scholarly journals De Moivre’s and Euler Formulas for Matrices of Hybrid Numbers

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 213
Author(s):  
Mücahit Akbıyık ◽  
Seda Yamaç Akbıyık ◽  
Emel Karaca ◽  
Fatih Yılmaz
Keyword(s):  
Dual Numbers ◽  
Trigonometric Identities

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

Sir Robert Stawell Ball and methodologies of modern screw theory

2002 ◽  
Vol 216 (1) ◽  
pp. 1-11 ◽  
Author(s):  
H Lipkin ◽  
J Duffy
Keyword(s):  
Computational Geometry ◽  
Rigid Body ◽  
Lie Algebra ◽  
Screw Theory ◽  
Mechanical Design ◽  
Ball Screw ◽  
Dual Numbers ◽  
Body Dynamics ◽  
Rigid Body Mechanics ◽  
Multi Body

The theory of screws was largely developed by Sir Robert Stawell Ball over 100 years ago to investigate general problems in rigid body mechanics. Nowadays, screw theory is applied in many different but related forms including dual numbers, Plilcker coordinates and Lie algebra. An overview of these methodologies is presented along with a perspective on Ball. Screw theory has re-emerged after a hiatus to become an important tool in robot mechanics, mechanical design, computational geometry and multi-body dynamics.

scholarly journals Fully Parameterized Reduced Order Models Using Hyper-Dual Numbers and Component Mode Synthesis

2015 ◽  
Author(s):  
Matthew S. Bonney ◽  
Daniel C. Kammer ◽  
Matthew R. W. Brake
Keyword(s):  
Truncation Error ◽  
Sampling Methods ◽  
Latin Hypercube Sampling ◽  
Component Mode Synthesis ◽  
Reduced Order Models ◽  
Dual Numbers ◽  
Order Model ◽  
Full System ◽  
Reduced Order ◽  
Computational Burden

The uncertainty of a system is usually quantified with the use of sampling methods such as Monte-Carlo or Latin hypercube sampling. These sampling methods require many computations of the model and may include re-meshing. The re-solving and re-meshing of the model is a very large computational burden. One way to greatly reduce this computational burden is to use a parameterized reduced order model. This is a model that contains the sensitivities of the desired results with respect to changing parameters such as Young’s modulus. The typical method of computing these sensitivities is the use of finite difference technique that gives an approximation that is subject to truncation error and subtractive cancellation due to the precision of the computer. One way of eliminating this error is to use hyperdual numbers, which are able to generate exact sensitivities that are not subject to the precision of the computer. This paper uses the concept of hyper-dual numbers to parameterize a system that is composed of two substructures in the form of Craig-Bampton substructure representations, and combine them using component mode synthesis. The synthesis transformations using other techniques require the use of a nominal transformation while this approach allows for exact transformations when a perturbation is applied. This paper presents this technique for a planar motion frame and compares the use and accuracy of the approach against the true full system. This work lays the groundwork for performing component mode synthesis using hyper-dual numbers.

Trigonometric Identities

1974 ◽  
Vol 47 (4) ◽  
pp. 226-227
Author(s):  
Andy R. Magid
Keyword(s):  
Trigonometric Identities

scholarly journals Parameterized reduced order models from a single mesh using hyper-dual numbers

2016 ◽  
Vol 371 ◽  
pp. 370-392 ◽  
Author(s):  
M.R.W. Brake ◽  
J.A. Fike ◽  
S.D. Topping
Keyword(s):  
Reduced Order Models ◽  
Dual Numbers ◽  
Reduced Order
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