carleman inequalities and the heat operator we prove a strong unique continuation theorem for functions u vanishing to infinite order in the space time variables at 0 0 and verifying the inequality 4u tu v x t u on rn 0 t when t1 n 2r 1 s v x t lt locl r x n 2r 1 s 1 n 2 r

a comparison of optimized higher order spectral detection techniques for non gaussian signals using the gaussian noise rejection property of higher order spectra hos hos based detectors have been proposed that outperform conventional second order techniques in certain scenarios based on statistical tests proposed by suhha rao and gabr 1980 as well as hinich 1982 kletter and messer 1990 and hinich and wilson 1990 have developed similar bifrequency domain detectors that are dependent on bispectral estimates of the observation process formalizing the estimate consistency requirements and the asymptotics for these detectors we derive a new f test statistic we consider the detrimental effects of using spectral estimates in the denominator of hinich s test we determine refined conditional distributions for third and fourth order versions of his detector we also modify his test for colored scenarios extending the bispectral detectors to their kth order counterparts we calculate the optimal smoothing bandwidth to use in constructing the hos estimates producing the best detection performances for both our f test and hinich s test with our refined distributions these new bandwidths yield significant improvements in detector performance over previous results for the finite sample case our calculations characterize the tradeoff between the two detectors and demonstrate that a larger smoothing bandwidth than the one suggested by previous researchers should be used our calculations are verified using simulations for both white and colored cases

detection and parameter estimation of a transient signal using order statistics detection and parameter estimation of a transient signal in noise is a problem of many applications it is characterized by the fact that some of the measurements consist of noise only modern statistical signal processing techniques are applied on a discrete version of the received data and are implemented by digital signal processing dsp in this correspondence we show how order statistics os based signal processing which is of a discrete nature can be used for simultaneous detection and estimation of parameters such as time of arrival and signal duration of a sampled transient signal in white noise we show that the resulting processors are more robust than the conventional processors whereas their performance is about the same at the cost of increased computational complexity

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

a faber krahn inequality for solutions of schrodinger s equation abstract we consider nontrivial solutions of u x v x u x where u 0 on the boundary of a bounded open region d r n and v x l d we prove a sharp relationship between v and the measure of d which generalizes the well known faber krahn theorem we also prove some geometric properties of the zero sets of the solution of the schrodinger equation u x v x u x

on minimal support properties of solutions of schrodinger equations abstract in this paper we obtain minimal support properties of solutions of schrodinger equations we improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small we also use these properties to obtain some new results on unique continuation for the schrodinger operator

detection of random signals by integrated polyspectral analysis we consider the problem of detecting an unknown random stationary signal gaussian or non gaussian in gaussian noise of known correlation structure no other assumptions are made about the signal to be detected we suggest two approaches utilizing both the power spectrum and the third and fourth order integrated polyspectra for the detection of random signals the detector structures of the proposed approaches are derived and their performance is evaluated via simulations and comparisons with the classical energy detector involving both gaussian and non gaussian signals it is shown that the power of one of the proposed tests is competitive with that of the energy detector for gaussian signals and it outperforms the energy detector for the non gaussian signals tested

decay and regularity in the inverse scattering problem abstract the regularity and decay properties for the potential q x in the schrodinger equation q k 2 on the line are characterized in terms of the decay and regularity of the reflection coefficients r and their fourier transforms

sharp counterexamples in unique continuation for second order elliptic equations we construct nontrivial solutions with compact support for the elliptic equation u v u with v lp p n 2 or v l w for n 3 and with v l1 for n 2 the same method also yields nontrivial solutions with compact support for the elliptic equation u w u with w lq p n or w lw for n 2 for the second order elliptic equations u v u in r 1 respectively u w u in r 2 we define the weak unique continuation property ucp respectively the strong unique continuation property sucp as follows let u be a solution to 1 which vanishes in an open set then u 0 ucp let u be a solution to 1 which vanishes of infinite order at some point x0 r then u 0 sucp research partially supported by the dfg and daad research partially supported by the nsf grants dms 9970297 int 9815286

solutions of the korteweg devries equation with steplike initial data no differentiability of u is assumed at all the goal is to find the range of n such that 1 1 and 1 2 has a classical solution it is easy to check that if u x t t 0 solves kdv so does u x t u x 6ht t h where h is in 02 similarly one verifies that if u x t solves kdv for t 0 with initial data u x 0 u x then w x t u x t solves kdv for t 0 and t 0 it is convenient to write b c2 c 0 our existence theorem is based on a construction suggested by the inverse scattering method if t 0 we prove existence of regular solutions for n 2 3 regularity and decay as x cc depend on n by using a new l inverse scattering result 9 we are also able to get control over our solution as x p co at least for initial data satisfying a rather mild 306 0022 0396186 3 00

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

unique continuation for δ and the c fefferman phong class we show that the strong unique continuation property holds for the inequality u v u where the potential v x satisfies the c fefferman phong condition in a certain range of p values we also deal with the situation of u x vanishing at infinity these are all consequences of appropriate carleman inequalities

on the use of asymptotics in detection and estimation we illustrate the importance of a finite dimensionality assumption when using functions of asymptotic statistics we also note that asymptotic distributions need to converge uniformly to facilitate algebraic manipulations finally we point to subtleties in using detection statistics stemming from the central limit theorem and taylor series without entering the large deviations regime

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

transient signal detection using higher order moments the asymptotic performance of transient detection based on higher order moments is theoretically derived invoking the asymptotic normality of the decision statistics and a detector based on the third order absolute moment is proposed the analysis shows that for a transient duration of less than half the observation window the proposed detector outperforms the detectors based on the second and fourth order moments for a wide range of snr values computer simulations assess the applicability of the theoretical analysis

time domain tests for gaussianity and time reversibility statistical signal processing algorithms often rely upon gaussianity and time reversibility two important notions related to the probability structure of stationary random signals and their symmetry parametric models obtained via second order statistics sos are appropriate when the available data is gaussian and time reversible on the other hand evidence of nonlinearity non gaussianity or time irreversibility favors the use of higher order statistics hos in order to validate gaussianity and time reversibility and quantify the tradeoffs between sos and hos consistent time domain chi squared statistical tests are developed exact asymptotic distributions are derived to estimate the power of the tests including a covariance expression for fourth order sample cumulants a modification of existing linearity tests in the presence of additive gaussian noise is discussed briefly the novel gaussianity statistic is computationally attractive leads to a constant false alarm rate test and is well suited for parametric modeling because it employs the minimal hos lags which uniquely characterize arma processes simulations include comparisons with an existing frequency domain approach and an application to real seismic data time reversibility tests are also derived and their performance is analyzed both theoretically and experimentally