characteristic submanifold theory and toroidal dehn filling the exceptional dehn filling conjecture of the second author concerning the re lationship between exceptional slopes on the boundary of a hyperbolic knot manifold m has been verified in all cases other than small seifert filling slopes in this paper we verify it whenis a small seifert filling slope andis a toroidal filling slope in the generic case where m admits no punctured torus fibre or semi fibre and there is no incompressible torus in m which intersects m in one or two components under these hypotheses we show that 5 our proof is based on an analysis of the relationship between the topology of m the combinatorics of the intersection graph of an immersed disk or torus in m and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in m

the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

optimized schwarz method with complete radiation transmission conditions for the helmholtz equation in waveguides in this paper we present a nonoverlapping optimized schwarz algorithm with a high order transmission condition for the helmholtz equation posed in waveguides we introduce the new high order transmission conditions based on the complete radiation boundary conditions crbcs that have been developed for high order absorbing boundary conditions we obtain the convergence rate of the algorithm in terms of reflection coefficients of crbcs which decrease exponentially with the order of crbcs it will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the schwarz algorithm also this algorithm can be employed efficiently for a preconditioner in gmres implementations based on the substructured form as in m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 finally numerical examples confirming the theory will be presented

poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

overlapping schwarz preconditioners for indefinite time harmonic maxwell equations time harmonic maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite a galerkin method using nedelec spaces can be employed to get approximate solutions numerically the problem of preconditioning the indefinite matrix arising from this method is discussed here specifically two overlapping schwarz methods will be shown to yield uniform preconditioners

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments

lens spaces and toroidal dehn fillings we show that if m is a hyperbolic 3 manifold with m a torus such that m r 1 is a lens space and m r 2 is toroidal then r 1 r 2 4

separation of multiple time delays using new spectral estimation schemes the problem of estimating multiple time delays in presence of colored noise is considered this problem is first converted to a high resolution frequency estimation problem then the sample lagged covariance matrices of the resulting signal are computed and studied in terms of their eigenstructure these matrices are shown to be as effective in extracting bases for the signal and noise subspaces as the standard autocorrelation matrix which is normally used in music and the pencil based methods frequency estimators are then derived using these subspaces the effectiveness of the method is demonstrated on two examples a standard frequency estimation problem in presence of colored noise and a real world problem that involves separation of multiple specular components from the acoustic backscattered from an underwater target

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

an analysis of the practical dpg method we give a complete error analysis of the discontinuous petrov galerkin dpg method accounting for all the approximations made in its practical implementation specifi cally we consider the dpg method that uses a trial space consisting of polynomials of degree p on each mesh element earlier works showed that there is a trial to test operator t which when applied to the trial space defines a test space that guarantees stability in dpg formulations this operator t is local it can be applied element by element however an infinite dimensional problem on each mesh element needed to be solved to apply t in prac tical computations t is approximated using polynomials of some degree r p on each mesh element we show that this approximation maintains optimal convergence rates provided that r 2 p n where n is the space dimension two or more for the laplace equation we also prove a similar result for the dpg method for linear elasticity remarks on the conditioning of the stiffness matrix in dpg methods are also included

annuli in generalized heegaard splittings and degeneration of tunnel number abstract we analyze how a family of essential annuli in a compact 3 manifold will induce from a strongly irreducible generalized heegaard splitting of the ambient manifold generalized heegaard splittings of the complementary components there are specific applications to the subadditivity of tunnel number of knots improving somewhat bounds of kowng kw for example in the absence of 2 bridge summands the tunnel number of the sum of n knots is no less than frac 2 5 the sum of the tunnel numbers

dehn twists and pseudo anosov diffeomorphisms we show that it is possible to obtain many pseudo anosov diffeomorphisms from dehn twists in particular we generalize a theorem of long and morton to obtain that iff is a pseudo anosov diffeomorphism of an oriented surface andt is the dehn twist around the simple closed curve then the isotopy class oft n f contains a pseudo anosov diffeomorphism except for at most 7 consecutive values ofn

the block processing ftf adaptive algorithm a new block type adaptive filtering algorithm is presented this new block adaptive filter differs from the frequency domain block adaptive filters of ferrara 1980 and of clark mitra and parker 1980 in that the new method applies a deterministic time domain least squares criteria within each of the data blocks information is carried from block to block via a weighted initial condition this new block fast transversal filters bftf algorithm is a numerically stable algorithm and can also be used to perform efficient least squares system identification on any one data block in which case it shows a moderate computational advantage over the previous most efficient single data block algorithms of morf et al 1977 of marple 1981 and of kalouptsidis manolakis and carayannis 1984 1985 mechanisms for tracking and varying block length from block to block are also presented and evaluated finally we indicate how the new algorithm could be pipelined for maximum throughput with delay proportional to the number of parameters after computation of the sample correlation lags

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

monopoles and lens space surgeries monopole floer homology is used to prove that real projective three space cannot be obtained from dehn surgery on a nontrivial knot in the three sphere to obtain this result we use a surgery long exact sequence for monopole floer homology together with a nonvanishing theorem which shows that monopole floer homology detects the unknot in addition we apply these techniques to give information about knots which admit lens space surgeries and to exhibit families of three manifolds which do not admit taut foliations

construction of dpg fortin operators for second order problems abstract the use of ideal optimal test functions in a petrov galerkin scheme guarantees the discrete stability of the variational problem however in practice the computation of the ideal optimal test functions is computationally intractable in this paper we study the effect of using approximate practical test functions on the stability of the dpg discontinuous petrov galerkin method and the change in stability between the ideal and practical cases is analyzed by constructing a fortin operator we highlight the construction of an optimal dpg fortin operator for h 1 and h div spaces the continuity constant of the fortin operator is a measure of the loss of stability between the ideal and practical dpg methods we take a two pronged approach first we develop a numerical procedure to estimate an upper bound on the continuity constant of the fortin operator in terms of the inf sup constant h of an auxiliary problem second we construct a sequence of approximate fortin operators and exactly compute the continuity constants of the approximate operators which provide a lower bound on the exact fortin continuity constant our results shed light not only on the change in stability by using practical test functions but also indicate how stability varies with the approximation order p and the enrichment order p the latter has important ramifications when one wishes to pursue local h p adaptivity

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

classifying torsion free subgroups of the picard group torsion free subgroups of finite index in the picard group are the fundamental groups of hyperbolic 3 manifolds the picard group is a polygonal product of finite groups recent work by karrass pietrowski and solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion free subgroups of any finite index this computation is carried out here for index 12 and 24 where there are respectively 2 and 17 nonisomorphic subgroups the manifolds are identified by using surgery

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

boundary structure of hyperbolic 3 manifolds admitting toroidal fillings at large distance we show that if a hyperbolic 3 manifold m has two toroidal dehn fillings with distance at least 3 then m consists of at most three tori as a result we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3 manifolds with boundary a union of at least 4 tori

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

dehn fillings of knot manifolds containing essential once punctured tori in this paper we study exceptional dehn fillings on hyperbolic knot manifolds which contain an essential once punctured torus let m be such a knot manifold and let be the boundary slope of such an essential once punctured torus we prove that if dehn filling m with slope produces a seifert fibred manifold then 5 furthermore we classify the triples m when 4 more precisely when 5 then m is the unique manifold w h 3 2 obtained by dehn filling one boundary component of the whitehead link exterior with slope 3 2 and is the pair of slopes 5 0 further 4 if and only if m is the triple w h 2n 1 n 4 0 for some integer n with n 1 combining this with known results we classify all hyperbolic knot manifolds m and pairs of slopes on m where is the boundary slope of an essential once punctured torus in m and is an exceptional filling slope of distance 4 or more from refined results in the special case of hyperbolic genus one knot exteriors in s3 are also given

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

tracking improvements in fast rls algorithms using a variable forgetting factor the concept of a variable forgetting factor vff is incorporated into fast recursive least squares frls algorithms compromises in the data matrix that are needed to do this are examined both prewindowed and growing memory covariance algorithms are presented in transversal and lattice structures forgetting factor adaptation schemes which improve tracking performance over conventional frls algorithms are suggested finally the bias introduced by the use of the vff is analyzed

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

a numerical study of the pollution error and dpg adaptivity for long waveguide simulations abstract high frequency wave propagation has many important applications in acoustics elastodynamics and electromagnetics unfortunately the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber it is critical to control these errors to obtain a stable and accurate method we study the effect of pollution for very long waveguide problems in the context of robust discontinuous petrov galerkin dpg finite element discretizations our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the dpg method while phase errors appear less significant we report results for 3d vectorial time harmonic maxwell problems in waveguides with more than 8000 wavelengths our results corroborate previous analysis for the galerkin discretization of the helmholtz operator by melenk and sauter 2011 additionally we discuss adaptive refinement strategies for multi mode fiber waveguides where the propagating transverse modes must be resolved sufficiently our study shows the applicability of the dpg error indicator to this class of problems finally we illustrate the importance of load balancing in these simulations for distributed memory parallel computing

characteristic subsurfaces and dehn filling let m be a simple knot manifold using the characteristic submanifold theory and the combinatorics of graphs in surfaces we develop a method for bounding the distance between the boundary slope of an essential surface in m which is not a fiber or a semi fiber and the boundary slope of a certain type of singular surface applications include bounds on the distances between exceptional dehn surgery slopes it is shown that if the fundamental group of m has no non abelian free subgroup and if m is a reducible manifold which is not homeomorphic to s 1 s 2 or p 3 p 3 then 5 under the same condition on m it is shown that if m is seifert fibered then 6 moreover in the latter situation character variety techniques are used to characterize the topological types of m and m in case the bound of 6 is attained

the maximal number of exceptional dehn surgeries thurston s hyperbolic dehn surgery theorem is one of the most important results in 3 manifold theory and it has stimulated an enormous amount of research if m is a compact orientable hyperbolic 3 manifold with boundary a single torus then the theorem asserts that for all but finitely many slopes s on m the manifold m s obtained by dehn filling along s also admits a hyperbolic structure the slopes s where m s is not hyperbolic are known as exceptional a major open question has been what is the maximal number of exceptional slopes on such a manifold m when m is the exterior of the figure eight knot the number of exceptional slopes is 10 and this was conjectured by gordon in 19 to be an upper bound that holds for all m in this paper we prove this conjecture

fast transversal filters with data sequence weighting a fast o n algorithm is presented for the adaptive recursive least squares design of transversal filters data sequence weighting is introduced to allow for arbitrarily time varying weighting strategies facilitating the tracking of arbitrary nonstationary phenomena a novel weighting adaptation mechanism is presented that has various desirable features and optimal performance under certain conditions the algorithm is derived for the prewindowed given data case and the underlying principles of the derivation are clearly exposed exact and soft constraint initialization are discussed and an improved restart procedure is proposed

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

exceptional and cosmetic surgeries on knots we show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and heegaard surfaces in the manifolds that result from non trivial dehn surgeries on the knot in particular knots with high bridge distance do not admit non trivial non hyperbolic surgeries or non trivial cosmetic surgeries we further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of seifert genus or indeed by the genus of almost any essential surface or heegaard surface in the surgered manifold

wavenumber explicit analysis of a dpg method for the multidimensional helmholtz equation we study the properties of a novel discontinuous petrov galerkin dpg method for acoustic wave propagation the method yields hermitian positive definite matrices and has good pre asymptotic stability properties numerically we find that the method exhibits negligible phase errors otherwise known as pollution errors even in the lowest order case theoretically we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber the mesh size h and the polynomial degree p but the current state of the theory does not fully explain the remarkably good numerical phase errors theoretically comparisons are made with several other recent works that gave wave number explicit estimates numerically comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures the new dpg method is designed following the previously established principles of optimal test functions in addition to the nonstandard test functions in this work we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates

a modified block ftf adaptive algorithm with applications to underwater target detection in this paper the problem of weighted block recursive least squares rls adaptive filtering is formulated in the context of a block fast transversal filter ftf algorithm this modified block ftf algorithm is derived by modifying the constrained block ls cost function to guarantee global optimality this new soft constrained algorithm provides an efficient way of transferring weight information between blocks of data the tracking ability of the algorithm can be controlled by varying the block length and or a soft constrained parameter this algorithm is computationally more efficient compared with other ls based schemes the effectiveness of this algorithm is tested on a real life problem dealing with underwater target identification from acoustic backscatter the process involves the identification of the presence of resonance in the acoustic backscatter from a target of unknown shape submerged in water

heegaard floer homology and fibred 3 manifolds given a closed 3 manifold y we show that the heegaard floer homology determines whether y fibres over the circle with a fibre of negative euler characteristic this is an analogue of an earlier result about knots proved by ghiggini and the author

a multigrid method for the helmholtz equation with optimized coarse grid corrections we study the convergence of multigrid schemes for the helmholtz equation focusing in particular on the choice of the coarse scale operators let g rm c denote the number of points per wavelength at the coarse level if the coarse scale solutions are to approximate the true solutions then the oscillatory nature of the solutions implies the requirement g rm c 2 however in examples the requirement is more like g rm c gtrsim 10 in a trade off involving also the amount of damping present and the number of multigrid iterations we conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators standard 5 point finite differences in two dimensions are our first example a new coarse scale 9 point operator is constructed to match the fine scale phase speeds we then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator the required g rm c is reduced from about 10 to about 3 5 with less dam

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

distance between toroidal surgeries on hyperbolic knots in the 3 sphere for a hyperbolic knot in the 3 sphere at most finitely many dehn surgeries yield non hyperbolic 3 manifolds as a typical case of such an ex ceptional surgery a toroidal surgery is one that yields a closed 3 manifold containing an incompressible torus the slope corresponding to a toroidal surgery called a toroidal slope is known to be integral or half integral we show that the distance between two integral toroidal slopes for a hyperbolic knot except the figure eight knot is at most four

sutured floer homology fibrations and taut depth one foliations for an oriented irreducible 3 manifold m with non empty toroidal boundary we describe how sutured floer homology sfh can be used to determine all fibred classes in h 1 m furthermore we show that the sfh of a balanced sutured manifold m gamma detects which classes in h 1 m admit a taut depth one foliation such that the only compact leaves are the components of r gamma the latter had been proved earlier by the first author under the extra assumption that h 2 m 0 the main technical result is that we can obtain an extremal spin c structure s i e one that is in a corner of the support of sfh via a nice and taut sutured manifold decomposition even when h 2 m not equal 0 assuming the corresponding group sfh m gamma s has non trivial euler characteristic

holomorphic disks and knot invariants abstract we define a floer homology invariant for knots in an oriented three manifold closely related to the heegaard floer homologies for three manifolds defined in an earlier paper we set up basic properties of these invariants including an euler characteristic calculation and a description of the behavior under connected sums then we establish a relationship with hf for surgeries along the knot applications include calculation of hf of three manifolds obtained by surgeries on some special knots in s 3 and also calculation of hf for certain simple three manifolds which fiber over the circle

essential laminations in 3 manifolds it is a well established principle in manifold topology that by studying codimension 1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself this is particularly true for 3 dimensional manifolds using incompressible surfaces a great many advances have been made during the last 30 years the most spectacular results were obtained by haken ha waldhausen w and thurston t1 more recently work t2 g14 using taut foliations in 3 manifolds has proven fruitful unfortunately in some sense most closed 3 manifolds do not contain incompressible surfaces and it is currently not known exactly which 3 manifolds have taut foliations the purpose of this paper is to study another codimension 1 object the essential lamination which is a generalization of the incompressible surface as well as the taut foliation we will show that the existence of an essential lamination in a 3 manifold m implies that m has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation we will also show that the universal cover of a manifold containing an essential lamination is r3 precise definitions of terms used in the following theorem will be given in section 1 here we give only a rough idea we begin with two alternative approximate definitions of essential lamination i an essential lamination in a 3 manifold m is a lamination satisfying four conditions the inclusion of leaves of the lamination into m should induce an injection on stj the complement of the lamination should be irreducible no leaf should be a sphere and the lamination should be end incompressible to say that a lamination is end incompressible means roughly that a folded leaf can be straightened using an isotopy there are no infinite folds ii alternatively

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

obstructing pseudoconvex embeddings and contractible stein fillings for brieskorn spheres abstract a question in low dimensional symplectic topology asks whether every compact contractible 4 manifold admits the structure of a stein domain a related conjecture due to gompf asserts that no nontrivial brieskorn homology sphere admits a pseudoconvex embedding in c 2 with either orientation we give the first known example of a contractible boundary irreducible 4 manifold that admits no stein structure with either orientation though its boundary has stein fillings with both orientations we also verify gompf s conjecture with one orientation for a family of brieskorn spheres of which some are known to admit a smooth embedding in c 2 a byproduct of our example is the construction of a cork that does not admit a stein structure with either orientation

dehn fillings of large hyperbolic 3 manifolds let m be a compact connected orientable hyperbolic 3 manifold whose boundary is a torus and which contains an essential closed surface s i t is conjectured that 5 is an upper bound for the distance between two slopes on m whose associated fillings are not hyperbolic manifolds in this paper we verify the conjecture when the first betti number of m is at least 2 by showing that given a pseudo anosov mapping class f of a surface and an essential simple closed curve in the surface then 5 is an upper bound for the diameter of the set of integers n for which the composition of f with the n th power of a dehn twist along is not pseudo anosov this sharpens an inequality of albert fathi for large manifolds m of first betti number 1 we obtain partial results set

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

fast subsampled updating stabilized fast transversal filter fsu sftf rls algorithm for adaptive filtering we present a new doubly fast algorithm for recursive least squares rls adaptive filtering that uses displacement structure and subsampled updating the fast subsampled updating stabilized fast transversal filter fsu sftf algorithm is mathematically equivalent to the classical fast transversal filter ftf algorithm the ftf algorithm exploits the shift invariance that is present in the rls adaptation of an fir filter the ftf algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the levinson 1947 algorithm in the subsampled updating approach we accumulate the rotation matrices over some time interval before applying them to the filters it turns out that the successive rotation matrices themselves can be obtained from a schur type algorithm that once properly initialized does not require inner products the various convolutions that appear in the algorithm are done using the fast fourier transform fft the resulting algorithm is doubly fast since it exploits ftf and ffts the roundoff error propagation in the fsu sftf algorithm is identical to that in the sftf algorithm a numerically stabilized version of the classical ftf algorithm the roundoff error generation on the other hand seems somewhat smaller for relatively long filters the computational complexity of the new algorithm is smaller than that of the well known lms algorithm rendering it especially suitable for applications such as acoustic echo cancellation

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

norms on the cohomology of hyperbolic 3 manifolds we study the relationship between two norms on the first cohomology of a hyperbolic 3 manifold the purely topological thurston norm and the more geometric harmonic norm refining recent results of bergeron engun and venkatesh as well as older work of kronheimer and mrowka we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3 manifold itself moreover we give families of examples showing that some but not all qualitative aspects of our estimates are sharp finally we exhibit closed hyperbolic 3 manifolds where the thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius

unknotting tunnels and seifert surfaces let k be a knot with an unknotting tunnel gamma and suppose that k is not a 2 bridge knot there is an invariant rho p q in mathbb q 2 mathbb z with p odd defined for the pair k gamma the invariant rho has interesting geometric properties it is often straightforward to calculate for example for k a torus knot and gamma an annulus spanning arc rho k gamma 1 although rho is defined abstractly it is naturally revealed when k cup gamma is put in thin position if rho neq 1 then there is a minimal genus seifert surface f for k such that the tunnel gamma can be slid and isotoped to lie on f one consequence is that if rho k gamma neq 1 then mathrm genus k 1 this confirms a conjecture of goda and teragaito for pairs k gamma with rho k gamma neq 1

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

dispersive and dissipative errors in the dpg method with scaled norms for helmholtz equation this paper studies the discontinuous petrov galerkin dpg method where the test space is normed by a modified graph norm the modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter the main finding is that as the parameter approaches zero better results are obtained under some circumstances when the method is applied to the helmholtz equation the main tool used is a dispersion analysis on the multiple interacting stencils that form the dpg method the analysis shows that the discrete wavenumbers of the method are complex explaining the numerically observed artificial dissipation in the computed wave approximations since the dpg method is a nonstandard least squares galerkin method its performance is compared with a standard least squares method having a similar stencil

reducing and annular dehn fillings if two dehn fillings on a simple manifold create a reducible manifold and an annular manifold respectively then the distance between those filling slopes is known to be at most two moreover eudave munoz and wu gave infinitely many examples of manifolds admitting reducing and annular dehn fillings at distance two in this paper we complement their examples to establish a complete list of simple manifolds admitting such a pair of dehn fillings

sutured manifold hierarchies essential laminations and dehn surgery x0 introduction a compact orientable surface f with nonnegative euler characteristic is either a sphere a disk a torus or an annulus if a 3 manifold m contains such an essential surface then it is said to be reducible reducible toroidal or annular respectively any such surface can be used to decompose the manifold further into simpler manifolds we say that m is a simple manifold if it has no such surfaces a simple manifold is expected to have a nice geometric structure if m has nonempty boundary then the geometrization theorem of thurston for haken manifolds says that m with boundary tori removed admits a nite volume hyperbolic structure with totally geodesic boundary when m has no boundaries thurston s geometrization conjecture asserts that m is either hyperbolic or is a seifert b er space with orbifold a sphere with at most 3 cone points suppose t is a torus boundary component of m we use m to denote the manifold obtained by dehn lling on t so that the slope on t bounds a disk in the dehn lling solid torus when m e k is the exterior of a knot k in s 3 denote m by k and call it the manifold obtained by surgery on the knot k it is well known that if m is simple then there are only nitely many dehn llings on t which produce non simple manifolds if m 1 and m 2 are non simple manifolds then the geometric intersection number between 1 and 2 denoted by 1 2 is proved to be at most 8 by gordon gor1 there are 10 dieren t cases according to the types of nonnegative euler characteristic surfaces in m i in many cases the upper bounds for 1 2 have been established see table 0 1

the method of subspace corrections abstract this paper gives an overview for the method of subspace corrections the method is first motivated by a discussion on the local behavior of high frequency components in a solution to an elliptic problem a simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more detail brief discussions are also given to some non linear examples including eigenvalue problems obstacle problems and liquid crystal modelings the relationship between the method of subspace correction and the method of alternating projects is observed and discussed

the topology of symplectic circle bundles we consider circle bundles over compact three manifolds with symplectic total spaces we show that the base of such a space must be irreducible or the product of the two sphere with the circle we then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases namely if the base is seifert bered has vanishing thurston norm or if the total space admits a lefschetz bration

an adaptive dpg method for high frequency time harmonic wave propagation problems abstract the discontinuous petrov galerkin dpg method for high frequency wave propagation problems is discussed the dpg method with its attractive uniform mesh and wavenumber independent pre asymptotic stability property allows for a fully automatic adaptive h p algorithm that can be initiated from very coarse meshes moreover dpg always delivers a hermitian positive definite system suggesting the use of the conjugate gradient algorithm for its solution we present a new iterative solution scheme which capitalizes on these attractive properties of dpg this novel solver is integrated within the adaptive procedure by constructing a two grid like preconditioner for the conjugate gradient method that exploits information from previous meshes the construction of our preconditioner is discussed and its efficacy is illustrated with an example of a 2d acoustics problem our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber