on the construction of good fully symmetric integration rules in this paper we review some classical theory of moments for polynomials this theory turns out to be a useful tool when we want to construct good fully symmetric integration rules fsg rules for fully symmetric regions we also show how to apply this theory by constructing several new fsg rules of degree 7 9 and 11 for the 3 cube

sequential testing of sorted and transformed data as an efficient way to implement long glrts it is often required to detect a long weak signal in gaussian noise and frequently the exact form of that signal is parameterized but not known a bank of matched filters provides an appropriate detector however in some practical applications there are very many matched filters and most are quite long the consequent computational needs may render the classical bank of filters approach infeasibly expensive one example and our original motivation is the detection of chirp gravitational waves by an earth based interferometer in this paper we provide a computational approach to this problem via sequential testing since the sequential tests to be used are not for constant signals we develop the theory in terms of average sample number asn for this case specifically we propose two easily calculable expressions for the asn one a bound and the other an approximation the sequential approach does yield moderate computational savings but we find that by preprocessing the data using short medium fast fourier transforms ffts and an appropriate sorting of these fft outputs such that the most informative samples are entered to a sequential test first quite high numerical efficiency can be realized the idea is simple but appears to be quite successful examples are presented in which the computational load is reduced by several orders of magnitude the fft is an example of an energy agglomerating transform but of course there are many others the point here is that the transform need not match the sought signal exactly in the sense that all energy becomes confined to a single sample it is enough that the energy becomes concentrated and the more concentrated the better

economical symmetrical quadrature rules for complete polynomials over a square domain it is of interest in numerical analysis to develop symmetrical quadrature rules for integration of complete polynomial functions over a square domain with minimum computational effort gaussian product quadrature rules integrate such functions with maximum effort symmetrical quadrature rules are developed and presented for integration of complete polynomial functions up to 21st order with minimum computational effort

on the identification of symmetric quadrature rules for finite element methods in this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals triangles tetrahedra prisms pyramids and hexahedra the methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points we also present polyquad which is an implementation of our methodology using polyquad v1 0 we proceed to derive a complete set of symmetric rules on the aforementioned domains all rules possess purely positive weights and have all points inside the domain many of the rules appear to be new and an improvement over those tabulated in the literature

c1 weierstrass for compact sets in hilbert space abstract the c 1 weierstrass approximation theorem is proved for any compact subset x of a hilbert space h the same theorem is also proved for whitney 1 jets on x when x satisfies the following further condition there exist finite dimensional linear subspaces h 1 h 2 h such that n 1 h n is dense in span x and n x x h n for each n 1 here n h h n is the orthogonal projection it is also shown that when x is compact convex with span x h and satisfies the above condition then c 1 x is complete if and only if the c 1 whitney extension theorem holds for x finally for compact subsets of h an extension of the c 1 weierstrass approximation theorem is proved for c 1 maps h h with compact derivatives

the topology of balls in riemannian surfaces and gromov hyperbolicity for each k 0 we find an explicit function f k such that the topology of s inside the ball b s p r is bounded by f k r for every complete riemannian surface compact or non compact s with k ge k 2 every p in s and every r 0 using this result we obtain a characterization simple to check in practical cases of the gromov hyperbolicity of a riemann surface s with its own poincare metric obtained by deleting from one original surface s any uniformly separated union of continua and isolated points

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

comparing the floyd and ideal boundaries of a metric space we discuss and compare the notions of ideal boundaries floyd boundaries and gromov boundaries of metric spaces the three types of boundaries at infinity are compared in the general setting of unbounded length spaces as well as in the special cases of cat 0 and gromov hyperbolic spaces gromov boundaries usually defined only for gromov hyperbolic spaces are extended to arbitrary metric spaces

variation of the modulus of a foliation abstract the modulus is a generalization of the extremal length of plane curves introduced by l ahlfors we study the variation t mod p f t of the p modulus mod p f of a foliation f on a riemannian manifold in particular we consider the product of moduli of orthogonal foliations

sequential cfar detectors using a dead zone limiter the performances of some proposed sequential constant false alarm rate cfar detectors are evaluated the observations are passed through a dead zone limiter the output of which is 1 0 or 1 depending on whether the input is less than c between c and c or greater than c where c is a constant the test statistic is the sum of the outputs the test is performed on a reduced set of data those with absolute value larger than c with the test statistic being the sum of the signs of the reduced set of data both constant and linear boundaries are considered numerical results show a significant reduction of the average number of observations needed to achieve the same false alarm and detection probabilities as compared to a fixed sample size cfar detector using the same kind of test statistic

opportunistic detection under a fixed sample size setting with a finite number of samples drawn from one of two possible distributions sequentially revealed an opportunistic detection rule is proposed which possibly makes an early decision in favor of the alternative hypothesis while always deferring the decision of the null hypothesis until collecting all the samples properties of this opportunistic detection rule are discussed and its key asymptotic behavior in the large sample size limit is established specifically a chernoff stein lemma type of characterization of the exponential decay rate of the miss probability under the neyman pearson criterion is established and consequently a performance metric of asymptotic exponential efficiency loss is proposed and discussed which is exactly the ratio between the kullback leibler distance and the chernoff information of the two hypotheses analytical results are corroborated by numerical experiments

constrained partially observed markov decision processes with probabilistic criteria for adaptive sequential detection dynamic programming equations are derived which characterize the optimal value functions for a partially observed constrained markov decision process problem with both total cost and probabilistic criteria more specifically the goal is to minimize an expected total cost subject to a constraint on the probability that another total cost exceeds a prescribed threshold the markov decision process is partially observed but it is assumed that the constraint costs are available to the controller i e they are fully observed the problem is motivated by an adaptive sequential detection application the application of the dynamic programming results to optimal adaptive truncated sequential detection is demonstrated using an example involving the optimization of a radar detection process

autocorrelation based decentralized sequential detection of ofdm signals in cognitive radios this paper introduces a simple and computationally efficient spectrum sensing scheme for orthogonal frequency division multiplexing ofdm based primary user signal using its autocorrelation coefficient further it is shown that the log likelihood ratio test llrt statistic is the maximum likelihood estimate of the autocorrelation coefficient in the low signal to noise ratio snr regime performance of the local detector is studied for the additive white gaussian noise awgn and multipath channels using theoretical analysis obtained results are verified in simulation the performance of the local detector in the face of shadowing is studied by simulations a sequential detection sd scheme where many secondary users cooperate to detect the same primary user is proposed user cooperation provides diversity gains as well as facilitates using simpler local detectors the sequential detection reduces the delay and the amount of data needed in identification of the underutilized spectrum the decision statistics from individual detectors are combined at the fusion center fc the statistical properties of the decision statistics are established the performance of the scheme is studied through theory and validated by simulations a comparison of the sd scheme with the neyman pearson fixed sample size fss test for the same false alarm and missed detection probabilities is also carried out

sequential detection of markov targets with trajectory estimation the problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered assuming a markov evolution of the system in particular the pair signal observation forms a hidden markov model hmm a sequential procedure is proposed wherein the detection part is a sequential probability ratio test sprt and the estimation part relies upon a maximum a posteriori probability map criterion gated by the detection stage the parameter to be estimated is the trajectory of the state evolution of the system itself a thorough analysis of the asymptotic behavior of the test in this new scenario is given and sufficient conditions for its asymptotic optimality are stated i e for almost sure minimization of the stopping time and for first order minimization of any moment of its distribution an application to radar surveillance problems is also examined

detecting small moving objects in image sequences using sequential hypothesis testing an algorithm is proposed for the solution of the class of multidimensional detection problems concerning the detection of small barely discernible moving objects of unknown position and velocity in a sequence of digital images a large number of candidate trajectories organized into a tree structure are hypothesized at each pixel in the sequence and tested sequentially for a shift in mean intensity the practicality of the algorithm is facilitated by the use of multistage hypothesis testing mht for simultaneous inference as well as the existence of exact closed form expressions for mht test performance in gaussian white noise gwn these expressions predict the algorithm s computation and memory requirements where it is shown theoretically that several orders of magnitude of processing are saved over a brute force approach based on fixed sample size tests the algorithm is applied to real data by using a robust preprocessing procedure to eliminate background structure and transform the image sequence into a residual representation modeled as gwn results are verified experimentally on a variety of video image sequences

efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube the fully symmetrical cubature rules developed and presented are at present the most economically available for numerical integration over the unit cube of complete polynomial functions up to and including nineteenth order conventional gaussian triple product integration formulae by comparison require more than twice as many integration points and weights

on the construction of higher degree three dimensional embedded integration rules we want to approximate i f equiv int r 3 f x dx for a given function f x defined on a fully symmetric fs region r 3 in e 3 let x x 1 x 2 x n be a fully symmetric set of points in r 3 let q a f and q b f be two fs rules of degree 2m 1 and 2m 1 respectively approximating i f and defined as follows eqalign q a left f right equiv sum nolimits i 1 n a i f left x i right text degree 2 m text 1 cr q a left f right equiv sum nolimits i 1 n b i f left x i right text degree 2 m text 1 cr

rotation invariant cubature formulas over the n dimensional unit cube we consider cubature formulas to approximate multivariate integrals that remain unchanged under the orientation preserving transformations of a cube the use of invariant theory molien series reynold operator for the construction of such cubature formulas is investigated some new cubature formulas for the unit cube that are obtained using this approach are presented

