analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

a variational approach to surface solitary waves abstract two dimensional flow of an incompressible inviscid fluid in a region with a horizontal bottom of infinite extent and a free upper surface is considered the fluid is acted on by gravity and has a non diffusive heterogeneous density which may be discontinuous it is shown that the governing equations allow both periodic and single crested progressing waves of permanent form the analogues respectively of the classical cnoidal and solitary waves these waves are shown to be critical points of flow related functionals and are proved to exist by means of a variational principle

equivariant higher analytic torsion and equivariant euler characteristic we show that j lott s equivariant higher analytic torsion only depends on the equivariant euler characteristic we give an explicit formula 1 introduction let g be a compact connected lie group with lie algebra g let 1 g denote the ring of ad g invariant complex polynomials on g then 1 g 1 f e 1 g f 0 0 is a maximal ideal of 1 g by 1 g we denote the

loop spaces and the hypoelliptic laplacian the purpose of this paper is to introduce some ideas that motivated the construction of the hypoelliptic laplacian as a deformation of hodge theory which interpolates between hodge theory and the geodesic flow results obtained with lebeau on the analysis of the hypoelliptic laplacian are also presented 2007 wiley periodicals inc

vortex rings in r 3 and rearrangements we study the existence of steady axisymmetric vortex rings in an ideal fluid a functional comprising a linear combination of kinetic energy and impulse is to be maximized subject to the constraint that a quantity related to vorticity belongs to a set of rearrangements of a given function generalized solutions of a quite specific type are shown to exist arising as extreme points of a convex extended constraint set in the case when the given function is the indicator of a set of finite measure the existence of proper maximizers and local maximizers is demonstrated

equivariant analytic torsion for compact lie group actions abstract we define the equivariant analytic torsion for a compact lie group action and study its dependence on the geometric data

differential k theory and localization formula for eta invariants in this paper we obtain a localization formula in differential k theory for s 1 actions we establish a localization formula for equivariant eta invariants by combining this result with our extension of goette s result on the comparison of two types of equivariant eta invariants an important step in our approach is to construct a pre lambda ring structure in differential k theory

a steady vortex ring in poiseuille flow and rearrangements of a function this study proves the existence of a steady vortex ring of an ideal fluid in poiseuille flow the method that was used is a variational method proposed by benjamin benjamin 1976 the alliance of practical and analytical insight into the nonlinear problems of fluid mechanics vol 503 pp 8 29 in which a steady vortex ring can be obtained as a maximizer of a functional that is related to kinetic energy and the impulse over the set of rearrangements of a prescribed function

on the eigenvalue estimation for solution to lyapunov equation abstract this paper deals with the problems of eigenvalue estimation for the solution to the perturbed matrix lyapunov equation we obtain some eigenvalue inequalities on condition that x is a positive semidefinite solution to the equation atxa x q which can be used in control theory and linear system stability

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

on the incompressible fluid limit and the vortex motion law of the nonlinear schrodinger equation the nonlinear schrodinger equation nls has been a fundamental model for understanding vortex motion in superfluids the vortex motion law has been formally derived on various physical grounds and has been around for almost half a century we study the nonlinear schrodinger equation in the incompressible fluid limit on a bounded domain with dirichlet or neumann boundary condition the initial condition contains any finite number of degree 1 vortices we prove that the nls linear momentum weakly converges to a solution of the incompressible euler equation away from the vortices if the initial nls energy is almost minimizing we show that the vortex motion obeys the classical kirchhoff law for fluid point vortices similar results hold for the entire plane and periodic cases and a related complex ginzburg landau equation we treat as well the semi classical wkb limit of nls in the presence of vortices in this limit sound waves propagate through steady vortices

low rank solution of lyapunov equations this paper presents the cholesky factor alternating direction implicit cf adi algorithm which generates a low rank approximation to the solution x of the lyapunov equation ax xat bbt the coefficient matrix a is assumed to be large and the rank of the right hand side bbt is assumed to be much smaller than the size of a the cf adi algorithm requires only matrix vector products and matrix vector solves by shifts of a hence it enables one to take advantage of any sparsity or structure in a this paper also discusses the approximation of the dominant invariant subspace of the solution x we characterize a group of spanning sets for the range of x a connection is made between the approximation of the dominant invariant subspace of x and the generation of various low order krylov and rational krylov subspaces it is shown by numerical examples that the rational krylov subspace generated by the cf adi algorithm where the shifts are obtained as the solution of a rational minimax problem often gives the best approximation to the dominant invariant subspace of x

hardy spaces and the two dimensional euler equations with nonnegative vorticity we prove that certain quadratic expressions involving the gradient of a weakly superharmonic function in r belong to a local hardy space as an application we provide a new proof of j m delort s convergence theorem for solutions of the two dimensional euler equations with vorticities of one sign department of mathematics university of california berkeley california 94720 sfb 256 institut fur angewandte mathematik beringstrasse 4 d5300 bonn 1 ger many this content downloaded from 207 46 13 101 on sat 08 oct 2016 05 44 35 utc all use subject to http about jstor org terms

desingularization of vortex rings and shallow water vortices by a semilinear elliptic problem steady vortices for the three dimensional euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments the construction is based on a study of solutions to the semilinear elliptic problem left begin aligned rm div left frac nabla u varepsilon b right frac 1 varepsilon 2 b f left u varepsilon log tfrac 1 varepsilon q right text in omega u varepsilon 0 text on partial omega end aligned right for small values of varepsilon 0

desingularizations of systems of point vortices abstract consider the evolution of an incompressible inviscid fluid in r 2 zabusky conjectured that given a steady system of point vortices there exists a one parameter family of steady system of vortex patches in which each patch shrinks to the corresponding point vortex while keeping the same circulation as the parameter tends to zero such a family is called a desingularization of the given system of point vortices in this article we establish that 1 a zabusky s conjecture holds under a mild condition on the system of point vortices 2 b the desingularizations of the system of point vortices preserve their type of stability 3 c desingularizations of an unfolding of a system of point vortices exist under a typical situation

on the blow up of solutions of the 3 d euler equations in a bounded domain it is shown that if 0 hat t is the maximal interval of existence of a smooth solutionu of the incompressible euler equations in a bounded simply connected domain subseteq r3 then int 0 hat t left omega cdot t right l infty omega dt infty where u is the vorticity crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

liouville type theorems for the euler and the navier stokes equations abstract we prove liouville type theorems for weak solutions of the navier stokes and the euler equations in particular if the pressure satisfies p l 1 0 t l 1 r n with r n p x t d x 0 then the corresponding velocity should be trivial namely v 0 on r n 0 t in particular this is the case when p l 1 0 t h q r n where h q r n q 0 1 the hardy space on the other hand we have equipartition of energy over each component if p l 1 0 t l 1 r n with r n p x t d x 0 similar results hold also for the magnetohydrodynamic equations

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

analytic torsion for group actions the reidemeister torsion is a classical topological invariant for nonsimply connected manifolds 14 let m be a closed oriented smooth manifold with the fundamental group m nontrivial let p x m o n be an orthogonal representation of nx m if the twisted real cohomology h m ep vanishes then one can define the reidemeister torsion p gr which is a homeomorphism invariant of m the original interest of p was that it is not a homotopy invariant and so can distinguish spaces which are homotopy equivalent but are not homeomorphic ray and singer asked whether as for many other topological quantities one can compute p by analytic methods 17 given a riemannian metric g on m they defined an analytic torsion t e r as a certain combination of the eigenvalues of the laplacian acting on twisted differential forms they showed that under the above acyclicity condition t is independent of the metric g and they conjectured that the analytic expression t equals the combinatorial expression this conjecture was proven to be true independently by cheeger 4 and m ller 15 one can look at the above situation in the following way the group n m acts freely on the universal cover m and so one has an invariant for free group actions a natural question is whether the reidemeister torsion can be extended to an invariant for more general group actions for a finite group acting not necessarily freely on a closed oriented pl manifold x a reidemeister torsion was defined algebraically by rothenberg 19 one can then ask whether there is a corresponding analytic torsion when x is smooth and whether the analytic torsion equals the combinatorial torsion in 11 we define the analytic torsion tp for a finite group action and show that if the relevant cohomology groups vanish then it is independent of the g invariant metric used in its definition this was shown previously in unpublished work by cheeger 5 the analysis involved to

renormalized solutions of the 2d euler equations in this paper we prove that solutions of the 2d euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized

planar vortex patch problem in incompressible steady flow abstract in this paper we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain of r 2 let k be a positive integer and let j be a positive constant j 1 k for any given non degenerate critical point x 0 x 0 1 x 0 k of the kirchhoff routh function defined on k corresponding to 1 k we prove the existence of a planar flow such that the vorticity of this flow equals a large given positive constant in each small neighborhood of x 0 j j 1 k and 0 elsewhere moreover as the vorticity set y y shrinks to j 1 k x 0 j and the local vorticity strength near each x 0 j approaches j j 1 k

complex equivariant intersection excess normal bundles and bott chern currents the purpose of this paper is to establish an intersection formula in equivariant complex geometry in the presence of an excess normal bundle the contribution of the excess normal bundle to the formula appears through an additive genus k r in a forthcoming paper an infinite dimensional analogue of this formula will be shown to be the result of bismut lebeau on the behaviour of quillen metrics under complex immersions

a criterion for the equivalence of the birkhoff rott and euler descriptions of vortex sheet evolution in this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible euler equations and as a weak solution of the birkhoff rott equations we begin by discussing the classical birkhoff rott equations with respect to arbitrary parametrizations of the sheet we introduce a notion of weak solution to the birkhoff rott system and we prove consistency of this notion with the classical formulation of the equations our main purpose in this paper is to present a sharp criterion for the equivalence of the weak euler and weak birkhoff rott descriptions of vortex sheet dynamics

equivariant de rham torsions the purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de rham theory this difference is the sum of the integral of a chern simons current and of a new invariant the v invariant of an odd dimensional manifold equipped with an action of a compact lie group the v invariant localizes on the critical manifolds of invariant morse bott functions the results in this paper are shown to be compatible with results of bunke and also our with previous results on analytic torsion forms

linear stability of compressible vortex sheets in two dimensional elastodynamics abstract the linear stability of rectilinear compressible vortex sheets is studied for two dimensional isentropic elastic flows this problem has a free boundary and the boundary is characteristic a necessary and sufficient condition is obtained for the linear stability of the rectilinear vortex sheets more precisely it is shown that besides the stable supersonic zone the elasticity exerts an additional stable subsonic zone we also find that there is a class of states in the interior of subsonic zone where the stability of such states is weaker than the stability of other states in the sense that there is an extra loss of tangential derivatives with respect to the source terms this is a new feature that the euler flow does not possess one of the difficulties for the elastic flow is that the non differentiable points of the eigenvalues may coincide with the roots of the lopatinskii determinant as a result the kreiss symmetrization cannot be applied directly instead we perform an upper triangularization of the system to separate only the outgoing modes at all points in the frequency space so that an exact estimate of the outgoing modes can be obtained moreover all the outgoing modes are shown to be zero due to the l 2 regularity of solutions the estimates for the incoming modes can be derived directly from the lopatinskii determinant this new approach avoids the lengthy computation and estimates for the outgoing modes when kreiss symmetrization is applied this method can also be applied to the euler flow and mhd flow

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

existence of vortex sheets with reflection symmetry in two space dimensions the main purpose of this work is to establish the existence of a weak solution to the incompressible 2d euler equations with initial vorticity consisting of a radon measure with distinguished sign in h 1 compactly supported in the closed right half plane superimposed on its odd reflection in the left half plane we make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis we prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2d euler equations we also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half plane finally we extend our existence result to odd l 1 perturbations without distinguished sign of our original initial vorticity

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

superconnection currents and complex immersions let i m m be an immersion of complex manifolds and let be a complex of holomorphic hermitian vector bundles on m which provides a projective resolution of the sheaf of sections of a holomorphic vector bundle on m we study the convergence as u of a class of superconnection currents u introduced by quillen and we calculate the limit microlocal estimates of the speed of convergence are also given

