My Course Materials
Prepared by:
Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451
Email: malkevitch@york.cuny.edu
Fall, 2008
Mathematics 479 (History and Philosophy of Mathematics)
Syllabus
Homework 1
Homework 2
Homework 3
Homework 4
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Problems about Determinants, Applied to the Euclidean and Real Projective Planes
Exercises about the Real Projective Plane
The Real Projective Plane
Writing Project
Examination Review
Final Examination Review
Mathematics 488 (Digital Codes)
Syllabus
Modular Arithmetic
Congruence Primer
Euclidean Algorithm and Solving Congruences
Practice 1: Congruences
Practice 2: Check Digit Systems
Practice 3: Zipcodes
Practice 4: Credit Card Numbers
Hamming Distance and Error Correction
Writing Project
Examination Review
Final Examination Review
Fall, 2007
General Information for Undergraduate Students: Fall, 2007
Mathematics 120 (Pre-Calculus)
Self Test
Review Exam I
Review Exam II
Writing Project
Review Exam III
Final Examination Review
Trig and Quadratic Equation Problems
Math 479 (History and Philosophy of Mathematics)
Course Information: Fall, 2007
Homework 1
Homework 2
Homework 3
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Writing Project
Problems about Determinants, Applied to the Euclidean and Real Projective Planes
Review for First Examination: Math 479
Review for the Final Examination: Math 479
(Graduate Center Course)
Math 80130 (Euler's Combinatorial and Geometrical Mathematics)
Brief Description
General Information
Information about the New York Area Geometry Seminar at the Courant Institute
Information about the CUNY Graduate Center Combinatorics Seminar
Springer maintains an on-line Encyclopedia of Mathematics which you may find a useful resource, subject to the observation that definitions are not always universally the same.
On Line Encyclopedia of Mathematics
Euler Archive (VERY RICH SET OF RESOURCES)
Biography of Euler
Another Biography of Euler
Sheet A: Basic Graph Theory
Sheet B: Basic Graph Theory II
Sheet C: (Degree sequences)
Research Frontier
An Introduction to Planar, Plane, and Non-Planar Graphs
Branko Grunbaum's comments on the career of the great geometer Victor Klee who recently died (August, 2007)
Sheet D: Plane Graphs
Tranforming Plane Graphs
Sheet E: Polyhedra and Plane Graphs
Sheet F: Face Vectors
Sheet G: Face Vectors of Graphs
Many Different Proofs of Euler's Polyhedral Formula
A Critique of Euler's Original "Proof" of his formula, in the April, 2007, American Mathematical Monthly
Bondy and Murty's Graph Theory book is available in the form of pdf files here.
Bondy and Murty's Graph Theory Book
You may find these articles I wrote for the Feature Column of AMS of interest. They deal with coloring problems and with Euler's Polyhedral Formula.
Colorful Mathematics: Part I
Colorful Mathematics: Part II
Colorful Mathematics: Part III
Colorful Mathematics: Part IV
Euler's Polyhedral Formula
Euler's Polyedral Formula: Part II
I wrote a skeletal paper of applications of Euler's Polyhedral Formula which appeared in Graph Theory Notes, and which is available in a slightly altered pdf version.
Gifts from Euler's Polyhedral Formula.
Here is a list of some of the journals that publish materials about the theory of graphs.
Graph Theory Journals
Here is a problem set to work on.
Problem Set I
Here are some notes about hamiltonian circuits in plane graphs, including a description and a proof of Grinberg's condition that a plane graph which has a hamiltonian circuit must obey.
Hamiltonian circuits in plane graphs (Grinberg equation)
Here are some further remarks about Hamiltonian circuits, Gray Codes, and Error Correction codes.
Hamiltonian circuits, Gray Codes, Error Correction
The distinguished American geometer Victor Klee died recently. He made a variety of contributions to the theory of polytopes and related to the Euler Characteristic. This brief item surveys some of his work that is especially elementary to describe.
The Mathematical Legacy of Victor Klee
This set of notes deals with various special and interesting classes or graphs, and poses relatively simple questions to try to make sure the concepts involved are understood.
Classes of Graphs
Final
Spring, 2007
General Information for Students: Spring, 2007
Mathematics 225 (Discrete Mathematics)
What is Discrete Matheamtics
Handout 1
Handout 2
Base b Arithmetic
Rational Numbers and Decimal Numbers
Modular Arithmetic
Congruences
Congruence Primer
Euclidean Algorithm and Solving Congruences
Review Exam I
Review Exam II
Review Exam III
Review for Final Examination
Mathematics 244 (Geometric Structures)
Sheet A-(Careful looking; words)
Sheet B-Polygons (4-gons)
Quadrilaterals
Sheet C (Polyominoes)
Sheet D (Polyominoes II)
Sheet E (Equilateral triangles)
Some Taxicab Geometry Investigations
Sheet F (Equidistance problems) Taxicab and Euclidean Planes
Sheet G (Polygons and Polyhedra)
Angles
Taxicab Geometry
Sheet H (Polyhedra and Graphs)
Sheet I (Finite Projective Plane)
The Real Projective Plane
Desargues' Theorem
Pappus' Theorem
Examples of Finite Affine, Finite Projective and Finite Bolyai Lobachevsky Planes
Error Correction and Geometry
Guards, Visibility and Polygons
Polygonal Guards
Art Galleries and Guards
Plane Graphs
Euler relations
Review Exam I
Review Exam II
Review Exam II (Part II; Guard Problems)
Review for the Final Examination
Writing Project
Math 479 (History and Philosophy of Mathematics)
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Examination Review
Humanities 320 (Honors Seminar: Fairness and Equity)
Sheet A: Fairness and Political vs. Economic Structure
Sheet B: Fairness, Randomization, and Lotteries
Sheet C: Emotional and Intellectual Aspects of Fairness
Example Illustrating Jefferson, Adams, and Websters Apportionment Methods
Final Review (Election, Weighted Voting, Apportionment, Bankruptcy)
Honors Seminar Take Home Final
Fall, 2006
General Information for Students: Fall, 2006
Mathematics 120 (Pre-Calculus)
Self Test
Review Exam I
Review Exam II
Review Exam III
Final Review
Writing Project
Math 225 (Discrete Mathematical Structures)
Handout 1
Handout 2
Base b Arithmetic
Rational Numbers and Decimal Numbers
Modular Arithmetic
Congruences
Congruence Primer
Euclidean Algorithm and Solving Congruences
Review Exam I
Review Exam II
Review Exam III
Final Review
Writing Project
Math 479 (History and Philosophy of Mathematics)
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Writing Project
Math 484 (Seminar in Contemporary Mathematics) (This class will deal with Fairness and Equity Questions)
Homework I
Writing Project
Examination I Review
Fairness Notes 1
Fairness Notes 2
Fairness Notes 3
Fairness Notes 4
Fairness Notes 5
Fairness Notes 6
Fairness Notes 7
Fairness Notes 8
Fairness Notes 9
Fairness Notes 10
Fairness Notes 11
Fairness Glossary
Spring, 2006
Math 225 (Discrete Mathematical Structures)
What is Discrete Matheamtics
Handout 1
Handout 2
Base b Arithmetic
Rational Numbers and Decimal Numbers
Modular Arithmetic
Congruences
Congruence Primer
Euclidean Algorithm and Solving Congruences
Summation and Product Problems
Review Exam I
Review Exam II
Review Exam III
Final Examination Review
Math 244 (Geometric Structures)
Syllabus
Sheet A (Careful looking; words)
Sheet B (4-gons)
Sheet C (Polyominoes)
Sheet D (Polyominoes II)
Quadrilaterals
Sheet E (Equilateral triangles)
Some Taxicab Geometry Investigations
Sheet F (Equidistance problems) Taxicab and Euclidean Planes
Sheet G (Polygons and Polyhedra)
Sheet H (Polyhedra and Graphs)
Sheet I (Finite Projective Plane)
The Real Projective Plane
Desargues' Theorem
Pappus' Theorem
Examples of Finite Affine, Finite Projective and Finite Bolyai Lobachevsky Planes
Error Correction and Geometry
Guards, Visibility and Polygons
Polygonal Guards
Art Galleries and Guards
Plane Graphs
Euler relations
Review: Examination I
Review: Examination II (Part I)
Review: Examination II (Part II)
Final Examination Review
Writing Project
Math 479 (History and Philosophy of Mathematics)
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Examination Review
Final Examination Review
Writing Project
Fall, 2005
Mathematics 120 (Pre-Calculus)
Self Test
Review Exam i
Review Exam i (Part 2)
Review Exam ii
Review Exam iii
Writing Project
Mathematics 243 (Combinatorial and Discrete Geometry)
Syllabus
Sheet A
Sheet B
Sheet C (Degree sequences)
Sheet D (Walks, Trails, Paths, Circuits and Cyles)
Sheet E (Deleting Vertices Or Edges From A Graph)
Sheet F (Eulerian circuits and trails)
Sheet G (Classes of graphs and more)
Sheet H (Complement of a Graph)
Sheet I (Eulerizing a Graph: The Chinese Postman Problem)
Sheet J (Hamiltonian cycles, paths; Hamiltonian connected)
Sheet K (Nearest neighbor and Sorted Edges Heuristics for the TSP)
Sheet L (Trees: Valence Sequences)
Sheet M (Matchings)
Sheet N (Prim, Kruskal, and Boruvka)
Sheet O (Weighted Matchings)
Sheet P (Systems of Distinct Representatives)
Sheet Q (Matchings in Bipartite Graphs)
Sheet X (Transforming Plane Graphs)
Sheet Y (Plane Graphs and Their Duals; Coloring)
Sheet Z (Vertex Colorings)
Colorings I
Review Exam I
Review Exam II
Review Exam II (Part II)
Take Home Problem Set I
Writing Project
Mathematics 483 (Game Theory)
What is Game Theory?
Homework Assignment I
Homework Assignment II
Homework Assignment III
Homework Assignment IV
Homework Assignment V
Homework Assignment VI
Homework Assignment VII
Midterm Review
Writing Project
Spring, 2005
Humanities 320 (Honors Seminar)
The theme for this course is the Digital Revolution
Poll
Binary Arithmetic
Pixel Pictures
Modular Arithmetic
Modular arithmetic and codes
Error-Correction Codes I
Error-Correction and Data Compression (Huffman Codes)
Writing Assignment I
Writing Assignment II
Review: Examination I
Review: Examination II
Final Examination Review
Mathematics 225 (Discrete Mathematics)
Handout 1
Handout 2
Rational Numbers and Decimal Numbers
Base b Arithmetic
Relations and Digraphs
Review Exam I
Review Exam II
Review Exam III
Final Examination Review
Writing Project
Mathematics 244 (Geoemtric Structures)
Homework 1
Sheet B (4-gons)
Sheet C (Polyominoes)
Sheet D (Polyominoes II)
Sheet E (Equilateral triangles)
Sheet F (Equidistance problems)
Contrasting the Euclidean and Taxicab Planes
Finite Fields and Finite Geometries
Construction of Finite Fields
Field Extensions
The Euler Line and the Nine-Point Circle
Desargues' Theorem
Pappus' Theorem
Examples of Finite Affine, Finite Projective and Finite Bolyai Lobachevsky Planes
Error Correction and Geometry
Polygons and Polyhedra
Polyhedra and Graphs
Guards, Visibility and Polygons
Art Galleries and Guards
Plane Graphs
Euler relations
Review: Examination I
Review: Examination II
Final Examination Review
Writing Project
Mathematics 479 (History of and Philosophy of Mathematics)
Homework 1
Homework 2
Homework 3
Homework 4
Roman Numerals
Rational Numbers and Decimal Numbers
Egyptian Fractions
Writing Project
Examination Review
Final Examination Review
Fall, 2004
Mathematics 241 (Combinatorial Geometry)
Sheet C
Sheet M
Plane Graph Coloring Sheet
Plane Graph Coloring Sheet II
Writing Project
Examination I: Review
Inductive Proof that a tree with k vertices has k-1 edges
Mathematics 479 (History of and Philosophy of Mathematics)
Homework 1
Homework 2
Homework 3
Midterm Review
Writing Project
Liberal Studies 400 (Seminar In Fairness and Equity)