THREE-DIMENSIONAL FRACTALS FROM QUATERNIONS AND DOUBLE COMPLEX NUMBERS

2020 ◽  
pp. 161-228
Keyword(s):  
Three Dimensional ◽  
Complex Numbers ◽  
Double Complex

Model Solutions to Quadratic Equations

1986 ◽  
Vol 79 (5) ◽  
pp. 332-336
Author(s):  
Alastair McNaughton
Keyword(s):  
Three Dimensional ◽  
Quadratic Functions ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Model Solutions

Here is a method of representing quadratic functions by three-dimensional wire models. It enables one to form a simple geometric concept of the location of the imaginary zeros. I have been using this material with my students and have been delighted with the ease with which they respond to it. As a result, their confidence in dealing with complex numbers has increased, their concept of functions has shown much improvement, and they are attacking problems with real insight.

scholarly journals Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

2018 ◽  
Vol 10 (6) ◽  
pp. 91
Author(s):  
Harry Wiggins ◽  
Ansie Harding ◽  
Johann Engelbrecht
Keyword(s):  
Complex Function ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Complex Polynomials ◽  
Singular Case ◽  
Hyperbolic Paraboloid ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Coefficients ◽  
Roots Of A Polynomial

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called "imaginary" roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.  We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

scholarly journals On Complex Numbers in Higher Dimensions

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 22
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Probability Distributions ◽  
Three Dimensional ◽  
Geometric Approach ◽  
Higher Dimensions ◽  
Space Structure ◽  
Complex Numbers ◽  
Vector Product ◽  
Multiplication Rule ◽  
Generalized Complex ◽  
General Vector

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

Surface Vibration of a Multilayered Elastic Medium due to Harmonic Concentrated Force

2011 ◽  
Author(s):  
Akindeji Ojetola ◽  
Hamid Hamidzadeh
Keyword(s):  
Fourier Transform ◽  
Elastic Medium ◽  
Three Dimensional ◽  
Concentrated Force ◽  
Double Complex ◽  
Concentrated Load ◽  
Propagator Matrix ◽  
Displacement Vectors ◽  
Bottom Interface ◽  
Similar Information

Dynamic response of a multi-layer elastic medium subjected to harmonic surface concentrated load is considered. In development of the analytical solution, the three-dimensional theory of elasto-dynamic is utilized for derivation of the governing partial differential equations for each layer. These equations are solved in the Fourier domain by employing the Double Complex Fourier Transform technique. In the analysis, each layer of the medium is assumed to be extended infinitely in the horizontal x and z directions and has uniform depth in the y direction and is considered to be linearly elastic, homogeneous, and isotropic. Utilizing the Integral Fourier Transform, displacements and stresses at any point in each layer can be determined in terms of boundary stresses for each layer. Also, the presented solution provides the relation between stress and displacement vectors for the top and bottom of each layer in matrix notation. By satisfying the compatibility of displacements and stresses for each interface, a propagator matrix relating displacements and stresses at the top of the medium to the bottom interface will be obtained. This relates the displacement and stress vector on the top surface to the bottom interface by eliminating similar information for the other interfaces. In this study, the displacements on the surface of the layered medium are computed for the two cases where the surface of the medium is subjected to a concentrated harmonic vertical or horizontal harmonic force.

The classification of three-dimensional Lie algeb­ras on complex field

2020 ◽  
pp. 143-155
Author(s):  
E. R. Shamardina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Structural Equations ◽  
Complex Numbers ◽  
Complex Field ◽  
Low Dimensional

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

On S-wave directivity patterns

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 822-825 ◽  
Author(s):  
Walter L. Pilant
Keyword(s):  
Free Surface ◽  
Plane Wave ◽  
Three Dimensional ◽  
Complex Numbers ◽  
S Wave ◽  
S Waves ◽  
Time Harmonic ◽  
Source Case ◽  
Directivity Patterns ◽  
Impulsive Source

Plane‐wave directivity patterns for both P- and S-waves approaching a free surface are well known (Knopoff et al., 1957, Figure 3–5). These have been shown to apply in a reciprocal manner to time‐harmonic S-waves emanating from vertical and horizontal sources (Miller and Pursey, 1954; Cherry, 1962) in both two‐dimensional (2-D) and three‐dimensional (3-D) cases. Knopoff and Gilbert (1959) showed that the plane‐wave directivity patterns also apply to the first motions seen in the impulsive‐source case (3-D) and Pilant (1979, sec. 9–6) showed that they held in the equivalent 2-D problem. Theoretical expressions for these patterns are given by Pilant (ibid) as [Formula: see text] and [Formula: see text] where [Formula: see text] is measured from the vertical and the positive z-axis is into the medium. The x-axis lies along the free surface and the quantity [Formula: see text]. For angles greater than critical [Formula: see text], the proper expression for the square root is given by [Formula: see text] Thus for angles of incidence (or take‐off) greater than [Formula: see text], both [Formula: see text] and [Formula: see text] become complex numbers and lead to phase‐shift induced waveform changes as the S-waves interact with the free surface. The functions [Formula: see text] and [Formula: see text] are shown in Figure 1 for the angular range 34–37 degrees which includes the angle [Formula: see text] degrees. For this example, [Formula: see text] corresponding to a Poisson’s ratio equal to one‐quarter. The null in [Formula: see text] and the maximum in [Formula: see text] are clearly seen.

The Inversive Distance Between Two Circles

1967 ◽  
Vol 19 ◽  
pp. 1149-1152
Author(s):  
O. Bottema
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Single Point ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Inversive Geometry ◽  
The One ◽  
Three Dimensional Space

H. S. M. Coxeter (3) has recently studied the correspondence between two geometries the isomorphism of which was well known, but to which he was able to add some remarkable consequences. The two geometries are the inversive geometry of a plane E (the Euclidean plane completed with a single point at infinity or, what is the same thing, the plane of complex numbers to which ∞ is added) on the one hand, and the hyperbolic geometry of three-dimensional space S.

AN EXTENDED ŠIL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS

2009 ◽  
Vol 19 (05) ◽  
pp. 1679-1693 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU ◽  
GUANRONG CHEN
Keyword(s):  
Theoretical Analysis ◽  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
The Other ◽  
Complex Numbers ◽  
New Chaotic System ◽  
Focus Type

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

scholarly journals On logarithmic residue of monogenic functions in a three-dimensional commutative algebra with one-dimensional radical

2017 ◽  
Vol 25 (3) ◽  
pp. 167-182
Author(s):  
Roman Pukhtaievych ◽  
Sergiy Plaksa
Keyword(s):  
Commutative Algebra ◽  
Three Dimensional ◽  
Singular Points ◽  
Complex Numbers ◽  
Monogenic Functions ◽  
One Dimensional ◽  
Hypercomplex Number ◽  
Real Subspace ◽  
Logarithmic Residue

Abstract We consider monogenic functions taking values in a three-dimensional commutative algebra A2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A2 into A2. It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A2, and, in general case, is a hypercomplex number.

scholarly journals Probabilistic theories and reconstructions of quantum theory

2021 ◽  
Author(s):  
Markus Müller
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Classical Probability ◽  
Quantum Bit ◽  
Classical Probability Theory ◽  
Generalized Probabilistic Theories ◽  
Usual Representation ◽  
Probabilistic Theories

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

Sign in / Sign up

Export Citation Format

Share Document