weighted shift matrices unitary equivalence reducibility and numerical ranges abstract an n by n n 3 weighted shift matrix a is one of the form 0 a 1 0 a n 1 a n 0 where the a j s called the weights of a are complex numbers assume that all a j s are nonzero and b is an n by n weighted shift matrix with weights b 1 b n we show that b is unitarily equivalent to a if and only if b 1 b n a 1 a n and for some fixed k 1 k n b j a k j a n j a j for all j next we show that a is reducible if and only if a j j 1 n is periodic that is for some fixed k 1 k n 2 n is divisible by k and a j a k j for all j 1 j n k finally we prove that a and b have the same numerical range if and only if a 1 a n b 1 b n and s r a 1 2 a n 2 s r b 1 2 b n 2 for all 1 r n 2 where s r s are the circularly symmetric functions

local linear estimators for the bioelectromagnetic inverse problem linear estimators have been used widely in the bioelectromagnetic inverse problem but their properties and relationships have not been fully characterized here we show that the most widely used linear estimators may be characterized by a choice of norms on signal space and on source space these norms depend in part on assumptions about the signal space and source space covariances we demonstrate that two estimator classes standardized and weight vector normalized yield unbiased estimators of source location for simple source models including only the noise free case but biased estimators of source magnitude in the presence of instrumental white noise we show that the nonadaptive standardized estimator is a biased estimator of source location while the adaptive weight vector normalized estimator remains unbiased a third class distortionless is an unbiased estimator of source magnitude but a biased estimator of source location

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

cancellation and direct summands in dimension 1 abstract let be a module finite algebra over a commutative noetherian ring of krull dimension 1 we extend roiter s direct summand theorem to arbitrary finitely generated modules obtaining a sharpened form of serre s direct summand theorem in this setting we also extend drozd s cancellation theorem to arbitrary finitely generated modules obtaining a sharpened form of bass s cancellation theorem in this setting a corollary is that over commutative reduced noetherian rings of dimension 1 direct sum cancellation holds in every genus of finitely generated modules this becomes false if the ring has nilpotent elements another corollary is that if direct sum cancellation holds in the genera of modules m and n then it holds in the genus of m n this seems to be new even for the modules that occur in integral representation theory the main thrust of this paper is to close the gap between integral representation theory and the rest of module theory by eliminating hypotheses concerning the existence of maximal orders of finite normalization in the commutative case and allowing our rings to have nilpotent ideals

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

hierarchical bayesian inference for the eeg inverse problem using realistic fe head models depth localization and source separation for focal primary currents the estimation of the activity related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill posed inverse problem this is especially true in the recovery of brain networks involving deep lying sources by means of eeg meg recordings which is still a challenging task for any inverse method recently hierarchical bayesian modeling hbm emerged as a unifying framework for current density reconstruction cdr approaches comprising most established methods as well as offering promising new methods our work examines the performance of fully bayesian inference methods for hbm for source configurations consisting of few focal sources when used with realistic high resolution finite element fe head models the main foci of interest are the correct depth localization a well known source of systematic error of many cdr methods and the separation of single sources in multiple source scenarios both aspects are very important in the analysis of neurophysiological data and in clinical applications for these tasks hbm provides a promising framework and is able to improve upon established cdr methods such as minimum norm estimation mne or sloreta in many aspects for challenging multiple source scenarios where the established methods show crucial errors promising results are attained additionally we introduce wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios

comparison of noise normalized minimum norm estimates for meg analysis using multiple resolution metrics noise normalization has been shown to partly compensate for the localization bias towards superficial sources in minimum norm estimation however it has been argued that in order to make inferences for the case of multiple sources localization properties alone are insufficient instead multiple measures of resolution should be applied to both point spread and cross talk functions psfs and ctfs here we demonstrate that noise normalization affects the shapes of psfs but not of ctfs we evaluated psfs and ctfs for the mne dspm and sloreta inverse operators on the metrics dipole localization error dle spatial dispersion sd and overall amplitude oa we used 306 channel meg configurations obtained from 17 subjects in a real experiment including individual noise covariance matrices and head geometries we confirmed that for psfs dle improved after noise normalization and is zero for sloreta however sd was generally lower for the unnormalized mne oa distributions were similar for all three methods indicating that all three methods may greatly underestimate some sources relative to others the reliability of differences between methods across subjects was demonstrated using distributions of standard deviations and p values from paired t tests as predicted the shapes of ctfs were the same for all methods reflecting the general resolution limits of the inverse problem this means that noise normalization is of no consequence where linear estimation procedures are used as spatial filters while low dle is advantageous for the localization of a single source or possibly a few spatially distinct sources the benefit for the case of complex source distributions is not obvious we suggest that software packages for source estimation should include comprehensive tools for evaluating the performance of different methods

minimum variance beamformer weights revisited abstract adaptive minimum variance beamformers are widely used analysis tools in meg and eeg when the target brain activity presents in the form of spatially localized responses the procedure usually involves two steps first positions and orientations of the sources of interest are determined second the filter weights are calculated and source time courses reconstructed this last step is the object of the current study despite different approaches utilized at the source localization stage basic expressions for the weights have the same form dictated by the minimum variance condition these classic expressions involve covariance matrix of the measured field which includes contributions from both the sources of interest and the noise background we show analytically that the same weights can alternatively be obtained if the full field covariance is replaced with that of the noise provided the beamformer points to the true sources precisely in practice however a certain mismatch is always inevitable we show that such mismatch results in partial suppression of the true sources if the traditional weights are used to avoid this effect the alternative weights based on properly estimated noise covariance should be applied at the second source time course reconstruction step we demonstrate mathematically and using simulated and real data that in many situations the alternative weights provide significantly better time course reconstruction quality than the traditional ones in particular they a improve source level snr and yield more accurately reconstructed waveforms b provide more accurate estimates of inter source correlations and c reduce the adverse influence of the source correlations on the performance of single source beamformers which are used most often importantly the alternative weights come at no additional computational cost as the structure of the expressions remains the same

multi core beamformers derivation limitations and improvements minimum variance beamformers are popular tools used in eeg and meg for analysis of brain activity in recent years new multi source beamformer methods were developed including the dual core beamformer dcbf and its enhanced version edcbf both techniques should allow modeling of correlated brain activity under a wide range of conditions however the mathematical justification given is based on single source results and computer simulations which do not provide an insight into the assumptions involved and the limits of their applicability current work addresses this problem analytical expressions relating actual source parameters to those obtained with the dcbf and edcbf are derived and rigorous conclusions regarding the accuracy of the dcbf edcbf reconstructions are made in particular it is shown that dcbf accurately identifies source coordinates but amplitudes and orientations are only correct for high snrs and fully correlated sources in contrast edcbf source localization is inaccurate but if the source positions are found precisely edcbf allows perfect reconstruction for arbitrary snrs if the source positions are approximate the reconstruction errors are generally larger for higher snr values the edcbf results can be improved by using global unbiased localizer functions and an alternative way of estimating source orientations

stable isomorphism of modules over one dimensional rings abstract two r modules m and n are said to be stably isomorphic provided m r n n r n for some n 1 in this paper r is always a commutative reduced one dimensional noetherian ring with finite normalization r and m and n are torsionfree all modules are assumed to be finitely generated there is a natural action of r c the group of units of r c where c is the conductor of r in r on torsionfree r modules and the main theorem of this paper is that the orbits of the induced action of r c are exactly the stable isomorphism classes as applications we show that stably isomorphic modules are actually isomorphic if r is a domain finitely generated as an r algebra with at most one singular real maximal ideal we give several examples of modules that are stably isomorphic but not isomorphic illustrating that these hypotheses cannot be significantly weakened we are grateful to l s levy and the referee for several suggestions that have improved the exposition of this work

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

ica based artifact correction improves spatial localization of adaptive spatial filters in meg abstract beamformers are one of the most common inverse models currently used in the estimation of source activity from magnetoencephelography meg data they rely on a minimization of total power while constraining the gain in the voxel of interest resulting in the suppression of background noise nonetheless in cases where background noise is strong compared to the source of interest or when many sources are present the ability of the beamformer to detect and accurately localize weak sources is reduced in visual paradigms two main background sources can substantially impact an accurate estimation of weaker sources ocular artifacts are orders of magnitude higher than neural sources making it difficult for the beamformer to effectively suppress them primary visual activations also result in strong signals that can impede localization of weak sources in this paper we systematically evaluated how neural visual and non neural eye heart sources affect the localization accuracy of frontal and medial temporal sources in visual tasks these sources are of tremendous interest in learning and memory studies as well as in clinical settings alzheimer s epilepsy and are typically difficult to localize robustly in meg empirical data from two tasks active learning and control were used to evaluate our analysis techniques global field power calculations showed multiple time periods where active learning was significantly different from response selection with dominant sources converging to the eyes extensive leakage of eye activity into frontal and visual that evoked responses into parietal cortices was also observed contributions from ocular activity to the reconstructed time series were indiscernible from task based recruitment of frontal sources in the original data removing artifacts eye movements cardiac and muscular by means of independent component analysis ica led to a significant improvement in detection and localization of frontal and medial temporal sources we verified our results by using simulations of sources placed in frontal and medial temporal regions with various types of background noise eye heart and visual we report that the detection and localization accuracy of frontal and medial temporal sources with beamformer techniques is highly dependent on the magnitude and location of background sources and that removing artifacts can substantially improve the beamformer s performance

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

reduction of the c numerical range to the classical numerical range abstract let a be an n n complex matrix and c c 1 c 2 c n a real n tuple the c numerical range of a is defined as the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis for c n when c 1 0 0 w c a becomes the classical numerical range of a which is often defined as the set w a x ax x c n x x 1 we show that for any n n complex matrix a and real n tuple c there exists a complex matrix b of size at most n such that w c a w b constructions of the matrix b for some matrices a and real n tuple c are provided

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

remarks on the lax conjecture for hyperbolic polynomials abstract a conjecture of lax p lax differential equations difference equations and matrix theory commun pure appl math 11 1958 175 194 says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix the conjecture has recently been proved by lewis parillo ramana based on previous work of dubrovin and helton vinnikov in this note we prove related results for polynomials in several space variables which have rotational symmetries

greater robustness of second order statistics than higher order statistics algorithms to distortions of the mixing matrix in blind source separation of human eeg implications for single subject and group analyses a mandatory assumption in blind source separation bss of the human electroencephalogram eeg is that the mixing matrix remains invariant i e that the sources electrodes and geometry of the head do not change during the experiment actually this is not often the case for instance it is common that some electrodes slightly move during eeg recording this issue is even more critical for group independent component analysis gica a method of growing interest in which only one mixing matrix is estimated for several subjects indeed because of interindividual anatomo functional variability this method violates the mandatory principle of invariance here using simulated experiments 1 and 2 and real experiment 3 eeg data we test how eleven current bss algorithms undergo distortions of the mixing matrix we show that this usual kind of perturbation creates non gaussian features that are virtually added to all sources impairing the estimation of real higher order statistics hos features of the actual sources by hos algorithms e g ext infomax fastica hos based methods are likely to identify more components with similar properties than actual neurological sources a problem frequently encountered by bss users in practice the quality of the recovered signal and the efficiency of subsequent source localization are substantially impaired performing dimensionality reduction before applying hos based bss does not seem to be a safe strategy to circumvent the problem second order statistics sos based bss methods belonging to the less popular sobi family class are much less sensitive to this bias

hyperbolic forms associated with cyclic weighted shift matrices abstract the singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined it is shown that the singular points of such a curve are real nodes some results related the numerical ranges of cyclic weighted shift matrices are presented in particular the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form further an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights

the boundary of higher rank numerical ranges abstract the rank k numerical range of an n n matrix a is defined as k a c pap p for some rank k orthogonal projection p this study presents a method to generate the boundary of k a and examines the flat portions of the boundary of the rank k numerical range of a matrix associated with a roulette curve a sufficient condition is given to ensure that the rank k numerical range is not attainable by a classical numerical range

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

source stability index a novel beamforming based localisation metric many experimental studies into human brain function now use magnetoencephalography meg to non invasively investigate human neuronal activity a number of different analysis techniques use the observed magnetic fields outside of the head to estimate the location and strength of the underlying neural generators one such technique a spatial filtering method known as beamforming produces whole head volumetric images of activation typically a differential power map throughout the head is generated between a time window containing the response to a stimulus of interest and a window containing background brain activity a statistical test is normally performed to reveal locations which show a significantly different response in the presence of the stimulus despite this being a widely used measure for both phase locked and non phase locked information it requires a number of assumptions namely that the baseline activity defined is stable and also that a change in total power is the most effective way of revealing the neuronal sources required for the task this paper introduces a metric which evaluates the consistency of the response at each location within a cortical volume such a method of localisation negates the need for a baseline period of activity to be defined and also moves away from simply considering the energy content of brain activity the paper presents both simulated and real data it demonstrates that this new metric of stability is able to more accurately and crucially more reliably draw inferences about neuronal sources of interest

truncated rap music trap music for meg and eeg source localization electrically active brain regions can be located applying multiple signal classification music on magneto or electroencephalographic meg eeg data we introduce a new music method called truncated recursively applied and projected music trap music it corrects a hidden deficiency of the conventional rap music algorithm which prevents estimation of the true number of brain signal sources accurately the correction is done by applying a sequential dimension reduction to the signal subspace projection we show that trap music significantly improves the performance of music type localization in particular it successfully and robustly locates active brain regions and estimates their number we compare trap music and rap music in simulations with varying key parameters e g signal to noise ratio correlation between source time courses and initial estimate for the dimension of the signal space in addition we validate trap music with measured meg data we suggest that with the proposed trap music method music type localization could become more reliable and suitable for various online and offline meg and eeg applications

numerical ranges of weighted shift matrices abstract we show that if a is an n by n n 3 matrix of the form 0 a 1 0 a n 1 a n 0 then the boundary of its numerical range contains a line segment if and only if the a j s are nonzero and the numerical ranges of the n 1 by n 1 principal submatrices of a are all equal for n 3 this is the case if and only if a 1 a 2 a 3 0 in which case w a the numerical range of a is the equilateral triangular region with vertices the three cubic roots of a 1 a 2 a 3 for n 4 the condition becomes a 1 a 3 0 and a 2 a 4 0 in which case w a is the convex hull of two degenerate or otherwise ellipses

failure of cancellation for direct sums of line bundles in answer to a question of murthy and wiegand examples are given of finitely generated projective modules l of rank 1 over a commutative ring r such that l l 1 is stably free but not free examples are also given of projective modules for which the determinant map det aut p r is not onto some related topological questions are also discussed

finite representation type and direct sum cancellation abstract consider the notion of finite representation type frt for short an integral domain r has frt if there are only finitely many isomorphism classes of indecomposable finitely generated torsion free r modules now specialize let r be of the form d c o where d is a principal ideal domain whose residue fields are finite c d is a nonzero nonunit and o is the ring of integers of some finite separable field extension of the quotient field of d if the d rank of r is at least four then r does not have frt in this case we show that cancellation of finitely generated torsion free r modules is valid if and only if every unit of o c o is liftable to a unit of o we also give a complete analysis of cancellation for some rings of the form d c o having frt we include some examples which illustrate the difficult cubic case

kharitonov s theorem and bezoutians abstract an elementary proof of the kharitonov theorem is presented the proof is based on the concept of a bezoutian matrix generally exploiting the special structure of such matrices e g bezoutians toeplitz hankel or vandermonde matrices etc can be interesting e g leading to unified approaches in different cases as well as to further generalizations here the concept of the bezoutian matrix is used to provide a unified derivation of the kharitonov like theorems for the continuous time and discrete time settings finally the block anderson jury bezoutians are used to propose a possible technique to attack an difficult open problem related to the robust stability in the mimo case

source localization using recursively applied and projected rap music a new method for source localization is described that is based on a modification of the well known music algorithm in classical music the array manifold vector is projected onto an estimate of the signal subspace errors in the estimate of the signal subspace can make localization of multiple sources difficult recursively applied and projected rap music uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the signal subspace estimate into its orthogonal complement the music projection to find the next source is then performed in this reduced subspace special assumptions about the array manifold structure such as vandermonde or shift invariance are not required using the metric of principal angles we describe a general form of the rap music algorithm for the case of diversely polarized sources through a uniform linear array simulation with two highly correlated sources we demonstrate the improved monte carlo error performance of rap music relative to music and two other sequential subspace methods s and ies music we then demonstrate the more general utility of this algorithm for multidimensional array manifolds in a magnetoencephalography meg source localization simulation

pencils of real symmetric matrices and real algebraic curves abstract let a rational real algebraic curve of degree n in the real projective plane x y z be given by its parametric equations let not containing the point o 0 0 1 have the property that i every line through the point o intersects in n real points counting multiplicities an explicit form is given for real symmetric matrices a and b of order n such that det xa yb zl 0 is an equation for for a diagonal b is a diagonal multiple of a loewner matrix the result partly proves a previous conjecture of the author about general real algebraic curves of degree n having the property i it is also shown that in this general case the conjecture is not true

numerical ranges of weighted shift matrices with periodic weights abstract let a be an n by n n 2 matrix of the form 0 a 1 0 a n 1 a n 0 we show that if the a j s are nonzero and their moduli are periodic then the boundary of its numerical range contains a line segment we also prove that w a contains a noncircular elliptic arc if and only if the a j s are nonzero n is even a 1 a 3 a n 1 a 2 a 4 a n and a 1 a 2 finally we give a criterion for a to be reducible and completely characterize the numerical ranges of such matrices

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

on the history of the bezoutian and the resultant matrix abstract it is shown how the bezoutian and the resultant matrix evolved from euler s work in elimination theory

sturm and sylvester algorithms revisited via tridiagonal determinantal representations first we show that sturm algorithm and sylvester algorithm which compute the number of real roots of a given univariate polynomial lead to two dual tridiagonal determinantal representations of the polynomial next we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

higher rank numerical ranges and kippenhahn polynomials abstract we prove that two n by n matrices a and b have their rank k numerical ranges k a and k b equal to each other for all k 1 k n 2 1 if and only if their kippenhahn polynomials p a x y z det x re a y im a zi n and p b x y z det x re b y im b zi n coincide the main tools for the proof are the li sze characterization of higher rank numerical ranges weyl s perturbation theorem for eigenvalues of hermitian matrices and bezout s theorem for the number of common zeros for two homogeneous polynomials

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

flat portions on the boundary of the indefinite numerical range of 3 3 matrices abstract we focus on complex 3 3 matrices whose indefinite numerical ranges have a flat portion on the boundary the results here obtained are parallel to those of keeler rodman and spitkovsky for the classical numerical range

singular points of cyclic weighted shift matrices abstract let s be an n by n cyclic weighted shift matrix and f s t x y det t i x s y s be a ternary form associated with s we investigate the number of singular points of the curve f s t x y 0 and show that the number of singular points of f s t x y 0 associated with a cyclic weighted shift matrix whose weights are neither 1 periodic nor 2 periodic is less than or equal to n n 3 2 furthermore we verify the upper bound n n 3 2 is sharp for 4 n 7

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

application of multi source minimum variance beamformers for reconstruction of correlated neural activity linearly constrained minimum variance beamformers are highly effective for analysis of weakly correlated brain activity but their performance degrades when correlations become significant multiple constrained minimum variance mcmv beamformers are insensitive to source correlations but require a priori information about the source locations besides the question whether unbiased estimates of source positions and orientations can be obtained remained unanswered in this work we derive mcmv based source localizers that can be applied to both induced and evoked brain activity they may be regarded as a generalization of scalar minimum variance beamformers for the case of multiple correlated sources we show that for arbitrary noise covariance these beamformers provide simultaneous unbiased estimates of multiple source positions and orientations and remain bounded at singular points we also propose an iterative search algorithm that makes it possible to find sources approximately without a priori assumptions about their locations and orientations simulations and analyses of real meg data demonstrate that presented approach is superior to traditional single source beamformers in situations where correlations between the sources are significant

maximum likelihood methods for direction of arrival estimation five methods of direction of arrival doa estimation which can be derived from the maximum likelihood ml principle are considered the ml method mlm results from the application of the ml principle to the statistics of the observed raw data the standard multiple signal classification music procedure called music 1 is obtained as a brute force approximation of the mlm an improved music procedure named music 2 is obtained by applying the ml principle to the statistics of certain linear combinations of the sample noise space eigenvectors a procedure which compromises between the good performance of the mlm and the computational simplicity of music is a method of direction estimation mode 1 which is derived as a large sample realization of the mlm a fifth method called mode 2 is obtained by using the ml principle on the statistics of certain linear combinations of the sample eigenvectors mode 2 is computationally less demanding than the mlm it is of the same complexity as mode 1 and statistically more efficient a numerical comparison of these five doa estimation methods is presented it confirms the analytic results on their theoretical performance levels

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

critical values for higher rank numerical ranges associated with roulette curves abstract in this paper the critical value is determined for the higher rank numerical ranges of matrices associated with a parameter of roulette curves for which the higher rank numerical range is a regular polygon for every parameter less than or equal to the critical value