robust analysis and control for a class of uncertain nonlinear discrete time systems abstract this paper deals with the problem of robust analysis and control of a class of nonlinear discrete time systems with constant uncertain parameters for the analysis problem we use a polynomial lyapunov function and we generalize for nonlinear systems the extended stability notion proposed by oliveira et al 1999 in the context of linear discrete time uncertain systems as a result we propose an lmi optimization problem to maximize an estimate of the domain of attraction and also extend this approach to the synthesis problem by considering parameter dependent lyapunov functions and nonlinear multipliers numerical examples illustrate the approach and show its potential for solving analysis and control problems of nonlinear discrete time systems

divisibility of reduction in groups of rational numbers given a multiplicative group of nonzero rational numbers and a positive integer m we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m in the case when the group is finitely generated the density is explicitly computed some examples of groups with infinite rank are considered

steady state performance evaluation of continuous mono t semiflow petri nets the number of states in discrete event systems can increase exponentially with respect to the size of the system a way to face this state explosion problem consists of relaxing the system model for example by converting it to a continuous one in the scope of petri nets the firing of a transition in a continuous petri net system is done in a real amount hence the marking state of the net system becomes a vector of non negative real numbers the main contribution of the paper lies in the computation of throughput bounds for continuous petri net systems with a single t semiflow for that purpose a branch and bound algorithm is designed moreover it can be relaxed and converted into a linear programming problem some conditions under which the system always reaches the computed bounds are extracted the results related to the computation of the bounds can be directly applied to a larger class of nets called mono t semiflow reducible

square free orders for cm elliptic curves modulo p let e be an elliptic curve defined over mathbb q of conductor n and with complex multiplication we prove unconditional and conditional asymptotic formulae for the number of ordinary primes p nmid n p x for which the group of points of the reduction of e modulo p has square free order these results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of e modulo p is cyclic first studied by serre 1977 they are also related to the stronger problem about primitive points on e modulo p formulated by lang and trotter bull am math soc 83 289 292 1977 and the one about the primality of the order of e modulo p formulated by koblitz pacific j math 131 1 157 165 1988

structure theory of equal conflict systems abstract equal conflict net systems are a weighted generalisation of the extended free choice subclass that keeps the total autonomy of choices a substantial part of the analytical branch of the structure theory of free choice systems including decomposition and duality the rank theorem characterisations of impartial sequences and prompt interfaces and the existence of home states is extended to the equal conflict case in so doing several familiar concepts and objects from the structure theory of place transition net systems are revisited from a linear algebraic perspective which is specially adequate to cope with weighted nets

robust stability of a class of uncertain markov jump nonlinear systems this note addresses the problem of robust stability analysis for a class of markov jump nonlinear systems subject to polytopic type parameter uncertainty a condition for robust local exponential mean square stability in terms of linear matrix inequalities is developed an estimate of a robust domain of attraction of the origin is also provided the approach is based on a stochastic lyapunov function with polynomial dependence on the system state and uncertain parameters a numerical example illustrates the proposed result

on euclidean ideal classes in certain abelian extensions in this article we show that certain abelian extensions k with unit rank greater than or equal to three have cyclic class group if and only if it has a euclidean ideal class this result improves an earlier result of murty and graves one can improve this result up to unit rank 2 if one assumes the elliott and halberstam conjecture see conjecture 1 in preliminaries these results are known under generalized riemann hypothesis by the work of lenstra j lond math soc 10 457 465 see also weinberger proc symp pure math 24 321 332

stability analysis of implicit polynomial systems this technical note develops a robust stability analysis method for a class of implicit polynomial systems subject to uncertain constant parameters a polynomial with respect to the state and uncertain parameters lyapunov function is computed to assess the robust local stability of the system and to provide a robust estimate of the domain of attraction the results are given in terms of linear matrix inequalities that are affinely dependent on the system state and uncertain parameters numerical examples illustrate the effectiveness of the developed approach

on the r rank artin conjecture we assume the generalized riemann hypothesis and prove an asymptotic formula for the number of primes for which f p can be generated by r given multiplicatively independent numbers in the case when the r given numbers are primes we express the density as an euler product and apply this to a conjecture of brown zassenhaus j number theory 3 1971 306 309 finally in some examples we compare the densities approximated with the natural densities calculated with primes up to 9 10 4

the sato tate law for drinfeld modules we prove an analogue of the sato tate conjecture for drinfeld modules using ideas of drinfeld j k yu showed that drinfeld modules satisfy some sato tate law but did not describe the actual law more precisely for a drinfeld module defined over a field l he constructs a continuous representation wl d of the weil group of l into a certain division algebra which encodes the sato tate law when has generic characteristic and l is finitely generated we shall describe the image of up to commensurability as an application we give improved upper bounds for the drinfeld module analogue of the lang trotter conjecture

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

the zeros of dedekind zeta functions and class numbers of cm fields let f f be a finite normal extension of number fields with galois group gal f f let be an irreducible character of gal f f of degree greater than one and l s x the associated artin l function assuming the truth of artin s conjecture we have explicitly determined a zero free region about 1 for l s x as an application we show that for a cm field k of degree 2n with solvable normal closure over q if n 370 as well as n 384 400 416 448 512 then the relative class number of k is greater than one

regional robust stabilisation and domain of attraction estimation for mimo uncertain nonlinear systems with input saturation abstractthe regional robust stabilisation by means of linear time invariant state feedback control for a class of uncertain mimo nonlinear systems with parametric uncertainties and control input saturation is investigated the nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain of attraction doa inside a given polytopic region in the state space a novel set of sufficient linear matrix inequality lmi conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger doas by considering the polytopic description of the saturated inputs a few examples are presented showing favourable comparisons with recently published similar control design methodologies

cyclicity of elliptic curves modulo p and elliptic curve analogues of linnik s problem let e be an elliptic curve defined over and of conductor n for a prime we denote by the reduction of e modulo p we obtain an asymptotic formula for the number of primes p x for which is cyclic assuming a certain generalized riemann hypothesis the error terms that we get are substantial improvements of earlier work of j p serre and m ram murty we also consider the problem of finding the size of the smallest prime p p e for which the group is cyclic and we show that under the generalized riemann hypothesis p e log n 4 if e is without complex multiplication and p e log n 2 if e is with complex multiplication for any 0 1

cyclicity of cm elliptic curves modulo p let e be an elliptic curve defined over q and with complex multiplication for a prime p of good reduction let e be the reduction of e modulo p we find the density of the primes p x for which e f p is a cyclic group an asymptotic formula for these primes had been obtained conditionally by j p serre in 1976 and unconditionally by ram murty in 1979 the aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula

guaranteed cost control of uncertain nonlinear systems via polynomial lyapunov functions in this paper we consider the problem of guaranteed cost control for a class of uncertain nonlinear systems we derive linear matrix inequality conditions for the regional robust stability and performance problems based on lyapunov functions which are polynomial functions of the state and uncertain parameters the performance index is calculated over a set of initial conditions also we discuss the synthesis problem for a class of affine control systems numerical examples illustrate our method

reductions of an elliptic curve and their tate shafarevich groups in this paper we study the tate shafarevich groups iii p of the reductions modulo primes p of an elliptic curve e considered as being defined over their function fields assuming grh when e has no cm we show that iii p is trivial for a positive proportion of primes p provided e has an irrational point of order two

frobenius fields for elliptic curves let e bbb q be an elliptic curve over the field of rational numbers with mathop rm end bar bbb q e bbb z let k be a fixed imaginary quadratic field over bbb q and x a positive real number for each prime p of good reduction for e let pi p e be a root of the characteristic polynomial of the frobenius endomorphism of e over the finite field bbb f p let pi e k x be the number of primes p leq x such that the field extension bbb q pi p e is the fixed imaginary quadratic field k we present upper bounds for pi e k x obtained using two different approaches the first one inspired from work of serre is to consider the image of frobenius in a mixed galois representation associated to k and to the elliptic curve e the second one inspired from work of cojocaru fouvry and murty is based on an application of the square sieve the bounds obtained using the first approach are better pi e k x ll x 4 5 log x 1 5 and are the best known so far the bounds obtained using the second approach are weaker but are independent of the number field k a property which is essential for other applications all results are conditional upon grh

control of rational systems using linear fractional representations and linear matrix inequalities abstract every system of the form x f x u y g x u where f and g are rational functions of the state x and linear functions of the input u possesses a linear fractional representation lfr in this lfr the system is viewed as an lti system connected with a diagonal feedback element linear in the state we devise an algorithm for computing lfrs based on this construction we give sufficient conditions for various properties to hold for the open loop system these include checking whether a given polytope is stable finding a lower bound on the decay rate of trajectories initiating in this polytope computing an upper bound on the l2 gain etc all these conditions are obtained by analyzing the properties of a differential inclusion related to the lfr and given as convex optimization problems over linear matrix inequalities lmis we show how to use this approach for static state feedback synthesis we then generalize the results to dynamic output feedback synthesis in the case when f and g are linear in every state coordinate that is not measured extensions towards a class of nonrational and uncertain nonliner systems are discussed

computation of lyapunov functions for smooth nonlinear systems using convex optimization linearly parameterized non quadratic lyapunov functions for smooth nonlinear systems are computed numerically using large scale linear or quadratic programming

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

counting primes groups and manifolds let lambda sl 2 z be the modular group and let c n lambda be the number of congruence subgroups of lambda of index at most n we prove that lim n infinity log c n lambda log n 2 log log n 3 2 sqrt 2 4 the proof is based on the bombieri vinogradov riemann hypothesis on the average and on the solution of a new type of extremal problem in combinatorial number theory similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple lie groups if g is such a lie group and gamma is an irreducible lattice of g it turns out that the subgroup growth of gamma is independent of the lattice and depends only on the lie type of the direct factors of g it can be calculated easily from the root system the most general case of this result relies on the generalized riemann hypothesis but many special cases are unconditional the proofs use techniques from number theory algebraic groups finite group theory and combinatorics

chebotarev density theorem in short intervals for extensions of _ an old open problem in number theory is whether the chebotarev density theorem holds in short intervals more precisely given a galois extension e of q with galois group g a conjugacy class c in g and a 1 e 0 one wants to compute the asymptotic of the number of primes x p x x e with frobenius conjugacy class in e equal to c the level of difficulty grows as e becomes smaller assuming the generalized riemann hypothesis one can merely reach the regime 1 e 1 2 we establish a function field analogue of the chebotarev theorem in short intervals for any e 0 our result is valid in the limit when the size of the finite field tends to and when the extension is tamely ramified at infinity the methods are based on a higher dimensional explicit chebotarev theorem and applied in a much more general setting of arithmetic functions which we name g factorization arithmetic functions

the explicit sato tate conjecture and densities pertaining to lehmer type questions let f z n 1 af n q n s k 0 n be a newform with squarefree level n that does not have complex multiplication for a prime p define p 0 to be the angle for which af p 2p k 1 2 cos p let i 0 be a closed subinterval and let d st 2 sin 2 d be the sato tate measure of i assuming that the symmetric power l functions of f satisfy certain analytic properties all of which follow from langlands functoriality and the generalized riemann hypothesis we prove that if x is sufficiently large then p x p i st i x 2 dt log t x log nkx log x with an implied constant of 3 33 by letting i be a short interval centered at 2 and counting the primes using a smooth cutoff we compute a lower bound for the density of positive integers n for which af n 6 0 in particular if is the ramanujan tau function then under the aforementioned hypotheses we prove that lim x n x n 6 0 x 1 1 54 10 13 we also discuss the connection between the density of positive integers n for which af n 6 0 and the number of representations of n by certain positive definite integer valued quadratic forms

