DOCSLIB.ORG

Cramer Benjamin PMET

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cramer Benjamin PMET Just-in-Time-Teaching and other gadgets Richard Cramer-Benjamin Niagara University http://faculty.niagara.edu/richcb The Class MAT 443 – Euclidean Geometry 26 Students 12 Secondary Ed (9-12 or 5-12 Certification) 14 Elementary Ed (1-6, B-6, or 1-9 Certification) The Class Venema, G., Foundations of Geometry , Preliminaries/Discrete Geometry 2 weeks Axioms of Plane Geometry 3 weeks Neutral Geometry 3 weeks Euclidean Geometry 3 weeks Circles 1 week Transformational Geometry 2 weeks Other Sources Requiring Student Questions on the Text Bonnie Gold How I (Finally) Got My Calculus I Students to Read the Text Tommy Ratliff MAA Inovative Teaching Exchange JiTT Just-in-Time-Teaching Warm-Ups Physlets Puzzles On-line Homework Interactive Lessons JiTTDLWiki JiTTDLWiki Goals Teach Students to read a textbook Math classes have taught students not to read the text. Get students thinking about the material Identify potential difficulties Spend less time lecturing Example Questions For February 1 Subject line WarmUp 3 LastName Due 8:00 pm, Tuesday, January 31. Read Sections 5.1-5.4 Be sure to understand The different axiomatic systems (Hilbert's, Birkhoff's, SMSG, and UCSMP), undefined terms, Existence Postulate, plane, Incidence Postulate, lie on, parallel, the ruler postulate, between, segment, ray, length, congruent, Theorem 5.4.6*, Corrollary 5.4.7*, Euclidean Metric, Taxicab Metric, Coordinate functions on Euclidean and taxicab metrics, the rational plane. Questions Compare Hilbert's axioms with the UCSMP axioms in the appendix. What are some observations you can make? What is a coordinate function? What does it have to do with the ruler placement postulate? What does the rational plane model demonstrate? List 3 statements about the reading. A statement can be something significant you learned or a question you have. At least two of the statements should be questions. One point geometry contains one point and no lines. Which Incidence Axioms does one point geometry satisfy? Justify your answer. I don't think it satisfies any of the incidence axioms because there aren't any lines; all of the incidence axioms involved lines and multiple points. One point geometry does not satisfy any of the incidence axioms. Incidence axiom 1 says "for every pair of distinct points", but one-point geometry only has one point, not a pair of points. Incidence axiom 2 says "for every line l", but one point geometry does not have any lines. Incidence axiom 3 says "there exist three points", but in one point geometry there do not exist three points. Therefore, it doesn't satisfy any of the incidence axioms. One point geometry satisfies Incidence Axiom 1 (For every pair of distinct points P and Q there exists exactly one line L such that both P and Q lie on L) and Incidence Axiom 2 (For every line L there exist at least two distinct points P and Q such that both P and Q lie on L). This can be illustrated by using If /Then statements. For example in Incidence Axiom 1 the If /Then statement would say: If for every pair of distinct points P and Q Then there exists exactly one line L such that both P and Q lie on L. Since both parts of this statement are false for one point geometry, a false/false statement results in a true statement which means that one point geometry does in fact satisfy Incidence Axiom 1. For Incidence Axiom 2, the If /Then statement would say: If for every line L there exist at least two distinct points P and Q then both P and Q lie on L. Again, both parts of this statement are false for one point geometry and again a false/false statement results in a true statement which means that one point geometry does satisfy Incidence Axiom 2. However, when it comes to Incidence Axiom 3 (There exist three points that do not all lie on any one line) it is not possible in one point geometry because in this cause there are 3 points . Compare Hilbert's axioms with the UCSMP axioms in the appendix. What are some observations you can make? There are more undefined terms in Hilbert's axioms than in the UCSMP axioms. Distance and angle measure are an integral part of the UCSMP axioms and are not utilized in Hilbert's axioms. Additionally, UCSMP's axioms deal with area and volume while Hilbert's do not. Both axioms are organized by categories. Hilbert's axioms fall under five categories of axioms: Incidence, Order, Congruence, Parallels, and Continuity. UCSMP's axioms fall under eight categories of postulates: Point-Line-Plane, Distance, Triangle Inequality, Angle Measure, Corresponding Angle, Reflection, Area, and Volume. (Notice Hilbert's axioms are called "axioms" while UCSMP's are called "postulates.") Hilbert's