Related papers: Complex Spaces and Nonstandard Schemes
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
Bayesian nonparametric space partition (BNSP) models provide a variety of strategies for partitioning a $D$-dimensional space into a set of blocks. In this way, the data points lie in the same block would share certain kinds of homogeneity.…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…
The "quantum-event / prime ideal in a category/ noncommutative-point" alternative to "classical-event / commutative prime ideal/ point" is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered…
In this essay, we immerse into the framework of normed division algebras as a suitable arena to accommodate the standard model of elementary particles, and we explore some applications to cosmology. Remarkably, they permit interesting…
We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Then we find height bounds in the polynomial ring over algebraic numbers to test…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
In cluster analysis, a common first step is to scale the data aiming to better partition them into clusters. Even though many different techniques have throughout many years been introduced to this end, it is probably fair to say that the…
We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified…
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of…
Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits…
The standard format of matrices belonging to Lie superalgebras consists of partitioning the matrices into even and odd blocks. In this paper, we study other possible matrix formats and in particular the so-called diagonal format which…
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical…