Related papers: A Program For Geometric Arithmetic
In this paper, we introduce a geometrically stylized arithmetic cohomology for number fields. Based on such a cohomology, we define and study new yet genuine non-abelian zeta functions for number fields, using an intersection stability.
Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the…
The set theory relations \in, \backslash, \Delta, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and \Delta…
We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As applications, we study moduli spaces of…
Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using…
As a part of our program for Geometric Arithmetic, we develop an arithmetic cohomology theory for number fields using theory of locally compact groups.
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…
Every algorithm which can be executed on a computer can at least in principle be realized in hardware, i.e. by a discrete physical system. The problem is that up to now there is no programming language by which physical systems can…
A nested coordinate system is a reassigning of independent variables to take advantage of geometric or symmetry properties of a particular application. Polar, cylindrical and spherical coordinate systems are primary examples of such a…
Computer algebra programs are presented for application in general relativity, in electrodynamics, and in gauge theories of gravity. The mathematical formalism used is the calculus of exterior differential forms, the computer algebra system…
This is an integrated part of our Geo-Arithmetic Program. In this paper we introduce and hence study non-abelian zeta functions and more generally non-abelian $L$-functions for number fields, based on geo-arithmetical cohomology,…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
The characteristic feature of the adeles is that they involve localizations of products (or equivalently restricted products of localizations). The point of this paper is to introduce an adelic style cohomological invariant of a partially…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes.…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms…
We give a language for geometry which makes curves and number fields look alike.
We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…