on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

wirtinger based multiple integral inequality for stability of time delay systems abstractnote that the conservatism of the delay dependent stability criteria can be reduced by increasing the integral terms in lyapunov krasovskii functional lkf this brief revisits the stability problem for a class of linear time delay systems via multiple integral approach the novelty of this brief lies in that a wirtinger based multiple integral inequality is employed to estimate the derivative of a class of lkf with multiple integral terms based on these innovations a new delay dependent stability criterion is derived in terms of linear matrix inequalities two numerical examples are exploited to demonstrate the effectiveness and superiority of the proposed method

stability analysis of linear systems with two delays of overlapping ranges in this paper a delay dependent stability criterion is derived for linear time delay systems with two constant delays having overlapping ranges this overlapping feature of the delays is exploited in delay dependent analysis of the system using lyapunov krasovskii approach the developed treatment of the delays reduces conservatism in analysis compared to the approaches treating the delays individually numerical examples are used to demonstrate the less conservativeness of the derived criterion

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

admissible delay upper bounds for global asymptotic stability of neural networks with time varying delays this paper is concerned with global asymptotic stability of a neural network with a time varying delay where the delay function is differentiable uniformly bounded with delay derivative bounded from above first a general reciprocally convex inequality is presented by introducing some slack vectors with flexible dimensions this inequality provides a tighter bound in the form of a convex combination than some existing ones second by constructing proper lyapunov krasovskii functional global asymptotic stability of the neural network is analyzed for two types of the time varying delays depending on whether or not the lower bound of the delay derivative is known third noticing that sufficient conditions on stability from estimation on the derivative of some lyapunov krasovskii functional are affine both on the delay function and its derivative allowable delay sets can be refined to produce less conservative stability criteria for the neural network under study finally two numerical examples are given to substantiate the effectiveness of the proposed method

technical communique absolute stability of time delay systems with sector bounded nonlinearity this paper deals with the problem of absolute stability of time delay systems with sector bounded nonlinearity some new delay dependent stability criteria are obtained and formulated in the form of linear matrix inequalities lmis neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria numerical examples show that the results obtained in this paper improve the estimate of the stability limit over some existing result

robust stabilization for uncertain saturated time delay systems a distributed delay dependent polytopic approach this technical note investigates the robust stabilization problem for uncertain linear systems with discrete and distributed delays under saturated state feedback different from the existing approaches a distributed delay dependent polytopic approach is proposed in this technical note and the saturation nonlinearity is represented as the convex combination of state feedback and auxiliary distributed delay feedback then by incorporating an appropriate augmented lyapunov krasovskii l k functional and some integral inequalities the less conservative stabilization and robust stabilization conditions are proposed in terms of linear matrix inequalities lmis the effectiveness and reduced conservatism of the proposed conditions are illustrated by numerical examples

stabilization of t s fuzzy system with time delay under sampled data control using a new looped functional this paper investigates the sampled data stabilization problem for a takagi sugeno t s fuzzy system with time delay by taking the information of states within the intervals from t k to t and t to t k 1 into account a new two side delay dependent looped functional is introduced which can not only relax the monotonic constraint of lyapunov krasovskii functional lkf but also make better use of the actual sampling pattern furthermore the sampled data fuzzy controller is designed to contain both the present and delayed state information thereby enhancing the control performance and design flexibility based on the novel augmented lkf and improved bounding technique less conservative stability criteria are derived in the form of linear matrix inequalities the superiority of proposed results is shown by two simulation examples

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

brief paper a delay decomposition approach to l2 l filter design for stochastic systems with time varying delay this paper investigates the problem of l 2 l filter design for a class of stochastic systems with time varying delay the addressed problem is the design of a full order linear filter such that the error system is asymptotically mean square stable and a prescribed l 2 l performance is satisfied in order to develop a less conservative filter design a new lyapunov krasovskii functional lkf is constructed by decomposing the delay interval into multiple equidistant subintervals and a new integral inequality is established in the stochastic setting then based on the lkf and integral inequality the delay dependent conditions for the existence of l 2 l filters are obtained in terms of linear matrix inequalities lmis the resulting filters can ensure that the error system is asymptotically mean square stable and the peak value of the estimation error is bounded by a prescribed level for all possible bounded energy disturbances finally two examples are given to illustrate the effectiveness of the proposed method

homoclinic bifurcations in reversible hamiltonian systems abstract we study the existence of homoclic solutions for reversible hamiltonian systems taking the family of differential equations u iv au u f u b 0 as a model where f is an analytic function and a b real parameters these equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others hamiltonian systems and successfully employed in situations where standard methods fail we reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations we also plot the homoclinic values configuration in parameters space giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations

parameter dependent lyapunov functional for stability of time delay systems with polytopic type uncertainties this note concerns the problem of the robust stability of a linear system with a time varying delay and polytopic type uncertainties in order to construct a parameter dependent lyapunov functional for the system we first devised a new method of dealing with a time delay system without uncertainties in this method the derivative terms of the state which is in the derivative of the lyapunov functional are retained and some free weighting matrices are used to express the relationships among the system variables and among the terms in the leibniz newton formula as a result the lyapunov matrices are not involved in any product terms of the system matrices in the derivative of the lyapunov functional this method is then easily extended to a system with polytopic type uncertainties numerical examples demonstrate the validity of the proposed criteria

on computing the maximal delay intervals for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems the authors consider delay systems of both retarded and neutral types expressed in state space forms the author s main goal is to provide a computation oriented method for computing the maximal delay intervals over which the systems under consideration maintain stability the author s results show that this can be accomplished by computing the generalized eigenvalues of certain frequency dependent matrices based on these results the author also states a necessary and sufficient condition concerning stability independent of delay for each of the retarded and neutral systems the author s results can be readily implemented and appear suitable for analyzing systems with high dimensions and many delay units

delay dependent robust stabilization of uncertain state delayed systems this paper concerns a problem of robust stabilization of uncertain state delayed systems a new delay dependent stabilization condition using a memoryless controller is formulated in terms of matrix inequalities an algorithm involving convex optimization is proposed to design a controller guaranteeing a suboptimal maximal delay such that the system can be stabilized for all admissible uncertainties

a delay dependent stability criterion for systems with uncertain time invariant delays this paper provides a new stability criterion for systems with time invariant uncertain delays based on an improved upper bound for the inner product of two vectors a new delay dependent robust stability criterion is derived which is shown by an example less conservative than existing stability criteria

chtoucas de drinfeld et correspondance de langlands on demontre la correspondance de langlands pour gl r sur les corps de fonctions la preuve generalise celle de drinfeld en rang 2 elle consiste a realiser la correspondance en rang r dans la cohomologie l adique des varietes modulaires de chtoucas de drinfeld de rang r

observer based finite time robust control of nonlinear time delay systems via hamiltonian function method this paper investigates the observer based finite time robust control problem of a general class of nonlinear time delay systems via hamiltonian function method and proposes some new finite time r

brief paper a discrete delay decomposition approach to stability of linear retarded and neutral systems this paper is concerned with stability of linear time delay systems of both retarded and neutral types by using some new simple quadratic lyapunov krasovskii functionals these lyapunov krasovskii functionals consist of two parts one part comes from some existing lyapunov krasovskii functionals employed in han q l 2005a absolute stability of time delay systems with sector bounded nonlinearity automatica 41 2171 2176 han q l 2005b a new delay dependent stability criterion for linear neutral systems with norm bounded uncertainties in all system matrices international journal of systems science 36 469 475 the other part is constructed by uniformly dividing the discrete delay interval into multiple segments and choosing proper functionals with different weighted matrices corresponding to different segments then using these new simple quadratic lyapunov krasovskii functionals some new discrete delay dependent stability criteria are derived for both retarded systems and neutral systems it is shown that these criteria for retarded systems and neutral systems are always less conservative than the ones in han 2005a and han 2005b cited above respectively numerical examples also show that the results obtained in this paper significantly improve the estimate of the discrete delay limit for stability over some other existing results

delay dependent state feedback stabilization for a networked control model with two additive input delays this paper is centered on delay dependent state feedback stabilization for a networked control model with two additive input delays firstly delay dependent stability is investigated by splitting the whole delay interval into subintervals according to the delays a lyapunov functional is constructed to reduce conservatism we handle the lyapunov functional in two ways more specifically we take the lyapunov functional as a whole to examine its positive definite rather than restrict each term of it to positive definite as usual in addition when estimating the derivative of the lyapunov functional we manage to get a fairly tighter upper bound by introducing different slack variables for the different subintervals the resulting stability results turn out dependent on the two delays separately and less conservative than some existing ones then based on the stability results state feedback stabilization is studied delay dependent conditions are formulated for the controller such that the closed loop system is asymptotically stable finally examples are given to show the less conservatism of the stability results and the effectiveness of the proposed stabilization method

a further refinement of discretized lyapunov functional method for the stability of time delay systems the discretized lyapunov functional method for the stability problem of time delay systems is further refined using a combination of integral inequality and variable elimination technique as a result the computational requirement is further reduced for the same discretization mesh for systems without uncertainty the convergence to the analytical solution is greatly accelerated for uncertain systems the new method is much less conservative numerical examples are presented to illustrate the effectiveness of the method

stability in linear delay equations abstract for linear autonomous differential difference equations of retarded or neutral type necessary and sufficient conditions are given for the zero solution to be stable hyperbolic for all values of the delays

a method for computing the interval of delay values for which a differential delay system is stable a test for determining the interval of delay values for which delay differential system is asymptotically stable is presented i e an explicit method is given for calculating h such that for 0

note on stability of linear systems with time varying delay this note considers the stability of linear systems with a time varying delay we are interested in a simple lyapunov krasovskii functional lkf approach without delay decomposition in this category all recent tractable results had a fixed bound on the allowable maximum size of the delay for years we propose a new simple lkf including the cross terms of variables and quadratic terms multiplied by a higher degree scalar function and present a new result expressed in the form of lmis we show by two well known examples that our result overcomes the previous allowable maximum size of delay and it is less conservative than the previous results having a relatively small upper bound in the derivative of time delay

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

co design of event generator and filtering for a class of t s fuzzy systems with stochastic sensor faults this paper is firstly concerned with reliable h filter design for a class of t s fuzzy systems with stochastic sensor faults under an event triggered scheme 1 a t s fuzzy model is used to approximate the nonlinear dynamics of the plant a set of stochastic variables are used to describe the sensor failure 2 an event triggered scheme which has some advantages over some existing ones is introduced to the networked control systems under the event triggered scheme the sensor data are transmitted only when the sampled measurements of the plant violate the specified event condition then an event based filter design model for t s fuzzy systems is constructed by taking the effect of event triggered scheme and the sensor faults into consideration sufficient conditions for the existence of the desired filter are established in terms of linear matrix inequalities and the explicit expression is given for the designed filter parameters a numerical example is provided to illustrate the design method

stability analysis for linear delayed systems via an optimally dividing delay interval approach this paper addresses the problem of stability for linear systems with time varying delay a novel augmented lyapunov krasovskii functional is constructed by using the idea of optimally dividing the delay interval 0 t into some variable sub intervals and line integral technology using the novel augmented functional the new delay dependent stability criteria are proposed for linear systems with time varying delay the gain is that this stability criterion can lead to much less conservative stability results compared to other methods for linear systems with delay two numerical examples are provided to verify the effectiveness of the proposed criteria

technical communique robust stability of uncertain delay differential systems of neutral type this paper focuses on the stability problem for neutral delay differential systems with norm bounded and possibly time varying uncertainty the time delay is assumed constant and known a stability criterion that is formulated in a linear matrix inequality lmi form is derived numerical examples show that the results obtained by this new criterion significantly improve the estimate of stability limit over some existing results in the literature

hierarchy of stability criterion for time delay systems based on multiple integral approach abstract taking a class of time delay systems as research object this brief aims at developing a theoretical support on the hierarchy of stability criterion which is derived by the multiple integral approach and free weighting matrix technique the hierarchy implies that the conservatism of stability criterion can be reduced by increasing the ply of integral terms in lyapunov krasovskii functional lkf together with three numerical experiments the hierarchy of stability criterion is further shown