yang mills equations on the two dimensional sphere we construct a 1 1 correspondence between the equivalence classes of yang mills fields overs2 and the conjugacy classes of closed geodesics of the structure group furthermore we give an explicit isolation theorem for any yang mills field overs2

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

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

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

conformal fields and stability lagrangiens naturels conformes deformations conformes theoremes d instabilite pour des spheres theoremes d annulation sur des espaces non compacts

on the holonomy of the coulomb connection over manifolds with boundary narasimhan and ramadas commun math phys 67 121 1979 showed that the restricted holonomy group of the coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle s3 su 2 s3 instead of a base manifold s3 we consider here a base manifold of dimension n 2 with a boundary and use dirichlet boundary conditions on connections as defined by marini commun pure appl math 45 1015 1992 a key step in the method of narasimhan and ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy lie algebra with respect to an appropriate sobolev norm our objective is to explore the effect of the presence of a boundary on this construction of the holonomy lie algebra fixing appropriate sobolev norms it will be shown that the space spanned linearly by the curvature form at any one connection is never dense in the holonomy lie algebra in contrast

yang mills fields on cr manifolds we study pseudo yang mills fields on a compact strictly pseudoconvex cr manifold m i e the critical points of the functional pym d 12 m hrd 2 d n where d is a connection in a hermitian cr holomorphic vector bundle e h m let 0 cn be a smoothly bounded strictly pseuodoconvex domain and g the bergman metric on we show that boundary values db of yang mills fields d on g are pseudo yang mills fields on provided that itrdb 0 and inrd 0 on h if s1 c m m is the canonical circle bundle and d is a yang mills field with respect to the fefferman metric f of m then d is a pseudo yang mills field on m the yang mills equations dr d 0 project on the euler lagrange equations bdrd 0 of the variational principle pym d 0 provided that itrd 0 when m has vanishing pseudohermitian ricci curvature the pullback d of the cr invariant tanaka connection d of e h is a yang mills field on c m we derive the second variation formula d2pym dt dt2 t 0 m sbd d n dt d

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

su n yang mills solutions with constant field strength on t 4 we study fort4 the class of solutions to the su n yang mills equations with constant field strength the fluctuation spectrum is explicitly calculated in terms of generalized riemann theta functions we show that if these solutions are stable they are necessarily anti selfdual in which case we verify the index theorem

the yang mills heat semigroup on three manifolds with boundary long time existence and uniqueness of solutions to the yang mills heat equation is proven over a compact 3 manifold with smooth boundary the initial data is taken to be a lie algebra valued connection form in the sobolev space h1 three kinds of boundary conditions are explored dirichlet type neumann type and marini boundary conditions the last is a nonlinear boundary condition specified by setting the normal component of the curvature to zero on the boundary the yang mills heat equation is a weakly parabolic nonlinear equation we use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation apriori estimates are developed by first establishing a gauge invariant version of the gaffney friedrichs inequality a gauge invariant regularization procedure for solutions is also established uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity this work is motivated by possible applications to quantum field theory

quaternionic reduction and quaternionic orbifolds on considere un processus de reduction pour des varietes de kahler quaternioniques on montre qu il existe dans ce cas une certaine application impulsion qui se reduit a celle classique quand la courbure scalaire k de l espace est zero et les metriques deviennent hyperkahler quand k 0 cette application impulsion est definie de facon unique pour un groupe compact d automorphismes h

multiple instantons representing higher order chern pontryagin classes ii this paper is a continuation of an earlier study on the generalized yang mills instantons over 4m dimensional spheres we will first present a discussion on the generalized yang mills equations the higher order chern pontryagin classes c 2m and the self dual or anti self dual equations we will then obtain some sharp asymptotic estimates for the self dual or anti self dual equations within the witten tchrakian framework which relates the integer value of c 2m to the number of vortices of the solution to a reduced 2 dimensional abelian higgs system over the poincare half plane we will prove that indeed for any integer n there exists a 2 n parameter family of the generalized self dual or anti self dual instantons realizing the topology c 2m n furthermore for the purpose of accommodating more general solutions we establish a removable singularity theorem which enables us to extend the solutions obtained on a 4m dimensional euclidean space with an integral bound to the holder continuous solutions on a 4m dimensional sphere

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

random matrix theory and portfolio optimization in moroccan stock exchange abstract in this work we use random matrix theory to analyze eigenvalues and see if there is a presence of pertinent information by using marcenko pastur distribution thus we study cross correlation among stocks of casablanca stock exchange moreover we clean correlation matrix from noisy elements to see if the gap between predicted risk and realized risk would be reduced we also analyze eigenvectors components distributions and their degree of deviations by computing the inverse participation ratio this analysis is a way to understand the correlation structure among stocks of casablanca stock exchange portfolio

well solvable special cases of the traveling salesman problem a survey the traveling salesman problem tsp belongs to the most basic most important and most investigated problems in combinatorial optimization although it is an cal np hard problem many of its special cases can be solved efficiently in polynomial time we survey these special cases with emphasis on the results that have been obtained during the decade 1985 1995 this survey complements an earlier survey from 1985 compiled by gilmore lawler and shmoys the traveling salesman problem a guided tour of combinatorial optimization wiley chichester pp 87 143

analysis on analytic spaces and non self dual yang mills fields on donne une description cohomologique de la generalisation de witten isenberg yasskin green au cas non autodual de la construction torseur de ward pour des champs de yang mills autoduaux

on a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth on etudie le comportement asymptotique de varietes de riemann on construit des coordonnees a l infini sous certaines conditions intrinseques

a conformally invariant gap theorem in yang mills theory we show a sharp conformally invariant gap theorem for yang mills connections in dimension 4 by exploiting an associated yamabe type problem

yang mills fields which are not self dual the purpose of this paper is to prove the existence of a new family of non self dual finite energy solutions to the yang mills equations on euclidean four space withsu 2 as a gauge group the approach is that of equivariant geometry attention is restricted to a special class of fields those that satisfy a certain kind of rotational symmetry for which it is proved that 1 a solution to the yang mills equations exists among them and 2 no solution to the self duality equations exists among them the first assertion is proved by an application of the direct method of the calculus of variations existence and regularity of minimizers and the second assertion by studying the symmetry properties of the linearized self duality equations the same technique yields a new family of non self dual solutions on the complex projective plane

characterizing yang mills fields by stochastic parallel transport abstract we give a new stochastic approach to yang mills fields on vector bundles by studying the derivative of the stochastic parallel transport in the vector bundle under a variation of connection we establish martingale criterions for variations transversal to the gauge orbit and for yang mills fields using variations induced by the flow of vector fields of gradient type on the base manifold in dimension four we link the local yang mills action to the quadratic variation of the derivative process defined by scalings of the driving brownian motion in a coordinate chart

pyramidal tours and the traveling salesman problem a traveling salesman problem is studied containing a shortest hamiltonian tour that is as long as a shortest pyramidal tour a tour is pyramidal if it consists of a path from city 1 to n with cities in between visited in ascending order and a path from n to 1 with cities in between visited in descending order if the distance matrix satisfies the so called demidenko conditions in which case it is called a demidenko matrix then there exists an optimal tour that is pyramidal a new proof of this theorem is given for symmetric demidenko matrices the proof cannot be used for the nonsymmetric case if however the demidenko conditions are replaced by somewhat stronger conditions it is possible to give a similar proof for the nonsymmetric case a method to construct demidenko matrices is presented and finally several tsp heuristics are tested on the class of demidenko matrices

the li yau hamilton estimate and the yang mills heat equation on manifolds with boundary the paper pursues two connected goals firstly we establish the li yau hamilton estimate for the heat equation on a manifold m with nonempty boundary results of this kind are typically used to prove monotonicity formulas related to geometric flows secondly we establish bounds for a solution t of the yang mills heat equation in a vector bundle over m the li yau hamilton estimate is utilized in the proofs our results imply that the curvature of t does not blow up if the dimension of m is less than 4 or if the initial energy of t is sufficiently small

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

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

quaternion structure on the moduli space of yang mills connections soit m h un 2 tore complexe avec une metrique plate ou une surface k3 a metrique de ricciplate soit p un fibre g principal sur m g est un groupe de lie compact simple et l espace des modules des connexions irreductibles anti auto duales sur p alors la variete de rieman admet une structure de quaternion constante de facon covariante son groupe d holonomie est contenu dans sp n 4n dim r

instability of solutions to the ginzburg landau equation on sn and cpn abstract we study critical points of the ginzburg landau gl functional and the abelian yang mills higgs ymh functional on the sphere and the complex projective space both equipped with the standard metrics for the gl functional we prove that on s n with n 2 and c p n with n 1 stable critical points must be constants in addition for gl critical points on s n for n 3 we obtain a lower bound on the morse index under suitable assumptions on the other hand for the abelian ymh functional we prove that on s n with n 4 there are no stable critical points unless the line bundle is isomorphic to s n c in which case the only stable critical points are the trivial ones our methods come from the work of lawson simons

on the structure of yang mills fields in compactified minkowski space in this work the yang mills functional with gauge group su 2 in compactified minkowski space is reduced to a mechanical lagrangian which enables one to find new classes of exact dual and nonself dual solutions the technique used in the reduction allows a simple interpretation of meron type solutions the formalism is then used to produce a good simple analytic approximation to static solutions of skyrme s field equation relevant to low energy hadronic interactions

anti self dual instantons with lagrangian boundary conditions ii bubbling we study bubbling phenomena of anti self dual instantons on open image in new window where is a closed riemann surface the instantons satisfy a global lagrangian boundary condition on each boundary slice open image in new window the main results establish the energy quantization and removal of singularities near such boundary slices this completes the analytic foundations for the definition of a new instanton floer homology for 3 manifolds with boundary in the interior case for anti self instantons on open image in new window our methods provide a new approach to the removable singularity theorem by sibner sibner for codimension 2 singularities with a holonomy condition

gromov s compactness of pseudo holomorphic curves and symplectic geometry on utilise certaines idees de schoerr et uhlenbeck sur les equations aux derivees partielles pour demontrer la compacite de gromov des courbes pseudoholomorphes

a quantum anharmonic oscillator model for the stock market abstract a financially interpretable quantum model is proposed to study the probability distributions of the stock price return the dynamics of a quantum particle is considered an analog of the motion of stock price then the probability distributions of price return can be computed from the wave functions that evolve according to schrodinger equation instead of a harmonic oscillator in previous studies a quantum anharmonic oscillator is applied to the stock in liquid market the leptokurtic distributions of price return can be reproduced by our quantum model with the introduction of mixed state and multi potential the trend following dominant market in which the price return follows a bimodal distribution is discussed as a specific case of the illiquid market

symmetric instantons and the adhm construction we construct allsu 2 yang mills instantons ons 4 that admit a certain symmetry quadrupole symmetry this is accomplished by an equivariant version of the adhm monad classification of instantons this work is part of an attempt to better understand the structure of non self dual yang mills connections with the same symmetry

the palais smale conditions for the yang mills functional on considere la fonctionnelle de yang mills definie sur des connexions sur un fibre principal sur une variete de riemann compacte de dimension 2 ou 3 on montre que si on considere la fonctionnelle de yang mills comme etant definie sur une variete de hilbert appropriee d orbites des connexions sous l action du groupe des automorphismes du fibre principal alors la fonctionnelle satisfait la condition de palais smale

non self dual yang mills connections with quadrupole symmetry we prove the existence of non self dual yang mills connections onsu 2 bundles over the four sphere specifically on all bundles with second chern number not equal 1 we study connections equivariant under ansu 2 symmetry group to reduce the effective dimensionality from four to one and then use variational techniques the existence of non self dualsu 2 ym connections on the trivial bundle second chern number equals zero has already been established by sibner sibner and uhlenbeck via different methods

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

seminaire de probabilites xxxii sous mesures symetriques sur un ensemble fini sur une inegalite de sobolev logarithmique pour une diffusion unidimensionnelle sur les minorations des constantes de sobolev et de sobolev logarithmiques pour les operateurs de jacobi et de laguerre quand l inegalite log sobolev implique l inegalite de trou spectral trous spectraux a basse temperature un contre exemple a un comportement asymptotique escompte some remarks on the optional decomposition theorem separation d une sur et d une sousmartingale par une martingale closedness of some spaces of stochastic integrals homogeneous diffusions on the sierpinski gasket almost sure path properties of branching diffusion processes criteria of regularity at the end of a tree normalized stochastic integrals in topological vector spaces pathwise uniqueness and approximation of solutions of stochastic differential equations stability of stochastic differential equations in manifolds propagation trajectorielle du chaos pour les lois de conservation scalaire some calculations for perturbed brownian motion perturbed bessel processes the maximum maximum of a martingale autour d un theoreme de tsirelson sur des filtrations browniennes et non browniennes sur un theoreme de tsirelson relatif a des mouvements browniens correles et a la nullite de certains temps locaux a remark on slutsky s theorem quelques calculs de compensateurs impliquant l injectivite de certains processus croissants the brownian burglar conditioning brownian motion by its local time process on the upcrossing chains of stopped brownian motion le theoreme de ray knight a temps fixe point le plus visite par un mouvement brownien avec derive brownian motion excursions and matrix factors sur les temps de coupure des marches aleatoires reflechies

econophysics financial time series from a statistical physics point of view in recent years physicists have started applying concepts and methods of statistical physics to study economic problems the word econophysics is sometimes used to refer to this work much recent work is focused on understanding the statistical properties of nancial time series one reason for this interest is that nancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that nancial time series viewed from a dierent perspective might yield new results this article reviews the results of three recent phenomenological studies i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to drastic events such as market crashes the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as eight orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by three orders of magnitude from 1 min up to approximately 10 d ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law and persist for several months iii correlations among dierent companies the third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two dierent stocks from a study of the eigenvalue statistics of the cross correlation matrix constructed from price uctuations of the leading 1000 stocks we nd that the largest 1 of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix whereas the rest of the eigenvalues conform to random matrix behavior suggesting that these 1 of the eigenvalues contain system specic information about correlated time evolution of dierent companies c 2000 published by elsevier science b v all rights reserved

a random matrix theory approach to financial cross correlations it is common knowledge that any two rms in the economy are correlated even rms belonging to dierent sectors of an industry may be correlated because of indirect correlations how can we analyze and understand these correlations this article reviews recent results regarding cross correlations between stocks specically we use methods of random matrix theory rmt which originated from the need to understand the interactions between the constituent elements of complex interacting systems to analyze the cross correlation matrix c of returns we analyze 30 min returns of the largest 1000 us stocks for the 2 year period 1994 1995 we nd that the statistics of approximately 20 of the largest eigenvalues 2 show deviations from the predictions of rmt to test that the rest of the eigenvalues are genuinely random we test for universal properties such as eigenvalue spacings and eigenvalue correlations and demonstrate that c shares universal properties with the gaussian orthogonal ensemble of random matrices the statistics of the eigenvectors of c conrm the deviations of the largest few eigenvalues from the rmt prediction we also nd that these deviating eigenvectors are stable in time in addition we quantify the number of rms that participate signicantly to an eigenvector using the concept of inverse participation ratio borrowed from localization theory c 2000 published by elsevier science b v all rights reserved

liouville theorem for the yang mills self duality equations it is shown that under certain conditions one may associatively matrix multiply lie algebra valued matrices with a componentwise lie bracket using this a simple algebraic constraint on a lie algebra valued antisymmetric n n matrix f which in n 4 is essentially self duality or anti self duality is described somewhat in analogy with liouville s theorem for the cauchy riemann equations in n 2 it is shown that for n 4 the constraint implies that the lie subalgebra generated by the matrix elements f decomposes into copies of s o n plus a few degenerate cases the result may be relevant to the structure of the quantum chromodynamic vacuum

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

the x and y axes travelling salesman problem the x and y axes travelling salesman problem forms a special case of the euclidean tsp where all cities are situated on the x axis and on the y axis of an orthogonal coordinate system of the euclidean plane by carefully analyzing the underlying combinatorial and geometric structures we show that this problem can be solved in polynomial time the running time of the resulting algorithm is quadratic in the number of cities

a branch and bound algorithm for the singly constrained assignment problem the singly constrained assignment problem scap is a linear assignment problem lap with one extra side constraint e g due to a time restriction the scap is in contrast to the lap difficult to solve a branch and bound algorithm is presented to solve the scap to optimality lower bounds are obtained by lagrangean relaxation computational results show that the algorithm is able to solve different types of scap instances up to size n 1000 within short running times on a standard personal computer

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

yang mills fields and random holonomy along brownian bridges we characterize yang mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of brownian bridges on the base manifold in particular we prove that a connection in a vector bundle e is yang mills if and only if the covariant derivative of parallel transport along brownian bridges in the direction of their drift is a local martingale when transported back to the starting point we present a taylor expansion up to order 3 for stochastic parallel transport in e along small rescaled brownian bridges and prove that the connection in e is yang mills if and only if all drift terms in the expansion up to order 3 vanish or equivalently if and only if the average rotation of parallel transport along small bridges and loops is of order 4

stationary points of the yang mills action we examine the structure of a recently discovered set of non self dual solutions of the yang mills equations these solutions have a symmetry that reduces the ym equations to a set of ode s the distinct solutions are indexed by two postive odd integers we develop a scheme to approximate on a computer the solutions for small values of the indexing integers and present some numerical results we then analyze the asymptotic behavior of the solutions as the indexing integers become large

collective behavior of stock price movements a random matrix theory approach we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt developed in physics to describe the energy levels ofcomplex systems rmt makes predictions f universal properties that do not depend on the interactions between the elements comprising the system deviations from rmt provide clues regarding system speci c properties we compare the statistics ofthe cross correlation matrix c whose elements cij are the correlation coe cients ofprice uctuations ofstock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks c 2001 published by elsevier science b v

stability of harmonic maps and eigenvalues of laplacian the index and nullity of the hessian of the energy for every harmonlc map are estimated above by a geometric quantity the stability theory of harmonic maps is developed and as an application the kahler version of the lichnerowicz obata theorem about the first eigenvalue of the laplacian is proved o introduction and statement of results in this paper we deal with the hessian the so called jacobi operator of the energy for a harmonic map between two riemannian manifolds as the analogue of morse theory for geodesics the morse theory for geodesics determines the homotopy type of the path space using the notion of the index and nullity of the hessian of the length of a geodesic the celebrated morse index theorem tells us that the index or nullity of a geodesic coincides with the number of conjugate points along the geodesic and this gives an upper estimate for the index and nullity in terms of the length of the geodesic under a curvature assumption giving a result known as the morse schoenberg theorem theorem morse schoenberg g k m assume that the sectional curvature nk

random matrix theory filters in portfolio optimisation a stability and risk assessment abstract random matrix theory rmt filters applied to covariance matrices of financial returns have recently been shown to offer improvements to the optimisation of stock portfolios this paper studies the effect of three rmt filters on the realised portfolio risk and on the stability of the filtered covariance matrix using bootstrap analysis and out of sample testing we propose an extension to an existing rmt filter based on krzanowski stability which is observed to reduce risk and increase stability when compared to other rmt filters tested we also study a scheme for filtering the covariance matrix directly as opposed to the standard method of filtering correlation where the latter is found to lower the realised risk on average by up to 6 7 we consider both equally and exponentially weighted covariance matrices in our analysis and observe that the overall best method out of sample was that of the exponentially weighted covariance with our krzanowski stability based filter applied to the correlation matrix we also find that the optimal out of sample decay factors for both filtered and unfiltered forecasts were higher than those suggested by riskmetrics j p morgan reuters riskmetrics technical document technical report 1996 http www riskmetrics com techdoc html with those for the latter approaching a value of 1 in conclusion rmt filtering reduced the realised risk on average and in the majority of cases when tested out of sample but increased the realised risk on a marked number of individual days in some cases more than doubling it

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

an asymmetric analog of van der veen conditions and the traveling salesman problem ii abstract j a a van der veen a new class of pyramidally solvable symmetric traveling salesman problems siam j discrete math 7 1994 585 592 proved that for the traveling salesman problem tsp which satisfies some symmetric conditions called van der veen conditions a shortest pyramidal tour is optimal that is an optimal tour can be computed in polynomial time in this paper we prove that a class satisfying an asymmetric analog of van der veen conditions is polynomially solvable an optimal tour of the instance in this class forms a tour which is an extension of pyramidal ones moreover this class properly includes some known polynomially solvable classes

the reverse yang mills higgs flow in a neighbourhood of a critical point the main result of this paper is a construction of solutions to the reverse yang mills higgs flow converging in the c topology to a critical point the construction uses only the complex gauge group action which leads to an algebraic classification of the isomorphis m classes of points in the unstable set of a critical point in terms of a filtration of the underlying higgs bundle analysing the compatibility of this filtration with the hard er narasimhan seshadri double filtration gives an algebraic criterion for two critical points to be connect ed by a flow line as an application we can use this to construct hecke modifications of higgs bundles via the yan g mills higgs flow when the higgs field is zero corresponding to the yang mills flow this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point which gives a precise description of broken and unbroken flow lines connec ting two critical points for non zero higgs field at generic critical points the analogous interpretation in volves the secant varieties of the spectral curve of the higgs bundle

random matrix theory for portfolio optimization a stability approach we apply random matrix theory rmt to an empirically measured financial correlation matrix c and show that this matrix contains a large amount of noise in order to determine the sensitivity of the spectral properties of a random matrix to noise we simulate a set of data and add different volumes of random noise having ascertained that the eigenspectrum is independent of the standard deviation of added noise we use rmt to determine the noise percentage in a correlation matrix based on real data from s p500 eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart finally we attempt to separate the noisy part from the non noisy part of c we apply an existing technique to cleaning c and then discuss its associated problems we propose a technique of filtering c that has many advantages from the stability point of view over the existing method of cleaning

wormholes in ach einstein manifolds we give a new construction of einstein manifolds which are asymptotically complex hyperbolic inspired by the work of mazzeo pacard in the real hyperbolic case the idea is to develop a gluing theorem for 1 handle surgery at infinity which generalizes the klein construction for the complex hyperbolic metric

index energy estimates for yang mills connections and einstein metrics we prove a conformally invariant estimate for the index of schrodinger operators acting on vector bundles over four manifolds related to the classical cwikel lieb rozenblum estimate applied to yang mills connections we obtain a bound for the index in terms of its energy which is conformally invariant and captures the sharp growth rate furthermore we derive an index estimate for einstein metrics in terms of the topology and the einstein hilbert energy lastly we derive conformally invariant estimates for the betti numbers of an oriented four manifold with positive scalar curvature