primes in short intervals contrary to what would be predicted on the basis of cramer s model con cerning the distribution of prime numbers we develop evidence that the distribution of x h x for 0 x n is approximately normal with mean h and variance h log n h when n h n 1

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

controllability and stabilizability of a class of continuous piecewise affine dynamical systems this paper studies controllability and stabilizability of continuous piecewise affine dynamical systems which can be considered as a collection of ordinary finite dimensional linear input state output systems together with a partition of the product of the state space and input space into full dimensional polyhedral regions each of these regions is associated with one particular linear system from the collection the main results of the paper are popov belevitch hautus type necessary and sufficient conditions for both controllability and stabilizability of such systems

tests of equal forecast accuracy and encompassing for nested models we examine the asymptotic and finite sample properties of tests for equal forecast accuracy and encompassing applied to 1 step ahead forecasts from nested linear models we first derive the asymptotic distributions of two standard tests and one new test of encompassing and provide tables of asymptotically valid critical values monte carlo methods are then used to evaluate the size and power of tests of equal forecast accuracy and encompassing the simulations indicate that post sample tests can be reasonably well sized of the post sample tests considered the encompassing test proposed in this paper is the most powerful we conclude with an empirical application regarding the predictive content of unemployment for inflation

small gaps between primes or almost primes let p n denote the n th prime goldston pintz and yildirim recently proved that li m in f pn 1 p n n log p n 0 we give an alternative proof of this result we also prove some corresponding results for numbers with two prime factors let q n denote the n th number that is a product of exactly two distinct primes we prove that lim inf qn 1 q n n 26 if an appropriate generalization of the elliott halberstam conjecture is true then the above bound can be improved to 6

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

higher correlations of divisor sums related to primes iii small gaps between primes we use divisor sums to approximate prime tuples and moments for primes in short intervals by connecting these results to classical moment problems we are able to prove that for any 0 a positive proportion of consecutive primes are within 4 times the average spacing between primes authors note this paper was written in 2004 prior to the solution in 8 of the problem considered here in 8 it is shown that 0 while the main result in theorem 1 has now been superseded we believe the method used here is both of interest and future utility in other applications in particular the work of green and tao 12 on arithmetic progressions of primes makes use of proposition 1 of this paper

dead beat control of simple hammerstein models dead beat controllers for simple hammerstein systems are investigated several designs for nonminimum time state dead beat controllers are given for certain classes of simple hammerstein systems a general minimum time state dead beat controller is presented for a class of simple hammerstein systems a design for a family of minimum time control laws is provided this enables to a certain extent shaping of transient response via choosing an appropriate control law finally the authors design an output dead beat controller for a class of hammerstein systems that are not necessarily state dead beat controllable

time series modeling by a second order hammerstein system in this paper we discuss the modeling of an observed time series as the output of a second order hammerstein model driven by an independent and identically distributed i i d gaussian noise the modeling is discussed from the point of view of the hypothesis testing ht of statistical indexes derived from higher order statistics hos three points of view can be adopted in ht the more general approach is to not make any assumption on the index statistics notably on the variance and thus to use the t 2 hotelling test the second point of view involves calculating a theoretical variance in order to define normalized variables whose statistics are known the last approach is a mixed version of the others two these three approaches are applied to statistical indexes derived from linear combination of slices of spectra from the third order up the fifth order some subcases need to use the derivatives of the polyspectra since general relationships are null and cannot be used for the ht the second part of this paper is devoted to linear and quadratic kernel identification using the relationship defined for ht definition as for the ht several subcases have to be discussed using either the polyspectra or the derivatives of polyspectra

small gaps between primes we introduce a renement of the gpy sieve method for studying prime k tuples and small gaps between primes this renement avoids previous limitations of the method and allows us to show that for each k the prime k tuples conjecture holds for a positive proportion of admissible k tuples in particular lim infn pn m pn 1 for every integer m we also show that lim inf pn 1 pn 600 and if we assume the elliott halberstam conjecture that lim infn pn 1 pn 12 and lim infn pn 2 pn 600

on limit points of the sequence of normalized prime gaps let pn denote the nth smallest prime number and let l denote the set of limit points of the sequence tppn 1 pnq logpnu 8 1 of normalized differences between consecutive primes we show that for k 9 and for any sequence of k nonnegative real numbers 1 2 k at least one of the numbers j i 1 i j k belongs to l it follows that at least 12 5 of all nonnegative real numbers belong to l

correlations of the von mangoldt and higher divisor functions ii divisor correlations in short ranges we study the problem of obtaining asymptotic formulas for the sums sum x n le 2x d k n d l n h and sum x n le 2x lambda n d k n h where lambda is the von mangoldt function d k is the k text th divisor function x is large and k ge l ge 2 are integers we show that for almost all h in h h with h log x 10000 k log k the expected asymptotic estimate holds in our previous paper we were able to deal also with the case of lambda n lambda n h and we obtained better estimates for the error terms at the price of having to take h x 8 33 varepsilon