a repeated significance test with applications to sequential detection in sensor networks in this paper we introduce a randomly truncated sequential hypothesis test using the framework of a repeated significance test rst we study a sequential test with truncation time based on a random stopping time using the functional central limit theorem fclt for a sequence of statistics we derive a general result that can be employed in developing a repeated significance test with random sample size we present effective methods for evaluating accurate approximations for the probability of type i error and the power function numerical results are presented to evaluate the accuracy of these approximations we apply the proposed test to a decentralized sequential detection problem in sensor networks sns with communication constraints finally a sequential detection problem with measurements at random times is investigated

asymptotic efficiencies of truncated sequential tests truncation of a sequential test with constant boundaries is considered for the problem of testing a location hypothesis f x theta 0 versus f x theta 1 a test design procedure is developed by using bounds for the error probabilities under the hypothesis and alternative by viewing the truncated sequential test as a mixture of a sequential probability ratio test and a fixed sample size test its boundaries and truncation point can be obtained once the degree of mixture is specified asymptotically correct approximations for the operating characteristic function and the average sample number function of the resulting test are derived numerical results show that an appropriately designed truncated sequential test performs favorably as compared to both the fixed sample size test and the sequential probability ratio test with the same error probabilities the average sample number function of the truncated test is uniformly smaller than that of the fixed sample size test and the truncated test maintains average sample sizes under the hypothesis and the alternative that are close to those optimum values achieved by wald s sequential probability ratio test moreover the truncated test is more favorable than the sequential probability ratio test in the sense that is has smaller average sample size when the actual location parameter is between theta 0 and theta 1 this behavior becomes more pronounced as the error probabilities become smaller implying that the truncated sequential test becomes more favorable as the error probabilities become smaller

opportunistic detection rules finite and asymptotic analysis opportunistic detection rules odrs are variants of fixed sample size detection rules in which the statistician is allowed to make an early decision on the alternative hypothesis opportunistically based on the sequentially observed samples from a sequential decision perspective odrs are also mixtures of one sided and truncated sequential detection rules several results regarding odrs are established in this paper in the finite regime the maximum sample size is modeled either as a fixed finite number or a geometric random variable with a fixed finite mean for both cases the corresponding bayesian formulations are investigated the former case is a slight variation of the well known finite length sequential hypothesis testing procedure in the literature whereas the latter case is new for which the bayesian optimal odr is shown to be a sequence of likelihood ratio threshold tests with two different thresholds a running threshold which is determined by solving a stationary state equation is used when future samples are still available and a terminal threshold simply the ratio between the priors scaled by costs is used when the statistician reaches the final sample and thus has to make a decision immediately in the asymptotic regime the tradeoff among the exponents of the false alarm and miss error probabilities and the normalized expected stopping time under the alternative hypothesis is completely characterized and proved to be tight via an information theoretic argument within the tradeoff region one noteworthy fact is that the performance of the stein chernoff lemma is attainable by odrs

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

a density problem for sobolev spaces on gromov hyperbolic domains abstract we prove that for a bounded domain r n which is gromov hyperbolic with respect to the quasihyperbolic metric especially when is a finitely connected planar domain the sobolev space w 1 is dense in w 1 p for any 1 p moreover if is also jordan or quasiconvex then c r n is dense in w 1 p for 1 p

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

the strong converse theorem for hypothesis testing the authors present the answer to a so called converse problem by giving the explicit form of the power exponent this strong converse should be regarded as completing a weak converse previously obtained by r e blahut 1974

the construction of numerical integration rules of degree three for product regions abstract numerical integration formulas in n dimensional euclidean space of degree three are discussed in this paper for the product regions a method is presented to construct numerical integration formulas of degree three with 2 n real points and positive weights the presented problem is a little different from those dealt with by other authors all the corresponding one dimensional integrals can be different from each other and they are also nonsymmetrical in this paper an n dimensional numerical integration problem is turned into n one dimensional moment problems which simplifies the construction process some explicit numerical formulas are given furthermore a more generalized numerical integration problem is considered which will shed light on the final solution to the third degree numerical integration problem

