on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

systemes entiers d equations sur un alphabet fini et conjecture d ehrenfeucht the following conjecture has been posed by a ehrenfeucht for each language l over a finite alphabet there exists a finite subset of f of l such that for each pair of morphisms of into a free monoid the equality is true for all in l if and only if it is true for all in f such a finite subset f of l is called a test set of l a system s of equations over the alphabet is called entire if there exists a morphism of into a free monoid such that s 1 we have introduced for each morphism of in a free monoid such that is finite and 1 1 1 a generalized automaton m called the shuffle automaton of which recognizes a rational language k over a finite alphabet and a pair of morphisms of into such that 1 k nivat s 1966 characterization of rational relations is a motivation for this construction and implies that all entire systems over a finite alphabet are rational culik ii and salomaa 1978 have proved that every rational language over a finite alphabet admits a test set culik ii and karhumaki 1983 have shown that if s is a system of equations over a finite alphabet and if there exists a language l over a finite alphabet and morphisms and from into such that s l then if l admits a test set f then t f is a finite equivalent subsystem of s this gives an algorithm which determines for each entire system of equations over a finite alphabet a finite subsystem this result can be improved by using the following theorem spehner 1984 each finitely generated submonoid m of a free monoid has a finite m presentation i e there exists a finite alphabet and a finite part of such that m is isomorphic to the quotient monoid of by the smallest malcev congruence containing is a malcev congruence of if the quotient monoid is embeddable in a group the algorithm for determining the shuffle automaton m and the finite equivalent subsystem of s 1 works with a maximum efficiency because m can be embedded in the minimal automaton of the language k

on test sets for checking morphism equivalence on languages with fair distribution of letters abstract a test set for a language l is a finite subset t of l with the property that each pair of morphisms that agrees on t also agrees on l some results concerning test sets for languages with fair distribution of letters are presented the first result is that every d0l language with fair distribution of letters has a test set the second result shows that every language l with fair distribution has a test set relative to morphisms g h which have bounded balance on l these results are generalizations of results of culik ii and karhumaki 1983

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

blind adaptive multiuser detection the decorrelating detector and the linear minimum mean square error mmse detector are known to be effective strategies to counter the presence of multiuser interference in code division multiple access channels in particular those multiuser detectors provide optimum near far resistance when training data sequences are available the mmse multiuser detector can be implemented adaptively without knowledge of signature waveforms or received amplitudes this paper introduces an adaptive multiuser detector which converges for any initialization to the mmse detector without requiring training sequences this blind multiuser detector requires no more knowledge than does the conventional single user receiver the desired user s signature waveform and its timing the proposed blind multiuser detector is made robust with respect to imprecise knowledge of the received signature waveform of the user of interest

on the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate by michel broniatowski paul deheuvels and luc devroyeiuniversite pierre et marie curie universite pierre et marie curie andmcgill universitylet f be a density on the real line and let f be the kernel estimate of fin which the smoothing factor is obtained by maximizing the cross validatedlikelihood product according to the method of duin and habbema hermansand vandenbroek under mild regularity conditions on the kernel and f weshow among other things that f jf f 0 almost surely if and only if thesample extremes of f are strongly stable

asymptotically optimal cells for a historgram on examine les proprietes de l histogramme lorsque les cellules sont arbitraires on obtient une expression asymptotique pour l infimum de l erreur quadratique moyenne integree mise sur toutes les partitions

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

mmse interference suppression for direct sequence spread spectrum cdma we consider interference suppression for direct sequence spread spectrum code division multiple access cdma systems using the minimum mean squared error mmse performance criterion the conventional matched filter receiver suffers from the near far problem and requires strict power control typically involving feedback from receiver to transmitter for acceptable performance multiuser detection schemes previously proposed mitigate the near far problem but are complex and require explicit knowledge or estimates of the interference parameters in this paper we present and analyze several new mmse interference suppression schemes which have the advantage of being near far resistant to varying degrees depending on their complexity and can be implemented adaptively when interference parameters are unknown and or time varying numerical results are provided that show that these schemes offer significant performance gains relative to the matched filter receiver we conclude that mmse detectors can alleviate the need for stringent power control in cdma systems and may be a practical alternative to the matched filter receiver

