exceptional units in a family of quartic number fields we determine all exceptional units among the elements of certain groups of units in quartic number fields these groups arise from a one parameter family of polynomials with two real roots

a parametric family of quintic thue equations for an integral parameter t z we investigate the family of thue equations f x y x 5 t 1 2 x 4 y 2t 3 4t 4 x 3 y 2 t 4 t 3 2t 2 4t 3 x 2 y 3 t 3 t 2 5t 3 xy 4 y 5 1 originating from emma lehmer s family of quintic fields and show that for t 3 28 10 15 the only solutions are the trivial ones with x 0 or y 0 our arguments contain some new ideas in comparison with the standard methods for thue families which gives this family a special interest

robust data fusion for multisensor detection systems minimax robust data fusion schemes for multisensor detection systems with discrete time observations characterized by statistical uncertainty are developed and analyzed block sequential and serial fusion rules are considered the performance measures used and made robust with respect to the uncertainties include the error probabilities of the hypothesis testing problem in the block fusion case and the error probabilities and expected numbers of samples or sensors in the sequential and serial fusion cases for different sensor observation statistics the minimax robust fusion rules are derived for two asymptotic cases of interest when the number of sensors is large and when the number of times the fusion center collects the local sensor decisions is large moreover for the case of identical sensor observation statistics and a large number of sensors it is shown that there is no loss in optimality if local tests using likelihood ratios and equal thresholds are used in the sequential fusion rule in all situations the robust decision rules at the sensors and the fusion center are shown to make use of likelihood ratios and thresholds that depend on the least favorable probability distributions of the uncertainty class describing the statistics of sensor observations

klein forms and the generalized superelliptic equation if f x y 2 z x y is an irreducible binary form of degree k 3 then a theorem of darmon and granville implies that the generalized superelliptic equation f x y z l has given an integer l maxf2 7 kg at most nitely many solutions in coprime integers x y and z in this paper for large classes of forms of degree k 3 4 6 and 12 including heuristically most cubic forms we extend this to prove a like result where the parameter l is now taken to be variable in the case of irreducible cubic forms this provides the rst examples where such a conclusion has been proven the method of proof combines classical invariant theory modular galois representations and properties of elliptic curves with isomorphic mod n galois representations

note on a polynomial of emma lehmer recently emma lehmer constructed a parametric family of units in real quintic fields of prime conductor p t 5t i5t 25 25 as translates of gaussian periods later schoof and washington showed that these units were fundamental units in this note we observe that lehmer s family comes from the covering of modular curves x 25 y0 25 this gives a conceptual explanation for the existence of lehmer s units they are modular units which have been studied extensively by relating lehmer s construction with ours one finds expressions for certain gauss sums as values of modular units on y 25 1 lehmer s polynomial throughout the discussion we fix a choice of primitive th roots of

robust matched filters the problem of designing robust matched filters for situations in which there is uncertainty in the signal structure or noise statistics is considered two general aspects of this problem are treated first maximin robust designs are characterized for a general hilbert space formulation of the matched filtering problem and explicit solutions are given for two intuitively appealing models for uncertainty these results are seen to generalize earlier results for specific matched filtering problems the second general aspect treated is the application of the theoretical maximin results to the particular problem of designing filters to combat uncertain nonlinear channel distortion channel distortion is modeled by considering an unknown received signal which may differ in l 2 norm from the transmitted or nominal signal by no more than a fixed amount the effect of the channel distortion is seen to be equivalent to that of adding a white noise to the channel whose spectral height depends on the degree of distortion general expressions are developed for the determination of the robust filter and its performance and numerical results are presented for the case of baseband detection in gauss markov noise

on minimax robust detection of stationary gaussian signals in white gaussian noise the problem of detecting a wide sense stationary gaussian signal process embedded in white gaussian noise in which the power spectral density of the signal process exhibits uncertainty is investigated the performance of minimax robust detection is characterized by the exponential decay rate of the miss probability under a neyman pearson criterion with a fixed false alarm probability as the length of the observation interval grows without bound a stochastic suppression condition is identified for the uncertainty set of spectral density functions and it is established that under the stochastic suppression condition the resulting minimax problem possesses a saddle point which is achievable by the likelihood ratio tests matched to a so called suppressing power spectral density in the uncertainty set no convexity condition on the uncertainty set is required to establish this result

on the minimization of convex functionals of probability distributions under band constraints the problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below a system of sufficient and necessary first order optimality conditions as well as a bound on the optimality gap of feasible candidate solutions are derived based on these results two numerical algorithms are proposed that iteratively solve the system of optimality conditions on a grid of discrete points both algorithms use a block coordinate descent strategy and terminate once the optimality gap falls below the desired tolerance while the first algorithm is conceptually simpler and more efficient it is not guaranteed to converge for objective functions that are not strictly convex this shortcoming is overcome in the second algorithm which uses an additional outer proximal iteration and which is proven to converge under mild assumptions two examples are given to demonstrate the theoretical usefulness of the optimality conditions as well as the high efficiency and accuracy of the proposed numerical algorithms

a class of optimum importance sampling strategies abstract the problem of estimating expectations of functionals of random vectors via simulation is investigated as an alternative to simple averaging an efficient form of monte carlo simulations known as importance sampling can be employed to produce arbitrarily accurate estimates of the expectation of interest the fundamental problem in importance sampling is to determine the appropriate statistics under which to conduct the simulation these statistics are typically determined from a constraint class by minimizing the variance of the importance sampling estimator over the class of admissible statistics in previous work the authors showed that minimizing the importance sampling variance is equivalent to finding the distribution which is closest to the unconstrained optimal distribution with respect to a specific information theoretic f divergence or ali silvey distance in this work the authors determine the optimal biasing density from a constraint class whose controlling parameter is fundamental in the performance analysis of importance sampling in addition it is shown that the constrained optimal distribution from this class minimizes every statistical distance measure to the global optimal distribution and as a consequence is optimal with respect to a large family of nonlinear cost functions salient features of this distribution are 1 unlike the unconstrained optimal solution this distribution can be made independent of the parameter to be estimated and thus can admit to implementation and 2 this distribution renders performance gains which can be made arbitrarily close to the optimal gains applications to estimating probabilities of rare events e g error rates in communication systems will be presented further analysis will show that in this case the savings over monte carlo simulations become unbounded as the probability of the rare event diminishes

robust wiener filtering for multiple inputs with channel distortion corresp robust wiener filtering has previously been considered for the single input scalar case where there is no channel distortion and where the signal to be estimated is the source signal itself here these results are extended to the multiple input vector case where linear channel distortion is allowed and the signal to be estimated is a linear filtered version of the source signal the results are obtained from those for the single input ease by modifying the constraints on signal and noise characteristics such a modification is motivated by examining the expression of the mean squared error for the optimum filter

connection between gaussian periods and cyclic units this paper finds that all known parametric families of units in real quadratic cubic quartic and sextic fields with prime conductor are linear combinations of gaussian periods and exhibits these combinations this approach is used to find new units in the real quintic field for prime conductors p n4 5n3 15n2 25n 25

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

minimax mse estimation of deterministic parameters with noise covariance uncertainties in this paper a minimax mean squared error mse estimator is developed for estimating an unknown deterministic parameter vector in a linear model subject to noise covariance uncertainties the estimator is designed to minimize the worst case mse across all norm bounded parameter vectors and all noise covariance matrices in a given region of uncertainty the minimax estimator is shown to have the same form as the estimator that minimizes the worst case mse over all norm bounded vectors for a least favorable choice of the noise covariance matrix an example demonstrating the performance advantage of the minimax mse approach over the least squares and weighted least squares methods is presented

uncertainty bounds for spectral estimation the purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second order statistics the family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the true power spectrum our aim is to quantify the size of this uncertainty set using suitable notions of distance and in particular to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the true power spectrum since the uncertainty set may contain power spectra with lines and discontinuities it is natural to quantify distances in the weak topology the topology defined by continuity of moments we provide examples of such weakly continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty we then consider certain high resolution techniques which utilize filter banks for preprocessing and compute worst case a priori uncertainty bounds solely on the basis of the filter dynamics this allows the a priori tuning of the filter banks for improved resolution over selected frequency bands

computing all power integral bases of cubic fields applying baker s effective method and the reduction procedure of baker and davenport we present several lists of solutions of index form equations in totally real and complex cubic algebraic number fields these solutions yield all power integral bases of these fields 1 introduction let k be a cubic algebraic number field and denote by zk the ring of integers of k a power integral basis of k is an integral basis of the form 1 2 with some a e zk if there exists such an a then we say that k is monogenic since zk z e obviously if a has this property then it holds also for a k with any k e z from a practical point of view it is important to know whether there exists a power integral basis of k and if so what are the numerical values of a if the galois group of k is cyclic then the discriminant of k is a full square the problem of monogeneity in cyclic cubic fields was considered by m n gras 12 13 archinard 1 and dummit and kisilevsky 4 m n gras and archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic moreover dummit and kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases for arbitrary algebraic number fields l it was proved by gyory 14 that up to obvious translations by elements of z there are only finitely many a e zl with zl z and he gave effective but rather large bounds for the sizes of these a in this paper we present a method which allows us to determine all possible values of a up to translation with rational integers such that 1 2 is a power integral

