Featured Researches

History And Overview

"Dear Kingos, It's all right to be noisy!" Why is it so hard to get them talking?

This paper discusses an effort to encourage student-instructor interactive engagement through active learning activities during class time. We do not only encouraged the Kingos to speak out when an opportunity arises but also required them to record their active participation in a student journal throughout the semester. In principle, any activities which constitute active learning can and should be recorded in the 'Student Journal'. These include, but not limited to, reading definition, theorem, problem, etc.; responding to questions and inquiries; asking questions; pointing out some mistakes during class time. Despite an incentive for this participation, our experience teaching of different mathematics courses in several consecutive semesters indicates that many Kingos resist in speaking out publicly, submitting an empty journal at the end of the semester. Students' feedback on teaching evaluation at the end of the semester reveals that many dislike and against the idea of active participation and recording it in a journal. This paper discusses the reason behind this resistance and provides some potential remedies to alleviate the situation.

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History And Overview

0-Pierced Triangles within a Poisson Overlay

Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is 0-pierced if no member of the line process intersects any side of T. Our starting point is Ambartzumian's 1982 joint density for angles of T; our exposition is elementary and raises several unanswered questions.

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History And Overview

19th century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

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History And Overview

A "right" path to cyclic polygons

It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges, which, surprisingly, has revealed to be far from simple. In this paper we investigate such a problem by following a new and elementary approach. We start from the simple observation that the incircle of a right triangle touches its hypothenuse in a point that splits it into two segments, the product of whose lengths equals the area of the triangle. From this curious fact we derive in a few lines: an unusual proof of the Pythagoras' theorem, Heron's theorem for right triangles, Heron's theorem for general triangles, and Brahmagupta's theorem for cyclic quadrangles. This suggests that cutting the edges of a cyclic polygon by means of suitable points should be the "right" working method. Indeed, following this idea, we obtain an explicit formula for the area of any convex cyclic polygon, as a symmetric function of the segments split on its edges by the incircles of a triangulation. We also show that such a symmetry can be rediscovered in Heron's and Brahmagupta's results, which consequently represent special cases of the general provided formula.

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History And Overview

A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords

The standard 2-dimensional Tonnetz describes parsimonious voice-leading connections between major and minor triads as the 3-dimensional Tonnetz does for dominant seventh and half-diminished seventh chords. In this paper, I present a geometric model for a 5-dimensional Tonnetz for parsimonious voice-leading between nearly symmetric hexachords of the mystic-Wozzeck genus. Cartesian coordinates for points on this discretized grid, generalized coordinate collections for 5-simplices corresponding to mystic and Wozzeck chords, and the geometric nearest-neighbors of a selected chord are derived.

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History And Overview

A Brief History of Algebra with a Focus on the Distributive Law and Semiring Theory

In this note, we investigate the history of algebra briefly. We particularly focus on the history of rings, semirings, and the distributive law.

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History And Overview

A Database of 2,500 Quasicrystal Cells

Here is a database of quasicrystal cells computed by the deBruijn Grand Dual Method. The database is in a form that can be converted and read by a variety of geometry programs. Proof of the accuracy of the computations is given by the consistency of the two values of the volumes of the cells. How the deBruijn algorithm works, and the possible use of the algorithm for modeling non-local phenomena is also discussed.

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History And Overview

A First Course in Linear Algebra: Study Guide for the Undergraduate Linear Algebra Course

In this book, there are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students practice additional sets of problems other than examples, and they can also check their solutions to some of these exercises by looking at Answers to Odd-Numbered Exercises section at the end of this book. This book is very useful for college students who studied Calculus I, and other students who want to review some linear algebra concepts before studying a second course in linear algebra. This book is available online for free in google books and ResearchGate in PDF format under a Creative Commons license.

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History And Overview

A Forgotten Theory of Proofs ?

Looking at MacLane's thesis on proof theory in the light of combinatory logic

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History And Overview

A Friendly Introduction to Differential Equations

In this book, there are five chapters: The Laplace Transform, Systems of Homogeneous Linear Differential Equations (HLDE), Methods of First and Higher Orders Differential Equations, Extended Methods of First and Higher Orders Differential Equations, and Applications of Differential Equations. In addition, there are exercises at the end of each chapter above to let students practice additional sets of problems other than examples, and they can also check their solutions to some of these exercises by looking at "Answers to Odd-Numbered Exercises" section at the end of this book. This book is a very useful for college students who studied Calculus II, and other students who want to review some concepts of differential equations before studying courses such as partial differential equations, applied mathematics, and electric circuits II. This book is available online for free in google books and ResearchGate in PDF format under a Creative Commons license.

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