kms states of quasi free dynamics on pimsner algebras abstract a continuous one parameter group of unitary isometries of a right hilbert c bimodule induces a quasi free dynamics on the cuntz pimsner c algebra of the bimodule and on its toeplitz extension the restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial and the corresponding kms states of the toeplitz cuntz pimsner and cuntz pimsner c algebras are characterized in terms of traces on the algebra of coefficients this generalizes and sheds light onto various earlier results about kms states of the gauge actions on cuntz algebras cuntz krieger algebras and crossed products by endomorphisms we also obtain a more general characterization in terms of kms weights for the case in which the inducing isometries are not unitary and accordingly the restriction of the quasi free dynamics to the algebra of coefficients is nontrivial

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

length scale estimates for the lans α equations in terms of the reynolds number abstract foias holm and titi c foias d d holm e s titi the three dimensional viscous camassa holm equations and their relation to the navier stokes equations and turbulence theory j dynam differential equations 14 2002 1 35 have settled the problem of existence and uniqueness for the 3d lans equations on periodic box 0 l 3 there still remains the problem first introduced by doering and foias c r doering c foias energy dissipation in body forced turbulence j fluid mech 467 2002 289 306 for the navier stokes equations of obtaining estimates in terms of the reynolds number r e whose character depends on the fluid response as opposed to the grashof number whose character depends on the forcing r e is defined as r e u l where u is a bounded spatio temporally averaged navier stokes velocity field and l the characteristic scale of the forcing it is found that the inverse kolmogorov length is estimated by l k 1 c l 1 4 r e 5 8 moreover the estimate of foias holm and titi for the fractal dimension of the global attractor in terms of r e comes out to be d f a c v v l 1 2 l 2 1 9 8 r e 9 4 where v l l 1 2 3 and v l l l 3 it is also shown that there exists a series of time averaged inverse squared length scales whose members n 0 2 are estimated as n 1 l 2 n 0 2 c n v n 1 n r e 11 4 7 4 n ln r e 1 n c 1 r e ln r e the upper bound on the first member of the hierarchy 1 0 2 coincides with the inverse squared taylor micro scale to within log corrections

crossed products of c correspondences by amenable group actions abstract let x a be a c correspondence with a continuous action by an amenable locally compact group g we will show that if x g a g is the induced c correspondence and is the induced action on o x then o x g o x g as applications we obtain b abadie takai duality for crossed products by hilbert c modules preprint arxiv 0709 1122v1 proposition 4 5 and a kumjian d pask c algebras of directed graphs and group actions ergodic theory dynam systems 19 6 1999 1503 1519 corollary 2 5

lp independence of spectral bounds of feynman kac semigroups by continuous additive functionals abstract we establish conditions for the l p independence of spectral bounds of feynman kac semigroup by continuous additive functionals whose revuz measures are smooth measures of kato class having non negative order green tightness our continuous additive functionals do not necessarily admit bounded variation in general examples of cauchy principal value and hilbert transform of brownian local time and for relativistic symmetric stable processes are presented

on the essential spectrum of the laplacian and the drifted laplacian abstract this paper concerns the l 2 essential spectrum of the laplacian and the drifted laplacian f on complete riemannian manifolds endowed with a weighted measure e f d vol g we prove that the essential spectrum of the drifted laplacian f is 0 provided the bakry emery curvature tensor ric f is nonnegative and f has sublinear growth when ric f 1 2 g and f 2 f we show that the essential spectrum of the laplacian is also 0 during the proofs of these results the f volume growth estimate plays an important role and may be of independent interest

discrete product systems and twisted crossed products by semigroups a product systemeover a semigrouppis a family of hilbert spaces es s p together with multiplicationses et est we vieweas a unitary valued cocycle onp and consider twisted crossed productsa e pinvolvingeand an action ofpby endomorphisms of ac algebraa whenpis quasi lattice ordered in the sense of nica we isolate a class of covariant representations ofe and consider a twisted crossed productbp e pwhich is universal for covariant representations ofewhenehas finite dimensional fibres and in general is slightly larger in particular whenp nand dim e1 our algebrabn e n is a new infinite analogue of the toeplitz cuntz algebras ton our main theorem is a characterisation of the faithful representations ofbp e p

bounds on moments of the energy spectrum for weak solutions of the three dimensional navier stokes equations we consider weak solutions of the incompressible navier stokes equations in a finite three dimensional volume without boundaries for flows driven by a time independent body force we focus on some long time averaged ratios of norms of derivatives of the velocity field corresponding to moments of the fourier energy spectrum of the solutions the rigorous bounds derived here without making any additional assumptions on regularity of solutions are consistent with an asymptotic k 8 3 energy spectrum in contrast if spatial fluctuations in velocity gradients are suppressed by assumption in the analysis then the bounds are consistent with the k 5 3 turbulent kolmogorov energy spectrum these results are interpreted and discussed in the context of spatial intermittency and the energy cascade picture of turbulent dynamics

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

comparison theorem for kahler manifolds and positivity of spectrum the first part of this paper is devoted to proving a comparison theorem for kahler manifolds with holomorphic bisectional curvature bounded from below the model spaces being compared to are m m and m in particular it follows that the bottom of the spectrum for the laplacian is bounded from above by m2 for a complete m dimensional kahler manifold with holomorphic bisectional curvature bounded from below by 1 the second part of the paper is to show that if this upper bound is achieved and when m 2 then it must have at most four ends

on the essential spectrum of complete non compact manifolds in this paper we prove that the l p essential spectra of the laplacian on functions are 0 on a noncompact complete riemannian manifold with non negative ricci curvature at infinity the similar method applies to gradient shrinking ricci soliton which is similar to non compact manifold with non negative ricci curvature in many ways 2010 elsevier inc all rights reserved

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

the radial curvature of an end that makes eigenvalues vanish in the essential spectrum i the concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the laplacian in particular this paper discusses the question of whether there is an eigenvalue of the laplacian embedded in the essential spectrum or not the borderline behavior of the radial curvatures for this problem will be determined we will assume that the radial curvature k rad of an end converges to a constant 1 at infinity with the decay order k rad 1 o r 1 and prove the absence of eigenvalues embedded in the essential spectrum furthermore in order to show that this decay order k rad 1 o r 1 is sharp we will construct a manifold with the radial curvature decay k rad 1 o r 1 and with an eigenvalue frac n 1 2 4 1 embedded in the essential spectrum frac n 1 2 4 infty of the laplacian

convexity and absolute continuity of the laplace beltrami operator on montre que l existence de bonnes fonctions convexes implique l absolue continuite par rapport a la mesure de lebesgue des mesures spectrales de l extension auto adjointe du laplacien c 0 m dans l 2 m

l p independence of spectral radius for generalized feynman kac semigroups under mild conditions on measures used in the perturbation we establish the l p independence of spectral radius for generalized feynman kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric markov processes these results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized feynman kac functionals developed by the second and third authors

dissipation in parabolic spdes the study of intermittency for the parabolic anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space in this paper we set up the equation on a finite spatial interval and study the other part of intermittency i e the part of the probability space on which the solution is close to zero this set has probability very close to one and we show that on this set the supremum of the solution over space is close to 0 as a consequence we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases we also show that if the noise term is very large then the probability of the set on which the supremum of the solution is very small has a very high probability

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

c envelopes of semicrossed products by lattice ordered abelian semigroups abstract a semicrossed product is a non selfadjoint operator algebra encoding the action of a semigroup on an operator or c algebra we prove that when the positive cone of a discrete lattice ordered abelian group acts on a c algebra the c envelope of the associated semicrossed product is a full corner of a crossed product by the whole group by constructing a c cover that itself is a full corner of a crossed product and computing the shilov ideal we obtain an explicit description of the c envelope this generalizes a result of davidson fuller and kakariadis from z n to the class of all discrete lattice ordered abelian groups

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

essential normality essential norms and hyperrigidity abstract let s s 1 s d denote the compression of the d shift to the complement of a homogeneous ideal i of c z 1 z d arveson conjectured that s is essentially normal in this paper we establish new results supporting this conjecture and connect the notion of essential normality to the theory of the c envelope and the noncommutative choquet boundary the unital norm closed algebra b i generated by s 1 s d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on v z i b d where z i is the variety corresponding to i consequently the essential norm of an element in b i is equal to the sup norm of its gelfand transform and the c envelope of b i is identified as the algebra of continuous functions on v b d which means it is a complete invariant of the topology of the variety determined by i in the ball motivated by this determination of the c envelope of b i we suggest a new more qualitative approach to the problem of essential normality we prove the tuple s is essentially normal if and only if it is hyperrigid as the generating set of a c algebra which is a property closely connected to arveson s notion of a boundary representation we show that most of our results hold in a much more general setting in particular for most of our results the ideal i can be replaced by an arbitrary not necessarily homogeneous invariant subspace of the d shift

on vanishing theorems for vector bundle valued p forms and their applications let f 0 0 be a strictly increasing c2 function with f 0 0 we unify the concepts of f harmonic maps minimal hypersurfaces maximal spacelike hypersurfaces and yang mills fields and introduce f yang mills fields f degree f lower degree and generalized yang mills born infeld fields with the plus sign or with the minus sign on manifolds when f t t frac 1p 2t frac p2 sqrt 1 2t 1 and 1 sqrt 1 2t the f yang mills field becomes an ordinary yang mills field p yang mills field a generalized yang mills born infeld field with the plus sign and a generalized yang mills born infeld field with the minus sign on a manifold respectively we also introduce the ef g energy functional resp f yang mills functional and derive the first variational formula of the ef g energy functional resp f yang mills functional with applications in a more general frame we use a unified method to study the stress energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed these stress energy tensors are naturally linked to f conservation laws and yield monotonicity formulae via the coarea formula and comparison theorems in riemannian geometry whereas a microscopic approach to some of these monotonicity formulae leads to celebrated blow up techniques and regularity theory in geometric measure theory a macroscopic version of these monotonicity inequalities enables us to derive some liouville type results and vanishing theorems for p forms with values in vector bundles and to investigate constant dirichlet boundary value problems for 1 forms in particular we obtain liouville theorems for f harmonic maps which include harmonic maps p harmonic maps exponentially harmonic maps minimal graphs and maximal space like hypersurfaces etc f yang mills fields extended born infeld fields and generalized yang mills born infeld fields with the plus sign and with the minus sign on manifolds etc as another consequence we obtain the unique constant solution of the constant dirichlet boundary value problems on starlike domains for vector bundle valued 1 forms satisfying an f conservation law generalizing and refining the work of karcher and wood on harmonic maps we also obtain generalized chern type results for constant mean curvature type equations for p forms on mathbb r m and on manifolds m with the global doubling property by a different approach the case p 0 and m mathbb r m is due to chern

a macroscopic multifractal analysis of parabolic stochastic pdes it is generally argued that the solution to a stochastic pde with multiplicative noise such as dot u frac12 u u xi where xi denotes space time white noise routinely produces exceptionally large peaks that are macroscopically multifractal see for example gibbon and doering arch ration mech anal 177 115 150 2005 gibbon and titi proc r soc a 461 3089 3097 2005 and zimmermann et al phys rev lett 85 17 3612 3615 2000 a few years ago we proved that the spatial peaks of the solution to the mentioned stochastic pde indeed form a random multifractal in the macroscopic sense of barlow and taylor j phys a 22 13 2621 2626 1989 proc lond math soc 3 64 125 152 1992 the main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio temporal peaks of the solution form infinitely many different multifractals on infinitely many different scales which we sometimes refer to as stretch factors a simpler though still complex such structure is shown to also exist for the constant coefficient version of the said stochastic pde

tensor algebras of c correspondences and their c envelopes abstract we show that the c envelope of the tensor algebra of an arbitrary c correspondence x coincides with the cuntz pimsner algebra o x as defined by katsura t katsura on c algebras associated with c correspondences j funct anal 217 2004 366 401 this improves earlier results of muhly and solel p s muhly b solel tensor algebras over c correspondences representations dilations and c envelopes j funct anal 158 1998 389 457 and fowler muhly and raeburn n fowler p muhly i raeburn representations of cuntz pimsner algebras indiana univ math j 52 2003 569 605 who came to the same conclusion under the additional hypothesis that x is strict and faithful

on the lp spectrum of uniformly elliptic operators on riemannian manifolds abstract we prove that the l p spectrum of uniformly elliptic divergence form operators on a complete riemannian manifold is independent of p 1 if the volume of the manifold grows uniformly subexponentially the latter condition is satisfied if the ricci curvature is bounded from below by a function k m r satisfying lim inf x k x 0 in particular if the ricci curvature is nonnegative on the other hand if we asume the existence of a globally defined volume density which grows exponentially in every direction then the l p spectrum depends on p this condition is satisfied if the manifold is simply connected and if the sectional curvature is bounded from above by a function k m r satisfying k 0 and lim sup x k x

on the spectrum of the laplacian in this article we prove a generalization of weyl s criterion for the essential spectrum of a self adjoint operator on a hilbert space we then apply this criterion to the laplacian on functions over open manifolds and get new results for its essential spectrum

complete manifolds with bounded curvature and spectral gaps abstract we study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius we give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps in particular for any noncompact covering of a compact manifold there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum also for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one also of bounded curvature and positive injectivity radius with an arbitrarily large finite number of gaps in its essential spectrum

tensor algebras of product systems and their c envelopes abstract let g p be an abelian lattice ordered group and let x be a compactly aligned product system over p with coefficients in a we show that the c envelope of the nica tensor algebra n t x coincides with both sehnem s covariance algebra a x p and the co universal c algebra n o x r for injective gauge compatible nica covariant representations of carlsen larsen sims and vittadello we give several applications of this result on both the selfadjoint and non selfadjoint operator algebra theory first we guarantee the existence of n o x r thus settling a problem of carlsen larsen sims and vittadello which was open even for abelian lattice ordered groups as a second application we resolve a problem posed by skalski and zacharias on dilating isometric representations of product systems to unitary representations as a third application we characterize the c envelope of the tensor algebra of a finitely aligned higher rank graph which also holds for topological higher rank graphs as a final application we prove reduced hao ng isomorphisms for generalized gauge actions of discrete groups on c algebras of product systems this generalizes recent results that were obtained by various authors in the case where g p z n

regularity and singularity in solutions of the three dimensional navier stokes equations higher moments of the vorticity field m t in the form of l 2 m norms graphic 1 are used to explore the regularity problem for solutions of the three dimensional incompressible navier stokes equations on the domain graphic 2 it is found that the set of quantities formula 3 provide a natural scaling in the problem resulting in a bounded set of time averages d m t on a finite interval of time 0 t the behaviour of d m 1 d m is studied on what are called good and bad intervals of 0 t which are interspersed with junction points neutral i for large but finite values of m with large initial data m 0 o gr 4 it is found that there is an upper bound formula 4 which is punctured by infinitesimal gaps or windows in the vertical walls between the good bad intervals through which solutions may escape while this result is consistent with that of leray leray 1934 acta math 63 193 248 doi 10 1007 bf02547354 5 and scheffer scheffer 1976 pacific j math 66 535 552 this estimate for m corresponds to a length scale well below the validity of the navier stokes equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed graphic 1 gif 4 embed graphic 2 gif 5 http dx doi org doi 10 1007 bf02547354

lp independence of spectral bounds of schrodinger type semigroups let l be an m symmetric markov generator and a signed measure in the kato class we consider a schrodinger type operator h l on lp m we prove that under certain conditions for the markov semigroup generated by l and the potential the lp spectral bound of h is independent of p if and only if the l2 spectral bound is non positive

a new look at crossed product correspondences and associated c algebras abstract when a locally compact group acts on a c correspondence it also acts on the associated cuntz pimsner algebra in a natural way hao and ng have shown that when the group is amenable the cuntz pimsner algebra of the crossed product correspondence is isomorphic to the crossed product of the cuntz pimsner algebra in this paper we have a closer look at this isomorphism in the case where the group is not necessarily amenable we also consider what happens at the level of toeplitz algebras

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

estimates for the two dimensional navier stokes equations in terms of the reynolds number the tradition in navier stokes analysis of finding estimates in terms of the grashof number gr whose character depends on the ratio of the forcing to the viscosity means that it is difficult to make comparisons with other results expressed in terms of reynolds number re whose character depends on the fluid response to the forcing the first task of this paper is to apply the approach of doering and foias c r doering and c foias j fluid mech 467 289 2002 to the two dimensional navier stokes equations on a periodic domain 0 l 2 by estimating quantities of physical relevance particularly long time averages in terms of the reynolds number re ul where u2 l 2 u 22 and l is the forcing scale in particular the constantin foias temam upper bound p constantin c foias and r temam physica d 30 284 1988 on the attractor dimension converts to al2re 1 lnre 1 3 while the estimate for the inverse kraichnan length is al2re 1 2 where al is the aspect ratio of the forcing other in

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

co universal algebras associated to product systems and gauge invariant uniqueness theorems let x be a product system over a quasi lattice ordered group under mild hypotheses we associate to x a c algebra which is co universal for injective nica covariant toeplitz representations of x which preserve the gauge coaction under appropriate amenability criteria this co universalc algebra coincides with the cuntz nica pimsner algebra introduced by sims and yeend we prove two key uniqueness theorems and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co universal algebras in each case it is an easy corollary that the cuntz nica pimsner algebra is isomorphic to the corresponding full crossed product

a class of c algebras generalizing both graph algebras and homeomorphism c algebras i fundamental results we introduce a new class of c algebras which is a generalization of both graph algebras and homeomorphism c algebras this class is very large and also very tractable we prove the so called gauge invariant unique ness theorem and the cuntz krieger uniqueness theorem and compute the k groups of our algebras

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

on complete gradient shrinking ricci solitons in this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons based on this we prove that a complete noncompact gradient shrinking ricci soliton has at most euclidean volume growth this latter result can be viewed as an analog of the well known volume comparison theorem of bishop that a complete noncompact riemannian manifold with nonnegative ricci curvature has at most euclidean volume growth

lp spectral theory and heat dynamics of locally symmetric spaces abstract in this paper we first derive several results concerning the l p spectrum of locally symmetric spaces with rank one in particular we show that there is an open subset of c consisting of eigenvalues of the l p laplacian if p 2 and that corresponding eigenfunctions are given by certain eisenstein series on the other hand if p 2 there is at most a discrete set of real eigenvalues of the l p laplacian these results are used in the second part of this paper in order to show that the dynamics of the l p heat semigroups for p 2 is very different from the dynamics of the l p heat semigroups if p 2

density and spectrum of minimal submanifolds in space forms let varphi m m rightarrow n n be a minimal proper immersion in an ambient space suitably close to a space form mathbb n n k of curvature k le 0 in this paper we are interested in the relation between the density function theta r of m and the spectrum of its laplace beltrami operator in particular we prove that if theta r has subexponential growth when k 0 or sub polynomial growth k 0 along a sequence then the spectrum of m m is the same as that of the space form mathbb n m k notably the result applies to anderson s smooth solutions of plateau s problem at infinity on the hyperbolic space independently of their boundary regularity we also give a simple condition on the second fundamental form that ensures m to have finite density in particular we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

the spectrum of the laplacian on forms over flat manifolds in this article we prove that the spectrum of the laplacian on k forms over a non compact flat manifold is always a connected closed interval of the non negative real line the proof is based on a detailed decomposition of the structure of flat manifolds

on c algebras associated with c correspondences abstract we study c algebras arising from c correspondences which were introduced by the author we prove the gauge invariant uniqueness theorem and obtain conditions for our c algebras to be nuclear exact or satisfy the universal coefficient theorem we also obtain a 6 term exact sequence of k groups involving the k groups of our c algebras

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

correlation functions in isotropic and anisotropic turbulence the role of the symmetry group the theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state real turbulent flows exhibit the effects of anisotropic forcing the analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations so 3 in this paper we first consider the needed general theory and explain why we expect different universal scaling exponents in the different sectors of the symmetry group we exemplify the theory context of isotropic turbulence for third order tensorial structure functions and in weakly anisotropic turbulence for the second order structure function the utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high reynolds number measurements of turbulence in the atmosphere

discrete product systems and their c algebras continuous product systems or product systems over r were recently introduced by arveson to develope an index theory for continuous semigroups of endomorphisms of b h in this paper we continue his work by studying discrete product systems or product systems over g where g is a countable dense subgroup of r we give examples of discrete product systems and compute their automorphism groups to each discrete product system e over g we associate a separable c algebra oe g whose non degenerate representations correspond bijectively to representations of e this family of c algebras may be regarded as analogues of cuntz algebras or car algebras in some degenerate cases these c algebras are the generalized toeplitz algebras we show that oe g is simple and study its gauge automorphisms quasi free automorphisms and fock state