A geometrical model for the synthesis of interval covers,

"Supported by Contract AT(11-1)-2118 with the U.S. Atomic Energy Commission."

Measurement-based teleportation along quantum spin chains

We examine the teleportation of an unknown spin-1/2 quantum state along a quantum spin chain with an even number of sites. Our protocol, using a sequence of Bell measurements, may be viewed as an iterated version of the 2-qubit protocol of C. H. Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)]. A decomposition of the Hilbert space of the spin chain into 4 vector spaces, called Bell subspaces, is given. It is established that any state from a Bell subspace may be used as a channel to perform unit fidelity teleportation. The space of all spin-0 many-body states, which includes the ground states of many known antiferromagnetic systems, belongs to a common Bell subspace. A channel-dependent teleportation parameter O is introduced, and a bound on the teleportation fidelity is given in terms of O.

On the Weight Distribution of the Coset Leaders of Constacyclic Codes

Constacyclic codes with one and the same generator polynomial and distinct length are considered. We give a generalization of the previous result of the first author [4] for constacyclic codes. Suitable maps between vector spaces determined by the lengths of the codes are applied. It is proven that the weight distributions of the coset leaders don’t depend on the word length, but on generator polynomials only. In particular, we prove that every constacyclic code has the same weight distribution of the coset leaders as a suitable cyclic code.

Indice de Point Fixe pour les Morphismes de Chaînes

2000 Mathematics Subject Classification: 54H25, 55M20.

Algebraic duality theorems for infinite LP problems

In this paper we consider a primal-dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.

Combining Taxonomies using Word2vec

Taxonomies have gained a broad usage in a variety of fields due to their extensibility, as well as their use for classification and knowledge organization. Of particular interest is the digital document management domain in which their hierarchical structure can be effectively employed in order to organize documents into content-specific categories. Common or standard taxonomies (e.g., the ACM Computing Classification System) contain concepts that are too general for conceptualizing specific knowledge domains. In this paper we introduce a novel automated approach that combines sub-trees from general taxonomies with specialized seed taxonomies by using specific Natural Language Processing techniques. We provide an extensible and generalizable model for combining taxonomies in the practical context of two very large European research projects. Because the manual combination of taxonomies by domain experts is a highly time consuming task, our model measures the semantic relatedness between concept labels in CBOW or skip-gram Word2vec vector spaces. A preliminary quantitative evaluation of the resulting taxonomies is performed after applying a greedy algorithm with incremental thresholds used for matching and combining topic labels.

Comparing the orthogonal and homotopy functor calculi

Goodwillie’s homotopy functor calculus constructs a Taylor tower of approximations toF , often a functor from spaces to spaces. Weiss’s orthogonal calculus provides a Taylortower for functors from vector spaces to spaces. In particular, there is a Weiss towerassociated to the functor V ÞÑ FpSVq, where SVis the one-point compactification of V .In this paper, we give a comparison of these two towers and show that when F isanalytic the towers agree up to weak equivalence. We include two main applications, oneof which gives as a corollary the convergence of the Weiss Taylor tower of BO. We alsolift the homotopy level tower comparison to a commutative diagram of Quillen functors,relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

“Varopoulos paradigm”: Mackey property versus metrizability in topological groups.

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Außenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Díaz Nieto and Martín Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

Development of unsupervised learning methods with applications to life sciences data

Machine Learning makes computers capable of performing tasks typically requiring human intelligence. A domain where it is having a considerable impact is the life sciences, allowing to devise new biological analysis protocols, develop patients’ treatments efficiently and faster, and reduce healthcare costs. This Thesis work presents new Machine Learning methods and pipelines for the life sciences focusing on the unsupervised field. At a methodological level, two methods are presented. The first is an “Ab Initio Local Principal Path” and it is a revised and improved version of a pre-existing algorithm in the manifold learning realm. The second contribution is an improvement over the Import Vector Domain Description (one-class learning) through the Kullback-Leibler divergence. It hybridizes kernel methods to Deep Learning obtaining a scalable solution, an improved probabilistic model, and state-of-the-art performances. Both methods are tested through several experiments, with a central focus on their relevance in life sciences. Results show that they improve the performances achieved by their previous versions. At the applicative level, two pipelines are presented. The first one is for the analysis of RNA-Seq datasets, both transcriptomic and single-cell data, and is aimed at identifying genes that may be involved in biological processes (e.g., the transition of tissues from normal to cancer). In this project, an R package is released on CRAN to make the pipeline accessible to the bioinformatic Community through high-level APIs. The second pipeline is in the drug discovery domain and is useful for identifying druggable pockets, namely regions of a protein with a high probability of accepting a small molecule (a drug). Both these pipelines achieve remarkable results. Lastly, a detour application is developed to identify the strengths/limitations of the “Principal Path” algorithm by analyzing Convolutional Neural Networks induced vector spaces. This application is conducted in the music and visual arts domains.

The combinatorics of Hilbert-Poincaré series of matroids

In this thesis we explore the combinatorial properties of several polynomials arising in matroid theory. Our main motivation comes from the problem of computing them in an efficient way and from a collection of conjectures, mainly the real-rootedness and the monotonicity of their coefficients with respect to weak maps. Most of these polynomials can be interpreted as Hilbert--Poincaré series of graded vector spaces associated to a matroid and thus some combinatorial properties can be inferred via combinatorial algebraic geometry (non-negativity, palindromicity, unimodality); one of our goals is also to provide purely combinatorial interpretations of these properties, for example by redefining these polynomials as poset invariants (via the incidence algebra of the lattice of flats); moreover, by exploiting the bases polytopes and the valuativity of these invariants with respect to matroid decompositions, we are able to produce efficient closed formulas for every paving matroid, a class that is conjectured to be predominant among all matroids. One last goal is to extend part of our results to a higher categorical level, by proving analogous results on the original graded vector spaces via abelian categorification or on equivariant versions of these polynomials.

Fixed point theorems in partially ordered spaces and aplications

[ES]Recientemente, en la Teoría del punto fijo, han aparecido muchos resultados que obtienen condiciones suficientes para la existencia de un punto fijo si trabajamos con aplicaciones en un conjunto dotado de un orden parcial. Generalmente, estos resultados combinan dos teoremas del punto fijo fundamentales: el Teorema de la contracción de Banach y el Teorema de Knaster-Tarski.

Three Spaces Problem for Lyapunov Theorem on Vector Measure

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it.

Uniform G-Convexity for Vector-Valued Lp Spaces

2000 Mathematics Subject Classification: 46B20.

Exotic phases induced by strong spin-orbit coupling in ordered double perovskites

We construct and analyze a microscopic model for insulating rocksalt ordered double perovskites, with the chemical formula A(2)BB'O(6), where the B' atom has a 4d(1) or 5d(1) electronic configuration and forms a face-centered-cubic lattice. The combination of the triply degenerate t(2g) orbital and strong spin-orbit coupling forms local quadruplets with an effective spin moment j=3/2. Moreover, due to strongly orbital-dependent exchange, the effective spins have substantial biquadratic and bicubic interactions (fourth and sixth order in the spins, respectively). This leads, at the mean-field level, to three main phases: an unusual antiferromagnet with dominant octupolar order, a ferromagnetic phase with magnetization along the [110] direction, and a nonmagnetic but quadrupolar ordered phase, which is stabilized by thermal fluctuations and intermediate temperatures. All these phases have a two-sublattice structure described by the ordering wave vector Q=2 pi(001). We consider quantum fluctuations and argue that in the regime of dominant antiferromagnetic exchange, a nonmagnetic valence-bond solid or quantum-spin-liquid state may be favored instead. Candidate quantum-spin-liquid states and their basic properties are described. We also address the effect of single-site anisotropy driven by lattice distortions. Existing and possible future experiments are discussed in light of these results.

Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c (q) as the minimum cardinality of a subset H of F 3 q which satisfies the following property: every vector in this space di ff ers in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c (q)