reductions of points on elliptic curves let e be an elliptic curve defined over mathbb q let be a subgroup of rank r of the group of rational points e mathbb q of e for any prime p of good reduction let bar gamma be the reduction of modulo p under certain standard assumptions we prove that for almost all primes p i e for a set of primes of density one we have bar gamma geq frac p f p where f x is any function such that f x at an arbitrary slow speed as x this provides additional evidence in support of a conjecture of lang and trotter from 1977

brief paper robust h filter design for a class of discrete time parameter varying systems this paper proposes a robust h filter design method for discrete time linear parameter varying systems with the state space model matrices depending rationally on time varying parameters the system is described by a difference algebraic representation and the parameters admissible values and variations are assumed to belong to given intervals a convex optimization approach in terms of linear matrix inequalities is devised for designing robust filters the filter design is based on lyapunov functions with rational dependence on the system parameters and incorporates information on available bounds on the parameters variation the proposed method can be also applied to the design of gain scheduled h filters numerical examples illustrate the effectiveness of the proposed filter design

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

finite horizon approximate optimal guaranteed cost control of uncertain nonlinear systems with application to mars entry guidance this paper studies the finite horizon optimal guaranteed cost control gcc problem for a class of time varying uncertain nonlinear systems the aim of this problem is to find a robust state feedback controller such that the closed loop system has not only a bounded response in a finite duration of time for all admissible uncertainties but also a minimal guaranteed cost a neural network nn based approximate optimal gcc design is developed initially by modifying the cost function to account for the nonlinear perturbation of system the optimal gcc problem is transformed into a finite horizon optimal control problem of the nominal system subsequently with the help of the modified cost function together with a parametrized bounding function for all admissible uncertainties the solution to the optimal gcc problem is given in terms of a parametrized hamilton jacobi bellman phjb equation then a nn method is developed to solve offline the phjb equation approximately and thus obtain the nearly optimal gcc policy furthermore the convergence of approximate phjb equation and the robust admissibility of nearly optimal gcc policy are also analyzed finally by applying the proposed design method to the entry guidance problem of the mars lander the achieved simulation results show the effectiveness of the proposed controller

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

robust observed state feedback design for discrete time systems rational in the uncertainties design of controllers in the form of a state feedback coupled to a state observer is studied in the context of uncertain systems the classical approach by luenberger is revisited results provide a heuristic design procedure that mimics the independent state feedback observer gains design by minimizing the coupling of observation error dynamics on the ideal state feedback dynamics the proposed design and analysis conditions apply to linear systems rationally dependent on uncertainties defined in the cross product of polytopes convex linear matrix inequality results are given thanks to the combination of a new descriptor multi affine representations of systems and the s variable approach stability and h performances are assessed by multi affine parameter dependent lyapunov matrices for both cases of constant and time varying uncertainties or combinations of the two numerical complexity issues and ways to keep it as limited as possible are discussed and illustrated on an academic example

optimal control of uncertain nonlinear quadratic systems abstract this paper addresses the problem of robust and optimal control for the class of nonlinear quadratic systems subject to norm bounded parametric uncertainties and disturbances by using an approach based on the guaranteed cost control theory a technique is proposed to design a state feedback controller ensuring for the closed loop system i the local exponential stability of the zero equilibrium point ii the inclusion of a given region into the domain of exponential stability of the equilibrium point iii the satisfaction of a guaranteed level of performance in terms of boundedness of some optimality indexes in particular a sufficient condition for the existence of a state feedback controller satisfying a prescribed integral quadratic index is provided followed by a sufficient condition for the existence of a state feedback controller satisfying a given l 2 gain disturbance rejection constraint by the proposed design procedures the optimal control problems dealt with here can be efficiently solved as linear matrix inequality lmi optimization problems

the number of hecke eigenvalues of same signs we give the best possible lower bounds in order of magnitude for the number of positive and negative hecke eigenvalues this improves upon a recent work of kohnen lau and shparlinski also we study an analogous problem for short intervals

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