level dynamics in pseudointegrable billiards an experimental study abstract the level dynamics of pseudointegrable systems with different genus numbers g is studied experimentally using microwave cavities for higher energies the distribution of the eigenvalue velocities is gaussian as it is expected for chaotic systems with time reversal symmetry and shows no dependence on g also the curvature distribution p k for large k is decaying as it is expected for chaotic systems i e p k k 3 for small k an intermediate behavior is found where p k changes from integrable towards chaotic behavior with growing g

two applications of inductive counting for complementation problems following the recent independent proofs of immerman siam j comput 17 1988 pp 935 938 and szelepcsenyi bull european assoc theoret comput sci 33 1987 pp 96 100 that nondeterministic space bounded complexity classes are closed under complementation two further applications of the inductive counting technique are developed first an errorless probabilistic algorithm for the undirected graph s t connectivity problem that runs in o log n space and polynomial expected time is given then it is shown that the class logcfl is closed under complementation the latter is a special case of a general result that shows closure under complementation of classes defined by semi unbounded fan in circuits or equivalently nondeterministic auxiliary pushdown automata or tree size bounded alternating turing machines as one consequence it is shown that small numbers of role switches in two person pebbling can be eliminated

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

partial order bounding a new approach to evaluation in game tree search abstract in computer game playing the established method for constructing an evaluation function uses a scalar value computed as a weighted sum of features this paper advocates the use of partial order evaluation and describes an efficient new search method called partial order bounding pob previous tree search algorithms using a partial order evaluation have attempted to propagate partially ordered values through the search tree which leads to many problems in practice such as the complexity of backing up sets of incomparable evaluations pob compares partially ordered values only in the leaves of a game tree and backs up boolean values through the tree a closely related new algorithm linear extension partial order bounding le pob uses a standard scalar alpha beta search with values from a suitably chosen linear extension of the partial order evaluation as an application the effectiveness of partial order evaluation is shown in the case of modeling capturing races called semeai in the game of go

fully dynamic point location in a monotone subdivision in this paper a dynamic technique for locating a point in a monotone planar subdivision whose current number of vertices is n is presented the complete set of update operations are insertion o

on random matrix averages involving half integer powers of goe characteristic polynomials correlation functions involving products and ratios of half integer powers of characteristic polynomials of random matrices from the gaussian orthogonal ensemble goe frequently arise in applications of random matrix theory rmt to physics of quantum chaotic systems and beyond we provide an explicit evaluation of the large n limits of a few non trivial objects of that sort within a variant of the supersymmetry formalism and via a related but different method as one of the applications we derive the distribution of an off diagonal entry k ab of the resolvent or wigner k matrix of goe matrices which among other things is of relevance for experiments on chaotic wave scattering in electromagnetic resonators

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

statistics of multiple avoided crossings the author considers the energy levels of a generic quantum mechanical system with hamiltonian h depending on a parameter x if the energy levels en are plotted as a function of x the curves do not cross the points of closest approach are called avoided crossings these have a distinctive geometry and are important because they determine the limits of applicability of the adiabatic theorem he describes some theoretical results on the parameter space density of avoided crossings for systems with the spectral statistics of the gaussian orthogonal ensemble goe these results are in good agreement with numerical experiments

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

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

local discriminative distance metrics ensemble learning the ultimate goal of distance metric learning is to incorporate abundant discriminative information to keep all data samples in the same class close and those from different classes separated local distance metric methods can preserve discriminative information by considering the neighborhood influence in this paper we propose a new local discriminative distance metrics lddm algorithm to learn multiple distance metrics from each training sample a focal sample and in the vicinity of that focal sample focal vicinity to optimize local compactness and local separability those locally learned distance metrics are used to build local classifiers which are aligned in a probabilistic framework via ensemble learning theoretical analysis proves the convergence rate bound the generalization bound of the local distance metrics and the final ensemble classifier we extensively evaluate lddm using synthetic datasets and large benchmark uci datasets

