scholarly journals Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 668
Author(s):  
Ji Eun Kim
Keyword(s):  
Real Function ◽  
Complex Number ◽  
Elementary Function ◽  
Complex Function ◽  
Elementary Functions ◽  
Analytical Studies ◽  
Definition Of ◽  
Derivatives Of

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

scholarly journals Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 206
Author(s):  
Ji-Eun Kim
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Complex Number ◽  
Complex Function ◽  
Taylor Series Expansion ◽  
Complex Numbers ◽  
Definition Of ◽  
Generalized Quaternion ◽  
Complex Valued ◽  
Complex Basis

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

scholarly journals Some remarks on incomplete gamma type function γ*(α, x_)

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Emin Özcağ ◽  
İnci Egeb
Keyword(s):  
Recurrence Relation ◽  
Real Line ◽  
Summable Function ◽  
Type Function ◽  
The Real ◽  
Definition Of ◽  
Derivatives Of

The incomplete gamma type function ?*(?, x_) is defined as locally summable function on the real line for ?>0 by ?*(?,x_) = {?x0 |u|?-1 e-u du, x?0; 0, x > 0 = ?-x_0 |u|?-1 e-u du the integral divergining ? ? 0 and by using the recurrence relation ?*(? + 1,x_) = -??*(?,x_) - x?_ e-x the definition of ?*(?, x_) can be extended to the negative non-integer values of ?. Recently the authors [8] defined ?*(-m, x_) for m = 0, 1, 2,... . In this paper we define the derivatives of the incomplete gamma type function ?*(?, x_) as a distribution for all ? < 0.

Identification of principal screws of two- and three-screw systems

2007 ◽  
Vol 221 (12) ◽  
pp. 1701-1715 ◽  
Author(s):  
J-S Zhao ◽  
F Chu ◽  
Z-J Feng
Keyword(s):  
Screw Theory ◽  
Coordinate Systems ◽  
Third Order ◽  
Analytical Methodology ◽  
Partial Derivatives ◽  
Spatial Mechanisms ◽  
Analysis Theory ◽  
Definition Of ◽  
Derivatives Of ◽  
Homogeneous Linear

The current paper proposes a unified analytical methodology to identify the principal screws of two- and three-screw systems. Based on the definition of the pitch of a screw, it first obtains an identical homogeneous quadric equation. According to functional analysis theory, it is known that the partial derivatives of an identical quadric equation with respect to its variables must be zero. Therefore, the paper deduces a set of linear homogeneous equations that are made up of the partial derivatives of the quadric equation. With the existing criteria of non-zero solutions for homogeneous linear algebra equations, it ultimately obtains the formulas of the principal pitches and the associated principal screws of the system. The most outstanding contribution of this methodology is that it proposes a unified analytical approach to identify the principal pitches and the principal coordinate systems of the second-order and the third-order screw systems. This should be a new contribution to the screw theory and will boost its applications to the kinematics analysis of robots and spatial mechanisms.

Gleason Parts of Real Function Algebras

1981 ◽  
Vol 33 (1) ◽  
pp. 181-200 ◽  
Author(s):  
S. H. Kulkarni ◽  
B. V. Limaye
Keyword(s):  
Real Function ◽  
Banach Algebras ◽  
Complex Function ◽  
Klein Surfaces ◽  
The Real ◽  
Uniform Algebras ◽  
Function Algebras ◽  
Complex Valued ◽  
Systematic Exposition

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.

A Remark on Disintegrations With Almost All Components Non -Additive

1979 ◽  
Vol 31 (4) ◽  
pp. 786-788 ◽  
Author(s):  
Nghiem Dang-Ngoc
Keyword(s):  
Real Function ◽  
Measure Space ◽  
Gibbs States ◽  
List Type ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Coin Tossing ◽  
Definition Of ◽  
Almost All

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:(i) ∀x ∊ X , σx(.) is a finitely additive measure .(ii) ∀A ∊ , σ. (A) is constant on each -atom.(iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and (iv)σx(B) = 1 if x ∊ B ∊ .

On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials

2020 ◽  
pp. 2150100
Author(s):  
Engi̇n Özkan ◽  
Bahar Kuloğlu
Keyword(s):  
Hankel Transform ◽  
Pell Numbers ◽  
Pascal’S Triangle ◽  
Narayana Numbers ◽  
Definition Of ◽  
Pascal's Triangle ◽  
The Relationship ◽  
Derivatives Of

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal’s triangle. We also explain the relationship between the new Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.

Finding Complex Roots: Can You Trust Your Calculator?

2006 ◽  
Vol 99 (5) ◽  
pp. 366-371
Author(s):  
John W. Watson ◽  
Barbara A. Ciesia
Keyword(s):  
Complex Number ◽  
Specific Instance ◽  
Complex Roots ◽  
Complex Power ◽  
Arbitrary Complex ◽  
The Difference ◽  
Definition Of

This article investigates a specific instance when the textbook answer for finding a root of a complex number differed with the answer given by the TI-83. After explaining the reason for the difference the article then expands the definition of the integral root of a complex number to an arbitrary complex power of a complex number. Read now to see where false assumptions might be made based on the results of a calculator and see explanations of how to overcome those assumptions with logic and proof.

scholarly journals 486. The Definition of a Complex Number

1916 ◽  
Vol 8 (124) ◽  
pp. 305
Author(s):  
G. W. Palmer
Keyword(s):  
Complex Number ◽  
Definition Of

scholarly journals Ablative Acceleration of Foils, Their Pulsations, and Interchange Instability

1997 ◽  
Vol 15 (4) ◽  
pp. 495-506
Author(s):  
N.A. Inogamov
Keyword(s):  
Inertial Confinement Fusion ◽  
Inertial Confinement ◽  
Inhomogeneous Distribution ◽  
New Approach ◽  
Promising Tool ◽  
Current State ◽  
Confinement Fusion ◽  
Interchange Instability ◽  
Definition Of ◽  
Derivatives Of

The problem of hydrodynamic stability is important for inertial confinement fusion (ICF) systems based upon high compression of fuel before its ignition. This problem for the case of complicated multilayer foils has been studied here by a new approach describing the development of Rayleigh-Taylor or interchange instability in compressible media with inhomogeneous distribution of “entropy”s = ρ/ρk, ∂ where K = (∂ In ρ/∂ In ρ)s is an adiabatic derivative taken in the local hydrostatic values of ρ and ρ. Inhomogeneous distribution of s simulates the dynamics of development of perturbations of multilayer flyer foils and shells. Besides instability, the same approach has been used for analysis of ID pulsations of a levitated foil. The problem of pulsations is real in the case of foils. Indeed, (1) an ablative acceleration is equivalent to an effective gravity field, which causes the appearance of an atmospheric-type distribution of thermodynamic functions, (2) the duration of ablative flight of foil is at least several times larger than the time that is necessary for an acoustic wave to travel from one side of the foil to another side, and (3) there is a strong initial impulse that initiates the motion of foil. This impulse together with (1, 2) is a reason for the powerful pulsations of foils. The period of pulsations is defined by the velocity of sound in the foil material, which is dependent on the derivatives of an equation of state (EOS). The check of the derivatives gives us finer information concerning the current state of matter and the EOS than the usual measurements of material velocity and pressure that are rougher measures. Therefore, an analysis of pulsations seems to be a promising tool for tracking the dynamics of flyer foil and for the definition of thermodynamic properties of matter.

Sign in / Sign up

Export Citation Format

Share Document