scholarly journals Fusion categories via string diagrams

2016 ◽  
Vol 18 (05) ◽  
pp. 1550080 ◽  
Author(s):  
Bruce Bartlett
Keyword(s):  
Complex Numbers ◽  
Algebraic Equations ◽  
Fusion Categories ◽  
String Diagram

We use the string diagram calculus to give graphical proofs of the basic results of Etingof, Nikshych and Ostrik [On fusion categories, Ann. Math. 162 (2005) 581–642; arXiv:math/0203060, doi:10.4007/annals.2005.162.581] on fusion categories. These results include: the quadruple dual is canonically isomorphic to the identity, positivity of the paired dimensions, and Ocneanu rigidity. We introduce the pairing convention as a convenient graphical framework for working with fusion categories. We use this framework to express the pivotal operators as a product of the apex associator monodromy and the pivotal indicators. We also characterize pivotal structures as solutions of an explicit set of algebraic equations over the complex numbers, refining a formula of Wang.

THE CASIMIR NUMBER AND THE DETERMINANT OF A FUSION CATEGORY

2020 ◽  
pp. 1-13
Author(s):  
ZHIHUA WANG ◽  
GONGXIANG LIU ◽  
LIBIN LI
Keyword(s):  
Fusion Category ◽  
Complex Numbers ◽  
Closed Field ◽  
Arbitrary Characteristic ◽  
Fusion Categories ◽  
Cauchy Theorem ◽  
Prime Factors ◽  
Numerical Invariants ◽  
Characteristic Two ◽  
Positive Integers

Abstract Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

scholarly journals How to solve third degree equations without moving to complex numbers

2020 ◽  
Vol 12 ◽  
pp. 123-131
Author(s):  
Antoni Leon Dawidowicz
Keyword(s):  
High School ◽  
16Th Century ◽  
Complex Numbers ◽  
Algebraic Equations ◽  
Current Version ◽  
Elementary Functions ◽  
Real Numbers ◽  
Fourth Degree ◽  
The Third ◽  
Algebraic Operations

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

Linear Computations over a Complex Field

1958 ◽  
Vol 62 (570) ◽  
pp. 451-455
Author(s):  
Josef Schmidtmayer
Keyword(s):  
Linear Algebra ◽  
Complex Numbers ◽  
Complex Field ◽  
Algebraic Equations ◽  
Effective Form ◽  
Linear Algebraic Equations ◽  
Linear Problems ◽  
Complex Coefficients

Two methods are given concerning the following problems of linear algebra over the field of complex numbers (or, less rigorously, linear problems with complex coefficients): the solving of a system of linear algebraic equations, the inversion of a matrix and the evaluation of a determinant. The second method is especially suitable for use with computers. In addition to the usual numerical checking, the second method also provides an effective form check.

In Favor of Imaginaries in Geometry

2020 ◽  
Vol 8 (2) ◽  
pp. 33-40
Author(s):  
A. Girsh
Keyword(s):  
Double Point ◽  
Complex Numbers ◽  
Open Door ◽  
Algebraic Equations ◽  
Real Numbers ◽  
Geometric Figures ◽  
Human Thought ◽  
Special Cases ◽  
Good For

“Complex numbers are something complicated”, as they are perceived in most cases. The expression “real numbers are also complex numbers” sounds strange as well. And for all that complex numbers are good for many areas of knowledge, since they allow solve problems, that are not solved in the field of real numbers. First and most important is that in the field of complex numbers all algebraic equations are solved, including the equation x2 + a = 0, which has long been a challenge to human thought. In the field of complex numbers, the problem solutions remain free from listing special cases in the form of "if ... then", for example, solving the problem for the intersection of the line g with the circle (O, r) always gives two points. And in the field of real numbers, three cases have to be distinguished: | Og | <r → there are two real points; | Og |> r → there is no intersection; | Og | = r → there is one double point. The benefit of complex numbers also lies in the fact that with their help not only problems that previously had no solutions are solved, they not only greatly simplify the solution result, but they also hold shown in this text further amazing properties in geometric figures, and open door to the amazing and colorful world of fractals.

scholarly journals NÚMEROS COMPLEXOS E REGIÕES NO PLANO

2018 ◽  
Vol 10 (Especial) ◽  
pp. 01-05
Author(s):  
Mariana Laura da Cruz da Costa ◽  
Antônio Carlos Tamarozzi
Keyword(s):  
Middle School Students ◽  
Education Program ◽  
Mathematics Teaching ◽  
Elementary Theory ◽  
Complex Numbers ◽  
Algebraic Equations ◽  
School Students ◽  
Graphic Display ◽  
Teaching Methodologies ◽  
Teaching Learning

This work results from the research practices of Mathematics teaching methodologies developed by the Group PETMAT-Tutorial education program UFMS, Três Lagoas-MS campus, where it is presented a proposal of application of complex numbers for conical and other parts of the plan cartesian for middle school students. With the development of this line of work, we intend to facilitate the obtaining of the algebraic equations of some curves and parts of the plan, as well as the corresponding graphic display. Are used concepts and techniques as real and imaginary parts, conjugates , modular and other, which are part of the elementary theory of complex numbers and, therefore, accessible to students. The work emphasizes the teaching-learning process in which mathematics is used as contextualization for the development of their own topics.

Complex numbers

2006 ◽  
Author(s):  
Stephen C. Roy
Keyword(s):  
Complex Numbers

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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