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In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for…
General relativity does not allow one to specify the topology of space, leaving the possibility that space is multiply rather than simply connected. We review the main mathematical properties of multiply connected spaces, and the different…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
The problem of topology change description in gravitation theory is analized in detailes. It is pointed out that in standard four-dimensional theories the topology of space may be considered as a particular case of boundary conditions (or…
This paper proposes a generalized framework with joint normalization which learns lower-dimensional subspaces with maximum discriminative power by making use of the Riemannian geometry. In particular, we model the similarity/dissimilarity…
The field equation of orthodox general relativity are written in the context of a geometry with non-vanishing torsion, the Absolute Parallelism (AP) geometry. An AP-structure, with homogeneity and isotropy, is used for cosmological…
A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie…
Most complex systems can be captured by graphs or networks. Networks connect nodes (e.g.\ neurons) through edges (synapses), thus summarizing the system's structure. A popular way of interrogating graphs is community detection, which…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that…
Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and the closely related exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We…
We study geometry and topology as complementary and dual aspects of the mathematical space. The same is used to get a better understanding of the Cosmological Constant. Having failed so far to include gravity in a proper unified framework…
In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
This note collects a number of standard statements in Riemannian geometry and in Sobolev-space theory that play a prominent role in analytic approaches to symplectic topology. These include relations between connections and complex…
Absolute parallelism (AP) geometry is frequently used for physical applications. Although it is wider than Riemannian geometry, it has two main defects. The first is that its path equation does not represent physical trajectories of any…
A new geometry, called General geometry, is constructed. It is proven that its the most simplest special case is geometry underlying Electromagnetism. Another special case is Riemannian geometry. Action for electromagnetic field and Maxwell…
In the realm of robotics, numerous downstream robotics tasks leverage machine learning methods for processing, modeling, or synthesizing data. Often, this data comprises variables that inherently carry geometric constraints, such as the…