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 Fractional Newton-Raphson Method

2021 ◽  
Vol 8 (1) ◽  
pp. 1-13
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz
Keyword(s):  
Fractional Calculus ◽  
Iterative Method ◽  
Complex Numbers ◽  
Initial Condition ◽  
The Real ◽  
Complex Roots ◽  
Newton Raphson ◽  
Roots Of A Polynomial ◽  
Raphson Method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ N. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

scholarly journals A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 317
Author(s):  
Diogo Freitas ◽  
Luiz Guerreiro Lopes ◽  
Fernando Morgado-Dias
Keyword(s):  
Iterative Methods ◽  
Fundamental Problem ◽  
Iterative Algorithms ◽  
Rounding Errors ◽  
Suggested Approach ◽  
Simultaneous Iterative Methods ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Comparative Results ◽  
The One

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

Locating complex eigenvalues for analytical eddy-current models used to detect flaws

2019 ◽  
Vol 38 (6) ◽  
pp. 1800-1809
Author(s):  
Grzegorz Tytko ◽  
Łukasz Dawidowski
Keyword(s):  
Eddy Current ◽  
Design Methodology ◽  
Complex Function ◽  
Precise Determination ◽  
Argument Principle ◽  
Content Type ◽  
Conductive Material ◽  
Complex Roots ◽  
Complex Eigenvalues

Purpose Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function. Design/methodology/approach The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them. Findings The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time. Originality/value The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

Complex Roots of a Polynomial: 10688

2000 ◽  
Vol 107 (2) ◽  
pp. 181
Author(s):  
Ioan Tomescu ◽  
Kee-Wai Lau ◽  
O. P. Lossers ◽  
K. F. Andersen ◽  
R. J. Chapman ◽  
...  
Keyword(s):  
Complex Roots ◽  
Roots Of A Polynomial

scholarly journals Hemp Cables, a Sustainable Alternative to Harmonic Steel for Cable Nets

Resources ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 70 ◽  
Author(s):  
Alberto Viskovic
Keyword(s):  
Three Dimensional ◽  
Structural Response ◽  
Building Design ◽  
Structural Elements ◽  
Nonlinear Analyses ◽  
Hyperbolic Paraboloid ◽  
Materials Engineering ◽  
Recent Developments ◽  
Vertical Displacements ◽  
Cable Nets

Recent developments in the field of materials engineering have allowed for the use of natural materials for common structural elements, instead of traditional materials, such as steel or concrete. In this context, hemp is a very interesting material for structural building design. This paper proposes the use of hemp cables for roofs with hyperbolic paraboloid cable nets, which sees the use of a sustainable material for structure, thus having a very low environmental impact, in terms of structural thickness and amount of material. The paper discusses five different plan sizes and two different hyperbolic paraboloid surface radius of curvatures. The cable traction, which gives the cable net stiffness, was varied in order to give a parametric database of structural response. Three dimensional geometrically nonlinear analyses were carried out on different geometries (i.e., 10), cable net stiffnesses (i.e., 8), and materials (i.e., 2). Traditional harmonic steel and hemp cables are compared, in terms of vertical displacements and natural periods under dead and permanent loads.

On the Singularities in Fracture and Contact Mechanics

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
Fazil Erdogan ◽  
Murat Ozturk
Keyword(s):  
Contact Mechanics ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Complex Function ◽  
Three Dimensional ◽  
Contact Problems ◽  
Cauchy Kernel ◽  
Square Root ◽  
Complex Function Theory ◽  
Graded Materials

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials.

Requirements for the Manufacturing of Scaffold Biomaterial With Features at Multiple Scales

2005 ◽  
Author(s):  
I. M. Sebastine ◽  
D. J. Williams
Keyword(s):  
Tissue Engineering ◽  
Multiple Scales ◽  
Cell Attachment ◽  
Structural Parameters ◽  
Complex Function ◽  
Three Dimensional ◽  
Cell Orientation ◽  
Length Scales

Tissue engineering aims to restore the complex function of diseased tissue using cells and scaffold materials. Tissue engineering scaffolds are three-dimensional (3D) structures that assist in the tissue engineering process by providing a site for cells to attach, proliferate, differentiate and secrete an extra-cellular matrix, eventually leading cells to form a neo-tissue of predetermined, three-dimensional shape and size. For a scaffold to function effectively, it must possess the optimum structural parameters conducive to the cellular activities that lead to tissue formation; these include cell penetration and migration into the scaffold, cell attachment onto the scaffold substrate, cell spreading and proliferation and cell orientation. In vivo, cells are organized in functional tissue units that repeat on the order of 100 μm. Fine scaffold features have been shown to provide control over attachment, migration and differentiation of cells. In order to design such 3D featured constructs effectively understanding the biological response of cells across length scales from nanometer to millimeter range is crucial. Scaffold biomaterials may need to be tailored at three different length scales: nanostructure (<1μm), microstructure (<20–100μm), and macrostructure (>100μm) to produce biocompatible and biofunctional scaffolds that closely resemble the extracellular matrix (ECM) of the natural tissue environment and promote cell adhesion, attachment, spreading, orientation, rate of movement, and activation. Identification of suitable fabrication techniques for manufacturing scaffolds with the required features at multiple scales is a significant challenge. This review highlights the effect and importance of the features of scaffolds that can influence the behaviour of cells/tissue at different length scales in vitro to increase our understanding of the requirements for the manufacture of functional 3D tissue constructs.

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