eigenfunctions of the weil representation of unitary groups of one variable in this paper we construct explicit eigenfunctions of the local weil representation on unitary groups of one variable in the p adic case when p is odd the idea is to use the lattice model and the results will be used to compute special values of certain hecke l functions in separate papers we also recover moen s results on when a local theta lifting from u 1 to itself does not vanish 0 introduction and notation let e f be a quadratic extension of local fields if v is an hermitian space over e and w is a skew hermitian space over e the unitary groups g g w and g g v form a reductive dual pair in sp w where w v w has the symplectic form 12 tre f over f according to a well known result this dual pair splits in the metaplectic cover of sp w and thus has a weil representation we consider the very special case in which dime v dimew 1 in this case g g u 1 e is compact and abelian where e is the kernel of the norm map n e f so irreducible representations of g are just characters and the weil representation has a direct sum decomposition

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

stable adaptive control for a class of nonlinear systems using a modified lyapunov function investigates the adaptive control design for a class of nonlinear systems using lyapunov s stability theory the proposed method is developed based on a novel lyapunov function which removes the possible controller singularity problem in some of the existing adaptive control schemes using feedback linearization techniques the resulting closed loop system is proven to be globally stable and the output tracking error converges to an adjustable neighborhood of zero

on the arithmetic fundamental lemma through lie algebras we prove that the arithmetic fundamental lemma conjecture afl of wei zhang is equivalent to a similar conjecture but for lie algebras in the case of non degenerate intersection we use this result to give a simplified proof of the afl for n 3 the idea for the reduction to the lie algebra is due to wei zhang

heegner divisors in the moduli space of genus three curves s kondo used periods of k3 surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6 ball in this paper we study heegner divisors in the ball quotient given by arithmetically defined hyperplane sections of the ball we show that the corresponding loci of genus three curves are given by hyperelliptic curves singular plane quartics and plane quartics admitting certain rational splitting curves

decentralized control design for large scale systems with strong interconnections using neural networks we propose a decentralized neural network nn controller for a class of large scale nonlinear systems with the strong interconnections the nns are used to approximate the unknown subsystems and interconnections due to the functional approximation capabilities of nns the additional precautions are not required to be made for avoiding the possible control singularity problems semiglobal asymptotic stability results are obtained and the tracking error converges to zero furthermore the issue of transient performance of the subsystems is also addressed under an analytical framework

adaptive neural pd control with semiglobal asymptotic stabilization guarantee this paper proves that adaptive neural plus proportional derivative pd control can lead to semiglobal asymptotic stabilization rather than uniform ultimate boundedness for a class of uncertain affine nonlinear systems an integral lyapunov function based ideal control law is introduced to avoid the control singularity problem a variable gain pd control term without the knowledge of plant bounds is presented to semiglobally stabilize the closed loop system based on a linearly parameterized raised cosine radial basis function neural network a key property of optimal approximation is exploited to facilitate stability analysis it is proved that the closed loop system achieves semiglobal asymptotic stability by the appropriate choice of control parameters compared with previous adaptive approximation based semiglobal or asymptotic stabilization approaches our approach not only significantly simplifies control design but also relaxes constraint conditions on the plant two illustrative examples have been provided to verify the theoretical results

relating invariant linear form and local epsilon factors via global methods we use the recent proof of jacquet s conjecture due to harris and kudla hk and the burger sarnak principle see bs to give a proof of the relationship between the existence of trilinear forms on representations of gl 2 k u for a non archimedean local field k u and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in p1 theorem 1 4 the method used is very flexible and gives a global proof of a theorem of saito and tunnell about characters of gl 2 using a theorem of waldspurger w theorem 2 about period integrals for gl 2 and also an extension of the theorem of saito and tunnell by this author in p3 theorem 1 2 which was earlier proved only in odd residue characteristic in the appendix to this article h saito gives a local proof of lemma 4 which plays an important role in the article

on the μ invariant of anticyclotomic p adic l functions for cm fields following hida s ideas we determine the invariant of the anticyclotomic katz p adic l functions for self dual characters when the branch conductor is not divisible by inert or ramified primes above 2 thus we generalize finis result to general cm fields in this case

local e factors and characters of gl 2 introduction the representation theory of gl 2 k for k a nonarchimedean local field is somewhat more complicated than the corresponding theory for gl 2 r in particular the formulas for the characters of discrete series representation of gl 2 r have a concise closed form while the published character formulas for supercuspidal representations of gl 2 k occupy several pages 10 14 the purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of gl 2 k the representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in gl 2 k and the subspace stabilized by an open compact subgroup of gl 2 k is finite dimensional irreducible admissible representations ir of gl 2 k satisfy schur s lemma so that the center k of gl 2 k acts by multiplication by a quasicharacter there exists a locally constant character function ch defined on a dense open subgroup of gl 2 k which determines the isomorphism class of ir 7 section 7 jacquet and langlands 7 section 2 introduced factors e 10 x attached to the twist of ir by quasicharacters x of k which describe the action of r in a specific model for each separable quadratic extension l of k there exists an irreducible admissible representation bcl k 1r of gl 2 l the base change of ir to l 9 section 2 this representation of gl 2 l enters into the expression for the character the main result of this paper is the following formula recall that conjugacy classes of nonsplit cartan subgroups are parameterized by quadratic separable field extensions l of k via embeddings of l in gl 2 k

the regularized siegel weil formula the second term identity and the rallis inner product formula in this paper we establish the second term identity of the siegel weil formula in full generality and derive the rallis inner product formula for global theta lifts for any dual pair as a corollary we resolve the non vanishing problem of global theta lifts initiated by steve rallis

the gan gross prasad conjecture for u n u n abstract we prove the gan gross prasad conjecture for u n u n under some local conditions using a relative trace formula we deduce some new cases of the gan gross prasad conjecture for u n 1 u n from the case of u n u n

vibration transmissibility of a unidirectional multi degree of freedom system with multiple dynamic absorbers an analytical and closed form vibration transmissibility of a general unidirectional multi degree of freedom system with multiple dynamic absorbers is studied in this paper the dynamics of the primary system and the dynamic absorbers are described as a two way state flow graph model to reduce the analysis work and simplify the representation of the derived results for analysis the entire graph model is divided into two parts the primary system and the dynamic absorbers based on the graph model the analytical and closed form expressions of the force and displacement transmissibility are derived using the topology scheme according to the developed results the vibration transmissibility for a uniform primary system can be represented as polynomial forms moreover the vibration transmissibility of a uniform mdof primary system with multiple non identical absorbers with multiple identical absorbers and with a single absorber are calculated and compared with those presented in other papers finally two numerical examples are investigated to show the implementation of this method

local theta correspondence of tempered representations and langlands parameters in this paper we give an explicit determination of the theta lifting for symplectic orthogonal and unitary dual pairs over a nonarchimedean field f of characteristic 0 we determine when theta lifts of tempered representations are nonzero and determine the theta lifts in terms of the local langlands correspondence

twisted doubling integrals for classical groups we describe the twisted doubling integrals of cai friedberg ginzburg kaplan in a conceptual way this also extends the construction to the quaternionic unitary groups we carry out the unfolding argument uniformly in this article to do so we define a family of degenerate whittaker coefficients that are suitable in this setup and study some of their properties we also prove certain related global and local results that use the same tools

the gross prasad conjecture and local theta correspondence we establish the fourier jacobi case of the local gross prasad conjecture for unitary groups by using local theta correspondence to relate the fourier jacobi case with the bessel case established by beuzart plessis to achieve this we prove two conjectures of prasad on the precise description of the local theta correspondence for almost equal rank unitary dual pairs in terms of the local langlands correspondence the proof uses arthur s multiplicity formula and thus is one of the first examples of a concrete application of this global reciprocity law

central values of hecke l functions of cm number fields 0 introduction it is well known that the zeta function of cm complex multiplication abelian varieties can be given in terms of l functions of associated hecke characters in this paper we prove a formula expressing the central special value of the l function of certain hecke characters in terms of theta functions the formula easily implies that the central value is nonnegative and yields a criterion for its positivity combining this criterion with the work of arthaud and rubin we show that certain cm elliptic curves have mordell weil rank zero over their field of definition let f be a totally real number field of degree t and let be a quadratic hecke character of f of conductor f such that 2f f 1 given a cm extension e of f we consider the twist can of by a canonical hecke character can of

an explicit formula for siegel series combining induction formulas for local densities with a functional equation for the siegel series we give an explicit formula for the siegel series by this formula we also give an explicit formula for the fourier coefficients of the siegel eisenstein series

endoscopy theta liftings and period integrals for the unitary group in three variables l packets for the quasi split unitary group in three variables u 3 we shall give an explicit parametrization of these l packets using theta liftings and describe some relations between the structure of l packets and certain period integrals we also prove that an endoscopic l packet contains a unique generic representation in local and global cases suppose that g is a reductive algebraic group over a local or global field

free vibrations of uniform timoshenko beams with attachments the problem of free transverse vibrations of timoshenko beams with attachments like translational and rotational springs concentrated mass including the moment of inertia linear undamped oscillators and additional supports is considered the frequency equation for the combined system is derived by means of the lagrange multiplier formalism the exact solution of the free vibration problem of the beam without attachments is taken into account for the formulation of the free vibration problem of the combined system numerical examples show the separate or coupling influences of the additional elements on the combined system s frequencies the comparison of results obtained by using the present approach with results of the exact solution indicates a good agreement

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

howe correspondence for real unitary groups roger howe proved that for any reductive dual pair g g in the symplectic groupsp 2n r there is a one to one correspondence of irreducible admissible representations of some two fold covers ofgandg we determine this correspondence explicitly for dual pairs of the form u p q u r s withr s p q and describe it in terms of langlands parameters in this case the correspondence may be understood in a natural way as a correspondence of representations of the linear groups rather than the appropriate covers we show that every irreducible admissible representation ofu p q occurs in the correspondence with precisely one unitary group of equal rank this result verifies a conjecture of harris kudla and sweet who investigated the correspondence for unitary groups of equal size overp adic fields the correspondence of discrete series representations was determined by j s li for induced representations the correspondence is obtained in a natural way from the corresponding discrete series on the respective levi factors of the parabolic subgroups ofu p q andu r s generalizing a result of li we show that under the correspondence representations with nonzero cohomology are matched in an interesting way with unitarity not necessarily preserved the proof uses the induction principle which is due to kudla and an argument involvingk types and the space of joint harmonics howe

stable adaptive control using fuzzy systems and neural networks stable direct and indirect adaptive controllers are presented which use takagi sugeno fuzzy systems conventional fuzzy systems or a class of neural networks to provide asymptotic tracking of a reference signal for a class of continuous time nonlinear plants with poorly understood dynamics the indirect adaptive scheme allows for the inclusion of a priori knowledge about the plant dynamics in terms of exact mathematical equations or linguistics while the direct adaptive scheme allows for the incorporation of such a priori knowledge in specifying the controller we prove that with or without such knowledge both adaptive schemes can learn how to control the plant provide for bounded internal signals and achieve asymptotically stable tracking of a reference input in addition for the direct adaptive scheme a technique is presented in which linguistic knowledge of the inverse dynamics of the plant may be used to accelerate adaptation the performance of the indirect and direct adaptive schemes is demonstrated through the longitudinal control of an automobile within an automated lane

a weak second term identity of the regularized siegel weil formula for unitary groups following w t gan and s takeda we obtain a weak second term identity of the regularized siegel weil formula for the unitary dual pair u n n u v where v is a split hermitian space of dimension 2r with r 1 le n le 2r 1 as an application we obtain a rallis inner product formula for theta lifts from u w to u v for a skew hermitian space w of dimension n

an improvement on stable adaptive control for a class of nonlinear systems in the work of zhang et al an excellent neural network controller is provided by introducing a modified lyapunov function to eliminate the singularity issue completely in this note we focus on the improvement on the work of zhang et al for a class of unknown nonlinear systems

feedback linearization using neural networks abstract for a class of single input single output continuous time nonlinear systems a multilayer neural network based controller that feedback linearizes the system is presented control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system the control structure consists of a feedback linearization portion provided by two neural networks plus a robustifying portion that keeps the control magniture bounded a stability proof is given in the sense of lyapunov it is shown that all the signals in the closed loop system are uniformly ultimately bounded no off line learning phase is needed initialization of the network weights is straightforward

adaptive tracking control for stochastic uncertain non linear systems satisfying short and long term cost criteria the tracking control problem for a class of stochastic and uncertain non linear systems is addressed the proposed controller uses suitable radial basis function neural network designs for the approximation of the unknown non linearities while it is arbitrarily regulated in order to effectively penalize the tracking error this regulation is implemented through a risk sensitivity parameter a stability analysis based on lyapunov functions obtained by the backstepping technique proves that all the error variables are bounded in probability simultaneously for any given risk sensitivity parameter the system performance is regulated with respect to both a desired small average tracking error and low long term average cost in accordance to a risk sensitive cost criterion moreover the larger this parameter is the mean square tracking error becomes semiglobally uniformly ultimately bounded in a smaller area while a lower level of a long term average cost is achieved the effectiveness of the design approac

l functions and theta correspondence for classical groups using the doubling method of piatetski shapiro and rallis we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the l functions and local theta correspondence

doubling constructions and tensor product l functions the linear case we present an integral representation for the tensor product l function of a pair of automorphic cuspidal representations one of a classical group the other of a general linear group our construction is uniform over all classical groups and is applicable to all cuspidal representations it does not require genericity the main new ideas of the construction are the use of generalized speh representations as inducing data for the eisenstein series and the introduction of a new global and local model which generalizes the whittaker model here we consider linear groups but our construction also extends to arbitrary degree metaplectic coverings this will be the topic of an upcoming work

a complex ball uniformization of the moduli space of cubic surfaces via periods of k3 surfaces in this paper we show that the moduli space of nodal cubic surfaces is iso morphic to a quotient of a 4 dimensional complex ball by an arithmetic subgroup of the unitary group this complex ball uniformization uses the periods of certain k3 surfaces which are naturally associated to cubic surfaces a similar uniformization is given for dif ferent covers of the moduli space corresponding to geometric markings of the picard group or a choice of a line on the surface we also give a detailed description of the boundary com ponents corresponding to singular surfaces

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

a regularized siegel weil formula for unitary groups for the quasi split unitary group u n n we prove a refined analogue of the result of s s kudla and s rallis on an identity a regularized siegel weil formula between a residue of an eisenstein series and the integral of a certain theta function

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

on the non vanishing of theta liftings of tempered representations of u p q abstract in this paper we give an explicit determination of the non vanishing of the theta liftings of tempered representations for unitary dual pairs u p q u r s for arbitrary non negative integers p q r s for discrete series representations in terms of harish chandra parameters we give a complete criterion when the theta lifts are nonzero for tempered representations we determine the non vanishing in terms of the local langlands correspondence assuming the local gan gross prasad conjecture

