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.

Refined First-Order Shear Deformation Theory Models for Composite Laminates

2003 ◽  
Vol 70 (3) ◽  
pp. 381-390 ◽  
Author(s):  
F. Auricchio ◽  
E. Sacco
Keyword(s):  
Shear Deformation ◽  
Analytical Solutions ◽  
Composite Laminates ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Quadratic Functions ◽  
Shear Stresses ◽  
Stress Profile ◽  
First Order ◽  
Out Of Plane

In the present work, new mixed variational formulations for a first-order shear deformation laminate theory are proposed. The out-of-plane stresses are considered as primary variables of the problem. In particular, the shear stress profile is represented either by independent piecewise quadratic functions in the thickness or by satisfying the three-dimensional equilibrium equations written in terms of midplane strains and curvatures. The developed formulations are characterized by several advantages: They do not require the use of shear correction factors as well as the out-of-plane shear stresses can be derived without post-processing procedures. Some numerical applications are presented in order to verify the effectiveness of the proposed formulations. In particular, analytical solutions obtained using the developed models are compared with the exact three-dimensional solution, with other classical laminate analytical solutions and with finite element results. Finally, we note that the proposed formulations may represent a rational base for the development of effective finite elements for composite laminates.

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.

ITERATION OF QUATERNION MAPS

1995 ◽  
Vol 05 (03) ◽  
pp. 877-881 ◽  
Author(s):  
STEPHEN BEDDING ◽  
KEITH BRIGGS
Keyword(s):  
Mandelbrot Set ◽  
Quadratic Functions ◽  
Complex Case ◽  
Complex Numbers ◽  
Four Dimensions

Quaternions are an extension of the idea of complex numbers to four dimensions. We discuss the iteration of linear and quadratic functions of the quaternions, and examine the rôle played by regularity (the analog of complex analyticity), in this context. In contrast to the complex case, regularity is not automatically preserved by iteration of quaternion functions. We find that demanding preservation of regularity is too restrictive, yielding very little new beyond the complex case. The quaternion generalisation of the Mandelbrot set is described.

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 LORENZ-LIKE DYNAMIC SYSTEMS WITH STRENGTHENED NONLINEARITY AND NEW PARAMETERS

2010 ◽  
Vol 08 (02) ◽  
pp. 293-311 ◽  
Author(s):  
BO LIAO ◽  
YUAN YAN TANG ◽  
LU AN
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Dynamic Systems ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Maximum Lyapunov Exponent ◽  
Preliminary Results ◽  
Quadratic Equations

This paper introduces two types of Lorenz-like three-dimensional quadratic autonomous chaotic systems with 7 and 8 new parameters free of choice, respectively. Both systems are investigated at the equilibriums to study their chaotic characteristics. We focus our attention on the second type of the introduced system which consists of three nonlinear quadratic equations. Predictably, coordinates of the equilibriums are prohibitively complex. Therefore, instead of directly analyzing their stability, we prove the asymptotical characterization of equilibriums by utilizing our preliminary results derived for the first type of system. Our result shows that, though the coordinates of equilibriums satisfy a ternary quadratic, the system still contains only three equilibriums in circumstances of chaos. Sufficient conditions for the chaotic appearance of systems are derived. Our results are further verified by numerical simulations and the maximum Lyapunov exponent for several examples. Our research takes a first step in investigating chaos in Lorenz-like dynamic systems with strengthened nonlinearity and general forms of parameters.

scholarly journals Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 340
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Diagonal Matrix ◽  
Invariance Property ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Euler’S Formula ◽  
Probability Densities

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Activities: A Graphical Approach to the Quadratic Formula

1996 ◽  
Vol 89 (1) ◽  
pp. 34-46
Keyword(s):  
Quadratic Functions ◽  
Quadratic Formula ◽  
Graphical Approach ◽  
Quadratic Equations

Introduction: Traditionally, the solution of quadratic equations has been taught before, and in isolation from, the study of quadratic functions. The quadratic formula itself has typically been derived by completing the square. Many teachers skip the derivation, and most students who see it do not fully understand it.

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