existence of solution of the logarithmic diffusion equation with bounded above gauss curvature abstract we give a simple proof of an extension of the existence results of ricci flow of giesen and topping 2010 2011 15 20 on incomplete surfaces with bounded above gauss curvature without using the difficult shi s existence theorem of ricci flow on complete non compact surfaces and the pseudolocality theorem of perelman 7 on ricci flow we will also give a simple proof of a special case of the existence theorem of topping 2010 16 without using the existence theorem of shi 1989 9

beyond time frequency analysis energy densities in one and many dimensions given a unitary operator a representing a physical quantity of interest we employ concepts from group representation theory to define two natural signal energy densities for a the first is invariant to a and proves useful when the effect of a is to be ignored the second is covariant to a and measures the a content of signals we also consider joint densities for multiple operators and in the process provide an alternative interpretation of cohen s see englewood cliffs nj prentice hall 1995 general construction for joint distributions of arbitrary variables

time frequency distributions for a wide sense stationary random signal it is shown that a wide class of time frequency distributions gives the power spectrum for the case of a wide sense stationary random signal a simple criterion on the cohen 1966 kernel for determining when a bilinear time frequency distribution gives the power spectrum of the signal is derived

the scale representation the authors considers scale a physical attribute of a signal and develop its properties he presents an operator which represents scale and study its characteristics and representation this allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal he obtains explicit expressions for the mean scale scale bandwidth instantaneous scale and scale group delay furthermore he derives expressions for mean time mean frequency duration frequency bandwidth in terms of the scale variable the short time transform is defined and used to obtain the conditional value of scale for a given time he shows that as the windows narrows one obtains instantaneous scale convolution and correlation theorems for scale are derived a formulation is devised for studying linear scale invariant systems he derives joint representations of time scale and frequency scale general classes for each are presented using the same methodology as for the time frequency case as special cases the joint distributions of marinovich altes 1978 1986 and bertrand bertrand 1984 are recovered also joint representations of the three quantities time frequency scale are devised a general expression for the local scale autocorrelation function is given uncertainty principles for scale and time and scale and frequency are derived

clustering based on gaussian processes in this letter we develop a gaussian process model for clustering the variances of predictive values in gaussian processes learned from a training data are shown to comprise an estimate of the support of a probability density function the constructed variance function is then applied to construct a set of contours that enclose the data points which correspond to cluster boundaries to perform clustering tasks of the data points an associated dynamical system is built and its topological invariant property is investigated the experimental results show that the proposed method works successfully for clustering problems with arbitrary shapes

frsvc towards making support vector clustering consume less abstract in spite of with great advantage of discovering arbitrary shapes of clusters support vector clustering svc is frustrated by large scale data especially on resource limited platform it is due to pricey storage and computation consumptions from solving dual problem and labeling clusters upon the pre computed kernel matrix and sampling point pairs respectively towards on it we first present a dual coordinate descent method to reformulate the solver that leads to a flexible training phase carried out on any runtime platform with without sufficient memory then a novel labeling phase who does connectivity analysis between two nearest neighboring decomposed convex hulls referring to clusters is proposed in which a new designed strategy namely sample once connected checking first tries to reduces the scope of sampling analysis by integrating them together a faster and reformulated svc frsvc is created with less consumption achieved according to comparative analysis of time and space complexities furthermore experimental results confirm a significant improvement on flexibility of selective efficiency without losing accuracy with which a balance can be easily reached on the basis of resources a platform equipped

kernel synthesis for generalized time frequency distributions using the method of alternating projections onto convex sets cohen s generalized time frequency distribution gtfr requires the choice of a two dimensional kernel the kernel directly affects many performance attributes of the gtfr such as time resolution frequency resolution realness and conformity to time and frequency marginals a number of different kernels may suffice for a given performance constraint high frequency resolution for example interestingly most sets of kernels satisfying commonly used performance constraints are convex we describe a method whereby kernels can be designed that satisfy two or more of these constraints if there exists a nonempty intersection among the constraint sets then the theory of alternating projection onto convex sets pocs guarantees convergence to a kernel that satisfies all of the constraints if the constraints can be partitioned into two sets each with a nonempty intersection then pocs guarantees convergence to a kernel that satisfies the inconsistent constraints with minimum mean square error we apply kernels synthesized using pocs to the generation of some example gtfrs and compare their performance to the spectrogram wigner distribution and cone kernel gtfr