vibrations of elastically mounted mass supported on symmetrically crossed beams this study presents a novel method to analyze the vibration of an elastically mounted concentrated mass supported on the joint of symmetrically crossed beams with flexible foundation analytical and exact solutions of the free and forced vibration responses of the system are also derived herein the dynamics of the mounted mass and the crossed beams are expressed as two way state flow twsf graph models in which the interactions between the components are considered based on the proposed model the frequency responses of the displacement of the mounted mass and every beam are derived using a topological method moreover the force transmissibility from the vibrating mass to the foundation and the frequency equation are obtained the derived results are expressed in both analytical and closed forms also presented herein are some special cases including identical structure properties for each beam simply supported boundary for each beam mass directly mounted on the beams and their combinations

a neural network based approximation method for discrete time nonlinear servomechanism problem a feedback controller that solves the discrete time nonlinear servomechanism problem relies on the solution of a set of nonlinear functional equations known as the discrete regulator equations the exact solution of the discrete regulator equations is usually unavailable due to the nonlinearity of the system the paper proposes to approximately solve the discrete regulator equations using a feedforward neural network this approach leads to an effective way to practically solve the discrete nonlinear servomechanism problem the approach has been illustrated using the well known inverted pendulum on a cart system the simulation shows that the control law designed by the proposed approach performs much better than the conventional linear control law

arithmetic torelli maps for cubic surfaces and threefolds it has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety we give a construction which works for cubics over an arithmetic base and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties this answers away from the prime 2 an old question of deligne and a recent question of kudla and rapoport

divisibility of anticyclotomic l functions and theta functions with complex multiplication the divisibility properties of dirichlet l functions in infinite families of characters have been studied by iwasawa ferrero and washington the families considered by them are obtained by twisting an arbitrary dirichlet character with all characters of p power conductor for some prime p one has to distinguish divisibility by p the case considered by iwasawa and ferrerowashington few and by a prime p considered by washington w1 w2 ferrero and washington proved the vanishing of the iwasawa invariant of any branch of the kubota leopoldt p adic l function this means that each of the power series which p adically interpolate the nontrivial l values of twists of a fixed dirichlet character by characters of p power conductor has some coefficient that is a p adic unit in the case p washington w2 obtained the following theorem on divisibility of l values by given an integer n 1 and a dirichlet character for all but finitely many dirichlet characters of p power conductor with 1 1 n

degenerate principal series and local theta correspondence following our previous paper lz which deals with the groupu n n we study the structure of certain howe quotients p q and p q 1 which are natural sp 2n r modules arising from the oscillator representation associated with the dual pair o p q sp 2n r by embedding them into the degenerate principal series representations of sp 2n r studied in l2

e factor of representations of classical groups abstract we let g be the isometry group of a nondegenerate symmetric or skew symmetric form h on a space v over some local or global field k i e g u v h we let lg be the l group of go the identity component of g and r be the natural representation of this group we develop here a theory of l functions and e factors attached to this r in particular we develop a theory of local and global zeta integrals for such g we use very special intertwining operators therein we require an analysis of the proper normalization of such operators with such techniques we determine the correct e factors associated to local representations the issue of the correct local l factor is discussed

half twists and the cohomology of hypersurfaces a hodge structure v of weight k on which a cm field acts defines under certain conditions a hodge structure of weight k 1 its half twist in this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a hodge substructure in the cohomology we determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces we use our results to prove the existence of a kuga satake correspondence for certain cubic 4 folds

projection based adaptive neurocontrol with switching logic deadzone tuning in this brief an adaptive neural network nn controller is proposed for multiple input multiple output mimo nonlinear systems with triangular control structure and unknown control directions deadzones are employed in the projection based nn weight learning laws and the nussbaum parameter update laws with levels tuned by an innovative switching logic tuning mechanism detailed analysis using a family of lyapunov like integral functions and the function approximation capability of nns proves that all the tracking errors are semiglobal uniform ultimate bounded in small neighborhoods of the origin while the closed loop system variables state vector nn weights nussbaum parameters and the control law remain bounded a simulation study confirms the theoretical results and verifies the effectiveness of the proposed design