on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

algorithm of adaptive fourier decomposition the present paper is a continuing work on the recently established adaptive fourier decomposition afd mainly stressing on the algorithm aspect including algorithm analysis and numerical examples afd is a variation and realization of greedy algorithm matching pursuit suitable for the hardy h2 and the l2 spaces applying afd to a given signal one obtains a series expansion in the basic signals called mono components that possess non negative analytic phase derivatives functions or equivalently meaningful analytic instantaneous frequencies afd is shown to be robust with computational complexity comparable with dft consistent to the greedy algorithm principle experiments show that afd produces pre mono component series with efficient energy decay that also leads to efficient pointwise convergence both in terms of computer running time

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

some sphere theorems in linear potential theory in this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities whose equality cases are saturated by domains with spherical sy

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

doubling inequality and nodal sets for solutions of bi laplace equations we investigate the doubling inequality and nodal sets for the solutions of bi laplace equations a polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for laplace eigenfunctions by logunov in addition we derive an implicit upper bound for the nodal sets of solutions we show two types of doubling inequalities for the solutions of bi laplace equations as a consequence the rate of vanishing is given for the solutions

strong unique continuation for higher order elliptic equations with gevrey coefficients article history we address the strong unique continuation problem for higher order elliptic partial differential equations in 2d with gevrey co efficients we provide a quantitative estimate of unique continu ation observability estimate and prove that the solutions satisfy the strong unique continuation property for ranges of the gevrey exponent strictly including non analytic gevrey classes as an ap plication we obtain a new upper bound on the hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients published by elsevier inc

the nodal line of the second eigenfunction of the robin laplacian in r2 can be closed abstract we construct a multiply connected domain in r 2 for which the second eigenfunction of the laplacian with robin boundary conditions has an interior nodal line in the process we adapt a bound of donnelly fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains which does not depend directly on any boundary behaviour this also gives a new proof of the nodal line property of the example in the dirichlet case

one or two frequencies the empirical mode decomposition answers this paper investigates how the empirical mode decomposition emd a fully data driven technique recently introduced for decomposing any oscillatory waveform into zero mean components behaves in the case of a composite two tones signal essentially two regimes are shown to exist depending on whether the amplitude ratio of the tones is greater or smaller than unity and the corresponding resolution properties of the emd turn out to be in good agreement with intuition and physical interpretation a refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments the analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the emd are assessed

nodal sets of thin curved layers abstract this paper is concerned with the location of nodal sets of eigenfunctions of the dirichlet laplacian in thin tubular neighbourhoods of hypersurfaces of the euclidean space of arbitrary dimension in the limit when the radius of the neighbourhood tends to zero it is known that spectral properties of the laplacian are approximated well by an effective schrodinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures by applying techniques of elliptic partial differential equations we strengthen the known perturbation results to get a convergence of eigenfunctions in holder spaces this enables us in particular to conclude that every nodal set has a non empty intersection with the boundary of the tubular neighbourhood

nonlinear chirp mode decomposition a variational method variational mode decomposition vmd a recently introduced method for adaptive data analysis has aroused much attention in various fields however the vmd is formulated based on the assumption of narrow band property of the signal model to analyze wide band nonlinear chirp signals ncss we present an alternative method called variational nonlinear chirp mode decomposition vncmd the vncmd is developed from the fact that a wideband ncs can be transformed to a narrow band signal by using demodulation techniques our decomposition problem is thus formulated as an optimal demodulation problem which is efficiently solved by the alternating direction method of multipliers our method can be viewed as a time frequency filter bank which concurrently extracts all the signal modes some simulated and real data examples are provided showing the effectiveness of the vncmd in analyzing ncss containing close or even crossed modes

decomposing a signal into short time narrow banded modes abstract an algorithm for nonparametric decomposition of a signal into the sum of short time narrow banded modes components is introduced specifically the signal data is augmented with its hilbert transform to obtain the analytic signal then the set of constituent amplitude and frequency modulated am fm analytic sinusoids each with slowly varying amplitude and frequency is sought the method for obtaining the short time narrow banded modes is derived by minimizing an objective function comprised of three criteria smoothness of the instantaneous amplitude envelope smoothness of the instantaneous frequency and complete reconstruction of the signal data a minimum of the objective function is approached using a sequence of suboptimal updates of amplitude and phase the updates are intuitive efficient and simple to implement for a given mode the amplitude and phase are extracted from the band pass filtered residual signal after the other modes are removed where the band pass filter is applied about the previous modal instantaneous frequency estimate the method is demonstrated by application to random output only vibration data and order tracking data it is demonstrated that vibration modal responses can be estimated from single channel data and order tracking can be performed without measured tachometer data

volume estimates on the critical sets of solutions to elliptic pdes in this paper we study solutions to elliptic linear equations l u i aij x ju bi x iu c x u 0 either on n or a riemannian manifold under the assumption that the coefficient functions aij are lipschitz bounded we focus our attention on the critical set c u x u 0 and the singular set c u x u 0 and more importantly on effective versions of these currently with just the lipschitz regularity of the coefficients the strongest results in the literature say that the singular set is n 2 dimensional however at this point it has not even been shown that hn 2 c unless the coefficients are smooth fundamentally this is due to the need of an regularity theorem that requires higher smoothness of the coefficients as the frequency increases we introduce new techniques for estimating the critical and singular set which avoids the need of any such regularity consequently we prove that if the frequency of u is bounded by then we have the estimates hn 2 c u c 2 and hn 2 s u c 2 depending on whether the equation is critical or not more importantly we prove corresponding estimates for the effective critical and singular sets even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature we also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem 2017 wiley periodicals inc

laplace operators with eigenfunctions whose nodal set is a knot we prove that given any knot in a compact 3 manifold m there exists a riemannian metric on m such that there is a complex valued eigenfunction u of the laplacian corresponding to the first nontrivial eigenvalue whose nodal set u 1 0 has a connected component given by higher dimensional analogs of this result will also be considered

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

on demodulation ridge detection and synchrosqueezing for multicomponent signals in this paper we present a novel technique for the retrieval of the modes of a multicomponent signal using a time frequency tf representation of the signal our approach is based on a novel ridge extraction method that takes into account the fact that the tf representation is both discrete in time and frequency followed by a demodulation procedure numerical results show the benefits of the proposed approach for mode reconstruction in comparison to similar techniques that do not make use of demodulation furthermore numerical investigations show that the proposed approach sharpens the tf representation on which it is built

finite criteria for conditional definiteness of quadratic forms abstract a real symmetric n n matrix q is a conditionally positivesemidefinite where a is a given m n matrix if x qx 0 whenever ax 0 and is a conditionally positive definite if strict inequality holds except when x 0 when a is the identity matrix these notions reduce to the well studied notions of copositivity and strict copositivity respectively this paper presents finite criteria involving only the solution of sets of linear equations constructed from the matrices q a for testing both types of conditional definiteness these criteria generalize known facts about copositive matrices and when q is invertible and all row submatrices of a have maximal rank can be very elegantly stated in terms of schur complements of the matrix aq 1 a

zero sets of solutions to semilinear elliptic systems of first order consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n dimensional manifold assume the operator has the strong unique continuation property we show that the zero set of the solution is contained in a countable union of smooth n 2 dimensional submanifolds hence it is countably n 2 rectifiable and its hausdorff dimension is at most n 2 moreover it has locally finite n 2 dimensional hausdorff measure we show by example that every real number between 0 and n 2 actually occurs as the hausdorff dimension for a suitable choice of operator we also derive results for scalar elliptic equations of second order

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

collapsing and the differential form laplacian the case of a smooth limit space we analyze the limit of the p form laplacian under a collapse with bounded sectional curvature and bounded diameter to a smooth limit space as an application we characterize when the p form laplacian has small positive eigenvalues in a collapsing sequence

cfar detection of range spread targets based on the time frequency decomposition feature of two adjacent returned signals radar returned signals are processed by stretch processing resulting in mixer outputs based on the time frequency decomposition of the cross s method csm of two adjacent mixer outputs a range spread target detector is proposed in this paper as a preparatory work we propose a signal synthesis method ssm based on the singular value decomposition the ssm synthesizes two signals in their normalized forms from their cross wigner distribution cwd and concentrates their energy on two singular values this detector consists of three steps first we derive the csm from the s method sm the csm is close to the sum of the cwds of the components in one mixer output and their counterparts in the other second we can decompose the csm by the ssm thereby obtaining singular values third the time frequency decomposition feature i e the ratio of the sum of several biggest singular values to the median or mean of the rest is defined to demonstrate the concentration of the singular values and used to detect the range spread target the proposed detector is evaluated by the raw radar data without range migration correction results show that it outperforms the conventional detectors in addition we prove that the proposed detector has the constant false alarm rate cfar property

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

on instantaneous amplitude and phase of signals in many questions of signal processing it is important to use the concepts of instantaneous amplitude or phase of signals this is especially the case in communication systems with amplitude or frequency modulation these concepts are often introduced empirically however it is well known that the correct approach for this purpose is to use the concept of analytic signal starting from this point we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted this introduces the problem of characterizing pure amplitude modulated or pure phase modulated signals it is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations this is no longer the case for phase modulated signals furthermore signals with constant amplitude have very specific properties which are analyzed in detail some consequences and extensions to random signals are finally discussed

the nodal set of solutions to some elliptic problems sublinear equations and unstable two phase membrane problem abstract we are concerned with the nodal set of solutions to equations of the form u u q 1 u q 1 in b 1 where 0 q 1 2 b 1 b 1 0 is the unit ball in r n n 2 and u max u 0 u max u 0 are the positive and the negative part of u respectively this class includes the unstable two phase membrane problem q 1 as well as sublinear equations for 1 q 2 we prove the following main results a the finiteness of the vanishing order at every point and the complete characterization of the order spectrum b a weak non degeneracy property c regularity of the nodal set of any solution the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having hausdorff dimension at most n 2 locally finite when n 2 d a partial stratification theorem ultimately the main features of the nodal set are strictly related with those of the solutions to linear or superlinear equations with two remarkable differences first of all the admissible vanishing orders can not exceed the critical value 2 2 q at threshold we find a multiplicity of homogeneous solutions yielding the non validity of any estimate of the n 1 dimensional measure of the nodal set of a solution in terms of the vanishing order as a byproduct we also prove the strong unique continuation property for the unstable obstacle problem corresponding to the case 0 the proofs are based on monotonicity formulae for a 2 parameter family of weiss type functionals blow up arguments and the classification of homogeneous solutions

tracking of unknown nonstationary chirp signals using unsupervised clustering in the wigner distribution space this paper is concerned with the problems of 1 detecting the presence of one or more fm chirp signals embedded in noise and 2 tracking or estimating the unknown time varying instantaneous frequency of each chirp component no prior knowledge is assumed about the number of chirp signals present the parameters of each chirp or how the parameters change with time a detection estimation algorithm is proposed that uses the wigner distribution transform to find the best piecewise cubic approximation to each chirp s phase function the first step of the wd based algorithm consists of properly thresholding the wd of the received signal to produce contours in the time frequency plane that approximate the instantaneous frequency of each chirp component these contours can then be approximated as generalized lines in the omega t t sup 2 space the number of chirp signals or equivalently generalized lines present is determined using maximum likelihood segmentation minimum mean square estimation techniques are used to estimate the unknown phase parameters of each chirp component the authors demonstrate that for the cases of i nonoverlapping linear or nonlinear fm chirp signals embedded in noise or ii overlapping linear fm chirp signals embedded in noise the approach is very robust highly reliable and can operate efficiently in low signal to noise environments where it is hard for even trained operators to detect the presence of chirps while looking at the wd plots of the overall signal for multicomponent signals the proposed technique is able to suppress noise as well as the troublesome cross wd components that arise due to the bilinear nature of the wd

improving the readability of time frequency and time scale representations by the reassignment method in this paper the use of the reassignment method first applied by kodera gendrin and de villedary 1976 to the spectrogram is generalized to any bilinear time frequency or time scale distribution this method creates a modified version of a representation by moving its values away from where they are computed so as to produce a better localization of the signal components we first propose a new formulation of this method followed by a thorough theoretical study of its characteristics its practical use for a large variety of known time frequency and time scale distributions is then addressed finally some experimental results are reported to demonstrate the performance of this method

boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions in this work we introduce the use of powerful tools from geometric measure theory gmt to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure omega of a domain omega omega subset mathbb r n and the harmonic measure omega of omega omega int omega c in dimension n ge 3

contact topology and hydrodynamics iii knotted orbits we employ the relationship between contact structures and beltrami fields derived in part i of this series to construct a steady nonsingular solution to the euler equations on a riemannian s3 whose flowlines trace out closed curves of all possible knot and link types using careful contacttopological controls we can make such vector fields real analytic and transverse to the tight contact structure on s3 sufficient review of concepts is included to make this paper independent of the previous works in this series

eigenfunctions with prescribed nodal sets in this paper we consider the problem of prescribing the nodal set of low energy eigenfunctions of the laplacian our main result is that given any separating closed hypersurface in a compact n manifold m there is a riemannian metric on m such that the nodal set of its first nontrivial eigenfunction is we present a number of variations on this result which enable us to show in particular that the first nontrivial eigenfunction can have as many non degenerate critical points as one wishes

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

vibration signal component separation by iteratively using basis pursuit and its application in mechanical fault detection abstract using different transform basis the vibration signal components such as impulses harmonics and modulated components can be effectively separated by basis pursuit for multicomponent vibration signals this paper proposes a new vibration signal component separation approach by iteratively using basis pursuit the signal is firstly denoised by basis pursuit denoising and the impulsive component can be separated by using identity matrix and redundant fourier basis then signal components with different frequency ranges can be separated by using short time fourier transform basis with different window lengths and the components with high frequency are separated after each iteration to choose the best basis from the dictionary the separation coefficient is proposed for evaluating the separating performance of the transform basis the proposed approach empirical mode decomposition emd and matching pursuit are respectively applied to analyzing a simulated faulty vibration and a simulated faulty gear vibration signal the comparison results demonstrate its superiority to emd and matching pursuit in separation accuracy finally the proposed approach is used for detecting the faults of a rotor and rolling bearings the results further show that this proposed method can be effectively applied to mechanical fault diagnosis

a remark on a curvature gap for minimal surfaces in the ball we extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional euclidean ball due to l ambrozio and i nunes

counting nodal lines which touch the boundary of an analytic domain we consider the zeros on the boundary of a neumann eigenfunction of a real analytic plane domain we prove that the number of its boundary zeros is o where 2 we also prove that the number of boundary critical points of either a neumann or dirichlet eigenfunction is o it follows that the number of nodal lines of components of the nodal set which touch the boundary is of order this upper bound is of the same order of magnitude as the length of the total nodal line but is the square root of the courant bound on the number of nodal components in the interior more generally the results are proved for piecewise analytic domains

on a decomposition of conditionally positive semidefinite matrices abstract a symmetric matrix c is said to be copositive if its associated quadratic form is nonnegative on the positive orthant recently it has been shown that a quadratic form x qx is positive for all x that satisfy more general linear constraints of the form ax 0 x 0 iff q can be decomposed as a sum q a ca s with c strictly copositive and s positive definite however if x qx is merely nonnegative subject to the constraints ax 0 it does not follow that q admits such a decomposition with c copositive and s positive semidefinite in this paper we give a characterization of those matrices a for which such a decomposition is always possible

lower bounds for nodal sets of dirichlet and neumann eigenfunctions let phi be a dirichlet or neumann eigenfunction of the laplace beltrami operator on a compact riemannian manifold with boundary we prove lower bounds for the size of the nodal set phi 0

variational mode decomposition during the late 1990s huang introduced the algorithm called empirical mode decomposition which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands emd is known for limitations like sensitivity to noise and sampling these limitations could only partially be addressed by more mathematical attempts to this decomposition problem like synchrosqueezing empirical wavelets or recursive variational decomposition here we propose an entirely non recursive variational mode decomposition model where the modes are extracted concurrently the model looks for an ensemble of modes and their respective center frequencies such that the modes collectively reproduce the input signal while each being smooth after demodulation into baseband in fourier domain this corresponds to a narrow band prior we show important relations to wiener filter denoising indeed the proposed method is a generalization of the classic wiener filter into multiple adaptive bands our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand the variational model is efficiently optimized using an alternating direction method of multipliers approach preliminary results show attractive performance with respect to existing mode decomposition models in particular our proposed model is much more robust to sampling and noise finally we show promising practical decomposition results on a series of artificial and real data

non parametric and adaptive modelling of dynamic periodicity and trend with heteroscedastic and dependent errors summary periodicity and trend are features describing an observed sequence and extracting these features is an important issue in many scientific fields however it is not an easy task for existing methods to analyse simultaneously the trend and dynamics of the periodicity such as time varying frequency and amplitude and the adaptivity of the analysis to such dynamics and robustness to heteroscedastic dependent errors are not guaranteed these tasks become even more challenging when there are multiple periodic components we propose a non parametric model to describe the dynamics of multicomponent periodicity and investigate the recently developed synchro squeezing transform in extracting these features in the presence of a trend and heteroscedastic dependent errors the identifiability problem of the non parametric periodicity model is studied and the adaptivity and robustness properties of the synchro squeezing transform are theoretically justified in both discrete and continuous time settings consequently we have a new technique for decoupling the trend periodicity and heteroscedastic dependent error process in a general non parametric set up results of a series of simulations are provided and the incidence time series of varicella and herpes zoster in taiwan and respiratory signals observed from a sleep study are analysed

iterative sparsification projection fast and robust sparse signal approximation in this paper we address recovery of sparse signals from compressed measurements and sparse signal approximation which have received considerable attention over the last decade first we revisit smoothed l0 sl0 a well known sparse recovery algorithm and give some insights into it that have not been noticed previously specifically we re derive the sl0 algorithm based on proximal methods and using recent results in solving nonconvex problems by proximal algorithms we provide a convergence guarantee for it in addition inspired by this derivation we propose a general family of algorithms which we call iterative sparsification projection isp having sl0 as a special member our algorithmic framework starts with an initial guess for the unknown sparse vector and then iteratively sparsifies it using a fixed threshold followed by projecting the result onto the admissible solution set the threshold is then decreased and the same process is repeated the algorithm terminates when the threshold becomes sufficiently small or another stopping criterion is satisfied we also propose a robust projection to handle the situations with observation noise or model uncertainties our extensive simulations confirm the promising performance of the isp algorithms compared with some well known algorithms

nodal sets of solutions of equations involving magnetic schrodinger operator in three dimensions it is well known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish the singular set of a real valued solution of a linear elliptic equation has been well investigated for a complex valued solution of a linear equation involving a magnetic schrodinger operator the structure of the nodal set has not been well investigated yet excepted in the two dimensional case in this paper we extend the arguments of garafalo and lin indiana univ math j 35 245 268 1986 and of han indiana univ math j 43 983 1002 1994 to show that the singular set of such a solution in a three dimensional domain is countably 1 rectifiable the functions considered in this paper include the order parameter in the ginzburg landau theory of superconductivity and the eigenfunctions of the magnetic schrodinger operator

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints abstract a symmetric matrix c is called copositive if the quadratic form x cx is nonnegative for all nonnegative values of the variables x 1 x 2 x n x a known sufficient condition for a quadratic form x qx to be positive unless x 0 subject to the linear inequality constraints ax 0 is that there should exist a copositive matrix c such that q a ca is positive definite the main result of this paper establishes the necessity of this condition for x qx to be merely nonnegative subject to ax 0 the situation is less straightforward the necessity of the existence of a copositive matrix c such that q a ca is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of a and counterexamples are given to show that in general no such matrix may exist even when slater s constraint qualification holds our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x qx subject to ax 0 in which specific symmetric matrices constructed by rational operations from a and q and depending upon a single real parameter v must be tested for positive definiteness or strict copositivity for large values of v this technique is illustrated by several examples

critical points of green s functions on complete manifolds we prove that the number of critical points of a li tam green s function on a complete open riemannian surface of finite type admits a topological upper bound given by the first betti number of the surface in higher dimensions we show that there are no topological upper bounds on the number of critical points by constructing for each nonnegative integer n a riemannian manifold diffeomorphic to r n n 3 whose minimal green s function has at least n non degenerate critical points variations on the method of proof of the latter result yield contractible n manifolds whose minimal green s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of r n

circularity of the stft and spectral kurtosis for time frequency segmentation in gaussian environment this paper investigates the circularity of short time fourier transform stft coefficients noise only and proposes a modified stft such that all coefficients coming from white gaussian noise are circular in order to use the spectral kurtosis sk as a gaussianity test to check if signal points are present in a set of stft points we consider the sk of complex circular random variables and its link with the kurtosis of the real and imaginary parts we show that the variance of the sk is smaller than the variance of the kurtosis estimated from both real and imaginary parts the effect of the noncircularity of gaussian variables upon the spectral kurtosis of stft coefficients is studied as well as the effect of signal presence finally a time frequency segmentation algorithm based on successive iterations of noise variance estimation and time frequency coefficients detection is proposed the iterations are stopped when the spectral kurtosis on nondetected points reaches zero examples of segmented time frequency space are presented on a dolphin whistle and on a simulated signal in nonwhite and nonstationary gaussian noise

intersection bounds for nodal sets of planar neumann eigenfunctions with interior analytic curves let r be a bounded piecewise smooth domain and be a neumann or dirichlet eigenfunction with eigenvalue and nodal set n x x 0 let h be an interior c curve consider the intersection number n h h n we first prove that for general piecewise analytic domains and under an appropriate goodness condition on h see theorem 1 1 n h oh as then using theorem 1 1 we prove in theorem 1 2 that the bound in is satisfied in the case of quantum ergodic qe sequences of interior eigenfunctions provided is convex and h has strictly positive geodesic curvature

a new algorithm for multicomponent signals analysis based on synchrosqueezing with an application to signal sampling and denoising in this paper we address the problem of the retrieval of the components from a multicomponent signal using ideas from the synchrosqueezing framework the emphasis is on the wavelet choice and we propose a novel algorithm based first on the detection of components followed by their reconstruction simulations illustrate how the proposed procedure compares with the empirical mode decomposition and other related methods in terms of mode mixing we conclude the paper by studying the sensitivity of the proposed technique to sampling and an application to signal denoising

upper bounds of nodal sets for eigenfunctions of eigenvalue problems the aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains the examples include biharmonic steklov eigenvalue problems buckling eigenvalue problems and champed plate eigenvalue problems the geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions it is done through analytic estimates of morrey nirenberg and carleman estimates

second order synchrosqueezing transform or invertible reassignment towards ideal time frequency representations this paper considers the analysis of multicomponent signals defined as superpositions of real or complex modulated waves it introduces two new post transformations for the short time fourier transform that achieve a compact time frequency representation while allowing for the separation and the reconstruction of the modes these two new transformations thus benefit from both the synchrosqueezing transform which allows for reconstruction and the reassignment method which achieves a compact time frequency representation numerical experiments on real and synthetic signals demonstrate the efficiency of these new transformations and illustrate their differences

on the geometry of the level sets of bounded static potentials in this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity in the case where the static potential is bounded we introduce new quantities which are proven to be monotone along the level set flow of the potential function we then show how to use these properties to detect the rotational symmetry of the static solutions deriving a number of sharp inequalities among these we prove the validity without any dimensional restriction of the riemannian penrose inequality as well as of a reversed version of it in the class of asymptotically flat static metrics with connected horizon as a consequence of our analysis a simple proof of the classical 3 dimensional black hole uniqueness theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m