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

geometric invariants for liaison of space curves let k be an algebraically closed field and let s k x x x2 x3 for a curve cc pi the hartshorne rao module m c bneie h p3 xc n is a graded s module of finite length and up to duals and shifts is a complete invariant of liaison cf 12 1 given a graded s module m z m of finite length the action of s p p 6 1 between two consecutive components i e ql s hom m m gives rise to a degeneracy locus which can be thought of as lying in psi p namely let i p les irk4 l r and let v v s where s max r i v 5 p3 or else v 0 these loci are isomo hism invariants and are preserved under duals and shifts for m n c these loci are related to the geometry of c itself the main philosophy that emerges section 2 is that the degeneracy locus generally corresponds to those planes h in p3 which meet c nongenerically either containing a component of c or having cn n impose an unusually small number of conditions on some plane curves on h we conclude section 2 by applying these ideas to derive some necessary conditions for m c to have components in negative degrees the remaining sections give various apphcations of the techniques introduced in section 2 first in section 3 we classify sets of skew lines up to liaison that is if c and c consist of t 2 3 and t skew lines respectively then we ask when c can be linked to c the answer in general is never but the situation changes signi cantly if c lies on a quadric surface cf theorem 3 1 a large part of the answer can be read from the degeneracy locus v

on the cohen macaulay and buchsbaum property for unions of planes in affine space the coordinate ring a of a union of planes in affine space is studied and it is asked when a is a cohen macaulay or buchsbaum ring these properties are related to the position of the planes via the notion of seminormality it is shown that a is cohen macaulay iff a is connected in codimension 2 and seminormal in an appropriate sense consideration of the cohen macaulification then yields a simple criterion for a to be buchsbaum methods with several examples are illustrated

flasque sheaves on posets and cohen macaulay unions of regular varieties the goal of this paper is to find new examples of cohen macaulay and non cohen macaulay rings among the coordinate rings of the unions of regular varieties the cohen macaulayness as well as other properties seminormality buchsbaumness etc was studied extensively for twodimensional varieties z 5 111 however for larger dimensions not much was known besides the reisner theorem lo which gives an equivalent combinatorial condition for the coordinate ring of the unions of coordinate linear subspaces stanley reisner or face rings 123 to be cohen macaulay the method used in this paper is to represent a ring as the section ring of a sheaf on a poset and then to apply theorem 6 4 of 131 more precisely for a finite set q of alline regular varieties we consider the poset x a of all intersections of elements of 0 and the sheaf a on x o of the coordinate rings of these intersections the restriction of regular functions defines a monomorphism p from the coordinate ring r of u u to the section ring s of d c the above mentioned result form 13 can be applied to r only if the two following restrictive conditions hold

parallel factor analysis in sensor array processing this paper links multiple invariance sensor array processing mi sap to parallel factor parafac analysis which is a tool rooted in psychometrics and chemometrics parafac is a common name for low rank decomposition of three and higher way arrays this link facilitates the derivation of powerful identifiability results for mi sap shows that the uniqueness of single and multiple invariance esprit stems from uniqueness of low rank decomposition of three way arrays and allows tapping on the available expertise for fitting the parafac model the results are applicable to both data domain and subspace mi sap formulations the paper also includes a constructive uniqueness proof for a special parafac model

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

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

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

a flexible and efficient algorithmic framework for constrained matrix and tensor factorization we propose a general algorithmic framework for constrained matrix and tensor factorization which is widely used in signal processing and machine learning the new framework is a hybrid between alternating optimization ao and the alternating direction method of multipliers admm each matrix factor is updated in turn using admm hence the name ao admm this combination can naturally accommodate a great variety of constraints on the factor matrices and almost all possible loss measures for the fitting computation caching and warm start strategies are used to ensure that each update is evaluated efficiently while the outer ao framework exploits recent developments in block coordinate descent bcd type methods which help ensure that every limit point is a stationary point as well as faster and more robust convergence in practice three special cases are studied in detail non negative matrix tensor factorization constrained matrix tensor completion and dictionary learning extensive simulations and experiments with real data are used to showcase the effectiveness and broad applicability of the proposed framework

blind parafac receivers for ds cdma systems this paper links the direct sequence code division multiple access ds cdma multiuser separation equalization detection problem to the parallel factor parafac model which is an analysis tool rooted in psychometrics and chemometrics exploiting this link it derives a deterministic blind parafac ds cdma receiver with performance close to non blind minimum mean squared error mmse the proposed parafac receiver capitalizes on code spatial and temporal diversity combining thereby supporting small sample sizes more users than sensors and or less spreading than users interestingly parafac does not require knowledge of spreading codes the specifics of multipath interchip interference doa calibration information finite alphabet constant modulus or statistical independence whiteness to recover the information bearing signals instead parafac relies on a fundamental result regarding the uniqueness of low rank three way array decomposition due to kruskal 1977 1988 and generalized herein to the complex valued case that guarantees identifiability of all relevant signals and propagation parameters these and other issues are also demonstrated in pertinent simulation experiments

parallel algorithms for constrained tensor factorization via alternating direction method of multipliers tensor factorization has proven useful in a wide range of applications from sensor array processing to communications speech and audio signal processing and machine learning with few recent exceptions all tensor factorization algorithms were originally developed for centralized in memory computation on a single machine and the few that break away from this mold do not easily incorporate practically important constraints such as non negativity a new constrained tensor factorization framework is proposed in this paper building upon the alternating direction method of multipliers admom it is shown that this simplifies computations bypassing the need to solve constrained optimization problems in each iteration and it naturally leads to distributed algorithms suitable for parallel implementation this opens the door for many emerging big data enabled applications the methodology is exemplified using non negativity as a baseline constraint but the proposed framework can incorporate many other types of constraints numerical experiments are encouraging indicating that admom based non negative tensor factorization ntf has high potential as an alternative to state of the art approaches

perturbed proximal primal dual algorithm for nonconvex nonsmooth optimization in this paper we propose a perturbed proximal primal dual algorithm pprox pda for an important class of linearly constrained optimization problems whose objective is the sum of smooth possibly nonconvex and convex possibly nonsmooth functions this family of problems can be used to model many statistical and engineering applications such as high dimensional subspace estimation and the distributed machine learning the proposed method is of the uzawa type in which a primal gradient descent step is performed followed by an approximate dual gradient ascent step one distinctive feature of the proposed algorithm is that the primal and dual steps are both perturbed appropriately using past iterates so that a number of asymptotic convergence and rate of convergence results to first order stationary solutions can be obtained finally we conduct extensive numerical experiments to validate the effectiveness of the proposed algorithm

tensor decomposition for signal processing and machine learning tensors or multiway arrays are functions of three or more indices i j k ldots similar to matrices two way arrays which are functions of two indices r c for row column tensors have a rich history stretching over almost a century and touching upon numerous disciplines but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing statistics data mining and machine learning this overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors as such it focuses on fundamentals and motivation using various application examples aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and or developing tensor algorithms and software some background in applied optimization is useful but not strictly required the material covered includes tensor rank and rank decomposition basic tensor factorization models and their relationships and properties including fairly good coverage of identifiability broad coverage of algorithms ranging from alternating optimization to stochastic gradient statistical performance analysis and applications ranging from source separation to collaborative filtering mixture and topic modeling classification and multilinear subspace learning

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

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

on the cuspidal cohomology of arithmetic subgroups of mathrm sl 2n and the first betti number of arithmetic 3 manifolds

seminormality and cohomology of projective varieties let x be a reduced closed subscheme of p proj r y k a field and r k x a r 2 2 let a be the homogeneous co ordinate ring of x in 2 3 51 the various authors have attempted to determine whether a is cohenmacaulay or seminormal or more generally to study the seminormalization or cohen macaulification of a the papers mentioned above work primarily with unions of straight lines in s corollary 5 91 it is proved for example that if x is a connected union of lines which have linearly independent directions at each intersection point then a is seminormal if and only if a is cohen macaulay the aim of this paper is to give a better understanding of the seminormalization of a and the relationship between seminormality and cohenmacaulayness hinted at by the above mentioned corollary 5 9 this we do in section 2 by interpreting the seminormalization of a in terms of sheaf cohomology we are then able to give a reasonable generalization our corollary 3 5 of s corollary 5 91 and also to resolve several computational problems left open in 2 31 and 4 the hilbert function of a graded ring a will be denoted by ha that is ha i dim ai if x proj a the hilbert polynomial of a is denoted hp x in some computations of section 6

exploiting efficient representations in large scale tensor decompositions decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data however storing the tensor and computing its

bundles and finite foliations by a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume we recall that if is a kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space otherwise is geometrically in nite if happens to be a surface group then we say it is quasi fuchsian if the limit set for the group action is a jordan curve c and preserves the components of s n c the starting point for this work is the following theorem which is a combination of theorems due to marden thurston and bonahon

optimization based algorithms for tensor decompositions canonical polyadic decomposition decomposition in rank l_r l_r 1 terms and a new generalization the canonical polyadic and rank l r l r 1 block term decomposition cpd and btd respectively are two closely related tensor decompositions the cpd and recently btd are important tools in psychometrics chemometrics neuroscience and signal processing we present a decomposition that generalizes these two and develop algorithms for its computation among these algorithms are alternating least squares schemes several general unconstrained optimization techniques and matrix free nonlinear least squares methods in the latter we exploit the structure of the jacobian s gramian to reduce computational and memory cost combined with an effective preconditioner numerical experiments confirm that these methods are among the most efficient and robust currently available for computing the cpd rank l r l r 1 btd and their generalized decomposition

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