a general approach for obtaining joint representations in signal analysis i characteristic function operator method a general approach for obtaining joint representations for arbitrary physical quantities is presented the characteristic function operator method of moyal 1949 and ville 1960 and generalized by cohen 1966 1976 and scully and cohen 1987 for the time frequency case is developed for arbitrary variables

an analytic approach to the normalized ricci flow like equation abstract this paper is concerned with a parabolic equation with a non local term defined on a compact two dimensional riemannian surface if the total mass of the solution is equal to 8 and is the standard sphere s 2 it is a hamilton s normalized ricci flow we obtain the global in time existence of the solution to this problem for 0 8 if 0 8 the orbit is compact while for 8 there is a time sequence along which the solution converges to a stationary solution

the development of a discrete transform for the wigner distribution and ambiguity function a theoretical basis for the development of the discrete wigner distribution and ambiguity function is presented that is based on the temporal and spectral properties of the two dimensional wigner kernel function this two dimensional description is helpful in visualizing the various functions and shows that the increase in the number of points required to compute the discrete wigner distribution is a natural consequence of the definition of the kernel and that it does not contradict the nyquist sampling theorem a generalized set of pseudofunctions is introduced that involves applying a two dimensional weighting function to the wigner kernel or its two dimensional fourier transform resulting in a wigner distribution that is weighted or smoothed in time or frequency two forms of the wigner distribution are derived the first is a pure wigner distribution without any windowing of the kernel that may be computed over the entire data set the second involves the general set of pseudo wigner distribution that may be calculated over a subset of the data set the discrete wigner distribution of the analytic signal is also developed and it is shown that the number of points may be reduced by a factor of 2 in the same way that the number of points required to represent the discrete analytic signal may be reduced the discrete ambiguity function is developed along with the discrete wigner distribution and it is shown that the four two dimensional functions are interrelated by discrete one and two dimensional fourier transforms

reciprocal transformation for one dimensional conservation equations one dimensional conservation equations oce of the form n t f x 0 with n n x t 0 and f f n n x 2n x2 admit a symmetric reciprocal transformation x x x t n n x t n 1 f f n 1f which produces an equivalent oce for n in x space certain oce of contemporary interest are reciprocal invariant in the sense that f f n n x 2n x 2 there also exists a class of essentially nonlinear oce for which the reciprocal transformation produces a linear oce and thus equations in this class are solvable analytically

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

hamilton s gradient estimates for fast diffusion equations under the ricci flow abstract let m be a complete noncompact riemannian manifold of dimension n we derive a hamilton s gradient estimate for positive solutions of the fast diffusion equations u t u m 1 4 n 8 m 1 on m 0 under the ricci flow

learning representative exemplars using one class gaussian process regression abstract an exemplar is an observation that represents a group of similar observations exemplars from data are examined to divide entire heterogeneous data into several homogeneous subgroups wherein each subgroup is represented by an exemplar with its inherent sparsity an exemplar based learning model provides a parsimonious model to represent or cluster large scale data a novel exemplar learning method using one class gaussian process gp regression is proposed in this study the proposed method constructs data distribution support from one class gp regression using automatic relevance determination prior and heterogeneous gp noise exemplars that correspond to the basis vectors of the constructed support function are then automatically located during the training process the proposed method is applied to various data sets to examine its operability characteristics of data representation and cluster analysis the exemplars of some real data generated by the proposed method are also reported

a high resolution data adaptive time frequency representation a data adaptive time frequency representation is developed that overcomes some limitations of the short time fourier transform while avoiding the cross terms that make the wigner distribution and other bilinear representations difficult to interpret the adaptive time frequency representations uses gaussian basis functions but varies their time width and chirp rate with time and frequency to achieve high signal concentration everywhere a measure of local signal concentration allows fully automated determination of the optimal basis parameters the adaptive method is computationally expensive but may provide much better performance than any currently known technique

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

existence of solutions of the very fast diffusion equation in bounded and unbounded domain we prove the existence of a unique solution of the following neumann problem u t phi m u xx u 0 in a b 0 t u x 0 u0 x 0 in a b phi m u x a t g t and phi m u x b t f t forall 0 0 int a bu 0dx int 0 t f g dt m 0 0 le f g in l loc infty 0 infty 0 le u 0 in l infty a b and the case 1 1 m we also obtain a similar result in higher dimensions as a corollary we will give a new proof of a result of a rodriguez and j l vazquez on the existence of infinitely many finite mass solutions of the above equation in mathbb r times 0 t for any 1 m 0 we also obtain the exact decay rate of the solution at infinity

the uncertainty principle global local or both we address the issue of the relation between local quantities and the uncertainty principle we approach the problem by defining local quantities as conditional standard deviations and we relate these to the uncertainty product appearing in the standard uncertainty principle we show that the uncertainty product for the average local standard deviations is always less than or equal to the standard uncertainty product and that it can be arbitrarily small we apply these results to the short time fourier transform spectrogram to explore the commonly held notion that the uncertainty principle somehow limits local quantities we show that indeed for the spectrogram there is a lower bound on the local uncertainty product of the spectrogram due to the windowing operation of this method this limitation is an inherent property of the spectrogram and is not a property of the signal or a fundamental limit we also examine the local uncertainty product for a large class of time frequency distributions that satisfy the usual uncertainty principle including the wigner distribution the choi williams distribution and many other commonly used distributions we obtain an expression for the local uncertainty product in terms of the signal and show that for these distributions the local uncertainty product is less than that of the spectrogram and can be arbitrarily small extension of our approach to an entropy formulation of the uncertainty principle is also considered

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

the mesa problem for neumann boundary value problem abstract in this paper we study the singular limit of the porous medium equation u t u m g x u as m in a bounded domain with neumann boundary condition

wide band proportional bandwidth wigner ville analysis the wigner ville w v distribution is a time frequency representation that yields a highly accurate estimate of instantaneous frequency it is related to the narrow band ambiguity function by an internal transform and it can be used in a variety of detection and estimation problems a spectrogram constructed with constant bandwidth filters can be obtained by convolving two w v distributions or by forming a magnitude squared narrow band cross ambiguity function the wide band ambiguity function represents the doppler effect with dilation or compression rather than with frequency shift as in the narrow band approximation the q distributions a modified w v representation that is related to the wide band ambiguity function by an integral transform is defined a spectrogram constructed with proportional bandwidth or constant q filters can be obtained by a convolution like operation involving two q distributions or by forming a magnitude squared wide band cross ambiguity function the q distribution is thus a wide band version of the w v distribution properties of the q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms

positivity properties of phase plane distribution functions the aim of this paper is to compare the members of cohen s class of phase plane distributions with respect to positivity properties it is known that certain averages which are in a sense compatible with heisenberg s uncertainty principle of the wigner distribution over the phase plane yield non negative values for all states it is shown in this paper that the wigner distribution is unique in this respect among the members of cohen s class that have correct marginals or that satisfy moyal s formula for all states the subset of members of cohen s class not necessarily satisfying one of these two conditions with positivity properties comparable with those for the wigner distribution is shown to be rather small

convergence of the dirichlet solutions of the very fast diffusion equation abstract for any 1 m 0 positive functions f g and u 0 0 we prove that under some mild conditions on f g and u 0 as r the solution u r of the dirichlet problem u t u m m x x in r r 0 u r t f t m r 1 m u r t g t m r 1 m for all t 0 u x 0 u 0 x in r r converges uniformly on every compact subset of r 0 t to the solution of the equation u t u m m x x in r 0 t u x 0 u 0 x in r which satisfies some mass loss formula on 0 t where t is the maximal time such that the solution u is positive we also prove that the solution constructed is equal to the solution constructed in hui 2007 15 using approximation by solutions of the corresponding neumann problem in bounded cylindrical domains

