scholarly journals Some results on dual third-order Jacobsthal quaternions

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1865-1876 ◽  
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Generating Functions ◽  
Lucas Numbers ◽  
Third Order ◽  
The Third

Dual Fibonacci and dual Lucas numbers are defined with dual Fibonacci and Lucas quaternions in Nurkan and G?ven [14]. In this study, we define the dual third-order Jacobsthal quaternion and the dual third-order Jacobsthal-Lucas quaternion. We derive the relations between the dual third-order Jacobsthal quaternion and dual third-order Jacobsthal-Lucas quaternion which connected the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers. In addition, we give the generating functions, the Binet and Cassini formulas for these new types of quaternions.

scholarly journals De Moivre-type identities for the Jacobsthal numbers

2021 ◽  
Vol 27 (3) ◽  
pp. 95-103
Author(s):  
Mücahit Akbiyik ◽  
◽  
Seda Yamaç Akbiyik ◽  
Keyword(s):  
Second Order ◽  
Lucas Numbers ◽  
Third Order ◽  
The Third

The main aim of this study is to obtain De Moivre-type identities for Jacobsthal numbers. Also, this paper presents a method for constructing the second order Jacobsthal and Jacobsthal third-order numbers and the third-order Jacobsthal and Jacobsthal–Lucas numbers. Moreover, we give some interesting identities, such as Binet’s formulas for some specific third-order Jacobsthal numbers that we derive from De Moivre-type identities.

scholarly journals Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences

2021 ◽  
Vol 53 ◽  
Author(s):  
Can Kızılateş ◽  
Wei-Shih Du ◽  
Feng Qi
Keyword(s):  
Lucas Numbers ◽  
Third Order ◽  
Explicit Formulas ◽  
The Third ◽  
Narayana Numbers ◽  
Tribonacci Numbers

In the paper, the authors present several explicit formulas for the $(p,q,r)$-Tribonacci polynomials and generalized Tribonacci sequences in terms of the Hessenberg determinants and, consequently, derive several explicit formulas for the Tribonacci numbers and polynomials, the Tribonacci--Lucas numbers, the Perrin numbers, the Padovan (Cordonnier) numbers, the Van der Laan numbers, the Narayana numbers, the third order Jacobsthal numbers, and the third order Jacobsthal--Lucas numbers in terms of special Hessenberg determinants.

scholarly journals The Third Order Jacobsthal Octonions: Some Combinatorial Properties

2018 ◽  
Vol 26 (3) ◽  
pp. 57-72 ◽  
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Lucas Numbers ◽  
Third Order ◽  
Combinatorial Properties ◽  
The Third ◽  
Number Sequences ◽  
Addition Formulas ◽  
Number Of Authors

AbstractVarious families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many di erent ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the third order Jacobsthal and third order Jacobsthal-Lucas numbers. We introduce the third order Jacobsthal octonions and the third order Jacobsthal-Lucas octonions and give some of their properties. We derive the relations between third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions.

scholarly journals On Third-Order Bronze Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2606
Author(s):  
Mücahit Akbiyik ◽  
Jeta Alo
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Matrix Representations ◽  
Third Order ◽  
The Third ◽  
Encryption And Decryption ◽  
Fibonacci Sequences

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

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