output dead beat control for a class of planar polynomial systems output dead beat control for a class of nonlinear discrete time systems which are described by a single input output i o polynomial difference equation is considered the class of systems considered is restricted to systems with a two dimensional state space description it is assumed that the highest degree with which the present input appears in the equation is odd necessary and sufficient conditions for the existence of output dead beat control and for the stability of the zero output constrained dynamics are presented we also design a minimum time output dead beat control algorithm feedback controller which yields stable zero dynamics whenever this is feasible a number of interesting phenomena are discussed and illustrated with examples

formulating wald tests of nonlinear restrictions on montre a partir de l evidence de monte carlo que les differences dans la forme fonctionnelle des restrictions non lineaires sont importantes dans les petites tailles d echantillons

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

algebraic necessary and sufficient conditions for the controllability of conewise linear systems the problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable this paper focuses on conewise linear systems i e systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions for this class of systems we present algebraic necessary and sufficient conditions for controllability we also show that the classical results of controllability of linear systems and input constrained linear systems can be recovered from our main result our treatment employs tools both from geometric control theory and mathematical programming

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

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

small gaps between prime numbers the work of goldston pintz yildirim in early 2005 dan goldston janos pintz and cem yildirim 12 made a spectacular breakthrough in the study of prime numbers resolving a long standing open problem they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes before their work it was known only that there were infinitely many gaps which were about a quarter the size of the average gap the new result may be viewed as a step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p 2 the gap here being 2 the smallest possible gap between primes perhaps most excitingly their work reveals a connection between the distribution of primes in arithmetic progressions and small gaps between primes assuming certain admittedly difficult conjectures on the distribution of primes in arithmetic progressions they are able to prove the existence of infinitely many prime pairs that differ by at most 16 the aim of this article is to explain some of the ideas involved in their work let us begin by explaining the main question in a little more detail the number of primes up to x denoted by x is roughly x log x for large values of x this is the celebrated prime number theorem therefore if we randomly choose an integer near x then it has about a 1 in log x chance of being prime in other words as we look at primes around size x the average gap between consecutive primes is about log x as x increases the primes get sparser and the gap between consecutive primes tends to increase here are some natural questions about these gaps between prime numbers do the gaps always remain roughly about size log x or do we sometimes get unexpectedly large gaps and sometimes surprisingly small gaps can we say something about the statistical distribution of these gaps that is can we quantify how often the gap is between say log x and log x given 0 except for the primes 2 and 3 clearly the gap between consecutive primes must be even does every even number occur infinitely often as a gap between consecutive primes for example the twin prime conjecture says that the

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

output deadbeat control of nonlinear discrete time systems with one dimensional zero dynamics global stability conditions output deadbeat control in one step is considered for a class of discrete time systems described by a nonlinear single input single output recursive representation global stability conditions are established for the particular subclass of systems with one dimensional zero dynamics the results are illustrated with applications to polynomial and neural dynamical systems

testing for inequality constraints in singular models by trimming or winsorizing the variance matrix there are many applications in which a statistic follows at least asymptotically a normal distribution with a singular or nearly singular variance matrix a classic example occurs in linear regre

on controllability of discrete time bilinear systems by near controllability abstract in this paper controllability of discrete time bilinear systems is studied by applying a recent result on near controllability a new sufficient condition for controllability of the systems is presented where controllability is proved by approximation with near controllability the new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete time bilinear systems which can be effective even when the classical result does not work furthermore the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near controllability therefore near controllability can be used to not only better characterize the system properties but also prove controllability with computable control inputs the new condition is then generalized to derive similar results on controllability and near controllability of the systems finally examples are given to illustrate the results of this paper

non parametric identification and estimation of the number of components in multivariate mixtures this article analyzes the identiability of the number of components in k variate mcomponent nite mixture models in which each component distribution has independent marginals including models in latent class analysis without making parametric assumptions on the component distributions we investigate how one can identify the number of components from the distribution function of the observed data when k 2 a lower bound on the number of components m is nonparametrically identiable from the rank of a matrix constructed from the distribution function of the observed variables building on this identication condition we develop a procedure to consistently estimate a lower bound on the number of components

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

on small controllability and controllability of a class of nonlinear systems in this paper small controllability and controllability of a class of nonlinear systems are studied both the discrete time case and continuous time case are considered by proposing an implicit function approach sufficient conditions for the systems respectively to be small controllable and controllable are obtained examples are also provided to illustrate the results of the paper

stabilizability and dead beat controllers for two classes of wiener hammerstein models two classes of block oriented models of the wiener hammerstein type are considered we prove that a generic condition is sufficient for a null controllable discrete time system of this form to have a stabilizing minimum time dead beat controller when the condition is violated we show how to design a nonminimum time stabilizing dynamic dead beat controller the result is used to obtain stabilizability conditions for these systems

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

output feedback stabilization of a class of wiener systems a globally stabilizing output feedback controller is designed for a class of continuous time wiener systems the wiener systems we consider consist of a linear dynamical block and an output polynomial nonlinearity connected in series the hybrid controller consists of three modes of operation which are periodically applied to the system the controller achieves a dead beat response of the closed loop system

a note on observability tests for general polynomial and simple wiener hammerstein systems we propose an algorithm based on symbolic computation packages for testing observability conditions of general polynomial systems which were formulated in sontag siam j control optim 17 1979 139 151 computational complexity of the observability test can be reduced and the test simplified for classes of polynomial systems we illustrate this by considering the class of simple wiener hammerstein systems which consist of a series of two linear dynamic blocks between which a static nonlinearity is sandwiched we consider the case when the nonlinearity is a monomial n q q 1 simple necessary and sufficient conditions for observability are given and they resemble but are subtly different from the well known result on observability for the series connection of two linear systems c 1998 elsevier science b v all rights reserved

exact and asymptotic tests for possibly non regular hypotheses on stochastic volatility models abstract we study the problem of testing hypotheses on the parameters of one and two factor stochastic volatility models sv allowing for the possible presence of non regularities such as singular moment conditions and unidentified parameters which can lead to non standard asymptotic distributions we focus on the development of simulation based exact procedures whose level can be controlled in finite samples as well as on large sample procedures which remain valid under non regular conditions we consider wald type score type and likelihood ratio type tests based on a simple moment estimator which can be easily simulated we also propose a c type test which is very easy to implement and exhibits relatively good size and power properties besides usual linear restrictions on the sv model coefficients the problems studied include testing homoskedasticity against a sv alternative which involves singular moment conditions under the null hypothesis and testing the null hypothesis of one factor driving the dynamics of the volatility process against two factors which raises identification difficulties three ways of implementing the tests based on alternative statistics are compared asymptotic critical values when available a local monte carlo or parametric bootstrap test procedure and a maximized monte carlo mmc procedure the size and power properties of the proposed tests are examined in a simulation experiment the results indicate that the c based tests built upon the simple moment estimator available in closed form have good size and power properties for regular hypotheses while monte carlo tests are much more reliable than those based on asymptotic critical values further in cases where the parametric bootstrap appears to fail for example in the presence of identification problems the mmc procedure easily controls the level of the tests moreover mmc based tests exhibit relatively good power performance despite the conservative feature of the procedure finally we present an application to a time series of returns on the standard and poor s composite price index

dead beat controllability of polynomial systems symbolic computation approaches state and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition qepcad symbolic computation program the method is unified for a very large class of systems and can handle one or two sided control constraints families of minimum time state output dead beat controllers are obtained the computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system we may beat the curse of complexity a computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented necessary and sufficient conditions are given they are still very difficult to check therefore a number of easier to check sufficient conditions are also provided the latter are based on the grobner basis method and qepcad it is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system

modified wald tests under nonregular conditions abstract under nonregular conditions wald tests are known to have incorrect size even asymptotically in part of the parameter space modifications are discussed which ensure an asymptotic 2 distribution of the wald statistic under h0 as an example wald tests for multi step causality are considered a variable yt is h step causal for another variable xt if the information in yt helps improving the j step forecasts of xt for some j 1 2 h if more than two variables are involved and are generated by a finite order vector autoregressive var process this type of multi step noncausality implies a set of highly nonlinear restrictions on the var coefficient matrices for this type of nonlinear restrictions standard wald tests fail to have limiting 2 distributions in general

controllability for a class of simple wiener hammerstein systems abstract controllability for a class of simple wiener hammerstein systems is considered necessary and sufficient conditions for dead beat and complete controllability for these systems are presented the controllability tests consist of two easy to check tests for the subsystems

diophantine equations in the primes let mathfrak p mathfrak p 1 ldots mathfrak p r be a system of r polynomials with integer coefficients of degree d in n variables mathbf x x 1 ldots x n for a given r tuple of integers say mathbf s a general local to global type statement is shown via classical hardy littlewood type methods which provides sufficient conditions for the solubility of mathfrak p mathbf x mathbf s under the condition that each of the x i s is prime

bounded gaps between primes it is proved that lim inf n 8 p n 1 p n 7 10 7 where p n is the n th prime our method is a refinement of the recent work of goldston pintz and yildirim on the small gaps between consecutive primes a major ingredient of the proof is a stronger version of the bombieri vinogradov theorem that is applicable when the moduli are free from large prime divisors only but it is adequate for our purpose

state deadbeat control of nonlinear systems construction via sets a geometric generalization of the discrete time linear deadbeat control problem is studied the proposed method to generate a deadbeat tracker for a given nonlinear system is constructive and makes use of sets that can be computed iteratively for demonstration derivations of the deadbeat feedback law and tracker dynamics are provided for an example system based on the method a simple algorithm that computes the deadbeat gain for a linear system with scalar input is given

popov belevitch hautus type controllability tests for linear complementarity systems it is well known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems this paper deals with the controllability problem of a class of piecewise linear systems known as linear complementarity systems by exploiting the underlying structure and employing the results on the controllability of the so called conewise linear systems we present a set of inequalitytype conditions as necessary and sufficient conditions for controllability of linear complementarity systems the presented conditions are of popov belevitch hautus type in nature 2006 elsevier b v all rights reserved

estimation and inference of dynamic structural factor models with over identifying restrictions abstract this paper develops a new estimator for the impulse response functions in structural factor models with a fixed number of over identifying restrictions the proposed identification scheme nests the conventional just identified recursive scheme as a special case we establish the asymptotic distributions of the new estimator and develop test statistics for the over identifying restrictions simulation results show that adding a few more over identifying restrictions can lead to a substantial improvement in estimation accuracy for impulse response functions at both zero and nonzero horizons we estimate the effects of a monetary policy shock using a u s data set the results show that our over identified scheme can help to detect incorrect specifications that lead to spurious impulse responses

a semi nonparametric estimator of regression discontinuity design with discrete duration outcomes abstract we consider the regression discontinuity rd design with the duration outcome which has discrete support the parameters of policy interest are treatment effects on unconditional duration effect and conditional hazard effect exiting probabilities for each discrete level we propose a novel semi nonparametric estimator which exploits a flexible separability structure of the underlying continuous time duration process simultaneous inference over discrete levels is nonstandard since the asymptotic variance matrix is singular with unknown rank the peculiarity is delivered by the nature of the rd estimand and we provide solutions random censoring and competing risks can also be allowed in our framework

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

a note on dead beat controllability of generalised hammerstein systems necessary and sufficient conditions for dead beat and complete controllability for a class of generalised hammerstein systems are presented since the system s structure is very close to linear only linear algebra is used for the controllability test the test is very simple and easy to use a closed loop minimum time dead beat controller can be designed for a subclass of generalised hammerstein systems using the grobner basis method

structure identification of nonlinear dynamic systems a survey on input output approaches abstract in the past several methods have been elaborated for the identification of nonlinear dynamic systems most of the methods assume that the structure of the system is given a priori therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation however in nonlinear system theory several methods are known to determine the structure of a system in this paper structure identification of block oriented especially cascade models of semi linear dynamic models with signal dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered different structure selection methods are summarized based on step and impulse tests frequency response measurements correlation analysis repeated reproducible tests and normal operating data

output dead beat control for nonlinear systems with one zero at infinity the problem of output dead beat control i e one step zeroing of the output is considered for nonlinear systems with discrete time controllers the nonlinear system is assumed to have one zero at infinity conditions for the existence of the dead beat control for small sampling intervals are derived the closed loop system is shown to be stable in the sense that all states converge to the origin if the controlled system is minimum phase

a unifying theory of tests of rank the general principles underlying tests of matrix rank are investigated it is demonstrated that statistics for such tests can be seen as implicit functions of null space estimators in turn the asymptotic behaviour of the null space estimators is shown to determine the asymptotic behaviour of the statistics through a plug in principle the theory simplifies the asymptotics under a variety of alternatives of empirical relevance as well as misspecification clarifies the relationships between the various existing tests makes use of important results in the numerical analysis literature and motivates numerous new tests a brief monte carlo study illustrates the results

testing causality between two vectors in multivariate autoregressive moving average models abstract in the analysis of economic time series a question often raised is whether a vector of variables causes another one in the sense of granger most of the literature on this topic is concerned with bivariate relationships or uses finite order autoregressive specifications the purpose of this article is to develop a causality analysis in the sense of granger for general vector autoregressive moving average arma models we give a definition of granger noncausality between vectors which is a natural and simple extension of the notion of granger noncausality between two variables in our context this definition is shown to be equivalent to a more complex definition proposed by tjostheim for the class of linear invertible processes we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis this result is then specialized to the class of stationary invertible arma process

zero dynamics of sampled nonlinear systems abstract the behaviour of the zero dynamics of a discrete time system obtained when sampling a continuous time nonlinear system is studied we give a result which generalizes the known behaviour of the zeros of a sampled linear system for small time intervals

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

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals