nodal line densities of chaotic quantum billiard modes satisfying mixed boundary conditions statistics of nodal lines for eigenmodes u in the stadium are computed and compared with previously derived formulae for monochromatic boundary adapted gaussian random waves in the plane these modes and random waves satisfy the helmholtz equation and mixed boundary conditions in which a linear combination of u and its normal derivative must vanish for the density of nodal lines and the excess density of nodal lines arising from the boundary the gaussian model accurately describes the statistics of the billiard eigenfunctions

centers of down up algebras abstract down up algebras which originated in the study of differential posets were recently defined and studied by benkart and roby benkart posed an open problem in her paper to determine the centers of all down up algebras here in this paper we completely solve this problem as an application we also get the centers of homogenizations of down up algebras

wave chaos in acoustics and elasticity interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area especially in elasticity and acoustics the theory has indeed been developed primarily in a quantum context it is increasingly becoming clear however that important applications lie in the field of mechanical vibrations and acoustics we provide an overview over basic concepts in this emerging field of wave chaos this ranges from ray approximations of the green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics both experimentally and from a theoretical numerical point of view

fuzzy based learning rate determination for blind source separation many independent component analysis ica algorithms have been proposed for blind source separation these algorithms belong to the lms type algorithm in natural hence the choice of the step size reflects a tradeoff between misadjustment and the speed of convergence based on the separation state of outputs of the neural network for ica the paper develops a fuzzy inference based step size selection algorithm the fuzzy inference system consists of two inputs the second and higher order correlation coefficients of output components and one output the fuzzy learning rate in this way the ica algorithms become more efficient which is verified by simulation results

on category o over triangular generalized weyl algebras abstract we analyze the bgg category o over a large class of generalized weyl algebras henceforth termed gwas given such a triangular gwa for which category o decomposes into a direct sum of subcategories we study in detail the homological properties of blocks with finitely many simples as consequences we show that the endomorphism algebra of a projective generator of such a block is quasi hereditary finite dimensional and graded koszul we also classify all tilting modules in the block as well as all submodules of all projective and tilting modules finally we present a novel connection between blocks of triangular gwas and young tableaux which provides a combinatorial interpretation of morphisms and extensions between objects of the block

on homological smoothness of generalized weyl algebras over polynomial algebras in two variables abstract homological smoothness and twisted calabi yau property of generalized weyl algebras over polynomial algebras in two variables is studied a necessary and sufficient condition to be homologically smooth is given the nakayama automorphisms of such algebras are also computed in terms of the jacobian determinants of defining automorphisms

blind separation of instantaneous mixture of sources via an independent component analysis in this paper we introduce a procedure for separating a multivariate distribution into nearly independent components based on minimizing a criterion defined in terms of the kullback leibner distance by replacing the unknown density with a kernel estimate we derive useful forms of this criterion when only a sample from that distribution is available we also compute the gradient and hessian of our criteria for use in an iterative minimization setting this gradient to zero yields a set of separating functions similar to the ones considered in the source separation problem except that here these functions are adapted to the observed data finally some simulations are given illustrating the good performance of the method

simple modules of the witten woronowicz algebra abstract it was proved in v v bavula d a jordan trans amer math soc 353 2 2001 769 794 that witten s second deformation and woronowicz s deformation are isomorphic algebras for a class of generalized weyl algebras including the two deformations above simple modules are being classified up to irreducible elements of certain noncommutative euclidean rings

nodal densities of gaussian random waves satisfying mixed boundary conditions nodal statistics are derived for boundary adapted gaussian random waves u i v in th ep lane satisfying the helmholtz equation and mixed boundary conditions in which a linear combination of and its normal derivative characterized by a parameter a v anishes on aline for both the density of nodal lines for re u and the density of nodal points for complex e ffects of the boundary persist surprisingly infinitely far from the boundary and also surprisingly are independent of a a sa increases from the dirichlet value a 0 nodal line and point structures migrate from the boundary and can be described analytically

down up algebras and their representations abstract the down up algebras were introduced in g benkart and t roby 1998 j algebra209 305 344 and g benkart 1998 contemp math 224 29 45 their representations were studied via category o and verma modules here we prove that when 0 they are hyperbolic rings and we study their representations via their left spectrum as defined in a l rosenberg 1995 noncommutative algebraic geometry and representations of quantized algebras kluwer academic dordrecht the center of these algebras is computed using results from hyperbolic rings which are also proved here the case when the down up algebra is a finite module over its center is considered in detail and its relation to the clifford algebra of a binary form of arbitrary degree is established

serial updating rule for blind separation derived from the method of scoring in the context of blind source separation the method of scoring based on the inverse of the fisher information matrix fim becomes the serial updating learning rule with an equivariant property this learning rule can be simplified to a low complexity algorithm by using the asymptotic form of the ftm around the equilibrium the simplified learning rule is still general enough to include some existing equivariant blind separation algorithms as its special cases

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

generalized weyl algebras are tensor krull minimal abstract formulae for calculating the krull dimension of noetherian rings obtained by the authors and their collaborators are used to calculate krull dimension for certain classes of algebras an f algebra t is said to be tensor krull minimal tkm with respect to a class of f algebras if k t b k t k b for each b we show that generalized weyl algebras over affine commutative f algebras where f is an uncountable algebraically closed field are tkm with respect to the class of countably generated left noetherian f algebras this simplifies the task of calculating many krull dimensions in addition we develop an improved formula for the krull dimension of a skew laurent extension d x x 1 where d is a polynomial algebra over an algebraically closed field and is an affine automorphism finally we calculate the krull dimension of the noetherian down up algebras introduced by benkart

representations of a clifford algebra that are azumaya algebras and generate the brauer group abstract clifford algebras of forms of degree d 2 are infinite dimensional see theorem 3 p 272 of l childs j lin mul alg 5 1978 267 278 and corollary 2 p 273 of m orzech and l small lecture notes in pure and applied mathematics vol 11 dekker new york 1975 in the case of a non singular binary cubic form over field f d haile amer math j 106 no 6 1984 1269 1280 demonstrated an injective representation of the clifford algebra in m 3 k where k is a transcendental extension of f he showed that the clifford algebra is thus an azumaya algebra over the coordinate ring of an elliptic curve and that for any algebraic extension f of f the group of f points on the curve map homomorphically into the brauer group of f

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

blind separation of signals with mixed kurtosis signs using threshold activation functions a parameterized activation function in the form of an adaptive threshold for a single layer neural network which separates a mixture of signals with any distribution except for gaussian is introduced this activation function is particularly simple to implement since it neither uses hyperbolic nor polynomial functions unlike most other nonlinear functions used for blind separation for some specific distributions the stable region of the threshold parameter is derived and optimal values for best separation performance are given if the threshold parameter is made adaptive during the separation process the successful separation of signals whose distribution is unknown is demonstrated and compared against other known methods

down up algebras abstract the algebra generated by the down and up operators on a differential or uniform partially ordered set poset encodes essential enumerative and structural properties of the poset motivated by the algebras generated by the down and up operators on posets we introduce here a family of infinite dimensional associative algebras called down up algebras we show that down up algebras exhibit many of the important features of the universal enveloping algebrau s l2 of the lie algebra s l 2including a poincare birkhoff witt type basis and a well behaved representation theory we investigate the structure and representations of down up algebras and focus especially on verma modules highest weight representations and category o modules for them we calculate the exact expressions for all the weights since that information has proven to be particularly useful in determining structural results about posets

nodal structure of chaotic eigenfunctions we investigate the distribution of nodes at and beyond the classical turning point of idealized chaotic quantum eigenfunctions a formula for the density of nodes is derived in the semiclassical limit and the rate at which this density falls off as one moves into the forbidden region is also studied the discussion is supported by numerical results corrections to the bessel function correlation in the classically allowed region are necessary for finite and are given here

distribution of nearest distances between nodal points for the berry function in two dimensions according to berry a wave chaotic state may be viewed as a superposition of monochromatic plane waves with random phases and amplitudes here we consider the distribution of nodal points associated with this state using the property that both the real and imaginary parts of the wave function are random gaussian fields we analyze the correlation function and densities of the nodal points using two approaches the poisson and bernoulli we derive the distribution of nearest neighbor separations furthermore the distribution functions for nodal points with specific chirality are found comparison is made with results from numerical calculations for the berry wave function

