condition numbers for the tensor rank decomposition abstract the tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low rank tensor these parameters are analyzed and interpreted in many applications as tensors are often perturbed by measurement errors in practice one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis the magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor this paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of terracini s matrix which represents the tangent space to the set of tensors of fixed rank a practical algorithm for computing this condition number is presented the latter s elementary properties such as scaling and orthogonal invariance are established rank 1 tensors are always well conditioned the class of weak 3 orthogonal tensors which includes orthogonally decomposable tensors contains both well conditioned and ill conditioned problems numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed

the ranks of m times n times mn 2 tensors it is shown that there is essentially only one m times n times mn 2 tensor of rank mn 1 it is also proved that except for this tensor all m times n times p tensors with p leqq mn 2 have rank at most mn 2 the main tool is kronecker s theory of matrix pencils which has already been applied directly by ja ja siam j comput 8 1979 pp 443 462 to study the ranks of m times n times 2 tensors we show that each nondegenerate m times n times mn 2 tensor is determined by a related m times n times 2 tensor and apply the kronecker theory to this related tensor

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

typical tensorial rank abstract upper bounds on the typical rank r n m l of tensors maximal border rank rank of almost all tensors of a given shape n m l are presented these improve previous results by a tkinson and lloyd for cubic shape tensors the typical rank is determined exactly r n n n n3 3n 2 n 3

secant varieties of toric varieties arising from simplicial complexes abstract motivated by the study of the secant variety of the segre veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes we prove that every such secant is toric which gives a way to use combinatorial tools to study singularities we focus on the segre veronese variety for which we completely classify their secants that give gorenstein or q gorenstein varieties we conclude providing the explicit description of the singular locus

spherical blow ups of grassmannians and mori dream spaces abstract in this paper we classify weak fano varieties that can be obtained by blowing up general points in prime fano varieties we also classify spherical blow ups of grassmannians in general points and we compute their effective cone these blow ups are in particular mori dream spaces furthermore we compute the stable base locus decomposition of the blow up of a grassmannian in one point and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the grassmannian at the blown up point and by the rational normal curves in the grassmannian passing through the blown up point

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

identification of non linear systems using multi scale ridges and skeletons of the wavelet transform abstract a new procedure of non linear system identification is presented the procedure employs the slowly varying time dependent amplitude and phase functions of the impulse response of the system these instantaneous characteristics are obtained from the ridges and skeletons of the wavelet transform the ridge extraction procedure uses the modulus of the transform and involves a combinatorial optimization algorithm based on simulated annealing the method is illustrated using two simple simulated examples it is shown that the procedure can be used for multi degree of freedom systems due to the frequency localization property of the continuous grossman morlet wavelets

damping identification using a continuous wavelet transform application to real data a continuous wavelet transform cwt based on the gabor wavelet function is used to identify the damping of a multi degree of freedom system the common procedures are already known especially the identification with a morlet cwt this study gives special attention to the following a description of the instantaneous noise the edge effect of the cwt the frequency shift of the cwt the bandwidth of the wavelet function and the selection of the parameter of the gabor wavelet function of the cwt the procedures are demonstrated using several numerical examples and on signals acquired from the lateral vibration of a uniform beam the study demonstrates the advantages of using the amplitude and phase methods both of which provide information about the instantaneous noise the procedures presented are appropriate for automating the identification process

local heaviside weighted mlpg meshless method for two dimensional solids using compactly supported radial basis functions abstract compactly supported radial basis functions csrbf are employed for constructing trial functions in the local heaviside weighted meshless local petrov galerkin method for stress analysis of two dimensional solids where the heaviside step function is used as the weighting function over a local sub domain the present method is a truly meshless method based only on a number of randomly located nodes no domain integration is needed no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions effects of the sizes of local sub domain and interpolation domain on the performance of the present method are investigated in this paper the size of the support of the basis function has been treated as a shape parameter and then the behaviour of this shape parameter has been systematically studied for six different csrbfs example problems in elastostatics are presented and compared with closed form solutions results show that the proposed method is highly accurate and possesses no numerical difficulties

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

the possible dimensions of the higher secant varieties let x c p be an irreducible closed subvariety of projective space the kth higher secant variety of x denoted x is the closure of the union of all linear spaces spanned by k points of x let a ix dim x dim x l it is known that the sequence ty x is nonincreasing and that there may be at most one 1 in such a sequence we prove that these two conditions are also sufficient conditions for an arbitrary sequence a of nonnegative integers to be the fl x s for an irreducible variety x to this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals this allows us to compute their dimensions showing that there is in fact always a rational normal scroll x such that ak x a for any sequence a satisfying the two conditions above adlandsvik al a2 and flenner and vogel f v much of this work has centered on the behavior of the differences ak x dimx dimx 1 of the dimensions of consecutive higher secant varieties it is well known that this sequence is nonincreasing and that there is at most one 1 in the sequence palatini claimed moreover that once ak x 1 were counterexamples a2 we answer here the question of what sequences actually appear as the ai x s of some variety x some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls explicit equations for the scrolls themselves have been known for some time cf acgh we give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in w this presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties x with a prescribed nonincreasing ak x with at most one 1

on the rationality of the moduli space of luroth quartics we prove that the moduli space mathfrak m l of luroth quartics in mathbb p 2 i e the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of mathrm pgl 3 mathbb c is rational as is the related moduli space of bateman seven tuples of points in mathbb p 2

spin curves and scorza quartics in the paper takagi and zucconi in on blow ups of the quintic del pezzo 3 fold and varieties of power sums of quartic hypersurfaces pp 1 40 preprint submitted 2008 we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces in this paper we show that these quartics coincide with the scorza quartics of general pairs of trigonal curves and ineffective theta characteristics among other applications we give an affirmative answer to the conjecture of dolgachev and kanev on the existence of the scorza quartics for any general pairs of curves and ineffective theta characteristics we also give descriptions of the moduli spaces of trigonal even spin curves

positivity of divisors on blown up projective spaces ii we construct log resolutions of pairs on the blow up of the projective space in an arbitrary number of general points and we discuss the semi ampleness of the strict transforms as an application we give an explicit proof that the abundance conjecture holds for an infinite family of such pairs for n 2 points these strict transforms are f nef divisors on the moduli space m0 n 3 in a kapranov s model we show that all of them are nef

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 fast high order finite difference algorithm for pricing american options we describe an improvement of han and wu s algorithm h han x wu a fast numerical method for the black scholes equation of american options siam j numer anal 41 6 2003 2081 2095 for american options a high order optimal compact scheme is used to discretise the transformed black scholes pde under a singularity separating framework a more accurate free boundary location based on the smooth pasting condition and the use of a non uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained comparisons with some recently proposed methods for the american options problem are carried out to show the advantage of our numerical method

on waringʼs problem for systems of skew symmetric forms abstract this paper is devoted to a problem of finding the smallest positive integer s m n k such that m 1 generic skew symmetric k 1 forms in n 1 variables as linear combinations of the same s m n k decomposable skew symmetric k 1 forms this problem is analogous to a well known problem called waring s problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry in this paper we will go through some basics of algebraic geometry describe how objects in algebraic geometry can be associated to systems of skew symmetric forms and discuss algebro geometric approaches to establish the existence of triples m n k where s m n k is more than expected

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

rationality of the moduli spaces of plane curves of sufficiently large degree we prove that the moduli space of plane curves of degree d is rational for all sufficiently large d

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

decomposition of homogeneous polynomials with low rank let f be a homogeneous polynomial of degree d in m 1 variables defined over an algebraically closed field of characteristic 0 and suppose that f belongs to the s th secant variety of the d uple veronese embedding of mathbb p m into mathbb p tiny left begin array c rm m d rm d end array right 1 but that its minimal decomposition as a sum of d th powers of linear forms m1 mr is f m 1 d cdots m r d with r s we show that if s r 2d 1 then such a decomposition of f can be split in two parts one of them is made by linear forms that can be written using only two variables the other part is uniquely determined once one has fixed the first part we also obtain a uniqueness theorem for the minimal decomposition of f if r is at most d and a mild condition is satisfied

identification of weak nonlinearities on damping and stiffness by the continuous wavelet transform we consider the free response of a nonlinear vibrating system using the ridges and skeletons of the continuous wavelet transform we identify weak nonlinearities on damping and stiffness and estimate their physical parameters the crucial choice of the son wavelet function is obtained using an optimization technique based on the entropy of the continuous wavelet transform the method is applied to simulated single degree of freedom systems and multi degree of freedom systems with nonlinearities on damping and stiffness experimental validation of the nonlinear identification and parameter estimation method is presented the experimental system is a clamped beam with nonlinearities on damping and stiffness and these nonlinearities are identified and quantified from a displacement sensor

non linear free vibration identification via the hilbert transform a time domain non parametric method for non linear vibration system identification based on the hilbert transform is introduced the method is demonstrated using computer simulations of different types of non linear elastic and damping dynamic systems

the moduli space of genus four even spin curves is rational abstract using the mori theory for threefolds we prove that the moduli space of pairs of smooth curves of genus four and theta characteristics without global sections is a rational variety

identification of weakly nonlinearities in multiple coupled oscillators this paper considers the analysis of free nonlinear response of multi degree of freedom mdof vibration systems demonstrating that system skeleton curves are significantly changed in the presence of coupling the hilbert transform vibration decomposition together with the modal spatial coordinate transform allow precise identification of initial nonlinear characteristics

on the branch curve of a general projection of a surface to a plane in this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible with only nodes and cusps this is a basic result in surface theory extremely useful in various applications however its proof in this general setting was so far lacking our approach substantially uses a powerful tool from projective differential geometry i e the concept of focal schemes

pricing real estate index options by compactly supported radial polynomial basis point interpolation abstract this paper presents a novel method to price the real estate index options which are modeled based on the framework proposed by fabozzi et al 2012 the cs pbf method combines compactly supported radial basis functions csrbf and polynomial basis functions pbf to yield the interpolation functions which can guarantee interpolation shape functions with kronecker property and overcome possible singularity associated with the pbf method compared with the csrbf method and the finite difference fd method the cs pbf method is more accurate and efficient for the real estate index option meanwhile a local mesh refinement technique is employed for dealing with the non smooth options payoffs which is very effective and stable to improve the computational accuracy for the cs pbf method finally the cs pbf method is extended to price american option of the real estate index

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

vanishing hessian wild forms and their border vsp wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank the discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero dimensional schemes and the span of their flat limit for concise forms of minimal border rank we show that the condition of vanishing hessian is equivalent to being wild this is proven by making a detour through structure tensors of smoothable and gorenstein algebras the equivalence fails in the non minimal border rank regime we exhibit an infinite series of minimal border rank wild forms of every degree d ge 3 as well as an infinite series of wild cubics inspired by recent work on border apolarity of buczynska and buczynski we study the border varieties of sums of powers underline mathrm vsp of these forms in the corresponding multigraded hilbert schemes

continuous wavelet transform for modal identification using free decay response abstract this paper deals with the use of the continuous wavelet transform for system identification purposes the wavelet analysis of the free responses of a linear mechanical system allows the estimation of its natural frequencies viscous damping ratios and mode shapes using either the modulus or the phase of the wavelet transform a complete procedure for modal identification from free responses based on the continuous wavelet transform is presented two difficulties during the implementation of this technique are highlighted the edge effect and the choice of the time frequency localization of the wavelet transform some upper and lower bounds for the mother wavelet s parameters are given in order to improve the numerical computation three complex valued mother wavelets are studied and the full procedure is applied to a damped discrete system the correct choice of the mother wavelet s parameters leads to an accurate identification of the modal parameters

a point interpolation method for two dimensional solids a point interpolation method pim is presented for stress analysis for two dimensional solids in the pim the problem domain is represented by properly scattered points a technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points the pim equations are then derived using variational principles in the pim the essential boundary conditions can be implemented with ease as in the conventional finite element methods the present pim has been coded in fortran the validity and efficiency of the present pim formulation are demonstrated through example problems it is found that the present pim is very easy to implement and very flexible for obtained displacements and stresses of desired accuracy in solids as the elements are not used for meshing the problem domain the present pim opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures copyright 2001 john wiley sons ltd

linear systems of plane curves with base points of equal multiplicity in this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m a conjecture of harbourne and hirschowitz implies that when d 3m the dimension is equal to the expected dimension given by the riemann roch theorem also systems for which the dimension is larger than expected should have a fixed part containing a multiple 1 curve we reformulate this conjecture by explicitly listing those systems which have unexpected dimension then we use a degeneration technique developed to show that the conjecture holds for all m 12

varieties with minimal secant degree and linear systems of maximal dimension on surfaces abstract in this paper we prove using a refinement of terracini s lemma a sharp lower bound for the degree of higher secant varieties to a given projective variety which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space moreover we study varieties for which the bound is attained proving some general properties related to tangential projections e g these varieties are rational in particular we completely classify surfaces and curves for which the bound is attained it turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus this is related to classical beautiful results of castelnuovo and enriques that we revise here in terms of adjunction theory

on apolarity and generic canonical forms abstract the notion of s algebra is introduced the theory of apolarity and generic canonical forms for polynomials is generalized to s algebras over the complex field c we apply this theory to the problem of finding the essential rank of general symmetric and skew symmetric tensors upper bounds for the essential ranks are found by different combinatorial coverings

postulation of general quintuple fat point schemes in p 3 we study the postulation of a general union y of double triple quartuple and quintuple points of p 3 in characteristic 0 we prove that y has good postulation in degree d 11 the proof is based on the combination of the horace differential lemma with a computer assisted proof we also classify the exceptions in degree 9 and 10

vanishing theorems for linearly obstructed divisors we study divisors on the blow up of pn at points in general position that are non special with respect to the notion of linear speciality introduced in 6 we describe the cohomology groups of their strict transforms via the blow up of the space along their linear base locus we extend the result to non effective divisors that sit in a small region outside the effective cone as an application we describe linear systems of divisors in pn blown up at points in star configuration and their strict transforms via the blow up of the linear base locus

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

on generic identifiability of symmetric tensors of subgeneric rank we prove that the general symmetric tensor in sdcn 1 of rank r is identifiable provided that r is smaller than the generic rank that is its waring decomposition as a sum of r powers of linear forms is unique only three exceptional cases arise all of which were known in the literature our original contribution regards the case of cubics d 3 while for d 4 we rely on known results on weak defectivity by ballico ciliberto chiantini and mella

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

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

penalty methods for the numerical solution of american multi asset option problems we derive and analyze a penalty method for solving american multi asset option problems a small non linear penalty term is added to the black scholes equation this approach gives a fixed solution domain removing the free and moving boundary imposed by the early exercise feature of the contract explicit implicit and semi implicit finite difference schemes are derived and in the case of independent assets we prove that the approximate option prices satisfy some basic properties of the american option problem several numerical experiments are carried out in order to investigate the performance of the schemes we give examples indicating that our results are sharp finally the experiments indicate that in the case of correlated underlying assets the same properties are valid as in the independent case

loci of curves with subcanonical points in low genus inside the moduli space of curves of genus three with one marked point we consider the locus of hyperelliptic curves with a marked weierstrass point and the locus of non hyperelliptic curves with a marked hyperflex these loci have codimension two we compute the classes of their closures in the moduli space of stable curves of genus three with one marked point similarly we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point these loci naturally arise in the study of minimal dimensional strata of abelian differentials

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

degenerations of planar linear systems fixing n general points pi in the plane what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i in this article we propose an approach to attack this problem and demonstrate it by successfully computing this dimension for all n and for mi constant at most 3 this application while previously known see hi1 demonstrates the utility of our approach which is based on an analysis of the corresponding linear system on a degeneration of the plane itself leading to a simple recursion for these dimensions we also obtain results in the quasi homogeneous case when all the multiplicities are equal except one this is the natural family to consider in the recursion

identification of damping in mdof systems using time scale decomposition abstract a method of damping identification in multi degree of freedom systems is presented the method is based on the time scale decomposition of the system impulse response the continuous wavelet transform is used to decompose the impulse response into the time scale domain three techniques based on the wavelet tranform are used to estimate damping these are the wavelet transform cross section procedure the impulse response recovery procedure based on wavelet domain filtering and the wavelet ridge detection procedure the methods are applied to simulated multi degree of freedom systems

a natural way for improving the accuracy of the continuous wavelet transforms aiming at further increasing the accuracy of the continuous wavelet transforms cwts of transient vibration signals a new coefficient trimming technique is proposed in this paper different from the conventional cwt analysis which treats different scales of daughter wavelets equally and without using any discrimination the proposed technique takes the role of the daughter wavelet into account in the cwt calculations in the concept of this new technique not only different scales of daughter wavelets play different roles in the calculations but also the daughter wavelet at the same scale also plays a different role at different time instants the role of the daughter wavelet is numerically evaluated through the approach of the correlation between the daughter wavelet and the instant signal within the time sliding window which window length is predefined by using a compromise strategy once the correlation coefficient indicating the role of a daughter wavelet at certain time instant is obtained it will be applied to trim the corresponding cwt coefficient that is derived at the corresponding wavelet scale and the current time instant the details of this operation are given in the following context of the paper extensive experiments show that with the aid of this proposed technique the false information in the cwt map is successfully removed while the true information of the vibration signal is retained sufficiently moreover different from the threshold based purification methods the continuity and the smoothness of the wavelet coefficients are maintained perfectly in the whole process so the results obtained by using the proposed technique are more explicit and reliable from which the signal features may be identified more easily and correctly this is very helpful for promoting understanding of the dynamic behaviour of the machines and furthermore giving a more reliable assessment of their running condition

extrapolation of difference methods in option valuation in the present investigation the fully implicit and crank nicolson difference schemes for solving option prices are analyzed it is proved that the error expansions for the difference methods have the correct form for applying richardson extrapolation to increase the order of accuracy of the approximations the difference methods are applied to european american and down and out knock out call options computational results indicate that richardson extrapolation significantly decreases the amount of computational work by as much as a factor of 16 in estimation of option prices

linear systems on the blow up of p1 n abstract in this note we study linear systems on the blow up of p 1 n at r points in very general position we prove that the fibers of the projections p 1 n p 1 s 1 s n 1 can give contribution to the speciality of the linear system this allows us to give a new definition of expected dimension of a linear system in p 1 n which we call fiber dimension finally we state a conjecture about linear systems on p 1 3

the computational complexity of continued fractions the knuth scho nhage algorithm for expanding a quolynomial into a continued fraction is shown to be essentially optimal with respect to the number of multiplications divisions used uniformly in the inputs

a proof of the set theoretic version of the salmon conjecture abstract we show that the irreducible variety of 4 4 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 the strassen commutative conditions of degree 6 the landsberg manivel polynomials and of degree 9 the symmetrization conditions

canonical curves and varieties of sums of powers of cubic polynomials in this note we show that the apolar cubic forms associated to codimension 2 linear sections of canonical curves of genus g 11 are special with respect to their presentation as sums of cubes

rank and optimal computation of generic tensors abstract the typical rank maximal border rank of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated for the size n n 3 with n odd the complement of the set of tensors of maximal border rank is a hypersurface its equation is given

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

birational aspects of the geometry of varieties of sums of powers abstract varieties of sums of powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms the study of these varieties dates back to sylvester and hilbert but only few of them for special degrees and number of variables are concretely identified in this paper we aim to understand a general birational behavior of vsp to do this we birationally embed these varieties into grassmannians and prove the rational connectedness of many vsp in arbitrary degrees and number of variables

secant varieties of segre veronese embeddings of p 1 r we use a double degeneration technique to calculate the dimension of the secant variety of any segre veronese embedding of mathbb p 1 r

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

weakly defective varieties a projective variety x is k weakly defective when its intersection with a general k 1 tangent hyperplane has no isolated singularities at the k 1 points of tangency if x is k defective i e if the k secant variety of x has dimension smaller than expected then x is also k weakly defective the converse does not hold in general a classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension we start this classification here describing all weakly defective irreducible surfaces our method also provides a new proof of the classical terracini s classification of k defective surfaces

koszul property of projections of the veronese cubic surface abstract let v p 9 be the veronese cubic surface we classify the projections of v to p 8 whose coordinate rings are koszul in particular we obtain a purely theoretical proof of the koszulness of the pinched veronese a result obtained originally by caviglia using filtrations deformations and computer assisted computations to this purpose we extend to certain complete intersections results of conca herzog trung and valla concerning homological properties of diagonal algebras