stable self similar blowup in energy supercritical yang mills theory we consider the cauchy problem for an energy supercritical nonlinear wave equation that arises in 1 5 dimensional yang mills theory a certain self similar solution w 0 of this model is conjectured to act as an attractor for generic large data evolutions assuming mode stability of w 0 we prove a weak version of this conjecture namely that the self similar solution w 0 is nonlinearly stable phrased differently we prove that mode stability of w 0 implies its nonlinear stability the fact that this statement is not vacuous follows from careful numerical work by bizon and chmaj that verifies the mode stability of w 0 beyond reasonable doubt

an lp analog to aak theory for p 2 we develop an lp analog to aak theory on the unit circle that interpolates continuously between the case p which classically solves for best uniform meromorphic approximation and the case p 2 which is equivalent to h2 best rational approximation we apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points as pointed out by a referee part of the theory extends to every p 1 when the definition of the hankel operator is suitably generalized this we discuss in connection with the recent manuscript by v a prokhorov submitted for publication

global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

blowup stability at optimal regularity for the critical wave equation abstract we establish strichartz estimates for the radial energy critical wave equation in 5 dimensions in similarity coordinates using these we prove the nonlinear asymptotic stability of the ode blowup in the energy space

fft formulations of adaptive fourier decomposition adaptive fourier decomposition afd has been found to be among the most effective greedy algorithms afd shows an outstanding performance in signal analysis and system identification as compensation of effectiveness the computation complexity is great that is especially due to maximal selections of the parameters in this paper we explore the discretization of the 1 d afd integration via with discrete fourier transform dft incorporating fast fourier transform fft we show that the new algorithm called fft afd reduces the computational complexity from o mn2 to o mnlogn the latter being the same as fft through experiments we verify the effectiveness accuracy and robustness of the proposed algorithm the proposed fft based algorithm for afd lays a foundation for its practical applications

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

radially symmetric wave maps from 1 2 dimensional minkowski space to the sphere by a blow up analysis as in 8 for a related problem we rule out concentration of energy for radially symmetric wave maps from the 1 2 dimensional minkowski space to the sphere when combined with the local existence and regularity results of christodoulou and tahvildar zadeh for this problem our result implies global existence of smooth solutions to the cauchy problem for radially symmetric wave maps for smooth radially symmetric data

on approximation of stable linear dynamical systems using laguerre and kautz functions approximation of stable linear dynamical systems by means of so called laguerre and kautz functions which are the laplace transforms of a class of orthonormal exponentials is studied since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials times polynomials it seems reasonable to use basis functions of the same type assuming that the transfer function of a system is bounded and analytic outside a given disc it is shown that laguerre basis functions are optimal in a mini max sense this result is extended to the two parameter kautz functions which can have complex poles while the poles of laguerre functions are restricted to the real axis by conformai mapping techniques the two parameter kautz approximation problem is recast as two laguerre approximation problems thus the well developed theory of laguerre functions can be applied to analyze kautz approximations unilateral shifts are used to further develop the connection between laguerre functions and kautz functions results on 2 and approximation using kautz models are given furthermore the weighted l2 kautz approximation problem is shown to be equivalent to solving a block toeplitz matrix equation

global regularity of wave maps from r3 1 to surfaces we consider wave maps with smooth compactly supported initial data of small norm from r3 1 to certain 2 dimensional riemannian manifolds and show that they stay smooth globally in time our methods are based on the introduction of a global coulomb gauge as in 17 followed by a dynamic separation as in 8 we then rely on an adaptation of t tao s methods used in his recent breakthrough result 24

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

strichartz estimates in similarity coordinates and stable blowup for the critical wave equation we establish strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time dependent self similar potential as an application we consider the critical wave equation and prove the asymptotic stability of the ode blowup profile in the energy space

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

on the stability of self similar solutions to nonlinear wave equations we consider an explicit self similar solution to an energy supercritical yang mills equation and prove its mode stability based on earlier work by one of the authors we obtain a fully rigorous proof of the nonlinear stability of the self similar blowup profile this is a large data result for a supercritical wave equation our method is broadly applicable and provides a general approach to stability problems related to self similar solutions of nonlinear wave equations

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

generic self similar blowup for equivariant wave maps and yang mills fields in higher dimensions we consider equivariant wave maps from the d 1 dimensional minkowski spacetime into the d sphere for d 4 we find a new explicit stable self similar solution and give numerical evidence that it plays the role of a universal attractor for generic blowup an analogous result is obtained for the so d symmetric yang mills field for d 6

system identification using kautz models in this paper the problem of approximating a linear time invariant stable system by a finite weighted sum of given exponentials is considered system identification schemes using laguerre models are extended and generalized to kautz models which correspond to representations using several different possible complex exponentials in particular linear regression methods to estimate this sort of model from measured data are analyzed the advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters the subsequent analysis is based on the result that the corresponding linear regression normal equations have a block toeplitz structure several results on transfer function estimation are extended to discrete kautz models for example asymptotic frequency domain variance expressions

forward backward spatial smoothing techniques for coherent signal identification a spatial smoothing scheme is further investigated in the context of coherent signal classification it is shown that by making use of a set of forward and complex conjugated backward subarrays simultaneously it is always possible to estimate any k directions of arrival using at most 3k 2 sensor elements this is achieved by creating a smoothed array output covariance matrix that is structurally identical to a covariance matrix in some noncoherent situation by incorporating eigenstructure based techniques on this smoothed covariance matrix it then becomes possible to correctly identify all directions of arrival irrespective of their correlation

statistical analysis of a high resolution spatial spectrum estimator utilizing an augmented covariance matrix in the context of multiple source location with multiple sensors it was shown 1 that by using an m element array with elements arranged in a minimum redundancy fashion and by augmenting the array output covariance matrix to create a covariance type matrix of larger dimensions it is possible to estimate the directions of arrival of as many as m m 1 2 uncorrelated sources the earlier work assumed perfectly estimated covariances this paper addresses the statistical properties and related issues of the augmented covariance matrix in the case when the array output sample covariances are estimated from finite data using the maximum likelihood method using a matrix factorization technique the distribution of the sample bartlett spatial spectrum estimator based on the augmented covariance matrix is shown to be a sum of weighted and dependent 2 distributed random variables the degradation in performance in making use of the augmented matrix is here found in terms of the variance of the bartlett estimator this is shown to be in the order of twice the variance obtained with a uniform array of actual elements the latter being equal in number to the dimension of the augmented array to match the performance of the uniform array of actual elements using the augmented array would then require about twice the number of data samples

a novel online mutual coupling compensation algorithm for uniform and linear arrays in this paper a novel online mutual coupling compensation algorithm especially tailored to uniform and linear arrays is presented it is conceived to simultaneously compensate for mutual coupling and estimate the direction of arrivals doas of signals impinging on the array since the estimated calibration matrix can be embedded within any classical super resolution direction finding method an alternating minimization procedure based on closed form solutions is performed to estimate the mutual coupling matrix in the field of complex symmetric toeplitz matrices unlike many existing array calibration methods it requires neither the presence of calibration sources nor previous calibration information as initialization computer simulations show the effectiveness of the proposed technique and prove that the nice statistical properties of classical super resolution doa estimation algorithms can be restored despite the presence of mutual coupling

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

on blowup of co rotational wave maps in odd space dimensions abstract we consider co rotational wave maps from the 1 d dimensional minkowski space into the d sphere for d 3 odd this is an energy supercritical model which is known to exhibit finite time blowup via self similar solutions based on a method developed by the second author and schorkhuber we prove the asymptotic nonlinear stability of the ground state self similar solution

weak convergence of wave maps from 1 2 dimensional minkowski space to riemannian manifolds the weak limit of a sequence of wave maps with uniformly bounded energy is again a wave map essential ingredients in the proof are hodge structures related to harmonic maps a concise description of the concentration set of a sequence of wave maps and the use of 1 bmo duality combined with an elliptic gauge condition in the hyperbolic context

