zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

performance analysis of the optimum beamformer in the presence of correlated sources and its behavior under spatial smoothing in this paper we first study the signal cancellation and interference rejection effects of the optimum constrained least squares or minimum variance beamformer in the presence of partially and fully correlated interfering sources in particular we derive expressions for the output power and the gain in the interference direction of the beamformer in terms of the source powers correlation and the sensor noise power and show quantitatively the penalties arising from increasing correlation in several scenarios of interest next we show that spatial smoothing progressively decorrelates the sources at a rate that depends on the spacing and directions of the sources and thus relate the degree of smoothing to the improvement in signal cancellation and interference rejection behavior provided by spatial smoothing results of computer simulations are included to support our analysis

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

on linear prediction models constrained to have unit modulus poles and their use for sinusoidal frequency estimation linear prediction models with their poles restricted to the unit circle can be efficiently determined by using the levinson or the split levinson algorithm a simple proof of this property is presented the prime application of linear prediction models constrained to have unit modulus poles is the estimation of sinusoidal frequencies the consistency properties of the corresponding frequency estimates are analyzed it is shown that in the presence of noise the estimates are inconsistent an explicit expression for the asymptotic bias is provided

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

on local and global conjugacy abstract local and global conjugacy issue is closely related to the multiplicity one property in representation theory and langlands program we classify all even dimensional orthogonal irreducible representations which are locally conjugate but not globally conjugate in image in so 2 n and with certain langlands functoriality these will lead to connected instances for the failure of multiplicity one for so 2 n named as lfmo in this note we gather most our local global results in a purely representation theoretic way

a note on minimizing the prediction error when the zeros are restricted to the unit circle it is shown that the problem of minimizing the prediction error under the constraint that the zeros of the predictor polynomial be on the unit circle can be solved by using a modification of the levinson algorithm consequences of this result are discussed

on the dimension datum of a subgroup and its application to isospectral manifolds the dimension datum of a subgroup of a compact lie group is a piece of spectral information about that subgroup we find some new invariants and phenomena of the dimension data and apply them to construct the first example of a pair of isospectral simply connected closed riemannian manifolds which are of different homotopy types we also answer questions proposed by langlands

on the number of signals resolvable by a uniform linear array an algebraic limitation on the maximum number of directions of arrival of plane waves that can be resolved by a uniform linear sensor array is studied achievable lower and upper bounds are derived on that number as a function of the number of elements in the array number of snapshots and the rank of the source sample correlation matrix the signals are assumed narrow band and of identical and known center frequency the results are also applicable in the coherent signal case and when directions of arrival are estimated from few snapshots while in the multiple snapshot case the lower bounds coincide with known asymptotic results the upper bounds indicate the potential for resolving more signals than by present methods of array processing

a digital method of modeling quadratically nonlinear systems with a general random input without assuming particular statistics of the input a practical digital method of estimating linear and quadratic transfer functions of a nonlinear time invariant system that can be described by volterra series of up to second order is presented the method is tested and validated by analyzing input output data of a known quadratically nonlinear system it is used when there is little knowledge about the input statistics or the input is non gaussian it is also noted that the ordinary coherence functions cannot be used in explaining the input output power transfer relationship of a quadratic system excited by a non gaussian input signal with respect to the practical application of the method the relationship between the mean square errors involved in the transfer function estimates and the number of averages taken from the spectral estimation is qualitatively discussed

effects of amplitude and phase errors on linear predictive array processors the effects of amplitude and phase errors are determined for symmetric and asymmetric linear predictive array processors the maximum entropy method mem corresponds to linear prediction in the asymmetric case of a line array of equispaced elements with an end reference a comparison to conventional beamforming and maximum likelihood processing shows that they have the same peak location as the symmetric linear predictive array processor

self inversive hilbert space operator polynomials with spectrum on the unit circle abstract hilbert space operator polynomials with self inversive structure are studied if the inner numerical radius of an associated polynomial is greater than or equal to one then the spectrum lies on the unit circle and consists of normal approximate characteristic values

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

relationships involving third moments and bispectra of a harmonic process relationships between third moments of a harmonic random process and its bispectrum are presented the skewness and asymmetry with respect to a vertical axis of the process and its derivative can be obtained by integrating the real imaginary and frequency scaled real and imaginary parts of the bispectrum respectively triads of fourier coefficients containing a high frequency component can contribute substantially to the third moments of the derivative of the process

the poles of symmetric linear prediction models lie on the unit circle symmetric linear prediction models have their poles on the unit circle a simple proof of this result is presented

sensor array signal processing via a procrustes rotations based eigenanalysis of the esprit data pencil the authors propose the procrustes rotation based esprit algorithm pro esprit for estimating characteristic parameters associated with a scene of radiating sources given data derived from two identical sensor arrays the x and y arrays which are displaced relative to one another pro esprit uses the inherent redundancy built into the esprit array system to compensate for deviations from the ideal array model this redundancy is manifested by two fundamental properties of the noiseless x and y data matrices comprising the esprit data pencil a they have the same row space the signal subspace and 2 they have the same column space the source subspace these properties are utilized by invoking two primary tools 1 the solution to the procrustes problem for optimally approximating an invariant subspace rotation and 2 the total least squares method for perturbing each of two estimates of a common subspace in a minimal fashion until the two perturbed spaces are the same

pattern recognition using invariants defined from higher order spectra one dimensional inputs a new approach to pattern recognition using in variant parameters based on higher order spectra is presented in particular invariant parameters derived from the bispec trum are used to classify one dimensional shapes the bispec trum which is translation invariant is integrated along straight lines passing through the origin in bifrequency space the phase of the integrated bispectrum is shown to be scale and amplifi cation invariant as well a minimal set of these invariants is selected as the feature vector for pattern classification and a minimum distance classifier using a statistical distance measure is used to classify test patterns the classification technique is shown to distinguish two similar but different bolts given their one dimensional profiles pattern recognition using higher or der spectral invariants is fast suited for parallel implementa tion and has high immunity to additive gaussian noise sim ulation results show very high classification accuracy even for low signal to noise ratios

retrieval of self coupled harmonics presence of coupled harmonics in the data is a symptom attributed to nonlinear mechanisms generating the available time series self coupling amounts to presence of frequency pairs spl omega sub 0 k spl omega sub 0 and perhaps phase pairs spl phi sub 0 k spl phi sub 0 as well with k an integer it appears in periodic signals or when harmonics undergo nonlinear transformations self coupled harmonics observed in additive noise are retrieved from the peaks of diagonal scaled polyperiodogram slices which contrary to existing higher order approaches do not increase dimensionality and thus computations when dealing with nonlinearities of increasing orders performance evaluation of the resulting frequency estimator reveals decreasing variance with slices of polyperiodograms of increasing order and ability to suppress stationary mixing noise irrespective of its color and distribution detection and estimation of self phase coupling via polyperiodogram slices obviates limitations of existing methods that require availability of multiple independent records theory and algorithms are illustrated with simulated and voiced speech data

cumulant based approach to harmonic retrieval and related problems a frequently encountered problem in signal processing is that of estimating the frequencies and amplitudes of harmonics observed in additive colored gaussian noise in practice the observed signals are contaminated with spatially and temporally colored noise of unknown power spectral density a cumulant based approach to these problems is proposed the cumulants of complex processes are defined and it is shown that specific 1 d slices of the fourth order cumulant of the noisy signal for the direction of arrival doa and retrieval of harmonics in noise rhn problems are identical to the autocorrelation of a related noiseless signal hence correlation based high resolution methods may be used with fourth order cumulants as well the effectiveness of the proposed methods is demonstrated through standard simulation examples

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

proper j holomorphic discs in stein domains of dimension 2 we prove the existence of global bishop discs in a strictly pseudoconvex stein domain in an almost complex manifold of complex dimension 2

on the dimension datum of a subgroup the dimension datum of a closed subgroup of a compact lie group is the sequence of invariant dimensions of irreducible representations by restriction in this article we classify connected closed subgroups of a given compact lie group with equal dimension data or linearly dependent dimension data we also study the equalities and linear relations among dimension data of closed subgroups of a unitary group which act irreducibly on the natural representation

detecting a transient signal by bispectral analysis a method for detecting a transient waveform of unknown shape in additive stationary noise is presented the method uses a statistic computed from the sample bispectrum of a sampled record of the signal plus noise of length t n tau where 1 tau is the sampling rate the key result underlying the method is that the bispectrum of the noise is zero in a triangle that is a proper subset of the principal domain triangle it is shown that the probability of detecting the signal is high if 3 16 sup 1 2 n sup 5 6 is larger than rho sup 1 where rho is the energy signal to noise ratio the result implies that the bispectrum based test may detect a weak signal of unknown form which evades detection by other methods

matrix based higher order spectral analysis for three wave coupling processes composite linear and quadratic systems produce three wave coupling when stimulated by random phase input sinusoids due to the nonlinearity of the system the output frequencies are arithmetically related to the input using third order cumulant statistics and their associated bispectrum techniques are devised based on phase insensitive matrix structures for detection and frequency estimation of coupling frequencies the separation of the third order cumulant series into symmetric and skew symmetric portions allows one to exploit their characteristic eigendecompositions for frequency estimation after symmetrization biphases can be easily extracted as coefficients of the cumulant sequence using a generalized eigenvector representation one can relate symmetric and skew symmetric bases by a subspace rotation algorithm biphases can be estimated directly from generalized eigenvalues of the matrix pencil formed by symmetric and skew symmetric matrices the dimensionality of the matrices can be reduced through the use of cumulant projections that yield a slice of the bispectrum the radon transform procedure is related to bispectral processing through an isotropic radial slice volterra filter the compact third order kronecker product matrix formulation and algorithms for coupling frequency estimation can also be converted for use in biphase estimation simulations showing the performance of the above procedures are also presented for both synthetic and biomedical time series

regular type of real hyper surfaces in almost complex manifolds the regular type of a real hyper surface m in an almost complex manifold at some point p is the maximal contact order at p of m with germs of non singular pseudo holomorphic disks the main purpose of this paper is to give two intrinsic characterizations the type one in terms of lie brackets of a complex tangent vector field on m the other in terms of some kind of derivatives of the levi form

low rank estimation of higher order statistics low rank estimators for higher order statistics are considered in this paper the bias variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants in general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator and the mean squared error can be significantly smaller this makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics the low rank estimators also offer considerable reductions in the computational complexity of such algorithms the design of subspaces to optimize the tradeoffs between bias variance and computation is discussed and a noisy input noisy output system identification problem is used to illustrate the results

z transform methods for the optimal design of one dimensional layered elastic media in this work we develop a finite trigonometric series representation for the stress in a multilayered goupillaud type elastic strip subjected to transient heaviside loading on one end while the other end is held fixed this representation is achieved by means of the z transform method and involves the so called base angles generally different layered designs could share the same set of base angles and the more layers the design has the more base angles are expected necessary conditions for the base angles and design parameters for any given design are described as a result of the stress representation we are able to identify optimal layered designs which provide the smallest stress amplitude for two and three layered designs for which the coefficients of the stress representation are easy to find the optimization results are achieved using a custom made discrete optimization technique applied in a p velo and g a gazonas int j solids structures 40 2003 pp 6417 6428 for other laye