universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

pendulum elliptic functions and relative cohomology classes revisiting canonical integration of the classical pendulum around its unstable equilibrium normal hyperbolic canonical coordinates are constructed and an identity between elliptic functions is found whose proof can be based on symplectic geometry and global relative cohomology alternatively it can be reduced to a well known identity between elliptic functions normal canonical action angle variables are also constructed around the stable equilibrium and a corresponding identity is exhibited

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

on linear prediction models constrained to have unit modulus poles and their use for sinusoidal frequency estimation linear prediction models with their poles restricted to the unit circle can be efficiently determined by using the levinson or the split levinson algorithm a simple proof of this property is presented the prime application of linear prediction models constrained to have unit modulus poles is the estimation of sinusoidal frequencies the consistency properties of the corresponding frequency estimates are analyzed it is shown that in the presence of noise the estimates are inconsistent an explicit expression for the asymptotic bias is provided

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

a note on minimizing the prediction error when the zeros are restricted to the unit circle it is shown that the problem of minimizing the prediction error under the constraint that the zeros of the predictor polynomial be on the unit circle can be solved by using a modification of the levinson algorithm consequences of this result are discussed

the zeroes of the partition function of the random energy model abstract the random energy model is one of the simplest disordered models containing some of the physics of spin glasses the zeroes of the partition function in the complex temperature plane are determined for this model they either lie on simple lines or are dense in some regions of the complex plane

on the shape of numerical range of matrix polynomials abstract the numerical range of an n n matrix polynomial p a m m a 1 a 0 is defined by w p c x p x 0 x c n x 0 in this paper we investigate the shape of w p by using the notion of local dimension the numerical range of first order matrix polynomials is always simply connected the special cases of diagonal matrix polynomials and 2 2 matrix polynomials are also considered

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

a circular stability test for general polynomials abstract we extend a new stability test proposed recently for discrete system polynomials 1 to polynomials with complex coefficients the method is based on a three term recursion of a conjugate symmetric sequence of polynomials the complex version has the same relative improved efficiency as the real version in comparison to the classical schur cohn formulation for counting the number of zeros of a polynomial with respect to the unit circle furthermore if desirable the complex text can be carried out using only real polynomials and arithmetic

self inversive hilbert space operator polynomials with spectrum on the unit circle abstract hilbert space operator polynomials with self inversive structure are studied if the inner numerical radius of an associated polynomial is greater than or equal to one then the spectrum lies on the unit circle and consists of normal approximate characteristic values

nonlinear heat conduction with absorption space localization and extinction in finite time additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions the statements answer the questions when the support of the solution is bounded in space and when the solution dies in finite time

on the polynomials with all their zeros on the unit circle abstract a necessary and sufficient condition for all the zeros of an algebraic polynomial to lie on the unit circle is given as a consequence of this some sufficient conditions are obtained the advantage is that these conditions on the coefficients are very easy to verify

the boundary of the numerical range of matrix polynomials abstract some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper we also consider isolated points of the numerical range and the location of the numerical range in a circular annulus

on computing the generalized crawford number of a matrix abstract we use methods of geometric computing combined with hermitean matrix eigenvalue eigenvector evaluations to find the generalized crawford number of a pair of hermitean matrices s 1 and s 2 quickly and to high precision the classical crawford number is defined as the minimal distance from zero in c to the field of values of the associated complex matrix a s 1 is 2 if zero lies outside the field of values of a we describe test and compare a geometry based matlab code for finding for the generalized crawford number that measures the smallest distance of zero from the boundary of the field of values of a even if zero lies inside

the poles of symmetric linear prediction models lie on the unit circle symmetric linear prediction models have their poles on the unit circle a simple proof of this result is presented

the numerical range of linear pencils of 2 by 2 matrices abstract let a b m n the numerical range of a linear pencil a b is defined by w a b t c tw aw w bw 0 for some non zero w c n the numerical range of a linear pencil of n by n matrices coincides with the union of the numerical ranges of linear pencils of 2 by 2 matrices we emphasize on computation of the numerical range of linear pencils of 2 by 2 matrices let a b m 2 with constant diagonals we show that the boundary of w a b lies on a rational curve of degree at most 4 by using lagrange s multiplier and implicit function we describe the boundary of w a b for general 2 by 2 matices

birkhoff james e orthogonality sets of vectors and vector valued polynomials abstract consider a complex normed linear space x and let x with 0 motivated by recent works on rectangular matrices and on normed linear spaces we study the birkhoff james e orthogonality set of with respect to give an alternative definition for this set and explore its rich structure we also introduce the birkhoff james e orthogonality set of polynomials in one complex variable whose coefficients are members of x and survey and record extensions of results on matrix polynomials to these vector valued polynomials

geometrical properties of numerical range of matrix polynomials abstract through the linearization of a matrix polynomial p the symmetry and the sharp points of the numerical range p are studied

detecting a definite hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem and associated nearness problems an important class of generalized eigenvalue problems ax lambdabx is those in which a and b are hermitian and some real linear combination of them is definite for the quadratic eigenvalue problem qep lambda 2 a lambdab c x 0 with hermitian a b and c and positive definite a particular interest focuses on problems in which x bx 2 4 x ax x cx is one signed for all non zero x for the positive sign these problems are called hyperbolic and for the negative sign elliptic the important class of overdamped problems arising in mechanics is a sub class of the hyperbolic problems for each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class for definite pairs a b the distance is the crawford number and we derive bisection and level set algorithms both for testing its positivity and for computing it testing hyperbolicity of a qep is shown to reduce to testing a related pair for definiteness the distance to the nearest non hyperbolic or non elliptic n x n qep is shown to be the solution of a global minimization problem with n 1 dearees of freedom numerical results are given to illustrate the theory and algorithms c 2002 published by elsevier science inc

existence results for doubly nonlinear higher order parabolic equations on unbounded domains on demontre un resultat d existence pour le probleme aux valeurs limites et initiales u q 1 sign u t a 1 u f sur 0 t q u satisfait des conditions de dirichlet sur 0 t u u 0 sur pour t 0 a 1 u 1 m m d d u p 1 sign d u pour p q 1 m n 1 et r n un ensemble ouvert

on the connectedness of numerical range of matrix polynomials abstract an investigation on nonconnectedness of numerical range for monic matrix polynomials l is undertaking here the distribution of eigenvalues of l to the components of numerical range and some other algebraic properties are also presented

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

on source type solutions and the cauchy problem for a doubly degenerate sixth order thin film equation i local oscillatory properties abstract as a key example the sixth order doubly degenerate parabolic equation from thin film theory u t u m u x x x x x n u x x x x x x in r r with two parameters n 0 and m n n 2 is considered in this first part of the research various local properties of its particular travelling wave and source type solutions are studied most complete analytic results on oscillatory structures of these solutions of changing sign are obtained for m 1 by an algebraic geometric approach with extension by continuity for m 1

manin products koszul duality loday algebras and deligne conjecture in this article we give a conceptual definition of manin products in any category endowed with two coherent monoidal products this construction can be applied to associative algebras non symmetric operads operads colored operads and properads presented by generators and relations these two products called black and white are dual to each other under koszul duality functor we study their properties and compute several examples of black and white products for operads these products allow us to define natural operations on the chain complex defining cohomology theories with these operations we are able to prove that deligne s conjecture holds for a general class of operads and is not specific to the case of associative algebras finally we prove generalized versions of a few conjectures raised by m aguiar and j l loday related to the koszul property of operads defined by black products these operads provide infinitely many examples for this generalized deligne s conjecture

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

structure of boundary blow up for higher order quasilinear parabolic equations singularity formation phenomena for 2 m th order quasilinear parabolic equations are studied by using energy estimates related to saint venant s principle sharp estimates of propagation of singularities generated by boundary global and regional blow up regimes are established

rigid motions action angles relative cohomology and polynomials with roots on the unit circle revisiting canonical integration of the classical solid near a hyperbolic or elliptic uniform rotation normal canonical coordinates p q are constructed so that the hamiltonian becomes a function normal form of x pq or of x p2 q2 the two cases are treated simultaneously distinguishing them respectively by a label a in terms of various power series with coefficients which are shown to be polynomials in a variable ra2 depending on the inertia moments the normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without reference to the construction of the normal coordinates via elliptic integrals unlike the derivation of the normal coordinates p q results and conjectures also emerge about the properties of the above polynomials and the location of their roots in particular a class of polynomials with all roots on the unit circle arises

finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption the energy solutions to the equation have finite speed of propagation if l or l if 1 vq t is uniformly bounded for t 0 localisation property and if q sharp upper bounds of the interface or free boundary are obtained we use a weighted energy method the weights being powers of the distance to a variable half space we also study decay rates as t and extinction in finite time for bounded and unbounded domains with null dirichlet boundary conditions our equation includes the porous media equation with absorption analogous results hold if m is replaced by an appropriate quasilinear elliptic operator

dialgebra cohomology as a g algebra it is well known that the hochschild cohomology h a a of an associative algebra a admits a g algebra structure in this paper we show that the dialgebra cohomology hy d d of an associative dialgebra d has a similar structure which is induced from a homotopy g algebra structure on the dialgebra cochain complex cy d d