the simple modules of certain generalized crossed products abstract we consider a dedekind domaindand az graded ring r i z r i withr0 dand eachri dvibeing a freed module of rank 1 the structure ofris described by an automorphism ofdand a generalized 2 cocyclec z z dnot necessarily taking its values in the units ofd the aim of this paper is to classify the simpler modules say in the case wherec i j 0 for alli j z we also deal with this problem in then graded case as a consequence we obtain a description of the simple modules of some classical algebras and of generalized weyl algebras

irreducible representations of witten s deformations ofu sl2 abstract using e witten s approach based on duality in conformal field theory one can obtain a 7 parameter family of deformations of u sl 2 called the witten s deformations le bruyn singled out a subfamily called conformal sl 2 algebras and studied their finite dimensional representations here we consider all of witten s deformations we compute the left spectrum of these algebras and thereby classify irreducible representations for generic values of parameters and finite dimensional representations in certain roots of unity cases

on azumaya algebras arising from clifford algebras in this paper we continue the investigation begun in haile 4 and tesser 12 into the structure of the clifford algebra of a form if f is a form of degree d in n variables over a field f then the clifford algebra off is the f algebra c r t where r fix x is the free associative f algebra in n variables and t is the ideal generated by a x x y f 1 cx jcxl cl f if d 2 this is the classical clifford algebra of a quadratic form if d 2 then this is also called the generalized clifford algebra and has been studied by various authors see roby 111 revoy lo childs 2 in haile 4 it was shown that if d 3 and n 2 i e iffis a binary cubic form and the characteristic of the field f is not 2 or 3 then the clifford algebra c of the form is azymaya with center the affine coordinate ring of an explicitly given elliptic curve over f the center was also determined by heerema in 6 moreover the algebra c can be extended to an azumaya algebra over the projective closure of the elliptic curve applying the pairing investigated by lichtenbaum 4 and manin 9 one obtains then for each field extension k f a homomorphism from the group of k rational points on the elliptic curve to the brauer group of k given by specialization in lo revoy using results of roby 111 exhibited an explicit f basis for the clifford algebra of an arbitrary form using this basis it is easy to

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

hochschild homology and cohomology of down up algebras abstract we present a detailed computation of the cyclic and the hochschild homology and cohomology of generic and 3 calabi yau homogeneous down up algebras this family was defined by benkart and roby in 3 in their study of differential posets our calculations are completely explicit by making use of the koszul bimodule resolution and some arguments similar to those used in 13 to compute the hochschild cohomology of yang mills algebras

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

curved boundary corrections to nodal line statistics in chaotic billiards a gaussian random wavefunction that satisfies dirichlet and neumann conditions locally on a convex circular boundary is introduced the average of the square of the wavefunction and its derivatives are computed and their asymptotics studied in the semi classical limit the mean nodal line length l is calculated and the first order boundary effect shown to be of order log k where k is the wavenumber in the limit of vanishing boundary curvature large boundary radius these results are shown to approach those for a straight wall boundary

statistics of nodal lines and points in chaotic quantum billiards perimeter corrections fluctuations curvature for real time reversal symmetric quantum billiards the mean length l of nodal line is calculated for the nth mode n 1 with wavenumber k using a gaussian random wave model adapted locally to satisfy dirichlet or neumann boundary conditions the leading term is of order k i e n and the first perimeter correction dominated by an unanticipated long range boundary effect is of order log k i e log n with the same sign negative for both boundary conditions the leading order state to state fluctuations l are of order log k for the curvature of nodal lines and 2 are of order k but 3 and higher moments diverge for complex e g aharonov bohm billiards the mean number n of nodal points phase singularities in the mode has a leading term of order k2 i e n the perimeter correction again a long range effect is of order klog k i e nlog n and positive notwithstanding nodal depletion near the boundary and the fluctuations n are of order k log k generalizations of the results for mixed robin boundary conditions are stated

signed zeros of gaussian vector fields density correlation functions and curvature we calculate correlation functions of the signed density of zeros of gaussian distributed vector fields we are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract riemann cartan or riemannian manifold as an application we discuss one and two point functions the zeros of a two dimensional gaussian vector field model the distribution of topological defects in the high temperature phase of two dimensional systems with orientational degrees of freedom such as superfluid films thin superconductors and liquid crystals

screening of charged singularities of random fields many types of point singularity have a topological index or charge associated with them for example the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple zero enabling the zeros to be assigned charges of 1 in random fields we can define a correlation function for the charge weighted density of singularities for many types of topologically charged singularity this correlation function satisfies an identity which shows that the singularities screen each other perfectly a positive singularity is surrounded by an excess of concentration of negatives which exactly cancel its charge and vice versa this paper gives a simple and widely applicable derivation of this result a counterexample where screening is incomplete is also exhibited

statistics of nodal points of in plane random waves in elastic media we consider the nodal points nps u 0 and v 0 of the in plane vectorial displacements u u v which obey the navier cauchy equation similar to the berry conjecture of quantum chaos we present the in plane eigenstates of chaotic billiards as the real part of the superposition of longitudinal and transverse plane waves with random phases by an average over random phases we derive the mean density and correlation function of nps consequently we consider the distribution of the nearest distances between nps

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

on the existence of universal nonlinearities for blind source separation many density based methods for blind signal separation employ one or more models for the unknown source distribution s this paper considers the issue of density model mismatch in maximum likelihood ml type blind signal separation algorithms we show that the score function nonlinearity which was previously derived from the standpoint of statistical efficiency is also the most robust in maintaining a separation solution for the ml algorithm class we also consider the existence of a universally applicable nonlinearity for separating all signal types deriving two results first among nonlinearities with a convergent taylor series a single fixed nonlinearity for universal separation using the natural gradient algorithm cannot exist second among nonlinearities with a single adjustable parameter a previously proposed threshold nonlinearity can separate all signals with symmetric amplitude distributions as long as the threshold parameter is properly chosen the design of difficult to separate signal distributions is also discussed

classification of factorial generalized down up algebras we determine when a generalized down up algebra is a noetherian unique factorisation domain or a noetherian unique factorisation ring

down up algebras and their representation theory abstract a class of algebras called down up algebras was introduced by g benkart and t roby 1998 j algebra 209 305 344 we classify the finite dimensional simple modules over noetherian down up algebras and show that in some cases every finite dimensional module is semisimple we also study the question of when two down up algebras are isomorphic

projective resolutions of associative algebras and ambiguities abstract the aim of this article is to give a method to construct bimodule resolutions of associative algebras generalizing bardzell s well known resolution of monomial algebras we stress that this method leads to concrete computations providing thus a useful tool for computing invariants associated to the considered algebras we illustrate how to use it by giving several examples in the last section of the article in particular we give necessary and sufficient conditions for noetherian down up algebras to be 3 calabi yau

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

self adjoint extensions of the laplace beltrami operator and unitaries at the boundary abstract we construct in this article a class of closed semi bounded quadratic forms on the space of square integrable functions over a smooth riemannian manifold with smooth compact boundary each of these quadratic forms specifies a semi bounded self adjoint extension of the laplace beltrami operator these quadratic forms are based on the lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible the corresponding quadratic forms are semi bounded below and closable finally the representing operators correspond to semi bounded self adjoint extensions of the laplace beltrami operator this family of extensions is compared with results existing in the literature and various examples and applications are discussed

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

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

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

correlations and screening of topological charges in gaussian random fields two point topological charge correlation functions of several types of geometric singularity in gaussian random fields are calculated explicitly using a general scheme zeros of n dimensional random vectors signed by the sign of their jacobian determinant critical points gradient zeros of real scalars in two dimensions signed by the hessian and umbilic points of real scalars in two dimensions signed by their index the functions in each case depend on the underlying spatial correlation function of the field these topological charge correlation functions are found to obey the first stillinger lovett sum rule for ionic fluids

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

adaptive online learning algorithms for blind separation maximum entropy and minimum mutual information there are two major approaches for blind separation maximum entropy me and minimum mutual information mmi both can be implemented by the stochastic gradient descent method for obtaining the demixing matrix the mi is the contrast function for blind separation the entropy is not to justify the me the relation between me and mmi is first elucidated by calculating the first derivative of the entropy and proving that the mean subtraction is necessary in applying the me and at the solution points determined by the mi the me will not update the demixing matrix in the directions of increasing the cross talking second the natural gradient instead of the ordinary gradient is introduced to obtain efficient algorithms because the parameter space is a riemannian space consisting of matrices the mutual information is calculated by applying the gram charlier expansion to approximate probability density functions of the outputs finally we propose an efficient learning algorithm that incorporates with an adaptive method of estimating the unknown cumulants it is shown by computer simulation that the convergence of the stochastic descent algorithms is improved by using the natural gradient and the adaptively estimated cumulants