selberg supertrace formula for super riemann surfaces iii bordered super riemann surfaces this paper is the third in a sequel to develop a super analogue of the classical selberg trace formula the selberg supertrace formula it deals with bordered super riemann surfaces the theory of bordered super riemann surfaces is outlined and the corresponding selberg supertrace formula is developed the analytic properties of the selberg super zeta functions on bordered super riemann surfaces are discussed and super determinants of dirac laplace operators on bordered super riemann surfaces are calculated in terms of selberg super zeta functions

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

geometric zeta functions of locally symmetric spaces the theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces we show that the zeta functions can be continued to meromorphic functions on the plane describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of patterson we also extend the range of zeta functions in considering higher dimensional flats

eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps we study eta invariants of dirac operators and regularized determinants of dirac lapla cians over hyperbolic manifolds with cusps and their relations with selberg zeta functions using the selberg trace formula and a detailed analysis of the unipotent orbital integral we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant we also show that the selberg zeta function of odd type has a meromorphic extension over c prove a relation of the eta invariant and a certain value of the selberg zeta function of odd type and derive a corresponding functional equation these results generalize the earlier work of john millson to hyperbolic manifolds with cusps we also prove that the selberg zeta function of even type has a meromorphic extension over c relate it to the regularized determinant and obtain a corresponding functional equation

adaptive sliding mode control in discrete time systems abstract discrete time sliding mode control is considered the case of known parameters is first analyzed leading to a new definition of the so called equivalent control as the piecewise constant control that reduces to zero in finite time the distance of the system state from the sliding manifold in the presence of bounded parametric uncertainties an adaptive control scheme that guarantees the asymptotic satisfaction of the same control objective is presented the main feature of this approach is the reduction of the order of the relevant error equation and the possibility of dealing with the nonmatched uncertainties introduced by the sampling process

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

sliding mode control of a discrete system abstract conventional sliding mode control designed on the basis of a continuous system is known to be robust to the plant uncertainties a realized digital system however not only yields chattering but also may become unstable by a long sampling interval this paper presents a stable discrete sliding mode control insensitive to the choice of sampling interval and not yielding chattering the control system is designed on the basis of a discrete lyapunov function and a sufficient condition of the control gain to make the system stable is given contrary to the continuous case the derived switching plane of the control law is different from the sliding mode and in its neighborhood the control law is given by the linear state feedback simulations show the effectiveness of the proposed method

on the stability of discrete time sliding mode control systems the stability of discrete time sliding mode control systems is investigated and a new sliding mode condition is suggested it is shown that the control must have upper and lower bounds a numerical example is discussed as an illustration

functional determinants on mandelstam diagrams we investigate the special properties of mandelstam metrics in regard to changing weights in path integrals and relations between determinants of different spins regularizations of determinants are discussed along the lines of sonoda weyl anomalies developing at zeroes of metrics in reparametrization invariant regularizations are evaluated in terms of arakelov metrics holomorphic forms are constructed and determinant identities for arakelov and mandelstam metrics rigorously established for any weight and generic even and odd spin structures

a local families index formula for overline partial operators on punctured riemann surfaces using heat kernel methods developed by vaillant a local index formula is obtained for families of overline partial operators on the teichmuller universal curve of riemann surfaces of genus g with n punctures the formula also holds on the moduli space mathcal m g n in the sense of orbifolds where it can be written in terms of mumford morita miller classes the degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the quillen connection a result originally obtained by takhtajan and zograf

numerical solutions of linear mixed boundary value problems using stochastic representations the application of ito s formula induces some probabilistic representations of solutions of deterministic linear problems with boundary conditions of dirichlet neumann fourier and mixed types these representations are used to establish some easily implementable algorithms which compute an approximate solution by means of simulation of reflected random walks they do not require selected configurations at the neighbourhood of the domain boundary nor a discretization mesh the associated simulation methods are obtained and applied to problems for each class of boundary conditions numerical experiments with distributed source in two or three dimensional geometries and computational results with estimation of error are reported 1997 by john wiley sons ltd

selberg trace formula for bordered riemann surfaces hyperbolic elliptic and parabolic conjugacy classes and determinants of maass laplacians the selberg trace formula for automorphic forms of weightme on bordered riemann surfaces is developed the trace formula is formulated for arbitrary fuchsian groups of the first kind with reflection symmetry which include hyperbolic elliptic and parabolic conjugacy classes in the case of compact bordered riemann surfaces we can explicitly evaluate determinants of maass laplacians for both dirichlet and neumann boundary conditions respectively some implications for the open bosonic string theory are mentioned

variable structure control design for uncertain discrete time systems the main differences between the designs of discrete time variable structure systems vss and continuous time vss are the determination of the switching hyperplane and the satisfaction of the hitting sliding condition these differences yield many difficulties in the design of sliding mode control for the discrete time vss this note tries to overcome these difficulties and design a simple sliding mode control for a single input discrete time vss such that not only the discrete time uncertain system is stabilized robustly but also the chattering along the sliding mode is reduced explicitly instead of the conventional switching hyperplane in the discrete time case a switching region which includes the hyperplane s sub 0 and its neighborhood is determined and used the determination of the switching region is to locate the equilibrium point of the nominal subsystem on each hyperplane s sub p outside the switching region such that the state motions inside and outside the switching region will occur alternately until the state of the closed loop system reaches the origin moreover we prove that the robustness property is retained whenever the uncertainty satisfies a standard matching condition

regularized products and determinants zeta regularized products are used to define determinants of operators in infinite dimensional spaces this article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination

discrete time output feedback sliding mode control design for uncertain systems using linear matrix inequalities an output feedback based sliding mode control design methodology for discrete time systems is considered in this article in previous work it has been shown that by identifying a minimal set of current and past outputs an augmented system can be obtained which permits the design of a sliding surface based upon output information only if the invariant zeros of this augmented system are stable in this work a procedure for realising discrete time controllers via a particular set of extended outputs is presented for non square systems with uncertainties this method is applicable when unstable invariant zeros are present in the original system the conditions for existence of a sliding manifold guaranteeing a stable sliding motion are given a procedure to obtain a lyapunov matrix which simultaneously satisfies both a riccati inequality and a structural constraint is used to formulate the corresponding control to solve the reachability problem a numerical method using linear matrix inequalities is suggested to obtain the lyapunov matrix finally the design approach given in this article is applied to an aircraft problem and the use of the method as a reconfigurable control strategy in the presence of sensor failure is demonstrated

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

comments on variable structure control design for uncertain discrete time systems states that the main result theorem 2 of the above paper wang wu and yang ieee trans autom contr vol 39 no 1 p 99 102 1994 is incorrect since the convergence of the system state onto the hyperplane s x cx 0 is guaranteed only in the region where the nonlinear control component is applied region a the authors also state that no stability properties can be inferred in a case when only linear equivalent control u sub eq is used region b or when the applied control provokes a permanent switching between a and b regions the latter case is illustrated by a counterexample showing the instability of the control scheme proposed in the paper

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

regularized determinants of the laplacian for cofinite kleinian groups with finite dimensional unitary representations for cofinite kleinian groups or equivalently finite volume three dimensional hyperbolic orbifolds with finite dimensional unitary representations we evaluate the regularized determinant of the laplacian using w muller s regularization we give an explicit formula relating the determinant to the selberg zeta function

determinant expression of selberg zeta functions ii we will prove that for psl 2 r and its cofinite subgroup the selberg zeta function is expressed by the determinant of the laplacian we will also give an explicit calculation in case of congruence subgroups and deduce that the part of the determinant of the laplacian composed of the continuous spectrum is expressed by dirichlet functions the first discovery of the relation between the selberg zeta function and the determinant of the laplacian was by physicists 3 4 7 sarnak 14 and voros 15 obtained the determinant expression of selberg zeta functions for compact riemann surfaces with torsionfree fundamental groups in those cases all the spectrum of the laplacians are discrete the determinant was defined via the holomorphy at the origin of the spectral zeta function of minakshisundaram and pleijel 13 for noncompact but finite riemann surfaces with torsion free fundamental groups these results are generalized by efrat 5 in this case there exist both discrete and continuous spectrum he constructs the spectral zeta function composed not only of eigenvalues but some values concerning continuous spectrum which are decided by all the poles of the scattering de terminant in the selberg trace formula the determinant of the laplacian is defined by the standard method with the holomorphy of the spectral zeta func tion at the origin the aim of the present paper is to generalize his results to the case with any fundamental group y 3 and to give some arithmetic examples of the determinant of the laplacians 4 in 4 we restrict ourselves to the case when y is a congruence subgroup of psl 2 z in this case the partial spectral zeta function composed of only eigenvalues is also holomorphic at the origin 11 theorem 3 3 hence we have a decomposition of the determinant into parts corresponding to the discrete and continuous spectrum the scatter ing determinant is expressed very explicitly by huxley 8 in terms of dirichlet 7 functions i x almost all the poles of the scattering determinant are de scribed by nontrivial zeros of l s x the continuous part of the determinant

determinants of laplacians on surfaces of finite volume determinants of the laplace and other elliptic operators on compact manifolds have been an object of study for many years see mp rs vor up until now however the theory of determinants has not been extended to non compact situations since these typically involve a mixture of discrete and continuous spectra recent advances in this theory which are partially motivated by developments in mathematical physics have led to a connection in the compact riemann surface case between determinants of laplacians on spinors and the selberg zeta function of the underlying surface see dp kie sar vor

comments on on the stability of discrete time sliding mode control systems with reply in the above paper by s z sarpturk and y istefanopulos see ieee trans automat contr vol ac 32 p 930 2 1987 stability conditions for discrete time sliding mode control systems are presented which require the stabilizing control to be upper and lower bounded the present author shows that these bounds depend on the distance of the system state from the sliding manifold and that they converge as the state approaches the manifold sarpturk and istefanopulos agree that the bounds of the discrete time control guaranteeing the stability of the sliding mode system depend also on the distance of the state from the sliding manifold therefore in order to guarantee the stability of the sliding mode system the controls must be chosen so as to satisfy the convergence and sliding bounds derived in the paper

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

determinant of the laplacian on a non compact three dimensional hyperbolic manifold with finite volume the functional determinant of laplace type operators on a three dimensional non compact hyperbolic manifold with invariant fundamental domain of finite volume is expressed via the selberg zeta function related to the picard group

on discrete time variable structure sliding mode control the purpose of this paper is to show the limitations of discrete time variable structure sliding mode control and that the equivalent control must be used in order to have sliding in a neighborhood of the switching surface conflicting requirements for the sliding mode controller behavior in the continuous and discrete time domains are revealed and analyzed a linear control law for an uncertain discrete time linear plant with bounded uncertainties is analyzed and its superiority over nonlinear controllers is demonstrated the conclusion of the obtained results is that in the discrete time variable structure sliding mode controller design unlike in the continuous time the designer may have limited flexibility in selecting controller architectures

explicit hard bounding functions for boundary value problems for elliptic partial differential equations abstract we obtain tight super and sub solutions or hard bounding functions for the probabilistic representation of the solution to the boundary value problem for second order elliptic partial differential equations in an explicit polynomial form unlike accurate approximate solutions by the existing discretization based methods our hard bounding polynomial functions act as pointwise 100 confidence intervals within which the solution is guaranteed to exist our methodology can deal with a wide range of boundary conditions even of fully mixed type with dirichlet neumann and robin components without increasing the complexity the quality of hard bounds can be improved by introducing piecewise polynomial functions and domain decomposition convergence of hard bounding functions is proved under technical conditions numerical results are presented throughout the paper to support our theoretical developments and to illustrate the effectiveness of our methodology

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points