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

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

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

energy fluctuations of pseudointegrable systems with growing surface roughness the eigenfrequencies of two dimensional systems with fractal boundaries and with nonscaling rough boundaries are calculated numerically by the lanczos algorithm and analyzed by means of level statistics the systems are pseudointegrable and the fluctuations of their eigenvalue spectra show a global statistical behavior between the poisson and the wigner distributions with increasing irregularity of the boundary the systems approach the wigner limit and the results seem to depend only on the genus number of the geometry and not on details such as the asymptotic shape of the geometry the type of roughness scaling or nonscaling and the boundary conditions neumann or dirichlet no transition between localized and extended states is found in fractal drums

deformation of varieties defined by sections in homogeneous vector bundles in the present paper we prove some generalizations of a theorem of k kodaira and d spencer they showed in 1958 that any smooth projective algebraic hypersurface which is not a k3 surface remains projective algebraic under small deformation in a previous paper 13 we have shown that the analogous statement remains valid for projective algebraic complete intersections with singularities too recently there has been studied an interesting and rather different generalization by borcea 2 from his point of view the complex projective space is the most simple example in the class of homogeneous rational k ihler manifolds of rank 1 borcea generalizes the result of kodaira and spencer to smooth complete intersections in homogeneous manifolds of this class which covers for instance all grassmannians all these results correspond to special cases of the following question consider a compact complex manifold z together with a vector bundle e and a section of it vanishing along a subvariety x of z of codimension equal to rank e in which case can every small deformation of x be obtained by varying the section in h e a sufficient criterion is the vanishing of the cohomology groups

geometrical theory of diffraction and spectral statistics we investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit by applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit 0 this result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space

periodic orbit theory in fractal drums abstract the level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory we find that the spectral rigidity 3 l which is a measure for the correlations between the eigenvalues decreases to quite small values as compared to systems with only small boundary roughness thereby approaching the behavior of chaotic systems the periodic orbit results are in good agreement with direct calculations of 3 l from the eigenvalues

level statistics and eigenfunctions of pseudointegrable systems dependence on energy and genus number we study the level statistics second half moment i 0 and rigidity ensuremath delta 3 and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g we find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval at low enough energies the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows poisson like behavior at higher energies the level statistics of most systems moves from poisson like toward wigner like behavior with increasing g on investigating the wave functions we find many chaotic functions that can be described as a random superposition of regular wave functions the amplitude distribution p ensuremath psi of these chaotic functions was found to be gaussian with the typical value of the localization volume v mathrm loc ensuremath approx 0 33 for systems with periodic boundaries we find several additional energy regimes where i 0 is relatively close to the poisson limit in these regimes the eigenfunctions are either regular or localized functions where p ensuremath psi is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case an interesting intermediate case between chaotic and localized eigenfunctions also appears

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

ls svm approximate solution to linear time varying descriptor systems this paper discusses a numerical method based on least squares support vector machines ls svms for solving linear time varying initial and boundary value problems in differential algebraic equations daes the method generates a closed form model based approximate solution the results of numerical experiments on different systems with index from 0 to 3 are presented and compared with analytic solutions to confirm the validity and applicability of the proposed method

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

numerical solution of differential algebraic equations with index 2 abstract in this paper numerical solution of differential algebraic equations daes with index 2 has been presented using pade approximation method an index reduction for differential algebraic equation with index 2 is suggested first index 2 system is reduced to index 1 system then we have applied the method for solving differential algebraic equations with index 2 numerically

approximate solutions to ordinary differential equations using least squares support vector machines in this paper a new approach based on least squares support vector machines ls svms is proposed for solving linear and nonlinear ordinary differential equations odes the approximate solution is presented in closed form by means of ls svms whose parameters are adjusted to minimize an appropriate error function for the linear and nonlinear cases these parameters are obtained by solving a system of linear and nonlinear equations respectively the method is well suited to solving mildly stiff nonstiff and singular odes with initial and boundary conditions numerical results demonstrate the efficiency of the proposed method over existing methods

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces