measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

a multiple step selection procedure with sequential protection of preferred treatments dunnett 1955 journal of the american statistical association 50 1096 1121 proposed a one step many one test to compare several experimental treatment means with a control however after the control treatment is rejected one would want to make comparisons among the experimental treatments we assume that there is an order of decreasing preference among the experimental treatments based on other considerations we propose in this paper a multiple step selection procedure an extension of dunnett s one step many one test that will give protection to the preferred treatments and guarantee a high selection probability of the correct treatment when that selection is appropriate we also provide methods to calculate the sample size required to carry out this test for normal binomial and exponential data with random censoring

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

a monte carlo method for estimating the correlation exponent we propose a monte carlo method for estimating the correlation exponent of a stationary ergodic sequence the estimator can be considered as a bootstrap version of the classical hill estimator a simulation study shows that the method yields reasonable estimates

chaos in rainfall the question is posed regarding possible chaotic dynamics in the temporal rainfall of storm events chaotic dynamics implies a nonlinear deterministic system very sensitive to initial conditions which yields outputs indistinguishable from a stochastic process by standard techniques the trajectories of these systems in the phase space are characterized by being contained in a strange attractor of fractal dimension a time series of 1990 points of 15 s rainfall for the storm of october 25 1980 in boston is analyzed in detail toward this objective it is found that both the characteristics of the correlation integral and the lyapunov exponents of the historical data give preliminary support to the presence of chaotic dynamics with a strange attractor the implications of these findings for the modeling of storm rainfall and for the limits of predictability of the process are discussed jointly with the research challenges lying ahead in this field

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

an adaptive confirmatory trial with interim treatment selection practical experiences and unbalanced randomization this article has been motivated by an ongoing international adaptive confirmatory trial at the interim analysis of this two stage trial none one or two active treatment regimens are selected for further study in the second stage a combination test approach is used in this practical setting with an extension of the theory to unbalanced randomization we show that a combination test with suitable weights can still preserve the overall type i error rate provided that the test statistic is chosen appropriately and the unpooled z test for proportions is not used the accuracy of stagewise p values is also discussed in a more general framework monte carlo simulations confirm the validity of the approach retained and evaluate the necessary sample size additional issues addressed during the design of the trial are also examined such as multiplicity due to testing hypotheses on key secondary endpoints a non inferiority comparison to an active treatment and covariate adjusted analyses for various types of outcome

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

two stage selection and testing designs for comparative clinical trials summary a two stage design which selects the best of several experimental treatments and compares it to a standard control is proposed the design allows early termination with acceptance of the global null hypothesis optimal sample size and cut off parameters are obtained by minimizing expected total sample size for fixed significance level and power with many other factors moreover the usual error rate computations associated with comparative testing do not account for the preliminary selection process consequently the overall procedure may be neither effective nor efficient for identifying an experimental treatment which is an improvement over c in this paper we propose a new approach to the problem of identifying the best of k experimental treatments and determining whether it is superior to a control we deal with the binomial setting where patient response may be characterized as either success or failure with ok the success probability for ek k 0 corresponds to c for ease of notation assume 01 ok we propose a two stage procedure which allows early termination with acceptance of h0 o 01 ok with design parameters chosen to minimize expected total sample size the design and accompanying generalized definitions of size a and power 1 3 are given in 2 the algorithm used for optimization is described in 3 numerical results are presented in 4 followed by a discussion of the relative merits of the proposed design

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

on cyclic higgs bundles in this paper we derive a maximum principle for a type of elliptic systems and apply it to analyze the hitchin equation for cyclic higgs bundles we show several domination results on the pullback metric of the possibly branched minimal immersion f associated to cyclic higgs bundles also we obtain a lower and upper bound of the extrinsic curvature of the image of f as an application we give a complete picture for maximal sp 4 mathbb r representations in the 2g 3 gothen components and the hitchin components

mirror symmetry for moduli spaces of higgs bundles via p adic integration we prove the topological mirror symmetry conjecture by hausel thaddeus for smooth moduli spaces of higgs bundles of type sl n and pgl n more precisely we establish an equality of stringy hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties our proof utilises p adic integration relative to the fibres and interprets canonical gerbes present on these moduli spaces as characters on the hitchin fibres using tate duality furthermore we prove for d prime to n that the number of rank n higgs bundles of degree d over a fixed curve defined over a finite field is independent of d this proves a conjecture by mozgovoy schiffmann in the coprime case

characteristic classes and representations of discrete subgroups of lie groups a volume invariant is used to characterize those representations of a countable group into a connected semisimple lie group g which are injective and whose image is a discrete cocompact subgroup of g let it be a discrete cocompact subgroup of g and consider the analytic variety hom 7r g consisting of homomorphisms 0 tt g denote by k a maximal compact subgroup of g and x k g the associated symmetric space let m be the orbit space x tt for convenience we shall henceforth assume that n is torsionfree by selberg s lemma 12 this may be accomplished by replacing tt by a subgroup of finite index this insures that m is a compact smooth manifold having tt as its fundamental group the case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here to every representation 0 g hom 7r g we associate a foliated bundle e over m with fibre x and structure group g see e g 6 if co is a closed ginvariant differential fc form on x then we may spread co over the fibres of e copies of x to obtain a closed fc form co on e we define co 0 fj f where is any section of e moreover co 0 is independent of the choice of section for example taking co to be the g invariant volume form on x we obtain a real number u 0 which depends on 0 when x is even dimensional the chern gauss bonnet theorem implies that u 0 may be described as an euler number i e the self intersection number of any section which is a topological invariant of e when x is odd dimensional this volume invariant is related to a more recent kind of topological invariant based on bounded cohomology and due to gromov 3 and is constant on the connected components of hom 7r g the volume invariant satisfies an inequality

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

a composite design for transition from a preliminary to a full scale study in drug development a sequence of studies are conducted to evaluate effectiveness or efficacy and safety such as a phase ii study to assess pharmacological activity or safety that is then followed by a definitive phase iii study to assess clinical effectiveness rather than conducting separate successive studies we describe a design in which the patients enrolled in a preliminary e g phase ii study are continued into a subsequent full scale e g phase iii study this design also applies to a study that uses an internal pilot with a preliminary assessment of efficacy or safety the combined preliminary to full scale design potentially reduces the total numbers of patients required and the total duration of the program the design allows a futility or safety stopping boundary for the preliminary study result that is specified in terms of a lower critical z value z l and the pursuant type ii error probability incurred under a specified alternative hypothesis of a beneficial effect or no toxicity at that stage this boundary also leads to a deflation of the type i error probability for the final test at the completion of the full scale study such that a critical value for the final test z f can be determined that provides the desired level of the type i error probability exactly thus it is possible to determine sample sizes at the two stages and critical values z l and z f that provide specified type i and ii error probabilities for the combined study we describe the design using large sample normally distributed z tests at the two phases including a test for means or proportions or survival times or combinations thereof such as a test for means at phase ii followed by a test for proportions at phase iii we compare the properties of this design versus the conduct of two successive studies and explore the factors that influence the operating characteristics of the design we also discuss the practical considerations in the implementation of the design

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

uniformly minimum variance conditionally unbiased estimation in multi arm multi stage clinical trials multi arm multi stage clinical trials compare several experimental treatments with a control treatment with poorly performing treatments dropped at interim analyses this leads to inferential challenges including the construction of unbiased treatment effect estimators a number of estimators unbiased conditional on treatment selection have been proposed but are specific to certain selection rules may ignore the comparison to the control and are not all minimum variance we obtain estimators for treatment effects compared to the control that are uniformly minimum variance unbiased conditional on selection with any specified rule or stopping for futility

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

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

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

quantization of sl 2 r chern simons theory we discuss chern simons gauge theory with ansl 2 r gauge group on an arbitrary 3 manifoldm thesl 2 r chern simons action is defined for gauge bundles overm of arbitrary topological type the geometric quantization ofsl 2 r chern simons theory is discussed and related to the quantization of teichmuller space the generalization to chern simons theory with ansl n r gauge group is also considered

predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach 1 perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one even though the sensitivity of initial conditions limits the predictability in chaotic systems an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent all of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions conceptually global models are ineffective in modeling the highly unstable structure of the chaotic attractor this paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters i e embedding dimension and delay time to quantify the uncertainty in initial conditions the ensemble approach is implemented through a local learning wavelet network model with a global feed forward neural network structure for the phase space prediction of chaotic streamflow series quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times the ensemble approach is proved to be 50 more efficient than the single prediction for both local approximation and wavelet network approaches the wavelet network approach has proved to be 30 50 more superior to the local approximation approach compared to the traditional local approximation approach with single initial condition the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times localization property of wavelets utilizing different dilation and translation parameters helps in capturing most of the statistical properties of the observed data the need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

a two stage design for choosing among several experimental treatments and a control in clinical trials in clinical trials where several experimental treatments are of interest the goal may be viewed as identification of the best of these and comparison of that treatment to a standard control therapy however it is undesirable to commit patients to a large scale comparative trial of a new regimen without evidence that its therapeutic success rate is acceptably high we propose a two stage design in which patients are first randomized among the experimental treatments and the single treatment having the highest observed success rate is identified if this highest rate falls below a fixed cutoff then the trial is terminated otherwise the best new treatment is compared to the control at a second stage locally optimal values of the cutoff and the stage 1 and stage 2 sample sizes are derived by minimizing expected total sample size the design has both high power and high probability of terminating early when no experimental treatment is superior to the control numerical results for implementing the design are presented and comparison to dunnett s 1984 in design of experiments ranking and selection t j santner and a c tamhane eds 47 66 new york marcel dekker optimal one stage procedure is made

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

biases of correlation dimension estimates of streamflow data in the canadian prairies the correlation dimension d2 of the unregulated streamflow data of six rivers in the canadian prairies were estimated using the hill estimate and grassberger procaccia algorithms from both procedures and using an embedding phase space up to 10 the d2 obtained for all cases are found to saturate at 3 0 however the saturated d2 generally increased to between 4 and 6 when the data were first randomly resampled prior to estimating the correlation dimension since the highest embedding dimension d used in this study is 10 theoretically randomly resampled data should have a d2 approximately equal to 10 therefore there is a consistent underestimation of d2 by an amount of 4 6 by accounting for the combined clustering effect of the streamflow data s underlying distribution which is gamma the effect of sample size and possibly the choice of time delay the d2 of the prairie streamflow data should be 7 9 the hypothesis that the clustering effect of unevenly distributed data contributed to the underestimation of d2 was confirmed from the different d2 obtained for random numbers of uniform gamma and poisson distributions in addition analysis of the uniformly distributed random numbers and streamflow data showed that increasing the sample size from say 6500 to 17 000 marginally improved the estimated d2 however for the range of time delay chosen for this study 40 180 days its effect on d2 is less obvious this may be partly because all the streamflow data tested exhibit long term persistence

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

bihermitian surfaces with odd first betti number compact bihermitian surfaces are considered that is compact oriented conformal four manifolds admitting two distinct compatible complex structures it is shown that if the first betti number is odd then with respect to either complex structure such a manifold belongs to class vii of the enriques kodaira classification moreover it must be either a special hopf or an inoue surface in the strongly bihermitian case or obtained by blowing up a minimal class vii surface with curves in the non strongly bihermitian case

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

low complexity methods for estimation after parameter selection statistical inference of multiple parameters often involves a preliminary parameter selection stage the selection stage has an impact on subsequent estimation for example by introducing a selection bias the post selection maximum likelihood psml estimator is shown to reduce the selection bias and the post selection mean squared error psmse compared with conventional estimators such as the maximum likelihood ml estimator however the computational complexity of the psml is usually high due to the multi dimensional exhaustive search for a global maximum of the post selection log likelihood psll function moreover the psll involves the probability of selection that in general does not have an analytical form in this paper we develop new low complexity post selection estimation methods for a two stage estimation after parameter selection architecture the methods are based on implementing the iterative maximization by parts mbp approach which is based on the decomposition of the psll function into easily optimized and complicated parts the proposed second best psml method applies the mbp psml algorithm with a pairwise probability of selection between the two highest ranked parameters w r t the selection rule the proposed sa psml method is based on using stochastic approximation sa and monte carlo integrations to obtain a non parametric estimation of the gradient of the probability of selection and then applying the mbp psml algorithm on this approximation for low complexity performance analysis we develop the empirical post selection cram acute text e r rao type lower bound simulations demonstrate that the proposed post selection estimation methods are tractable and reduce both the bias and the psmse compared with the ml estimator while only requiring moderate computational complexity

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c