scholarly journals Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
E. Ballico
Keyword(s):  
Homogeneous Polynomial ◽  
Linear Span ◽  
Veronese Embedding ◽  
Complex Variety ◽  
Uniqueness Set

Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q   ∈  〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals.

scholarly journals Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 140
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear Span ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
General Point ◽  
Symmetric Tensor Rank ◽  
Maximum Rank ◽  
Veronese Embedding ◽  
Finite Set

Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | ≤ ( n + ⌊ d / 2 ⌋ n ) . For the tensor rank, we describe the cases with | S | ≤ 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d).

scholarly journals Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
E. Ballico
Keyword(s):  
Linear Form ◽  
Homogeneous Polynomial ◽  
Linear Subspace ◽  
Symmetric Tensor ◽  
Minimal Cardinality ◽  
Symmetric Tensors ◽  
Veronese Embedding ◽  
Integral Variety

Let νd:ℙm→ℙn, n:=(n+dn)-1, denote the degree d Veronese embedding of ℙm. For any P∈ℙn, let sr(P) be the minimal cardinality of S⊂νd(ℙm) such that P∈〈S〉. Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯(S) powers Ld with L a linear form (or as a sum of ♯(S) d-powers of vectors). Here we fix an integral variety T⊊ℙm and P∈〈νd(T)〉 and study a similar decomposition with S⊈T and ♯(S) minimal. For instance, if T is a linear subspace, then we prove that ♯(S)≥♯(S∩T)+d+1 and classify all (S,P) such that ♯(S)-♯(S∩T)≤2d-1.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

Rethinking Iconoclasm in Early Modern England: the Case of Cheapside Cross

2000 ◽  
Vol 4 (3-4) ◽  
pp. 379-404 ◽  
Author(s):  
Joel Budd
Keyword(s):  
Early Modern ◽  
Early Modern England ◽  
Traditional Religion ◽  
Traditional Beliefs ◽  
Beliefs And Practices ◽  
The Reformation ◽  
Complex Variety ◽  
City Of London ◽  
The City

AbstractProtestant iconoclasm has generally been understood as an assault on the beliefs and practices of traditional religion. This article challenges that understanding through a detailed study of Cheapside Cross, a large monument that was repeatedly attacked by iconoclasts in the sixteenth and seventeenth centuries. It draws on contemporary pamphlets and the manuscripts records of the City of London to reveal the complex variety of associations that Cheapside Cross acquired before and during the Reformation era. It argues that perceptions of the monument were shaped not only by its iconography but also by its involvement in ceremonies and rituals, including royal coronation processions. The iconoclastic attacks on Cheapside Cross should be interpreted not merely as a challenge to traditional beliefs but as attempts to restructure the monument's associations. The paper concludes that attacks on other images may be understood in a similar manner.

scholarly journals A uniqueness result on the decompositions of a bi-homogeneous polynomial

2016 ◽  
Vol 65 (4) ◽  
pp. 677-698 ◽  
Author(s):  
Edoardo Ballico ◽  
Alessandra Bernardi
Keyword(s):  
Homogeneous Polynomial ◽  
Uniqueness Result
Sign in / Sign up

Export Citation Format

Share Document