isospectral conformal metrics on 3 manifolds theorem let g uj go be a sequence of conformal metrics satisfying the following conditions i vol m gj ao for some positive constant ao ii f r2 gj lp gj 12 dvj a 0 i e for each q defined on m we have

isospectral sets of conformally equivalent metrics il existe des varietes m a metriques conformement equivalentes g et g telles que g soit isospectrale a g

compactness of conformal metrics with integral bounds on curvature in this thesis we show that a sequence of conformal metrics on a compact n dimensional riemannian manifold n 4 which has an upper bound on volume and an upper bound on the lp norm of the curvature tensor for fixed p n 2 has a subsequence which converges in c if n 3 we have the same result if we assume in addition that the scalar curvature has an l2 bound as corollaries we have the compactness of a sequence of conformal metrics on a compact three manifold which are isospectral with respect to either the standard or conformal laplacian and the result of lelong ferrand that any compact manifold with non compact conformal group is conformally equivalent to the standard sphere

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

subspace clustering of categorical and numerical data with an unknown number of clusters in clustering analysis data attributes may have different contributions to the detection of various clusters to solve this problem the subspace clustering technique has been developed which aims at grouping the data objects into clusters based on the subsets of attributes rather than the entire data space however the most existing subspace clustering methods are only applicable to either numerical or categorical data but not both this paper therefore studies the soft subspace clustering of data with both of the numerical and categorical attributes also simply called mixed data for short specifically an attribute weighted clustering model based on the definition of object cluster similarity is presented accordingly a unified weighting scheme for the numerical and categorical attributes is proposed which quantifies the attribute to cluster contribution by taking into account both of intercluster difference and intracluster similarity moreover a rival penalized competitive learning mechanism is further introduced into the proposed soft subspace clustering algorithm so that the subspace cluster structure as well as the most appropriate number of clusters can be learned simultaneously in a single learning paradigm in addition an initialization oriented method is also presented which can effectively improve the stability and accuracy of k means type clustering methods on numerical categorical and mixed data the experimental results on different benchmark data sets show the efficacy of the proposed approach

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

numerical computation of e entropy for parabolic equations with analytic solutions abstract we report on numerical experiments computing the e entropy for parabolic partial differential equations the e entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors the e entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is asymptotically independent of domain size we consider the complex ginzburg landau equation as an example where a number of analytic results exist and the kuramoto sivashinsky equation where the accompanying theory has yet to be fully developed our numerical results for the kuramoto sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size

optimal bounds on the kuramoto sivashinsky equation in this paper we consider solutions u t x of the one dimensional kuramoto sivashinsky equation i e tu x 12u2 x2u x4u 0 which are l periodic in x and have vanishing spatial average numerical simulations show that for l 1 solutions display complex spatio temporal dynamics the statistics of the pattern in particular its scaled power spectrum is reported to be extensive i e not to depend on l for l 1 more specifically after an initial layer it is observed that the spatial quadratic average x u 2 of all fractional derivatives x u of u is bounded independently of l in particular the time space average x u 2 is observed to be bounded independently of l the best available result states that x u 2 1 2 o l for all 0 2 in this paper we prove that x u 2 1 2 o ln5 3l for 1 3 2 to our knowledge this is the first result in favor of an extensive behavior albeit only up to a logarithm and for a restricted range of fractional derivatives as a corollary we obtain u2 1 2 o l1 3 which improves the known bounds

analyticity for the kuramoto sivashinsky equation abstract we study the analyticity properties of solutions of the kuramoto sivashinsky equation tu x t 2x 4x u x t u x t xu x t for initial data which are periodic with period l numerical expers are presented which show that the solutions of the ks equation are analytic in a strip around the real axis whose width is independent of l a rigorous lower bound o l 16 25 is given for this width

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

oscillations of solutions of the kuramoto sivashinsky equation abstract oscillation properties of graphs of solutions of the kuramoto sivashinsky equation are studied it is shown that every solution either converges to the stationary solution 0 with an exponential rate or after some time the cardinalities of the sets of zeros in the space variable of all the spatial derivatives remain bounded by certain constants independent of the particular solution

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

on the well posedness of an anisotropically reduced two dimensional kuramoto sivashinsky equation abstract the kuramoto sivashinsky equations kse arise in many diverse scientific areas and are of much mathematical interest due in part to their chaotic behavior and their similarity to the navier stokes equations however very little is known about their global well posedness in the 2d case moreover regularizations of the system e g adding large diffusion etc do not seem to help due to the lack of any control over the l 2 norm in this work we propose a new reduced 2d model that modifies only the linear part of the vector form of the 2d kse in only one component this new model shares much in common with the 2d kse it is 4th order in space it has an identical nonlinearity which does not vanish in energy estimates it has low mode instability and it lacks a maximum principle however we prove that our reduced model is globally well posed we also examine its dynamics computationally moreover while its solutions do not appear to be close approximations of solutions to the kse the solutions do seem to hold many qualitative similarities with those of the kse we examine these properties via computational simulations comparing solutions of the new model to solutions of the 2d kse

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

a new distance metric for unsupervised learning of categorical data distance metric is the basis of many learning algorithms and its effectiveness usually has a significant influence on the learning results in general measuring distance for numerical data is a tractable task but it could be a nontrivial problem for categorical data sets this paper therefore presents a new distance metric for categorical data based on the characteristics of categorical values in particular the distance between two values from one attribute measured by this metric is determined by both the frequency probabilities of these two values and the values of other attributes that have high interdependence with the calculated one dynamic attribute weight is further designed to adjust the contribution of each attribute distance to the distance between the whole data objects promising experimental results on different real data sets have shown the effectiveness of the proposed distance metric

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

efficient dual approach to distance metric learning distance metric learning is of fundamental interest in machine learning because the employed distance metric can significantly affect the performance of many learning methods quadratic mahalanobis metric learning is a popular approach to the problem but typically requires solving a semidefinite programming sdp problem which is computationally expensive the worst case complexity of solving an sdp problem involving a matrix variable of size d d with o d linear constraints is about o d6 5 using interior point methods where d is the dimension of the input data thus the interior point methods only practically solve problems exhibiting less than a few thousand variables because the number of variables is d d 1 2 this implies a limit upon the size of problem that can practically be solved around a few hundred dimensions the complexity of the popular quadratic mahalanobis metric learning approach thus limits the size of problem to which metric learning can be applied here we propose a significantly more efficient and scalable approach to the metric learning problem based on the lagrange dual formulation of the problem the proposed formulation is much simpler to implement and therefore allows much larger mahalanobis metric learning problems to be solved the time complexity of the proposed method is roughly o d3 which is significantly lower than that of the sdp approach experiments on a variety of data sets demonstrate that the proposed method achieves an accuracy comparable with the state of the art but is applicable to significantly larger problems we also show that the proposed method can be applied to solve more general frobenius norm regularized sdp problems approximately

investigation of the analyticity of dissipative dispersive systems via a semigroup method abstract in this work we study the analyticity of kuramoto sivashinsky type equations and related systems by exploring the applicability of the semigroup method which was developed in collet et al 5 we establish the analyticity with respect to the spatial variable in a strip around the real axis for a variety of dissipative dispersive systems which possess universal attractors we also provide lower bounds for the width of the strip of analyticity

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

categorical and numerical attribute data clustering based on a unified similarity metric without knowing cluster number most of the existing clustering approaches are applicable to purely numerical or categorical data only but not the both in general it is a nontrivial task to perform clustering on mixed data composed of numerical and categorical attributes because there exists an awkward gap between the similarity metrics for categorical and numerical data this paper therefore presents a general clustering framework based on the concept of object cluster similarity and gives a unified similarity metric which can be simply applied to the data with categorical numerical and mixed attributes accordingly an iterative clustering algorithm is developed whose outstanding performance is experimentally demonstrated on different benchmark data sets moreover to circumvent the difficult selection problem of cluster number we further develop a penalized competitive learning algorithm within the proposed clustering framework the embedded competition and penalization mechanisms enable this improved algorithm to determine the number of clusters automatically by gradually eliminating the redundant clusters the experimental results show the efficacy of the proposed approach

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

space analyticity for the navier stokes and related equations with initial data inlp abstract we introduce a method of estimating the space analyticity radius of solutions for the navier stokes and related equations in terms of l p and l norms of the initial data the method enables us to express the space analyticity radius for 3d navier stokes equations in terms of the reynolds number of the flow also for the kuramoto sivashinsky equation we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period

metric and kernel learning using a linear transformation metric and kernel learning arise in several machine learning applications however most existing metric learning algorithms are limited to learning metrics over low dimensional data while existing kernel learning algorithms are often limited to the transductive setting and do not generalize to new data points in this paper we study the connections between metric learning and kernel learning that arise when studying metric learning as a linear transformation learning problem in particular we propose a general optimization framework for learning metrics via linear transformations and analyze in detail a special case of our framework that of minimizing the logdet divergence subject to linear constraints we then propose a general regularized framework for learning a kernel matrix and show it to be equivalent to our metric learning framework our theoretical connections between metric and kernel learning have two main consequences 1 the learned kernel matrix parameterizes a linear transformation kernel function and can be applied inductively to new data points 2 our result yields a constructive method for kernelizing most existing mahalanobis metric learning formulations we demonstrate our learning approach by applying it to large scale real world problems in computer vision text mining and semi supervised kernel dimensionality reduction

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

computational study of the dispersively modified kuramoto sivashinsky equation we analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified kuramoto sivashinsky equations time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization the general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long wave instability negative diffusion short wave damping fourth order stabilization nonlinear saturation burgers nonlinearity and dispersive effects the solutions of such systems typically converge to compact absorbing sets of finite dimension i e global attractors and are characterized by chaotic behavior our objective is to employ schemes which capture faithfully these chaotic dynamics in the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the kuramoto sivashinsky equation in such instances we show that first and second order time stepping schemes are appropriate and provide convergence proofs for the schemes in physical situations when the dispersion is of lower order than the fourth order stabilization term for example a hybrid kuramoto sivashinsky korteweg devries equation also known as the kawahara equation in hydrodynamics higher order time stepping schemes can be used and we analyze and implement schemes of order six or less we derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors various numerical diagnostic tools are implemented such as the projection of the infinite dimensional dynamics to one dimensional return maps that enable us to probe the geometry of the attractors quantitatively such results are possible only if computations are carried out for very long times we provide examples where integrations are carried out for 10 8 time units and it is shown that the schemes used here are very well suited for such tasks for illustration computations are carried out for third order dispersion the kawahara equation as well as fifth order dispersion the benney lin equation but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics

the variation of the de rham zeta function calcul de la variation des valeurs speciales de la fonction zeta de de rham par rapport a la variation de la metrique etablissement de l equation d euler lagrange pour les variations conformes

bubble tree convergence for conformal metrics with m r n2dvg bounds abstract given a sequence of conformal metrics g k u k 4 n 2 g 0 on a smooth compact boundaryless riemannian manifold m n g 0 assume the volume of g k and l n 2 norm of scalar curvatures both are bounded we prove that after passing to a subsequence u k weakly converges to the bubble tree limit u u 1 1 u i u l l 1 i l 1 i in w 2 p for some p n 2 where u w 2 p m g 0 and u i w 2 p r n g r n moreover after passing to a subsequence the sequence of metric spaces m d k defined by g k converges to a connected metric space m d in the gromov hausdorff topology sense and lim k vol m g k h n m d where h n is the n dimensional hausdorff measure defined by d

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

on the non homogeneous stationary kuramoto sivashinsky equation abstract in this note we give some explicit estimates for the l norm of the periodic solutions of the time independent non homogeneous kuramoto sivashinsky equation in particular we give an estimate of the michelson s upper bound of all periodic solutions of the time independent homogeneous kuramoto sivashinsky equation

symbolic clustering using a new dissimilarity measure abstract a new dissimilarity measure based on position span and content of symbolic objects is proposed for symbolic clustering the dissimilarity measure is new in the sense that it is not just another aspect of a similarity measure in the proposed hierarchical agglomerative clustering methodology composite symbolic objects are formed using a cartesian join operator whenever a mutual pair of symbolic objects is selected for agglomeration based on minimum dissimilarity the minimum dissimilarity values of different merging levels are used to compute the cluster indicator values and hence to determine the number of clusters in the data the results of the application of the algorithm on numeric data of known number of classes are described first so as to show the efficacy of the method subsequently the results of the experiments on two data sets of assertion type of symbolic objects drawn from the domains of fat oil and microcomputers are presented

space analyticity for the nonlinear heat equation in a bounded domain 1 introductionin fot foias and temam introduced a method for estimating thespace analyticity radius of solutions of the navier stokes equation withperiodic boundary conditions for the numerous applications seee g cees crt dth ft gr k lo in our previous paper gk we did not use the fourier series and were thus able to treat the navier stokes and other semilinear parabolic equations with singular l