axioms seem to use spatial relationships of angles, lines, and points. UCSMP's seem to branch off into many perspectives other than spatial. What does the rational plane model demonstrate? The rational plane model demonstrates that every rational number is also a real number. The rational plane model demonstrates the existence of rational numbers through ordered pairs. Not only does the rational plane demonstrate rational numbers but it also demonstrates real numbers since all rational numbers are part of the set of real numbers. Although, the rational plane model demonstrates that the rational plane satisfies all five of Euclid's postulates, it also shows how the proof of Euclid's very first proposition breaks down in the rational plane. This means that the Euclid was using unstated hypotheses in his proofs, which also means that there is a gap in Euclid's proofs. Questions Every Warm Up assignment will have one area for “Give three questions or important things you learned from the reading.” Questions must be stated as questions. Stating “I didn’t understand…” is not a question. Also, your question should not be something that can easily be looked up in the index. What indications or hints were Euclid given in order to know the importance of his postulates and the remarkable role they would play in geometry? How did he pick and choose the axioms as well? I was under the impression that a model satisfies a postulate or a proof. The book states that there are two models used to show Euclid?s fifth postulate yet one of them satisfies it and the other model does not sitisfy it. How is then a model for Euclid?s fifth postulate? In the section involving indirect proof it seems as if they tell you the exact manner in which to do a RAA and then show cases in which it is considered sloppy when it seems as if they are following the steps listed above in the description of the method of an indirect proof, why is this considered sloppy? Are certain elements of geometry more worthy or do they lend themselves better to a RAA? How do we know when to use an RAA, when is it most effective? When we are writing proofs in class, should we try and use all of the six kinds of reason? Would they all apply at once or would that rarely be the case? Why is it not enough to provide experimental evidence that a theorm is true? When writing a proof, how would one know to write a proof using the logic formula, induction, proof by contradiction, etc.? what are some hints to tell when to use each proof process? ‘Loud Thinking H. A. Peelle, “Alternative Modes for Teaching Mathematical Problem Solving: An Overview”, The Journal of Mathematics and Science: Collaborative Explorations 4 (Spring 2001) 119-142 ‘Loud Thinking Students work in pairs. Solver works on the problem Should verbalize thoughts while trying to solve problem. Recorder records solving process Can give hints and read back previous attempts. Should not be paired problem solving ‘Loud Thinking Recorders found it difficult not to take part in the problem solving Students liked that it forced them to be aware of the problem solving process. Many students are afraid to ‘write down or say something incorrect.’ This is counter to what we think of as good problem solving. Weak students got to observe strong problem solvers. Collaborative Oral Take Home Exam MAA Notes #49 - Annalisa Crannell Students work in groups of 3 or 4 4-5 challenging problems 1 week to solve the problems Each group schedules a half-hour exam time Each member selects a problem at random and solves the problem at the board with no notes. Group is graded on written solutions, individuals are graded on oral solutions Collaborative Oral Take Home Exam Originally made the grade 100% oral. Too much pressure for students Most groups did very well. Some difficulty with student schedules. Nice way to make collaborative learning part of a higher stakes assessment. Reading Out Loud Students pair up Take turns reading a sentence from the text. The listener then asks a question about the passage just read. Students read very quickly and have trouble asking probing questions. Circle Time Groups of three Each person in the group gets a unique problem to work outside of class. Students then exchange solutions and evaluate groups members solutions Circle Time Students have very little ability to evaluate the correctness of a proof. Students can not follow the logical arguments of another student Students can comment on notation Web pages My Home Page http://facutly.niagara.edu/richcb JiTT http://134.68.135.1/JiTTDLwiki/index.php/Mai n_Page MAA Inovative Teaching Exchange http://www.maa.org/t_and_l/exchange/exchan ge.html Requirements Secondary Elementary Calc I, II, and III Intro to Statistics Foundations Elem.
Load more

Recommended publications
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • COMBINATORICS, Volume
    http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R.
    [Show full text]
  • Collineations in Perspective
    Collineations in Perspective Now that we have a decent grasp of one-dimensional projectivities, we move on to their two di- mensional analogs. Although they are more complicated, in a sense, they may be easier to grasp because of the many applications to perspective drawing. Speaking of, let's return to the triangle on the window and its shadow in its full form instead of only looking at one line. Perspective Collineation In one dimension, a perspectivity is a bijective mapping from a line to a line through a point. In two dimensions, a perspective collineation is a bijective mapping from a plane to a plane through a point. To illustrate, consider the triangle on the window plane and its shadow on the ground plane as in Figure 1. We can see that every point on the triangle on the window maps to exactly one point on the shadow, but the collineation is from the entire window plane to the entire ground plane. We understand the window plane to extend infinitely in all directions (even going through the ground), the ground also extends infinitely in all directions (we will assume that the earth is flat here), and we map every point on the window to a point on the ground. Looking at Figure 2, we see that the lamp analogy breaks down when we consider all lines through O. Although it makes sense for the base of the triangle on the window mapped to its shadow on 1 the ground (A to A0 and B to B0), what do we make of the mapping C to C0, or D to D0? C is on the window plane, underground, while C0 is on the ground.
    [Show full text]
  • Finite Projective Geometry 2Nd Year Group Project
    Finite Projective Geometry 2nd year group project. B. Doyle, B. Voce, W.C Lim, C.H Lo Mathematics Department - Imperial College London Supervisor: Ambrus Pal´ June 7, 2015 Abstract The Fano plane has a strong claim on being the simplest symmetrical object with inbuilt mathematical structure in the universe. This is due to the fact that it is the smallest possible projective plane; a set of points with a subsets of lines satisfying just three axioms. We will begin by developing some theory direct from the axioms and uncovering some of the hidden (and not so hidden) symmetries of the Fano plane. Alternatively, some projective planes can be derived from vector space theory and we shall also explore this and the associated linear maps on these spaces. Finally, with the help of some theory of quadratic forms we will give a proof of the surprising Bruck-Ryser theorem, which shows that if a projective plane has order n congruent to 1 or 2 mod 4, then n is the sum of two squares. Thus we will have demonstrated fascinating links between pure mathematical disciplines by incorporating the use of linear algebra, group the- ory and number theory to explain the geometric world of projective planes. 1 Contents 1 Introduction 3 2 Basic Defintions and results 4 3 The Fano Plane 7 3.1 Isomorphism and Automorphism . 8 3.2 Ovals . 10 4 Projective Geometry with fields 12 4.1 Constructing Projective Planes from fields . 12 4.2 Order of Projective Planes over fields . 14 5 Bruck-Ryser 17 A Appendix - Rings and Fields 22 2 1 Introduction Projective planes are geometrical objects that consist of a set of elements called points and sub- sets of these elements called lines constructed following three basic axioms which give the re- sulting object a remarkable level of symmetry.
    [Show full text]
  • Matroid Enumeration for Incidence Geometry
    Discrete Comput Geom (2012) 47:17–43 DOI 10.1007/s00454-011-9388-y Matroid Enumeration for Incidence Geometry Yoshitake Matsumoto · Sonoko Moriyama · Hiroshi Imai · David Bremner Received: 30 August 2009 / Revised: 25 October 2011 / Accepted: 4 November 2011 / Published online: 30 November 2011 © Springer Science+Business Media, LLC 2011 Abstract Matroids are combinatorial abstractions for point configurations and hy- perplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinear- ity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of inci- dence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algo- rithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in R3 and that of Motzkin on ≤12 lines in R2, extend- ing previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of ori- ented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n ≤ 12 elements and rank 4 on n ≤ 9 elements. We further list several new minimal non-orientable matroids. Y. Matsumoto · H. Imai Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan Y.
    [Show full text]
  • Projective Planes
    Projective Planes A projective plane is an incidence system of points and lines satisfying the following axioms: (P1) Any two distinct points are joined by exactly one line. (P2) Any two distinct lines meet in exactly one point. (P3) There exists a quadrangle: four points of which no three are collinear. In class we will exhibit a projective plane with 57 points and 57 lines: the game of The Fano Plane (of order 2): SpotIt®. (Actually the game is sold with 7 points, 7 lines only 55 lines; for the purposes of this 3 points on each line class I have added the two missing lines.) 3 lines through each point The smallest projective plane, often called the Fano plane, has seven points and seven lines, as shown on the right. The Projective Plane of order 3: 13 points, 13 lines The second smallest 4 points on each line projective plane, 4 lines through each point having thirteen points 0, 1, 2, …, 12 and thirteen lines A, B, C, …, M is shown on the left. These two planes are coordinatized by the fields of order 2 and 3 respectively; this accounts for the term ‘order’ which we shall define shortly. Given any field 퐹, the classical projective plane over 퐹 is constructed from a 3-dimensional vector space 퐹3 = {(푥, 푦, 푧) ∶ 푥, 푦, 푧 ∈ 퐹} as follows: ‘Points’ are one-dimensional subspaces 〈(푥, 푦, 푧)〉. Here (푥, 푦, 푧) ∈ 퐹3 is any nonzero vector; it spans a one-dimensional subspace 〈(푥, 푦, 푧)〉 = {휆(푥, 푦, 푧) ∶ 휆 ∈ 퐹}.
    [Show full text]
  • Arxiv:1310.0196V1 [Math.CO]
    GRAPHS EMBEDDED INTO FINITE PROJECTIVE PLANES KEITH MELLINGER, RYAN VAUGHN, AND OSCAR VEGA Abstract. We introduce and study embeddings of graphs in finite projec- tive planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs. 1. Introduction We are interested in studying how graphs may be embedded in finite planes. These embeddings are injective functions which preserve graph-incidence within the incidence relation of a finite projective plane and are analogous to the notion of embedding used in planar graphs and topological graph theory. Although planar graphs have been well-studied, embeddings of graphs into finite projective planes are relatively new. Thus, embeddings may give insight into the structure of finite projective planes. We begin by introducing necessary terminology, and establish counting lemmas used in the later constructions and proofs. In Section 3 we prove our main results, determining which complete bipartite graphs can be embedded in a given plane, and by counting the number of embeddings of complete bipartite graphs of large order. Finally, in Section 4, we establish bounds on the number of vertices and edges an embedded complete bipartite graph can have in any given Baer subplane. We also establish conditions when the image of a complete bipartite graph must contain a non-Baer subplane. We now give some terminology and well-known results necessary for later sec- tions. Most of the content of this section is ‘folklore’; any terminology not found here may be found in [1] (graph theory), or [4] (geometry).