robust sequential detection in distributed sensor networks we consider the problem of sequential binary hypothesis testing with a distributed sensor network in a non gaussian noise environment to this end we present a general formulation of the consensus innovations sequential probability ratio test mathcal ci sprt furthermore we introduce two different concepts for robustifying the mathcal ci sprt and propose four different algorithms namely the least favorable density mathcal ci sprt the median mathcal ci sprt the m mathcal ci sprt and the myriad mathcal ci sprt subsequently we analyze their suitability for different binary hypothesis tests before verifying and evaluating their performance in a shift in mean and a shift in variance scenario for different network connectivities and amounts of noise contamination

a family of cyclic quartic fields arising from modular curves we study a family of cyclic quartic fields arising from the covering of modular curves x1 16 xo 16 an integral basis and a fundamental system of units are found it is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field recently 0 lecacheux 9 10 and h darmon 4 showed how to use coverings of modular curves to obtain cyclic extensions of q in particular they were able to give a geometric construction of a family of cyclic quintic fields discovered by e lehmer 11 the covering x1 n x0 n for n 2 has degree q n 2 and group z nz x i 1 for the quintic case they took n 25 which gave a cyclic covering of degree 10 then took the subcovering of degree 5 an important ingredient in the construction was the fact that xo 25 has genus 0 this also occurs for n 1 10 12 13 16 18 these all give trivial or quadratic coverings except for n 7 9 13 16 18 the values n 7 9 18 yield cubic extensions and can be shown to yield the family of polynomials x3 ax2 a 3 x 1 namely the simplest cubic fields 17 however it should be remarked that every cyclic cubic extension of q comes from a polynomial of this form if a is allowed to be rational similarly we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form x2 ax 1 with a rational the case n 13 is treated by lecacheux 9 it might be suspected that the sextic fields she obtains are the same as the simplest sextics constructed by m n gras 6 however these latter fields were found by taking the fixed field of an element of order 6 in pgl2 q aut q x therefore they come from a covering of curves of genus 0 but x1 13 has genus 2 alternatively these sextic fields must be different because the discriminants of the quadratic and cubic subfields are different in the present paper we study the case n 16 as above it might be hoped that this case would give a geometric construction of the quartic fields studied received march 14 1990 revised october 22 1990 1980 mathematics subject classification 1985 revision primary 1 i r 16 research supported in part by n s f 1991 american mathematical society 0025 5718 91 1 00 25 per page

block interpolation a framework for tight exponential time counting complexity we devise a framework for proving tight lower bounds under the counting exponential time hypothesis mathsf eth introduced by dell et al our framework allows to convert many known mathsf p hardness results for counting problems into results of the following type if the given problem admits an algorithm with running time 2 o n on graphs with n vertices and mathcal o n edges then mathsf eth fails as exemplary applications of this framework we obtain such tight lower bounds for the evaluation of the zero one permanent the matching polynomial and the tutte polynomial on all non easy points except for two lines

complete solutions to families of quartic thue equations using a method due to e thomas we prove that if lai 9 9 1027 then the diophantine equations 4 3 2 2 3 4 x ax y x y axy y and 4 3 2 2 3 4 x ax y 3x y axy y i1 have exactly twelve solutions namely x y 0 1 1 0 1 1 f 1 1 a 1 l fa and eight solutions x y o 1 i 0 i 1 i f1 respectively

optimal and robust memoryless discrimination from dependent observations discrimination is considered between two possible sources based on dependent observations of their output the discrimination problem is modeled by means of a general binary hypothesis test the main emphasis being on situations that cannot be modeled as signals in additive noise the observations are modeled as stationary m dependent or rho mixing processes the structure of the discriminator is such that the observations are passed through a memoryless nonlinearity summed up to form a test statistic which is then compared to a threshold only fixed sample size tests are considered four different performance measures which resemble the signal to noise ratios encountered in the signal in additive noise problems are derived under different problem formulations the optimal nonlinearities for each of the performance measures are derived as solutions to various integral equations for three of the four performance measures the authors have successfully obtained robust nonlinearities for uncertainty in the marginal and the joint probability density functions of the observations computer simulation results that demonstrate the advantage of using these nonlinearities over the i i d nonlinearity under the probability of error criterion are presented

