scholarly journals HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials

10.37236/5324 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Kemal Taşköprü ◽  
İsmet Altıntaş
Keyword(s):  
Homfly Polynomial ◽  
Matrix Representations ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Fundamental Properties ◽  
Conway Polynomial ◽  
The Matrix ◽  
Fibonacci Polynomial ◽  
Homfly Polynomials ◽  
Torus Links

The focus of this paper is to study the HOMFLY polynomial of $(2,n)$-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of $ (2,n) $-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental properties. We also show that the HOMFLY polynomial of $ (2,n) $-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial. We finally give the matrix representations and prove important identities, which are similar to the Fibonacci identities, for the our generalized Fibonacci polynomial and the HOMFLY polynomial of $ (2,n) $-torus link.

scholarly journals Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix

2021 ◽  
Vol 13 (2) ◽  
pp. 21
Author(s):  
Chung-Chuan Chen ◽  
Lin-Ling Huang
Keyword(s):  
Fibonacci Polynomials ◽  
New Approach ◽  
Lucas Polynomials ◽  
Type Formula ◽  
Type Identity ◽  
Fibonacci Polynomial

We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix. These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which generalize the previous results in references.

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

scholarly journals EXTENSION OF THE POINCARÉ GROUP AND NON-ABELIAN TENSOR GAUGE FIELDS

2010 ◽  
Vol 25 (31) ◽  
pp. 5765-5785 ◽  
Author(s):  
GEORGE SAVVIDY
Keyword(s):  
Gauge Group ◽  
Gauge Transformation ◽  
Space Time ◽  
Gauge Fields ◽  
Poincaré Group ◽  
Matrix Representations ◽  
Poincare Group ◽  
Yang Mills ◽  
The Matrix ◽  
Transversal Plane

In the recently proposed generalization of the Yang–Mills theory, the group of gauge transformation gets essentially enlarged. This enlargement involves a mixture of the internal and space–time symmetries. The resulting group is an extension of the Poincaré group with infinitely many generators which carry internal and space–time indices. The matrix representations of the extended Poincaré generators are expressible in terms of Pauli–Lubanski vector in one case and in terms of its invariant derivative in another. In the later case the generators of the gauge group are transversal to the momentum and are projecting the non-Abelian tensor gauge fields into the transversal plane, keeping only their positively definite spacelike components.

scholarly journals Computing mu-values for Representations of Symmetric Groups in Engineering Systems

2018 ◽  
Vol 11 (3) ◽  
pp. 774-792
Author(s):  
Mutti-Ur Rehman ◽  
M. Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Symmetric Groups ◽  
Singular Values ◽  
Linear Control ◽  
Linear Control Systems ◽  
Complex Numbers ◽  
Engineering Systems ◽  
Matrix Representations ◽  
The Matrix

In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv.

scholarly journals Hashing Based Answer Selection

2020 ◽  
Vol 34 (05) ◽  
pp. 9330-9337
Author(s):  
Dong Xu ◽  
Wu-Jun Li
Keyword(s):  
Question Answering ◽  
Matrix Representation ◽  
Matrix Representations ◽  
Binary Matrix ◽  
Large Memory ◽  
The Matrix ◽  
Novel Method ◽  
Online Prediction ◽  
Memory Cost ◽  
Vector Representations

Answer selection is an important subtask of question answering (QA), in which deep models usually achieve better performance than non-deep models. Most deep models adopt question-answer interaction mechanisms, such as attention, to get vector representations for answers. When these interaction based deep models are deployed for online prediction, the representations of all answers need to be recalculated for each question. This procedure is time-consuming for deep models with complex encoders like BERT which usually have better accuracy than simple encoders. One possible solution is to store the matrix representation (encoder output) of each answer in memory to avoid recalculation. But this will bring large memory cost. In this paper, we propose a novel method, called hashing based answer selection (HAS), to tackle this problem. HAS adopts a hashing strategy to learn a binary matrix representation for each answer, which can dramatically reduce the memory cost for storing the matrix representations of answers. Hence, HAS can adopt complex encoders like BERT in the model, but the online prediction of HAS is still fast with a low memory cost. Experimental results on three popular answer selection datasets show that HAS can outperform existing models to achieve state-of-the-art performance.

scholarly journals Construction of generalized rotations and quasi-orthogonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 107-113
Author(s):  
Luis Verde-Star
Keyword(s):  
Image Processing ◽  
Hadamard Matrices ◽  
Complex Numbers ◽  
Data Communications ◽  
Matrix Representations ◽  
Orthogonal Matrices ◽  
Orthogonal Designs ◽  
The Matrix ◽  
The One

Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

THE HOMFLY POLYNOMIAL FOR TORUS LINKS FROM CHERN–SIMONS GAUGE THEORY

1995 ◽  
Vol 10 (07) ◽  
pp. 1045-1089 ◽  
Author(s):  
J. M. F. LABASTIDA ◽  
M. MARIÑO
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Homfly Polynomial ◽  
Fundamental Representation ◽  
Torus Knots ◽  
Polynomial Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Torus Links

Polynomial invariants corresponding to the fundamental representation of the gauge group SU(N) are computed for arbitrary torus knots and links in the framework of Chern–Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

Sign in / Sign up

Export Citation Format

Share Document