and least eigenvalues of a hermitian operator corresponding to the subregion the problem is solved exactly in the case of an arbitrary elliptical region these bounds provide checks on experimentally measured quasiprobability distributions

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

quantum mechanics as an approximation to classical mechanics in hilbert space classical mechanics is formulated in complex hilbert space with the introduction of a commutative product of operators an antisymmetric bracket and a quasidensity operator that is not positive definite these are analogues of the star product the moyal bracket and the wigner function in the phase space formulation of quantum mechanics quantum mechanics is then viewed as a limiting form of classical mechanics as planck s constant approaches zero rather than the other way around the forms of semiquantum approximations to classical mechanics analogous to semiclassical approximations to quantum mechanics are indicated

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

neutrino billiards time reversal symmetry breaking without magnetic fields a dirac hamiltonian describing massless spin half particles neutrinos moving in the plane r x y under the action of a 4 scalar not electric potential v r is in position representation h i h c v r z where x y and z are the pauli matrices acts on two component column spinor wavefunctions r 1 2 and has eigen values ck n does not possess time reversal symmetry t if v r describes a hard wall bounding a finite domain d billiards this is equivalent to a novel boundary condition for 2 1 t breaking is interpreted semiclassically as a difference of between the phases accumulated by waves travelling in opposite senses round closed geo desics in d with odd numbers of reflections the semiclassical large k asymptotics of the eigenvalue counting function spectral staircase n k are shown to have the weyl leading term ak 2 4 where a is the area of d but zero perimeter correction the dirac equation is transformed to an integral equation round the boundary of d and forms the basis of a numerical method for computing the k n when d is the unit disc geodesics are integrable and the eigenvalues which satisfy j l k n j l 1 k n are locally poisson distributed when d is an africa shape cubic conformal map of the unit disc the eigenvalues are locally distributed according to the statistics of the gaussian unitary ensemble of random matrix theory as predicted on the basis of t breaking and lack of geometric symmetry

a class of parameter dependent commuting matrices we present a novel class of real symmetric matrices in arbitrary dimension d linearly dependent on a parameter x the matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom this class of matrices generically violates the wigner von neumann non crossing rule and is argued to be intimately connected with finite dimensional hamiltonians of quantum integrable systems

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

the n level spectral correlations for chaotic systems we study the n level spectral correlation functions of classically chaotic quantum systems without time reversal symmetry according to bohigas giannoni and schmit s universality conjecture it is expected that the correlation functions are in agreement with the prediction of the circular unitary ensemble cue of random matrices a semiclassical resummation formalism allows us to express the correlation functions as sums over pseudo orbits using an extended version of the diagonal approximation on the pseudo orbit sums we derive the n level correlation functions identical to the n n determinantal correlation functions of the cue

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

the δn statistic for the β hermite ensemble abstract the fluctuation n of the n th unfolded eigenvalue was recently characterized for the classical gaussian ensembles of n n random matrices goe gue gse it is investigated here for the hermite ensemble as a function of the reciprocal of the temperature by monte carlo simulations the ensemble averaged fluctuation n 2 and the autocorrelation function vary logarithmically with n for any 0 1 n n the simple logarithmic behavior of the higher order moments of n reported in the literature for the goe 1 and the gue 2 holds for any 0 and is accounted for by gaussian distributions whose variances depend linearly on ln n

the weight hierarchy of product codes the weight of a code is the number of coordinate positions where no codeword is zero the rth minimum weight d sub r is the least weight of an r dimensional subcode wei and yang 1993 gave a conjecture about the minimum weights for some product codes in this article we find a relation between product codes and the segre embedding of a pair of projective systems and we use this to prove the conjecture

proposed solution of the quantum measurement problem a discussion of quantum measurement theory is presented it is argued that the quantum measurement problem consists in the failure of any formalism based on unitary evolution to describe fully the observed behaviour in measurement like processes it is shown that the problem is essentially concerned with the phase of the coherence between orthogonal eigenstates of an object measuring device system it is then proposed that the observed behaviour is fully described if the value of this phase is formally undecidable

the pointwise product in weyl quantization we study the product of bracken 1 which is the weyl quantized version of the pointwise product of functions in phase space we prove that it is not compatible with the algebras of finite rank and hilbert schmidt operators by solving the linearization problem for the special hermite functions we are able to express the product in terms of the component operators mediated by the linearization coefficients this is applied to finite rank operators and their matrices and operators whose symbols are radial and angular distributions

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

revisiting universality of the liquid gas critical point in two dimensions critical point of liquid gas lg transition does not conform with the paradigm of spontaneous symmetry breaking because there is no broken symmetry in both phases we revisit the conjecture that this critical point belongs to the ising class by performing large scale monte carlo simulations in two dimensional free space in combination with the numerical flowgram method our main result is that the critical indices do agree with the onsager values within the error of 1 2 this significantly improves the accuracy reported in the literature the related problem about the role of higher order odd terms in the real ensuremath varphi 4 field model as a mapping of the lg transition is addressed too the scaling dimension of the ensuremath varphi 5 term at criticality is shown to be the same as that of the linear one ensuremath varphi we suggest that the role of all higher order odd terms at criticality is simply in generating the linear field operator with the critical dimension consistent with the ising universality class

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

the quantum state vector in phase space and gabor s windowed fourier transform representations of quantum state vectors by complex phase space amplitudes complementing the description of the density operator by the wigner function have been defined by applying the weyl wigner transform to dyadic operators linear in the state vector and anti linear in a fixed window state vector here aspects of this construction are explored and a connection is established with gabor s windowed fourier transform the amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations generalized bargmann representations of the state vector appear as special cases associated with gaussian windows for every choice of window amplitudes lie in a corresponding linear subspace of square integrable functions on phase space a generalized born interpretation of amplitudes is described with both the wigner function and a generalized husimi function appearing as quantities linear in an amplitude and anti linear in its complex conjugate schrodinger s time dependent and time independent equations are represented on phase space amplitudes and their solutions described in simple cases

critical finite size scaling of the free energy the privman fisher universal amplitude characterizing the free energy of a finite sized system at its critical point is identified as the critical counterpart of the number of stable phases on the coexistence curve its value is shown to be determined by the critical point form of the order parameter distribution directly accessible to monte carlo studies

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

multifractal eigenstates of quantum chaos and the thue morse sequence we analyze certain eigenstates of the quantum baker s map and demonstrate using the walsh hadamard transform the emergence of the ubiquitous thue morse sequence a simple sequence that is at the border between quasiperiodicity and chaos and hence is a good paradigm for quantum chaotic states we show a family of states that are also simply related to the thue morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions we give approximate expressions for these states and provide evidence that these and other generic states are multifractal

two point spectral correlations for the square billiard we investigate the two point correlations in the quantum spectrum of the square billiard this system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states hence for example the standard level spacings distribution does not tend to the poissonian limit expected for integrable systems in this paper we calculate the leading order asymptotic form of a degeneracy weighted two point correlation function using a combination of probabilistic techniques and classical number theory the result exhibits number theoretical fluctuations about a mean which is a sum of two terms one having the usual constant poissonian form and the second representing a small correction which decays as the inverse of the correlation distance this is confirmed by numerical computations

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

weight hierarchy of a product code formulas to compute the first members of the weight hierarchy of a product code are given and proved basic patterns for the supports of subcodes in which the minimum cardinalities can be found are presented hence formulas for every term of the hierarchy can be given and the proof for each case will be a generalization of the proofs shown here

periodic orbit action correlations in the baker map periodic orbit action correlations are studied for the piecewise linear area preserving baker map semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum baker map suggest the existence of universal periodic orbit correlations the calculation of periodic orbit sums for the baker map can be performed with the help of a perron frobenius type operator this makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation

disheveled arnold s cat and the problem of quantum classic correspondence abstract quantum arnold s cat map is studied for a case of perfect square inverse planck s constant n m 2 the classic limit is analyzed on a subset of numbers n increasing as 4 k the quantum problem in this case allows exact reduction to the classic cat map defined on a discrete lattice of size m m and supplemented by evolution of a phase variable a link between the classic periodic orbits and spectrum of eigenvalues of the quantum evolution operator is outlined for m growing as 2 k genetic analysis is developed for periodic orbits and they are classified by means of a tree like graph a phase shift accumulated over a period of the orbits evolves from level to level of the graph according to a certain rule governed by non periodic binary code representation of a localized gaussian wave packet in a basis of eigenvectors of the evolution operator gives rise to a probability measure distributed on a unit circle where the eigenvalues are located this measure looks like spectrum of a finite time sample of a stationary random process periodogram 1 majority of the eigenstates have intensities of comparable order of magnitude 2 the spectral distribution is of locally random like nature i e statistical variance of the amplitudes has the same order as the amplitudes themselves this combination of properties in very straightforward manner follows from chaotic nature of the classic map and is conjectured to be the most fundamental attribute of quantum chaos

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

ising universality in three dimensions a monte carlo study we investigate three ising models on the simple cubic lattice by means of monte carlo methods and finite size scaling these models are the spin 1 2 ising model with nearest neighbour interactions a spin 1 2 model with nearest neighbour and third neighbour interactions and a spin 1 model with nearest neighbour interactions the results are in accurate agreement with the hypothesis of universality analysis of the finite size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field we find that the correction to scaling amplitudes are strongly dependent on the introduction of further neighbour interactions or a third spin state in a spin 1 ising model these corrections appear to be very small this is very helpful for the determination of the universal constants of the ising model the renormalization exponents of the ising model are determined as yt 1 587 2 yh 2 4815 15 and yi 0 82 6 the universal ratio q m2 2 m4 is equal to 0 6233 4 for periodic systems with cubic symmetry the critical point of the nearest neighbour spin 1 2 model is kc 0 2216546 10

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

discrete symmetries and spectral statistics we calculate the 2 point spectral statistics associated with a given irreducible representation i e symmetry class for time reversal invariant systems possessing discrete symmetries using semiclassical periodic orbit theory when the representation in question is real or pseudo real our results conform to those of the gaussian orthogonal ensemble goe of random matrices when it is complex we find instead gaussian unitary ensemble gue behaviour this provides a direct semiclassical explanation for the recent observation by leyvraz et al 1996 of gue correlations in the desymmetrized spectra of certain symmetric billiards in the absence of any time reversal invariance breaking e g magnetic fields

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

time delay resonances riemann zeros and chaos in a model quantum scattering system the quantum treatment of an intrinsically chaotic model scattering system originally studied by gutzwiller 1983 is extended to include the time delay and to make explicit the zeros of the riemann zeta function in the mathematical expressions for the scattering matrix s and the time delay the system consists of a particle moving on a two dimensional surface of constant negative curvature the scattering in this unusual system is dominated by resonances associated with poles of s in the complex momentum plane the real parts of these poles are one half of the imaginary parts of the riemann zeros the resonances have a constant width but their average spacing varies with the momentum the focal point is the manifestation of chaotic behaviour in the time delay features considered in this regard include the momentum dependence of the time delay in regions of overlapping and isolated resonances the decomposition of the time delay into an average dynamical component and a fluctuating chaotic component and characterisation of the fluctuating component by its autocorrelation function

on the quantum reconstruction of the riemann zeros we discuss a possible spectral realization of the riemann zeros based on the hamiltonian h xp perturbed by a term that depends on two potentials which are related to the berry keating semiclassical constraints we find perturbatively the potentials whose jost function is given by the zeta function it for 1 for 1 2 we find the potentials that yield the smooth approximation to the zeros we show that the existence of potentials realizing the zeta function at 1 2 as a jost function would imply that the riemann zeros are point like spectra embedded in the continuum resolving in that way the emission spectral interpretation