Related papers: An algebraic geometry question and wave turbulence
In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of…
We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
In the first part of this article, I summarise two centuries of research on turbulence. I also critically discuss some of the interpretations that are still in use, as turbulence remains an inherently non-linear problem that is still…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
In this paper, the analog of Maxwell electromagnetism for hydrodynamic turbulence, the metafluid dynamics, is extended in order to reformulate the metafluid dynamics as a gauge field theory. That analogy opens up the possibility to…
We argue that the mathematical structure, enabling certain cascading and emergent phenomena to intuitively emerge, coincides with Galois connections. We introduce the notion of generative effects to formally capture such phenomena. We…
Recent advances in twistor theory are applied to geometric optics in ${\Bbb{R}}^3$. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the…
This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…
Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of…
A new theory of edge waves over a slowly varying depth.
We discuss an approach to gravitational waves based on Geometric Algebra and Gauge Theory Gravity. After a brief introduction to Geometric Algebra (GA), we consider Gauge Theory Gravity, which uses symmetries expressed within the GA of flat…
This paper is the third in a series of papers, the aim of which is to construct algebraic geometry over metabelian Lie algebras.
We present a novel derivation of the elastic theory of shells. We use the language of Geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools…
During direct numerical simulation of the isotropic turbulence of surface gravity waves in the framework of Hamiltonian equations formation of the long wave background or condensate was observed. Exponents of the direct cascade spectra at…
General relativity describes the dynamics of gravitational waves, which can feature nonlinear interactions, such as those underlying turbulent processes. Theoretical and numerical explorations have demonstrated the existence of…
The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
Waves play an essential role in many aspects of plasma science, such as plasma manipulation and diagnostics. Due to the complexity of the governing equations, approximate models are often necessary to describe wave dynamics. In this…