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We consider the problem of data classification where the training set consists of just a few data points. We explore this phenomenon mathematically and reveal key relationships between the geometry of an AI model's feature space, the…

This is a first stab at a mathematical framework in which one can study quantum field theories on spacetimes with quite general geometries. We will study these theories via their factorization algebras. The aim is to identify a minimalist…

Model-based approaches bear great promise for decision making of agents interacting with the physical world. In the context of spatial environments, different types of problems such as localisation, mapping, navigation or autonomous…

We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.

Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case…

We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the…

The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, making them impractical…

Marginal models involve restrictions on the conditional and marginal association structure of a set of categorical variables. They generalize log-linear models for contingency tables, which are the fundamental tools for modelling the…

We discuss supernear spaces.

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate…

The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence…

We prove a characterization for BLD-mappings between locally complete locally compact path-metric spaces. As a corollary we obtain a sharp limit theorem for BLD-mappings.

We show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices. Some related bounds for the spread and condition number of Hermitian matrices are also…

We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric…

Linear polarimetric transformations of light polarization states by the action of material media are fully characterized by the corresponding Mueller matrices, which contain in an implicit and intricate manner all measurable information on…

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems…

This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the…

We establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended…

We argue that parameterized complexity is a useful tool with which to study global constraints. In particular, we show that many global constraints which are intractable to propagate completely have natural parameters which make them…

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its…