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

a natural gradient descent algorithm for the solution of discrete algebraic lyapunov equations based on the geodesic distance a new framework to calculate the numerical solution of the discrete algebraic lyapunov equation is proposed by using the geometric structures on the riemannian manifold specifically two algorithms based on the manifold of positive definite symmetric matrices are provided one is a gradient descent algorithm with an objective function of the classical euclidean distance the other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved riemannian manifold furthermore these two algorithms are compared with a traditional iteration method simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms

an elliptic boundary value problem with a discontinuous nonlinearity ii soit un domaine borne dans r 2 on etudie le probleme aux valeurs limites pour k h k dans r 2 0 sur h est la fonction echelon de heaviside

functional and measure valued solutions of the euler equations for flows of incompressible fluids we consider the notion of a functional solution of the euler equations for incompressible fluid flows we show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation an estimate uniform in lloc1 together with weak consistency with the equation is sufficient to construct a solution we prove that if we have an estimate uniform in lloc2 available for the approximate solution sequence then the structured functional solution just described becomes a measure valued solution in the sense of diperna majda we also show that a functional solution can be obtained from a measure valued solution we give an example showing that a much higher concentration of energy than in the case of measure valued solutions is allowed by the approximation procedure of a functional solution

renormalization and energy conservation for axisymmetric fluid flows we study vanishing viscosity solutions to the axisymmetric euler equations without swirl with relative vorticity in l p with p 1 we show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions moreover we show that the kinetic energy is preserved provided that p 3 2 and the vorticity is nonnegative and has finite second moments

global existence of weak solutions to the three dimensional euler equations with helical symmetry abstract in this paper we mainly investigate the weak solutions of the three dimensional incompressible euler equations with helical symmetry in the whole space when the helical swirl vanishes specifically we establish the global existence of weak solutions when the initial vorticity lies in l 1 l p with p 1 our result extends the previous work 2 where the initial vorticity is compactly supported and belongs to l p with p 4 3 the key ingredient in this paper involves the explicit analysis of biot savart law with helical symmetry in domain r 2 via the theories of singular integral operators and second order elliptic equations

the atiyah singer index theorem for families of dirac operators two heat equation proofs the purpose of this paper is to give two heat equation proofs of the index theorem of atiyah singer for a family of dirac operators

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

on strong convergence to 3d steady vortex sheets in this paper we first establish a strong convergence criterion of approximate solutions for the 3d steady incompressible euler equations for axisymmetric flows under the assumption that the vorticity is of one sign and uniformly bounded in l 1 space we obtain a sufficient and necessary condition for the strong convergence in l 2 r 3 of approximate solutions furthermore for one sign and l 1 bounded vorticity it is shown that if a sequence of approximate solutions concentrates at an isolated point in r z plane then the concentration point can appear neither in the region near the axis including the symmetry axis itself nor in the region far away from the axis finally we present an example of approximates solutions which converge strongly in l 2 r 3 by using hill s spherical vortex

corotating steady vortex flows with n fold symmetry in this paper we study certain vertical flows of an ideal fluid in two dimensions which are steady relative to a uniformly rotating frame in particular we confine our analysis to those flows for which the vorticity relative to the rotating frame possesses n fold rotational symmetry for some given n 2 3 that is we require that the relative vorticity be invariant under rotation through arbitrary multiples of 2zjn as in our earlier related work 19 20 these relative steady solutions of the euler fluid dynamical equations are obtained as extremals for a certain variational problem the solutions then represent steady vortex flows in the sense defined in 19 namely each such flow has a prescribed distribution of relative vorticity and is separated into regions of zero and of positive relative vorticity in 1883 thomson 18 considered the flow due to a configuration of n unit point vortices spaced equally around the circumference of a circle the relative vorticity of such a configuration on a circle of radius a is given by

