analysis of the response of a strongly nonlinear damped system using a differential transformation technique the differential transformation technique is applied to solve a nonlinear differential equation that describes the underdamped and the overdamped motion of a system subjected to external excitations numerical results are compared to those obtained by the runge kutta method to illustrate the preciseness and effectiveness of the proposed method

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

spacing distributions for rhombus billiards we show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the wigner dyson family of random matrix theory and the poisson distribution some of the distributions find explanation in a recent work of bogomolny gerland and schmit for the irrational billiards despite ergodicity we get the same distribution for the examples considered once again distinct from the wigner dyson distributions all the results are obtained numerically by a method that allows us to reach very high energies

rademacher penalties and structural risk minimization we suggest a penalty function to be used in various problems of structural risk minimization this penalty is data dependent and is based on the sup norm of the so called rademacher process indexed by the underlying class of functions sets the standard complexity penalties used in learning problems and based on the vc dimensions of the classes are conservative upper bounds in a probabilistic sense uniformly over the set of all underlying distributions for the penalty we suggest thus for a particular distribution of training examples one can expect better performance of learning algorithms with the data driven rademacher penalties we obtain oracle inequalities for the theoretical risk of estimators obtained by structural minimization of the empirical risk with rademacher penalties the inequalities imply some form of optimality of the empirical risk minimizers we also suggest an iterative approach to structural risk minimization with rademacher penalties in which the hierarchy of classes is not given in advance but is determined in the data driven iterative process of risk minimization we prove probabilistic oracle inequalities for the theoretical risk of the estimators based on this approach as well

models of intermediate spectral statistics based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the anderson model at the transition point a simple theoretical model where such statistics can be obtained analytically is briefly discussed a few models with similar behavior are considered in particular we analytically found the eigenvalue statistics of a poisson distributed matrix perturbed by a rank one matrix which is a good model for spectral statistics of a singular billiard

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

intermediate statistics in quantum maps we present a one parameter family of quantum maps whose spectral statistics are of the same intermediate type as observed in polygonal quantum billiards our central result is the evaluation of the spectral two point correlation form factor at small argument which in turn yields the asymptotic level compressibility for macroscopic correlation lengths

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

range of spectral correlations in pseudointegrable systems gaussian orthogonal ensemble statistics in a rectangular membrane with a point scatterer conventional wisdom holds that a finite reverberant system with chaotic ray trajectories will have at high frequencies eigenvalue statistics identical to those of the gaussian orthogonal ensemble goe of random matrices it also holds that a nonchaotic system will have simple poissonian statistics recent experiments on the eigenvalues of elastic blocks with angled cuts and recent calculations of the eigenfrequencies of membranes with staircaselike jagged boundaries and the eigenfrequencies of a rectangular domain with a single isotropic point scatterer have however found goe statistics even in these pseudointegrable systems even though all rays in such systems are nonchaotic in this work the rectangular domain with a single isotropic point scatterer is studied further in contrast to recent related work the scatterer is characterized here by its t matrix and scattering cross section it is shown that the long range level repulsion in this system is not in precise accord with the predictions of the goe nor is the long range spectral rigidity goe does though correctly describe the short range statistics a quantitative prediction for the range in which goe applies is advanced based upon the lifetime of a ray against mixing i e based upon the scattering cross section of the scatterer this prediction is corroborated by numerical calculations of the eigenfrequencies

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

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

selective rademacher penalization and reduced error pruning of decision trees rademacher penalization is a modern technique for obtaining data dependent bounds on the generalization error of classifiers it appears to be limited to relatively simple hypothesis classes because of computational complexity issues in this paper we nevertheless apply rademacher penalization to the in practice important hypothesis class of unrestricted decision trees by considering the prunings of a given decision tree rather than the tree growing phase this study constitutes the first application of rademacher penalization to hypothesis classes that have practical significance we present two variations of the approach one in which the hypothesis class consists of all prunings of the initial tree and another in which only the prunings that are accurate on growing data are taken into account moreover we generalize the error bounding approach from binary classification to multi class situations our empirical experiments indicate that the proposed new bounds outperform distribution independent bounds for decision tree prunings and provide non trivial error estimates on real world data sets

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

rectangular billiard in the presence of a flux line we analyze the quantum spectrum of a rectangular billiard coupled to flux line the nearest neighbor spacing distribution shows a transition from poisson to almost wigner distribution as the flux parameter is varied even though classically the system remains pseudointegrable and nonchaotic in fact the level statistics do not display any generic behavior the calculations presented here are simple and avoid the complexity involved in modeling the singular quantum billiard problem

an efficient algorithm for optimal pruning of decision trees abstract pruning decision trees is a useful technique for improving the generalization performance in decision tree induction and for trading accuracy for simplicity in other applications in this paper a new algorithm called opt 2 for optimal pruning of decision trees is introduced the algorithm is based on dynamic programming in its most basic form the time and space complexities of opt 2 are both nc where n is the number of test nodes in the initial decision tree and c is the number of leaves in the target pruned decision tree this is an improvement over the recently published opt algorithm of bohanec and bratko which is the only known algorithm for optimal decision tree pruning especially in the case of heavy pruning and when the tests of the given decision tree have many outcomes if so desired the space required by opt 2 can further be reduced by a factor of r at the cost of increasing the execution time by a factor that is bounded above by r 1 2 this is a considerable overestimate however from a practical point of view opt 2 enjoys considerable flexibility in various aspects and is easy to implement

zeta functions of the dirac operator on quantum graphs we construct spectral zeta functions for the dirac operator on metric graphs we start with the case of a rose graph a graph with a single vertex where every edge is a loop the technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices the regularized spectral determinant of the dirac operator is also obtained as the derivative of the zeta function at a special value in each case the zeta function is formulated using a contour integral method which extends results obtained for laplace and schrodinger operators on graphs

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

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

level spacing distribution of a singular billiard in this paper we examine the level spacing distribution p s of the rectangular billiard with a single point like scatterer which is known as pseudointegrable it is shown that the observed p s is a new type which is quite different from the previous conclusion even in the strong coupling limit the poisson like behavior rather than wigner like is seen for s 1 although the level repulsion still remains in the small s region the difference from the previous works is analyzed in detail

wave chaos in quantum systems with point interaction we study perturbations of the quantized version 0 of integrable hamiltonian systems by point interactions we relate the eigenvalues of to the zeros of a certain meromorphic function assuming the eigenvalues of 0 are poisson distributed we get detailed information on the joint distribution of the zeros of and give bounds on the probability density for the spacings of eigenvalues of our results confirm the wave chaos phenomenon as different from the quantum chaos phenomenon predicted by random matrix theory

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

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

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

periodic orbits contribution to the 2 point correlation form factor for pseudo integrable systems using heuristic arguments based on the trace formulas we relate the 2 point correlation form factor k2 at small values of with sums over classical periodic orbits for typical examples of pseudo integrable systems the later sums have been explicitly calculated for the following models i plane billiards in the form of right triangles with one angle n and ii rectangular billiards with the aharonov bohm flux line in the first model using the properties of the veech structure it is shown that k2 0 n e n 3 n 2 where e n 0 for odd n e n 2 for even n not divisible by 3 and e n 6 for even n divisible by 3 for completeness we also recall informally the main features of the veech construction in the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle when these ratios are non commensurable irrational numbers k2 0 1 3 where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when the comparison of these results with numerical calculations of the form factor is discussed in detail the above values of k2 0 differ from all known examples of spectral statistics thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo integrable systems

comment on models of intermediate spectral statistics in this comment we point out that the semi poisson is well suited only as a reference point for the so called intermediate statistics which cannot be interpreted as a universal ensemble like the gaussian orthogonal ensemble or the poissonian statistics in ref 2 it was proposed that the nearest neighbor distribution p s of the spectrum of a poissonian distributed matrix perturbed by a rank one matrix is similar to the semi poisson distribution we show however that the p s of this model differs considerably in many aspects from the semi poisson in addition we give an asymptotic formula for p s as s 0 which gives p 0 pisqrt 3 2 for the slope at s 0 this is different not only from the goe case but also from the semi poisson prediction

point perturbations of circle billiards the spectral statistics of the circular billiard with a point scatterer is investigated in the semiclassical limit the spectrum is demonstrated to be composed of two uncorrelated level sequences the first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point scatterer the nearest neighbour spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular

complexity regularization via localized random penalties in this article model selection via penalized empirical loss minimization in nonparametric classification problems is studied data dependent penalties are constructed which are based on estimates of the complexity of a small subclass of each model class containing only those functions with small empirical loss the penalties are novel since those considered in the literature are typically based on the entire model class oracle inequalities using these penalties are established and the advantage of the new penalties over those based on the complexity of the whole model class is demonstrated

nearest neighbor distribution for singular billiards the exact computation of the nearest neighbor spacing distribution p s is performed for a rectangular billiard with a pointlike scatterer inside for periodic and dirichlet boundary conditions and it is demonstrated that when s infinity this function decreases exponentially together with the results of bogomolny gerland and schmit phys rev e 63 036206 2001 it proves that spectral statistics of such systems is of intermediate type characterized by level repulsion at small distances and exponential fall off of the nearest neighbor distribution at large distances the calculation of the nth nearest neighbor spacing distribution p n s and its asymptotics is performed as well for any boundary conditions