measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

an a posteriori parameter choice for tikhonov regularization in hilbert scales leading to optimal convergence rates an a posteriori parameter choice for tikhonov regularization in hilbert scales is proposed that leads to optimal convergence rates the advantage of this method is that knowledge of the smoothness parameters of both the operator and the exact solution is not needed the finite dimensional realization of this method is also discussed numerical examples confirm the theoretical results

monitoring changing dynamics with correlation integrals case study of an epileptic seizure abstract we describe a procedure and the motivation behind it which rapidly and accurately tracks the onset and progress of an epileptic seizure roughly speaking one monitors changes in the relative dispersion of a re embedded time series the results are robust with respect to variation of adjustable parameters such as embedding dimension lag time and critical distances moreover the general method is virtually unaffected when the data are significantly corrupted by external noise when the information computed for the individual channels is displayed in an appropriate space time plot the progress and geometric location of the seizure are easily seen an interpretation of these results in terms of a cloud of particles moving in an abstract phase space is examined

a posteriori parameter choice with an efficient discretization scheme for solving ill posed problems abstract recently rajan m p rajan an efficient discretization scheme for solving the ill posed problems journal of mathematical analysis and applications 313 2 2006 654 677 considered a discretization scheme for solving ill posed problems and obtained optimal error estimates under an a priori parameter choice strategy with a particular smoothness assumption on the solution it is shown that the computational information for solving the system is far less than the traditional projection schemes neubauer a neubauer an a posteriori parameter choice for tikhonov regularization in the presence of modeling error applied numerical mathematics 4 1988 507 519 and engl and gfrerer h w engl h gfrerer a posteriori parameter choice for general regularization methods for solving ill posed problems applied numerical mathematics 4 1988 395 417 suggested an a posteriori method for choosing the regularization parameter which does not require any knowledge about the exact solution respectively under finite dimensional as well as general setting in this paper we consider an a posteriori parameter choice for finite dimensional approximation which does not require any information about the smoothness of the exact solution and apply to the discretization scheme considered by rajan the computational efficiency of the scheme is illustrated through numerical examples

a remark on the regularity criterion for the 3d navier stokes equations involving the gradient of one velocity component abstract in this paper we consider the regularity criterion for weak solutions to the 3d navier stokes equations we show that if u 3 belongs to some multiplier spaces then the solution actually is smooth on 0 t in particular we have the regularity condition in the bmo spaces u 3 l 4 3 0 t bmo which improves previous results

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

recycling subspace information for diffuse optical tomography we discuss the efficient solution of a long sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging the reconstruction of three dimensional absorption and scattering information by matching computed solutions from a parameterized model to measured data leads to a nonlinear least squares problem that we solve using the gauss newton method with a line search this algorithm requires the solution of a long sequence of linear systems each choice of parameters in the nonlinear least squares algorithm results in a different matrix describing the optical properties of the medium these matrices change slowly from one step to the next but may change significantly over many steps for each matrix we must solve a set of linear systems with multiple shifts and multiple right hand sides for this problem we derive strategies for recycling krylov subspace information that exploit properties of the application and the nonlinear optimization algorithm to significantly reduce the total number of iterations over all linear systems furthermore we introduce variants of gcro that exploit symmetry and that allow simultaneous solution of multiple shifted systems using a single krylov subspace in combination with recycling although we focus on a particular application and optimization algorithm our approach is applicable generally to problems where sequences of linear systems must be solved this may guide other researchers to exploit the opportunities of tunable solvers we provide results for two sets of numerical experiments to demonstrate the effectiveness of the resulting method

generic solvability of the axisymmetric 3 d euler equations and the 2 d boussinesq equations abstract in this paper we first prove a new criterion of global solvability for smooth axisymmetric solutions of the 3 d euler equations for a cylindrical type of domain which can be whole r3 then using this result we obtain generic solvability of the equations in the cylindrical domain which does not include the axis of symmetry by a similar method we also obtain generic solvability of the 2 d boussinesq system