a computational method of solving free boundary problems in vortex dynamics abstract a general method for computing two dimensional and axisymmetric three dimensional solutions of the steady euler equations is developed the method is based upon variational principles which are formulated entirely in terms of the natural conserved quantities for the unsteady euler equations the vorticity fields considered have a free boundary and may be defined either in a domain with boundary or in free space moreover the common restriction to the case of constant vorticity regions is not imposed the global convergence of the resulting iterative solver is established the numerical implementation of the method is discussed and several computational examples are given

the uniqueness of hill s spherical vortex the mathematical description of steady vortex rings in an ideal fluid occupying the whole space r3 can be approached in various ways the physical basis of the problem its history up to 1973 and several formulations are outlined in 12 pp 14 21 another quite different formulation and the plan for a corresponding existence theory are presented in 6 further existence theorems variational principles and results are to be found in 3 7 11 13 and 20 here we state only definitions and equations that seem relevant to our immediate purpose

desingularization of vortices for the euler equation we study the existence of stationary classical solutions of the incompressible euler equation in the planes that approximate singular stationary solutions of this equation the construction is performed by studying the asymptotics of equation varepsilon 2 delta u varepsilon u varepsilon q frac kappa 2 pi log frac 1 varepsilon p with dirichlet boundary conditions and q a given function we also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices

lagrangian solutions to the 2d euler system with l1 vorticity and infinite energy we consider solutions to the two dimensional incompressible euler system with only integrable vorticity thus with possibly locally infinite energy with such regularity we use the recently developed theory of lagrangian flows associated to vector fields with gradient given by a singular integral in order to define lagrangian solutions for which the vorticity is transported by the flow we prove strong stability of these solutions via strong convergence of the flow under the only assumption of l1 weak convergence of the initial vorticity the existence of lagrangian solutions to the euler system follows for arbitrary l1 vorticity relations with previously known notions of solutions are established

a remark on vorticity maximal function in this note the following inequality is proved for any nonnegative measure h 1 r2 x r2 and 0 r 1 there holds 1 b x r c ln1r 12 h 1 where c is a positive constant using 1 an estimate for the vorticity maximal function similar to the estimate in majda a majda remarks on weak solutions for vortex sheets with a distinguished sign indiana univ math j 42 1993 921 939 is established without assuming the initial vorticity having compact support from this a more simple proof of the delort s celebrated theorem j m delort existence de mappes de fourbillon en dimension deux j amer math soc 4 1991 553 586 is presented

on diperna majda concentration sets for two dimensional incompressible flow abstract the existence of weak solutions to the incompressible euler equations for vortex sheet initial data is an open problem two approaches to this problem are the smoothing of vortex sheets by approximated identifies which provides a sequence of smooth initial conditions converging to vortex sheet initial conditions for which classical solutions exist and viscous smoothing of vortex sheets these ideas are among the principal motivations for the recent series of papers in which the nature of the limiting behavior of sequences of solutions of euler s equations are examined the concept of generalized young measure valued solution of the euler equations is introduced in two dimensions approximate solution sequences always have subsequences which converge in an appropriate sense to non oscillatory generalized young measure valued solutions of euler s equations infinitesimal vortices of zero circulation were described in and named phantom vortices the vortices in our example are somewhat different in that they have two length scales the two length scales are important for one cannot pack phantom vortices on successively finer lattices as above keeping the vorticity absolutely summable without at the same time obtaining strong convergence in l sq reprints

some inequalities for the eigenvalues of the product of positive semidefinite hermitian matrices let 1 a n a denote the eigenvalues of a hermitian n by n matrix a and let 1 i1 ik n our main results are t 1k t gh t 1k it g n it 1 h and t 1k it gh t 1k it g n t 1 h here g and h are n by n positive semidefinite hermitian matrices these results extend marshall and olkin s inequality t 1k t gh t 1k t g n t 1 h we also present analogous results for singular values

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

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

concentrations in the one dimensional vlasov poisson equations ii screening and the necessity for measure valued solutions in the two component case abstract weak and measure valued solutions for the two component vlasov poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two dimensional euler equations with general vorticity of two signs from numerical experiments and mathematical theory it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non negative vorticity here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration for the two component vlasov poisson equations the concepts of measure valued and weak solution are introduced explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure valued solution in some cases rather than the anticipated weak solution i e no concentration cancellation occurs the limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure valued solutions and also for new examples with non unique weak solutions the authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure valued and or non unique weak solutions

on strong convergence to 3 d axisymmetric vortex sheets abstract we consider the 3 d axisymmetric incompressible euler equations without swirls with vortex sheets initial data it is proved that the approximate solutions generated by smoothing the initial data converge strongly in l 2 0 t l loc 2 r 3 provided that they have strong convergence in the region away from the symmetry axis this implies that if there would appear singularity or energy lost in the process of limit for the approximate solutions it then must happen in the region away from the symmetry axis there is no restriction on the signs of initial vorticity here in order to exclude the possible concentrations on the symmetry axis we use the special structure of the equations for axisymmetric flows and careful choice of test functions

extended hamiltonian algorithm for the solution of discrete algebraic lyapunov equations in this paper we use a second order learning algorithm for solving the numerical solution of the discrete algebraic lyapunov equation specifically extended hamiltonian algorithm based on the manifold of positive definite symmetric matrices is provided furthermore this algorithm is compared with the euclidean gradient algorithm the riemannian gradient algorithm and the two traditional iteration methods simulation examples show that the convergence speed of the extended hamiltonian algorithm is the fastest one among these algorithms

smooth approximations and exact solutions of the 3d steady axisymmetric euler equations in this paper we prove that a class of c1 smooth approximate solutions ue pe to the 3d steady axisymmetric euler equations will converge strongly to 0 in l 2 loc r 3 the main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields we also show a liouville type theorem that there are no non trivial c1 smooth exact solutions with finite energy and uniform constant state at far fields

the suboptimal nehari problem for well posed linear systems we solve the suboptimal nehari problem for a transfer function that has a state space realization as a system stable input output and input output stable well posed linear system we obtain an explicit solution in terms of the state space parameters

factored forms for solutions of ax xb c and x axb c in companion matrices abstract the main concern of this paper is linear matrix equations with block companion matrix coefficients it is shown that general matrix equations ax xb c and x axb c can be transformed to equations whose coefficients are block companion matrices c l x xc m diag i 0 0 and x c l xc m diag i 0 0 respectively where l and cm stand for the first and second block companion matrices of some monic r r matrix polynomials l si s 1j 0 jlj and m ti t7minus 1j 0 jmj the solution of the equat with block companion coefficients is reduced to solving vector equations sx where the matrix s is r2l r2l l max s t and enjoys some symmetry properties

on the gluing problem for the eta invariant we solve the gluing problem for the invariant consider a generalized dirac operator d over a compact riemannian manifold m that is partitioned by a compact hypersurface n such that m m un m2 we assume that the riemannian metric of m and d have a product structure near n i e d i d d dn with some dirac operator dn on n using boundary conditions of atiyah patodi singer type parametrized by lagrangian subspaces l of kerd we define selfadjoint extensions d i 1 2 over m we express the invariant of d in terms of the invariants of di an invariant m l l2 of the pair of the lagrangian subspaces l l2 which is related to the maslov index and an integer valued term j in the adiabatic limit i e if a tubular neighborhood of n is long enough the vanishing of j is shown under certain regularity conditions we apply this result in order to prove cutting and pasting formulas for the invariant a wall nonadditivity result for the index of atiyah patodi singer boundary value problems and a splitting formula for the spectral flow 1 the gluing problem and applications 1 1 introduction we solve the gluing problem for the invariant of generalized dirac operators consider a closed compact riemannian manifold carrying a dirac bundle with a generalized dirac operator assume that this manifold is separated into two pieces by a compact hypersurface the gluing problem for the invariant consists in expressing the invariant of the original dirac operator in terms of the invariants of the dirac operators living on the pieces the selfadjoint operators on the two components with boundary depend on a boundary condition given by lagrangian subspaces lx l2 of a certain symplectic vector space the gluing formula also contains an additional real valued term m l l2 which is nicely related to the symplectic geometry and the maslov index