Mathematica Journal

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  1. https://doi.org/10.3888/tmj.21-3 We study the distribution of eigenspectra for operators of the form with self-adjoint 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 […]
  2. https://doi.org/10.3888/tmj.21-2 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 […]
  3. https://doi.org/10.3888/tmj.21-1 A comprehensive discussion is presented of the closed-form solutions for the responses of single-degree-of-freedom systems subject to swept-frequency harmonic excitation. The closed-form solutions for linear and octave swept-frequency excitation are presented and these are compared to results obtained by direct numerical integration of the equations of motion. Included is an in-depth discussion of the […]
  4. dx.doi.org/doi:10.3888/tmj.20-8 This article presents a numerical pseudo-dynamic approach to solve a nonlinear stationary partial differential equation (PDE) with bifurcations by passing from to a pseudo-time-dependent 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 […]
  5. dx.doi.org/doi:10.3888/tmj.20-7 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, […]
  6. dx.doi.org/doi:10.3888/tmj.20-6 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 […]
  7. dx.doi.org/doi:10.3888/tmj.20-5 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 […]
  8. dx.doi.org/doi:10.3888/tmj.20-4 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 […]
  9. dx.doi.org/doi:10.3888/tmj.20-3 The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element and an element of infinite order . Viewing the action of the group elements […]
  10. dx.doi.org/doi:10.3888/tmj.20-2 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 […]
   

   
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