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

signal detection under mismatch corresp a binary detection problem of the neymann pearson type in which the probability density functions used are inaccurate versions of the true ones are considered the performance of the above suboptimal detection scheme as the number of observations increases is investigated a necessary and sufficient condition is given for the exponential convergence to zero of the two error probabilities as the number of observations increases the condition is in terms of an inequality between differences of asymptotic per sample informational divergence expressions

on the solutions of a family of quartic thue equations in this paper we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields in fact we prove that for n 5 10 6 and n n 1 191 x 10 19 with n n 2 n 2 4 square free the thue equation n x y x 4 n 2 x 3 y n 3 2n 2 4n 2 x 2 y 2 n 2 xy 3 y 4 1 has no integral solution except the trivial ones 1 0 1 0 0 1 0 1

units in some families of algebraic number fields multi dimensional continued fractions associated with gl n z are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature 2 1

minimax robust coding for channels with uncertainty statistics the problem of minimax robust coding for classes of channels with uncertainty in their statistical description is addressed specific consideration is given to 1 discrete memoryless channels with uncertainty in the probability transition matrices 2 discrete time stationary gaussian channels with spectral uncertainty and to uncertainty with classes determined by 2 alternating choquet capacities both block codes and convolutional codes are considered a robust maximum likelihood decoding rule is derived the rule guarantees that for all channels in the uncertainty class and all rates smaller than a critical rate the average probability of decoding error for the ensemble of random block codes and the ensemble of random time varying convolutional codes converges to zero exponentially with increasing block length or constraint length respectively the channel capacity and cut off rate of the class are then evaluated

the rate distortion function on classes of sources determined by spectral capacities the quantity sup a in cal r r a d is considered where cal r is a class of homogeneous n parameter sources and r a d denotes the single letter mean square error mse rate distortion function for the individual source a in particular the case in which the class cal r is specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities in the sense of choquet alternating of order two this type of class includes many common models for spectral uncertainty such as mixture models spectral band models and neighborhoods generated by variation and prohorov metrics it is shown that each such class contains a worst case source whose rate distortion function achieves the supremum over the class for each value of distortion this source is characterized as having a spectral density that is a derivative in the sense of huber and strassen of the upper spectral measure with respect to lebesgue measure on pi pi n moreover it is shown that the spectral measure of the worst case source is closest in a sense defined by directed divergence to lebesgue measure which corresponds to a memoryless source numerical results are presented for the particular case in which the source spectral measure is a mixture of a gauss markov spectrum and an unknown contaminating component

robust competitive estimation with signal and noise covariance uncertainties robust estimation of a random vector in a linear model in the presence of model uncertainties has been studied in several recent works while previous methods considered the case in which the uncertainty is in the signal covariance and possibly the model matrix but the noise covariance is assumed to be completely specified here we extend the results to the case where the noise statistics may also be subjected to uncertainties we propose several different approaches to robust estimation which differ in their assumptions on the given statistics in the first method we assume that the model matrix and both the signal and the noise covariance matrices are uncertain and develop a minimax mean squared error mse estimator that minimizes the worst case mse in the region of uncertainty the second strategy assumes that the model matrix is given and tries to uniformly approach the performance of the linear minimum mse estimator that knows the signal and noise covariances by minimizing a worst case regret measure the regret is defined as the difference or ratio between the mse attainable using a linear estimator ignorant of the signal and noise covariances and the minimum mse possible when the statistics are known as we show earlier solutions follow directly from our more general results however the approach taken here in developing the robust estimators is considerably simpler than previous methods

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

on the robust discrimination of poisson random counting measures corresp statistical decisions are considered between two hypotheses which consist of classes of poisson distributions for random counting measures each poisson distribution is generated by an intensity measure on a general observation space the classes are specified by choquet capacity bounds on the intensity measures this problem was first posed and studied by geraniotis and poor a minimax neyman pearson result for error probability performance is the main contribution recursive computation of the minimax test statistic is also investigated

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

robust deconvolution of deterministic and random signals we consider the problem of designing an estimation filter to recover a signal x n convolved with a linear time invariant lti filter h n and corrupted by additive noise our development treats the case in which the signal x n is deterministic and the case in which it is a stationary random process both formulations take advantage of some a priori knowledge on the class of underlying signals in the deterministic setting the signal is assumed to have bounded weighted energy in the stochastic setting the power spectra of the signal and noise are bounded at each frequency the difficulty encountered in these estimation problems is that the mean squared error mse at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized beginning with the deterministic setting we develop a minimax mse estimation filter that minimizes the worst case point wise mse between the true signal x n and the estimated signal over the class of bounded norm inputs we then establish that the mse at the output of the minimax mse filter is smaller than the mse at the output of the conventional inverse filter for all admissible signals next we treat the stochastic scenario for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties this filter is designed to minimize the worst case difference between the mse in the presence of power spectrum uncertainties and the mse of the wiener filter that knows the correct power spectra the minimax regret filter takes the entire uncertainty interval into account and as demonstrated through an example can often lead to improved performance over traditional minimax mse approaches for this problem

power spectral density bounds corresp the determination of a power density spectrum from a finite set of correlation samples is an ill posed problem furthermore it is not possible even to bound the values that consistent power density spectra can take on at a particular point a more reasonable problem is to try to determine the total spectral power in some frequency interval although this power cannot be determined exactly upper and lower bounds on its possible values can be determined this observation leads to a unified treatment of certain classical and modern spectral estimation techniques and to new interpretations for two data adaptive spectral estimators maximum likelihood method mlm and data adaptive spectral estimator dase according to these new interpretations mlm and dase provide upper bounds on spectral power in a specified frequency region subject to the assumption that the spectral density is constant in that region these methods make no use of an extendibility constraint that can be used to obtain tight upper bounds as well as nontrivial lower bounds on power cybenko has studied a related problem of bounding windowed power for an arbitrary window with no assumptions about the form of the spectral density a new type of classical resolution limit for these bounds is derived and a numerical example is presented

on the method of thue siegel the methods of thue and siegel based on explicit pade approximations to algebraic functions are used to examine diophantine approximations to roots of

an adaptive control algorithm for linear systems having unknown time delay abstract this paper presents an algorithm for adaptive control of linear systems having unknown time delay a rational approximation is used for the delay and it is argued that within this framework the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics a global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms the analysis is supported by simulation results

robust matched filters for noise uncertainty within two alternating capacity classes the problem of designing matched filters that are robust against uncertainty in the statistics of a noise process modeled by two alternating choquet capacities is addressed the robust design is based on the maximum signal to noise ratio the problem is formulated and solved for both discrete time and continuous time matched filters with uncertainty in either the autocorrelation function or the spectral measure of the noise explicit solutions that are characterized by the hyber strassen derivative of the capacity generating the class with respect to a lebesgue like measure on a suitable interval are obtained

bounds from noisy linear measurements an estimation problem in which a finite number of linear measurements of an unknown function is available and in which the only prior information available concerning the unknown function consists of inequality constraints on its magnitude is ill posed in that insufficient information is available from which point estimates of the unknown function can be made with any reliability even with exact measurements an alternative to point estimation involves the computation of bounds on linear functionals of the unknown function in terms of the measurements a generalization is described of the bounding technique to problems in which the measurements are inexact the bounds are defined in terms of a primal optimization problem a deterministic interpretation of the bounds is given as well as a probabilistic one for the case of additive gaussian measurement noise an unconstrained dual optimization problem is derived that has an interesting data adaptive filtering interpretation and provides an attractive basis for computation several aspects of the primal and dual optimization problems are investigated that have important implications for the reliable computation of the bounds

a new interpretation of capon s maximum likelihood method of frequency wavenumber spectral estimation the maximum likelihood method of frequency wavenumber spectral estimation mlm is a high resolution alternative to conventional fourier analysis this paper provides a new interpretation for mlm the point of view taken is that at each wavenumber we attempt to estimate the power of a discrete planewave at that wavenumber a problem complicated by the presence of interfering energy at other wavenumbers noise with an unknown spectral density matrix it is shown that subject to some assumptions the problem of choosing the variance of the planewave complex amplitude that is the plane wave power and the noise covariance matrix jointly to maximize the likelihood function for the observations has no unique solution there is a continuous range of values for the estimates all of which have the same maximum likelihood the lower bound on the planewave power estimate being zero and the upper bound being the mlm power estimate some important differences between the problem of estimating the planewave complex amplitude for each independent data segment and the problem of estimating the variance of the planewave complex amplitude are indicated

minimax discrimination for observed poisson processes with uncertain rate functions the problem of robust design is considered in the context of testing hypotheses concerning the rate function of an observed point process designs that are insensitive to uncertainty in the rate functions are developed by applying a minimax formulation to two different measures of signal to noise ratio uncertainty in the rate is modeled by using general classes of rate measures generated by choquet 2 alternating capacities and solutions are characterized for this case by a radon nikodym type derivative between such classes it is shown that for uncertainty within capacity classes the robust decision design developed for the signal to noise ratio is also robust in a weaker sense for the chernoff upper bounds on the error probabilities furthermore the use of such a test guarantees the exponential convergence of these bounds to zero with increasing length of the observation interval for all rates in the uncertainty class

the computational complexity of tutte invariants for planar graphs for each pair of algebraic numbers x y the complexity of computing the tutte polynomial t g x y of a planar graph g is determined this computation is found to be overline rm p complete except when x 1 y 1 1 2 or when x y is one of 1 1 1 1 j j 2 or j 2 j where j e 2 pi i 3 in which case it is polynomial time computable a corollary gives the computational complexity of various enumeration problems for planar graphs

on mordell s equation y2 k x3 a problem of stolarsky on determine l ensemble complet des solutions en nombres entiers de l equation de stolarsky y 2 999 x 3

approximating the partition function of the ferromagnetic potts model we provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q state potts model when q 2 specifically we show that the partition function is hard for the complexity class rhpi under approximation preserving reducibility thus it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems including that of determining the number of independent sets in a bipartite graph our proof exploits the first order phase transition of the random cluster model which is a probability distribution on graphs that is closely related to the q state potts model

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

monte carlo methods for the ferromagnetic potts model using factor graph duality normal factor graph duality offers new possibilities for monte carlo algorithms in graphical models specifically we consider the problem of estimating the partition function of the ferromagnetic ising and potts models by monte carlo methods which are known to work well at high temperatures but to fail at low temperatures we propose monte carlo methods uniform sampling and importance sampling in the dual normal factor graph and demonstrate that they behave differently they work particularly well at low temperatures by comparing the relative error in estimating the partition function we show that the proposed importance sampling algorithm significantly outperforms the state of the art deterministic and monte carlo methods for the ferromagnetic ising model in an external field we show the equivalence between the valid configurations in the dual normal factor graph and the terms that appear in the high temperature series expansion of the partition function following this result we discuss connections with jerrum sinclair s polynomial randomized approximation scheme the subgraphs world process for evaluating the partition function of ferromagnetic ising models

robust large deviations performance analysis for large sample detectors large deviations theory is used to analyze the exponential rate of decrease of error probabilities for a sequence of decisions based on a test statistics sequence t sub n it is assumed that for a given statistical hypothesis the distributions of t sub n are determined by some unknown member of a class of probability distributions the worst case or least favorably exponential rate of error probability decrease over this class is sought it is shown that the legendre fenchel transform of the maximized cumulant function yields a lower bound for the minimized large deviations rate function and that in many cases this bound is tight application of the result is illustrated by a detailed consideration of i i d memoryless detection with an epsilon contamination distribution family

robust coding for multiple access channels the problem of minimax robust coding for classes of multiple access channels with uncertainty in their statistical description is addressed we consider 1 discrete memoryless multiple access channels with uncertainty in the probability transition matrices and 2 discrete time stationary additive gaussian multiple access channels with spectral uncertainty the uncertainty is modeled using classes determined by two alternating choquet capacities both block codes and tree codes are considered a robust maximum likelihood decoding rule is derived which guarantees that for ali two user channels in the uncertainty class and all pairs of code rates in a critical rate region the average probability of decoding error for the ensemble of pairs of random block codes and the ensemble of pairs of random tree codes converges to zero exponentially with increasing block length or constraint length respectively the channel capacity and cutoff rate regions of the class are then evaluated

complete solutions of a family of quartic thue and index form equations continuing the recent work of the second author we prove that the diophantine equation f a x y x 4 ax 3 y x 2 y 2 axy 3 y 4 1 for a 3 has exactly 12 solutions except when a 4 when it has 16 solutions if a denotes one of the zeros of f a x 1 then for a 4 we also find all z a with z z

robust matched filters for optical receivers the problem of designing optical receivers that are robust against uncertainty in the statistics of the observation process in photodetection is investigated in particular a modification in the design of the postdetection matched filter is proposed to account for possible uncertainty in the rate function of the incident light the rate of the dark current and in the statistics of the additive noise present at the input to the optical receiver this design is based on a game theoretic approach in which a filter is sought that has the maximum worst case output signal to noise ratio possible over the class of allowable statistics that is the design criterion is maximin signal to noise ratio a general characterization of maximin robust matched filters for observed poisson processes is presented in this context and specific solutions for several useful uncertainty models are obtained numerical results are presented for a specific example to illustrate the performance of the proposed technique

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

effective irrationality measure for certain algebraic numbers a result of chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given more particularly pade approximants to the function 1 x 113 are employed to show for certain integers a and b that j a b 1 3 p q cq k when q 0 here c and k are given as functions of a and b only in 1964 baker 1 improving a technique used by siegel 8 was able to obtain effective irrationality measures for the function 1 x m n evaluated at certain rational points in particular he was able to show that for integers p q we have 1 121 3 p q 10 6q 2955 when q 0 the technique was further refined by chudnovsky 2 whose results when applied to 21 3 imply that for any 0 there exists a positive integer qo c such that for integers p q we have 2 121 3 p qi q 2 429 e when q q0 chudnovsky s result is effective in the sense that it is possible in principle to work through the proof and compute for any particular value of a qo c for which 2 holds however chudnovsky does not undertake such computations in this article we rework baker s proof using chudnovsky s refinement together with a chebyshev type result for primes in arithmetical progressions due to mccurley 6 and obtain the following quantitative result theorem let a b be integers with 0 b a define d by 0 if 3 4 a b 3 d i1 if 311 a b 13 2 otherwise further define a k c and q0 by 4 x 2328 3d al1 2 b1 2 2 received may 9 1985 1980 mathematics subject classification primary iof25 1986 american mathematical society 0025 5718 86 1 00 25 per page 613 this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 36 01 utc all use subject to http about jstor org terms