    [Show full text]
  • Masterarbeit
    Masterarbeit Solution Methods for the Social Golfer Problem Ausgef¨uhrt am Institut f¨ur Informationssysteme 184/2 Abteilung f¨ur Datenbanken und Artificial Intelligence der Technischen Universit¨at Wien unter der Anleitung von Priv.-Doz. Dr. Nysret Musliu durch Markus Triska Kochgasse 19/7, 1080 Wien Wien, am 20. M¨arz 2008 Abstract The social golfer problem (SGP) is a combinatorial optimisation problem. The task is to schedule g ×p golfers in g groups of p players for w weeks such that no two golfers play in the same group more than once. An instance of the SGP is denoted by the triple g−p−w. The original problem asks for the maximal w such that the instance 8−4−w can be solved. In addition to being an interesting puzzle and hard benchmark problem, the SGP and closely related problems arise in many practical applications such as encoding, encryption and covering tasks. In this thesis, we present and improve upon existing approaches towards solving SGP instances. We prove that the completion problem correspond- ing to the SGP is NP-complete. We correct several mistakes of an existing SAT encoding for the SGP, and propose a modification that yields consid- erably improved running times when solving SGP instances with common SAT solvers. We develop a new and freely available finite domain constraint solver, which lets us experiment with an existing constraint-based formula- tion of the SGP. We use our solver to solve the original problem for 9 weeks, thus matching the best current results of commercial constraint solvers for this instance.
    [Show full text]
  • Polynomial Methods and Incidence Theory
    Polynomial Methods and Incidence Theory Adam Sheffer This document is an incomplete draft from July 26, 2020. Several chapters are still missing. Acknowledgements This book would not have been written without Micha Sharir, from whom I learned much of the Discrete Geometry that I know, and Joshua Zahl, from whom I learned much of the Real Algebraic Geometry that I know. I am indebted to Frank de Zeeuw for carefully reading and commenting on earlier versions of this book, and to Nets Katz for being patient with me spending a large amount of time on it. I would like to thank the many mathematicians who helped improving this book. These include Boris Aronov, Abdul Basit, Zachary Chase, Ana Chavez Caliz, Alan Chang, Daniel Di Benedetto, Jordan Ellenberg, Evan Fink, Davey Fitzpatrick, Nora Frankl, Alex Iosevich, Ben Lund, Bob Krueger, Brett Leroux, Shachar Lovett, Michael Manta, Brendan Murphy, Jason O'Neill, Yumeng Ou, Cosmin Pohoata, Piotr Pokora, Anurag Sahay, Steven Senger, Olivine Silier, Shakhar Smorodinsky, Noam Solomon, Samuel Spiro, Jonathan Tidor, and Bartosz Walczak. iii Contents Introduction vii How to read this book............................. ix Notation and inequalities............................x 1 Incidences in Classical Discrete Geometry1 1.1 Introduction................................1 1.2 First proofs................................2 1.3 The crossing lemma............................4 1.4 Szemer´edi-Trotter via the crossing lemma................7 1.5 The unit distances problem.......................8 1.6 The distinct distances problem...................... 10 1.7 A problem about unit area triangles................... 12 1.8 The sum-product problem........................ 13 1.9 Rich points................................ 15 1.10 Exercises.................................. 16 1.11 Open problems.............................
    [Show full text]
  • Frobenius Collineations in Finite Projective Planes
    Frobenius Collineations in Finite Projective Planes Johannes Ueberberg 1 Introduction Given a finite field F = GF (qn)oforderqn it is well-known that the map f : F ! F , f : x 7! xq is a field automorphism of F of order n, called the Frobenius automor- phism.IfV is an n-dimensional vector space over the finite field GF (q), then V can be considered as the vector space of the field GF (qn)overGF (q). Therefore the Frobenius automorphism induces a linear map over GF (q) R : V ! V R : x 7! xq of order n on V . It follows that R induces a projective collineation ' on the (n − 1)- dimensional projective space PG(n−1;q). We call ' and any projective collineation conjugate to ' a Frobenius collineation. In the present paper we shall study the case n = 3, that is, the Frobenius collineations of the projective plane PG(2;q). Let P = PG(2;q2). Then every Singer cycle σ (see Section 3) of P defines a partition P(σ)ofthepointsetofP into pairwise disjoint Baer subplanes. These partitions are called linear Baer partitions or, equivalently, Singer Baer partitions [17]. If % is a Frobenius collineation of P , then we define E% to be the set of Baer subplanes of P fixed by %. It turns out that for q ≡ 2mod3wehavejP(σ) \E%j2 2 f0; 1; 3g with jP(σ) \E%j =3ifandonlyif% 2 NG(<σ>), where G = PGL3(q ) (see 3.5). Received by the editors March 1996. Communicated by J. Thas. 1991 Mathematics Subject Classification : 51E20, 51E24. Key words and phrases : diagram geometry, Baer partitions, Frobenius collineations.
    [Show full text]
  • Oriented Hypergraphs I: Introduction and Balance Arxiv:1210.0943V1
    Oriented Hypergraphs I: Introduction and Balance Lucas J. Rusnak∗ Department of Mathematics Texas State University San Marcos, Texas, U.S.A. [email protected] Submitted: Sept 30, 2012; Accepted: MMM DD, YYYY; Published: XX Mathematics Subject Classifications: 05C22, 05C65, 05C75 Abstract An oriented hypergraph is an oriented incidence structure that extends the con- cept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families { balanced, bal- anceable, and unbalanceable { and we obtain a complete classification of the bal- anced circuits of oriented hypergraphs. Keywords: Oriented hypergraph; balanced hypergraph; balanced matrix; signed hypergraph 1 Introduction Oriented hypergraphs have recently appeared in [6] as an extension of the signed graphic incidence, adjacency, and Laplacian matrices to examine walk counting. This paper fur- ther expands the theory of oriented hypergraphs by examining the extension of the cycle arXiv:1210.0943v1 [math.CO] 2 Oct 2012 space of a graph to oriented hypergraphs, and we obtain a classification of the balanced minimally dependent columns of the incidence matrix of an oriented hypergraph. It is known that the cycle space of a graph characterizes the dependencies of the graphic matroid and the minimal dependencies, or circuits, are the edge sets of the simple cycles of the graph. Oriented hypergraphs have a natural division into three categories: balanced, balanceable, and unbalanceable. The family of balanced oriented hypergraphs contain graphs, so a characterization of the balanced circuits of oriented hypergraphs can be regarded as an extension of the following theorem: ∗A special thanks to Thomas Zaslavsky and Gerard Cornu´ejolsfor their feedback.
    [Show full text]
  • Kirkman's Schoolgirls Wearing Hats and Walking Through Fields Of
    ARTICLES Kirkman’s Schoolgirls Wearing Hats and Walking through Fields of Numbers EZRA BROWN Virginia Polytechnic Institute and State University Blacksburg, VA 24061 [email protected] KEITH E. MELLINGER University of Mary Washington Fredericksburg, VA 22401 [email protected] Fifteen young ladies at school Imagine fifteen young ladies at the Emmy Noether Boarding School—Anita, Barb, Carol, Doris, Ellen, Fran, Gail, Helen, Ivy, Julia, Kali, Lori, Mary, Noel, and Olive. Every day, they walk to school in the Official ENBS Formation, namely, in five rows of three each. One of the ENBS rules is that during the walk, a student may only talk with the other students in her row of three. These fifteen are all good friends and like to talk with each other—and they are all mathematically inclined. One day Julia says, “I wonder if it’s possible for us to walk to school in the Official Formation in such a way that we all have a chance to talk with each other at least once a week?” “But that means nobody walks with anybody else in a line more than once a week,” observes Anita. “I’ll bet we can do that,” concludes Lori. “Let’s get to work.” And what they came up with is the schedule in TABLE 1. TABLE 1: Walking to school MON TUE WED THU FRI SAT SUN a, b, e a, c, f a, d, h a, g, k a, j, m a, n, o a, i, l c, l, o b, m, o b, c, g b, h, l b, f, k b, d, i b, j, n d, f, m d, g, n e, j, o c, d, j c, i, n c, e, k c, h, m g, i, j e, h, i f, l, n e, m, n d, e, l f, h, j d, k, o h, k, n j, k, l i, k, m f, i, o g, h, o g, l, m e, f, g TABLE 1 was probably what T.
    [Show full text]