one dimensional behavior of singular n dimensional solutions of semilinear heat equations we consider u x t a solution of u t u u p 1 u that blows up at time t where u n 0 t p 1 n 2 p 0 in particular if in addition p 3 then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow up set finally we prove that the mere hypothesis that the blow up set is continuous implies that it is c 1 1 2 for any 0

profile for the imaginary part of a blowup solution for a complex valued semilinear heat equation abstract in this paper we consider the following complex valued semilinear heat equation t u u u p u c in the whole space r n where p n p 2 we aim at constructing for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a with the following estimates for the final profile u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a as x a note that the imaginary part is non zero and that it blows up also at point a our method relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion up to our knowledge this is the first time where the blowup behavior of the imaginary part is derived in multi dimension

a novel spike distance the discrimination between two spike trains is a fundamental problem for both experimentalists and the nervous system itself we introduce a measure for the distance between two spike trains the distance has a time constant as a parameter depending on this parameter the distance interpolates between a coincidence detector and a rate difference counter the dependence of the distance on noise is studied with an integrate and fire model for an intermediate range of the time constants the distance depends linearly on the noise this property can be used to determine the intrinsic noise of a neuron

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

a reproducing kernel hilbert space framework for spike train signal processing this letter presents a general framework based on reproducing kernel hilbert spaces rkhs to mathematically describe and manipulate spike trains the main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes moreover because many inner products can be formulated a particular definition can be crafted to best fit an application these ideas are illustrated by the definition of a number of spike train inner products to further elicit the advantages of the rkhs framework a family of these inner products the cross intensity ci kernels is analyzed in detail this inner product family encapsulates the statistical description from the conditional intensity functions of spike trains the problem of their estimation is also addressed the simplest of the spike train kernels in this family provide an interesting perspective to others work as will be demonstrated in terms of spike train distance measures finally as an application example the rkhs framework is used to derive a clustering algorithm for spike trains from simple principles

neural decoding with kernel based metric learning in studies of the nervous system the choice of metric for the neural responses is a pivotal assumption for instance a well suited distance metric enables us to gauge the similarity of neural responses to various stimuli and assess the variability of responses to a repeated stimulus exploratory steps in understanding how the stimuli are encoded neurally here we introduce an approach where the metric is tuned for a particular neural decoding task neural spike train metrics have been used to quantify the information content carried by the timing of action potentials while a number of metrics for individual neurons exist a method to optimally combine single neuron metrics into multineuron or population based metrics is lacking we pose the problem of optimizing multineuron metrics and other metrics using centered alignment a kernel based dependence measure the approach is demonstrated on invasively recorded neural data consisting of both spike trains and local field potentials the experimental paradigm consists of decoding the location of tactile stimulation on the forepaws of anesthetized rats we show that the optimized metrics highlight the distinguishing dimensions of the neural response significantly increase the decoding accuracy and improve nonlinear dimensionality reduction methods for exploratory neural analysis

a unified framework for quadratic measures of independence this paper proposes a unified framework for several available measures of independence by generalizing the concept of information theoretic learning itl the key component of itl is the use of inner product between two density functions as a measure of similarity between two random variables we show that by generalizing the inner product using a symmetric strictly positive definite kernel and by choosing appropriate kernels it is possible to reproduce a number of popular measures of independence this unified framework also allows the design of new strictly positive definite kernels and corresponding measures of independence following this framework we explore a new measure of independence and apply it in the context of linear independent component analysis ica an attractive property of the proposed method is that it does not involve any free parameter and we demonstrate that it performs equally well compared to the existing methods for ica

generic solutions for some perturbed optimization problem in non reflexive banach spaces abstract let z be a closed boundedly relatively weakly compact nonempty subset of a banach space x and j z r a lower semicontinuous function bounded from below if x 0 is a convex subset in x and x 0 has approximatively z property k then the set of all points x in x 0 z for which there exists z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x contains a subsequence strongly convergent to an element of z is a dense g subset of x 0 z moreover under the assumption that x 0 is approximatively z strictly convex we show more namely that the set of all points x in x 0 z for which there exists a unique point z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x converges strongly to z 0 is a dense g subset of x 0 z here x inf j z x z z z these extend s cobzas s result j math anal appl 243 2000 344 356

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

the blow up problem for a semilinear parabolic equation with a potential abstract let be a bounded smooth domain in r n we consider the problem u t u v x u p in 0 t with dirichlet boundary conditions u 0 on 0 t and initial datum u x 0 m x where m 0 is positive and compatible with the boundary condition we give estimates for the blow up time of solutions for large values of m as a consequence of these estimates we find that for m large the blow up set concentrates near the points where p 1 v attains its maximum

binless strategies for estimation of information from neural data we present an approach to estimate information carried by experimentally observed neural spike trains elicited by known stimuli this approach makes use of an embedding of the observed spike trains into a set of vector spaces and entropy estimates based on the nearest neighbor euclidean distances within these vector spaces l f kozachenko and n n leonenko probl peredachi inf 23 9 1987 using numerical examples we show that this approach can be dramatically more efficient than standard bin based approaches such as the direct method s p strong r koberle r r de ruyter van steveninck and w bialek phys rev lett 80 197 1998 for amounts of data typically available from laboratory experiments

