subband energy based reduced complexity spectrum sensing under noise uncertainty and frequency selective spectral characteristics the present work proposes a subband energy detection method that performs efficiently under noise uncertainty nu and frequency selective channels the critical impact of detrimental modeling uncertainties such as nu is analytically quantified and it is shown that the introduced method is robust to both nu and frequency selectivity conditions this is also the case for eigenvalue based sensing techniques in contrast to traditional energy detector based sensing connections of the subband energy based approach and existing eigenvalue based methods are established analytically which leads to a novel reduced complexity processing technique based on the difference between maximum and minimum subband energies the proposed method is capable of providing accurate and robust performance with low signal to noise ratios snr in the presence of nu closed form expressions are derived for the corresponding probability of false alarm and probability of detection under frequency selectivity due to the primary signal spectrum and or the transmission channel the validity of the offered expressions is justified through comparisons with respective results from computer simulations the sensing performance is evaluated in different communication scenarios with different frequency selective channel models and primary user waveforms the offered results indicate that the proposed methods provide quite significant savings in complexity e g 78 reduction in the considered example case while also improving the detection performance at low snrs and in the presence of nu

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

a rate of convergence result for the largest eigenvalue of complex white wishart matrices it has been recently shown that if x is an n n matrix whose entries are i i d standard complex gaussian and l1 is the largest eigenvalue of x x there exist sequences mn n and sn n such that l1 mn n sn n converges in distribution to w2 the tracy widom law appearing in the study of the gaussian unitary ensemble this probability law has a density which is known and computable the cumulative distribution function of w2 is denoted f2 in this paper we show that under the assumption that n n 0 we can find a function m continuous and nonincreasing and sequences n n and n n such that for all real s0 there exists an integer n s0 for which if n n n s0 we have with ln n l1 n n n n s s0 n n 2 3 p ln n s f2 s m s0 exp s the surprisingly good 2 3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution w2 is a good approximation to the empirical distribution of ln n in simulations an important fact from the point of view of e g statistical applications

quantum hydrodynamic models from a maximum entropy principle we use a density matrix formalism to derive a kinetic theory for a quantum gas generalized kinetic fields are introduced and employing the wigner function a certain hierarchy of quantum hydrodynamic qhd equations for the corresponding macroscopic variables is obtained we assert a maximum entropy principle to obtain closure of the qhd system for the explicit incorporation of statistics a proper quantum entropy is analyzed in terms of the reduced density matrix the determination of the reduced wigner function for equilibrium and non equilibrium conditions is found to become possible only by assuming that the lagrange multipliers can be expanded in powers of 2 quantum contributions are expressed in powers of 2 while classical results are recovered in the limit 0

boundary element methods for optimization of distributed parameter systems a numerical method of solution is proposed for optimization problems of distributed parameter systems two model problems from continuum mechanics are investigated by means of constructing the problems as the steady state optimal control problems governed by elliptic partial differential equations the basis of the suggested method of solution lies in the space discretization of the necessary conditions for optimality by the boundary element method and the minimization of the performance indices by the conjugate gradient method of optimization

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

multigrid acceleration of the flux split euler equations multigrid acceleration is applied to a flux split algorithm for solving the euler equations in three dimensions the basic algorithm is an implicit spatially split approximate factorization method the stability and performance of the scheme in comparison to other factorizations are examined results are presented for three dimensional flows that demonstrate substantially unproved convergence with the multigrid algorithm

a classical bound on quantum entropy a classical upper bound for quantum entropy is identified and illustrated 0 sq ln e 2 2 involving the variance 2 in phase space of the classical limit distribution of a given system a fortiori this further bounds the corresponding information theoretical generalizations of the quantum entropy proposed by renyi

three dimensional aerodynamic shape optimization using discrete sensitivity analysis an aerodynamic shape optimization procedure based on discrete sensitivity analysis is extended to treat three dimensional geometries the function of sensitivity analysis is to directly couple computational fluid dynamics cfd with numerical optimization techniques which facilitates the construction of efficient direct design methods the development of a practical three dimensional design procedures entails many challenges such as 1 the demand for significant efficiency improvements over current design methods 2 a general and flexible three dimensional surface representation and 3 the efficient solution of very large systems of linear algebraic equations it is demonstrated that each of these challenges is overcome by 1 employing fully implicit newton methods for the cfd analyses 2 adopting a bezier bernstein polynomial parameterization of two and three dimensional surfaces and 3 using preconditioned conjugate gradient like linear system solvers whereas each of these extensions independently yields an improvement in computational efficiency the combined effect of implementing all the extensions simultaneously results in a significant factor of 50 decrease in computational time and a factor of eight reduction in memory over the most efficient design strategies in current use the new aerodynamic shape optimization procedure is demonstrated in the design of both two and three dimensional inviscid aerodynamic problems including a two dimensional supersonic internal external nozzle two dimensional transonic airfoils resulting in supercritical shapes three dimensional transport wings and three dimensional supersonic delta wings each design application results in realistic and useful optimized shapes

position momentum correlations in the moshinsky atom position momentum correlation in the moshinsky atom is studied by examining the mutual information between wigner functions and position momentum space densities at the one and two particle levels we show that this correlation is independent of the strength of the interparticle potential at the two particle level at the one particle level the correlation decreases as the potential is increased shannon entropies of one and two particle wigner functions are also discussed as measures of the localization in phase space distributions

stability analysis and aerodynamic design optimization of euler equations using variational methods the one and two dimensional inviscid euler equations are formulated in an integro differential form for shape design sensitivity analysis and optimization the principal tool employed to derive the performance derivative sensitivity equations is the variational method which is a continuous alternative to the discrete sensitivity analysis along with the sensitivity equations the co state equations and their boundary conditions are derived thereafter based on the modal analysis of von neumann theory the stability limits of the co state equations are investigated the stability characteristics of the co state equations are compared with those of the euler equations finally using the criteria from the stability analysis of the state and co state equations some results of shape design optimization and sensitivity analysis for both quasi one and two dimensional euler equations are presented copyright 1999 john wiley sons ltd

spectrum sensing algorithms via finite random matrices we address the primary user pu detection spectrum sensing problem relevant to cognitive radio from a finite random matrix theoretical rmt perspective specifically we employ recently derived closed form and exact expressions for the distribution of the standard condition number scn of uncorrelated and semi correlated random dual central wishart matrices of finite sizes in the design hypothesis testing algorithms to detect the presence of pu signals in particular two algorithms are designed with basis on the scn distribution in the absence h0 and in the presence h1 of pu signals respectively due to an inherent property of the scn s the h0 test requires no estimation of snr or any other information on the pu signal while the h1 test requires snr only further attractive advantages of the new techniques are a due to the accuracy of the finite scn distributions superior performance is achieved under a finite number of samples compared to asymptotic rmt based alternatives b since expressions to model the scn statistics both in the absence and presence of pu signal are used the statistics of the spectrum sensing problem in question is completely characterized and c as a consequence of a and b accurate and simple analytical expressions for the receiver operating characteristic roc both in terms of the probability of detection as a function of the probability of false alarm pd versus pf and in terms of the probability of acquisition as a function of the probability of miss detection pa versus pm are yielded it is also shown that the proposed finite rmt based algorithms outperform all similar alternatives currently known in the literature at a substantially lower complexity in the process several new results on the distributions of eigenvalues and scns of random wishart matrices are offered including a closed form of the marchenko pastur s cumulative density function cdf and extensions of the latter as well as variations of asymptotic the distributions of extreme eigenvalues tracy widom and their ratio tracy widom curtiss which are simpler than those obtained with the spiked population model

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

numerical methods for second order shape sensitivity analysis with applications to heat conduction problems in this paper two finite element based schemes for second order shape sensitivity analysis are presented in the first formulation the av dd method the first order shape sensitivity equation is derived and expressed in terms of state and adjoint variables the resultant equation is then directly differentiated to obtain the second order shape sensitivity equation in the second formulation the dd av method the functional of concern is differentiated twice to yield the second order sensitivity equation in which the second order shape derivatives can be eliminated by introducing a proper adjoint equation a thermal fin problem and a thermal insulation layer problem have been studied to validate the proposed schemes it is shown that both methods yield identical results though the dd av method is computationally more efficient

performance of statistical tests for single source detection using random matrix theory this paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver the generalized maximum likelihood test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace using recent results from random matrix theory a practical way to evaluate the threshold and the p value of the test is provided in the asymptotic regime where the number k of sensors and the number n of observations per sensor are large but have the same order of magnitude the theoretical performance of the test is then analyzed in terms of receiver operating characteristic roc curve it is in particular proved that both type i and type ii error probabilities converge to zero exponentially as the dimensions increase at the same rate and closed form expressions are provided for the error exponents these theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

fourth moment analysis for wave propagation in the white noise paraxial regime in this paper we consider the ito schrodinger model for wave propagation in random media in the paraxial regime we solve the equation for the fourth order moment of the field in the regime where the correlation length of the medium is smaller than the initial beam width in terms of applications we prove that the centered fourth order moments of the field satisfy the gaussian summation rule we derive the covariance function of the intensity of the transmitted beam and the variance of the smoothed wigner transform of the transmitted field the second application is used to explicitly quantify the scintillation of the transmitted beam and the third application to quantify the statistical stability of the wigner transform

entropy and wigner functions the properties of an alternative definition of quantum entropy based on wigner functions are discussed such a definition emerges naturally from the wigner representation of quantum mechanics and can easily quantify the amount of entanglement of a quantum state it is shown that smoothing of the wigner function induces an increase in entropy this fact is used to derive some simple rules to construct positive definite probability distributions which are also admissible wigner functions

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

the largest eigenvalues of sample covariance matrices for a spiked population diagonal case we consider large complex random sample covariance matrices obtained from spiked populations that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to 1 we investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large under some conditions on moments of the sample distribution we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex gaussian sample with the same true covariance the real setting is also considered