 |
 |
[5] viXra:2505.0155 [pdf] submitted on 2025-05-22 20:38:01
Authors: Egor Sibrin, Vladimir Orlov
Comments: 23 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This work presents the framework of Nonliquent Mathematics (NL), a commutative extension of the real numbers that defines division by zero through the introduction of an idempotent element denoted by ї=1/0 and the distinction between a true zero and a special zero. The algebraic operations of addition, multiplication, division, conjugation, and root extraction are formulated in this system, along with Taylor expansions for basic functions and a consistent differential and integral calculus adapted to NL variables. A Hilbert space of NL-valued functions is defined with a corresponding inner product, and a modified form of the Navier—Stokes equation is derived within this framework. Two particular solutions of the modified equation are obtained for the cases of zero and nonzero viscosity, demonstrating the consistency and potential applications of NL in analysis and fluid dynamics.
Category: Algebra
[4] viXra:2505.0101 [pdf] submitted on 2025-05-15 20:35:33
Authors: Michael Reed
Comments: 6 Pages. (2019)
The Grassmann.jl package provides tools for computations based on multilinear algebra and spin groups using the extended geometric algebra known as Leibniz-Grassmann-Clifford-Hestenes algebra. Combinatorial products include exterior, regressive, inner, and geometric; along with the Hodge star, adjoint, reversal, and boundary operators. The kernelized operations are built up from composite sparse tensor products and Hodge duality, with high dimensional support for up to 62 indices using staged caching and pre-compilation. Code generation enables concise yet highly extensible definitions. DirectSum.jl multivector parametric type polymorphism is based on tangent vector spaces and conformal projective geometry. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.
Category: Algebra
[3] viXra:2505.0100 [pdf] submitted on 2025-05-15 20:40:27
Authors: Michael Reed
Comments: 42 Pages.
Initially, Grassmann.jl and Cartan.jl introduced the DirectSum formalism into computational language design for differential geometric algebra, thus enabling the construction of custom algebras through metaprogramming.Grassmann.jl pioneered a novel type system design along with its syntax and semantics, which has undergone significant refinement through many years of development and continuous improvement.Cartan.jl represents a groundbreaking extension of capabilities offered by Grassmann.jl, marking a pioneering fully realized implementation of numerical differential geometric algebra, based on TensorField representations over a FrameBundle and the ImmersedTopology of the FiberBundle.Grassmann.jl and Cartan.jl build on Julia's multiple dispatch and metaprogramming capabilities, presenting a new computational language design approach to interfacing with differential geometric algebra based on a new sector integral theorem.This pioneering design not only actualizes but also elevates computational language syntax to new heights using the foundations of Grassmann.jl and Cartan.jl.The Grassmann.jl and Cartan.jl packages introduce pioneering computational language designs, having inspired imitation projects and thereby validating the project's relevance as significant advance in computational mathematics.
Category: Algebra
[2] viXra:2505.0068 [pdf] submitted on 2025-05-10 20:50:02
Authors: Oliver Couto
Comments: 5 Pages.
In math literature there are not many examples of parametrization for the equation:(a^3+b^3+c^3+d^3=e^3). There are two Identities mentioned in ref. # (3). In this paper the author has parametrized the above equation by means of Algebra. By applying certain conditions to the above equation, we have avoided the use of elliptic curve theory in-order to solve the problem.----------------
Category: Algebra
[1] viXra:2505.0015 [pdf] submitted on 2025-05-01 06:24:29
Authors: Jose Acevedo Jimenez
Comments: 1 Page.
We prove the elementary identity for any real number. The identity reveals a symmetric structure between shifted quadratic terms.
Category: Algebra
| Third-Party Links: |
|