conditions imposed by tacnodes and cusps the study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress the horace method introduced by a hirschowitz has been successfully exploited to prove many h1 vanishing theorems even in higher dimension other specialization techniques which include degenerations of the plane are due to z ran and c ciliberto and r miranda g m greuel c lossen and e shustin use a local specialization procedure together with the horace method to give the first asymptotically proper general existence criterion for singular curves of low degree in this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by greuel lossen and shustin for curves with tacnodes and cusps

a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

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

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

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

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

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

projective transformations for interior point algorithms and a superlinearly convergent algorithm for the w center problem the purpose of this study is to broaden the scope of projective transformation methods in mathematical programming both in terms of theory and algorithms we start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w center of a polyhedral system which stands for weighted center where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronx we prove basic results regarding contained and containing ellipsoids centered at the w center of the systemx we next shift attention to projective transformations and we exhibit an elementary projective transformation that transforms the polyhedronx to another polyhedronz and that transforms the current interior point to the w center of the transformed polyhedronz we work throughout with a polyhedral system of the most general form namely both inequality and equality costraints

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

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

optimization of log x entropy over linear equality constraints in this paper we develop a special purpose iterative algorithm of the row action type for solving the problem of maximizing the log x entropy functional over linear equality constraints the algorithm employs projections onto hyperplanes which we call log x entropy projections a complete proof of convergence is given

a continuation method for strongly monotone variational inequalities we consider the variational inequality problem denoted by vip x f wheref is a strongly monotone function and the convex setx is described by some inequality and possibly equality constraints this problem is solved by a continuation or interior point method which solves a sequence of certain perturbed variational inequality problems these perturbed problems depend on a parameter 0 it is shown that the perturbed problems have a unique solution for all values of 0 and that any sequence generated by the continuation method converges to the unique solution of vip x f under a well known linear independence constraint qualification licq we also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method 1998 the mathematical programming society inc published by elsevier science b v

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

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

a potential function reduction algorithm for solving a linear program directly from an infeasible warm start this paper develops an algorithm for solving a standard form linear program directly from an infeasible warm start i e directly from a given infeasible solution hat x that satisfies a hat x b but hat x ngeqslant 0 the algorithm is a potential function reduction algorithm but the potential function is somewhat different than other interior point method potential functions and is given by f x b q ln c t x b sum limits j 1 n ln x j h j c t x b where q n sqrt n is a given constant h is a given strictly positive shift vector used to shift the nonnegativity constaints andb is a lower bound on the optimal value of the linear program the duality gapc t x b is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints the algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap and hence also in the infeasibility in o n iterations under more restrictive assumptions regarding the dual feasible region this algorithm is modified by the addition of a dual barrier term and will achieve a constant decrease in the duality gap and in the infeasibility in o sqrt n iterations

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

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

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

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

a method for the parametric center problem with a strictly monotone polynomial time algorithm for linear programming given a system of linear inequalities ax b dt and equalities mx g ht where the right hand sides are parametrically deformed over the scalar t the parametric center problem is to trace the parametric family of approximate solutions x t to the center problems p t where p t is the problem maximize i 1m ln bi dit aix subject to ax b dt and mx g ht we present an algorithm for tracing the parametric family of solutions x t over some given range at the kth iterate of the algorithm the point x tk is an approximate solution to the center problem p tk in an appropriate measure of approximation the value of tk is increased to tk 1 and a newton step is used to generate x tk 1 which is an approximate solution to the center problem p tk 1 the sequence of values of t exhibit at least one of the following geometric rates of change tmax tk 1 tmax tk 1 1 128m tk 1 tmin tk tmin 1 1 128m where tmin tmax is the maximum minimum number t such t

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

geometric descriptions of entangled states by auxiliary varieties the aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry in the context of this paper geometric means each stratum of the hilbert space corresponding to an entangled state is an open subset of an algebraic variety built by classical geometric constructions tangent lines secant lines from the set of separable states in this setting we describe well known classifications of multipartite entanglement such as 2 2 n 1 for n 1 quantum systems and a new description with the 2 3 3 quantum system our results complete the approach of miyake and make stronger connections with recent work of algebraic geometers moreover for the quantum systems detailed in this paper we propose an algorithm based on the classical theory of invariants to decide to which subvariety of the hilbert space a given state belongs

a globally convergent lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds we consider the global and local convergence properties of a class of lagrangian barrier methods for solving nonlinear programming problems in such methods simple bound constraints may be treated separately from more general constraints the objective and general constraint functions are combined in a lagrangian barrier function a sequence of such functions are approximately minimized within the domain defined by the simple bounds global convergence of the sequence of generated iterates to a first order stationary point for the original problem is established furthermore possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero this paper is a companion to previous work of ours on augmented lagrangian methods

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

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

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

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

on singularities of third secant varieties of veronese embeddings abstract in this paper we study singularities of third secant varieties of veronese embedding v d p n which corresponds to the variety of symmetric tensors of border rank at most three in c n 1 d

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

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

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

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

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

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

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

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

asymptotic mesh independence of newton galerkin methods via a refined mysovskii theorem the paper presents a theoretical characterization of the often observed asymptotic mesh independence of newton s method which means that newton s method applied to discretized operator equations behaves essentially the same for all sufficiently fine discretizations unlike previous theoretical approaches the theory presented here does not need any uniform lipschitz assumptions which may be hard to verify in realistic problems or in some cases may not hold at all moreover this approach leads to a significantly simpler presentation the new results are obtained by means of a new refined newton mysovskii theorem in affine invariant formulation this theorem will be of interest in a wider context because it gives both existence and uniqueness of the solution and quadratic convergence for sufficiently good starting points attention is restricted to galerkin approximations even though similar results should hold for finite difference methods but corresponding proofs would certainly be more technical

a combined homotopy interior point method for convex nonlinear programming abstract in this paper we present a new interior point method combined homotopy interior point method chip method for convex nonlinear programming without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set we prove that for any 0 an solution of the problem can be obtained by the chip method to our knowledge strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well known center path following method therefore the chip method essentially reduces the assumptions of the center path following method and can be applied to more general problems

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

following a balanced trajectory from an infeasible point to an optimal linear programming solution with a polynomial time algorithm this paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem a class of trajectories for the problem is defined based on the notion of a balanced solution to the problem given a prespecified positive balancing constant an infeasible solution x is said to be balanced if the optimal value gap is less than or equal to times the infeasibility gap mathematically this can be written as ctx z tx where the linear form tx is the phase i objective function and z is the optimal objective value of the linear program the concept of a balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program each trajectory has the property that all points on or near the trajectory in a suitable metric are balanced the main thrust of the paper is the development of an algorithm that traces a given balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time more specifically the algorithm allows for fixed improvement in the bound on the phase i and the phase ii objectives in on newton steps

on weakly nonlinear inverse problems in this paper the class of weakly nonlinear inverse problems is introduced these problems are characterized by the property that the second derivative of the parameter to observation mapping can be bounded by the square of the first derivative of that mapping using geometric techniques it is shown that weakly nonlinear inverse problems behave similarly to linear inverse problems in particular their tikhonov regularization leads to a family of quadratically well posed problems examples involving the determination of source terms in semilinear reaction diffusion equations are given

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

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

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

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

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

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

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

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

a combined phase i phase ii projective algorithm for linear programming we devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution inclusion of such a constraint allows the algorithm to be applied to a possibly infeasible standard form linear program without the addition of any bigm terms or conversion to a primal dual problem

a potential reduction algorithm with user specified phase i phase ii balance for solving a linear program from an infeasible warm start this paper develops a potential reduction algorithm for solving a linear programming problem directly from a warm start initial point that is neither feasible nor optimal the algorithm is an interior point variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement phase i and objective value improvement phase ii the key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense given a prespecified balancing parameter beta 0 the algorithm maintains the following phase i phase ii beta balancing constraint throughout left c t x z right beta xi t x where c t x is the objective function z is the unknown optimal objective value of the linear program and xi t x measures the infeasibility of the current iterate x this balancing constraint can be used to either emphasize rapid attainment of feasibility set beta large

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

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

theoretical efficiency of a shifted barrier function algorithm for linear programming abstract this paper examines the theoretical efficiency of solving a standard form linear program by solving a sequence of shifted barrier problems of the form minimize x c t x e j 1 n ln x j eh j s t a x b x eh 0 for a given and fixed shift vector h 0 and for a sequence of values of 0 that converges to zero the resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program the advantage of using the shifted barrier approach is that a starting feasible solution is unnecessary and there is no need for a phase i phase ii approach to solving the linear program either directly or through the addition of an artificial variable furthermore the algorithm can be initiated with a warm start i e an initial guess of a primal solution x that need not be feasible the number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible and optimal the number of iterations will also depend on the judicious choice of the shift vector h if an approximate center of the dual feasible region is known in advance then h can be chosen so that the guaranteed fractional decrease in at each iteration is 1 1 6 n which contributes afactor of 6 n to the number of iterations needed to solve the problem

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

geometric aspects of iterated matrix multiplication abstract this paper studies geometric properties of the iterated matrix multiplication polynomial and the hypersurface that it defines we focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial the dual variety and the jacobian loci of the hypersurface that are computed with the aid of representation theory of quivers

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

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