robust mse equalizer design for mimo communication systems in the presence of model uncertainties this paper considers a robust mean square error mse equalizer design problem for multiple input multiple output mimo communication systems with imperfect channel and noise information at the receiver when the channel state information csi and the noise covariance are known exactly at the receiver a minimum mean square error mmse equalizer can be employed to estimate the transmitted signal however in actual systems it is necessary to take into account channel and noise estimation errors we consider here a worst case equalizer design problem where the goal is to find the equalizer minimizing the equalization mse for the least favorable channel model within a neighborhood of the estimated model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and estimated channel models lagrangian optimization is used to convert this min max problem into a convex min min problem over a convex domain which is solved by interchanging the minimization order the robust mse equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem simulation results are presented to demonstrate the mse and bit error rate ber performance of robust equalizers when applied to the least favorable channel model

optimal designs for space time linear precoders and decoders we introduce a new paradigm for the design of transmitter space time coding that we refer to as linear precoding it leads to simple closed form solutions for transmission over frequency selective multiple input multiple output mimo channels which are scalable with respect to the number of antennas size of the coding block and transmit average peak power the scheme operates as a block transmission system in which vectors of symbols are encoded and modulated through a linear mapping operating jointly in the space and time dimension the specific designs target minimization of the symbol mean square error and the approximate maximization of the minimum distance between symbol hypotheses under average and peak power constraints the solutions are shown to convert the mimo channel with memory into a set of parallel flat fading subchannels regardless of the design criterion while appropriate power bits loading on the subchannels is the specific signature of the different designs the proposed designs are compared in terms of various performance measures such as information rate ber and symbol mean square error

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

p adic properties of division polynomials and elliptic divisibility sequences for a fixed rational point p e k on an elliptic curve we consider the sequence of values f n p n 1 of the division polynomials of e at p for a finite field open image in new window we prove that the sequence is periodic for a local field open image in new window we prove under certain hypotheses that there is a power q p e so that for all m 1 the limit of open image in new window exists in k and is algebraic over open image in new window we apply this result to prove an analogous p adic limit and algebraicity result for elliptic divisibility sequences

lower bounds on the mean squared error of low rank matrix reconstruction we investigate the behavior of the mean square error mse of low rank matrix reconstruction and its special case matrix completion we first derive the constrained cramer rao bound crb on the mse matrix of any locally unbiased estimator and then analyze the behavior of the constrained crb when a subset of entries of the underlying matrix is randomly observed we design an alternating minimization procedure to compute the maximum likelihood estimator mle for the low rank matrix and demonstrate through numerical simulations that the performance of the mle approaches the constrained crb when the signal to noise ratio is high applying a chapman robbins type barankin bound allows us to derive lower bounds on the worst case scalar mse we demonstrate that the worst case scalar mse is infinite even if the model is identifiable however the infinite scalar mse is achieved only on a set of low rank matrices with measure zero we discuss the implications of these bounds and compare them with the empirical performance of the matrix lasso estimator and the existing bounds in the literature

annihilating filter based low rank hankel matrix approach for image inpainting in this paper we propose a patch based image inpainting method using a low rank hankel structured matrix completion approach the proposed method exploits the annihilation property between a shift invariant filter and image data observed in many existing inpainting algorithms in particular by exploiting the commutative property of the convolution the annihilation property results in a low rank block hankel structure data matrix and the image inpainting problem becomes a low rank structured matrix completion problem the block hankel structured matrices are obtained patch by patch to adapt to the local changes in the image statistics to solve the structured low rank matrix completion problem we employ an alternating direction method of multipliers with factorization matrix initialization using the low rank matrix fitting algorithm as a side product of the matrix factorization locally adaptive dictionaries can be also easily constructed despite the simplicity of the algorithm the experimental results using irregularly subsampled images as well as various images with globally missing patterns showed that the proposed method outperforms existing state of the art image inpainting methods

robust least squares estimation with a relative entropy constraint given a nominal statistical model we consider the minimax estimation problem consisting of finding the best least squares estimator for the least favorable statistical model within a neighborhood of the nominal model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and nominal models for a gaussian nominal model and a finite observations interval or for a stationary gaussian process over an infinite interval the usual noncausal wiener filter remains optimal however the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty on the other hand standard causal least squares estimators are not optimal and a characterization is provided for the causal estimator and the corresponding least favorable model the causal estimator takes the form of a risk sensitive estimator with an appropriately selected risk sensitivity coefficient

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

optimal transmitter eigen beamforming and space time block coding based on channel mean feedback optimal transmitter designs obeying the water filling principle are well documented they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems we design in this paper an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams the optimality pertains to minimizing an upper bound on the symbol error rate which leads to better performance than maximizing the expected signal to noise ratio snr at the receiver coupled with orthogonal space time block codes two directional eigen beamforming emerges as a more attractive choice than conventional one directional beamforming with uniformly improved performance without rate reduction and without essential increase in complexity with multiple receive antennas and reasonably good feedback quality the two directional eigen beamformer is also capable of achieving the best possible performance in a large range of transmit power to noise ratios without a rate penalty

on the discrete logarithm problem in class groups of curves we study the discrete logarithm problem in degree 0 class groups of curves over finite fields with particular emphasis on curves of small genus we prove that for every fixed g 2 the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of o q 2 2 g where f q is the ground field this result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group and just as this previous result it is obtained via an index calculus algorithm with double large prime variation generalizing this result we prove that for fixed g 0 2 the discrete logarithm problem in class groups of all curves c f q of genus g g 0 can be solved in an expected time of o q g 2 g 0 1 1 g 0 and in an expected time of o cl 0 c 2 g 0 1 1 g0 as a complementary result we prove that for any fixed n with n 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields f q n q a prime power can be solved in an expected time of o q 2 2 n furthermore we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree given the curve and its l polynomial

on taking square roots without quadratic nonresidues over finite fields we present a novel idea to compute square roots over finite fields without being given any quadratic nonresidue and without assuming any unproven hypothesis the algorithm is deterministic and the proof is elementary in some cases the square root algorithm runs in o log 2 q bit operations over finite fields with q elements as an application we construct a deterministic primality proving algorithm which runs in o log 3 n for some integers n

combining beamforming and orthogonal space time block coding multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication efficient space time codes have been developed that utilize a large portion of the available capacity these codes are designed under the assumption that the transmitter has no knowledge about the channel in this work on the other hand we consider the case when the transmitter has partial but not perfect knowledge about the channel and how to improve a predetermined code so that this fact is taken into account a performance criterion is derived for a frequency nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code the resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods in addition a particularly efficient solution method is developed for the special case of independently fading channel coefficients the proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space time block coding simulation results for a narrow band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge

efficient non iterative estimator for imaging contrast agents with spectral x ray detectors an estimator to image contrast agents and body materials with x ray spectral measurements is described the estimator is usable with the three or more basis functions that are required to represent the attenuation coefficient of high atomic number materials the estimator variance is equal to the cramer rao lower bound crlb and it is unbiased its parameters are computed from measurements of a calibration phantom with the clinical x ray system and it is non iterative the estimator is compared with an iterative maximum likelihood estimator the estimator first computes a linearized maximum likelihood estimate of the line integrals of the basis set coefficients corrections for errors in the initial estimates are computed by interpolation with calibration phantom data the final estimate is the initial estimate plus the correction the performance of the estimator is measured using a monte carlo simulation random photon counting with pulse height analysis data are generated the mean squared errors of the estimates are compared to the crlb the random data are also processed with an iterative maximum likelihood estimator previous implementations of iterative estimators required advanced physics instruments not usually available in clinical institutions the estimator mean squared error is essentially equal to the crlb the estimator outputs are close to those of the iterative estimator but the computation time is approximately 180 times shorter the estimator is efficient and has advantages over alternate approaches such as iterative estimators

robust optimization for amplify and forward mimo relaying from a worst case perspective in this paper we consider robust optimization of amplify and forward af multiple input multiple output mimo relay precoders in presence of deterministic imperfect channel state information csi when the csi uncertainty lies in a norm bounded region two widely used performance metrics mutual information mi and mean square error mse are adopted as design objectives according to the philosophy of worst case robustness the robust optimization problems with respect to maximizing the worst case mi and minimizing the worst case mse are formulated as maximin and minimax problems respectively due to the fact that these two problems do not have a concave convex or convex concave structure we cannot rely on the conventional saddle point theory to find the robust solutions nevertheless by exploiting majorization theory we show that the formulated maximin and minimax problems both admit saddle points we further analytically characterize the saddle points and provide closed form solutions to robust relay precoder designs interestingly we find that under both mi and mse metrics the robust relay optimization leads to a channel diagonalizing structure meaning that eigenmode transmission is optimal from the worst case robustness perspective the proposed robust designs can improve the spectral efficiency and reliability of af mimo relaying against csi uncertainties at the similar cost of computational complexity as the existing non robust schemes

optimum linear joint transmit receive processing for mimo channels with qos constraints this paper considers vector communications through multiple input multiple output mimo channels with a set of quality of service qos requirements for the simultaneously established substreams linear transmit receive processing also termed linear precoder at the transmitter and linear equalizer at the receiver is designed to satisfy the qos constraints with minimum transmitted power the exact conditions under which the problem becomes unfeasible are given although the original problem is a complicated nonconvex problem with matrix valued variables with the aid of majorization theory we reformulate it as a simple convex optimization problem with scalar variables we then propose a practical and efficient multilevel water filling algorithm to optimally solve the problem for the general case of different qos requirements the optimal transmit receive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols for situations in which the resulting transmit power is too large we give the precise way to relax the qos constraints in order to reduce the required power based on a perturbation analysis we also propose a robust design under channel estimation errors that has an important interest for practical systems numerical results from simulations are given to support the mathematical development of the problem

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

bezout space time precoders and equalizers for mimo channels the transmitter receiver diversities in p input q output multiple input multiple output mimo channels will play a key role in future high rate wireless communication the major challenge in mimo signal recovery is the mitigation of the inevitable intersymbol interference isi and interchannel interference ici in multipath mimo channels the bezout system theory provides a simple solution for isi ici cancellation and thus can serve as a powerful mathematical foundation for mimo systems if p q and the mimo system h d is left coprime the system is perfectly recoverable via a bezout precoder assuming the channel information is available at the transmitter for a robust channel design a quantitative analysis of signal to noise ratio snr of the optimal bezout equalizer precoder can be derived from the bezout null space compared with the alamouti see ieee j select areas commun vol 16 p 1451 58 oct 1998 space time coding the bezout precoder is shown to be more appealing in slowly time varying channels where the feedback of the channel information induces minor overhead the bezout system can work well jointly with the space time block coding stbc the stbc can be adopted to artificially construct a larger q spl times p q p virtual transfer function without the channel information a necessary and sufficient condition is provided for perfect recoverability pr for the stbc induced virtual mimo systems based on the virtual transfer function both channel independent and channel dependent precoding strategies are feasible via a singular value decomposition svd analysis the bezout precoder can be shown to outperform the orthogonal frequency division multiplexing ofdm precoder in bit error rate ber transmission rate and receiver implementation the interplay of the two design parameters n and spl rho is analyzed to provide a simple guideline for an optimal configuration the combined stbc and bezout equalization techniques offer a broader spectrum of transceiver system configuration to achieve optimal design tradeoff among ber transmission rate and implementation complexity

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

robust transceiver optimization in downlink multiuser mimo systems we study robust transceiver optimization in a downlink multiuser wireless system where the transmitter and the receivers are equipped with antenna arrays the robustness is defined with respect to imperfect knowledge of the channel at the transmitter the errors in the channel state information are assumed to be bounded and certain quality of service targets in terms of mean square errors mses are guaranteed for all channels from the uncertainty regions iterative algorithms are proposed for the transceiver design the iterations perform alternating optimization of the transmitter and the receivers and have equivalent semidefinite programming representations with efficient numerical solutions the framework supports robust counterparts of several mse optimization problems including transmit power minimization with per user or per stream mse constraints sum mse minimization min max fairness etc although the convergence to the global optimum cannot be claimed due to the intricacy of the problems numerical examples show good practical performance of the presented methods we also provide various possibilities for extensions in order to accommodate a broader set of scenarios regarding the precoder structure the uncertainty modeling and a multicellular setup

