decision theoretic optimality of the cusum procedure suppose x 1 x 2 are independent random variables such that for some unknown each of x 1 x 1 is distributed according to f 0 while x x 1 are all distributed according to f 1 we prove a result of moustakides that claims that the cusum procedures are optimal in the sense of lorden we do that by proving that the procedures are bayes for some stochastic mechanism of generating

on thermal stability of a reactive third grade fluid in a channel with convective cooling the walls in this paper the thermal stability of a reactive third grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated the system is assumed to exchange heat with the ambient following newton s cooling law and the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed it is observed that a combined increase in non newtonian parameter and convective cooling enhances the thermal stability of the material

the asymptotic behaviour of algebraic approximants we study the convergence of algebraic approximants to a function represented by a power series we consider for an arbitrary but fixed degree approximant sequences which can be generated recursively by use of sergeyev s algorithm for the exponential function a logarithmic function and a power of a binomial we find explicit formulae for the coefficients that appear in a resulting linear recurrence relation we assume that the error equation may be linearized for small errors analysis then yields the generic dominant term in the asymptotic behaviour of the error when a large number of terms of the series are used extensive numerical results confirm the behaviour finally we compare this behaviour with that for the closely related method of drazin tourigny in which the degree of the approximants grows without bound

nonanticipating estimation applied to sequential analysis and changepoint detection suppose a process yields independent observations whose distributions belong to a family parameterized by theta e theta when the process is in control the observations are i i d with a known parameter value theta 0 when the process is out of control the parameter changes we apply an idea of robbins and siegmund proc sixth berkeley symp math statist probab 4 1972 37 41 to construct a class of sequential tests and detection schemes whereby the unknown post change parameters are estimated this approach is especially useful in situations where the parametric space is intricate and mixture type rules are operationally or conceptually difficult to formulate we exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a gamma distribution in both a univariate and a multivariate setting

detection of gauss markov random fields with nearest neighbor dependency the problem of hypothesis testing against independence for a gauss markov random field gmrf is analyzed assuming an acyclic dependency graph an expression for the log likelihood ratio of detection is derived assuming random placement of nodes over a large region according to the poisson or uniform distribution and nearest neighbor dependency graph the error exponent of the neyman pearson detector is derived using large deviations theory the error exponent is expressed as a dependency graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals the exponent is analyzed for different values of the variance ratio and correlation it is found that a more correlated gmrf has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

optimal detection of a change in distribution abstract document discusses a stopping rule that is a limit of bayes rules derived then an almost minimax rule is presented

quadratic twists of elliptic curves in this paper we give the method of constructing non torsion points on elliptic curves which generalizes the classical birch lemma as an application we get more quadratic twist families of the elliptic curve x 0 49 which have rank one this report is a combination of the two joint works coates li tian zhai proc lond math soc 110 2 357 394 2015 4 cai li wang sci china math 59 7 1307 1326 2016 2

congruent numbers with many prime factors mohammed ben alhocain in an arab manuscript of the 10th century stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number m becomes a square dickson le 1971 history of the theory of numbers chelsea new york vol 2 chap 16 in modern language this object is to find a rational point of infinite order on the elliptic curve heegner constructed such rational points in the case that m are primes congruent to 5 7 modulo 8 or twice primes congruent to 3 modulo 8 monsky p 1990 math z 204 45 68 we extend heegner s result to integers m with many prime divisors and give a sketch in this report the full details of all the proofs will be given in ref 1 tian y 2012 congruent numbers and heegner points arxiv 1210 8231

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

thermal stability of a reactive third grade fluid in a cylindrical pipe an exploitation of hermite pade approximation technique abstract this study is devoted to investigate the thermal stability of a reactive third grade fluid flowing steadily through a cylindrical pipe with isothermal wall it is assumed that the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

quickest detection for changes in maximal knn coherence of random matrices this paper addresses the problem of quickest detection of a change in the maximal coherence between columns of an n times p random matrix based on a sequence of matrix observations having a single unknown change point the random matrix is assumed to have identically distributed rows and the maximal coherence is defined as the largest of the p atopwithdelims 2 correlation coefficients associated with any row likewise the k nearest neighbor knn coherence is defined as the k th largest of these correlation coefficients the forms of the pre and postchange distributions of the observed matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown a nonparametric stopping rule is proposed that is based on the maximal knn sample coherence between columns of each observed random matrix this is a summary statistic that is related to a test of existence of a hub vertex in a sample correlation graph having degree at least k performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained in the purely high dimensional regime where p rightarrow infty and n is fixed when the prechange dispersion matrix is diagonal it is shown that among all functions of the proposed summary statistic the proposed stopping rule is asymptotically optimal under a minimax quickest change detection model as the stopping threshold approaches infinity the theory developed also applies to sequential hypothesis testing and fixed sample size tests

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

the p adic gross zagier formula for elliptic curves at supersingular primes let p be a prime number and let e be an elliptic curve defined over of conductor n let k be an imaginary quadratic field with discriminant prime to pn such that all prime factors of n split in k b perrin riou established the p adic gross zagier formula that relates the first derivative of the p adic l function of e over k to the p adic height of the heegner point for k when e has good ordinary reduction at p in this article we prove the p adic gross zagier formula of e for the cyclotomic p extension at good supersingular prime p our result has an application for the full birch and swinnerton dyer conjecture suppose that the analytic rank of e over is 1 and assume that the iwasawa main conjecture is true for all good primes and the p adic height pairing is not identically equal to zero for all good ordinary primes then our result implies the full birch and swinnerton dyer conjecture up to bad primes in particular if e has complex multiplication and of analytic rank 1 the full birch and swinnerton dyer conjecture is true up to a power of bad primes and 2

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria