minimal resolutions of algebras abstract a method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself the method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible with some additional finiteness restrictions in the latter case in particular it applies to any finite dimensional algebra over an algebraically closed field the method is illustrated by a number of examples viz truncated algebras monomial algebras and koszul algebras with the aim of unifying existing treatments of these in the literature

knots sutures and excision we develop monopole and instanton floer homology groups for balanced sutured manifolds in the spirit of 12 applications include a new proof of property p for knots

stochastic nonlinear minimax dynamic games with noisy measurements this note is concerned with nonlinear stochastic minimax dynamic games which are subject to noisy measurements the minimizing players are control inputs while the maximizing players are square integrable stochastic processes the minimax dynamic game is formulated using an information state which depends on the paths of the observed processes the information state satisfies a partial differential equation of the hamilton jacobi bellman hjb type the hjb equation is employed to characterize the dissipation properties of the system to derive a separation theorem between the design of the estimator and the controller and to introduce a certainty equivalence principle along the lines of whittle finally the separation theorem and the certainty equi valence principle are applied to solve the linear quadratic gaussian minimax game the results of this note generalize the l sup 2 gain of deterministic systems to stochastic analogs they are related to the controller design of stochastic systems which employ risk sensitive performance criteria and to the controller design of deterministic systems which employ minimax performance criteria

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

certain nonlinear partially observable stochastic optimal control problems with explicit control laws equivalent to leqg lqg problems this paper is concerned with partially observed stochastic optimal control problems when nonlinearities enter the dynamics of the unobservable state and the observations as gradients of potential functions explicit representations for the information state are derived in terms of a finite number of sufficient statistics consequently the partially observed problem is recast as one of complete information with a new state generated by a modified version of the kalman filter when the terminal cost is quadratic in the unobservable state and includes the integral of the nonlinearities the optimal control laws are explicitly computed similar to linear exponential quadratic gaussian leqg and linear quadratic gaussian lqg tracking problems the results are applicable to filtering and control of hamiltonian systems

geometric composition in quilted floer theory abstract we prove that floer cohomology of cyclic lagrangian correspondences is invariant under transverse and embedded composition under a general set of assumptions

partially observable nonlinear risk sensitive control problems dynamic programming and verification theorems in this paper we consider continuous time partially observable optimal control problems with exponential of integral cost criteria we derive a rigorous verification theorem when the state and control enter nonlinear in the dynamics in addition we show that the quadratic sensor problem is estimation solvable with respect to a certain cost criterion the framework relies on dynamic programming and the hamilton jacobi theory

risk sensitive control and dynamic games for partially observed discrete time nonlinear systems solves a finite horizon partially observed risk sensitive stochastic optimal control problem for discrete time nonlinear systems and obtains small noise and small risk limits the small noise limit is interpreted as a deterministic partially observed dynamic game and new insights into the optimal solution of such game problems are obtained both the risk sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states dynamic programming and associated separated policies a certainty equivalence principle is also discussed the authors results have implications for the nonlinear robust stabilization problem the small risk limit is a standard partially observed risk neutral stochastic optimal control problem

a finite dimensional risk sensitive control problem a partially observed stochastic control problem with exponential running cost is considered the dynamics are linear and the running cost is quadratic although the control may enter nonlinearly explicit solutions are found to a modified zakai equation and a backward adjoint equation this enables the problem to be expressed in terms of observable finite dimensional dynamics and a separation principle to be applied

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

an overview of sequence comparison time warps string edits and macromolecules a wide variety of different applications lead to problems in which sequences of different lengths must be compared to see how different they are and to see which elements in one sequence correspond to which elements in the other sequence successful methods for handling these problems been repeatedly reinvented which incorporate two basic ideas a systematic concept of distance between sequences and elegant recursive algorithms for doing the necessary computations the major application areas are speech processing and macromolecular biology computer science is another significant application area and applications have also been made to bird song handwriting analysis gas chromatography geological strata and text collation this paper surveys the applications methods and theory of sequence comparison

precision edge contrast and orientation estimation the contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive gaussian noise the test image is an ideal step edge that has been sampled with a square aperture grid the effects of subpixel translations and rotations of the edge on the performance of the operators are studied it is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels methods with improved results are presented for sobel angle estimates and the nevatia babu operator and theoretical noise performance evaluations are also provided an edge operator based on two dimensional spatial moments is presented all methods are compared according to worst case and rms error in an ideal noiseless situation and rms error under various noise levels

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

h control for non linear stochastic systems the output feedback case the h output feedback control problem for non linear stochastic systems is considered a solution for a large class of non linear stochastic systems is introduced including non linear diffusion systems as a subclass this solution is based on a bounded real lemma for non linear stochastic systems that was previously established via a stochastic dissipativity concept the theory yields sufficient conditions for the closed loop system to possess a prescribed l 2 gain bound in terms of two hamilton jacobi inequalities one that is associated with the state feedback part of the problem is n dimensional where n is the underlying system s state dimension and the other inequality that stems from the estimation part is 2n dimensional both stationary and non stationary systems are considered stability of the closed loop system is established both in the mean square and the in probability senses as the solution to the hamilton jacobi inequalities may in general lead to a non realisable state estimator

the role of information state and adjoint in relating nonlinear output feedback risk sensitive control and dynamic games this paper employs logarithmic transformations to establish relations between continuous time nonlinear partially observable risk sensitive control problems and analogous output feedback dynamic games the first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state the second logarithmic transformation is applied to the risk sensitive cost function using the laplace varadhan lemma in the small noise limit this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

guaranteed cost lqg control for uncertain systems with a normalized coprime factor uncertainty structure the paper extends the robust minimax lqg control design methodology to stochastic uncertain systems with a general uncertainty structure which includes normalized coprime factor uncertainty passive uncertainty and sector norm bounded uncertainty as special cases the derivation of the result uses a special parameter dependent girsanov measure transformation the efficacy of the proposed methodology is illustrated using the problem of frequency locking of an optical cavity which occurs in the area of experimental quantum optics

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

toric structures on the moduli space of flat connections on a riemann surface volumes and the moment map abstract in earlier papers we constructed a hamiltonian torus action on an open dense set in the moduli space of flat su 2 connections on a compact riemann surface where the dimension of the torus is half the dimension of the moduli space this torus action shows that this set can be viewed symplectically as a noncompact toric variety the number of integral points of the moment map for the torus action turns out to be identical to the verlinde dimension d g k as an application we furnish a new proof of the relation between the large k limit of d g k and the volume of the moduli space from our point of view this relation follows from the equality between the symplectic volume of a toric variety and the euclidean volume of the image of the moment map similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a riemann surface knowledge of these volumes has been shown to allow a proof of the verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space

improving classifier performance through repeated sampling abstract in traditional pattern recognition the classification decision is based on a single observation of the input in this paper we show that by relaxing this assumption the performance of the classifier can be improved substantially we present a detailed analysis of one particular method for achieving this taking a consensus vote on the classifier s output for repeated samples of the input we prove that this approach always yields a net improvement in recognition accuracy for common distributions of interest upper and lower bounds on the improvement are also discussed under certain conditions it is even possible to beat the bayes error bound associated with the classifier we conclude by presenting results from three sets of experiments examining the effectiveness of the idea

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

robust h sub spl infin filtering for nonlinear stochastic systems this paper describes the robust h sub spl infin filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance dependent noise we assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by ito spl circ type stochastic differential equations for general nonlinear stochastic systems the h sub spl infin filter can be obtained by solving second order nonlinear hamilton jacobi inequalities when the worst case disturbance is considered in the design procedure a mixed h sub 2 h sub spl infin filtering problem is also solved by minimizing the total estimation error energy it is found that for a class of special nonlinear stochastic systems the h sub spl infin filtering design can be given via solving several linear matrix inequalities instead of hamilton jacobi inequalities a few examples show that the proposed methods are effective