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

thinned coprime array for second order difference co array generation with reduced mutual coupling in this work we present a new coprime array structure termed thinned coprime array tca which exploits the redundancy in the structure of existing coprime array and achieves the same virtual aperture and degrees of freedom dofs as the conventional coprime array with much fewer number of sensors in comparison to other sparse arrays thinned coprime arrays possess more unique lags total number of difference co arrays than the nested arrays while the number of consecutive lags connected co arrays generated is close to 75 of the consecutive lags of the nested arrays with hole free co arrays the resulting structure is much sparser and the number of sensor pairs with small separation is significantly reduced theoretical properties and proofs are provided and simulations are presented to demonstrate its robustness against heavy levels of mutual coupling using compressive sensing based direction of arrival estimation as well as certain additional desirable characteristics

a hybrid genetic algorithm for estimating the optimal time scale of linear systems approximations using laguerre models we deal with the problem of finding the optimal time scale of the truncated laguerre series using numerical search techniques we develop a hybrid genetic algorithm ga to search the nonlinear multimodal squared error function that results from least squares approximations of the impulse response of causal linear time invariant stable systems the hybrid ga incorporates a newton raphson nr local optimizer for fast convergence to the global minimum point the proposed method competes favorably with the pure ga in solution accuracy the number of function evaluations being the same and with an established gradient directed optimization algorithm in number of function evaluations the solution accuracy being the same

uniqueness for critical nonlinear wave equations and wave maps via the energy inequality we show uniqueness of sufficiently regular solutions to critical semilinear wave equations and wave maps in the a priori much larger class of distribution solutions with finite energy assuming only that the energy is nonincreasing in time 1999 john wiley sons inc

frequency domain identification with generalized orthonormal basis functions a method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain to this end use is made of the empirical transfer function estimate etfe identified from time domain data as a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions being generalizations of the classical pulse and laguerre and kautz types of bases an asymptotic analysis of the estimated models is provided and conditions for consistency are formulated explicit and transparent bias and variance expressions are established the latter ones also valid in a situation of undermodeling

sparse sensing with co prime samplers and arrays this paper considers the sampling of temporal or spatial wide sense stationary wss signals using a co prime pair of sparse samplers several properties and applications of co prime samplers are developed first for uniform spatial sampling with m and n sensors where m and n are co prime with appropriate interelement spacings the difference co array has o mn freedoms which can be exploited in beamforming and in direction of arrival estimation an m point dft filter bank and an n point dft filter bank can be used at the outputs of the two sensor arrays and their outputs combined in such a way that there are effectively mn bands i e mn narrow beams with beamwidths proportional to 1 mn a result following from co primality the ideas are applicable to both active and passive sensing though the details and tradeoffs are different time domain sparse co prime samplers also generate a time domain co array with o mn freedoms which can be used to estimate the autocorrelation at much finer lags than the sample spacings this allows estimation of power spectrum of an arbitrary signal with a frequency resolution proportional to 2 mnt even though the pairs of sampled sequences xc ntn and xc mtn in the time domain can be arbitrarily sparse in fact from the sparse set of samples xc ntn and xc mtn one can estimate o mn frequencies in the range t it will be shown that the co array based method for estimating sinusoids in noise offers many advantages over methods based on the use of chinese remainder theorem and its extensions examples are presented throughout to illustrate the various concepts

nonlinear stability of blow up solutions to the hyperbolic mean curvature flow abstract we study the so called hyperbolic mean curvature hmc flow introduced by lefloch and smoczyk in 2008 for the evolution of a closed hypersurface moving in the direction of its mean curvature vector this flow stems from a geometrically natural action consisting of a kinetic energy and an internal energy we study the initial value problem for this flow in the case of an entire graph in arbitrary dimension and we establish the existence of a singular self similar solution and its nonlinear stability in a suitably weighted sobolev space by relying on nash moser iterations

on backward shift algorithm for estimating poles of systems abstract in this paper we present an algorithm for estimating poles of linear time invariant systems by using the backward shift operator we prove that poles of rational functions including zeros and multiplicities are solutions to an algebraic equation which can be obtained by taking backward shift operator to the shifted cauchy kernels in the unit disc case the algorithm is accordingly developed for frequency domain identification we also prove the robustness of this algorithm some illustrative examples are presented to show the efficiency in systems with distinguished and multiple poles

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

further results on the cramer rao bound for sparse linear arrays sparse linear arrays such as co prime and nested arrays can identify up o m 2 uncorrelated sources with only o m sensors by utilizing their difference coarray model in our previous work we derived closed form asymptotic mean squared error mse expressions for two coarray based music algorithms and analyzed the cramer rao bound crb in high signal to noise ratio snr regions under the assumption of uncorrelated sources in this paper we provide further analysis of the crb presented in our previous work we first establish the connection between two crbs the crb derived with the assumption that the sources are uncorrelated and the classical stochastic crb derived without this assumption we show that they are asymptotically equal in high snr regions for uncorrelated sources next we analyze the behavior of the former crb for co prime and nested arrays with a large number of sensors we investigate the effect of configuration parameters on this crb we show that for co prime and nested arrays this crb can decrease at a rate of o m 5 for large values of m while this rate is only o m 3 for an m sensor ula this finding analytically demonstrates that co prime and nested arrays can achieve better asymptotic estimation performance when the number of sensors is a limiting factor we also show that for a fixed aperture co prime and nested arrays require more snapshots to achieve the same performance as ulas showing the trade off between the number of spatial samples and the number of temporal samples finally we demonstrate our analytical results with numerical experiments

optimality conditions for truncated kautz networks with two periodically repeating complex conjugate poles presents optimality conditions for the approximation of a siso system by a truncated kautz network with two repeating complex conjugate poles both the continuous and the discrete time cases are discussed the author approaches the problem in a system approximation framework and he does not assume that the input signal is white or an impulse the results obtained generalize the results already known for truncated laguerre networks

equivariant wave maps in two space dimensions singularities of corotational wave maps from 1 2 dimensional minkowski space into a surface n of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into in consequence for noncompact target surfaces of revolution the cauchy problem for wave maps is globally well posed 2003 wiley periodicals inc

a class of large global solutions for the wave map equation in this paper we consider the equation for equivariant wave maps from r3 1 to s 3 and we prove global in forward time existence of certain c infinity smooth solutions which have infinite critical sobolev norm h overdot 3 4 r 3 x h overdot 1 2 r 3 our construction provides solutions which can moreover satisfy the additional size condition parallel to u 0 parallel to l infinity vertical bar vertical bar 1 m for arbitrarily chosen m 0 these solutions are also stable under suitable perturbations our method strongly inspired by work of krieger and schlag is based on a perturbative approach around suitably constructed approximate self similar solutions

system identification by discrete rational atoms a novel adaptive frequency domain system identification method for linear time invariant systems is proposed in this paper it finds poles for discrete rational atoms with discrete frequency responses the theoretical foundation including adaptive decomposition principle and decomposition convergence rate is established the algorithm of the adaptive pursuit is also provided in this paper

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

parametric identification of transfer functions in the frequency domain a survey this paper gives a survey of frequency domain identification methods for rational transfer functions in the laplace s or z domain the interrelations between the different approaches are highlighted through a study of the equivalent cost functions the properties of the various estimators are discussed and illustrated by several examples

brief paper frequency domain identification an algorithm based on an adaptive rational orthogonal system this paper presents a new adaptive algorithm for frequency domain identification the algorithm is related to the rational orthogonal system takenaka malmquist system this work is based on an adaptive decomposition algorithm previously proposed for decomposing the hardy space functions in which a greedy sequence is obtained according to the maximal selection criterion we modify the algorithm through necessary changes for system identification

direction finding algorithms based on high order statistics two direction finding algorithms are presented for nongaussian signals which are based on the fourth order cumulants of the data received by the array the first algorithm is similar to music while the second is asymptotically minimum variance in a certain sense the first algorithm requires singular value decomposition of the cumulant matrix while the second is based on nonlinear minimization of a certain cost function the performance of the minimum variance algorithm can be assessed by analytical means at least for the case of discrete probability distributions of the source signals and spatially uncorrelated gaussian noise the numerical experiments performed seem to confirm the insensitivity of these algorithms to the gaussian noise parameters

on worst case approximation of feasible system sets via orthonormal basis functions this note deals with the approximation of sets of linear time invariant systems via orthonormal basis functions this problem is relevant to conditional set membership identification where a set of feasible systems is available from observed data and a reduced complexity model must be estimated the basis of the model class is made of impulse responses of linear filters the objective of the note is to select the basis function poles according to a worst case optimality criterion suboptimal conditional identification algorithms are introduced and tight bounds are provided on the associated identification errors

a fast adaptive model reduction method based on takenaka malmquist systems abstract an adaptive model reduction method is proposed for linear time invariant systems based on the continuous time rational orthogonal basis takenaka malmquist basis the method is to find an adaptive approximation in the energy sense by selecting optimal points for the rational orthogonal basis the stability of the reduced models holds and the steady state values of step responses are kept to be equal furthermore the method automatically ensures the reduced system to be in the hardy space h 2 the existence of the best approximation in the hardy space h 2 by n blaschke forms is proved in the proposed approach the effectivity of this method is illustrated through three well known examples

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

detection of the number of signals a predicted eigen threshold approach a novel method for detecting the number of signals incident upon an array of sensors is described the method is based on finding upper thresholds for the observed eigenvalues of the correlation matrix of the array output the asymptotic normality of the multifold eigenvalue estimate is used to derive a prediction formula for the thresholds due to the fact that the detection error of the method can be controlled by a parameter t the performance of the method is superior to the mdl minimum description length under low snr signal to noise ratio and superior to an aic akaike information criterion under high snr the distribution functions of both missing and false alarm errors are evaluated so that the performance of the method can be analyzed and so that t can be chosen simulation results are presented to confirm the analysis

global regularity of wave maps from r 2 1 to h 2 small energy we demonstrate that wave maps with smooth initial data and small energy from r2 1 to the lobatchevsky plane stay smooth globally in time our method is similar to the one employed in 18 however the multilinear estimates required are considerably more involved and present novel technical challenges in particular we shall have to work with a modification of the functional analytic framework used in 30 33 18

stable blow up dynamics for energy supercritical wave equations we study the semilinear wave equation displaystyle partial t 2 psi delta psi vert psi vert p 1 psi for p 3 with radial data in three spatial dimensions there exists an explicit solution which blows up at t t 0 given by displaystyle psi t t x c p t t frac 2 p 1 where c p is a suitable constant we prove that the blow up described by psi t is stable in the sense that there exists an open set in a topology strictly stronger than the energy of radial initial data that leads to a solution which converges to psi t as t to t in the backward lightcone of the blow up point t r t 0

nested arrays a novel approach to array processing with enhanced degrees of freedom a new array geometry which is capable of significantly increasing the degrees of freedom of linear arrays is proposed this structure is obtained by systematically nesting two or more uniform linear arrays and can provide o n2 degrees of freedom using only n physical sensors when the second order statistics of the received data is used the concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically it is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors this cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than the number of physical sensors in minimum input minimum output mimo radar the degrees of freedom are increased by constructing a longer virtual array through active sensing the method proposed here however does not require active sensing and is capable of providing increased degrees of freedom in a completely passive setting to utilize the degrees of freedom of the nested co array a novel spatial smoothing based approach to doa estimation is also proposed which does not require the inherent assumptions of the traditional techniques based on fourth order cumulants or quasi stationary signals as another potential application of the nested array a new approach to beamforming based on a nonlinear preprocessing is also introduced which can effectively utilize the degrees of freedom offered by the nested arrays the usefulness of all the proposed methods is verified through extensive computer simulations

super nested arrays linear sparse arrays with reduced mutual coupling part ii high order extensions in array processing mutual coupling between sensors has an adverse effect on the estimation of parameters e g doa sparse arrays such as nested arrays coprime arrays and minimum redundancy arrays mra have reduced mutual coupling compared to uniform linear arrays ulas these arrays also have a difference coarray with o left n 2 right virtual elements where n is the number of physical sensors and can therefore resolve o left n 2 right uncorrelated source directions but these well known sparse arrays have disadvantages mras do not have simple closed form expressions for the array geometry coprime arrays have holes in the coarray and nested arrays contain a dense ula in the physical array resulting in significantly higher mutual coupling than coprime arrays and mras in a companion paper a sparse array configuration called the second order super nested array was introduced which has many of the advantages of these sparse arrays while removing most of the disadvantages namely the sensor locations are readily computed for any n unlike mras and the difference coarray is exactly that of a nested array and therefore hole free at the same time the mutual coupling is reduced significantly unlike nested arrays in this paper a generalization of super nested arrays is introduced called the q th order super nested array this has all the properties of the second order super nested array with the additional advantage that mutual coupling effects are further reduced for q 2 many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

on stable self similar blowup for equivariant wave maps reference epfl article 166023doi 10 1002 cpa 20366 url http arxiv org abs 1003 0707 record created on 2011 05 23 modified on 2017 05 12