Related papers: Structural Highness Notions
We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…
We discuss the notion of linearization through examples, which include the Price map, PageRank, representation theory, the Euler characteristic and quantum invariants. We also review categorification, which adds an additional layer of…
In this paper we suggest how the mathematical concept of hyperstructures may be a useful tool in the study of the higher, hierachical structure of languages.
We show that on an arbitrary collection of objects there is a wide variety of higher order architectures governed by hyperstructures. Higher order gluing, local to global processes, fusion of collections, bridges and higher order types are…
This dissertation builds a compositional cyber-physical systems theory to develop concrete semantics relating the above diverse views necessary for safety and security assurance. In this sense, composition can take two forms. The first is…
After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were…
We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about…
We introduce a theory of multigraded Cayley-Chow forms associated to subvarieties of products of projective spaces. Two new phenomena arise: first, the construction turns out to require certain inequalities on the dimensions of projections;…
The concept of configuration was first introduced to give a characterization for the amenability of groups. Then the concept of two-sided configuration was suggested to provide normality to study the group structures more efficiently. It…
Structural causal models provide a formalism to express causal relations between variables of interest. Models and variables can represent a system at different levels of abstraction, whereby relations may be coarsened and refined according…
We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces…
There are different ways to define similarity for grouping similar texts into clusters, as the concept of similarity may depend on the purpose of the task. For instance, in topic extraction similar texts mean those within the same semantic…
This paper is devoted to the study of metric subregularity and strong subregularity of any positive order $q$ for set-valued mappings in finite and infinite dimensions. While these notions have been studied and applied earlier for $q=1$…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
We relate the notions of spectral gap for unitary representations and subfactors with definability of certain important sets in the corresponding structures. We give several applications of this relationship.
We propose a possible relation between complex networks and gravity. Our guide in our proposal is the power-law distribution of the node degree in network theory and the information approach to gravity. The established bridge may allow us…
We present a notion of precompactness, and study some of its properties, in the context of apartness spaces whose apartness structure is not necessarily induced by any uniform one. The presentation lies entirely with a Bishop-style…
The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Gr\"obner bases pass to the quotient and when they can be lifted. The main…
Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level…
Complex numbers define the relationship between entities in many situations. A canonical example would be the off-diagonal terms in a Hamiltonian matrix in quantum physics. Recent years have seen an increasing interest to extend the tools…