Related papers: Generalised vector products in three-dimensional g…
We give necessary and sufficient conditions for a family of inner products in a finite-dimensional vector space $V$ over an arbitrary field $\mathbb{K}$ to have an orthogonal basis relative to all the inner products. Some applications to…
Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned…
We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in…
Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed.
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated…
In this paper, we introduce the Grassmann tensor by tensor product of vectors and some basic terminology in tensor theory. Some basic properties of the Grassmann tensors are investigated and the tensor language is used to rewrite some…
Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…
Vector fields are a highly abstract physical concept that is often taught using visualizations. Although vector representations are particularly suitable for visualizing quantitative data, they are often confusing, especially when…
We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional…
In textbooks and historical literature, the cross product has been defined only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product of only two vectors has been defined only in the high dimensional Euclidean space…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…
We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities.…
As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a…
Every ordered collection of sets in Euclidean space can be associated to a combinatorial code, which records the regions cut out by the sets in space. Given two ordered collections of sets, one can form a third collection in which the…
This paper revisits the little-known Gibbs-Rodrigues representation of rotations in a three-dimensional space and demonstrates a set of algorithms for handling it. In this representation the rotation is itself represented as a…
In models of emergent gravity the metric arises as the expectation value of some collective field. Usually, many different collective fields with appropriate tensor properties are candidates for a metric. Which collective field describes…
The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After…
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that…
We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector space over Q generated by these angles.…