scholarly journals Mathematics and Poetry • Unification, Unity, Union

Sci ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 72 ◽  
Author(s):  
Florin F. Nichita
Keyword(s):  
Differential Geometry ◽  
Baxter Equation ◽  
Dual Numbers ◽  
Euler’S Formula ◽  
Euler’S Identity

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

scholarly journals Mathematics and Poetry • Unification, Unity, Union

Sci ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 84
Author(s):  
Florin Felix Nichita
Keyword(s):  
Differential Geometry ◽  
Baxter Equation ◽  
Dual Numbers ◽  
Euler’S Formula ◽  
Euler’S Identity

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

scholarly journals Mathematics and Poetry • Unification, Unity, Union

Sci ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 58
Author(s):  
Florin F. Nichita
Keyword(s):  
Differential Geometry ◽  
Baxter Equation ◽  
Dual Numbers ◽  
Euler’S Formula ◽  
Euler’S Identity

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

scholarly journals Unification Theories: Means and Generalized Euler Formulas

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 144
Author(s):  
Radu Iordanescu ◽  
Florin Felix Nichita ◽  
Ovidiu Pasarescu
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Current Paper ◽  
Baxter Equation ◽  
Quadratic Mean ◽  
Euler’S Formula ◽  
Associative Structures ◽  
Generalized Mean

The main concepts in this paper are the means and Euler type formulas; the generalized mean which incorporates the harmonic mean, the geometric mean, the arithmetic mean, and the quadratic mean can be further generalized. Results on the Euler’s formula, the (modified) Yang–Baxter equation, coalgebra structures, and non-associative structures are also included in the current paper.

scholarly journals THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250102
Author(s):  
A. L. AGORE ◽  
S. CAENEPEEL ◽  
G. MILITARU
Keyword(s):  
Differential Geometry ◽  
Commutative Ring ◽  
Baxter Equation ◽  
Flat Connection ◽  
Dual Basis ◽  
Monoidal Categories ◽  
Noncommutative Rings ◽  
Descent Data ◽  
Weak Center ◽  
Braided Monoidal Categories

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.

scholarly journals Dual Numbers Approach in Multiaxis Machines Error Modeling

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jaroslav Hrdina ◽  
Petr Vašík
Keyword(s):  
Differential Geometry ◽  
Linear Algebra ◽  
Error Modeling ◽  
Geometric Errors ◽  
Dual Numbers ◽  
Algebraic Properties ◽  
Classes Of Matrices ◽  
Multiaxis Machines ◽  
Special Classes

Multiaxis machines error modeling is set in the context of modern differential geometry and linear algebra. We apply special classes of matrices over dual numbers and propose a generalization of such concept by means of general Weil algebras. We show that the classification of the geometric errors follows directly from the algebraic properties of the matrices over dual numbers and thus the calculus over the dual numbers is the proper tool for the methodology of multiaxis machines error modeling.

scholarly journals On Jordan Algebras and Some Unification Results

2020 ◽  
Author(s):  
Florin Nichita
Keyword(s):  
Differential Geometry ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
International Workshop ◽  
Jordan Algebras ◽  
Baxter Equation ◽  
Associative Algebras ◽  
Associative Structures

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxter equation and Yang-Baxter systems, with the goal to unify constructions from Differential Geometry.

scholarly journals Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

2020 ◽  
Vol 51 (1) ◽  
Author(s):  
Luiz C. B. Da Silva
Keyword(s):  
Differential Geometry ◽  
Linear Equation ◽  
Frenet Frame ◽  
Dual Numbers ◽  
Trigonometric Functions ◽  
Imaginary Unit ◽  
Isotropic Space ◽  
Klein Geometry ◽  
Frame Motion ◽  
Plane At Infinity

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

The Synthesis of Cam-Oscillating Roller-Follower Mechanisms: A Unified Approach

1992 ◽  
Author(s):  
Max Antonio González-Palacios ◽  
Jorge Angeles
Keyword(s):  
Differential Geometry ◽  
Contact Line ◽  
Contact Surface ◽  
Dual Numbers ◽  
Sliding Velocity ◽  
Unified Approach ◽  
Pressure Angle

Abstract A unified approach to the synthesis of the contact surface of cam-oscillating roller-follower mechanisms is presented. This is achieved using dual numbers. The surfaces of both cam and roller are obtained so that, at their contact line, a minimum-magnitude sliding velocity is produced. The differential geometry of the pitch surface and the pressure angle are both analyzed.

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