numerical range of linear pencils abstract consider a linear pencil a b where a and b are n n complex matrices the numerical range of a b is defined as w a b c x a b x 0 for some nonzero x c n in this paper we study the geometrical properties of w a b with emphasis to its boundary an answer to the problem of the numerical approximation of w a b when one of the coefficients a and b is hermitian is presented the numerical range of a matrix on an indefinite inner product space is also considered

matrix based higher order spectral analysis for three wave coupling processes composite linear and quadratic systems produce three wave coupling when stimulated by random phase input sinusoids due to the nonlinearity of the system the output frequencies are arithmetically related to the input using third order cumulant statistics and their associated bispectrum techniques are devised based on phase insensitive matrix structures for detection and frequency estimation of coupling frequencies the separation of the third order cumulant series into symmetric and skew symmetric portions allows one to exploit their characteristic eigendecompositions for frequency estimation after symmetrization biphases can be easily extracted as coefficients of the cumulant sequence using a generalized eigenvector representation one can relate symmetric and skew symmetric bases by a subspace rotation algorithm biphases can be estimated directly from generalized eigenvalues of the matrix pencil formed by symmetric and skew symmetric matrices the dimensionality of the matrices can be reduced through the use of cumulant projections that yield a slice of the bispectrum the radon transform procedure is related to bispectral processing through an isotropic radial slice volterra filter the compact third order kronecker product matrix formulation and algorithms for coupling frequency estimation can also be converted for use in biphase estimation simulations showing the performance of the above procedures are also presented for both synthetic and biomedical time series

birkhoff james approximate orthogonality sets and numerical ranges abstract in this paper the notion of birkhoff james approximate orthogonality sets is introduced for rectangular matrices and matrix polynomials the proposed definition yields a natural generalization of standard numerical range and q numerical range and also of recent extensions sharing with them several geometric properties

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

the nearest definite pair for the hermitian generalized eigenvalue problem the generalized eigenvalue problem ax lambda bx has special properties when a b is a hermitian and definite pair given a general hermitian pair a b it is of interest to find the nearest definite pair having a specified crawford number delta 0 we solve the problem in terms of the inner numerical radius associated with the field of values of a ib we show that once the problem has been solved it is trivial to rotate the perturbed pair a da b db to a pair widetilde a widetilde b for which lambda min widetilde b achieves its maximum value delta which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by inverting b numerical examples are given to illustrate the analysis

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

point equation of the boundary of the numerical range of a matrix polynomial abstract the numerical range of an n n matrix polynomial p a m m a m 1 m 1 a 1 a 0 is defined by w p c x p x 0 x c n x 0 for the linear pencil p i a the range w p coincides with the numerical range of matrix a f a x ax x c n x x 1 in this paper we obtain necessary conditions for the origin to be a boundary point of f a as a consequence an algebraic curve of degree at most 2 n n 1 m which contains the boundary of w p is constructed

canonical partition functions of hamiltonian systems and the stationary phase formula we show that there is a symplectic action of a torus associated to harmonic flows on the cotangent bundle of a semi simple lie algebra this allows to obtain a completely classical proof of the gallavotti marchioro formula by the method of the stationary phase

matrix polynomials with spectral radius equal to the numerical radius in this paper we study a class of matrix polynomials with the property that spectral radius and numerical radius coincide special attention is paid to the spectrum on the boundary of the numerical range

estimates on the support of the solutions of some nonlinear elliptic and parabolic problems we obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one dimensional motion of a non newtonian fluid precise estimates for the support of these solutions are obtained and the optimality of our hypotheses is discussed

z transform methods for the optimal design of one dimensional layered elastic media in this work we develop a finite trigonometric series representation for the stress in a multilayered goupillaud type elastic strip subjected to transient heaviside loading on one end while the other end is held fixed this representation is achieved by means of the z transform method and involves the so called base angles generally different layered designs could share the same set of base angles and the more layers the design has the more base angles are expected necessary conditions for the base angles and design parameters for any given design are described as a result of the stress representation we are able to identify optimal layered designs which provide the smallest stress amplitude for two and three layered designs for which the coefficients of the stress representation are easy to find the optimization results are achieved using a custom made discrete optimization technique applied in a p velo and g a gazonas int j solids structures 40 2003 pp 6417 6428 for other laye

the successive derivatives of the period function of a plane vector field previously we provided an expression which generalized the classical melnikov function to any order for the first nonzero derivative of a return mapping our method relied on the decomposition of a 1 form associated to the relative cohomology of the perturbed hamiltonian with the same techniques we give a formula for the first nonzero derivative of a period function we focus on the particular example ofh 1 2 x2 y2 and then we define a class of hamiltonians for which the same computation remains valid finally we investigate relations with birkhoff normal form