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

strictly positive definite spike train kernels for point process divergences exploratory tools that are sensitive to arbitrary statistical variations in spike train observations open up the possibility of novel neuroscientific discoveries developing such tools however is difficult due to the lack of euclidean structure of the spike train space and an experimenter usually prefers simpler tools that capture only limited statistical features of the spike train such as mean spike count or mean firing rate we explore strictly positive definite kernels on the space of spike trains to offer both a structural representation of this space and a platform for developing statistical measures that explore features beyond count or rate we apply these kernels to construct measures of divergence between two point processes and use them for hypothesis testing that is to observe if two sets of spike trains originate from the same underlying probability law although there exist positive definite spike train kernels in the literature we establish that these kernels are not strictly definite and thus do not induce measures of divergence we discuss the properties of both of these existing nonstrict kernels and the novel strict kernels in terms of their computational complexity choice of free parameters and performance on both synthetic and real data through kernel principal component analysis and hypothesis testing

discrimination with spike times and isi distributions we study the discrimination capability of spike time sequences using the chernoff distance as a metric we assume that spike sequences are generated by renewal processes and study how the chernoff distance depends on the shape of interspike interval isi distribution first we consider a lower bound to the chernoff distance because it has a simple closed form then we consider specific models of isi distributions such as the gamma inverse gaussian ig exponential with refractory period er and that of the leaky integrate and fire lif neuron we found that the discrimination capability of spike times strongly depends on high order moments of isi and that it is higher when the spike time sequence has a larger skewness and a smaller kurtosis high variability in terms of coefficient of variation cv does not necessarily mean that the spike times have less discrimination capability spike sequences generated by the gamma distribution have the minimum discrimination capability for a given mean and variance of isi we used series expansions to calculate the mean and variance of isis for lif neurons as a function of the mean input level and the input noise variance spike sequences from an lif neuron are more capable of discrimination than those of ig and gamma distributions when the stationary voltage level is close to the neuron s threshold value of the neuron

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

porosity of perturbed optimization problems in banach spaces let x be a banach space and z a nonempty closed subset of x l etj z r be a lower semicontinuous function bounded from below this paper is concerned with the perturbed optimization problem infz z j z x z denoted by x j inf for x x in the case when x is compactly fully 2 convex it is proved in the present paper that the set of all points x in x for which there does not exist z0 z such that j z0 x z0 infz z j z x z is a porous set in x f urthermore ifx is assumed additionally to be compactly locally uniformly convex we verify that the set of all points x x z 0 such that the problem x j inf fails to be approximately compact is a porous set in x z 0 w herez 0 denotes the set of all z z such that z pz z moreover a counterexample to which some results of ni r x ni generic solutions for some perturbed optimization problem in nonreflexive banach space j math anal appl 302 2005 417 424 fail is provided

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

construction of a blow up solution for the complex ginzburg landau equation in a critical case we construct a solution for the complex ginzburg landau equation in some critical case which blows up in finite time t only at one blow up point we also give a sharp description of its profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude the interpretation of the parameters of the finite dimension problem in terms of the blow up point and time allows to prove the stability of the constructed solution

construction of a stable periodic solution to a semilinear heat equation with a prescribed profile we construct a periodic solution to the semilinear heat equation with power nonlinearity in one space dimension which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude thanks to the geometrical interpretation of the finite dimensional parameters in terms of the blow up time and blow up point we derive the stability of the constructed solution with respect to initial data

existence and porosity for a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty closed subset of x let j z r be an upper semicontinuous function bounded from above this paper is concerned with the perturbed optimization problem sup z z j z x z which is denoted by x j sup we shall prove in the present paper that if z is a closed boundedly relatively weakly compact nonempty subset then the set of all x x for which the problem x j sup has a solution is a dense g subset of x in the case when x is uniformly convex and j is bounded we will show that the set of all points x in x for which there does not exist z 0 z such that j z 0 x z 0 sup z z j z x z is a porous subset of x and the set of all points x x z 0 such that there exists a maximizing sequence of the problem x j sup which has no convergent subsequence is a porous subset of x z 0 where z 0 denotes the set of all z z such that z is in the solution set of z j sup

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

generic existence of solutions for some perturbed optimization problems abstract let x be a real banach space z a closed nonvoid subset of x and j z r a lower semicontinuous function bounded from below if x is reflexive and has the kadets property then the set of all x x for which there exists z0 z such that j z0 x z0 inf j z x z z z is g and dense in x this extends a result of k s lau 1978 indiana univ math j 27 791 795 on nearest points the case j 0