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

on the number of elliptic pseudoprimes for an elliptic curve e with complex multiplication by an order in k q v d a point p of infinite order on e and any prime p with d i p 1 we have that p 1 p o mod p where 0 is the point at infinity and calculations are done using the addition law for e any composite number which satisfies these conditions is called an elliptic pseudoprime in this paper it is shown that assuming the generalized riemann hypothesis elliptic pseudoprimes are less numerous than primes in particular on the grh the number of elliptic pseudoprimes less than x is o x log log x log2 x for certain curves it is shown that infinitely many elliptic pseudoprimes exist

note on taking square roots modulo n in this article it is shown how gauss 1981 famous cyclotomic sum formula can be used for extracting square roots modulo n

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

worst case mse precoder design for imperfectly known mimo communications channels this paper considers a worst case precoder design problem for multiple input multiple output mimo wireless communication systems with imperfect channel knowledge at the transmitter when the mimo channel is a full row rank matrix which arises generically when the number n sub t of transmit antennas is greater or equal to the number n sub r of receive antennas and when channel state information is known perfectly at the transmitter the channel can be equalized exactly by employing a precoder equal to the channel pseudo inverse however in actual systems it is necessary to take into account the channel estimation error we consider here a worst case precoder design problem where the goal is to find the precoder minimizing the equalization mean square error for the least favorable channel located in a ball centered about the estimated channel lagrangian optimization is used to convert this min max problem into a min min convex minimization problem over a convex domain which can be solved in closed form the robust precoder and associated least favorable channel have an intuitive interpretation since the least favorable channel zeroes out the weakest subchannels and the robust precoder implements a pseudo inverse only for the remaining subchannels

using partial smoothness of p 1 for factoring polynomials modulo p let an arbitrarily small positive constant less than 1 and a poly nomial f with integer coefficients be fixed we prove unconditionally that j modulo p can be completely factored in deterministic polynomial time if p 1 has a lnp o 1 smooth divisor exceeding p we also address the issue of factoring f modulo p over finite extensions of the prime field f p and show that p 1 can be replaced by p k 1 k n for explicit classes of primes p

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

estimation of basis line integrals in a spectral distortion modeled photon counting detector using low rank approximation based x ray transmittance modeling k edge imaging application photon counting detectors pcds provide multiple energy dependent measurements for estimating basis line integrals however the measured spectrum is distorted from the spectral response effect sre via charge sharing k fluorescence emission and so on thus in order to avoid bias and artifacts in images the sre needs to be compensated for this purpose we recently developed a computationally efficient three step algorithm for pcd ct without contrast agents by approximating smooth x ray transmittance using low order polynomial bases it compensated the sre by incorporating the sre model in a linearized estimation process and achieved nearly the minimum variance and unbiased mvu estimator in this paper we extend the three step algorithm to k edge imaging applications by designing optimal bases using a low rank approximation to model x ray transmittances with arbitrary shapes i e smooth without the k edge or discontinuous with the k edge the bases can be used to approximate the x ray transmittance and to linearize the pcd measurement modeling and then the three step estimator can be derived as in the previous approach estimating the x ray transmittance in the first step estimating basis line integrals including that of the contrast agent in the second step and correcting for a bias in the third step we demonstrate that the proposed method is more accurate and stable than the low order polynomial based approaches with extensive simulation studies using gadolinium for the k edge imaging application we also demonstrate that the proposed method achieves nearly mvu estimator and is more stable than the conventional maximum likelihood estimator in high attenuation cases with fewer photon counts

worst case robust mimo transmission with imperfect channel knowledge in this paper we consider robust transmit strategies against the imperfectness of the channel state information at the transmitter csit for multi input multi output mimo communication systems following a worst case deterministic model the actual channel is assumed to be inside an ellipsoid centered at a nominal channel the objective is to maximize the worst case received signal to noise ratio snr or to minimize the worst case chernoff bound of the error probability thus leading to a maximin problem moreover we also consider the qos problem as a complement of the maximin design which minimizes the transmit power consumption and meanwhile keeps the received snr above a given threshold for any channel realization in the ellipsoid it is shown that for a general class of power constraints both the maximin and qos problems can be equivalently transformed into convex problems or even further into semidefinite programs sdps thus efficiently solvable by the numerical methods the most interesting result is that the optimal transmit directions i e the eigenvectors of the transmit covariance are just the right singular vectors of the nominal channel under some mild conditions this result leads to a channel diagonalizing structure as in the cases of perfect csit and statistical csit with mean or covariance feedback and reduces the complicated matrix valued problem to a scalar power allocation problem then we provide the closed form solution to the resulting power allocation problem

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

robust minimax mse equalizer designs for mimo wireless communications with time varying channel uncertainties in this paper we propose robust equalizers based on a minimax mean squares error mse scheme for wireless multi input multi output mimo communications subject to time varying channel uncertainties we consider channel uncertainties within a neighborhood of the estimated channel matrix formed by placing a bound on the spectral matrix norm of channel estimation errors first we derive a linear matrix inequality lmi based solution to the targeted problem next we modify the mse cost function of the targeted problem and derive a guaranteed cost based solution which may save computational cost subsequently channel uncertainty is partitioned into finite markov transitioned channel uncertainty states based on the least upper bound of the set of the tightest upper bounds on the matrix spectral norm of channel uncertainty this leads to a multiple model based minimax mse approach on this basis a feasible equalizer can be obtained from a weighted combination of multiple over guaranteed cost based equalizers each of which is designed with respect to a channel uncertain state simulation results show that a weighted combination of a moderate number of multiple over guaranteed cost based equalizers can achieve robust equalization effectively with superior mse and bit error rate ber performance

optimal transmitter eigen beamforming and space time block coding based on channel correlations optimal transmitter designs obeying the water filling principle are well documented and widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information csi may be costly or impossible to acquire in rapidly varying wireless environments we develop in this paper statistical water filling approaches for stationary random fading channels these approaches require only knowledge of the channel correlations that do not necessitate frequent updates and can be easily acquired applied to a multiple transmit antenna paradigm our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel s correlation matrix and with proper power loading across the beams the optimality pertains to minimizing a tight bound on the symbol error rate the resulting loaded eigen beamforming outperforms not only the equal power allocation across all antennas but also the conventional beamformer that transmits the available power along the strongest direction coupled with orthogonal space time block codes two dimensional 2 d eigen beamforming emerges as a more attractive choice than conventional one dimensional 1 d beamforming with uniformly better performance without rate reduction and without complexity increase

robust mmse precoding in mimo channels with pre fixed receivers in this paper we design robust precoders under the minimum mean square error mmse criterion for different types of channel state information csi in multiple input multiple output mimo channels we consider low complexity pre fixed receivers that may adapt to the channel but are oblivious to the existence of a precoder at the transmitter in particular three types of csi are taken into account i perfect csi ii statistical csi in the form of mean feedback and iii deterministic imperfect csi assuming that the actual channel is within the neighborhood of a nominal channel which leads to the worst case robust design that is the focus of this paper interestingly it is found that under some mild conditions the optimal transmit directions i e the left singular vectors of the precoder are equal to the right singular vectors of the channel the channel mean and the nominal channel for perfect csi statistical csi and the worst case design respectively consequently the matrix valued problems can be simplified to scalar power allocation problems that either admit closed form solutions or can be efficiently solved by the proposed algorithm

the distribution of lucas and elliptic pseudoprimes let l x denote the counting function for lucas pseudoprimes and e x denote the elliptic pseudoprime counting function we prove that for large x l x x l x 1 2 and e x x l x 1 3 where l x exp log xlogloglog x loglog x

counting points on hyperelliptic curves in average polynomial time let g 1 and let q2 z x be a monic squarefree polynomial of degree 2g 1 for an odd prime p not dividing the discriminant of q let zp t denote the zeta function of the hyperelliptic curve of genus g over the nite eld fp obtained by reducing the coecients of the equation y 2 q x modulo p we present an explicit deterministic algorithm that given as input q and a positive integer n computes zp t simultaneously for all such primes p n whose average complexity per prime is polynomial in g logn and the number of bits required to represent q