Mathematics Commons^™

King, John (Sc 594), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full-text (click on "Additional Files" below) for Manuscripts Small Collection 594. Ciphering book of John King including mathematical exercises, numeration of money, simple and compound reduction, weights and measures, and word problems.

Curd, Spencer, 1788-1832 (Sc 966), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full text (click on "Additional Files" below) for Manuscripts Small Collection 966. Ciphering book (94 p.), focusing chiefly on surveying and navigating rules for solving problems as well as giving specific problems, of Spencer Curd, Logan County, Kentucky, with accompanying genealogical data.

Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support …

A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong

All HMC Faculty Publications and Research

This activity is an investigation of a special nonregular tetrahedron that can be arranged to fill space without leaving any internal gaps in the same way that certain planar figures tessellate the plane. These tetrahedra can be connected together with hinges to make fun and interesting puzzles. More background information can be found in the paper "An Amazing, Space-Filling, Non-Regular Tetrahedron" by Joyce Frost and Peg Cagle, published by the IAS/Park City Mathematics Institute (available at mathforum.org/pcmi/hstp/resources/dodeca/).

A Locus Construction In The Hyperbolic Plane For Elliptic Curves With Cross-Ratio On The Unit Circle, Lyudmila Shved

Theses Digitization Project

This project demonstrates how an elliptic curve f defined by invariance under two involutions can be represented by the locus of circumcenters of isosceles triangles in the hyperbolic plane, using inversive model.

Geodesics Of Surface Of Revolution, Wenli Chang

Theses Digitization Project

The purpose of this project was to study the differential geometry of curves and surfaces in three-dimensional Euclidean space. Some important concepts such as, Curvature, Fundamental Form, Christoffel symbols, and Geodesic Curvature and equations are explored.

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Erica Flapan, Blake Mellor, Ramin Naimi

Pomona Faculty Publications and Research

We determine for which m the complete graph K_m has an embedding in S³ whose topological symmetry group is isomorphic to one of the polyhedral groupsA₄, A₅ or S₄.

A Topological Constructions For All Two-Row Springer Varieties, Heather M. Russell

Department of Math & Statistics Faculty Publications

Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification, Khovanov provides a topological construction of (m,m) Springer varieties. Here we extend his construction to all two-row Springer varieties. Using the combinatorial and diagrammatic properties of this construction we provide a particularly useful homology basis and construct the Springer representation using this basis. We also provide a skein-theoretic formulation of the representation in this case.

The Mathematical Landscape, Antonio Collazo

CMC Senior Theses

The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject.  By no means is the following a complete formulation of any of the topics covered.  I want to give the reader a tour of the mathematical landscape.  There are plenty of further details to explore in each section, I have just touched the tip the iceberg.  The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math.  The first sections are a exposition into the mathematical objects and their …

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Eric Flapan, Blake Mellor, Ramin Naimi

Mathematics, Statistics and Data Science Faculty Works

We determine for which m the complete graph K_m has an embedding in S³ whose topological symmetry group is isomorphic to one of the polyhedral groups A₄, A₅ or S₄.

Numerical Calculation Of Lyapunov Exponents For Three-Dimensional Systems Of Ordinary Differential Equations, Clyde-Emmanuel Estorninho Meador

Theses, Dissertations and Capstones

We consider two algorithms for the computation of Lyapunov exponents for systems of ordinary dierential equations: orbit separation and continuous Gram-Schmidt orthonormal-ization. We also consider two Runge-Kutta methods for the solution of ordinary dierential equations. These algorithms and methods are applied to four three-dimensional systems of ordinary dierential equations, and the results are discussed.

Interval Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the notion of interval semirings are introduced. The authors study and analyse semirings algebraically. Methods are given for the construction of non-associative semirings using loops and interval semirings or interval loops and semirings. Another type of non-associative semirings are introduced using groupoids and interval semirings or interval groupoids and semirings. Examples using integers and modulo integers are given. Also infinite semirings which are semifields are given using interval semigroups and semirings or semigroups and interval semirings or using groups and interval semirings. Interval groups are introduced to construct interval group interval semirings, and properties related with them …

Interval Semigroups, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce the notion of interval semigroups using intervals of the form [0, a], a is real. Several types of interval semigroups like fuzzy interval semigroups, interval symmetric semigroups, special symmetric interval semigroups, interval matrix semigroups and interval polynomial semigroups are defined and discussed. This book has eight chapters. The main feature of this book is that we suggest 241 problems in the eighth chapter. In this book the authors have defined 29 new concepts and illustrates them with 231 examples. Certainly this will find several applications. The authors deeply acknowledge Dr. Kandasamy for the proof reading …

Problems With And Without … Problems!, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This book is addressed to College honor students, researchers, and professors. It contains 136 original problems published by the author in various scientific journals around the world. The problems could be used to preparing for courses, exams, and Olympiads in mathematics. Many of these have a generalized form. For each problem we provide a detailed solution.

I was a professeur coopérant between 1982-1984, teaching mathematics in French language at Lycée Sidi EL Hassan Lyoussi in Sefrou, Province de Fès, Morocco. I used many of these problems for selecting and training, together with other Moroccan professors, in Rabat city, of the …

Balance Systems And The Variational Bicomplex, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental "pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the "pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947–948] and, later, asserted as …

Slider-Pinning Rigidity: A Maxwell-Laman-Type Theorem, Ileana Streinu, Louis Theran

Computer Science: Faculty Publications

We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley’s direction networks.

An Analytical Technique For Solving Nonlinear Heat Transfer Equations, Hossein Aminikhah, Milad Hemmatnezhad

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, an analytic technique, namely the New Homotopy Perturbation Method (NHPM) is applied for solving the nonlinear differential equations arising in the field of heat transfer. In this method, the solution is considered as an infinite series expansion where converges rapidly to the exact solution. The nonlinear convective–radioactive cooling equation and nonlinear equation of conduction heat transfer with the variable physical properties are chosen as illustrative examples and the exact solutions have been found for each case.

Rethinking Geometrical Exactness, Marco Panza

MPP Published Research

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in …

Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke

Computer Science: Faculty Publications

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.

Curvedland: An Applet For Illustrating Curved Geometry Without Embedding, Gary Felder, Stephanie Erickson

Physics: Faculty Publications

We have written a Java applet to illustrate the meaning of curved geometry. The applet provides a mapping interface similar to MapQuest or Google Maps; features include the ability to navigate through a space and place permanent point objects and/or shapes at arbitrary positions. The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from different relative angles and distances.

Coarser Connected Metrizable Topologies, Lynne Yengulalp

Mathematics Faculty Publications

We show that every metric space, X, with w(⩾) c has a coarser connected metrizable topology.

On Folding A Polygon To A Polyhedron, Joseph O'Rourke

Computer Science: Faculty Publications

We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.

On Flat Polyhedra Deriving From Alexandrov's Theorem, Joseph O'Rourke

Computer Science: Faculty Publications

We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n³) time for polygons whose gluings are specified by n labels.

Star Unfolding Convex Polyhedra Via Quasigeodesic Loops, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron ℘ to a simple (nonoverlapping) planar polygon: cut along one shortest path from each vertex of ℘ toQ, and cut all but one segment of Q.

Morphological Study Of Surface Corrosion In Carbon Steel Using Fractal Methods., Abdul Mohaimen W Fagri

Student Works (2010-2019)

Corrosion can be defined as the disintegration of a material into its constituent atoms due to chemical reactions with its surroundings. There are various chemical and physical factors that affect the rate and mechanisms of corrosion processes. In this study, we focus on the corrosion of carbon steel in mineral water at room temperature. The morphologies of corroded surface are characterized using fractal analysis. The steel plates of 10cmx10cmx0.2cm are first polished using abrasive papers of different coarseness until smooth surfaces are obtained. The samples are placed in glass dish filled with mineral water and a simple imaging procedure using …

Wavelet Transform Of Fractional Integrals For Integrable Boehmians, Deshna Loonker, P. K. Banerji, S. L. Kalla

Applications and Applied Mathematics: An International Journal (AAM)

The present paper deals with the wavelet transform of fractional integral operator (the Riemann- Liouville operators) on Boehmian spaces. By virtue of the existing relation between the wavelet transform and the Fourier transform, we obtained integrable Boehmians defined on the Boehmian space for the wavelet transform of fractional integrals.

The Yao Graph Y6 Is A Spanner, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that Y₆ is a spanner. Y₆ is the Yao graph on a set of planar points, which has an edge from each point x to a closest point y within each of the six angular cones of 60^◦ surrounding x .

Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp

Dartmouth Scholarship

In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool …

Symmetry From Nature To The Classroom, Rochelle Lueders

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

Numerical Investigation Of Pre-Detonator Geometries For Pde Applications, Robert T. Fievisohn

Theses and Dissertations

A parametric study was performed to determine optimal geometries to allow the successful transition of a detonation from a pre-detonator into the thrust tube of a pulse detonation engine. The study was performed using a two-dimensional Euler solver with progress variables to model the chemistry. The geometrical configurations for the simulations look at the effect of shock reflections, flow obstructions, and detonation diffraction to determine successful geometries. It was observed that there are success and failure rates associated with pre-detonators. These success rates appear to be determined by the transverse wave structure of a stably propagating detonation wave and must …