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

constructing cubature formulae of degree 5 with few points this paper is devoted to construct a family of fifth degree cubature formulae for n cube with symmetric measure and n dimensional spherically symmetrical region the formula forn cube contains at most n 2 5n 3 points and for n dimensional spherically symmetrical region contains only n 2 3n 3 points moreover the numbers can be reduced to n 2 3n 1 and n 2 n 1 if n 7 respectively the latter of which is minimal

hypothesis testing with the general source the asymptotically optimal hypothesis testing problem with general sources as the null and alternative hypotheses is studied under exponential type error constraints on the first kind of error probability our fundamental philosophy is to convert all of the hypothesis testing problems to the pertinent computation problems in the large deviation probability theory this methodologically new approach enables us to establish compact general formulas of the optimal exponents of the second kind of error and correct testing probabilities for the general sources including all nonstationary and or nonergodic sources with arbitrary abstract alphabet countable or uncountable these general formulas are presented from the information spectrum point of view

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

noncoherent sequential acquisition of pn sequences for ds ss communications with without channel fading studies sequential acquisition schemes of m sequences based on the sequential probability ratio test sprt and a truncated sprt tsprt with noncoherent demodulation most reported results on sequential acquisition schemes assume that independent samples are available for the decision process the assumption of independent and identically distributed i i d samples requires the integrator in the receiver to be reset periodically this introduces loss in the effective signal to noise ratio degrading the performance in the present paper on the contrary the authors two sequential schemes use continuous integration it can be shown that the likelihood ratio is monotonic consequently the tests can be easily implemented in realtime methods are proposed for designing the decision thresholds to achieve the desired false alarm and miss probabilities performances of the proposed schemes are obtained and they suggest that the tsprt is more desirable the effect of slowly varying channel fading is also investigated results show that fading does not affect the false alarm probabilities but it can drastically reduce the probability of detecting the alignment of the two pn sequences especially when the fading is severe

a 2 sprt based approach to multiple model hypothesis testing for multi distribution detection this paper presents an approach to multiple model hypothesis testing mmht based on 2 sprt 2 mmsrpt for detecting unknown events that may have multiple possible distributions the sequential probability ratio test sprt based mmht method mmsprt is promising because of its efficiency and theoretical validity however it may suffer from sprt s lack of an upper bound on its stopping time especially in the mis specified case the proposed 2 mmsprt algorithm not only copes with this problem but also is in a setting that can provide efficient detection in the sense of minimizing the maximum expected sample size subject to error probability constraints specifically we prove the theoretical validity of 2 sprt for the problem of testing hypotheses with multivariate normal densities moreover we present a technique of forced independence and identical distribution i i d to map the non i i d likelihood ratio sequence to an i i d one which enables us to apply the sprt and 2 sprt effectively to the dynamic case under the linear gaussian assumption with a non identical distribution also 2 mmsprt s detection efficiency under some assumptions approximations is verified performance of 2 mmsprt is evaluated for signal detection and model set selection problems in several scenarios simulation results demonstrate the detection efficiency of the proposed 2 mmsprt compared with the mmsprt and some traditional tests

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

multihypothesis sequential probability ratio tests i asymptotic optimality the problem of sequential testing of multiple hypotheses is considered and two candidate sequential test procedures are studied both tests are multihypothesis versions of the binary sequential probability ratio test sprt and are referred to as msprts the first test is motivated by bayesian optimality arguments while the second corresponds to a generalized likelihood ratio test it is shown that both msprts are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution when the error probabilities or more generally risks associated with incorrect decisions are small the results are first derived for the discrete time case of independent and identically distributed i i d observations and simple hypotheses they are then extended to general possibly continuous time statistical models that may include correlated and nonhomogeneous observation processes it also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters where the composite hypotheses due to nuisance parameters can be reduced to simple ones by using the principle of invariance these results provide a complete generalization of the results given by veeravalli and baum see ibid vol 41 p 1994 97 1995 where it was shown that the quasi bayesian msprt is asymptotically efficient with respect to the expected sample size for i i d observations

conformal metrics and size of the boundary we establish a lower bound for the hausdorff dimension of the boundary associated with a conformal deformation of the euclidean metric on the unit ball in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the deformations we consider are motivated by quasiconformal maps our abstract approach leads to new results on the boundary behavior of these maps we prove a conjecture by hanson on the compression of sets and obtain an improved version of the so called wall theorem we also establish a riesz privalov theorem in higher dimensions

geometric characterizations of gromov hyperbolicity we prove the equivalence of three different geometric properties of metric measure spaces with controlled geometry the first property is the gromov hyperbolicity of the quasihyperbolic metric the second is a slice condition and the third is a combination of the gehring hayman property and a separation condition

improved cubature formulae of high degrees of exactness for the square the method of constructing minimal cubature rules with high algebraic degrees of exactness is developed by adapting a powerful algorithm for solving the system of nonlinear equations as a result new cubature formulae of degrees 15 17 19 21 and 23 are derived for the square they lead to lower numbers of knots and or to better quality with respect to those known previously the formulae obtained should be considered as the most efficient for the calculation of two dimensional integrals with a high precision