unique continuation for sublinear elliptic equations based on carleman estimates abstract in this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth under suitable regularity assumptions we prove the weak and the strong unique continuation property moreover we also discuss the unique continuation property from measurable sets which shows that nodal domains to these equations must have vanishing lebesgue measure our methods rely on suitable carleman estimates for which we include the sublinear potential into the main part of the operator

carleman estimates and absence of embedded eigenvalues let l w be a schrodinger operator with a potential w in l frac n 1 2 mathbb r n n geq 2 we prove that there is no positive eigenvalue the main tool is an l p l p prime carleman type estimate which implies that eigenfunctions to positive eigenvalues must be compactly supported the carleman estimate builds on delicate dispersive estimates established in 7 we also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials

on treewidth and minimum fill in of asteroidal triple free graphs we present o n5r n3r3 time algorithms to compute the treewidth pathwidth minimum fill in and minimum interval graph completion of asteroidal triple free graphs where n is the number of vertices and r is the number of minimal separators of the input graph this yields polynomial time algorithms for the four np complete graph problems on any subclass of the asteroidal triple free graphs that has a polynomially bounded number of minimal separators as e g cocomparability graphs of bounded dimension and d trapezoid graphs for any fixed d 1

a note on eigenvalue bounds for schrodinger operators abstract we obtain a new bound on the location of eigenvalues for a non self adjoint schrodinger operator with complex valued potentials by obtaining a weighted l 2 estimate for the resolvent of the laplacian

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

schrodinger operators with ll 2w rl potentials we consider the schrodinger operators v with v in the weak lq class with q 1 2 the underlying dimension which is the borderline for a definition to be possible we concentrate first on optimal bounds on how large the weak norm of v can be and then on spectral properties on lp spaces

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

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

an integrated model for evaluating the amount of data required for reliable recognition many recognition procedures rely on the consistency of a subset of data features with a hypothesis as the sufficient evidence to the presence of the corresponding object we analyze here the performance of such procedures using a probabilistic model and provide expressions for the sufficient size of such data subsets that if consistent guarantee the validity of the hypotheses with arbitrary confidence we focus on 2d objects and the affine transformation class and provide for the first time an integrated model which takes into account the shape of the objects involved the accuracy of the data collected the clutter present in the scene the class of the transformations involved the accuracy of the localization and the confidence we would like to have in our hypotheses increasingly it turns out that most of these factors can be quantified cumulatively by one parameter denoted effective similarity which largely determines the sufficient subset size the analysis is based on representing the class of instances corresponding to a model object and a group of transformations as members of a metric space and quantifying the variation of the instances by a metric cover

the periodic schrodinger operators with potentials in the morrey class we consider the periodic schrodinger operator v x in rd d 3 with potential v in the morrey class let be a periodic cell for v we show that for p d 1 2 d 2 there exists a positive constant e depending only on the shape of p and d such that iflim supr 0supx r21 b x r b x r v y pdy1 p e then the spectrum of v is purely absolutely continuous we obtain this result as a consequence of certain weighted l2 sobolev inequalities on the d torus it improves an early result by the author for potentials in ld 2 or weak ld 2 space

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

global unique continuation from a half space for the schrodinger equation abstract we obtain a global unique continuation result for the differential inequality i t u v x u in r n 1 this is the first result on global unique continuation for the schrodinger equation with time independent potentials v x in r n our method is based on a new type of carleman estimates for the operator i t on r n 1 as a corollary of the result we also obtain a new unique continuation result for some parabolic equations

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

on spectral stability of the nonlinear dirac equation abstract we study the point spectrum of the nonlinear dirac equation in any spatial dimension linearized at one of the solitary wave solutions we prove that in any dimension the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds we then prove that the birth of point eigenvalues with nonzero real part the ones which lead to linear instability from the essential spectrum is only possible from the embedded eigenvalues or thresholds and therefore can not take place beyond the embedded thresholds we also prove that in the nonrelativistic limit m the point eigenvalues can only accumulate to 0 and 2 m i

on quantitative uniqueness for elliptic equations we address the question of quantitative uniqueness for the equation delta u v u with either periodic or dirichlet boundary conditions in a disk we construct solutions u corresponding to potential functions v such that u vanishes of order mathrm const vert v vert l infty 2 3 the example also shows sharpness of recently obtained bounds in the case of a parabolic equation u t delta u v u

restriction theorems for orthonormal functions strichartz inequalities and uniform sobolev estimates we generalize the theorems of stein tomas and strichartz about surface restrictions of fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions we deduce the corresponding strichartz bounds for solutions to schr odinger equations up to the endpoint thereby solving an open problem of frank lewin lieb and seiringer we also prove uniform sobolev estimates in schatten spaces extending the results of kenig ruiz and sogge we finally provide applications of these results to a limiting absorption principle in schatten spaces to the well posedness of the hartree equation in schatten spaces to lieb thirring bounds for eigenvalues of schr odinger operators with complex potentials and to schatten properties of the scattering matrix

listing all potential maximal cliques of a graph a potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph it is known that if these objects can be listed in polynomial time for a class of graphs the treewidth and the minimum fill in are polynomially tractable for these graphs we show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph thus the treewidth and the minimum fill in are polynomially tractable for all classes of graphs with a polynomial number of minimal separators

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

weighted doubling properties and unique continuation theorems for the degenerate schrodinger equations with singular potentials abstract let u be the weak solution to the degenerate schrodinger equation with singular coefficients in lipschitz domain as following div w x a x u x v x u x w x 0 where a x is a real symmetric matrix function satisfying the elliptic condition and the lipschitz continuity w x is an a 2 weight function of muckenhoupt class and v x is the fefferman phong s potential the weighted doubling properties and unique continuations for the weak solution u in the interior of any domains as well as at the boundary of some lipschitz domains are derived in this paper

treewidth and minimum fill in on permutation graphs in linear time permutation graphs form a well studied subclass of cocomparability graphs permutation graphs are the cocomparability graphs whose complements are also cocomparability graphs a triangulation of a graph g is a graph h that is obtained by adding edges to g to make it chordal if no triangulation of g is a proper subgraph of h then h is called a minimal triangulation the main theoretical result of the paper is a characterisation of the minimal triangulations of a permutation graph that also leads to a succinct and linear time computable representation of the set of minimal triangulations we apply this representation to devise linear time algorithms for various minimal triangulation problems on permutation graphs in particular we give linear time algorithms for computing treewidth and minimum fill in on permutation graphs

recognizing solid objects by alignment with an image in this paper we consider the problem of recognizing solid objects from a single two dimensional image of a three dimensional scene we develop a new method for computing a transformation from a three dimensional model coordinate frame to the two dimensional image coordinate frame using three pairs of model and image points we show that this transformation always exists for three noncollinear points and is unique up to a reflective ambiguity the solution method is closed form and only involves second order equations we have implemented a recognition system that uses this transformation method to determine possible alignments of a model with an image each of these hypothesized matches is verified by comparing the entire edge contours of the aligned object with the image edges using the entire edge contours for verification rather than a few local feature points reduces the chance of finding false matches the system has been tested on partly occluded objects in highly cluttered scenes

remark on unique continuation for higher powers of the laplace operator abstract there are unique continuation results 2 5 for the differential inequality u x v x u x n when v l loc d 2 r d d 2 in this paper we obtain a result for the case d 2

treewidth and minimum fill in grouping the minimal separators we use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph we show that if these objects can be listed in polynomial time for a class of graphs the treewidth and the minimum fill in are polynomially tractable for these graphs we prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill in exist we can list their potential maximal cliques in polynomial time our approach unifies these algorithms finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs for which the treewidth and the minimum fill in problems were open

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

schrodinger operators and unique continuation towards an optimal result abstract in this article we prove the property of unique continuation also known for c functions as quasianalyticity for solutions of the differential inequality u v u for v from a wide class of potentials including l loc d 2 r d class and u in a space of solutions y v containing all eigenfunctions of the corresponding self adjoint schrodinger operator motivating question is it true that for potentials v for which self adjoint schrodinger operator is well defined the property of unique continuation holds

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

grouping based nonadditive verification verification is the final decision stage in many object recognition processes it is carried out by evaluating a score for every hypothesis and choosing the hypotheses associated with the highest score this paper suggests a grouping based verification paradigm relying on the observation that a group of data features belonging to a hypothesized object instance should be a good group therefore it should support perceptual grouping information available from the image by grouping relations the proposed score which is the joint likelihood of these grouping cues quantifies this observation in a probabilistic framework experiments with synthetic and real images show that the proposed method performs better in difficult cases

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds