analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

realizations of conformal current type lie algebras in this paper we obtain the realizations of some infinite dimensional lie algebras named conformal current type lie algebras in terms of a two dimensional novikov algebra and its deformations furthermore ovsienko and roger s loop cotangent virasoro algebra which can be regarded as a nice generalization of the virasoro algebra with two space variables is naturally realized as an affinization of the tensor product of a deformation of the two dimensional novikov algebra and the laurent polynomial algebra these realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite dimensional lie algebras

stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

energy gap for yang mills connections ii arbitrary closed riemannian manifolds abstract we prove an l d 2 energy gap result for yang mills connections on principal g bundles p over arbitrary closed riemannian smooth manifolds of dimension d 2 we apply our version of the ojasiewicz simon gradient inequality 16 19 to remove a positivity constraint on a combination of the ricci and riemannian curvatures in a previous l d 2 energy gap result due to gerhardt 23 theorem 1 2 and a previous l energy gap result due to bourguignon lawson and simons 10 theorem c 11 theorem 5 3 as well as an l 2 energy gap result due to nakajima 42 corollary 1 2 for a yang mills connection over the sphere s d but with an arbitrary riemannian metric

iterative learning control synthesis based on 2 d system theory an algorithm is presented for iterative learning of the control input for a linear discrete time multivariable system necessary and sufficient conditions are stated for convergence of the proposed algorithm the algorithm synthesis and analysis are based on two dimensional 2 d system theory a numerical example is given

on d type and p type ilc designs and anticipatory approach many schemes of iterative learning control ilc have been developed for continuous time non linear dynamic systems to improve tracking performance two schemes d type and p type have been the bases for many ilc designs recently the anticipatory ilc scheme has been introduced on the basis of a different approach wang 1998a 1999 in this paper these basic schemes are compared from both analysis and implementation view points the anticipatory ilc scheme is designed on the basis of a causal pair of the action taken and its resulting state variables this approach has the anticipatory characteristics of the d type ilc and the simplicity for implementation of p type ilcs a sampled data ilc scheme is presented as another form of this anticipatory ilc scheme furthermore control device saturation is taken into account and tracking error convergence results are established with proofs the convergence results are also provided in the presence of uncertainties disturbances and measurement noises ex

a direct method for minimizing the yang mills functional over 4 manifolds a direct method is employed to minimize the yang mills functional over a 4 dimensional manifold the limiting connection is shown to be yang mills but in a possibly new bundle we show that a topological invariant of the bundle is preserved by the minimizing process this implies the existence of an absolute minimum of the yang mills functional in a wide class of bundles

analytic fields on riemann surfaces ii the properties of analytic fields on a riemann surface represented by a branch covering of 1 are investigated in detail branch points are shown to correspond to the vertex operators with simple conformal properties as applications we compute determinants of bar partial j operators forzn symmetric surfaces and obtain various representations for the two loop measure in the bosonic string theory together with various identities for theta functions of hyperelliptic surfaces we also present an integral representation for the quantum part of the twist field correlation functions which describe propagation of the string on the orbifold background we also calculate the quantum part of the structure constants of the twist field operator algebra generalizing the results of dixon friedan martinec and shenker

higher adeles and non abelian riemann roch abstract we show a riemann roch theorem for group ring bundles over an arithmetic surface this is expressed using the higher adeles of beilinson parshin and the tame symbol via a theory of adelic equivariant chow groups and chern classes the theorem is obtained by combining a group ring coefficient version of the local riemann roch formula as in kapranov vasserot with results on k groups of group rings and an explicit description of group ring bundles over p 1 our set up provides an extension of several aspects of the classical frohlich theory of the galois module structure of rings of integers of number fields to arithmetic surfaces

optimal selection of the forgetting matrix into an iterative learning control algorithm a recursive optimal algorithm based on minimizing the input error covariance matrix is derived to generate the optimal forgetting matrix and the learning gain matrix of a p type iterative learning control ilc for linear discrete time varying systems with arbitrary relative degree this note shows that a forgetting matrix is neither needed for boundedness of trajectories nor for output tracking in particular it is shown that in the presence of random disturbances the optimal forgetting matrix is zero for all learning iterations in addition the resultant optimal learning gain guarantees boundedness of trajectories as well as uniform output tracking in presence of measurement noise for arbitrary relative degree

determinant bundles and virasoro algebras we consider the interplay of infinite dimensional lie algebras of virasoro type and moduli spaces of curves suggested by string theory we will see that the infinitesimal geometry of determinant bundles is governed by virasoro symmetries the mumford forms are just invariants of these symmetries the representations of virasoro algebra define twisted d modules on moduli spaces thesed modules are equations on correlators in conformal field theory

hitchin s connection and differential operators with values in the determinant bundle for a family of smooth curves we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves there exists a so called theta line bundle on the family of moduli spaces when the kodaira spencer map of the family of curves is an isomorphism we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves as an application the so called hitchin connection on the direct image of powers of the theta line bundle is derived naturally from the identification theorem a logarithmic extension to certain singular stable curves is also presented in this paper

optimal design of an optical lens we wish to construct an optimal optical lens in the sense that it focuses all rays from a given source onto a given focal point the construction can be reduced mathematically to a global study of a system of two nonlinear functional differential equations we prove that this system has precisely two analytic solutions

intervalized iterative learning control for monotonic convergence in the sense of sup norm in this paper the monotonic convergence in the sense of sup norm is studied first it is pointed out that when typical iterative learning control ilc algorithms are applied to lti systems some huge overshoot in the sense of sup norm may be observed even though monotonic convergence in the sense of norm is guaranteed then new types of ilc algorithms are proposed for continuous time systems and discrete time systems to resolve such an undesirable phenomenon by adopting an intervalized learning scheme

on the p type learning control sufficient conditions for the robustness and convergence of p type learning control algorithms for a class of time varying nonlinear systems are presented the authors prove the uniform boundedness of the system state and the input control with respect to the existence of errors of initialization measurement noises and fluctuations of system dynamics furthermore the system output converges uniformly to the desired one in absence of all disturbances finally specialization of the results to linear systems are presented

brief paper convergence and robustness of discrete time nonlinear systems with iterative learning control in this paper iterative learning control design of a class of discrete time nonlinear dynamic systems with disturbances are considered an iterative learning control law is proposed to overcome the uncertainties in system parameters and disturbances it is shown that the system outputs states and control inputs can be guaranteed to converge to desired trajectories in the absence of state output disturbances and repeatability uncertainty in the presence of these disturbances and initial state uncertainty the tracking errors will be bounded experiment is carried out to verify the theory and results are presented

two dimensional model and algorithm analysis for a class of iterative learning control systems the iterative learning control system ilcs has attracted considerable research interest in recent years however the theoretical results on some fundamental properties of the ilcs have not yet been established one of the common drawbacks shared by current approaches is lack of a suitable mathematical model which can clearly describe both the dynamics of the control system itself and the behaviour of the learning process as well in this paper a class of iterative learning control systems is analysed from the point of view of two dimensional 2 d system theory the 2 d model for a class of ilcs is established in the form of roessor s model based on which a general type of learning controller is proposed analysis of the 2 d error equation shows that the 2 d asymptotic stability of the 2 d model guarantees the learning convergence of an ilcs design criteria for a learning controller is also suggested the learning gain matrices are obtained from a recursive 2 d identifier using the input and output

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

energy gap for yang mills connections i four dimensional closed riemannian manifolds we extend an l2l2 energy gap result due to min oo 28 theorem 2 and parker 30 proposition 2 2 for yang mills connections on principal g bundles p over closed connected four dimensional oriented smooth manifolds x from the case of positive riemannian metrics to the more general case of good riemannian metrics including metrics that are generic and where the topologies of p and x obey certain mild conditions and the compact lie group g is su 2 su 2 or so 3 so 3

an optical lens for focusing two pairs of points we construct an optical lens in the x y plane which focuses two pairs of points i e all the rays from a given point x i are focused by the lens at a given point y i for i 1 2 the points x 1 x 2 y 1 y 2 lie on the x axis and the lens has the form left gamma text 1 text text f text 1 text y text underline leqslant text x text underline leqslant text gamma text 2 text text f text 2 text y text text left y right text underline leqslant text y text 0 right where 1 2 are given and f i 0 0 f i y f i y we then let x 2 x 1 y 2 y 1 and investigate the limiting lens we show that this limit is generally not a symmetric lens i e f 1 f 2 0

flat holomorphic connections on principal bundles over a projective manifold let g be a connected complex linear algebraic group and r u g its unipotent radical a principal g bundle e g over a projective manifold m will be called polystable if the associated principal g r u g bundle is so a g bundle e g over m is polystable with vanishing characteristic classes of degrees one and two if and only if e g admits a at holomorphic connection with the property that the image in g r u g of the monodromy of the connection is contained in a maximal compact subgroup of g r u g

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

a unified approach to string scattering amplitudes 1 physics in the calculation of g loop string tachyon amplitudes withn scattering points the distinguished polyakov measure d g n on the moduli spacem g n of riemann surfaces of genus g withn punctures arises we give an interpretation of this measure as the modulus squared of a holomorphic section g n the mumford form of a certain holomorphic line bundle i e we prove an analog of the belavin knizhnik theorem d g n g n 2 in the amplitudic case we give an expression for this measure through the determinants of the laplace operators over ghosts and over multivalued fields with monodromy prescribed by momenta at the scattering points we show also that the form g n n 0 n 0 for the partition function andn point amplitudes can be obtained from a unified over alln universal mumford form 2 mathematics the following new concepts from the theory of complex algebraic curves are investigated divisors with complex coefficients complex powers of holomorphic line bundles determinants of laplace operators over multivalued functions etc the corresponding generalizations of the determinant line bundles the weil deligne pairings the quillen and the arakelov deligne metrics are constructed a suggested by string amplitude considerations analog of the mumford theorem on holomorphic triviality of the bundle 2 1 13 over the moduli space is given this analog asserts the existence of a canonical flat metric on a certain line bundle lambda 2 otimes lambda 1 13 otimes left mathop otimes limits v 1 13 left langle mathcal o d v mathcal o d v right rangle 1 right see the main body of the text there exist two differences the latter bundle is not holomorphically trivial but has a canonical flat metric and being defined on the teichmuller spacet g n this bundle can be pulled down only on an infinite sheeted covering of the moduli spacem g n the universal isometries and the relative curvatures from the second part of the paper may be interesting too

on holomorphic principal bundles over a compact riemann surface admitting a flat connection let g be a connected reductive linear algebraic group over mathbb c and x a compact connected riemann surface let l subset g be a levi factor of some parabolic subgroup of g with l 0 l l l its maximal abelian quotient we prove that a holomorphic g bundle e g over x admits a flat connection if and only if for every such l and every reduction e l subset e g of the structure group of e g to l the l 0 bundle obtained by extending the structure group of e l is topologically trivial for g rm gl n mathbb c this is a well known result of a weil

the pfaffian line bundle we analyze the holomorphic pfaffian line bundle defined over an infinite dimensional isotropic grassmannian manifold using the infinite dimensional relative pfaffian we produce a fock space structure on the space of holomorphic sections of the dual of this bundle on this fock space an explicit and rigorous construction of the spin representations of the loop groupslo n is given we also discuss and prove some facts about the connection between the pfaffian line bundle over the grassmannian and the pfaffian line bundle of a dirac operator

a lens focusing light at two different wavelengths abstract a single element lens capable of focusing light from a source at two different wavelengths to two distinct foci in the image space is studied in this paper results are obtained concerning the existence and uniqueness of the lens the investigation is centred around a system of functional differential equations derived from fermat s principle and an asymptotic method is developed for the special case when the image space foci are close and the source nearly monochromatic

curvature of determinant bundles for degenerate families we calculate the 1 1 curvature of the beilinson schechtman connection for the determinant bundle associated to a family of riemann surfaces with ordinary singularities as consequences we obtain generalizations of theorems of bismut and bost

the chern classes of sobolev connections assumef is the curvature field of a connection potential onr4 with finitel2 norm left int limits r 4 left f right 2 dx infty right we show the chern number c 2 1 mathord left vphantom 1 8 right kern nulldelimiterspace 8 pi 2 int limits r 4 f wedge f topological quantum number is an integer this generalizes previous results which showed that the integrality holds forf satisfying the yang mills equations we actually prove the natural general result in all even dimensions larger than 2

new bosonization and conformal field theory over z new formulation of bosonization is given so that it is defined over the ring z of integers the charge zero sector of the new boson fock space is the completion of the coordinate ring of the universal witt scheme by using new bosonization conformal field theory of free fermions over z is given

gerbes of chiral differential operators ii vertex algebroids introduction 605 0 recollections and notation 612 1 vertex algebroids 615 2 from vertex algebras to vertex algebroids 622 3 category of vertex algebroids 628 4 cofibered structure 633 5 computation of a cocycle 637 6 characteristic classes of vector bundles 643 7 torseurs and gerbes 647 8 gerbes of vertex algebroids 653 9 chiral differential operators on curves and sheaves of virasoro algebras 657 10 enveloping algebra of a vertex algebroid 664 11 calabi yau structures 668 12 virasoro and calabi yau structures 671 references 680

solutions to yang mills equations that are not self dual abstract the yang mills functional for connections on principle su 2 bundles over s4 is studied critical points of the functional satisfy a system of second order partial differential equations the yang mills equations if in particular the critical point is a minimum it satisfies a first order system the self dual or anti self dual equations here we exhibit an infinite number of finite action nonminimal unstable critical points they are obtained by constructing a topologically nontrivial loop of connections to which min max theory is applied the construction exploits the fundamental relationship between certain invariant instantons on s4 and magnetic monopoles on h3 this result settles a question in gauge field theory that has been open for many years

twisted wess zumino witten models on elliptic curves investigated is a variant of the wess zumino witten model called a twisted wzw model which is associated to a certain lie group bundle on a family of elliptic curves the lie group bundle is a non trivial bundle with flat connection and related to the classical elliptic r matrix the usual non twisted wzw model is associated to a trivial group bundle with trivial connection on a family of compact riemann surfaces and a family of its principal bundles the twisted wzw model on a fixed elliptic curve at the critical level describes the xyz gaudin model the elliptic knizhnik zamolodchikov equations associated to the classical elliptic r matrix appear as flat connections on the sheaves of conformal blocks in the twisted wzw model

selection of the learning gain matrix of an iterative learning control algorithm in presence of measurement noise arbitrary high precision output tracking is one of the most desirable control objectives found in industrial applications regardless of measurement errors the main purpose of this paper is to supply to the iterative learning control ilc designer guidelines to select the corresponding learning gain in order to achieve this control objective for example if certain conditions are met then it is necessary for the learning gain to converge to zero in the learning iterative domain in particular this paper presents necessary and sufficient conditions for boundedness of trajectories and uniform tracking in presence of measurement noise and a class of random reinitialization errors for a simple ilc algorithm the system under consideration is a class of discrete time affine nonlinear systems with arbitrary relative degree and arbitrary number of system inputs and outputs the state function does not need to satisfy a lipschitz condition this work also provides a recursive algorithm that generates the appropriate learning gain functions that meet the arbitrary high precision output tracking objective the resulting tracking output error is shown to converge to zero at a rate inversely proportional to square root of the number of learning iterations in presence of measurement noise and a class of reinitialization errors two illustrative numerical examples are presented

automorphism group of k t applications to the bosonic string this paper is concerned with the formulation of a non pertubative theory of the bosonic string we introduce a formal group g which we propose as the universal moduli space for such a formulation this is motivated because g establishes a natural link between representations of the virasoro algebra and the moduli space of curves

local conformal field algebras the local conformal field algebras with the multiplication corresponding to the regularized pointwise product of fields in the operator algebras of the quantum conformal field theory are investigated

a discrete time learning control algorithm for a class of linear time invariant systems a discretized version of the d type learning control algorithm is presented for a mimo linear discrete time system a necessary and sufficient condition for uniform convergence of the proposed learning algorithm is presented then we prove that the same condition is sufficient for the global robustness of the proposed learning algorithm to state disturbances measurement noise at the output and reinitialization error are present at each iteration a numerical example is given to illustrate the results

model based iterative learning control with a quadratic criterion for time varying linear systems in this paper iterative learning control ilc based on a quadratic performance criterion is revisited and generalized for time varying linear constrained systems with deterministic stochastic disturbances and noises the main intended area of application for this generalized method is chemical process control where excessive input movements are undesirable and many process variables are subject to hard constraints it is shown that within the framework of the quadratic criterion based ilc q ilc various practical issues such as constraints disturbances measurement noises and model errors can be considered in a rigorous and systematic manner algorithms for the deterministic case the stochastic case and the case with bounded parameter uncertainties are developed and relevant properties such as the asymptotic convergence are established under some mild assumptions numerical examples are provided to demonstrate the performance of the proposed algorithms

an average operator based pd type iterative learning control for variable initial state error this note studies the effect of variable initial state error in iterative learning control ilc systems and proposes a new ilc algorithm based on an average operator then it is shown that when the proposed algorithm is applied to linear time invariant lti systems the effect of the initial state error can be exactly estimated under a specific condition while the existing algorithms guarantee only the boundness of the error or the convergence from stochastic point of view to show the validity of the proposed algorithm a numerical example is given

geometric realization of conformal field theory on riemann surfaces conformal field theory on a family of riemann surfaces is formulated we derive equations of motion of vacua which are parametrized by moduli of riemann surfaces and show that these vacua are characterized uniquely by these equations our theory has a deep connection with sato s theory of kp equations

an identification of the connections of quillen and beilinson schechtman given a family of riemann surfaces and a holomorphic vector bundle beilinson and schechtman construct a canonical connection on the associated determinant bundle we prove the conjecture which states that their connection coincides with the quillen connection this is done by reducing to the case where bar partial along fibers are invertible both connection forms become more accessible in this case

opers and theta functions abstract we construct natural maps the klein and wirtinger maps from moduli spaces of semistable vector bundles over an algebraic curve x to affine spaces as quotients of the nonabelian theta linear series we prove a finiteness result for these maps over generalized kummer varieties moduli space of torus bundles leading us to conjecture that the maps are finite in general the conjecture provides canonical explicit coordinates on the moduli space the finiteness results give low dimensional parametrizations of jacobians in p 3g 3 for generic curves described by 2 functions or second logarithmic derivatives of theta we interpret the klein and wirtinger maps in terms of opers on x opers are generalizations of projective structures and can be considered as differential operators kernel functions or special bundles with connection the matrix opers analogues of opers for matrix differential operators combine the structures of flat vector bundle and projective connection and map to opers via generalized hitchin maps for vector bundles off the theta divisor the szego kernel gives a natural construction of matrix oper the wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the szego kernel this generalizes the wirtinger projective connections associated to theta characteristics and the associated klein bidifferentials

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

an iterative learning control theory for a class of nonlinear dynamic systems abstract an iterative learning control scheme is presented for a class of nonlinear dynamic systems which includes holonomic systems as its subset the control scheme is composed of two types of control methodology a linear feedback mechanism and a feedforward learning strategy at each iteration the linear feedback provides stability of the system and keeps its state errors within uniform bounds the iterative learning rule on the other hand tracks the entire span of a reference input over a sequence of iterations the proposed learning control scheme takes into account the dominant system dynamics in its update algorithm in the form of scaled feedback errors in contrast to many other learning control techniques the proposed learning algorithm neither uses derivative terms of feedback errors nor assumes external input perturbations as a prerequisite the convergence proof of the proposed learning scheme is given under minor conditions on the system parameters

optimality of first order ilc among higher order ilc higher order iterative learning control ho ilc algorithms use past system control information from more than one past iterative cycle this class of ilc algorithms have been proposed aiming at improving the learning efficiency and performance this paper addresses the optimality of ho ilc in the sense of minimizing the trace of the control error covariance matrix in the presence of a class of uncorrelated random disturbances it is shown that the optimal weighting matrices corresponding to the control information associated with more than one cycle preceding the current cycle are zero that is an optimal ho ilc does not add to the optimality of standard first order ilc in the sense of minimizing the trace of the control error covariance matrix the system under consideration is a linear discrete time varying systems with different relative degree between the input and each output

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

the riemannian geometry of the yang mills moduli space the moduli space of self dual connections over a riemannian 4 manifold has a natural riemannian metric inherited from thel 2 metric on the space of connections we give a formula for the curvature of this metric in terms of the relevant green operators we then examine in great detail the moduli space 1 ofk 1 instantons on the 4 sphere and obtain an explicit formula for the metric in this case in particular we prove that 1 is rotationally symmetric and has finite geometry it is an incomplete 5 manifold with finite diameter and finite volume

on a numerical minimization problem for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics and beamforming lenticular antennas 11 abstract an improved mixed numerical and analytical procedure for solving the problem of determining the surfaces i and i of a three dimensional aperture or volume extremized and ellipticized bifocal nonspherical singlet optical radar or sonar lens antenna defined by a system of four nonlinear partial differential equations with specified symmetry and boundary conditions subject to a minimizing side condition on the antenna focal length of f number is presented the index of refraction n0 the maximum lens diameter 2b0 and the bifocal off axis angle 0 are assumed given the results obtained are of approximately twice the accuracy of those computed in part i

cyclic homology of differential operators the virasoro algebra and a q analogue we show how methods from cyclic homology give easily an explicit 2 cocycle on the lie algebra of differential operators of the circle such that restricts to the cocycle defining the virasoro algebra the same methods yield also aq analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity we use an algebra ofq difference operators andq analogues of koszul and the rham complexes to construct these quantum cocycles

a stochastic iterative learning control algorithm with application to an induction motor a recursive optimal algorithm based on minimizing the input error covariance matrix is derived to generate the learning gain matrix of a p type ilc for linear discrete time varying systems with arbitrary relative degree it is shown that in the case where the number of inputs is not greater than the number of outputs the input error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of uncorrelated random state disturbance reinitialization errors and measurement noise the state error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of measurement noise in the case where the number of inputs is greater than the number of outputs then the system output error converges to zero at a rate inversely proportional to the number of iterations in presence of measurement noise another suboptimal recursive algorithm is also proposed based on unknown system dynamics and unknown disturb

moduli of curves grobner bases and the krichever map abstract we study moduli spaces of possibly non nodal curves c p 1 p n of arithmetic genus g with n smooth marked points equipped with nonzero tangent vectors such that o c p 1 p n is ample and h 1 o c a 1 p 1 a n p n 0 for given integer weights a a 1 a n such that a i 0 and a i g we show that each such moduli space u g n n s a is an affine scheme of finite type and the krichever map identifies it with the quotient of an explicit locally closed subscheme of the sato grassmannian by the free action of the group of changes of formal parameters we study the git quotients of u g n n s a by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of m g n with projective coarse moduli spaces more generally using similar techniques we construct moduli spaces of curves with chains of divisors supported at marked points with prescribed number of sections which in the case n 1 corresponds to specifying the weierstrass gap sequence at the marked point