Mathematica Journal
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From Discrete to Continuous Spectra
https://doi.org/10.3888/tmj.213 We study the distribution of eigenspectra for operators of the form with selfadjoint boundary conditions on both bounded and unbounded interval domains. With integrable potentials , we explore computational methods for calculating spectral density functions involving cases of discrete and continuous spectra where discrete eigenvalue distributions approach a continuous limit as the domain becomes […] 
The Arithmetic of Points on a Conic and Projectivities
https://doi.org/10.3888/tmj.212 H. S. M. Coxeter wrote several geometry film scripts that were produced between 1965 and 1971 [1]. In 1992, Coxeter gave George Beck mimeographs of two scripts that had not been made. Beck wrote Mathematica code for the stills and animations. This material was added to the third edition of Coxeter’s The Real Projective […] 
Peak Response of SingleDegreeofFreedom Systems to SweptFrequency Excitation
https://doi.org/10.3888/tmj.211 A comprehensive discussion is presented of the closedform solutions for the responses of singledegreeoffreedom systems subject to sweptfrequency harmonic excitation. The closedform solutions for linear and octave sweptfrequency excitation are presented and these are compared to results obtained by direct numerical integration of the equations of motion. Included is an indepth discussion of the […] 
PseudoDynamic Approach to the Numerical Solution of Nonlinear Stationary Partial Differential Equations
dx.doi.org/doi:10.3888/tmj.208 This article presents a numerical pseudodynamic approach to solve a nonlinear stationary partial differential equation (PDE) with bifurcations by passing from to a pseudotimedependent PDE . The equation is constructed so that the desired nontrivial solution of represents a fixed point of . The numeric solution of is then obtained as the solution of […] 
A Wolfram Language Approach to Real Numerical Algebraic Plane Curves
dx.doi.org/doi:10.3888/tmj.207 This article is a summary of my book A Numerical Approach to Real Algebraic Curves with the Wolfram Language [1]. 1. Introduction The nineteenth century saw great progress in geometric (real) and analytic (complex) algebraic plane curves. In the absence of an ability to do the large number of computations for a concrete theory, […] 
Improving the Kruskal—Katona Bounds for Complete Subgraphs of a Graph
dx.doi.org/doi:10.3888/tmj.206 An important problem in graph theory is to find the number of complete subgraphs of a given size in a graph. If the graph is very large, it is usually only possible to obtain upper bounds for these numbers based on the numbers of complete subgraphs of smaller sizes. The Kruskal–Katona bounds are often […] 
Computational Aspects of Quaternionic Polynomials
dx.doi.org/doi:10.3888/tmj.205 This article explores the numerical mathematics and visualization capabilities of Mathematica in the framework of quaternion algebra. In this context, we discuss computational aspects of the recently introduced Newton and Weierstrass methods for finding the roots of a quaternionic polynomial. Introduction Since Niven proved in his pioneering work [1] that every nonconstant polynomial of […] 
Computational Aspects of Quaternionic Polynomials
dx.doi.org/doi:10.3888/tmj.204 This article discusses a recently developed Mathematica tool––a collection of functions for manipulating, evaluating and factoring quaternionic polynomials. relies on the package , which is available for download at w3.math.uminho.pt/QuaternionAnalysis. Introduction Some years ago, the first two authors of this article extended the standard Mathematica package implementing Hamilton’s quaternion algebra—the package —endowing it with […] 
The Modular Group
dx.doi.org/doi:10.3888/tmj.203 The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper halfplane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an ordertwo element and an element of infinite order . Viewing the action of the group elements […] 
Symbolic Solutions of Simultaneous FirstOrder PDEs in One Unknown
dx.doi.org/doi:10.3888/tmj.202 We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on the Bour–Mayer method to determine compatibility conditions via Jacobi–Mayer brackets. We solve compatible systems recursively by imitating what one would do with pen and paper: Solve one equation, substitute its solution into […]