the integration of multidimensional scaling and multivariate analysis with optimal transformations the recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines in multidimensional scaling mds the available data typically describe the relationships among a set of objects in terms of similarity dissimilarity or pseudo distances in multivariate analysis mva data usually result from observation on a collection of variables over a common set of objects this paper starts from a very general multidimensional scaling task defined on distances between objects derived from one or more sets of multivariate data particular special cases of the general problem following familiar notions from mva will be discussed that encompass a variety of analysis techniques including the possible use of optimal variable transformation throughout it will be noted how certain data analysis approaches are equivalent to familiar mva solutions when particular problem specifications are combined with particular distance approximations

vortices in holomorphic line bundles over closed kahler manifolds we apply a modified yang mills higgs functional to unitary bundles over closed kahler manifolds and study the equations which govern the global minima the solutions represent vortices in holomorphic bundles and are direct analogs of the vortices overr2 we obtain a complete description of the moduli space of these new vortices where the bundle is of rank one the description is in terms of a class of divisors in the base manifold there is also a dependence on a real valued parameter which can be attributed to the compactness of the base manifold

stochastic cellular automata with gibbsian invariant measures a geometric characterization is given of a class of stochastic cellular automata whose invariant probability measures are gibbsian distributions i e markovian random fields this class whose members may be considered as the cartesian product of convex polytopes with a finite number of vertices contains all known automata of this type e g the metropolis gibbs sampler and heat bath algorithms and also a new family which is characterized by having nonreversible dynamic behavior the fact that there is a complete geometric structure instead of isolated points opens the possibility of using classical techniques such as convex programming for the design of optimal automata some examples of applications are given oriented towards the solution of image restoration problems

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

coupled vortex equations and moduli deformation theoretic approach and kahler geometry we investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact kahler manifold x these solutions are known to be related to polystable triples via a kobayashi hitchin type correspondence using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex we construct a hermitian structure on these moduli spaces this hermitian structure is proved to be kahler the proof involves establishing a fiber integral formula for the hermitian form we compute the curvature tensor of this kahler form when x is a riemann surface the holomorphic bisectional curvature turns out to be semi positive it is shown that in the case where x is a smooth complex projective variety the kahler form is the chern form of a quillen metric on a certain determinant line bundle

on the h theorem for polyatomic gases the h theorem for a classical gas of polyatomic molecules of arbitrarily complex structure is examined a simple use of time reversal invariance of the equations of dynamics is used to circumvent the objections which were raised by lorentz against boltzmann s proof nonexistence of inverse collisions

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

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

gauged vortices in a background we discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism at critical self coupling and for low temperatures it has been argued that the configuration space for vortex dynamics in each topological class of the abelian higgs model approximately truncates to a finite dimensional moduli space with a kahler structure for the case where the vortices live on a 2 sphere we explain how localization formulae on the moduli spaces can be used to compute exactly the partition function of the vortex gas interacting with a background potential the coefficients of this analytic function provide geometrical data about the kahler structures the simplest of which being their symplectic volume computed previously by manton using an alternative argument we use the partition function to deduce simple results on the thermodynamics of the vortex system in particular the average height on the sphere is computed and provides an interesting effective picture of the ground state

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

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

a second generation nonlinear factor analysis nonlinear common factor models with polynomial regression functions including interaction terms are fitted by simultaneously estimating the factor loadings and common factor scores using maximum likelihood ratio and ordinary least squares methods a monte carlo study gives support to a conjecture about the form of the distribution of the likelihood ratio criterion

maxallent maximizers of all entropies and uncertainty of uncertainty a b s t r a c t the entropy maximum approach maxent was developed as a minimization of the subjective uncertainty measured by the boltzmann gibbs shannon entropy many new entropies have been invented in the second half of the 20th century now there exists a rich choice of entropies for fitting needs this diversity of entropies gave rise to a maxent anarchism the maxent approach is now the conditional maximization of an appropriate entropy for the evaluation of the probability distribution when our information is partial and incomplete the rich choice of non classical entropies causes a new problem which entropy is better for a given class of applications we understand entropy as a measure of uncertainty which increases in markov processes in this work we describe the most general ordering of the distribution space with respect to which all continuous time markov processes are monotonic the markov order for inference this approach results in a set of conditionally most random distributions each distribution from this set is a maximizer of its own entropy this uncertainty of uncertainty is unavoidable in the analysis of nonequilibrium systems surprisingly the constructive description of this set of maximizers is possible two decomposition theorems for markov processes provide a tool for this description

local equivalence of reversible and general markov kinetics a b s t r a c t we consider continuous time markov kinetics with a finite number of states and a positive equilibrium p this class of systems is significantly wider than the systems with detailed

the cohomology of the space of magnetic monopoles denote byx q the reduced space ofsu 2 monopoles of chargeq in 3 in this paper the cohomology ofx q the cohomology with compact supports ofx q and the image of the latter in the former are all calculated as representations of q which acts onx 2 this provides a non trivial lower bound for thel 2 cohomology ofx q which is compatible with some conjectures of sen it is also shown that granted some assumptions about the metric onx q itsl 2 cohomology does not exceed this bound in the situation referred to in the paper as the coprime case

simultaneous classification and multidimensional scaling with external information for the exploratory analysis of a matrix of proximities or dis similarities between objects one often uses cluster analysis ca or multidimensional scaling mds solutions resulting from such analyses are sometimes interpreted using external information on the objects usually the procedures of ca mds and using external information are carried out independently and sequentially although combinations of two of the three procedures ca and mds or multidimensional scaling and using external information have been proposed in the literature the present paper offers a procedure that combines all three procedures in one analysis using a model that describes a partition of objects with cluster centroids represented in a low dimensional space which in turn is related to the information in the external variables a simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure also to illustrate the method it is applied to two empirical data sets

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

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

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

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

irreversible monte carlo algorithms for efficient sampling abstract equilibrium systems evolve according to detailed balance db this principle guided the development of monte carlo sampling techniques of which the metropolis hastings mh algorithm is the famous representative it is also known that db is sufficient but not necessary we construct irreversible deformation of a given reversible algorithm capable of dramatic improvement of sampling from known distribution our transformation modifies transition rates keeping the structure of transitions intact to illustrate the general scheme we design an irreversible version of metropolis hastings imh and test it on an example of a spin cluster standard mh for the model suffers from critical slowdown while imh is free from critical slowdown

semistability of complex balanced kinetic systems with arbitrary time delays abstract in this letter we introduce a class of delayed kinetic systems derived from mass action type reaction network models we define the time delayed positive stoichiometric compatibility classes and the notion of complex balanced time delayed kinetic systems we prove the uniqueness of equilibrium solutions within the time delayed positive stoichiometric compatibility classes for such models in our main result we prove the semistability of the equilibrium solutions for complex balanced systems with arbitrary time delays using an appropriate lyapunov krasovskii functional and lasalle s invariance principle as a consequence we obtain that every positive complex balanced equilibrium solution is locally asymptotically stable relative to its positive stoichiometric compatibility class

l 2 cohomology of hyperkahler quotients using an argument of jost and zuo we give a criterion which implies that the l 2 harmonic forms on a complete noncompact hyperkahler manifold lie in the middle dimension and are invariant under the isometry group this is applied to various examples and in particular gives a verification of some of the predictions of sen on monopole moduli spaces

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

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

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

on the curvature of vortex moduli spaces we use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed riemann surface after computing the homotopy type of the universal cover of the moduli spaces which are symmetric products of the surface we prove that for genus g 1 the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case and this property extends to all kahler metrics on certain symmetric products our result rules out an established and natural conjecture on the geometry of the moduli spaces

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

extending multivariate distance matrix regression with an effect size measure and the asymptotic null distribution of the test statistic person centered methods are useful for studying individual differences in terms of dis similarities between response profiles on multivariate outcomes multivariate distance matrix regression mdmr tests the significance of associations of response profile dis similarities and a set of predictors using permutation tests this paper extends mdmr by deriving and empirically validating the asymptotic null distribution of its test statistic and by proposing an effect size for individual outcome variables which is shown to recover true associations these extensions alleviate the computational burden of permutation tests currently used in mdmr and render more informative results thus making mdmr accessible to new research domains

clustering objects on subsets of attributes with discussion read before the royal statistical society at a meeting organized by the research section on wednesday may 5th 2004 professor j t kent in the chair summary a new procedure is proposed for clustering attribute value data when used in conjunction with conventional distance based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously the relevant attribute subsets for each individual cluster can be different and partially or completely overlap with those of other clusters enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed applications in different domains including gene expression arrays are presented

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative

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