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

dynamic reachability in planar digraphs with one source and one sink abstract in this paper we investigate the reachability problem on spherical st graphs which are planar acyclic digraphs with exactly one source and exactly one sink these digraphs find specific applications to visibility representations planarity testing planar graph embedding graph drawing and motion planning let n be the number of vertices first we show that reachability queries in a spherical st graph can be answered in o 1 time using an o n space data structure that can be constructed in o n time next we present a dynamic data structure which uses o n space and supports reachability queries and updates in o log n time finally we show how to extend these results to planar digraphs that have one source and one sink and may have directed cycles

adaptive relevance matrices in learning vector quantization we propose a new matrix learning scheme to extend relevance learning vector quantization rlvq an efficient prototype based classification algorithm toward a general adaptive metric by introducing a full matrix of relevance factors in the distance measure correlations between different features and their importance for the classification scheme can be taken into account and automated and general metric adaptation takes place during training in comparison to the weighted euclidean metric used in rlvq and its variations a full matrix is more powerful to represent the internal structure of the data appropriately large margin generalization bounds can be transferred to this case leading to bounds that are independent of the input dimensionality this also holds for local metrics attached to each prototype which corresponds to piecewise quadratic decision boundaries the algorithm is tested in comparison to alternative learning vector quantization schemes using an artificial data set a benchmark multiclass problem from the vci repository and a problem from bioinformatics the recognition of splice sites for c elegans

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

distance learning in discriminative vector quantization discriminative vector quantization schemes such as learning vector quantization lvq and extensions thereof offer efficient and intuitive classifiers based on the representation of classes by prototypes the original methods however rely on the euclidean distance corresponding to the assumption that the data can be represented by isotropic clusters for this reason extensions of the methods to more general metric structures have been proposed such as relevance adaptation in generalized lvq glvq and matrix learning in glvq in these approaches metric parameters are learned based on the given classification task such that a data driven distance measure is found in this letter we consider full matrix adaptation in advanced lvq schemes in particular we introduce matrix learning to a recent statistical formalization of lvq robust soft lvq and we compare the results on several artificial and real life data sets to matrix learning in glvq a derivation of lvq like learning based on a heuristic cost function in all cases matrix adaptation allows a significant improvement of the classification accuracy interestingly however the principled behavior of the models with respect to prototype locations and extracted matrix dimensions shows several characteristic differences depending on the data sets

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

two familiar transitive closure algorithms which admit no polynomial time sublinear space implementations any boolean straight line program which computes the transitive closure of an n times n boolean matrix by successive squaring requires time exceeding any polynomial in n if the space used is o n this is the first demonstration of a natural algorithm which 1 has a polynomial time implementation and 2 has a small e g o log 2 n space implementation but 3 has no implementation running in polynomial time and small space simultaneously it is also shown that any implementation of warshall s transitive closure algorithm requires omega n space and that many familiar sorting algorithms exhibit similar behavior

level curvature distribution in a model of two uncoupled chaotic subsystems we study distributions of eigenvalue curvatures for a block diagonal random matrix perturbed by a full random matrix the most natural physical realization of this model is a quantum chaotic system with some inherent symmetry such that its energy levels form two independent subsequences subject to a generic perturbation which does not respect the symmetry we describe analytically a crossover in the form of a curvature distribution with a tunable parameter namely the ratio of intersubsystem intrasubsystem coupling strengths we find that the peak value of the curvature distribution is much more sensitive to the changes in this parameter than the power law tail behavior this observation may help to clarify some qualitative features of the curvature distributions observed experimentally in acoustic resonances of quartz blocks

an efficient data structure for lattice operations in this paper we consider the representation and management of an element set on which a lattice partial order relation is defined in particular let n be the element set size we present an nradn space implicit data structure for performing the following set of basic operations 1 test the presence of an order relation between two given elements in constant time 2 find a path between two elements whenever one exists in o l steps where l is the path length 3 compute the successors and or predecessors set of a given element in o h steps where h is the size of the returned set 4 given two elements find all elements between them in time o k log d where k is the size of the returned set and d is the maximum in degree or out degree in the transitive reduction of the order relation 5 given two elements find the least common ancestor and or the greatest common successor in o sqrt n time 6 given k elements find the least common ancestor and or the greatest common successor in o sqrt n k log n time unless stated otherwise all logarithms are to the base 2 the preprocessing time is o n2 focusing on the first operation representing the building box for all the others we derive an overall nradn space times time bound which beats the order n2 bottleneck representing the present complexity for this problem moreover we will show that the complexity bounds for the first three operations are optimal with respect to the worst case additionally a stronger result can be derived in particular it is possible to represent a lattice in space o n sqrt t where t is the minimum number of disjoint chains which partition the element set

gaussian random matrix process and universal parametric correlations in complex systems we introduce the framework of the gaussian random matrix process as an extension of dyson s gaussian ensembles and use it to discuss the statistical properties of complex quantum systems that depend on an external parameter we classify the gaussian processes according to the short distance diffusive behavior of their energy levels and demonstrate that all parametric correlation functions become universal upon the appropriate scaling of the parameter the class of differentiable gaussian processes is identified as the relevant one for most physical systems we reproduce the known spectral correlators and compute eigenfunction correlators in their universal form numerical evidence from both a chaotic model and weakly disordered model confirms our predictions

distance metric learning for large margin nearest neighbor classification the accuracy of k nearest neighbor knn classification depends significantly on the metric used to compute distances between different examples in this paper we show how to learn a mahalanobis distance metric for knn classification from labeled examples the mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes knn classification using euclidean distances in our approach the metric is trained with the goal that the k nearest neighbors always belong to the same class while examples from different classes are separated by a large margin as in support vector machines svms the margin criterion leads to a convex optimization based on the hinge loss unlike learning in svms however our approach requires no modification or extension for problems in multiway as opposed to binary classification in our framework the mahalanobis distance metric is obtained as the solution to a semidefinite program on several data sets of varying size and difficulty we find that metrics trained in this way lead to significant improvements in knn classification sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster we show how to learn and combine these local metrics in a globally integrated manner

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

regularization in matrix relevance learning in this paper we present a regularization technique to extend recently proposed matrix learning schemes in learning vector quantization lvq these learning algorithms extend the concept of adaptive distance measures in lvq to the use of relevance matrices in general metric learning can display a tendency towards oversimplification in the course of training an overly pronounced elimination of dimensions in feature space can have negative effects on the performance and may lead to instabilities in the training we focus on matrix learning in generalized lvq glvq extending the cost function by an appropriate regularization term prevents the unfavorable behavior and can help to improve the generalization ability the approach is first tested and illustrated in terms of artificial model data furthermore we apply the scheme to benchmark classification data sets from the uci repository of machine learning we demonstrate the usefulness of regularization also in the case of rank limited relevance matrices i e matrix learning with an implicit low dimensional representation of the data

sparsification a technique for speeding up dynamic graph algorithms we provide data strutures that maintain a graph as edges are inserted and deleted and keep track of the following properties with the following times minimum spanning forests graph connectivity graph 2 edge connectivity and bipartiteness in time o n 1 2 per change 3 edge connectivity in time o n 2 3 per change 4 edge connectivity in time o n n per change k edge connectivity for constant k in time o n log n per change 2 vertex connectivity and 3 vertex connectivity in the o n per change and 4 vertex connectivity in time o n n per change further results speed up the insertion times to match the bounds of known partially dynamic algorithms all our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph which we call sparsification

distribution of the spacing between two adjacent avoided crossings we consider the frequency at which avoided crossings appear in an energy level structure when an external field is applied to a quantum chaotic system the distribution of the spacing in the parameter between two adjacent avoided crossings is investigated using a random matrix model we find that the distribution of these spacings is well fitted by a power law distribution for small spacings the powers are 2 and 3 for the gaussian orthogonal ensemble and gaussian unitary ensemble respectively we also find that the distributions decay exponentially for large spacings the distributions in concrete quantum chaotic systems agree with those of the random matrix model

quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles he quantum limited line width of a laser cavity is enhanced above the schawlow townes value by the petermann factor k due to the non orthogonality of the cavity modes we derive the relation between the petermann factor and the residues of poles of the scattering matrix and investigate the statistical properties of the petermann factor for cavities in which the radiation is scattered chaotically for a single scattering channel we determine the complete probability distribution of k and find that the average petermann factor depends non analytically on the area of the opening and greatly exceeds the most probable value for an arbitrary number n of scattering channels we calculate as a function of the decay rate i of the lasing mode we find for n 1 that for typical values of i the average petermann factor propto sqrt n 1 is parametrically larger than unity

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

time reversal symmetry breaking and the statistical properties of quantum systems the breaking of the generalized time reversal symmetry of a quantum chaotic system corresponds to a transition from orthogonal to unitary ensembles of random matrices investigating this transition for the circular ensembles applicable for time dependent periodic systems the authors demonstrate and explain that there exists a relevant difference of the transition rate in comparison with the gaussian ensembles appropriate for quantum conservative systems the above supposition is supported by a numerical study of the eigenvalues and eigenvectors of the periodically kicked top

dimensionality dependent generalization bounds for k dimensional coding schemes the k dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k dimensional vectors and include nonnegative matrix factorization dictionary learning sparse coding k means clustering and vector quantization as special cases previous generalization bounds for the reconstruction error of the k dimensional coding schemes are mainly dimensionality independent a major advantage of these bounds is that they can be used to analyze the generalization error when data are mapped into an infinite or high dimensional feature space however many applications use finite dimensional data features can we obtain dimensionality dependent generalization bounds for k dimensional coding schemes that are tighter than dimensionality independent bounds when data are in a finite dimensional feature space yes in this letter we address this problem and derive a dimensionality dependent generalization bound for k dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error the bound is of order where m is the dimension of features k is the number of the columns in the linear implementation of coding schemes and n is the size of sample when n is finite and when n is infinite we show that our bound can be tighter than previous results because it avoids inducing the worst case upper bound on k of the loss function the proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality dependent bound is an indispensable complement to the dimensionality independent generalization bounds

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

a data structure for lattice representation in this paper we present an implicit data structure for the representation of a partial lattice l s n which allows to test the partial order relation among two given elements in constant time the data structure proposed has an overall o n n space complexity where n is the size of ground set n which we will prove to be optimal in the worst case hence we derive an overall o n n space time bound for the relation testing problem thus beating the o n2 bottle neck representing the present complexity the overall pre processing time is o n2

soft learning vector quantization learning vector quantization lvq is a popular class of adaptive nearest prototype classifiers for multiclass classification but learning algorithms from this family have so far been proposed on heuristic grounds here we take a more principled approach and derive two variants of lvq using a gaussian mixture ansatz we propose an objective function based on a likelihood ratio and derive a learning rule using gradient descent the new approach provides a way to extend the algorithms of the lvq family to different distance measure and allows for the design of soft lvq algorithms benchmark results show that the new methods lead to better classification performance than lvq 2 1 an additional benefit of the new method is that model assumptions are made explicit so that the method can be adapted more easily to different kinds of problems

distribution of multiple avoided crossings numerical evaluation the authors compute numerically the distribution rho delta in of delta in the local minimum distance avoided crossing gap size between neighbouring energy levels of two model systems they find that limit delta in to 0 rho approximately delta in the result differs from that expected for a goe model for which wilkinson 1988 has shown that in the small delta in limit rho delta in is constant they suggest that the discrepancy arises from parametric correlations which are known to exist but are absent from the goe

zeta functions of nearly circular domains we study the zeta functions of negative integer powers of eigenvalues of the laplacian on two dimensional domains which are close to the unit disc d for closed form expressions are obtained which describe when the technique developed is applied to derive an asymptotic expansion for zeta functions of regular n sided polygons in the limit

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