oriented distance function and its evolution equation for initial sets with thin boundary the central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function b omega of a set omega with thin boundary under the influence of a velocity field we relate it to equations and constructions used in the context of level set methods we further introduce a new moving narrow band method which not only can be readily implemented to solve our evolution equation but could also be used for equations of motion by curvatures in the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach e g local semiconvexity in an extended sense of the oriented distance function of the closure of the set for w2 p sobolev domains a new characterization and a compactness theorem are given in terms of the laplacian of the oriented distance function rather than its whole hessian matrix

on maximum typical and generic ranks we show that for several notions of rank including tensor rank waring rank and generalized rank with respect to a projective variety the maximum value of rank is at most twice the generic rank we show that over the real numbers the maximum value of the real rank is at most twice the smallest typical rank which is equal to the complex generic rank

induction for secant varieties of segre varieties this paper studies the dimension of secant varieties to segre varieties the problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry an inductive procedure is built around the ideas of successive specializations of points and projections this reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases as applications of the technique we give a complete classification of defective p secant varieties to segre varieties for p 6 we generalize a theorem of catalisano geramita gimigliano on non defectivity of tensor powers of pn we determine the set of p for which unbalanced segre varieties have defective p secant varieties in addition we completely describe the dimensions of the secant varieties to the deficient segre varieties p1 x p1 x pn x pn and p2 x p3 x p3 in the final section we propose a series of conjectures about defective segre varieties

on approximate diagonalization of third order symmetric tensors by orthogonal transformations abstract in this paper we study the approximate orthogonal diagonalization problem of third order symmetric tensors we define several classes of approximately diagonal tensors including the ones corresponding to the stationary points of this problem we study the relationships between these classes and other well known objects such as tensor z eigenvalue and z eigenvector we also prove results on convergence of the cyclic jacobi or jacobi com2 algorithm

artin level modules we introduce level modules and show that these form a natural class of modules over a polynomial ring we prove that there exist compressed level modules i e level modules with the expected maximal hilbert function given socle type and the number of generators we also show how to use the theory of level modules to compute minimal free resolutions of the coordinate ring of points from the back which reveals new examples where random sets of points fail to satisfy the minimal resolution conjecture

boundary regularity for maxwell s equations with applications to shape optimization abstract the dynamic maxwell equations with a strictly dissipative boundary condition is considered sharp trace regularity for the electric and the magnetic field are established for both weak and differentiable solutions as an application a shape optimization problem for maxwell s equations is considered in order to characterize the shape derivative as a solution to a boundary value problem the aforementioned sharp regularity of the boundary traces is critical

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

construction of quasi cyclic ldpc codes for awgn and binary erasure channels a finite field approach in the late 1950s and early 1960s finite fields were successfully used to construct linear block codes especially cyclic codes with large minimum distances for hard decision algebraic decoding such as bose chaudhuri hocquenghem bch and reed solomon rs codes this paper shows that finite fields can also be successfully used to construct algebraic low density parity check ldpc codes for iterative soft decision decoding methods of construction are presented ldpc codes constructed by these methods are quasi cyclic qc and they perform very well over the additive white gaussian noise awgn binary random and burst erasure channels with iterative decoding in terms of bit error probability block error probability error floor and rate of decoding convergence collectively particularly they have low error floors since the codes are qc they can be encoded using simple shift registers with linear complexity

on minimal decompositions of low rank symmetric tensors abstract in this paper we study the set of decompositions of symmetric tensors of low rank of any dimension and of arbitrary order and the waring loci of points which appear in a decomposition of the tensor we describe a new and complete stratification of the set of symmetric tensors of rank less than 5 by studying hilbert function and regularity of ideals of points contained in the apolar ideal for each stratum we describe a procedure to compute a minimal apolar set of points exploiting the algebraic properties of the waring loci

shape analysis via oriented distance functions the object of this paper is twofold we first present constructions which induce topologies on subsets of a fixed domain or hold all d in rn by using set parameterized functions in an appropriate function space second we study the role of the family of oriented distance functions also known as algebraic or signed distance functions in the analysis of shape optimization problems they play an important role in the introduction of topologies which retain the classical geometric properties associated with sets convexity exterior normals mean curvature ck boundaries etc

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

on equations defining coincident root loci abstract we revisit an old problem in classical invariant theory viz that of giving algebraic conditions for a binary form to have linear factors with assigned multiplicities we construct a complex of sl2 representations such that the desired algebraic conditions are expressible as a specific cohomology group of this complex

efficient sequence detection for intersymbol interference channels with run length constraints this paper addresses the data detection problem of intersymbol interference isi channels with a specific modulation code constraint known as the d k run length limited rll constraint a popular modulation code constraint for data storage channels as well as certain communication channels a computationally efficient sequence detection algorithm is proposed which yields a performance close to that of the maximum likelihood sequence detector when applied to such isi channels the proposed detector is derived as a high signal to noise ratio approximation to the delay constrained optimum detector one which minimizes the symbol error probability given a fixed decision delay constraint the proposed algorithm is essentially a fixed delay tree search fdts algorithm with systematic ambiguity checking and is closely related to existing finite depth tree search algorithms it is observed that long critical error events common in uncoded isi channels are eliminated by the rll constraint based on this observation we show that for some important rll constrained channels the proposed fdts algorithm yields the same minimum euclidean distance between distinguishable channel output sequences as the unconstrained maximum likelihood sequence detector

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

singularities of linear systems and the waring problem the waring problem for homogeneous forms asks for additive decomposition of a form f into powers of linear forms a classical problem is to determine when such a decomposition is unique in this paper we answer this question when the degree of f is greater than the number of variables to do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of p with assigned singularities

koszul young flattenings and symmetric border rank of the determinant abstract we present new lower bounds for the symmetric border rank of the n n determinant for all n further lower bounds are given for the 3 3 permanent

fat points inverse systems and piecewise polynomial functions abstract we explore the connection between ideals of fat points which correspond to subschemes of p nobtained by intersecting mixed powers of ideals of points and piecewise polynomial functions splines on ad dimensional simplicial complex embedded inrd using the inverse system approach introduced by macaulay 11 we give a complete characterization of the free resolutions possible for ideals ink x y generated by powers of homogeneous linear forms we allow the powers to differ we show how ideals generated by powers of homogeneous linear forms are related to the question of determining for some fixed the dimension of the vector space of splines on of degree less than or equal tok we use this relationship and the results above to derive a formula which gives the number of planar mixed splines in sufficiently high degree

real rank with respect to varieties abstract we study the real rank of points with respect to a real variety x this is a generalization of various tensor ranks where x is in a specific family of real varieties like veronese or segre varieties the maximal real rank can be bounded in terms of the codimension of x only we show constructively that there exist varieties x for which this bound is tight the same varieties provide examples where a previous bound of blekherman teitler on the maximal x rank is tight we also give examples of varieties x for which the gap between maximal complex and the maximal real rank is arbitrarily large to facilitate our constructions we prove a conjecture of reznick on the maximal real symmetric rank of symmetric bivariate tensors finally we study the geometry of the set of points of maximal real rank in the case of real plane curves

convergence of a brinkman type penalization for compressible fluid flows abstract we show convergence of a brinkman type penalization of the compressible navier stokes equation in particular the existence of weak solutions for the system in domains with boundaries varying in time is established

inner approximations for polynomial matrix inequalities and robust stability regions following a polynomial approach many robust fixed order controller design problems can be formulated as optimization problems whose set of feasible solutions is modeled by parametrized polynomial matrix inequalities pmis these feasibility sets are typically nonconvex given a parametrized pmi set we provide a hierarchy of linear matrix inequality lmi problems whose optimal solutions generate inner approximations modeled by a single polynomial superlevel set those inner approximations converge in a well defined analytic sense to the nonconvex original feasible set with asymptotically vanishing conservatism one may also impose the hierarchy of inner approximations to be nested or convex in the latter case they do not converge any more to the feasible set but they can be used in a convex optimization framework at the price of some conservatism finally we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations

on the real rank of monomials in this paper we study the real rank of monomials and we give an upper bound for it we show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one

vector bundles give equations of cactus varieties abstract we prove that vector bundles give equations of cactus varieties we derive from it that equations coming from vector bundles are not enough to define secant varieties of veronese varieties in general

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

waring like decompositions of polynomials 1 abstract let f be a homogeneous form of degree d in n variables a waring decomposition of f is a way to express f as a sum of d th powers of linear forms in this paper we consider the decompositions of a form as a sum of expressions each of which is a fixed monomial evaluated at linear forms

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

a link between quantum entanglement secant varieties and sphericity in this paper we shed light on the relations between three concepts studied in representation theory algebraic geometry and quantum information theory first spherical actions of reductive groups on projective spaces second secant varieties of homogeneous projective varieties and the related notions of rank and border rank third quantum entanglement our main result concerns the relation between the problem of the state reconstruction from its reduced one particle density matrices and the minimal number of separable summands in its decomposition more precisely we show that sphericity implies that states of a given rank cannot be approximated by states of a lower rank we call states for which such an approximation is possible exceptional states for three important from a quantum entanglement perspective cases of distinguishable fermionic and bosonic particles we also show that non sphericity implies the existence of exceptional states remarkably the exceptional states belong to non bipartite entanglement classes in particular we show that the w type states and their appropriate modifications are exceptional states stemming from the second secant variety for three cases above we point out that the existence of the exceptional states is a physical obstruction for deciding the local unitary equivalence of states by means of the one particle reduced density matrices finally for a number of systems of distinguishable particles with a known orbit structure we list all exceptional states and discuss their possible importance in entanglement theory

metric normal and distance function in the heisenberg group we introduce a notion which is equivalent in the heisenberg group mathbb h to that of segment normal to a surface then we study some regularity properties of the function measuring the carnot caratheodory distance from an euclidean surface s in mathbb h in terms of the regularity of s

gale duality and free resolutions of ideals of points what is the shape of the free resolution of the ideal of a general set of points in p this question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals there is a lower bound for the resolution computable from the known hilbert function and it seemed natural to conjecture that this lower bound would be achieved this is the minimal resolution conjecture lorenzini 1987 1993 although the conjecture has been shown to hold in many cases three examples discovered computationally by frank olaf schreyer in 1993 strongly suggested that it would fail in general in this paper we shall describe a novel structure inside the free resolution of a set of points which accounts for the failure and provides a counterexample in p for every r 6 r 6 9 we begin by reviewing the conjecture and its status consider a set of c points in the projective r space over a eld k say c pk let s k x0 xr let ic be the homogeneous ideal of c and let sc denote the homogeneous coordinate ring of c let

a level set method for topology optimization of continuum structures with bounded design domains in practice a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary the design domains of practical structures are often limited and significantly affect the final optimal design of the structures structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods firstly to overcome the limitations of current level set methods and the stopping issue of structural boundary movements this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains a set of new normal velocities required by level set movements is given a level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques which include the level set regularization algorithm gradient projection algorithm nonlinear velocity mapping algorithm and return mapping algorithm secondly in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers a mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints the validity and effectiveness of the proposed method are demonstrated with two examples

varieties of completely decomposable forms and their secants abstract this paper is devoted to the study of higher secant varieties of varieties of completely decomposable forms the main goal is to develop methods to inductively verify the non defectivity of such secant varieties as an application of these methods we will establish the existence of large families of non defective secant varieties of small varieties of completely decomposable forms

waring decompositions of monomials abstract a waring decomposition of a polynomial is an expression of the polynomial as a sum of powers of linear forms where the number of summands is minimal possible we prove that any waring decomposition of a monomial is obtained from a complete intersection ideal determine the dimension of the set of waring decompositions and give the conditions under which the waring decomposition is unique up to scaling the variables

typical rank and indscal dimensionality for symmetric three way arrays of order i 2 2 or i 3 3 a peculiar property of three way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank given their order typical tensorial rank has much been studied over algebraically closed fields however very few results have been found pertaining to the typical rank of three way arrays over the real field these results refer to arrays sampled randomly from continuous distributions arrays that consist of symmetric slices do not fit into this sampling scheme the present paper offers typical rank results over the real field for arrays containing symmetric slices of order 2 2a nd 3 3 symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts this paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of indscal for all cases considered this is indeed true thus a full indscal solution may require fewer dimensions for a symmetric array than a full cp decomposition applied to an asymmetric array of the same size the reverse situation does not seem to arise next we examine in which cases cp solutions inevitably are indscal solutions finally the rank reducing impact of double standardizing the slices is discussed

k3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds the main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the fano variety of lines of a cubic 4 fold f in general different from f z f a general k3 surface s of genus 8 determines uniquely a pair of cubic 4folds the apolar cubic f s and the dual pfaffian cubic f s or for simplicity f and f as beauville and donagi have shown the fano variety tf of lines on the cubic f is isomorphic to the hilbert scheme hilb2 s of length two subschemes of s the first main result of this paper is that hilb2 s parametrizes the variety vsp f 10 of presentations of the cubic form f with f z f as a sum of 10 cubes which yields an isomorphism between nff and vsp f 10 furthermore we show that vsp f 10 sets up a 6 10 correspondence between f and ff the main result follows by a deformation argument 1 pfaffian and apolar cubic 4 folds associated to k3 surfaces of genus 8 1 1 let v be a 6 dimensional vector space over c fix a basis e e5 for v then ei a ej for 0 i j aijei aej e a2v a skewsymmetric matrix m g aij with aji aj with pliicker coordinates xij the embedding of the grassmannian g g 2 v in p14 p a2v is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices 0 x01 x02 x03 x04 x05 x0i 0 x12 x13 x14 x15 m x02 x12 0 x23 x24 x25 x03 x13 x23 0 x34 x35 x04 x14 x24 x34 0 x45 x05 x15 x25 x35 x45 0 since the sum of two rank 2 matrices has rank at most 4 and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices the secant variety of g is the cubic hypersurface k defined by the 6 x 6 pfaffian m of the matrix m the dual variety of g in p14 p a2v is a cubic hypersurface k k cf 10 k is the secant variety of g g v 2 the grassmannian of rank 2 quotient spaces of v and of course g c g received by the editors july 5 1999 2000 mathematics subject classification primary 14j70 secondary 14m15 14n99 2000 american mathematical society

power sum decompositions of elementary symmetric polynomials abstract we bound the tensor ranks of elementary symmetric polynomials and we give explicit decompositions into powers of linear forms the bound is attained when the degree is odd

secant varieties of the varieties of reducible hypersurfaces in pn given the space v p d n 1 n 1 1 of forms of degree d in n variables and given an integer l 1 and a partition of d d1 dr it is in general an open problem to obtain the dimensions of the l secant varieties l xn 1 for the subvariety xn 1 v of hypersurfaces whose defining forms have a factorization into forms of degrees d1 dr modifying a method from intersection theory we relate this problem to the study of the weak lefschetz property for a class of graded algebras based on which we give a conjectural formula for the dimension of l xn 1 for any choice of parameters n l and this conjecture gives a unifying framework subsuming all known results moreover we unconditionally prove the formula in many cases considerably extending previous results as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of segre varieties in the special case of a partition with two parts i e r 2 we also relate this problem to a conjecture by froberg on the hilbert function of an ideal generated by general forms

manifold derivative in the laplace beltrami equation abstract this paper is concerned with the derivative of the solution with respect to the manifold more precisely with theshape tangentialsensitivity analysis of the solution to the laplace beltrami boundary value problem with homogeneous dirichlet boundary conditions the domain is an open subset of a smooth compact manifold of r n the flow of a vector fieldv t changes in t and in t the relative boundary tof tin tis smooth enough andy t is the solution in tof the laplace dirichlet problem with zero boundary value on t the shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem that elementy v can be simply defined by the restriction to ofy y vwhereyis the material derivative ofyand yis the tangential gradient ofy the study splits in two parts whether the relative boundary of is empty or not in both cases the shape derivative depends on the deviatoric part of the second fundamental form of the surface on the fieldv 0 through its normal component on and on the tangential fieldv 0 through its normal component on the relative boundary we extend the structure results for theshape tangentialderivative making use ofintrinsic geometryapproach and intensive use of extension operators

an rll constrained ldpc coded recording system using deliberate flipping and flipped bit detection in this paper a low density parity check ldpc coded recording system is investigated for which the run length limited rll constraint is satisfied by deliberate flipping at the write side and by estimating the flipped bits at the read side two approaches are proposed for enhancing the error performance of such a system the first approach is to alleviate the negative effect of incorrect estimation of the flipped bits by adjusting the soft information the second approach is to increase the likelihood of the correct detection of flipped bits by designing a flipped bit detection algorithm that utilizes both the rll constraint and the parity check constraint of the ldpc code these two approaches can be combined to obtain significant improvement in performance over previously proposed methods

on the secant varieties to the tangent developable of a veronese variety abstract let v n d p n for n n d n 1 be the order d veronese embedding of p n x n d t v n d p n the tangent developable of v n d and s s 1 x n d p n the s secant variety of x n d i e the closure in p n of the union of all s 1 linear spaces spanned by s points of x n d s s 1 x n d has expected dimension min n 2 n 1 s 1 catalisano geramita and gimigliano conjectured that s s 1 x n d has always the expected dimension except when d 2 n 2 s or d 3 and n 2 3 4 in this paper we prove their conjecture when n 2 and n 3

free boundary problems and density perimeter we introduce a new concept the density perimeter which substitutes the measure of length of the perimeter of a set in free boundary problems we deduce some links between the hausdorff convergence and the char convergence for a family of domains with bounded density perimeter if this perimeter is considered as a penalty term we give existence results for the variational problem which appears in computer vision and in a bernoulli like free boundary problem we also make some considerations concerning a convergence property of the density perimeter

most secant varieties of tangential varieties to veronese varieties are nondefective we prove a conjecture stated by catalisano geramita and gimigliano in 2002 which claims that the secant varieties of tangential varieties to the dth veronese embedding of the projective n space pn have the expected dimension modulo a few wellknown exceptions it is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of alexander and hirschowitz in 1995 as bernardi catalisano gimigliano and ida demonstrated that the proof of this conjecture may be reduced to the case of cubics i e d 3 the main contribution of this work is the resolution of this base case the proposed proof proceeds by induction on the dimension n of the projective space via a specialization argument this reduces the proof to a large number of initial cases for the induction which were settled using a computer assisted proof the individual base cases were computationally challenging problems indeed the largest base case required us to deal with the tangential variety to the third veronese embedding of p79 in p88559

on the algebraic boundaries among typical ranks for real binary forms abstract we describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight showing that they are dual varieties of suitable coincident root loci

on the waring rank of binary forms abstract the k rank of a binary form f in k x y k c is the smallest number of d th powers of linear forms over k of which f is a k linear combination we provide lower bounds for the c rank waring rank and for the r rank real waring rank of binary forms depending on their factorization we study binary forms with unique c minimal representation

a bound for the waring rank of the determinant via syzygies abstract we show that the waring rank of the 3 3 determinant previously known to be between 14 and 18 is at least 15 we use syzygies of the apolar ideal which have not been used in this way before additionally we show that the symmetric cactus rank of the 3 3 permanent is at least 14

ranks of tensors and a generalization of secant varieties abstract we introduce subspace rank as a tool for studying ranks of tensors and x rank more generally we derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in c 2 c b c b we review the literature from a geometric perspective

secant varieties to osculating varieties of veronese embeddings of p n a well known theorem by alexander hirschowitz states that all the higher secant varieties of v n d the d uple embedding of pp n have the expected dimension with few known exceptions we study here the same problem for t n d the tangential variety to v n d and prove a conjecture which is the analogous of alexander hirschowitz theorem for n leq 9 moreover we prove that it holds for any n d if it holds for d 3 then we generalize to the case of o k n d the k osculating variety to v n d proving for n 2 a conjecture that relates the defectivity of sigma s o k n d to the hilbert function of certain sets of fat points in pp n

on the rank of a symmetric form abstract we give a lower bound for the degree of a finite apolar subscheme of a complex symmetric form f in terms of the degrees of the generators of the annihilator ideal f in the special case when f is a monomial x 0 d 0 x 1 d 1 x n d n with d 0 d 1 d n 1 d n we deduce that the minimal length of an apolar subscheme of f is d 0 1 d n 1 1 and if d 0 d n then this minimal length coincides with the rank of f

sensitivity interpretations of the costate variable for optimal control problems with state constraints in optimal control theory it is well known that the costate arc and the associated maximized hamiltonian function can be interpreted in terms of gradients of the value function evaluated along the optimal state trajectory such relations have been referred to as sensitivity relations in the literature we provide in this paper new sensitivity relations for state constrained optimal control problems for the class of optimal control problems considered there is no guarantee that the costate arc is unique a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid the proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem the introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer the extra information thereby obtained yields the desired set of sensitivity relations

topology of tensor ranks abstract we study path connectedness and homotopy groups of sets of tensors defined by tensor rank border rank multilinear rank as well as their symmetric counterparts for symmetric tensors we show that over c the set of rank r tensors and the set of symmetric rank r symmetric tensors are both path connected if r is not more than the complex generic rank these results also extend to border rank and symmetric border rank over c over r the set of rank r tensors is path connected if it has the expected dimension but the corresponding result for symmetric rank r symmetric d tensors depends on the order d connected when d is odd but not when d is even border rank and symmetric border rank over r have essentially the same path connectedness properties as rank and symmetric rank over r when r is greater than the complex generic rank we are unable to discern any general pattern for example we show that border rank three tensors in r 2 r 2 r 2 fall into four connected components for multilinear rank the manifold of d tensors of multilinear rank r 1 r d in c n 1 c n d is always path connected and the same is true in r n 1 r n d unless n i r i j i r j for some i 1 d beyond path connectedness we determine over both r and c the fundamental and higher homotopy groups of the set of tensors of a fixed small rank and taking advantage of bott periodicity those of the manifold of tensors of a fixed multilinear rank we also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank

the power spectrum of run length limited codes a novel method is developed for computing formulas for power spectra associated with run length limited rll codes explicit use is made of a compact description of the run length process associated with the rll code this association simplifies the general derivation of the power spectrum the calculation of the spectra of several rll codes popular in data storage applications is presented some of the closed form expressions for the spectra of these widely used codes are new

the solution to the waring problem for monomials and the sum of coprime monomials abstract in this paper we compute the waring rank of any polynomial of the form f i 1 r m i where the m i are pairwise coprime monomials i e gcd m i m j 1 for i j in particular we determine the waring rank of any monomial as an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form as a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

an inverse cavity problem for maxwellʼs equations abstract consider the scattering of a time harmonic electromagnetic plane wave by an open cavity embedded in a perfect electrically conducting infinite ground plane where the electromagnetic wave propagation is governed by the maxwell equations the upper half space is filled with a lossless homogeneous medium above the flat ground surface while the interior of the cavity is assumed to be filled with a lossy homogeneous medium accounting for the energy absorption the inverse problem is to determine the cavity structure or the shape of the cavity from the tangential trace of the electric field measured on the aperture of the cavity in this paper results on a global uniqueness and a local stability are established for the inverse problem a crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the cavity

waring loci and the strassen conjecture the waring locus of a form f is the collection of the degree one forms appearing in some minimal sum of powers decomposition of f in this paper we give a complete description of waring loci for several family of forms such as quadrics monomials binary forms and plane cubics we also introduce a waring loci version of strassen s conjecture which implies the original conjecture and we prove it in many cases

the asymptotic leading term for maximum rank of ternary forms of a given degree abstract let r max n d be the maximum waring rank for the set of all homogeneous polynomials of degree d 0 in n indeterminates with coefficients in an algebraically closed field of characteristic zero to our knowledge when n d 3 the value of r max n d is known only for n d 3 3 3 4 3 5 4 3 we prove that r max 3 d d 2 4 o d as a consequence of the upper bound r max 3 d d 2 6 d 1 4

a comparison of different notions of ranks of symmetric tensors abstract we introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers namely rank border rank catalecticant rank generalized rank scheme length border scheme length extension rank and smoothable rank we analyze the stratification induced by these ranks the mutual relations between these stratifications allow us to describe the hierarchy among all the ranks we show that strict inequalities are possible between rank border rank extension rank and catalecticant rank moreover we show that scheme length generalized rank and extension rank coincide