self similar set identification in the time scale domain we study the wavelet transform of deterministic self similar signals and derive their properties as well as a new algorithm for identification of the self similarity parameter we also include such applications as characterization and analysis of real chaotic signals in the presence of additive noise

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

similarity of matrices over local rings abstract wasow investigated the problem of when for a pair of matrices of analytic functions pointwise similarity implies analytic similarity by using the topology of spec r this is studied for an arbitrary commutative ring results of wasow ohm schneider and friedland are generalized in this setting a canonical form is determined for matrices satisfying the wasow condition

convergence from divergence we show how to convert divergent series which typically occur in many applications in physics into rapidly convergent inverse factorial series this can be interpreted physically as a novel resummation of perturbative series being convergent these new series allow rigorous extrapolation from an asymptotic region with a large parameter to the opposite region where the parameter is small we illustrate the method with various physical examples and discuss how these convergent series relate to standard methods such as borel summation and also how they incorporate the physical stokes phenomenon we comment on the relation of these results to dyson s physical argument for the divergence of perturbation theory this approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions and klein gordon and dirac determinants

on optimal truncation of divergent series solutions of nonlinear differential systems we prove that for divergent series solutions of nonlinear or linear differential systems near a generic irregular singularity the common prescription of summation to the least term is if properly interpreted meaningful and correct and we extend this method to trans series solutions in every direction in the complex plane at the singularity stokes9 directions not excepted there exists a non empty set of solutions whose difference from the optimally i e near the least term truncated asymptotic series is of the same exponentially small order of magnitude as the least term of the series there is a family of generalized borel summation formulae b which commute with the usual algebraic and analytic operations addition multiplication differentiation etc we show that there is exactly one of them b 0 such that for any formal series solution f b 0 f differs from the optimal truncation of f by at most the order of the least term of f we show in addition that the berry smoothing phenomenon is universal within this class of differential systems whenever the terms beyond all orders change in crossing a stokes line these terms vary smoothly on the berry scale x x 1 2 and the transition is always given by the error function under the same conditions we show that dingle9 rule of signs for stokes9 transitions holds

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices a b under a simultaneous similarity roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set then we construct a finite number of rational functions in the entries of a and b whose values are constant on all pairs similar to a b the values of the functions phi sub i a b i equals 1 s determine a finite number of similarity classes

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

optical uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations for first order differential equations of the form y9 p p x y p and second order homogeneous linear differential equations y a x y9 b x y 0 with locally integrable coefficients having asymptotic possibly divergent power series when x oo on a ray arg x const under some further assumptions it is shown that on the given ray there is a one to one correspondence between true solutions and complete formal solutions the correspondence is based on asymptotic inequalities which are required to be uniform in x and optimal with respect to certain weights

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

nonnegative solutions of the initial dirichlet problem for generalized porous medium equations in cylinders in this paper we study nonnegative solutions u of a class of nonlinear evolutions au at a u whose best known example is the porous medium equation u u m 1 see 9 for a recent survey on these equations we study our solutions u subject to dirichlet boundary conditions in d x 0 t where d is a bounded smooth domain in rn i e uiadx 0 t 0 the corresponding theory for d rn was studied by the authors in 3 we find that there is one special solution x t the friendly giant which tends to infinity as t 1 0 and that all other solutions u x t are in one to one correspondence with suitable pairs of measures u on d and a ou ad we also show that any solution on d x 0 t extends to a solution in d x 0 oo and that the friendly giant f x t governs the asymptotic behavior as t t oo the nonlinearity is assumed to be continuous increasing with p 0 0 and also verifies the growth conditions

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

time development of exponentially small non adiabatic transitions optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors in this paper we rigorously determine the leading order non adiabatic corrections to these approximations for a particular family of two level analytic hamiltonian functions our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function our results confirm the physics predictions of sir michael berry in the sense that the switching function for this family of hamiltonians has the form that he argues is universal

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

the fourier transform of the quartic gaussian exp ax4 hypergeometric functions power series steepest descent asymptotics and hyperasymptotics and extensions to exp ax2n the fourier transform of a quartic gaussian f k exp ikx exp x 4 dx is important in the theory of radial basis functions with f r exp r 4 we show that the transform is the exact sum of two generalized hypergeometric functions 0f 2 this implies that f k is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree using the method of steepest descent for integrals we derive the first five terms in an asymptotic series in powers of k 4 3 through a simple hyperasymptotic analysis we show that the magnitude of the error of the optimally truncated series superasymptotic error is oexp 0 818k 4 3 the lowest order approximation f k 2 7 6 p 3k 1 3exp 2 1 33 16 k 4 3 cos 3 3 22 1 3 16 k 4 3 p 6 shows that the transform f k is an exponentially decaying oscillation we show that the steepest descent method 2 27 bender and orszag 1978 miller 2006 26 33 lauwerier 1966 ursell 1965 can be applied to the transform of exp ax 2 n for general integer n and give the first two terms another theme is that modern computer technology has simplified the steepest descent method the steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function j t where z j t is the logarithm of the integrand and z is the large parameter here proportional to k 4 3 the task of inverting w t to obtain a series in powers of w for the metric factor dt dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete maple code

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

normal forms of holomorphic matrix valued functions and corresponding forms for singular differential operators abstract normal forms of a holomorphic matrix valued function a z under different classes of similarity transformations and corresponding forms of the singular differential operator z r d dz a z r n r 1 are investigated the problem of holomorphic similarity of matrix valued functions is given special consideration

unfolding the high orders of asymptotic expansions with coalescing saddles singularity theory crossover and duality we study the leading behaviour of the late coefficients high orders r of asymptotic expansions in a large parameter k for contour integrals involving a cluster of coalescing saddles and thereby establish the form of the divergence of the expansions the two principal cases are saddle to cluster where the integral is through a simple saddle and its expansion diverges because of a distant cluster and cluster to saddle where the integral is through a cluster and its expansion diverges because of a distant simple saddle in both the large r coefficients are dominated by the factorial divided by power familiar in asymptotics but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters a a 1 a 2 in the integrand the crossover between different forms is described by a series of canonical integrals built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence the arguments of these integrals involve not only the a but also fractional powers of r which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals a by product of the cluster to saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t