center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

invariant manifolds for impulsive equations and nonuniform polynomial dichotomies for impulsive differential equations in banach spaces we construct stable and unstable invariant manifolds for sufficiently small perturbations of a polynomial dichotomy we also consider the general case of nonuniform polynomial dichotomies moreover we introduce the notions of polynomial lyapunov exponent and of regularity coefficient for a linear impulsive differential equation and we show that when the lyapunov exponent never vanishes the linear equation admits a nonuniform polynomial dichotomy

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

polynomial growth rates abstract we consider linear equations v a t v with a polynomial asymptotic behavior that can be stable unstable and central we show that this behavior is exhibited by a large class of differential equations by giving necessary and sufficient conditions in terms of generalized polynomial lyapunov exponents for the existence of polynomial behavior in particular any linear equation in block form in a finite dimensional space with three blocks having polynomial lyapunov exponents respectively negative positive and zero has a nonuniform version of polynomial trichotomy which corresponds to the usual notion of trichotomy but now with polynomial growth rates we also obtain sharp bounds for the constants in the notion of polynomial trichotomy in addition we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

linear differential equations on the riemann sphere and representations of quivers our interest in this article is a generalization of the additive deligne simpson problem which was originally defined for fuchsian differential equations on the riemann sphere we extend this problem to differential equations having an arbitrary number of unramified irregular singular points and we determine the existence of solutions of the generalized additive deligne simpson problems moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the riemann sphere namely open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces furthermore we detail the connectedness of the moduli spaces

lyapunov functions for trichotomies with growth rates abstract we consider linear equations x a t x that may exhibit stable unstable and central behaviors in different directions with respect to arbitrary asymptotic rates e c t determined by a function t for example the usual exponential behavior with t t is included as a very special case and when t log t we obtain a polynomial behavior we emphasize that we also consider the general case of nonuniform exponential behavior which corresponds to the existence of what we call a nonuniform exponential trichotomy this is known to occur in a large class of nonautonomous linear equations our main objective is to give a complete characterization in terms of strict lyapunov functions of the linear equations admitting a nonuniform exponential trichotomy this includes criteria for the existence of a nonuniform exponential trichotomy as well as inverse theorems providing explicit strict lyapunov functions for each given exponential trichotomy in the particular case of quadratic lyapunov functions we show that the existence of strict lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the lyapunov functions as an application of the characterization of nonuniform exponential trichotomies in terms of strict lyapunov functions we establish the robustness of nonuniform exponential trichotomies under sufficiently small linear perturbations we emphasize that in comparison with former works our proof of the robustness is much simpler even when t t

double poisson algebras in this paper we develop poisson geometry for non commutative algebras this generalizes the bi symplectic geometry which was recently and independently introduced by crawley boevey etingof and ginzburg our quasi poisson brackets induce classical quasi poisson brackets on representation spaces as an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by crawley boevey and shaw carry a natural poisson structure

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

wavelet based parameter estimation for polynomial contaminated fractionally differenced processes we consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component trend i e large scale variations in the series that are best modeled outside of a stochastic framework is often confounded with low frequency stochastic fluctuations this problem is particularly evident in models such as fractionally differenced fd processes which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components we use the discrete wavelet transform dwt to estimate parameters for stationary and nonstationary fd processes in a model of polynomial trend plus fd noise using daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the dwt decorrelates fd processes approximately we consider this decorrelation in detail examining the between and within scale wavelet correlations separately better between scale decorrelation can be achieved by increasing the length of the wavelet filter whereas the within scale correlations can be handled via explicit modeling by a low order autoregressive process we demonstrate our methodology by applying it to a popular climate dataset

asymptotic decorrelation of wavelet packet transform for certain long memory processes the asymptotic decorrelation of the discrete wavelet transform for long memory processes such as the fractionally differenced fd process the autoregressive fractionally integrated moving average arfima process and the gegenbauer autoregressive moving average garma process as well as the statistical inference techniques based on this property have received much attention nowadays in this paper we investigate the asymptotic decorrelation property of the discrete wavelet packet transform dwpt for two classes of discrete time long memory processes containing the arfima and the garma processes especially we prove theoretically that the covariance across between packet dwpt coefficients decays hyperbolically or exponentially fast as the width of the underlying daubechies scaling and wavelet filters to generate the dwpt gets large meanwhile we show that the covariance between within packet dwpt coefficients converges hyperbolically fast to its corresponding counterpart when the underlying scaling and wavelet filters to generate the dwpt are the shannon s ideal low and high pass filters

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

stokes phenomenon gelfand zeitlin systems and relative ginzburg weinstein linearization abstract in 2007 alekseev meinrenken proved that there exists a ginzburg weinstein diffeomorphism from the dual lie algebra u n to the dual poisson lie group u n compatible with the gelfand zeitlin integrable systems in this paper we explicitly construct such diffeomorphisms via stokes phenomenon and boalch s dual exponential maps then we introduce a relative version of the ginzburg weinstein linearization motivated by irregular riemann hilbert correspondence and generalize the results of enriquez etingof marshall to this relative setting in particular we prove the connection matrix for a certain irregular riemann hilbert problem satisfies a relative gauge transformation equation of the alekseev meinrenken dynamical r matrices this gauge equation is then derived as the semiclassical limit of the relative drinfeld twist equation

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

signal processing with fractals a wavelet based approach wavelet transmission statistically self similar signals detection and estimation with 1 processes deterministically self similar signals fractal modulation linear self similar signals

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

multiple hypothesis mapping of functional mri data in orthogonal and complex wavelet domains we are interested in methods for multiple hypothesis testing that optimize power to refute the null hypothesis while controlling the false discovery rate fdr the wavelet transform of a spatial map of brain activation statistics can be tested in two stages to achieve this objective first a set of possible wavelet coefficients to test is reduced and second each hypothesis in the remaining subset is formally tested we show that a bayesian bivariate shrinkage operator baybishrink for the first step provides a powerful and expedient alternative to a subband adaptive chi squared test or an enhanced fdr algorithm based on the generalized degrees of freedom we also investigate the dual tree complex wavelet transform cwt as an alternative basis to the orthogonal discrete wavelet transform dwt we design and validate a test for activation based on the magnitude of the complex wavelet coefficients and show that this confers improved specificity for mapping spatial signals the methods are applied to simulated and experimental data including a pharmacological magnetic resonance imaging mri study we conclude that using baybishrink to define a reduced set of complex wavelet coefficients and testing the magnitude of each complex pair to control the fdr represents a competitive solution for multiple hypothesis mapping in fmri

on the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes in this correspondence we develop an asymptotic theory for the correlations of wavelet coefficients of the discrete wavelet transform dwt for fractionally differenced processes it provides a theoretical justification for the approximate decorrelation property of the dwt for fractionally differenced processes in addition it provides insights on how the length of the wavelet filter affects the within scale correlations and the between scale correlations differently for within scale correlations increasing the length of the wavelet filter increases the rate of decay as the two wavelet coefficients get further apart while for between scale correlations using a wavelet filter that is long enough can reduce the between scale correlations even for wavelet coefficients that are close together

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

the monodromy of meromorphic projective structures we study projective structures on a surface having poles of prescribed orders we obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed mathrm pgl 2 mathbb c local systems on the associated marked bordered surface we prove that the image of this map is contained in the union of the domains of the cluster charts we discuss a number of open questions concerning this monodromy map

on weak exponential stability of evolution operators in banach spaces abstract the aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in banach spaces variants for weak exponential stability of some well known results in uniform stability theory buse and niculescu 2009 1 daleckij and krein 1974 2 datko 1973 3 rolewicz 1986 7 stoica and megan 2009 8 are obtained

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

determining significant connectivity by 4d spatiotemporal wavelet packet resampling of functional neuroimaging data an active area of neuroimaging research involves examining functional relationships between spatially remote brain regions when determining whether two brain regions exhibit significant correlation due to true functional connectivity one must account for the background spatial correlation inherent in neuroimaging data we define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing we develop a 4d spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice across axial slices and through each voxel time series while excluding the specific correlations due to true functional relationships we also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data our method improves upon existing wavelet based methods and extends them to 4d we apply our resampling technique to determine significant functional connectivity from resting state and motor task fmri datasets

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

on the digital filter associated with daubechies wavelets the magnitude of the filters associated with daubechies wavelets sub n spl psi is shown to monotonically converge to an ideal highpass filter when n spl rarr spl infin the rate of the convergence is also given the magnitude of each filter is shown to be monotonically increasing from 0 spl pi the convergence does not have gibbs phenomenon

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow

conjugacy classes of n tuples in lie algebras and algebraic groups let g be a reductive algebraic group over an algebraically closed field f of characteristic zero and let l g be the lie algebra of g then g acts on g by inner automorphisms and it acts on l g by the adjoint representation by taking the diagonal actions we get actions of g on the spaces of n tuples g n and l g in this paper we will prove a number of geometric properties of the orbits of g on these spaces of n tuples