on the p point uncertainty class corresp the p point uncertainty class also known as sakrison s class b is considered although the p point class is not a 2 alternating choquet capacity class it is shown how many of the robust signal processing and communication theory results which have been developed for capacity classes can be applied to the p point class by embedding it in a slightly larger extended p point class

primes in arithmetic progressions strengthening work of rosser schoenfeld and mccurley we establish explicit chebyshev type estimates in the prime number theorem for arithmetic progressions for all moduli k 72 and other small moduli

the complexity of computing the sign of the tutte polynomial we study the complexity of computing the sign of the tutte polynomial of a graph as there are only three possible outcomes positive negative and zero this seems at first sight more like a decision problem than a counting problem surprisingly however there are large regions of the parameter space for which computing the sign of the tutte polynomial is actually p hard as a trivial consequence approximating the polynomial is also p hard in this case thus approximately evaluating the tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a cnf boolean formula for most other points in the parameter space we show that computing the sign of the polynomial is in fp whereas approximating the polynomial can be done in polynomial time with an np oracle as a special case we completely resolve the complexity of computing the sign of the chromatic polynomial this is easily computable at q 2 and when q leq 32 27 and is np hard to compute for all other va

brief paper an analysis of the effects of spectral uncertainty on wiener filtering a representative selection of results from an extensive study of the performance of wiener filtering under spectral uncertainty is presented for a variety of spectral uncertainty models the wiener filter is shown to have undesirable sensitivity to even small deviations from those signal and noise spectral densities which were used to design the filter performance of a robust filter designed to have the best possible performance when the uncertainty is worst is also examined in most cases the robust filter s insensitivity to spectral uncertainty makes it preferable to the traditional wiener filter

minimax linear smoothing for capacities the case of correlated signals and noise results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties are extended to the case in which the signal and noise fields are arbitrarily correlated as before spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two alternating choquet capacities it is demonstrated that this problem admits a general solution in terms of the huber strassen derivative between the capacities that generate the uncertainty sets and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise the resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in nonparametric minimax hypothesis testing these new results extend the applicability of minimax robust smoothing techniques to application involving signal dependent noise phenomena such as multipath and clutter which are usually difficult to model precisely

constrained solutions in importance via robust statistics the problem of estimating estimating expectations of functions of random vectors via simulation is investigated monte carlo simulations also known as simple averaging have been used as a direct means of estimation a technique known as importance sampling can be used to modify the simulation via weighted averaging in the hope that the estimate will converge more rapidly to the expected value than standard monte carlo simulations a constrained optimal solution to the problem of minimizing the variance of the importance sampling estimator is derived this is accomplished by finding the distribution which is closest to the unconstrained optimal solution in the ali silvey sense s ali et al 1966 the solution from the constraint class is shown to be the least favorable density function in terms of bayes risk against the optimal density function examples of constraint classes which include epsilon mixture show that the constrained optimal solution can be made arbitrarily close to the optimal solution applications to estimating probability of error in communication systems are presented

power integral bases in a parametric family of totally real cyclic quintics we consider the totally real cyclic quintic fields k n q v n generated by a root v n of the polynomial f n x x 5 n 2 x 4 2n 3 6n 2 10n 10 x 3 n 4 5n 3 11n 2 15n 5 x 2 n 3 4n 2 10n 10 x 1 assuming that m n 4 5n 3 15n 2 25n 25 is square free we compute explicitly an integral basis and a set of fundamental units of k n and prove that k n has a power integral basis only for n 1 2 for n 1 2 both values presenting the same field all generators of power integral bases are computed

quintic polynomials and real cyclotomic fields with large class numbers we study a family of quintic polynomials discoverd by emma lehmer we show that the roots are fundamental units for the corresponding quintic fields these fields have large class numbers and several examples are calculated as a consequence we show that for the prime p 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401 in an appendix we give a characterization of the simplest quadratic cubic and quartic fields

robust wiener filters for random signals in correlated noise corresp minimax robust wiener filtering is considered for the case in which the signal and noise spectral density matrix is not completely specified results are obtained for spectral density matrix classes which are defined by upper and lower bounds on the components of the matrix these results form an extension of earlier results on robust wiener filtering for the case of uncorrelated signals and noise

minimax robust discrete time matched filters the problem of designing finite length discrete time matched filters is considered for situations in which exact knowledge of the input signal and or noise characteristics is not available such situations arise in many applications due to channel distortion incoherencies nonlinear effects and other modeling uncertainties in such cases it is often of interest to design a minimax robust matched filter i e a nonadaptive filter with an optimum level of worst case performance for the expected uncertainty class this problem is investigated here for three types of uncertainty models for the input signal namely the mean absolute mean square and maximum absolute distortion classes and for a wide generality of norm deviation models for the noise covariance matrix some numerical examples illustrate the robustness properties of the proposed designs

minimax linear smoothing for capacities abstract minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field a minimax game with the quadratic mean estimation error as an objective function is used to formulate this problem uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by huber and strassen for the problem of minimax hypothesis testing these classes which are described in terms of choquet alternating capacities of order 2 include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem author

robust linear filtering for multivariable stationary time series we consider the problem of asymptotic noncausal linear filtering for multivariable second order stationary time series under spectral uncertainty in both the signal and the noise processes the spectral uncertainty is modeled by e contaminated and p point classes for the case where both the signal and noise spectra are e contaminated we assume that the eigenvectors of the corresponding nominal spectral density matrices are identical the problem is formulated as a game whose saddle point solutions are found and analyzed

old bands new tracks revisiting the band model for robust hypothesis testing the density band model proposed by kassam 1 for robust hypothesis testing is revisited first a novel criterion for the general characterization of least favorable distributions is proposed that unifies existing results this criterion is then used to derive an implicit definition of the least favorable distributions under band uncertainties in contrast to the existing solution it only requires two scalar values to be determined and eliminates the need for case by case statements based on this definition a generic fixed point algorithm is proposed that iteratively calculates the least favorable distributions for arbitrary band specifications finally three different types of robust tests that emerge from band models are discussed and a numerical example is presented to illustrate their potential use in practice

performance bounds for discrimination problems with uncertain statistics corresp decision designs that are insensitive to modeling uncertainty are developed for the chernoff bounds on the performance of binary hypothesis testing problems these designs are based on observations with statistical uncertainty modeled by using general classes generated by 2 alternating capacities the results are illustrated for the two cases of independent identically distributed observations with uncertainty in the probability distribution and discrete time stationary gaussian observations with spectral uncertainty and they are applicable to several other cases as well for the chernoff upper bounds on the error probabilities a robust decision design based on the i kel hood ratio test between a least favorable pair of probability distributions or spectral measures respectively is derived it is then shown that for all elements in the uncertainty class this choice of likelihood ratio guarantees the exponential convergence of the aforementioned chernoff bounds to zero as the number of observations or the length of the observation interval increases

the solution of triangularly connected decomposable form equations an algorithm is given to solve the equations of the title it generalizes an earlier algorithm to solve discriminant form equations an application is given to finding curves of genus 2 with good reduction outside a given finite set of primes and weierstrass points in given number fields

robust eckart filters for time delay estimation minimax or robust filters which optimize worst case performance have been considered previously for matched and wiener filtering in this paper we show how similar ideas can be applied to obtain robust versions of one particular filter the eckart filter for time delay estimation the use of a robust filter can eliminate the requirement of estimation of the signal and noise power spectral densities psd s and cross psd s robust filters being less sensitive in their performance to variations of the input psd s within specified classes of psd s explicit solutions for specific classes of psd s are derived and several numerical examples and simulation results are given similar formulations for robust filters should also be possible for other types of filters used in time delay estimation

on the maximum entropy method for interval covariance sequences given an interval covariance sequence the existence of a maximum entropy spectral estimate is considered it is shown that a maximum entropy spectral estimate does exist and moreover it is unique

spectral estimation for sensor arrays the array processing problem is briefly discussed and an abstract spectral estimation problem is formulated this problem involves the estimation of a multidimensional frequency wave vector power spectrum from measurements of the correlation function and knowledge of the spectral support the investigation of correlation matching spectral estimates leads to the extendibility question does there exist any positive spectrum on the spectral support that exactly matches a given set of correlation samples in answering this question a mathematical framework is developed in which to analyze and design spectral estimation algorithms pisarenko s method of spectral estimation which models the spectrum as a sum of impulses plus a noise component is extended from the time series case to the more general array processing case pisarenko s estimate is obtained as the solution of a linear optimization problem which can be solved using a linear programming algorithm such as the simplex method