a refined regularity criterion for the navier stokes equations involving one non diagonal entry of the velocity gradient abstract this paper concerns about the regularity criteria for the three dimensional navier stokes equations by establishing a more subtle estimate of the crucial convective term in navier stokes equations we improve 4 and 10 a lot

flexible and deflated variants of the block shifted gmres method abstract the solution of linear systems with multiple shifts and multiple right hand sides given simultaneously is required in many large scale scientific and engineering applications in this paper we introduce new flexible and deflated variants of the shifted block gmres method for this problem class the proposed methods solve the whole sequence of linear systems simultaneously detecting effectively the linear systems convergence and allowing the use of variable preconditioning which may be particularly useful in some applications numerical experiments are illustrated to show the overall significant robustness of the iterative method for solving general sparse multi shifted and multiple right hand side systems and in realistic pagerank calculations to the best of our knowledge this is the first krylov subspace method that combines deflation techniques and variable preconditioning for solving sequences of multi shifted linear systems with multiple right hand sides simultaneously

galerkin projection methods for solving multiple linear systems in this paper we consider using conjugate gradient cg methods for solving multiple linear systems a i x i b i for 1 le i le s where the coefficient matrices a i and the right hand sides b i are different in general in particular we focus on the seed projection method which generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems onto the generated krylov subspace to get the approximate solutions the whole process is repeated until all the systems are solved most papers in the literature t f chan and w l wan it siam j sci comput 18 1997 pp 1698 1721 b parlett it linear algebra appl 29 1980 pp 323 346 y saad it math comp 48 1987 pp 651 662 v simoncini and e gallopoulos it siam j sci comput 16 1995 pp 917 933 c smith a peterson and r mittra it ieee trans antennas and propagation 37 1989 pp 1490 1493 considered only the case where the coefficient matrices a i are the same but the right hand sides are different we extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right hand sides a theoretical error bound is given for the approximation obtained from a projection process onto a krylov subspace generated from solving a previous linear system finally numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method

enriched krylov subspace methods for ill posed problems we describe a modification of the conjugate gradient method for the normal equations cgnr that allows us to enrich the krylov subspaces in which the iterates are determined with vectors containing pertinent information about the desired solution the enriched cgnr method easily can be adapted to the solution of linear systems arising from penalized least squares problems and tikhonov regularization applications to the solution of linear discrete ill posed problems illustrate that enrichment of the krylov subspaces can improve the quality of the computed approximate solutions and reduce the computational effort required for their determination

reduced sqp methods for parameter identification problems parameter estimation problems are formulated as constrained regularized optimization problems reduced sqp methods with bfgs update are analyzed to solve these infinite dimensional optimization problems rate of convergence results are given and numerical feasibility of the resulting algorithms is demonstrated

components reduction regularity results for the navier stokes equations in general dimensions abstract we consider the cauchy problem of the navier stokes equations in arbitrary dimensions and establish several new components reduction regularity criteria

approximation accuracy of the krylov subspaces for linear discrete ill posed problems abstract for the large scale linear discrete ill posed problem min a x b or a x b with b contaminated by gaussian white noise the lanczos bidiagonalization based krylov solver lsqr and its mathematically equivalent cgls the conjugate gradient cg method implicitly applied to a t a x a t b are most commonly used and cgme the cg method applied to min a a t y b or a a t y b with x a t y and lsmr which is equivalent to the minimal residual minres method applied to a t a x a t b have also been choices these methods exhibit typical semi convergence feature and the iteration number k plays the role of the regularization parameter however there has been no definitive answer to the long standing fundamental question can lsqr and cgls find 2 norm filtering best possible regularized solutions the same question is for cgme and lsmr too at iteration k lsqr cgme and lsmr compute different iterates from the same k dimensional krylov subspace a first and fundamental step towards answering the above question is to accurately estimate the accuracy of the underlying k dimensional krylov subspace approximating the k dimensional dominant right singular subspace of a assuming that the singular values of a are simple we present a general sin theorem for the 2 norm distances between these two subspaces and derive accurate estimates on them for severely moderately and mildly ill posed problems we also establish some relationships between the smallest ritz values and these distances numerical experiments justify the sharpness of our results

regularization of mellin type inverse problems with an application to oil engineering abstract in this paper we present a novel method for the stable inversion of mellin convolution operators although the inverse operator is explicitly known for this class of operators regularization techniques have to be used due to the ill posedness of the problem considering operators which are continuously invertible between the space l 2 0 and a mellin sobolev space h s 0 we propose a combination of a data smoothing algorithm and an exact inversion for the order optimal regularization of the equation as application we derive the mellin transform of the capillary pressure operator and analyze its mapping properties finally we present some numerical results

restarted gmres for shifted linear systems shifted matrices which differ by a multiple of the identity only generate the same krylov subspaces with respect to any fixed vector this fact has been exploited in lanczos based methods like cg qmr and bicg to simultaneously solve several shifted linear systems at the expense of only one matrix vector multiplication per iteration here we develop a variant of the restarted gmres method exhibiting the same advantage and we investigate its convergence for positive real matrices in some detail we apply our method to speed up multiple masses calculations arising in lattice gauge computations in quantum chromodynamics one of the most time consuming supercomputer applications

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

remarks on the regularity criterion to the navier stokes equations via the gradient of one velocity component abstract in this paper we consider the classical three dimensional 3d incompressible navier stokes equations by making use of a new anisotropic sobolev inequality we improve a regularity criterion for the weak solution to the 3d navier stokes equations involving the gradient of one velocity component this result improves the previous work of zhou and pokorn 2010

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

an iterative lagrange method for the regularization of discrete ill posed inverse problems in this paper an iterative method is presented for the computation of regularized solutions of discrete ill posed problems in the proposed method the regularization problem is formulated as an equality constrained minimization problem and an iterative lagrange method is used for its solution the lagrange iteration is terminated according to the discrepancy principle the relationship between the proposed approach and classical tikhonov regularization is discussed results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method

quantum and classical mechanics on homogeneous riemannian manifolds starting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous riemannian manifolds the classical theory is derived as a consequence complete with its phase space realized as the space of pure classical states a generalized version of the wigner moyal correspondence rule the jordan and lie algebra structures of functions on the cotangent bundle given by point wise multiplication and poisson bracket and the momentum map a comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

some serrin type regularity criteria for weak solutions to the navier stokes equations we consider the regularity criteria for a weak solution u u1 u2 u3 to the navier stokes equations in r3 denoting by 1 2 3 curlu the vorticity we then prove that u is smooth provided u3 3 or 3u3 3 are in some serrin type integrability classes

quantum anosov flows a new family of examples a quantum version is presented for the anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the poincare half plane while the canonical weyl algebra does not close under time evolution the symplectic structure of these classical systems can be exploited to produce objects akin to the ccr algebras encountered in quantum field theory this construction allows one to lift both the geodesic and the horocyclic flows to a weyl algebra describing the quantum dynamics corresponding to the systems under consideration the anosov relations as proposed in ref reference 1 are found to be valid for these models a quantum version of the classical ergodicity of these systems is discussed in the last section

krylov subspace algorithms and circulant embedding method for efficient wideband single carrier equalization wider bandwidth allows higher data rate by transmitting narrower pulses however doing so would also increase the discrete channel memory length for single carrier communication systems this results in higher computational burden at the receiver we are concerned with single carrier nonblock transmission schemes with receiver oversampling as they can provide higher spectral efficiency than block transmission schemes in the presence of large delay spreads we first propose a simple finite impulse response fir equalizer that is based on the circulant embedding ce method and analyze its performance by investigating the relationship between solutions of various finite dimensional models and the original infinite dimensional problem we show that under proper conditions the ce fir equalizer converges exponentially fast to the iir equalizer we then focus on the conjugate gradient cg algorithm as an efficient means for equalization that is specifically well suited for dealing with large delay spread channels we discuss the importance of stopping the iterations for the cg algorithm at the right time in the presence of noise and present several reliable low cost stopping criteria it turns out that the cg algorithm equipped with appropriate stopping criteria can outperform mmse equalizers since both the ce and the cg methods can be efficiently implemented via fast fourier transforms equalization complexity is only in the order of n log n for n data symbols several numerical experiments demonstrate the performance of the proposed methods

an efficient discretization scheme for solving ill posed problems in this paper we consider a finite dimensional approximation scheme combined with tikhonov regularization for solving ill posed problems error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite dimensional approach

a regularity criterion for the solutions of 3d navier stokes equations abstract in terms of two partial derivatives of any two components of velocity fields we give a new criterion for the regularity of solutions of the navier stokes equation in r 3 more precisely let u u 1 u 2 u 3 be a weak solution in 0 t r 3 then u becomes a classical solution if any two functions of 1 u 1 2 u 2 and 3 u 3 belong to l 0 t l r r 3 provided with 2 3 r 2 3 2 r

fast multiscale galerkin methods for solving ill posed integral equations via a coupled system under general source conditions abstract fast multiscale regularized galerkin methods for solving fredholm integral equations of the first kind are investigated we apply a multiscale galerkin method with a matrix compression strategy to discretize the integral equation of the first kind and obtain a sparse system to dealing with the ill posedness and keeping sparsity of the discrete system we use an equivalent coupled system form of the tikhonov regularization then the multilevel augmentation method is applied to solve the coupled system a priori and a posteriori regularization parameter choice strategies are proposed convergence rates of the regularized solutions are established under general source conditions numerical results are presented to demonstrate the efficiency and accuracy of the proposed methods

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

anosov actions on noncommutative algebras an axiomatic framework for a quantum mechanical extension to the theory of anosov systems is constructed and it is shown that this retains some of the characteristic features of its classical counterpart e g nonvanishing lyapunov exponents a vectorial k property and exponential clustering the effects of quantization are investigated on two prototype examples of anosov systems namely the iterations of an automorphism of the torus the arnold cat model and the free dynamics of a particle on a surface of negative curvature it emerges that the anosov property survives quantization in the case of the former model but not of the latter one finally we show that the modular dynamics of a relativistic quantum field on the rindler wedge of minkowski space is that of an anosov system

a fast truncated lagrange method for large scale image restoration problems in this work we present a new method for the restoration of images degraded by noise and spatially invariant blur in the proposed method the original image restoration problem is replaced by an equality constrained minimization problem a quasi newton method is applied to the first order optimality conditions of the constrained problem in each quasi newton iteration the hessian of the lagrangian is approximated by a circulant matrix and the fast fourier transform is used to compute the quasi newton step the quasi newton iteration is terminated according to the discrepancy principle results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method

a regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces abstract we consider the regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces it is proved that if u l p 0 t f q 2 q 3 0 with 2 p 3 q 2 3 2 q then the solution remains smooth in 0 t as a corollary we obtain the classical beal kato majda criterion that is the condition u l 1 0 t b 0 ensures the smoothness of the solution

regularity criterion of weak solutions to the 3d navier stokes equations via two velocity components abstract this paper is concerned with the regularity criterion of leray hopf weak solutions to the 3d navier stokes equations with respect to serrin type condition on two velocity filed components it is shown that the weak solution u u 1 u 2 u 3 is regular on 0 t if there exist two solution components for example u 2 and u 3 satisfying the condition u 2 u 3 l p 1 0 t l q 1 r 3 for 2 p 1 3 q 1 2 3 2 q 1

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

an adaptive discretization for tikhonov phillips regularization with a posteriori parameter selection summary the aim of this paper is to describe an efficient adaptive strategy for discretizing ill posed linear operator equations of the first kind we consider tikhonov phillips regularization x alpha delta left a ast a alpha i right 1 a ast y delta with a finite dimensional approximation a n instead of a we propose a sparse matrix structure which still leads to optimal convergences rates but requires substantially less scalar products for computing a n compared with standard methods

an optimal regularized projection method for solving ill posed problems via dynamical systems method a new regularized projection method was developed for numerically solving ill posed equations of the first kind this method consists of combining the dynamical systems method with an adaptive projection discretization scheme optimality of the proposed method was proved on wide classes of ill posed problems

geometric quantization modular reduction theory and coherent states the natural role played by coherent states in the geometric quantization program is brought out by studying the mathematical equivalence between two physical interpretations that have recently been proposed for this program these interpretations are based respectively on the modular algebra structure of prequantization and the reproducing kernel structure of phase space quantization the arguments are presented in this paper for the particular case where the phase space of the system considered is the cotangent bundle t m of a homogeneous manifold m and for didactic reasons the latter is taken to be a real vector space

fast cg based methods for tikhonov phillips regularization tikhonov phillips regularization is one of the best known regularization methods for inverse problems a posteriori criteria for determining the regularization parameter alpha require solving a a alpha i x a y delta for different values of alpha we investigate two methods for accelerating the standard cg algorithm for solving the family of systems the first one utilizes a stopping criterion for the cg iterations which depends on alpha and delta the second method exploits the shifted structure of the linear systems which allows us to solve simultaneously for different values of alpha we present numerical experiments for three test problems which illustrate the practical efficiency of the new methods the experiments as well as theoretical considerations show that run times are accelerated by a factor of at least 3

data driven optimal filtering for phase and frequency of noisy oscillations application to vortex flow metering a method for measuring the phase of oscillations from noisy time series is proposed to obtain the phase the signal is filtered in such a way that the filter output has minimal relative variation in the amplitude over all filters with complex valued impulse response the argument of the filter output yields the phase implementation of the algorithm and interpretation of the result are discussed we argue that the phase obtained by the proposed method has a low susceptibility to measurement noise and a low rate of artificial phase slips the method is applied for the detection and classification of mode locking in vortex flow meters a measure for the strength of mode locking is proposed

regularity criterion via two components of vorticity on weak solutions to the navier stokes equations in r 3 we consider the regularity of weak solutions to the navier stokes equations in r3 let u be a weak solution in r3 0 t w curlu and w w1 w2 0 it is proved that u becomes a classical solution if w lq 0 t b r 0 for 2 q 3 r 2 32 r and 2r 3 this is an improvement of the result given by kozono yatsu extension criterion via two components of vorticity on strong solution to the 3d navier stokes equations math z 246 2003 55 68

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

comparing different types of approximators for choosing the parameters in the regularization of ill posed problems regularized approximations to the solutions of ill posed problems typically vary from over smoothed inaccurate reconstructions to under smoothed and unstable solutions as the regularization parameter varies about its optimal value it thus makes sense to compare two or more regularized approximations and seek the parametervalues where the distance between the approximations is a minimum this paper advances the theory and practice of this methodology the method appears to workvery well and be very stable particularly in the presence of extreme error levels

comparison of three nonlinear seizure prediction methods by means of the seizure prediction characteristic epilepsy is characterized by the spontaneous and unforeseeable occurrence of seizures during which the perception or behavior of patients is disturbed the predictability of these seizures would render novel therapeutic approaches possible several prediction methods have claimed to be able to predict seizures based on eeg recordings minutes in advance however the term seizure prediction is not unequivocally defined different criteria to assess prediction methods exist and only little attention has been paid to issues of sensitivity and false prediction rate we introduce an assessment criterion called the seizure prediction characteristicthat incorporates the assessment of sensitivity and false prediction rate within this framework three nonlinear seizure prediction methods were evaluated on a large eeg data pool of 21 patients altogether 582 h intracranial eeg data and 88 seizures were examined with a rate of 1 3 6 false predictions per day the dynamical similarity index achieves a sensitivity between 21 and 42 which was the best result of the three methods sensitivity was between 18 and 31 for the extended prospective version of the accumulated energy and between 13 and 30 for the effective correlation dimension these results still are not sufficient for clinical applications

