Quadratic Iteration Research Articles

Approximating roots by quadratic iteration

We apply a coctel of elementary methods to the problem of finding the roots of an arbitrary polynomial. Specifically, we combine properties of the iteration z → z2 + c with rudimentary Galois theory in order to justify an algorithm to find the roots of a complex polynomial.

Imaginary scators quadratic mapping in 1+2D dynamic space

The elliptic scator algebra quadratic iteration is evaluated in 1+2 dimensions in dynamic space. There exists a non divergent K set in the scator three dimensional space with a highly complex boundary J=∂K. Two and three dimensional renderings of the sets in dynamic space are published for the first time. The sets exhibit a rich fractal boundary in all three directions. Some of the salient features of the sets can be described in terms of square nilpotent iterations. The Julia and filled in Julia sets are identically reproduced at two perpendicular planes where only one non-vanishing hypercomplex director component is present. The fixed points of K in S1+2 with real constant c, can be obtained from the roots of a scator quadratic polynomial equation. In S1+2 there can be, in addition to the usual complex roots, hypercomplex roots that give rise to four additional fixed points. The inverse orbits of the hypercomplex roots reveal a rich complex structure, that allow for the evaluation of an infinite set of points in ∂K. The ∂K set of the origin is equal to the unit magnitude scator surface, named a cusphere. The J set exhibits self similar structures in 3D at different scales, typical of fractal phenomena. The ix set, is the three dimensional equivalent of the M-set in three dimensions. It is conjectured that the ix-set with some restrictions, is the set of parameters where the J set is connected.

Residual life prediction of a deep-rolled axle with flaws based on the extended finite element method

<p indent="0mm">Railway axles are subjected to many load cycles during an operation period, and the axle integrity may be destroyed, so railway axles risk fatigue failure. To prevent axle fatigue failure, the residual life of an axle with defects must be evaluated. The residual stress field of axle depth direction was reconstructed using SIGINI and the quadratic iteration method to study the fatigue crack propagation behavior of an EA4T alloy steel axle after deep rolling treatment. Simultaneously, the axle press-fitting process and the actual load spectrum of an axle in service were considered comprehensively, and the extended finite element method was adopted to calculate the stress intensity factor of an EA4T axle with an initial crack. The residual life of a full-size EA4T axle was predicted using the fracture mechanics evaluation method. The results show that the crack will not propagate when the crack depth direction is within the range of the residual stress layer. When the crack is located in the middle of the axle body, the critical failure sizes of the power axle and nonpower axle are 7 and <sc>9 mm,</sc> respectively, and the corresponding residual lives are 471300 and <sc>1013500 km,</sc> respectively. When the crack is located at the inner arc transition of the wheel, the critical failure sizes of the power and nonpower axles are 4.5 and <sc>5 mm,</sc> respectively. The results of this study can provide a reference for optimizing the maintenance interval of an EA4T axle after deep rolling treatment.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

FRACTALS WITH HYPERBOLIC SCATORS IN 1 + 2 DIMENSIONS

A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled [Formula: see text](s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.

An Intrinsically Three-Dimensional Fractal

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

A cellular model for spatial population dynamics

Interacting populations exhibit complex behavior in nature. Classic quadratic iteration models with two or three populations exhibit some of the features seen in nature, but fail to account for spatial variation. Indeed, the diversity paradox is that many classic population models predict one species dominates while nature exhibits diversity. While various schemes have been presented to address this dilemma, we present a simple, deterministic cellular model that incorporates classic iteration schemes and a spatial migration component that provides for self-organizing and rich behavior. Our visualization method allows us to observe dynamically changing predominance of species, global diversity, waves of species progression, and highly organized spiral structures.

&gt; Mandelbrot BB: Fractals and Chaos

Benoit Mandelbrot has produced a comprehensive, well-presented review of essential topics related to Mandelbrot set theory and applications. last part of the title The Mandelbrot set and beyond fully describes its potential allowing the reader to navigate through pictures, hard-to-find early papers and important and effective chapters on the historical background. All chapters are assembled in a way that the overall mix becomes a very well integrated source of know-how and knowledge bringing the readers into the Mandelbrot set world. spirit of the book is well summarized in a sentence on page 34: When seeking new insights, I look, look, look, and play with many pictures. (One picture is never enough). It is certainly true that in the last twenty years, mathematics has changed so deeply that to younger persons some chapter's stories might be simply incredible (p.36), as well, one should admit that after Mandelbrot's sets, initially describing trees, coastlines' shapes or allowing measuring the length of the Britain coast, and after the seminal book on The Fractal Geometry of Nature our way of looking at the world changed. Mandelbrot wrote: Why is geometry often described as 'cold' and 'dry'? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. I think our vision of the world, from the atom to the higher length scales, is still changing using those concepts clearly illustrated in the current Mandelbrot's book. Selected notes and papers make this book unique within the several books published on this topic. It is clear the touch of the author under all aspects: a touch of pure genius. There are five main topics dominating the book, namely: Quadratic iteration and its Mandelbrot set – Quadratic Julia and Mandelbrot sets; Nonquadratic iterations – Nonquadratic rational dynamics; Kleinian groups' limit set – Iterated nonlinear function systems and the fractal limit sets of Kleinian groups; Multifractal invariant measures – Exponentially vanishing multifractal measures; Background and History. Cumulative bibliography is impressive and well done. It is clearly pointed out, following the pathway through the book, how fractal geometry played an important role in offering a quantitative tool in several areas. Circumstances and facts are put together also to bring important lessons for young scientists. author made a serious and effective effort to realize a book that contains more than history, more than mathematics... it is a sort of ideal book for stimulating new ideas, new concepts, and new discoveries. So far, it is an excellent book also for supporting courses at University, PhD and Post doc level. Moreover, it is indispensable for scientists not only as a lesson of a pathway in science but also as an important source for science of tomorrow. This is a valuable reference source to researchers from these and related areas including bioengineering, biophysics, nanobiosciences and, of course, applied mathematics.

Jacobi identities, modular lattices, and modular towers

We give first a simple proof of a generalized Jacobi identity for n-dimensional odd diagonal lattices which specializes to the classical Jacobi identity for the lattice Z 2. For Z + ℓ Z , it recovers a one-parameter family of Jacobi identities discovered recently by Chan, Chua and Solé, used to deduce two quadratically converging algorithms for computing π corresponding to elliptic functions for the cubic and septic bases. Next, motivated by strongly modular lattices for the ten special levels ℓ, where σ 1(ℓ)∣24, we derive quadratic iterations in these ten special levels generalizing the cubic and septic cases. This also gives a uniform proof of the equations used by N.D. Elkies for 13 of his explicit modular towers. They correspond exactly to the case where all eta terms occur to the same power in his list. This provides a link between strongly modular lattices and modular towers.

Quadratic iterations to ${\pi }$ associated with elliptic functions to the cubic and septic base

In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.

A dynamical system for biological development: The case of Caenorhabditis elegans

We show how a simple nonlinear dynamical system (the discrete quadratic iteration on the unit segment) can be the basis for modelling the embryogenesis process. Such an approach, even though being crude, can nevertheless prove to be useful when looking with the two main involved processes: (i) on one hand the cell proliferation under successive divisions; (ii) on the other hand, the differentiation between cell lineages. We illustrate this new approach in the case of Caenrhabditis elegans by looking at the early stages of embryogenesis, up to several hundreds of cells ("lima bean" larval stage). We show how the many results that have been obtained by several groups can be interpreted in terms of values for the parameters controlling the dynamical system. Furthermore, we can extend the model to the cases of genetic mutations. More precisely the teratogenetic and lethal effects are associated with abnormal variation of teh control parameters with time.

A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration

We introduce a new three-stage process for calculating the zeros of a polynomial with real coefficients. The algorithm finds either a linear or quadratic factor, working completely in real arithmetic. In the third stage the algorithm uses one of two variable-shift iterations corresponding to the linear or quadratic case. The iteration for a linear factor is a real arithmetic version of the third stage of the algorithm for complex polynomials which we studied in an earlier paper. A new variable-shift iteration is introduced in this paper which is suitable for quadratic factors. If the complex algorithm and the new real algorithm are applied to the same real polynomial, then the real algorithm is about four times as fast. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. The problem and algorithm may be recast into matrix form. The third stage is a quadratic form of shifted inverse powering and a quadratic form of generalized Rayleigh iteration. The results of extensive testing are summarized. For an ALGOL W program run on an.IBM 360/67 we found that for polynomials ranging in degree from 20 to 50, the time required to calculate all zeros averaged $2n^2 $ milliseconds. An ALGOL 60 implementation of the algorithm and a program which calculates a posteriors bounds on the zeros may be found in Jenkins’ 1969 Stanford dissertation [2].