# complex

This document is an instruction manual for the complex analysis portion of the course on Mathematical Methods.

(Last version: April 20, 2001)

Harold V. McIntosh

This paper is available PS format (gzip compressed) (complex.ps.gz) Size: 19.2 MB.

# Funciones de doble periodo

Proyecto final de la materia Análisis Numérico II, realizado por Meneses Viveros Amilcar, Roldán Palacios Marisol y Zárate Vázquez Saul
(Last version: March 26, 2001)

This paper is available in Acrobat PDF (an2.pdf) Size: 13 MB.

# Julia Curves, Mandelbrot Set

Julia curves, and their relation to the Mandelbrot set, are explained.
(Last version: Feb 26, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (julia.pdf) Size: 1 MB.

# Aproximación de ez a través de un polinomio de Taylor

Sabemos que si una función es continua y derivable se puede aproximar por un polinomio de Taylor. Las preguntas que surgen son, ¿qué tan precisa es la aproximación y si el polinomio se comporta en forma parecida a la función que aproximan?. Se responde a estas preguntas estudiando la aproximación de ez a través de un polinomio de Taylor.
(Last version: Feb 17, 2001)

José Pedro Hernández Enríquez

This paper is available in Acrobat PDF (an1.pdf) Size: 950 KB.

# REC/C for Complex Arithmetic

REC/C is one of a series of specialized REC programs; in this case one which acts like a hand held graphing calculator for complex arithmetic.
(Last version: Feb 17, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (rec-c.pdf) Size: 1.7 MB.

# Contours for <PLOT>

Techniques used to plot contours in the package <PLOT> are described. Aside from its using traditional interpolation formulas, <PLOT> is characterized by taking pairs of triangles rather than rectangles for primitive domains, and for tracing out one entire contour before starting another, to minimize pen raising and lowering.
(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (contours.pdf) Size: 181 KB.

# Resonance in the Dirac Harmonic Oscillator

Detailed examination of the resonances in the Dirac harmonic oscillator, using results from the differential equation solver SERO.

(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (dirhosc.pdf.gz) Size: 9.3 MB.

# Periodic Potentials in One Dimension

As part of a collection of samples illustrating the use of the second order real differential equation solver SERO, some variants on the theme of periodic potentials are discussed. Constructing the stability chart for a periodic potential is one of the most important uses of the program, although details of specific solutions are also frequently called for.

(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (mathieu.pdf.gz) Size: 1.44 MB.

# REC-R for Second Order Real Differential Equations

REC-R is one of a series of specialized REC programs; in this case one which provides assistance in graphing the solutions of real, second order differential equations. It is therefore an adjunct to the program SERO which serves the same purpose.

(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (rec-r.pdf) Size: 835 KB.

# Representations of the magnetic Symmetry groups 1 2 3

In studying the symmetry groups of a crystal lattice in the presence of a magnetic field, the projective representations of these groups must be considered, because of the phase change in the wave functions which is due to the gauge transformations which accompany the geometric symmetries. These projective representations are ordinary representations of a group which is isomorphic to the generalized Dirac groups. The irreducible representations of the symmetry adapted functions of these groups can be obtained by means of techniques arising from the method of induced representations for those groups which can be exhibited as semi-direct products.

Para estudiar los grupos de simetría de una red cristalográfica en presencia de un campo magnético deben considerarse las representaciones proyectivas de dichos grupos, debido al cambio de fase que sufren las funciones de onda y que proviene de las transformaciones de norma que acompañan a las simetrías geométricas. Estas representaciones proyectivas son representaciones ordinarias de un grupo que es isomórfico a los grupos generalizados de Dirac. Las representaciones irreducibles y las funciones adaptadas a la simetría de dichos grupos se puede obtener mediante técnicas que resultan del método de representaciones inducidas para los grupos que se pueden exhibir como productos semi-directos.

(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (magnetic.pdf) Size: 118 KB.

# Quantization and a Green's Function for Systems of Linear Ordinary Differential Equations

One of the traditional puzzles for students of quantum mechanics is the reconciliation of the quantification principle that wave functions must be square integrable with the reality that continuum wave functions do not respect this requirement. By protesting that neither are they quantized the problem can be sidestepped, although some ingenuity may still be required to find suitable boundary conditions. Further difficulties await later on when particular systems are studied in more detail. Sometimes all the solutions, and not just some of them, are square integrable. Then it is necessary to resort to another principle, such as continuity or finiteness of the wave function, to achieve quantization. Examples where such steps have to be taken can be found both in the Schrödinger equation and the Dirac equation. The ground state of the hydrogen atom, the hydrogen atom in Minkowski space, the theta component of angular momentum, all pose problems for the Schrödinger equation. Finiteness of wavefunction alone is not a reliable principle because it fails in the radial equation of the Dirac hydrogen atom. Thus the quantizing conditions which have been invoked for one potential or another seen to be quite varied.
(Last version: Oct 21, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (green.pdf) Size: 145 KB.

# Quantization as an Eigenvalue Problem

(Last version: Oct 21, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (quant.pdf) Size: 1 MB.

# as it rules relates to the presence of gliders.

Recent correspondence (Fall, 1998) in LifeMail dealt with the possiblity of ``universal computation'' using Wolfram's (2,1) cellular automaton Rule 110. While awaiting further details participants in the list were referred to an eight page prospectus written by Matthew Cook cataloging the known gliders for the rule. Some of the commentary surrounding his introduction is reproduced and elaborated here, namely the results of the survey of the properties of Rule 110 carried out via the cellular automaton program NXLCAU21. Since then, we have played with glider collisions, and examined Rule 110 from the point of view of tiling the plane with isosceles right triangles.

Last version: May 31, 2000

Harold V. McIntosh

This paper is available in Acrobat PDF (rule110.pdf) Size: 24.5 MB.

# A concordance for rule 110

The Typical places where various triangles can be found in Rule 110.

Harold V. McIntosh .

This paper is available in Acrobat PDF (ccord.pdf) Size: 2.9 MB.

# summer99

The Ninth Reasearch Summer was held in Mexico during July and August, 1999. The participants in my portion of this enterprise prepared individual reports of their summer's work, which can be consulted elsewhere. So here we survey some of the background ideas, topics discussed, and mention a few of the results obtained.

Harold V. McIntosh

This paper is available in Acrobat PDF (summer99.pdf) Size: 2.7 MB.

# REC-F for Flexagons

REC-F is one of a series of specialized REC programs; in this case one which provides assistance in designing flexagons. Flexagons were popularized by Martin Gardner in Scientific American in the early 1960's. It is still possible to read his descriptions inasmuch as the original articles have since been reprinted in various collections of his articles. Basically, a flexagon is a fractal derived from a stack of polygons which can be folded back and forth, preferably coming to rest in a plane; so they resemble dragon curves and Lindenmayer L-systems. The purpose of REC-F is to draw the frieze of polygons which will eventually be folded up to make a given flexagon.
(Last version: Jan 29, 2001)

Harold V. McIntosh

This paper is available in Acrobat PDF (rec-f.pdf) Size: 145 KB.

# Octagonal Flexagons

Maps and cutouts for a variety of flexagons are presented, emphasizing those which can be cut out, mostly from single sheets of paper. Since TeX may not align front and back images, and in any event if cutting up the booklet is not desired, the .eps files can be printed directly to get sheets suitable for cutting. In the same spirit, only those sheets which are going to be used right away need be printed.

(Last version: Nov 18, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (octagon.pdf) Size: 193 KB.

# Heptagon Flexagons

Maps and cutouts for a variety of flexagons are presented, emphasizing those which can be cut out, mostly from single sheets of paper. Since TeX may not align front and back images, and in any event if cutting up the booklet is not desired, the .eps files can be printed directly to get sheets suitable for cutting. In the same spirit, only those sheets which are going to be used right away need be printed.

(Last version: Nov 18, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (heptagon.pdf) Size: 106 KB.

# Hexagon Flexagons

Maps and cutouts for a variety of flexagons are presented, emphasizing those which can be cut out, mostly from single sheets of paper. Since TeX may not align front and back images, and in any event if cutting up the booklet is not desired, the .eps files can be printed directly to get sheets suitable for cutting. In the same spirit, only those sheets which are going to be used right away need be printed.

(Last version: Oct 31, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (hexagon.pdf) Size: 186 KB.

# Tetragonal Flexagons

Maps and cutouts for a variety of flexagons are presented, emphasizing those which can be cut out, mostly from single sheets of paper. Since TeX may not align front and back images, and in any event if cutting up the booklet is not desired, the .eps files can be printed directly to get sheets suitable for cutting. In the same spirit, only those sheets which are going to be used right away need be printed.

(Last version: Oct 31, 2000)

Harold V. McIntosh

This paper is available in Acrobat PDF (tetragon.pdf) Size: 200 KB.