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

binary equality sets are generated by two words we show that the equality set eq g h of two non periodic binary morphisms g h a is generated by at most two words if the rank of eq g h is two then and begin and end with different letters this in particular implies that any binary language has a test set of cardinality at most two

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

on polynomial based projection indices for exploratory projection pursuit we develop asymptotic theory for two polynomial based methods of estimating orientation in projection pursuit density approximation one of the techniques uses legendre polynomials and has been proposed and implemented by friedman 1 the other employs hermite functions issues of smoothing parameter choice and robustness are addressed it is shown that each method can be used to construct sqrt n consistent estimates of the projection which maximizes distance from normality although the former can only be employed in that manner when the underlying distribution has extremely light tails the former can be used very generally to measure low frequency departure from normality

blind multiuser detection a subspace approach a new multiuser detection scheme based on signal subspace estimation is proposed it is shown that under this scheme both the decorrelating detector and the linear minimum mean square error mmse detector can be obtained blindly i e they can be estimated from the received signal with the prior knowledge of only the signature waveform and timing of the user of interest the consistency and asymptotic variance of the estimates of the two linear detectors are examined a blind adaptive implementation based on a signal subspace tracking algorithm is also developed it is seen that compared with the previous minimum output energy blind adaptive multiuser detector the proposed subspace based blind adaptive detector offers lower computational complexity better performance and robustness against signature waveform mismatch two extensions are made within the framework of signal subspace estimation first a blind adaptive method is developed for estimating the effective user signature waveform in the multipath channel secondly a multiuser detection scheme using spatial diversity in the form of an antenna array is considered a blind adaptive technique for estimating the array response for diversity combining is proposed it is seen that under the proposed subspace approach blind adaptive channel estimation and blind adaptive array response estimation can be integrated with blind adaptive multiuser detection with little attendant increase in complexity

performance analysis of the linearly constrained constant modulus algorithm based multiuser detector this paper quantifies the adaptive performance of a blind adaptive multiuser detector mud based on the linearly constrained constant modulus algorithm lccma in both a stationary and nonstationary channel a framework is developed to apply the feedback analysis method to analyzing adaptive mud schemes a closed form expression for the excess mean square error emse of lccma blind adaptive mud in a cdma communications system is derived for both of the steady state and tracking cases the effects of additive noise and multiple access interference are considered a transient analysis is performed that predicts the learning curve of the adaptive filter computer simulation is used to verify the accuracy of the analysis

nonparametric estimation of a probability density on a riemannian manifold using fourier expansions supposing a given collection y 1 cdots y n of i i d random points on a riemannian manifold we discuss how to estimate the underlying distribution from a differential geometric viewpoint the main hypothesis is that the manifold is closed and that the distribution is sufficiently smooth under such a hypothesis a convergence arbitrarily close to the n 1 2 rate is possible both in the l 2 and the l infty senses

convergence analysis of chip and fractionally spaced lms adaptive multiuser cdma detectors this paper analyzes the convergence behavior of the least mean square lms filter when used in an adaptive code division multiple access cdma detector consisting of a tapped delay line with adjustable tap weights the sampling rate may be equal to or higher than the chip rate and these correspond to chip spaced cs and fractionally spaced fs detection respectively it is shown that cs and fs detectors with the same time span exhibit identical convergence behavior if the baseband received signal is strictly bandlimited to half the chip rate even in the practical case when this condition is not met deviations from this observation are imperceptible unless the initial tap weight vector gives an extremely large mean squared error mse this phenomenon is carefully explained with reference to the eigenvalues of the correlation matrix when the input signal is not perfectly bandlimited the inadequacy of the eigenvalue spread of the tap input correlation matrix as an indicator of the transient behavior and the influence of the initial tap weight vector on convergence speed are highlighted specifically a initialization within the signal subspace or to the origin leads to very much faster convergence compared with initialization in the a noise subspace