nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

existence and stability of standing waves for a planar gauged nonlinear schrodinger equation abstract in this article we prove the existence stability and mass collapse behavior of standing waves for the chern simons schrodinger model in the plane this is a covariant nonlinear schrodinger type problem which is mass critical and arises from the chern simons theory in the defocusing case 1 we prove the existence of stable standing waves with arbitrary mass however in the focusing case 1 stable standing waves with small mass exist and we give a mass collapse behavior of the standing waves

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

chern simons limit of the standing wave solutions for the schrodinger equations coupled with a neutral scalar field abstract we show the existence of standing wave solutions to the schrodinger equation coupled with a neutral scalar field we also verify the chern simons limit for these solutions more precisely we prove that solutions to eqs 1 3 1 4 converge to the unique positive radially symmetric solution of the nonlinear schrodinger equation 1 6 as the coupling constant q goes to infinity

existence and concentration of semiclassical ground state solutions for the generalized chern simons schrodinger system in h1 r2 abstract this paper is concerned with the following singularly perturbed problem in h 1 r 2 e 2 u v x u a 0 u x u j 1 2 a j 2 u x u f u e 1 a 2 u x 2 a 1 u x 1 2 u 2 1 a 1 u x 2 a 2 u x 0 e a 0 u 1 a 2 u 2 2 a 1 u 2 where e is a small parameter v c r 2 r and f c r r by using some new variational and analytic techniques joined with the manifold of pohozaev nehari type we prove that there exists a constant e 0 0 determined by v and f such that for e 0 e 0 the above problem admits a semiclassical ground state solution v e with exponential decay at infinity we also establish a new concentration behaviour of v e as e 0 in particular our results are available to the nonlinearity f u u s 2 u for s 4 6 which extend the existing results concerning the case f u u s 2 u for s 6

bagging for linear classifiers abstract classifiers built on small training sets are usually biased or unstable different techniques exist to construct more stable classifiers it is not clear which ones are good and whether they really stabilize the classifier or just improve the performance in this paper bagging bootstrapping and aggregating l breiman bagging predictors machine learning j 24 2 123 140 1996 is studied for a number of linear classifiers a measure for the instability of classifiers is introduced the influence of regularization and bagging on this instability and the generalization error of linear classifiers is investigated in a simulation study it is shown that in general bagging is not a stabilizing technique it is also demonstrated that one can consider the instability of the classifier to predict how useful bagging will be finally it is shown experimentally that bagging might improve the performance of the classifier only for very unstable situations

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

lieb thirring constant on the sphere and on the torus abstract we prove on the sphere s 2 and on the torus t 2 the lieb thirring inequalities with improved constants for orthonormal families of scalar and vector functions

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

standing waves of nonlinear schrodinger equations with the gauge field abstract we study standing waves for nonlinear schrodinger equations with the gauge field some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field a in terms of complex scalar field we also show that there exists no standing wave for certain range of parameters by establishing a new inequality of sobolev type

averaging maximum penalized likelihood and bayesian estimation for improving gaussian mixture probability density estimates we apply the idea of averaging ensembles of estimators to probability density estimation in particular we use gaussian mixture models which are important components in many neural network applications we investigate the performance of averaging using three data sets for comparison we employ two traditional regularization approaches i e a maximum penalized likelihood approach and a bayesian approach in the maximum penalized likelihood approach we use penalty functions derived from conjugate bayesian priors such that an expectation maximization em algorithm can be used for training in all experiments the maximum penalized likelihood approach and averaging improved performance considerably if compared to a maximum likelihood approach in two of the experiments the maximum penalized likelihood approach outperformed averaging in one experiment averaging was clearly superior our conclusion is that maximum penalized likelihood gives good results if the penalty term in the cost function is appropriate for the particular problem if this is not the case averaging is superior since it shows greater robustness by not relying on any particular prior assumption the bayesian approach worked very well on a low dimensional toy problem but failed to give good performance in higher dimensional problems

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

numerical study of blow up and dispersive shocks in solutions to generalized korteweg de vries equations abstract we present a detailed numerical study of solutions to general korteweg de vries equations with critical and supercritical nonlinearity both in the context of dispersive shocks and blow up we study the stability of solitons and show that they are unstable against being radiated away and blow up in the l 2 critical case the blow up mechanism by martel merle and raphael can be numerically identified in the limit of small dispersion it is shown that a dispersive shock always appears before an eventual blow up in the latter case always the first soliton to appear will blow up it is shown that the same type of blow up as for the perturbations of the soliton can be observed which indicates that the theory by martel merle and raphael is also applicable to initial data with a mass much larger than the soliton mass we study the scaling of the blow up time t in dependence of the small dispersion parameter and find an exponential dependence t and that there is a minimal blow up time t 0 greater than the critical time of the corresponding hopf solution for 0 to study the cases with blow up in detail we apply the first dynamic rescaling for generalized korteweg de vries equations this allows to identify the type of the singularity

combining multiple classifiers by averaging or by multiplying abstract in classification tasks it may be wise to combine observations from different sources not only it decreases the training time but it can also increase the robustness and the performance of the classification combining is often done by just weighted averaging of the outputs of the different classifiers using equal weights for all classifiers then results in the mean combination rule this works very well in practice but the combination strategy lacks a fundamental basis as it cannot readily be derived from the joint probabilities this contrasts with the product combination rule which can be obtained from the joint probability under the assumption of independency in this paper we will show differences and similarities between this mean combination rule and the product combination rule in theory and in practice

sign changing solutions to a gauged nonlinear schrodinger equation abstract we study the existence and asymptotic behavior of the least energy sign changing solutions to a gauged nonlinear schrodinger equation u u h 2 x x 2 x h s s u 2 s d s u u p 2 u x r 2 u h r 1 r 2 where 0 p 6 and h s 1 2 0 s r u 2 r d r combining constraint minimization method and quantitative deformation lemma we prove that the problem possesses at least one least energy sign changing solution u which changes sign exactly once moreover we show that the energy of u is strictly larger than two times of the ground state energy finally the asymptotic behavior of u as 0 is also analyzed

exact solutions of the gross pitaevskii equation in periodic potential in the presence of external source exact periodic solutions solitonlike solutions singular solitary and singular trigonometric wave solutions of the time dependent gross pitaevskii equation gpe with elliptic function potential in the presence of external source are analyzed a simple approach that applies equally to both attractive and repulsive time dependent gpe and allows one to find an extensive list of explicit periodic solutions of the gpe in terms of the jacobian elliptic functions is developed in the limit as the elliptic modulus tends to unity or to zero the linear solutions in either the jacobian elliptic cosine or the jacobian elliptic function of third order give solitonlike solutions while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions the stability of these solutions is investigated numerically

the asymptotic value in finite stochastic games in 1976 bewley and kohlberg proved that the discounted values v of finite zero sum stochastic games have a limit as tends to 0 using the tarski seidenberg elimination theorem from real algebraic geometry this was a fundamental step in the development of the theory of stochastic games the current paper provides a new and direct proof for this result relying on the explicit description of asymptotically optimal strategies both approaches can be used to obtain the existence of the uniform value using the construction from mertens and neyman 1981

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

a bayesian committee machine the bayesian committee machine bcm is a novel approach to combining estimators that were trained on different data sets although the bcm can be applied to the combination of any kind of estimators the main foci are gaussian process regression and related systems such as regularization networks and smoothing splines for which the degrees of freedom increase with the number of training data somewhat surprisingly we find that the performance of the bcm improves if several test points are queried at the same time and is optimal if the number of test points is at least as large as the degrees of freedom of the estimator the bcm also provides a new solution for on line learning with potential applications to data mining we apply the bcm to systems with fixed basis functions and discuss its relationship to gaussian process regression finally we show how the ideas behind the bcm can be applied in a non bayesian setting to extend the input dependent combination of estimators

nodal standing waves for a gauged nonlinear schrodinger equation in r2 abstract this paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear schrodinger equation u u h 2 x x 2 x h s s u 2 s d s u u p 2 u x r 2 u x u x h 1 r 2 where 0 p 6 and h s 1 2 0 s r u 2 r d r is the so called chern simons term we prove that for any positive integer k the problem has a sign changing solution u k which changes sign exactly k times moreover the energy of u k is strictly increasing in k and for any sequence n n there exists a subsequence n s such that n s 1 p 2 u k n s converges in h 1 r 2 to w k as s where w k also changes sign exactly k times and solves the following equation u u u p 2 u u h 1 r 2

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

on fusers that perform better than best sensor in a multiple sensor system sensor s sub i i 1 2 n outputs y sup i spl isin 0 1 according to an unknown probability distribution p y sup i x in response to input x spl isin 0 1 we choose a fuser that combines the outputs of sensors from a function class f f 0 1 sup n spl rarr 0 1 by minimizing empirical error based on an i i d sample if f satisfies the isolation property we show that the fuser performs at least as well as the best sensor in a probably approximately correct sense several well known fusers such as linear combinations special potential functions and certain feedforward networks satisfy the isolation property

multiple normalized solutions for quasi linear schrodinger equations abstract in this paper we prove the existence of two solutions having a prescribed l 2 norm for a quasi linear schrodinger equation one of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global to overcome the lack of differentiability of the associated functional we rely on a perturbation method developed in 25

standing waves for a schrodinger chern simons higgs system abstract we consider a system arising from a nonrelativistic chern simon higgs model in which a charged field is coupled with a gauge field we prove an existence result for small coupling constants

definable zero sum stochastic games definable zero sum stochastic games involve a finite number of states and action sets and reward and transition functions that are definable in an o minimal structure prominent examples of such games are finite semi algebraic or globally subanalytic stochastic games we prove that the shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure definability in the same structure does not hold systematically we provide a counterexample of a stochastic game with semi algebraic data yielding a non semi algebraic but globally subanalytic shapley operator our definability results on shapley operators are used to prove that any separable definable game has a uniform value in the case of polynomially bounded structures we also provide convergence rates using an approximation procedure we actually establish that general zero sum games with separable definable transition functions have a uniform value these results highlight the key role played by the tame structure of transition functions as particular cases of our main results we obtain that stochastic games with polynomial transitions definable games with finite actions on one side and definable games with perfect information or switching controls have a uniform value applications to nonlinear maps arising in risk sensitive control and perron frobenius theory are also given

standing waves for the chern simons schrodinger equation with critical exponential growth abstract in this paper by combing the variational methods and trudinger moser inequality we study the existence and multiplicity of the positive standing waves for the following chern simons schrodinger equation 0 1 u u 0 h s s u 2 s d s h 2 x x 2 u f x u k x in r 2 where h s 0 s l 2 u 2 l d l 0 and the nonlinearity f r 2 r r behaves like exp u 2 as u for the case 0 we can get a mountain pass type solution

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

standing waves of the schrodinger equation coupled with the chern simons gauge field we prove the existence of infinitely many radially symmetric standing wave solutions of the chern simons schrodinger system our result is established by applying the mountain pass theorem to the functional which is obtained by representing gauge fields a in terms of a scalar field

averaging regularized estimators we compare the performance of averaged regularized estimators we show that the improvement in performance that can be achieved by averaging depends critically on the degree of regularization which is used in training the individual estimators we compare four different averaging approaches simple averaging bagging variance based weighting and variance based bagging in any of the averaging methods the greatest degree of improvement if compared to the individual estimators is achieved if no or only a small degree of regularization is used here variance based weighting and variance based bagging are superior to simple averaging or bagging our experiments indicate that better performance for both individual estimators and for averaging is achieved in combination with regularization with increasing degrees of regularization the two bagging based approaches bagging and variance based bagging outperform the individual estimators simple averaging and variance based weighting bagging and variance