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Elena Rubei

Publications and source records attributed to Elena Rubei.

32 records · Page 2Linked to original sources
arXiv2013

On graphlike k-dissimilarity vectors

Let {\cal G}=(G,w) be a positive-weighted simple finite graph, that is, let G be a simple finite graph endowed with a function w from the set of the edges of G to the set of the positive real numbers. For any subgraph G' of G, we define w(G') to be the sum of the weights of the edges of G'. For any i_1,..., i_k vertices of G, let D_{i_1,.... i_k}({\cal G}) be the minimum of the weights of the subgraphs of G connecting i_1,..., i_k. The D_{i_1,.... i_k}({\cal G}) are called k-weights of {\cal G}. Given a family of positive real numbers parametrized by the k-subsets of {1,..., n}, {D_I}_{I k-subset of {1,...,n}}, we can wonder when there exist a weighted graph {\cal G} (or a weighted tree) and an n-subset {1,..., n} of the set of its vertices such that D_I({\cal G}) =D_I for any I k-subset of {1,...,n}. In this paper we study this problem in the case k=n-1.

math.CO↗
arXiv2012

On the weights of simple paths in weighted complete graphs

Consider a weighted graph G with n vertices, numbered by the set {1,...,n}. For any path p in G, we call w_G(p) the sum of the weights of the edges of the path and we define the multiset {\cal D}_{i,j} (G) = {w_G(p) | p simple path between i and j} We establish a criterion to say when, given a multisubset of the set of the real numbers there exists a weighted complete graph G such that the multisubset is equal to {\cal D}_{i,j} (G) for some i,j vertices of G. Besides we establish a criterion to say when, given for any i, j in {1,...,n} a multisubset of the set of the real numbers,{\cal D}_{i,j}, there exists a weighted complete graph G with vertices {1,...,n} such that {\cal D}_{i,j} (G)= {\cal D}_{i,j} for any i,j.

math.CO↗
arXiv2011

Stability of homogeneous bundles on P^3

We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver representation is a parallelepiped are stable, for instance the bundles E whose minimal free resolution is of the kind 0 --> S^{l_1, l_2, l_3} V (t) --> S^{l_1 +s, l_2, l_3} V (t+s) --> E --> 0 are stable.

math.AG↗
arXiv2011

Sets of double and triple weights of trees

Let T be a weighted tree with n leaves. Let D_{i,j} be the distance between the leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will call such numbers "triple weights" of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a $n$-set to be double weights of a tree. By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an $n$-set to be the set of the triple weights of a tree with $n$ leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_{i,j}.

math.AG↗
arXiv2010

On dissimilarity vectors of (not necessarily positive) weighted trees

Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1,...,n}. Define the k-weights of the tree D_{i_1,....,i_k}(T) as the sum of the lengths of the edges of the minimal subtree connecting i_1,....,i_k. We will call such numbers "k-weights" of the tree. In this paper, we characterize the sets of real numbers indexed by the subsets of any cardinality >= 2 of a n-set to be the weights of a tree with n leaves.

math.CO↗
arXiv2009

The 4x4 minors of a 5xn matrix are a tropical basis

We compute the space of 5x5 matrices of tropical rank at most 3 and show that it coincides with the space of 5x5 matrices of Kapranov rank at most 3, that is, the space of five labeled coplanar points in TP4. We then prove that the Kapranov rank of every 5xn matrix equals its tropical rank; equivalently, that the 4x4 minors of a 5xn matrix of variables form a tropical basis. This answers a question asked by Develin, Santos, and Sturmfels.

math.CO↗
arXiv2005

Resolutions of homogeneous bundles on P^2

In this paper we characterize minimal free resolutions of homogeneous bundles on P^2. Besides we study stability and simplicity of homogeneous bundles on P^2 by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal free resolution in the case the first bundle of the resolution is irreducible.

math.AG↗
arXiv2005

Quivers and the cohomology of homogeneous vector bundles

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver ${\cal Q}_X$ with relations, whose vertices are the dominant weights of the reductive part of $P$. This equivalence was found in some cases by Bondal, Kapranov and Hille and we find the appropriate relations for any Hermitian symmetric variety, computing them explicitly for Grassmannians.

math.AG↗
arXiv2003

A result on resolutions of Veronese embeddings

This paper deals with syzygies of the ideals of the Veronese embeddings. We prove that O(3) on P^n satisfies Property N_4 for every n. Besides we prove that O(d) on P^n satisfies N_p for all n >= p iff O(d) on P^p satisfies N_p.

math.AG↗
arXiv2001

A strange example concerning Property N_p

We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfy Property N_p for some p, the p-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M^2 does not satisfy Property N_p.

math.AG↗
arXiv2001

On syzygies of Segre embeddings

The main result of this paper is the following: let d be a natural number >= 3; the line bundle O(1,...,1) on the product of d copies of the projective line P^1 satisfies Property N_p of Green-Lazarsfeld if and only if p <= 3.

math.AG↗
arXiv1998

Lazzeri's Jacobian of oriented compact riemannian manifolds

The subject of this paper is a Jacobian, introduced by F. Lazzeri, (unpublished), associated to every compact oriented riemannian manifold of dimension twice an odd number. We start the investigation of Torelli type problems and Schottky type problem for Lazzeri's Jacobian; in particular we examine the case of tori with flat metrics. Besides we study Lazzeri's Jacobian for Kahler manifolds and its relationship with other Jacobians. Finally we examine Lazzeri's Jacobian of a bundle.

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