battle outcome prediction for an extended system of lanchester type differential equations abstract battle outcome prediction conditions are given for an extended system of lanchester type differential equations for two different types of battle termination conditions a fixed force level breakpoint battles and b fixed force ratio breakpoint battles necessary and sufficient conditions for predicting battle outcome are given in the former case for a fight to the finish while sufficient conditions are given in the latter case the former results are equivalent to those for the problem of classical analysis of determining explicitly as a function of the initial conditions the occurrence of a zero point for the solution to this extended system although such results as given here have not appeared previously for nonoscillatory in the strict sense solutions

longer than average intervals containing no primes we present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes the first method is based on an argument of erdos which uses a sieve upper bound for prime twins to bound the density function for gaps between primes the second method uses known results about the first three moments for the distribution of intervals with a given number of primes better results are obtained by assuming that the first n moments are poisson the related problem of longer than average gaps between primes is also considered

the dirichlet problem for a quasilinear singularly perturbed second order system we study an equivalent form of the dirichlet problem for a quasilinear singularly perturbed second order system which is a singular singularly perturbed boundary value problem in this way we have not only eliminated the usual assumption of the existence of a vector potential function but also proved the existence and uniqueness of the exact solution of the studied problem meanwhile an asymptotic analysis and a remainder estimate of the solution are also presented

distance measures and estimation performance bounds for continuous time data corresp recursive expressions for certain distance measures between continuous time stationary vector gaussian processes are derived and used to derive upper bounds to the mean square error performance of the bayes and maximum likelihood estimates of a parameter when only a finite valued parameter set is used the question of convergence when the true parameter value does not belong to the finite set is also answered

wkbj type approximation for finite moments perturbations of the differential equation y 0 and the analogous difference equation abstract wkbj or liouville green type approximation theorems are proved for differential equations like y g x y 0 x 1 and their discrete counterparts 2 y n g n y n 0 n v the coefficients g x g n having a finite second moment precise error bounds are obtained as in the already existing theory due to f w j olver for differential equations in other classes and in a previous work by the authors concerning certain difference equations when only the first moment is finite the same asymptotic approximation with bound still holds for recessive solutions an effort has been made to present both aspects the continuous and the discrete in a unified way basing the error analysis on a volterra stieltjes integral equation examples are given to illustrate the theory

sums of kloosterman sums in arithmetic progressions and the error term in the dispersion method we prove a bound for quintilinear sums of kloosterman sums with congruence conditions on the smooth summation variables this generalizes classical work of deshouillers and iwaniec and is key to obtaining power saving error terms in applications notably the dispersion method as a consequence assuming the riemann hypothesis for dirichlet l functions we prove a power saving error term in the titchmarsh divisor problem of estimating sum p leq x tau p 1 unconditionally we isolate the possible contribution of siegel zeroes showing it is always negative extending work of fouvry and tenenbaum we obtain power saving in the asymptotic formula for sum n leq x tau k n tau n 1 reproving a result announced by bykovski u i and vinogradov by a different method the gain in the exponent is shown to be independent of k if a generalized lindel of hypothesis is assumed

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

decoupling of recessive and nonrecessive solutions for a second order system abstract a riccati transformation is used to obtain liouville green approximations along with error bounds for a vector system u f t u on a real interval a subject to suitable assumptions on the n n matrix valued function f f t

decoupling and order reduction via the riccati transformation a program that dates back to riccati 1724 and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use

liouville green approximations via the riccati transformation the riccati transformation is used to obtain liouville green approximations along with error bounds possessing a particularly simple and explicit form for the equation u f t u the resulting error bounds involve the function f and its first derivative whereas the earlier bounds of olver 1961 1974 and taylor 1978 1982 involve also the second derivative of f the theorems given here resemble corresponding results of olver 1974 chapter 6 and indeed the present work owes a great debt to olver the riccati transformation used here is based on work of y sibuya k w chang and w a harris and can be used in a more general study of linear ordinary differential equations and systems of such equations cf smith 1984 1985

a new version of the chang transformation a new decoupling transformation is proposed for singularly perturbed linear systems this transformation has the advantage over the previous one in that in order to find such a transformation only one algorithm is required and the computation can be done in parallel because the transformation matrices are obtained from two independent equations that are identical in form

prime numbers in logarithmic intervals let x be a large parameter we will first give a new estimate for the i ntegral moments of primes in short intervals of the type p p h where p x is a prime number and h o x then we will apply this to prove that for every 1 2 there exists a positive proportion of primes p x such that the interval p p log x contains at least a prime number as a consequence we improve cheer and goldston s result on the size of real numbers 1 with the property that there is a positive proportion of int egers m x such that the interval m m log x contains no primes we also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers m x such that the interval m m log x contains at least a prime number the last application of these techniques are two theorems the fi rst one unconditional and the second one in which we assume the validity of the riemann hypothesis and of a form of the montgomery pair correlation conjecture on the positive proportion of primes p x such that the interval p p log x contains no primes ams classification 11n05 11a41

more on asymptotic approximation for matrix differential equations an asymptotic approximation is obtained for solutions of a matrix differential equation with symmetric matrix coefficient analytic on the real line our contribution consists of an asymptotic approximation that is valid even for various singularities near infinity

new convergence bounds for bayes estimators using some recently derived spectral expressions for distance measures between vector gaussian stationary processes upper bounds are given for the mean square error performance of the bayes estimate of a parameter on the basis of vector gaussian observations the result is directly applicable to evaluating the performance of certain adaptive estimation schemes and to finite state markov chain systems

primes in tuples i we introduce a method for showing that there exist prime numbers which are very close together the method depends on the level of distribution of primes in arithmetic progressions assuming the elliott halberstam conjecture we prove that there are infinitely often primes differing by 16 or less even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart unconditionally we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing that is lim inf n pn 1 pn log pn log 0 we will quantify this result further in a later paper

singularly perturbed linear boundary value problems abstract in this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed the method consists of developing a non singular linear transformation that transforms an arbitrary n time scale system into diagonal form this fast and slow mode decomposition provides a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system corresponding to the decoupled system furthermore the decoupled system provides a useful mechanism to relate the asymptotic behavior of the solution of the original system and the solution of the degenerate system relative to the original system

le probleme de tchebychev pour le douzieme polynome cyclotomique let p n denote the largest prime factor of the integer n and 12 n n 4 n 2 1 we prove that for x large enough we have p x x 1 c with c 10 26531

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

asymptotic approximations for second order linear difference equations in banach algebras ii abstract a liouville green wkb asymptotic approximation theory is developed for some classes of linear second order difference equations in banach algebras the special case of linear matrix difference equations or equivalently of second order systems is emphasized rigorous and explicitly computable bounds for the error terms are obtained and this when both the sequence index and some parameter that may enter the coefficients go to infinity a simple application is made to orthogonal matrix polynomials in the nevai class

unified canonical forms for matrices over a differential ring abstract linear differential equations with variable coefficients of the vector form i x a t x and the scalar form ii y n n t y n 1 2 t y 1 t y 0 can be studied as operators on a differential module over a differential ring using this differential algebraic structure and a classical result on differential operator factorizations developed by cauchy and floquet a new variable eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring in particular eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion jordan diagonal and generalized vandermonde canonical forms as traditionally used for matrices over a number ring field therefore these new matrix canonical forms can be viewed as unifications of these classical ones two additional canonical matrices that perform order reductions of a cauchy floquet factorization of a linear differential operator are also introduced general and explicit formulas for obtaining the new canonical forms are derived in terms of the new variable eigenvalues the results obtained here have immediate applications in diverse engineering and scientific fields such as physics control systems communications and electrical networks in which linear differential equations i and ii are used as mathematical models

liouville green asymptotic approximation for a class of matrix differential equations and semi discretized partial differential equations abstract a liouville green or wkb asymptotic approximation theory is developed for the class of linear second order matrix differential equations y f t a g t y on a where a and g t are matrices and f t is scalar this includes the case of an asymptotically constant not necessarily diagonalizable coefficient a when f t 1 an explicit representation for a basis of the right module of solutions is given and precise computable bounds for the error terms are provided the double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown several examples some concerning semi discretized wave and convection diffusion equations are given

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

comments and corrections to new convergence bounds for bayes estimators by d kazakos a conjecture and corrects some inaccuracies of the convergence bounds that were derived in a previous paper

primes in tuples ii we introduce a method for showing that there exist prime numbers which are very close together the method depends on the level of distribution of primes in arithmetic progressions assuming the elliott halberstam conjecture we prove that there are infinitely often primes differing by 16 or less even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart unconditionally we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing that is lim inf n 1 pnc1 pn logpn d0

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

on the kuznetsov trace formula for pgl2 c abstract in this note using a representation theoretic method of cogdell and piatetski shapiro we prove the kuznetsov trace formula for an arbitrary discrete group in pgl 2 c that is cofinite but not cocompact an essential ingredient is a kernel formula recently proved by the author on bessel functions for pgl 2 c this approach avoids the difficult analysis in the existing method due to bruggeman and motohashi

autour de la conjecture de sato tate pour les sommes de kloosterman i keywords almost primes sieve identity of buchstab type kloosterman sum sato tate conjecture distribution function sato tate distribution estimates for multiple exponential sums reference tan article 1995 001doi 10 1007 bf01884290 record created on 2008 11 14 modified on 2017 05 12

spectral distance measures between continuous time vector gaussian processes corresp a new expression for the chernoff distance between two continuous time stationary vector gaussian processes that contain a common white noise component and have equal means is derived the expression is given in terms of the spectral density matrices for large observation interval t the expression is then used for deriving upper and lower bounds to the bayes probability of error both bounds converge to zero exponentially in t it is also shown that the i divergence and j divergence can be easily evaluated in the frequency domain by differentiation of the chernoff distance

exact slow fast decomposition of the singularly perturbed matrix differential riccati equation in this paper the hamiltonian matrix formulation of the riccati equation is used to derive the reduced order pure slow and pure fast matrix differential riccati equations of singularly perturbed systems these pure slow and pure fast matrix differential riccati equations are obtained by decoupling the singularly perturbed matrix differential riccati equation of dimension n 1 n 2 into the pure slow regular matrix differential riccati equation of dimension n 1 and the pure fast stiff matrix differential riccati equation of dimension n 2 a formula is derived that produces the solution of the original singularly perturbed matrix differential riccati equation in terms of solutions of the pure slow and pure fast reduced order matrix differential riccati equations and solutions of two reduced order initial value problems in addition to its theoretical importance the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time invariant systems independently in pure slow and pure fast time scales

improved error bounds for the liouville green or wkb approximation a variant of the gronwall bellman inequality is used to develop new bounds on solutions to the fundamental singular integral equations that arise in the error analysis of the liouville green or wkb approximation consequently new improved error bounds are obtained from the liouville green approximation to the solution of the differential equation d2wdt2 q t w 0 where the function q is real and twice continuously differentiable and does not vanish

conformally flat riemannian metrics schrodinger operators and semiclassical approximation abstract a relationship between laplace beltrami and schrodinger operators on euclidean domains is analyzed and exploited for several purposes we use the schrodinger equation to analyze the spectra of laplace beltrami operators with periodic metrics on r v and use geometric notions and nonlinear differential equations to bound spectra and green functions of schrodinger operators in various ways we also have a new more operator theoretic analysis of the semiclassical limit and the liouville green or jwkb approximation in one dimension

asymptotic analysis of wave propagation along weakly non uniform repetitive systems we consider monochromatic wave propagation along a long finite one dimensional slightly non uniform waveguide whose ends are connected to uniform semi infinite waveguides the non uniformity in the system parameters which is assumed slowly varying and deterministic can be tuned to produce a desired scattered wave field or reflection transmission properties for a broad range of incident wave fields with this objective in mind we obtain an analytic solution for wave propagation along repetitive systems asymptotic in the slowness of the variation of the system parameters we consider systems governed by a second order finite difference equation and apply the wkb method allowing the index variable to be complex this allows complex turning points to be considered the coefficients of the difference equation are represented by their discrete fourier modes for complex turning points we obtain exponentially small reflection a new result in the context of difference equations the asymptotic solution besides revealing how the non uniformity in the parameters affects wave propagation furnishes an analytic expression for the system scattering matrix as a function of the system parameters it also sheds light on the mechanism of localization phenomena for this class of repetitive systems we also compare the asymptotic results with numerical experiments for large finite one dimensional non uniform chains of coupled pendula

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

a lower bound for the counting function of lucas pseudoprimes etablissement d une borne inferieure pour le denombrement des nombres pseudopremiers de lucas

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1