Related papers: Lexicographic cones and the ordered projective ten…

The natural lexicographic semigroupoids associated with Cantor product spaces indexed by countable linear orders are classified. Applications are given to the classification of triangular operator algebras which are direct limits of upper…

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either…

We describe a new way to construct finite geometric objects. For every k we obtain a symmetric configuration E(k-1) with k points on a line. In particular, we have a constructive existence proof for such configurations. The method is very…

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power…

Classification is a common AI problem, and vector search is a typical solution. This transforms a given body of text into a numerical representation, known as an embedding, and modern improvements to vector search focus on optimising speed…

Coecke, Sadrzadeh, and Clark (arXiv:1003.4394v1 [cs.CL]) developed a compositional model of meaning for distributional semantics, in which each word in a sentence has a meaning vector and the distributional meaning of the sentence is a…

We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex…

What is a good vector representation of an object? We believe that it should be generative in 3D, in the sense that it can produce new 3D objects; as well as be predictable from 2D, in the sense that it can be perceived from 2D images. We…

For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order…

Tensor diagrams are a handy way to depict complicated relationships between objects in projective geometry. One of the simpler ones takes two copies of a $3\times 3$ matrix and computes its adjugate. In this paper, we give a geometric…

Human-annotated datasets with explicit difficulty ratings are essential in intelligent educational systems. Although embedding vector spaces are widely used to represent semantic closeness and are promising for analyzing text difficulty,…

We propose a new data mining approach in ranking documents based on the concept of cone-based generalized inequalities between vectors. A partial ordering between two vectors is made with respect to a proper cone and thus learning the…

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are…

We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…

This paper investigates the learning of 3rd-order tensors representing the semantics of transitive verbs. The meaning representations are part of a type-driven tensor-based semantic framework, from the newly emerging field of compositional…

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension…

We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is…

Sequential data such as time series, video, or text can be challenging to analyse as the ordered structure gives rise to complex dependencies. At the heart of this is non-commutativity, in the sense that reordering the elements of a…

Most recent approaches to bilingual dictionary induction find a linear alignment between the word vector spaces of two languages. We show that projecting the two languages onto a third, latent space, rather than directly onto each other,…

We introduce an extended tangent cone of high order to a set and study its properties. Then we use this local approximation for deriving high-order necessary conditions for local minimizers of constrained optimization problems.