computing the hilbert transform on the real line we introduce a new method for computing the hilbert transform on the real line it is a collocation method based on an expansion in rational eigenfunctions of the hilbert transform operator and implemented through the fast fourier transform an error analysis is given and convergence rates for some simple classes of functions are established numerical tests indicate that the method compares favorably with existing methods

improvement and comparison of heuristics for solving the uncapacitated multisource weber problem the multisource weber problem is to locate simultaneouslym facilities in the euclidean plane to minimize the total transportation cost for satisfying the demand ofn fixed users each supplied from its closest facility many heuristics have been proposed for this problem as well as a few exact algorithms heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms we compare various heuristics i e alternative location allocation cooper 1964 projection bongartz et al 1994 tabu search brimberg and mladenovic 1996a p median plus weber hansen et al 1996 genetic search and several versions of variable neighbourhood search based on empirical tests that are reported it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that variable neighbourhood search gives consistently best results on average in moderate computing time

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

on the design of mimo block fading channels with feedback link capacity constraint in this paper we propose a combined adaptive power control and beamforming framework for optimizing multiple input multiple output mimo link capacity in the presence of feedback link capacity constraint the feedback channel is used to carry channel state information only it is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block we show that the hybrid design could achieve the optimal mimo link capacity and we derive a computationally efficient algorithm to search for the optimal design under a specific average power constraint finally we shall illustrate that a minimum mean square error spatial processor with a successive interference canceller at the receiver could be used to realize the optimal capacity we found that feedback effectively enhances the forward channel capacity for all signal to noise ratio snr values when the number of transmit antennas n sub t is larger than the number of receive antennas n sub r the snr gain with feedback is contributed by focusing transmission power on active eigenchannel and temporal power waterfilling the former factor contributed at most 10log sub 10 n sub t n sub r db snr gain when n sub t n sub r while the latter factor s snr gain is significant only for low snr values

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

approximation of semigroups and related operator functions by resolvent series we consider the approximation of semigroups e tau a and of the functions varphi j tau a that appear in exponential integrators by resolvent series the interesting fact is that the resolvent series expresses the operator functions e tau a and varphi j tau a respectively in efficiently computable terms this is important for semigroups where the new approximation is different from well known approximations by rational functions as well as for the application of exponential integrators which are currently of high interest and which are usually studied in a semigroup setting on banach spaces the approximation of the operator functions varphi j tau a in a general strongly continuous semigroup setting has not been discussed in the literature so far but this is crucial for an application of these integrators with unbounded operators or bounded operators like discretization matrices with large norm and eigenvalues somewhere in the left half plane

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

computing the hilbert transform and its inverse we construct a new method for approximating hilbert transforms and their inverse throughout the complex plane both problems can be formulated as riemann hilbert problems via plemelj s lemma using this framework we re derive existing approaches for computing hilbert transforms over the real line and unit interval with the added benefit that we can compute the hilbert transform in the complex plane we then demonstrate the power of this approach by generalizing to the half line combining two half lines we can compute the hilbert transform of a more general class of functions on the real line than is possible with existing methods

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

limiting performance of block fading channels with multiple antennas we derive the performance limits of a radio system consisting of a transmitter with t antennas and a receiver with r antennas a block fading channel with additive white gaussian noise awgn delay and transmit power constraints and perfect channel state information available at both the transmitter and the receiver because of a delay constraint the transmission of a codeword is assumed to span a finite and typically small number m of independent channel realizations therefore the relevant performance limits are the information outage probability and the delay limited or nonergodic capacity we derive the coding scheme that minimizes the information outage probability this scheme can be interpreted as the concatenation of an optimal code for the awgn channel without fading to an optimal beamformer for this optimal scheme we evaluate minimum outage probability and delay limited capacity among other results we prove that for the fairly general class of regular fading channels the asymptotic delay limited capacity slope expressed in bits per second per hertz b s hz per decibel of transmit signal to noise ratio snr is proportional to min t r and independent of the number of fading blocks m since m is a measure of the time diversity induced by interleaving or of the frequency diversity of the system this result shows that if channel state information is available also to the transmitter very high rates with asymptotically small error probabilities are achievable without the need of deep interleaving or high frequency diversity moreover for a large number of antennas delay limited capacity approaches ergodic capacity

computations of the complex error function rational expansions for computing the complex error function w z e z 2 text erfc iz are presented these expansions have the following attractive properties 1 they can be evaluated using a polynomial evaluation routine such as homer s method 2 the polynomial coefficients can be computed once and for all by a single fast fourier transform fft and 3 high accuracy is achieved uniformly in the complex plane with only a small number of terms comparisons reveal that in some parts of the complex plane certain competitors may be more efficient however the difference in efficiency is never great and the new algorithms are simpler than existing ones a complete program takes eight lines of matlab code

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

approximating cauchy type singular integral by an automatic quadrature scheme an automatic quadrature scheme is developed for the approximate evaluation of the product type indefinite integral q f x y c x y r t k t c f t dt 1 x y 1 1

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

an efficient method for approximate solution of a singular integral equation with cauchy kernel abstract in this paper the least squares method and the reproducing kernel space are employed to obtain approximate solution of a general form of singular integral equations with cauchy kernel this approach is based on the construction of the subspace which is dense in the reproducing kernel space the numerical examples illustrate the high accuracy of the method

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

the computation of the worst conditional expectation abstract recent advances in risk theory identify risk as a measure related to the tail of a probability distribution function since it represents the worst outcomes of the random variable measures like value at risk var conditional var expected shortfall and so on have become familiar operational tools for many financial applications in this paper one of these measures the worst conditional expectation with threshold of a discrete random variable z shortly wce z is considered it has been found that its computation can be formulated as a fractional integer programming problem with a single linear constraint but its complexity is np hard therefore it must be solved by implicit enumeration due to its similarity with the knapsack problem it has been found that a good upper bound and a sharp data structure allow the implementation of a branch bound that is able to solve realistic size problems in less than one hundredth of a second

uniform convergence of optimal order quadrature rules for cauchy principal value integrals abstract for the numerical evaluation of cauchy principal value integrals of the form e 1 1 f e c s 1 1 we consider a quadrature method based on spline interpolation of odd degree 2 k 1 k n 0 we show that these rules converge uniformly for 1 1 in particular we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded s th derivative if s e s 2 k 1 2 k 2 finally we compare the rule to the well known quadrature rule of elliott and paget which only converges pointwise

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

peano kernels and bounds for the error constants of gaussian and related quadrature rules for cauchy principal value integrals we show that if f in c k 1 1 k ge 2 the error term of every modified positive interpolatory quadrature rule for cauchy principal value integrals of the type mathop int nolimits 1 1 w x f x over x lambda dx lambda in 1 1 fulfills r n f lambda o n k ln n uniformly for all lambda in 1 1 and hence it is of optimal order of magnitude in the classes c k 1 1 k 2 3 4 ldots here w is a weight function with the property 0 le w x sqrt 1 x 2 le c we give explicit upper bounds for the peano type error constants of such rules this improves and completes earlier results by criscuolo and mastroianni calcolo 22 1985 391 441 and numer math 54 1989 445 461 and ioakimidis math comp 44 1985 191 198 for the special case of the gaussian rule we show that the restriction k ge 2 can be dropped the results are based on a new representation of the peano kernels of these formulae via the peano kernels of the underlying classical quadrature formulae this representation may also be useful in connection with some different problems

numerical and perturbative computations of solitary waves of the benjamin ono equation with higher order nonlinearity using christov rational basis functions computation of solitons of the cubically nonlinear benjamin ono equation is challenging first the equation contains the hilbert transform a nonlocal integral operator second its solitary waves decay only as o 1 x 2 to solve the integro differential equation for waves traveling at a phase speed c we introduced the artificial homotopy h u x x c u 1 d u 2 du 3 0 d 0 1 and solved it in two ways the first was continuation in the homotopy parameter d marching from the known benjamin ono soliton for d 0 to the cubically nonlinear soliton at d 1 the second strategy was to bypass continuation by numerically computing perturbation series in d and forming pade approximants to obtain a very accurate approximation at d 1 to further minimize computations we derived an elementary theorem to reduce the two parameter soliton family to a parameter free function the soliton symmetric about the origin with unit phase speed solitons for higher order benjamin ono equations are also computed and compared to their korteweg devries counterparts all computations applied the pseudospectral method with a basis of rational orthogonal functions invented by christov which are eigenfunctions of the hilbert transform

error estimates for interpolatory quadrature formulae in this paper we study the remainder of interpolatory quadrature formulae for this purpose we develop a simple but quite general comparison technique for linear functionals applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one by repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all with the help of this method we obtain asymptotically best possible error bounds for the clenshaw curtis quadrature and other polya type formulae our comparison technique can also be applied to the problem of definiteness i e the question whether the remainderr f of a formula of orderm can be represented asc f m by successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases finally we ask for good and worst quadrature formulae within certain classes we shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the gauss type formulae are worst interpreted in terms of peano kernels our theorem yields results on monosplines which may be of interest in themselves

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

the maximum capture problem with random utilities problem formulation and algorithms abstract a model for the optimal location of new facilities in a competitive market is introduced under the hypothesis that customers behavior can be modeled by random utility functions it means that the company that wished to locate uses a random utility model to forecast the market share of a location therefore the company cannot forecast the behavior of every customer in a deterministic fashion but is able to embed him by a probability distribution three formulations are proposed to compute upper bounds of the objective function and compared in a numerical simulation a branch and bound method is developed and tested on examples with up to 50 potential locations and a variable neighborhood search heuristic is proposed to solve larger instances

outer approximation and submodular cuts for maximum capture facility location problems with random utilities abstract we consider a family of competitive facility location problems in which a newcomer company enters the market and has to decide where to locate a set of new facilities so as to maximize its market share the multinomial logit model is used to estimate the captured customer demand we propose a first branch and cut approach for this family of difficult mixed integer non linear problems our approach combines two types of cutting planes that exploit particular properties of the objective function the first one are the outer approximation cuts and the second one are the submodular cuts the approach is computationally evaluated on three datasets from the recent literature the obtained results show that our new branch and cut drastically outperforms state of the art exact approaches both in terms of the computing times and in terms of the number of instances solved to optimality

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

functions of the second kind for freud weights and series expansions of hilbert transforms abstract let w e q where q r r is even continuous and of smooth polynomial growth at infinity then we call w 2 e 2 q a freud weight the most typical examples being w 2 x exp x 1 corresponding to the weight w 2 we can form the sequence of orthonormal polynomials p j w 2 x j 0 the functions of the second kind are q j w 2 x h p j w 2 x j 0 where h denotes the hilbert transform that is for g l r h g x p v g t t x dt here p v denotes principal value for a large class of freud weights we obtain bounds on q j j 0 in the l and l p norms we also estimate the generalized function of the second kind q j w 2 v x h p j w v x for a fixed function v we then apply these estimates to investigate the convergence of series of the second kind which form the basis of henrici s method of approximating cauchy principal value integrals

classification methods for random utility models with i i d disturbances under the most probable alternative rule in this paper random utility maximization based on maximization of correct classification of the choice decisions over a given data set is considered it is shown that if the disturbance vector in the random utility model is independent and identically distributed then preference determination based on the most probable alternative reduces to deterministic utility maximization as a consequence of the above equivalence the form of the error distribution normal weibull uniform etc plays no role in the determination of the preferred alternative parameter estimation under the most probable alternative rule is carried out using two methods the first is based on the solution of an appropriately defined system of linear inequalities and the second one is based on the function optimization of a newly proposed function whose optimum is achieved when the number of correctly classified individuals is maximized the ability to use these algorithms in the framework of pattern recognition and machine learning is pointed out simulations and a real case study involving intercity travel behavior are employed to assess the proposed methods

a definiteness criterion for linear functionals and its application to cauchy principle value quadrature abstract let g be a linear functional on a k 1 1 the class of all functions with an absolutely continuous k 1 th derivative let r and let g be given by g g in this situation a theo bras and schmeiser 1981 states that g is definite if g is definite in this paper we give conditions on g which ensure the definiteness of g an application of our result yields error bounds for a quadrature rule for cauchy principal value integrals

maximizing a submodular function by integer programming polyhedral results for the quadratic case abstract we give a new mixed integer programming mip formulation for the quadratic cost partition problem that is derived from a mip formulation for maximizing a submodular function several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron facet defining conditions and separation algorithms are discussed and computational results are reported

a comparison of linear reformulations for multinomial logit choice probabilities in facility location models abstract in the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model three linear reformulations of the original non linear model have been proposed so far in this paper we discuss these models in terms of solvability we present empirical findings based on synthetic data

a method for the practical evaluation of the hilbert transform on the real line abstract we investigate a method for the numerical evaluation of the weighted hilbert transform over the entire real line exp x 2 f x x t 1 d x where the integral is taken in the cauchy principal value sense the method is based on a combination of two polynomial approximation processes in this way we obtain a procedure that is numerically stable and efficient the algorithm can be implemented in a fast way and existing well known software packages may be used from the point of view of theoretical error bounds the method is competitive generalizations to other approximation methods are also possible

a branch and bound algorithm for the maximum capture problem with random utilities abstract the maximum capture problem with random utilities seeks to locate new facilities in a competitive market such that the captured demand of users is maximized assuming that each individual chooses among all available facilities according to the well know a random utility model namely the multinomial logit the problem is complex mostly due to its integer nonlinear objective function currently the most efficient approaches deal with this complexity by either using a nonlinear programing solver or reformulating the problem into a mixed integer linear programing milp model in this paper we show how the best milp reformulation available in the literature can be strengthened by using tighter coefficients in some inequalities we also introduce a new branch and bound algorithm based on a greedy approach for solving a relaxation of the original problem extensive computational experiments are presented benchmarking the proposed approach with other linear and non linear relaxations of the problem the computational experiments show that our proposed algorithm is competitive with all other methods as there is no method which outperforms the others in all instances we also show a large scale real instance of the problem which comes from an application in park and ride facility location where our proposed branch and bound algorithm was the most effective method for solving this type of problem

exit criteria and monotonicity in compound quadrature of gaussian type in this short note for compound quadrature rules of gaussian type we prove stopping rules and monotonicity results based on peano kernel methods

new error bounds for modified quadrature formulas for cauchy principal value integrals abstract for the numerical approximation of cauchy principal value integrals we consider the so called modified quadrature formulas i e formulas obtained by first subtracting out the singularity and then applying a classical quadrature formula we are interested in error bounds holding uniformly for all possible positions of the singular point the standard error bounds are based on suprema of derivatives but they often overestimate the true errors by a factor that grows with the number of nodes of the quadrature formula we give new bounds involving the total variation var s and l p norms t s t p of some derivative of the integrand function these bounds give additional possibilities for sharper estimations of the error

on the information throughput and optimized power allocation for mimo wireless systems with imperfect channel estimation in this paper we focus on the throughput analysis outage evaluation and optimized power allocation for multiple input multiple output mimo pilot based wireless systems subject to short term constraints on the radiated power and equipped with a feedback path for communicating back to the transmitter the imperfect mimo channel estimates available at the receiver the case of the ergodic throughput for gaussian distributed input signals is analyzed and the conditions for the asymptotical achievement of the shannon capacity are pointed out the main contributions of this work may be so summarized first we develop closed form analytical expressions for the computation of the ergodic information throughput conveyed by the considered mimo system for the case of ideal feedback link second we present an iterative algorithm for the optimized power allocation over the transmit antennas that explicitly accounts for the imperfect mimo channel estimates available at the receiver third after relaxing the assumption of ideal feedback link we test the sensitivity of the proposed power allocation algorithm on errors possibly introduced by the feedback channel and then we numerically evaluate the resulting throughput loss finally we develop closed form upper and lower bounds on the outage probability that are asymptotically tight

on the variations in mutual information of mimo communication systems due to perturbed channel state information at transmitter in this paper we address the issue of perturbed information at the transmitter in a multiple input multiple output mimo system two schemes correlation matrix feedback and channel information feedback are proposed to convey information necessary for achieving capacity to the transmitter this information is perturbed due to various impairments a perturbation analysis is carried out to study the variations in mutual information for each of the proposed schemes for ergodic channels this analysis is used to design a mimo system with a limited rate feedback using a codebook based approach vector quantizers are designed to minimize the loss in ergodic capacity for each of the proposed schemes and their performance is studied a non codebook method parameter based feedback is also presented and the underlying rate performance tradeoff is highlighted

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals in a previous paper the authors proposed a modified gaussian rule m wf t to compute the integral wf t in the cauchy principal value sense associated with the weightw and they proved the convergence in closed sets contained in the integration interval the main purpose of the present work is to prove uniform convergence of the sequence m wf t on the whole integration interval and to give estimates for the remainder term the same results are shown for particular subsequences of the gaussian rules m wf t for the evaluation of cauchy principal value integrals a result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made