Related papers: Structural Highness Notions
This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…
Network topologies can be non-trivial, due to the complex underlying behaviors that form them. While past research has shown that some processes on networks may be characterized by low-order statistics describing nodes and their neighbors,…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
The current ISO standards pertaining to the Concepts of System and Architecture express succinct definitions of these two key terms that lend themselves to practical application and can be understood through elementary mathematical…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
Over the past two decades the notion of a strong monad has found wide applicability in computing. Arising out of a need to interpret products in computational and semantic settings, different approaches to this concept have arisen. In this…
Topological statistical theory provides the foundation for a modern mathematical reformulation of classical statistical theory: Structural Statistics emphasizes the structural assumptions that accompany distribution families and the set of…
The purpose of this paper is to describe and elaborate the philosophical ideas behind hyperstructures and structure formation in general and emphasize the key ideas of the Hyperstructure Program.
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then…
A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…
We investigate the problem of type isomorphisms in the presence of higher-order references. We first introduce a finitary programming language with sum types and higher-order references, for which we build a fully abstract games model…
We define various height functions for motives over number fields. We compare these height functions with classical height functions on algebraic varieties, and also with analogous height functions for variations of Hodge structures on…
We study the notion of geometric structures for toposes: This generalizes the notion of (X,G) manifolds. We give some applications to algebraic geometry
In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this…
We propose a computationally efficient and high-performance classification algorithm by incorporating class structural information in analysis dictionary learning. To achieve more consistent classification, we associate a class…
Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…
We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that…
We are studying the degrees in which a computable structure is relatively computably categoricity, i.e., computably categorcial among all non-computable copies of the structure. Unlike the degrees of computable categoricity we can bound the…