nonlinear similarity analysis for epileptic seizures prediction abstract the prediction of epileptic seizures can promise a new diagnostic application and a novel approach for seizure control this paper proposes an improved dynamical similarity measure to predict epileptic seizures in electroencephalographic eeg first mutual information and cao s method are employed to reconstruct a phase space of preprocessed eeg recordings by using the positive zero crossing method second a gaussian function replaces the heavyside function within correlation integral at calculating a similarity index the crisp boundary of the heavyside function is eliminated because of the gaussian function s smooth boundary third an adaptive detection method based on the similarity index is proposed to predict the epileptic seizures in light of test results of eeg recordings of rats it is found that the new dynamical similarity index is insensitive to the selection of the radius value of gaussian function and the size of segmented eeg recordings comparing with the dynamical similarity index proposed by le van quyen et al anticipating epileptic seizures in real time by a non linear analysis of similarity between eeg recordings neuroreport 10 1999 2149 2155 the tests of twelve rats show the new dynamical similarity index is better to predict the epileptic seizures

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

electroencephalograms in epilepsy analysis and seizure prediction within the framework of lyapunov theory abstract epileptic seizures are defined as the clinical manifestation of excessive and hypersynchronous activity of neurons in the cerebral cortex and represent one of the most frequent malfunctions of the human central nervous system therefore the search for precursors and predictors of a seizure is of utmost clinical relevance and may even guide us to a deeper understanding of the seizure generating mechanisms we extract chaos indicators such as lyapunov exponents and kolmogorov entropies from different types of electroencephalograms eegs this covers mainly intracranial eegs semi invasive and invasive recording techniques but also scalp eegs from the surface of the skin among the analytical methods we tested up to now we find that the spectral density of the local expansion exponents is best suited to predict the onset of a forthcoming seizure we also evaluate the time evolution of the dissipation in these signals it exhibits strongly significant variations that clearly relate to the time relative to a seizure onset this article is mainly devoted to an assessment of these methods with respect to their sensitivity to eeg changes e g prior to a seizure further we investigate interictal eegs i e far away from a seizure in order to characterize their more general properties such as the convergence of the reconstructed quantities with respect to the number of phase space dimensions generally we use multichannel reconstruction but we also present a comparison with the delay embedding technique

on the regularization of projection methods for solving iii posed problems in this paper we consider a class of regularization methods for a discretized version of an operator equation which includes the case that the problem is ill posed with approximately given right hand side we propose an a priori as well as an a posteriori parameter choice method which is similar to the discrepancy principle of ivanov morozov from results on fractional powers of selfadjoint operators we obtain convergence rates which are in many cases the same for both parameter choices

a fast multiscale galerkin method for the first kind ill posed integral equations via iterated regularization in this paper we develop a fast multiscale galerkin method to solve the fredholm integral equations of the first kind via tikhonov regularization the method leads to fast solutions of discrete regularization methods we obtain optimal convergence rates for approximate solutions with an a priori parameter choice and a kind of discrepancy principle finally numerical experiments are given to illustrate the efficiency of the method

cotangent bundle quantization entangling of metric and magnetic field for manifolds of noncompact type endowed with an affine connection for example the levi civita connection and a closed 2 form magnetic field we define a hilbert algebra structure in the space and construct an irreducible representation of this algebra in this algebra is automatically extended to polynomial in momenta functions and distributions under some natural conditions this algebra is unique the non commutative product over is given by an explicit integral formula this product is exact not formal and is expressed in invariant geometrical terms our analysis reveals that this product has a front which is described in terms of geodesic triangles in the quantization of functions induces a family of symplectic reflections in and generates a magneto geodesic connection on this symplectic connection entangles on the phase space level the original affine structure on and the magnetic field in the classical approximation the 2 part of the quantum product contains the ricci curvature of and a magneto geodesic coupling tensor