Related papers: Geometrical Representation for Number-theoretic Tr…

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. The time-frequency analysis of bivariate signals is usually carried out by analyzing two separate…

Graph representations of solid state materials that encode only interatomic distance lack geometrical resolution, resulting in degenerate representations that may map distinct structures to equivalent graphs. Here we propose a hypergraph…

The combinatorial interpretation of the persistence diagram as a M\"obius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a…

We discuss a geometrical method to define a preferred reference frame for precessing binary systems and the gravitational waves they emit. This minimal-rotation frame is aligned with the angular-momentum axis and fixes the rotation about…

In the presence of boundaries the integrated conformal anomaly is modified by the boundary terms so that the anomaly is non-vanishing in any (even or odd) dimension. The boundary terms are due to extrinsic curvature whose exact structure in…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT)…

We show that each irreducible tensor representation of weight 2 of the rotation group of three-dimensional space in the space of rank 3 covariant tensors gives rise to an associative algebra with unity. We find the algebraic relations that…

As an extension of projective homology, stereohomology is proposed via an extension of Desargues theorem and the extended Desargues configuration. Geometric transformations such as reflection, translation, central symmetry, central…

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…

Kanerva's Binary Spatter Codes are reformulated in terms of geometric algebra. The key ingredient of the construction is the representation of XOR binding in terms of geometric product.

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…

It is shown how to introduce a geometric description of the algebraic approach to the non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but also Hermitian realization of a `quantum…

Aim. This paper (Paper D) unifies the ideal-theoretic, computational, and homological layers developed in Papers A (Rao 2025), B (Rao 2025), and C (Rao 2025) into a geometric framework that includes fuzzy and computational geometries on the…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…

Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence.…

Many datasets in scientific and engineering applications are comprised of objects which have specific geometric structure. A common example is data which inhabits a representation of the group SO$(3)$ of 3D rotations: scalars, vectors,…

Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely…

We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation…

General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its…