kernel design for reduced interference distributions the authors present a class of time frequency signal representations tfrs called the reduced interference distribution rid an overview of commonly used tfrs is given and desirable distribution properties are introduced particular attention is paid to the interpretation of cohen s class of time frequency distributions of tfrs in the ambiguity temporal correlation spectral correlation and time frequency domains based on the desirable kernel requirements the rid is discussed and further defined a systematic procedure to create rid kernels or equivalently compute rids is proposed some aspects and properties of the rid are discussed the authors estimate design considerations for rids and compare various selections of the primitive window some experimental results demonstrating the performance of the rid are presented

hamilton s gradient estimates for a nonlinear partial differential equation under the yamabe flow abstract in this paper we derive two hamilton s gradient estimates for positive solutions of the nonlinear partial differential equation t u t u p h u q under the yamabe flow

fast slow diffusion and collapsing sandpiles we regard the limit as p of the flow governed by the p laplacian as providing a simplistic model for the collapse of an initially unstable sandpile upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion in particular we note that such models entail instantaneous mass transfer governed by monge kantorovich theory

transition kernels for bilinear time frequency distributions the author proposes a sequence of time frequency signal representations which offer a continuous transition from time smoothed pseudo wigner ville distributions pwvd to a specified class of positive representations this allows the selection of a representation which offers the desired tradeoff between the inherent localization properties of the pwvd and the lack of interference terms present in the class of positive representations a cause of some criticism of the method the sequence converges uniformly to the specified positive distribution

single channel nonstationary stochastic signal separation using linear time varying filters separability of signal mixtures given only one mixture observation is defined as the identification of the accuracy to which the signals can be separated the paper shows that when signals are separated using the generalized wiener filter the degree of separability can be deduced from the signal structure to identify this structure the processes are represented on an general spectral domain and a sufficient solution to the wiener filter is obtained the filter is composed of a term independent of the signal values corresponding to regions in the spectral domain where the desired signal components are not distorted by interfering noise components and a term dependent on the signal correlations corresponding to the region where components overlap an example of determining perfect separability of modulated random signals is given with application in radar and speech processing

nonlinear dynamic projection for noise reduction of dispersed manifolds the search for a low dimensional structure in high dimensional data is one of the fundamental tasks in machine learning and pattern recognition manifold learning algorithms have recently emerged as alternatives to traditional linear dimension reduction techniques in this paper we propose a novel projection method that can be combined with any manifold learning methods to improve their dimension reduction performance when applied to high dimensional data with a high level of noise the method first builds a dispersion function that describes the distribution of dispersed manifold where the data lie it then projects the noisy data onto a region wrapping the true manifold sufficiently close to it by applying a dynamical projection system associated with the constructed dispersion function the effectiveness of the proposed projection method is validated by applying it to some real world data sets with promising results

unitary equivalence a new twist on signal processing unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time frequency and scale implementation of these new tools involves simply preprocessing the signal by a unitary transformation performing standard processing on the transformed signal and then in some cases transforming the resulting output the resulting unitarily equivalent systems can focus on the critical signal characteristics in large classes of signals and hence prove useful for representing and processing signals that are not well matched by current techniques as specific examples of this procedure we generalize linear time invariant systems orthonormal basis and frame decompositions and joint time frequency and time scale distributions these applications illustrate the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature

some properties of the generalized time frequency representation with cone shaped kernel the cone shaped kernel generalized time frequency representation gtfr of zhao atlas and marks zam has been shown empirically to generate quite good time frequency representation in comparison to other approaches the authors analyze some specific properties of this gtfr and compare them to other tfrs asymptotically the gtfr is shown to produce results identical to that of the spectrogram for stationary signals interference terms normally present in many gtfrs are shown to be attenuated drastically by the use of the zam gtfr the ability of the zam gtfr to track frequency hopping is shown to be close to that of the wigner distribution when a signal is subjected to white noise the zam gtfr produces an unbiased estimate of the zam gtfr of the signal without noise in many other gtfrs the power spectral density of the noise is superimposed on the gtfr of the signal it is also shown that in discrete form the zam gtfr is generally invertible

positive time frequency distribution functions we demonstrate the existence of positive joint distributions of time and frequency for arbitrary signals a method is given to readily generate an infinite number of them for any signal general properties of these distribution functions are derived and specific examples for some common signals are presented

cross spectral analysis of nonstationary processes consideration is given to the generalization of stationary cross spectral analysis methods to a class of nonstationary processes specifically the class of semistationary finite energy processes possessing sample functions that are of finite energy almost surely a new quantity called the time frequency coherence tfc is defined and it is demonstrated that its properties are analogous to those possessed by the stationary coherence function the problem of estimating the tfc by using elements of l cohen s 1966 class of joint time frequency representations is investigated it is shown that the only admissible estimators are those based on the class of time frequency smoothed periodograms thus the familiar procedure of segmentation and smoothed short time fourier analysis cannot be improved upon within the framework considered by the use of the higher resolution nonparametric time frequency methods procedures for selection of the appropriate estimators and a possible application are suggested

existence of singular solutions of a degenerate equation in r2 let a 1 a 2 a m 2 2 f c 0 g i c 0 be such that 0 g i t 2 on 0 i 1 m for any open image in new window p 1 we prove the existence and uniqueness of solutions of the equation u t logu u 0 in open image in new window satisfying open image in new window and logu x t log x f t as x logu x t log x a i g i t as x a i 0 uniformly on every compact subset of 0 t for any i 1 m under a mild assumption on u 0 where open image in new window we also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of 2 with prescribed singularities at a finite number of points in the domain

the use of cone shaped kernels for generalized time frequency representations of nonstationary signals generalized time frequency representations gtfrs which use cone shaped kernels for nonstationary signal analysis are presented the cone shaped kernels are formulated for the gtfrs to produce good resolution simultaneously in time and frequency specifically for a gtfr with a cone shaped kernel finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension and the cross terms are smoothed out experimental results on simulated data and real speech show the advantages of the gtfrs with the cone shaped kernels through comparisons to the spectrogram and the pseudo wigner distribution

dynamic pattern denoising method using multi basin system with kernels in this paper we propose a novel pattern denoising method that utilizes the topological property of a support that describes the distribution of normal patterns to denoise noisy patterns the method first trains a support function which captures the domain of normal patterns and then construct a so called multi basin system associated with the trained support function by moving noisy patterns along the trajectories of the multi basin system noise is removed while the pattern recovers its normality the denoised pattern is obtained when the noisy pattern arrives at the attracting manifold generated by a set of normal patterns and this is the most similar normal pattern with the noisy pattern in the topological sense through simulations on some toy dataset and real image datasets we show that the proposed framework effectively removes the noise while preserving the information contained in the noisy pattern

a general approach for obtaining joint representations in signal analysis ii general class mean and local values and bandwidth for pt i see ibid vol 44 no 5 p 1080 90 1996 using the method developed in cohen 1996 part i the concepts of instantaneous frequency and group delay are generalized to arbitrary variables in addition new expressions for mean values and bandwidths are obtained the kernel method is used to define a general class for arbitrary variables as in the time frequency case the general class generates all possible distributions the method is also formulated in terms of the local autocorrelation function

nonradial solvability structure of super diffusive nonlinear parabolic equations we study the solvability of the cauchy problem for the nonlinear parabolic equation u div u m 1 u at when m 0 then there is no local solution of the initial value problem on the other hand if such an interval does not exist then the initial value problem is locally solvable and the time of existence can be estimated explicitly

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

wigner higher order moment spectra definition properties computation and application to transient signal analysis the wigner higher order moment spectra whos are defined as extensions of the wigner ville distribution wd to higher order moment spectra domains a general class of time frequency higher order moment spectra is also defined in terms of arbitrary higher order moments of the signal as gen eralizations of the cohen s general class of time frequency rep resentations the properties of the general class of time fre quency higher order moment spectra can be related to the properties of whos which are in fact extensions of the prop erties of the wd discrete time and frequency wigner higher order moment spectra dtf whos distributions are intro duced for signal processing applications and are shown to be implemented with two fft based algorithms one application is presented where the wigner bispectrum wb which is a whos in the third order moment domain is utilized for the detection of transient signals embedded in noise the wb is compared with the wd in terms of simulation examples and analysis of real sonar data it is shown that better detection schemes can be derived in low signal to noise ratio when the wb is applied

large time behaviour of solutions of a singular diffusion equation in rn abstract let a 0 n 2 q n 2 p n q u 0 u 0 0 where 0 u 0 l loc p r n and l 1 r n l p r n satisfy a 1 1 k x q u 0 x a 1 1 k x q x r 0 k and a 1 1 k x q x 0 r 0 k x r 0 k 1 k 2 3 for some constants 0 r 0 k r 0 k 1 k 2 3 suppose u is the maximal solution of u t log u u 0 in r n 0 u x 0 u 0 x in r n and is the unique self similar solution of the equation with initial value a x q then the rescaled function v x t t q 2 q u t 1 2 q x t will converge uniformly on every compact subset of r n to x 1 as t

behaviour of solutions of a singular diffusion equation near the extinction time abstract we prove that if 2 and 0 u0 l1 r2 lp r2 for some constant p 1 is a radially symmetric function u0 0 and u is the unique solution of the equation u t log u u 0 in r 2 0 t u x 0 u 0 x in r2 which satisfies r 2 u x t d x r 2 u 0 d x 2 t 0 and rur x t u x t uniformly on a b as r x for any 0 t r 2 u 0 d x 2 then there exist unique constants 0 1 2 2 1 such that the rescaled function v y s u y t t t t t with s log t t will converge uniformly on every compact subset of r2 to the solution y of the ode r r r 0 in 0 with r 0 0 0 1 for some constant 0 as s

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

time varying filtering and signal estimation using wigner distribution synthesis techniques the short time fourier transform stft the ambiguity function af and the wigner distribution wd are mixed time frequency signal representations that use fourier transform techniques to map a one dimensional function of time into a two dimensional function of time and frequency these mixed time frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems although much work has also been done to develop stft and af synthesis algorithms that can be used to implement a variety of time varying signal processing operations no such synthesis techniques have thus far been developed for the wd in this paper a signal synthesis algorithm that works directly with the real valued high resolution wd will be derived examples of how this wd synthesis procedure can be used to perform time varying filtering operations or signal separation will be given

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

bilinear time frequency representations new insights and properties an analysis of the interference terms of cohen class bilinear time frequency representations tfr of multicomponent signals is presented constraints for achieving new interference properties are derived imposing these interference time and interference frequency concentration constraints on a tfr guarantees that the tfr will be zero everywhere the signal s t is zero and the tfr will contain only those frequencies that occur in the signal thus these new constraints guarantee strong finite support in a tfr when these interference concentration properties are combined with interference attenuation tradeoffs between finite support the marginals and the interference properties are shown to be unavoidable the useful class of product kernels are considered and generalized further to allow tfr with potentially superior interference properties the interference frequency concentration and attenuation properties allow tfr with spectrogram like interference suppression but without the spectrogram s inherent time frequency resolution tradeoff other useful combinations of properties are discussed and analyzed and properties and tradeoffs are illustrated by examples

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

singular limit of solutions of the very fast diffusion equation we prove that the distribution solutions of the very fast diffusion equation u t um m u t um m u 0u 0 in rn 0 rn 0 u x 0 u0 x u x 0 u0 x in rnrn where m n 2p n 2 and u0 x e x 2 u0 x e x 2 for any x r1 x r1 where e 0e 0 r1 0r1 0 m0 1 m0 n 2p 1 m0 n 2 we prove that the solution of the above problem will converge uniformly on every compact subset of rn 0 rn 0 to the maximal solution of the equation vt logvvt logv v x 0 u0 x v x 0 u0 x as m 0 m 0 for any smooth bounded domain rn rn m0 1 m0 max 1 n 2 p 1 m0 max 1 n 2 we prove the existence and uniqueness of solutions of the dirichlet problem u t um m u t um m u 0u 0 in 0 0 u u0u u0 in u gu g on 0 0 with either finite or infinite positive boundary value gg we also prove a similar convergence result for the solutions of the above dirichlet problem as m 0m 0

fast support based clustering method for large scale problems in many support vector based clustering algorithms a key computational bottleneck is the cluster labeling time of each data point which restricts the scalability of the method in this paper we review a general framework of support vector based clustering using dynamical system and propose a novel method to speed up labeling time which is log linear to the size of data we also give theoretical background of the proposed method various large scale benchmark results are provided to show the effectiveness and efficiency of the proposed method

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

a novel kernel method for clustering kernel methods are algorithms that by replacing the inner product with an appropriate positive definite function implicitly perform a nonlinear mapping of the input data into a high dimensional feature space in this paper we present a kernel method for clustering inspired by the classical k means algorithm in which each cluster is iteratively refined using a one class support vector machine our method which can be easily implemented compares favorably with respect to popular clustering algorithms like k means neural gas and self organizing maps on a synthetic data set and three uci real data benchmarks iris data wisconsin breast cancer database spam database

dynamic characterization of cluster structures for robust and inductive support vector clustering a topological and dynamical characterization of the cluster structures described by the support vector clustering is developed it is shown that each cluster can be decomposed into its constituent basin level cells and can be naturally extended to an enlarged clustered domain which serves as a basis for inductive clustering a simplified weighted graph preserving the topological structure of the clusters is also constructed and is employed to develop a robust and inductive clustering algorithm simulation results are given to illustrate the robustness and effectiveness of the proposed method

existence of solutions of the very fast diffusion equation abstract let m u um m if 1 m u log u if m 0 for any m 1 r t 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x and 0 f bv 0 such that 0 f d s for some constant 0 we prove the existence of solution of the equation ut m u xx in r 0 t 0 u x u0 in r for some constant t0 t such that 0 t u m 1 u x 0 s d s 0 u y t d y m 1 t t f d s 0 0 which satisfies a b u m 1 u x x s d s a b f d s as x for all 0 log u x as x uniformly on a b if m 0 for all 0 0 u x t d x and 0 u x t d x for all 0 log u x f t as x uniformly on a b if m 0 for all 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x we prove the existence of infinitely many finite mass solutions of the above equation with prescribed mass and line integral at time t and show that the solution will satisfy some uniform decay rate as x

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

aronson benilan estimates for the fast diffusion equation under the ricci flow abstract we study the fast diffusion equation with a linear forcing term 0 1 u t div u p 1 u r u under the ricci flow on a complete manifold m such that m r 2 has bounded curvature and nonnegative isotropic curvature where 0 p 1 and r r x t is the evolving scalar curvature of m at time t we prove aronson benilan and li yau hamilton type differential harnack estimates for positive solutions of 0 1 in addition we use similar method to prove certain li yau hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in cao and hamilton 2009 cao 2008 and kuang and zhang 2008 to the noncompact setting

optimum smoothing of the wigner ville distribution a compromise is found between the different requirements that we would like to be fulfilled by a time frequency distribution namely positivity and obtention of a distribution close to the dirac one for the unimodular signal s t exp i phi t the fulfillment of the marginal conditions being of less interest in signal theory starting from the usual wigner ville distribution we define an optimum smoothing by minimizing the width of the different functions approximating the desired dirac distribution the smoothing is obtained by a convolution through a double gaussian of width t and such that t 1 2 two possibilities appear in the first one we do not introduce any correlation between t and in the convolution kernel and obtain a simple result in the second one extrapolating the frequency variation and still using a gaussian we obtain a better result although the smoothing process becomes more complex these results to be physically meaningful impose inequalities on the successive derivatives of which are equivalent to those used for the obtention of the classical limit for the corresponding quantum problem

probabilistic generative ranking method based on multi support vector domain description as the volume of database grows retrieval and ordering of information according to relevance has become an important and challenging task ranking problem has recently been considered and formulated as a machine learning problem among the various learning to rank methods the ranking support vector machines svms have been widely applied in various applications because of its state of the art performance in this paper we propose a novel ranking method based on a probabilistic generative model approach the proposed method utilizes multi support vector domain description multi svdd and constructs pseudo conditional probabilities for data pairs thus enabling the construction of an efficient posterior probability function of relevance judgment of data pairs results of experiments on both synthetic and real large scale datasets show that the proposed method can efficiently learn ranking functions better than ranking svms

time scale energy distributions a general class extending wavelet transforms the theory of a new general class of signal energy representations depending on time and scale is developed time scale analysis has been introduced recently as a powerful tool through linear representations called continuous wavelet transforms wts a concept for which an exhaustive bilinear generalization is given although time scale is presented as an alternative method to time frequency strong links relating the two are emphasized thus combining both descriptions into a unified perspective the authors provide a full characterization of the new class the result is expressed as an affine smoothing of the wigner ville distribution on which interesting properties may be further imposed through proper choices of the smoothing function parameters not only do specific choices allow recovery of known definitions but they also provide via separable smoothing a continuous transition from wigner ville to either spectrograms or scalograms squared modulus of the wt this property makes time scale representations a very flexible tool for nonstationary signal analysis

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

position regularized support vector domain description support vector domain description svdd is an effective method for describing a set of objects as a basic tool several application oriented extensions have been developed such as support vector clustering svc svdd based k means svddk means and support vector based algorithm for clustering data streams svstream despite its significant success one inherent drawback is that the description is very sensitive to the selection of the trade off parameter which is hard to estimate in practice and affects the extensive approaches significantly to tackle this problem we propose a novel position regularized support vector domain description psvdd in the proposed psvdd the complexity of the sphere surface is adaptively regularized by assigning a position based weighting to each data point which is computed according to the distance between the corresponding feature space image and the mean of feature space images to demonstrate the effectiveness of the proposed psvdd we apply the position based weighting to improve two important clustering extensions i e svc and svddk means which respectively result in two new clustering approaches termed psvc and psvddk means experimental results on several real world data sets validate the significant improvement achieved by psvc and psvddk means

inductive manifold learning using structured support vector machine most manifold learning techniques are used to transform high dimensional data sets into low dimensional space in the use of such techniques after unseen data samples are added to the data set retraining is usually necessary however retraining is a time consuming process and no guarantee of the transformation into the exactly same coordinates thus presenting a barrier to the application of manifold learning as a preprocessing step in predictive modeling to solve this problem learning a mapping from high dimensional representations to low dimensional coordinates is proposed via structured support vector machine after training a mapping low dimensional representations of unobserved data samples can be easily predicted experiments on several datasets show that the proposed method outperforms the existing out of sample extension methods highlightswe propose a general framework for out of sample extensions for any manifold learning methods we learn a mapping from original space to its manifolds by using structured svm experiments on several datasets show that the proposed method outperforms the existing ones

from fast to very fast diffusion in the nonlinear heat equation we study the asymptotic behavior of the sign changing solution of the equation u t u u f when the diffusion becomes very fast i e as 1 we prove that a solution u t converges in l 1 uniformly for t in subsets with compact support in 0 t to a solution of u t u 1 u f in contrast with the case of 0 the limiting solutions are either positive or negative or identically equal to 0 in all this causes the limit to be singular in the sense that a boundary layer appears at t 0 when one lets 1

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc