role of the edge orbits in the semiclassical quantization of the stadium billiard in the periodic orbit quantization of the stadium billiard we show that important contributions may be due to the edge orbits i e to orbits bouncing between points where the curvature of the boundary is discontinuous we explicitly show that these edge orbits are necessary to reproduce several amplitudes of the length spectrum defined by the fourier transform of the staircase function in this way we explain some features overlooked in recent experiments on microwave cavities

false singularities in partial sums over closed orbits the contribution to the spectral density from all the repetitions of a single classical closed orbit can have singularities that are not eigenenergies of the system only by summing over all different topologies of orbits do these false singularities get replaced by 8 peaks at the true eigenenergies we illustrate this with an example where the expansion and sum over repetitions are exact rather than asymptotic in studies of the energy spectra of quantum systems much use has been made of the semiclassical expansion of the density of states as a sum over the closed orbits of the corresponding classical system gutzwiller 1967 1969 1970 1971 1978 balian and bloch 1972 berry and tabor 1976 berry 1983 it is known that this series has peculiar convergence properties and that all the periodic orbits must be included if there is to be any hope of getting the correct structure of any one singularity in this letter we consider a case for which the expansion is exact i e there are no asymptotic approxima tions non trivial i e there are infinitely many topologically different primitive periodic orbits and for which the sum over repetitions of a given primitive orbit can be performed analytically the system is the billiard on a torus a spinless particle moving freely in a rectangular cell with periodic boundary conditions this is integrable and has eigenenergies given in suitable units by

the canonical sheaf of du bois singularities we prove that a cohen macaulay normal variety x has du bois singularities if and only if x g x for a log resolution x x where g is the reduced exceptional divisor of many basic theorems about du bois singularities become transparent using this characterization including the fact that cohen macaulay log canonical singularities are du bois we also give a straightforward and self contained proof that generalizations of semi log canonical singularities are du bois in the cohen macaulay case it also follows that the kodaira vanishing theorem holds for semi log canonical varieties and that cohen macaulay semi log canonical singularities are cohomologically insignificant in the sense of dolgachev

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

nearest neighbor spacing distribution of prime numbers and quantum chaos we give heuristic arguments and computer results to support the hypothesis that after appropriate rescaling the statistics of spacings between adjacent prime numbers follows the poisson distribution the scaling transformation removes the oscillations in the nearest neighbor spacing distribution of primes these oscillations have the very profound period of length six we also calculate the spectral rigidity 3 for prime numbers by two methods after suitable averaging one of these methods gives the poisson dependence 3 l l 15

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

a relation between billiard geometry and the temperature of its eigenvalue gas according to a conjecture of yukawa the parametric motion of the eigenvalues of a chaotic system leads to a phase space distribution proportional to where e is the energy of the eigenvalue gas and is its reciprocal temperature to test the conjecture in a first step correspondence between the well known pechukas yukawa level dynamics and that of a billiard with variable length is established next is expressed in terms of the billiard geometry thus fixing the only free parameter of the model finally experimental distributions of eigenvalue velocities curvatures etc obtained from sinai microwave billiards are analysed in terms of the model in all cases a quantitative agreement was found apart from some small deviations caused by the dominating bouncing ball orbit

on the accuracy of the semiclassical trace formula the semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit when the dimensionality of the system increases the mean level spacing decreases as while the semiclassical approximation is commonly believed to provide an accuracy of order independently of d if this was true the semiclassical trace formula would be limited to systems in only in this work we set out to define proper measures of the semiclassical spectral accuracy and to propose theoretical and numerical evidence to the effect that the semiclassical accuracy measured in units of the mean level spacing depends only weakly if at all on the dimensionality detailed and thorough numerical tests were performed for the sinai billiard in two and three dimensions substantiating the theoretical arguments

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

bouncing ball orbits and symmetry breaking effects in a three dimensional chaotic billiard we study the classical and quantum mechanics of a three dimensional stadium billiard it consists of two quarter cylinders that are rotated with respect to each other by 90 ifmmode circ else textdegree fi and it is classically chaotic the billiard exhibits only a few families of nongeneric periodic orbits we introduce an analytic method for their treatment the length spectrum can be understood in terms of the nongeneric and unstable periodic orbits for unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory for equal radii the billiard exhibits an additional symmetry we investigated the effects of symmetry breaking on spectral properties moreover for equal radii we observe a small deviation of the level statistics from random matrix theory this led to the discovery of stable and marginally stable orbits which are absent for unequal radii

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

theory of a curved planar waveguide with robin boundary conditions a model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative robin boundary condition is analyzed theoretically within the framework of the linear schrodinger equation and is applied to the study of the processes in the bent magnetic multilayers superconducting films and metallic ferrite filled waveguides in particular subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the robin parameter 1 lambda with lambda being an extrapolation length entering robin boundary condition for the arbitrary robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest mode energy of the continuously curved duct is always smaller than its straight counterpart accordingly the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend in terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart localized level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius conditions of the bound state existence for the different robin parameters on the opposite edges are analyzed too in particular it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one in the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode for example for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths lambda at the opposite interfaces various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the dirichlet to the neumann case as the extrapolation length lambda sweeps the positive axis

f injectivity and buchsbaum singularities let r mathfrak m k be a local ring that contains a field we show that when r has equal characteristic p 0 and when h mathfrak m i r has finite length for all i dim r then r is f injective if and only if every ideal generated by a system of parameters is frobenius closed as a corollary we show that such an r is in fact a buchsbaum ring this answers positively a question of s takagi that f injective singularities with isolated non cohen macaulay locus are buchsbaum we also study the characteristic 0 analogue of this question and we show that du bois singularities with isolated non cohen macaulay locus are buchsbaum in the graded case

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

exact distributions of eigenvalue curvatures for time reversal invariant chaotic systems the parametric motion of eigenvalues of chaotic quantum systems is studied by means of the distribution of eigenvalue curvatures k second derivative with respect to a perturbation parameter using supersymmetric integral representations this distribution is computed exactly for the orthogonal ensuremath beta 1 and the symplectic ensuremath beta 4 ensemble it is found that p k ensuremath propto 1 mathit k 2 mathrm ensuremath 2 mathrm ensuremath beta 2 in agreement with a recent conjecture by zakrzewski and delande phys rev e 47 1650 1993

near integrable systems a two dimensional circular quantum billiard with unusual boundary conditions introduced by berry and dennis 2008 j phys a math theor 41 135203 is considered in detail it is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with neumann boundary conditions deviations from strong localization are also discussed these results agree well with numerical calculations

saturation of number variance in embedded random matrix ensembles we study fluctuation properties of embedded random matrix ensembles of noninteracting particles for ensemble of two noninteracting particle systems we find that unlike the spectra of classical random matrices correlation functions are nonstationary in the locally stationary region of spectra we study the number variance and the spacing distributions the spacing distributions follow the poisson statistics which is a key behavior of uncorrelated spectra the number variance varies linearly as in the poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths and the number variance approaches saturation values these results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory we conjecture that the interacting particle cases which exhibit the characteristics of classical random matrices for short correlation lengths will also show saturation effects for large correlation lengths

square free grobner degenerations let i be a homogeneous ideal of s k x 1 ldots x n and let j be an initial ideal of i with respect to a term order we prove that if j is radical then the hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of s i and s j coincide in particular text depth s i text depth s j and text reg s i text reg s j

the convective eigenvalues of the one dimensional p laplacian as p 1 abstract this paper studies the limit behavior as p 1 of the eigenvalue problem u x p 2 u x x c u x p 2 u x u p 2 u 0 x 1 u 0 u 1 0 we point out that explicit expressions for both the eigenvalues n and associated eigenfunctions are not available see 16 in spite of this hindrance we obtain the precise values of the limits lim p 1 n in addition a complete description of the limit profiles of the eigenfunctions is accomplished moreover the formal limit problem as p 1 is also addressed the results extend known features for the special case c 0 6 28

non universal spectral rigidity of quantum pseudo integrable billiards we obtain the dyson mehta statistic for pseudo integrable billiards and show that it is non universal with a universal trend also that this trend is similar to the one for integrable billiards we present a formula based on exact semiclassical calculations and the proliferation law of periodic orbits which gives rigidity for the entire range of l to consolidate our theory we discuss several examples finding complete agreement with the numerical results and also the underlying fundamental reasons for the non universality

in out decomposition of boundary integral equations we propose a reformulation of the boundary integral equations for the helmholtz equation in a domain in terms of incoming and outgoing boundary waves we obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling these effects are absent in the well known semiclassical transfer operators in the sense of bogomolny it has long been established that transfer operators are equivalent to the boundary integral approach within semiclassical approximation exact treatments have been restricted to specific boundary conditions such as dirichlet or neumann the approach we propose is independent of the boundary conditions and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary as an application we show how the decomposition may be used to calculate goos haenchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities

classical diffusion anderson localization and spectral statistics in billiard chains we study spectral properties of quasi one dimensional extended systems that show deterministic diffusion on the classical level and anderson localization in the quantal description using semi classical arguments we relate universal aspects of the spectral fluctuations to features of the set of classical periodic orbits expressed in terms of the probability to perform periodic motion which are likewise universal this allows us to derive an analytical expression for the spectral form factor which reflects the diffusive nature of the corresponding classical dynamics it defines a novel spectral universality class which covers the transition between goe statistics in the limit of a small ratio of the system size to the localization length corresponding to the ballistic regime of disordered systems to poissonian level fluctuations in the opposite limit our semi classical predictions are illustrated and confirmed by a numerical investigation of aperiodic chains of chaotic billiards

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

global parameter test ideals abstract this paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings we do this for cohen macaulay localizations resp completions of non local rings for generalized cohen macaulay rings and for non local rings with isolated non cohen macaulay points each being an isolated non f rational point the tools used to prove these results are constructive in nature and as a consequence our results yield algorithms for the computation of these global parameter test ideals finally we illustrate the power of our methods by analyzing the hsl numbers of local cohomology modules with support at any prime ideal

nilpotence of frobenius action and the hodge filtration on local cohomology abstract an f nilpotent local ring is a local ring r m of prime characteristic defined by the nilpotence of the frobenius action on its local cohomology modules h m i r a singularity in characteristic zero is said to be of f nilpotent type if its modulo p reduction is f nilpotent for almost all p in this paper we give a hodge theoretic interpretation of three dimensional normal isolated singularities of f nilpotent type in the graded case this yields a characterization of these singularities in terms of divisor class groups and brauer groups

statistical analysis of scars in stadium billiard in this paper by using our improved plane wave decomposition method we study the scars in the eigenfunctions of the stadium billiard from a very low state to as high as about the 1 millionth state in the systematic searching for scars of various types we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar we have analysed the profile of the integrated probability density along the orbit we found that the maximal integrated intensity of different types of scars scales in different way with the which confirms qualitatively and quantitatively the existing theories of scars such as that of bogomolny and robnik

boundary condition varying circle billiards and gratings the dirichlet singularity waves in a two dimensional domain with robin mixed boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena if the variation includes a d point where the boundary condition is dirichlet vanishing wavefunction a variety of arguments indicate that the system is singular for a circle billiard the boundary condition fails to determine a discrete set of levels so the spectrum is continuous for a diffraction grating defined by periodically varying boundary conditions on the edge of a half plane the phase of a diffracted beam amplitude remains undetermined in both cases the wavefunction on the boundary has a singularity at a d point described by the polylogarithm function

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

periodic orbit theory and spectral rigidity in pseudointegrable systems we calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers g that arise from rectangular systems with one or two salient corners from the periodic orbits we calculate the spectral rigidity delta3 l using semiclassical quantum mechanics with l reaching up to quite large values we find that the diagonal approximation is applicable when averaging over a suitable energy interval comparing systems of various shapes we find that our results agree well with delta3 calculated directly from the eigenvalues by spectral statistics therefore additional terms such as e g diffraction terms seem to be small in the case of the systems investigated in this work by reducing the size of the corners the spectral statistics of our pseudointegrable systems approaches that of an integrable system whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large both types of behavior can be well understood by the properties of the periodic orbits in the system

exact s matrix for n disc systems and various boundary conditions ii determination and partial classification of resonances scattering of waves and particles from two and three disc systems with discrete and symmetries is studied for various boundary conditions resonances are numerically determined and partially classified by using the irreducible representations of the symmetry groups new physical effects are expected splitting up of resonances and resonances of interaction between the scatterers some of these effects can be observed on the far field form functions

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

the dirichlet to robin transform a simple transformation converts a function satisfying a dirichlet boundary condition to a function satisfying a robin generalized neumann condition under sufficiently restrictive conditions one can arrange that the old function is the solution of a solvable differential equation and the new one solves an interesting differential equation in the simplest cases this observation enables the exact construction of the green functions for the wave heat and schrodinger problems with a robin boundary condition the resulting physical picture is that the field can exchange energy with the boundary and a delayed reflection from the boundary results in more general situations the method allows at least approximate and local construction of the appropriate reflected solutions and hence a classical path analysis of the green functions and the associated spectral information by this method we solve the wave equation on an interval with one robin and one dirichlet endpoint and thence derive several variants of a gutzwiller type expansion for the density of eigenvalues the variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighbourhood of zero in the frequency variable

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

f injectivity and frobenius closure of ideals in noetherian rings of characteristic p 0 abstract the main aim of this article is to study the relation between f injective singularity and the frobenius closure of parameter ideals in noetherian rings of positive characteristic the paper consists of the following themes including many other topics 1 we prove that if every parameter ideal of a noetherian local ring of prime characteristic p 0 is frobenius closed then it is f injective 2 we prove a necessary and sufficient condition for the injectivity of the frobenius action on h m i r for all i f m r where f m r is the finiteness dimension of r as applications we prove the following results a if the ring is f injective then every ideal generated by a filter regular sequence whose length is equal to the finiteness dimension of the ring is frobenius closed it is a generalization of a recent result of ma and which is stated for generalized cohen macaulay local rings b let r m k be a generalized cohen macaulay ring of characteristic p 0 if the frobenius action is injective on the local cohomology h m i r for all i dim r then r is buchsbaum this gives an answer to a question of takagi 3 we consider the problem when the union of two f injective closed subschemes of a noetherian f p scheme is f injective using this idea we construct an f injective local ring r such that r has a parameter ideal that is not frobenius closed this result adds a new member to the family of f singularities 4 we give the first ideal theoretic characterization of f injectivity in terms the frobenius closure and the limit closure we also give an answer to the question about when the frobenius action on the top local cohomology is injective

on the number of bouncing ball modes in billiards we study the number of bouncing ball modes in a class of two dimensional quantized billiards with two parallel walls using an adiabatic approximation we show that asymptotically for where depends on the shape of the billiard boundary in particular for the class of two dimensional sinai billiards which are chaotic one can get arbitrarily close from below to which corresponds to the leading term in weyl s law for the mean behaviour of the counting function of eigenstates this result shows that one can come arbitrarily close to violating quantum ergodicity we compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement

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

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

local cohomology and f stability abstract we study the relationship between the frobenius stability of an artinian module over an f injective ring and its stable part

spectral duality for planar billiards for a bounded open domain with connected complement inr2 and piecewise smooth boundary we consider the dirichlet laplacian on and the s matrix on the complement c we show that the on shell s matricessk have eigenvalues converging to 1 ask k0 exactly when has an eigenvalue at energyk02 this includes multiplicities and proves a weak form of transparency atk k0 we also show that stronger forms of transparency such assk0 having an eigenvalue 1 are not expected to hold in general

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

the behaviour of resonances in hecke triangular billiards under deformation the right hand boundary of artin s billiard on the poincare half plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the hecke groups 2 n at certain values of the deformation parameter the quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated the transitions to arithmetic chaos in the cases n 4 6 are closely examined and the explicit analytic form for the scattering matrix is given together with the fourier coefficients for the scattered wavefunction the n 4 and 6 cases have an additional set of regular equally spaced resonances compared to artin s billiard n 3 for a general deformation a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution for deformations away from the non generic arithmetic cases including that of the tiling hecke triangular billiard n 5 the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory

necessary condition for eigensolutions of the 1 laplace operator by means of inner variations eigenfunctions of the p laplace operator for p 1 are defined to be critical points of an associated variational problem or equivalently to be solutions of the corresponding euler lagrange equation in the highly degenerated limit case of the 1 laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope however the associated euler lagrange equation has many solutions that are not critical points and thus it cannot be used for an equivalent definition the present paper provides a new necessary condition for eigenfunctions of the 1 laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the euler lagrange equation that are not eigenfunctions

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

level curvature distribution for diffusive aharonov bohm systems analytical results we calculate analytically the distributions of level curvatures lc the second derivatives of eigenvalues with respect to a magnetic flux for a particle moving in a white noise random potential we find that the zakrzewski delande conjecture is still valid even if the lowest weak localization corrections are taken into account the ratio of mean level curvature modulus to mean dissipative conductance is proved to be universal and equal to 2 pi in agreement with available numerical data

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

a characteristic p analogue of plt singularities and adjoint ideals we introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure as a corollary we prove that a log pair x is plt if and only if the modulo p reduction of x is divisorially f regular for all large p 0 here divisorially f regular pairs are a class of singularities in positive characteristic introduced by hara and watanabe j algebra geom 11 363 392 2002 in terms of frobenius splitting

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

seminormality and local cohomology of toric face rings abstract we characterize the toric face rings that are normal respectively seminormal extending results about local cohomology of brun bruns ichim li and romer of seminormal monoid rings and stanley toric face rings we prove the vanishing of certain graded parts of local cohomology of seminormal toric face rings the combinatorial formula we obtain generalizes hochster s formula we also characterize all necessarily seminormal toric face rings that are f pure or f split over a field of characteristic p 0 an example is given to show that f injectivity does not behave well with respect to face projections of toric face rings finally it is shown that weakly f regular toric face rings are normal affine monoid rings

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

ballistic dynamics of a convex smooth wall billiard with finite escape rate along the boundary we focus on the problem of an impurity free billiard with a random position dependent boundary coupling to the environment the response functions of such an open system can be obtained non perturbatively from a supersymmetric generating functional the derivation of this functional is based on averaging over the escape rates and results in a nonlinear ballistic model characterized by system specific parameters particular emphasis is placed on the whispering gallery modes as the origin of surface diffusion modes in the limit of large dimensionless conductance

semiclassical quantization of billiards with mixed boundary conditions the semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived the parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards it is also used to check in detail some recently proposed parametric spectral statistics the methods are illustrated in the analysis of the spectrum of the sinai billiard and its parametric dependence on the boundary condition on the dispersing arc

bogomolny section for the stadium i quantum theory the quarter stadium can be decomposed into a rectangle and a quarter circle in each of these the helmholtz equation is separable we thus construct green s functions for both regions and a fully quantum mechanical bogomolny matrix this results in a very efficient algorithm for calculating eigenvalues and eigenfunctions we discuss the relation of the family of periodic bouncing ball orbits to the contribution of the non oscillating modes