**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)

Discussion Questions 1

Writing Project 1

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

The following web pages have materials that are related to fairness and equity questions:

Basic Fairness:

Introduction to Basic Fairness Ideas

Elections:

Primer About Elections

Elections and Voting

Voting games and weighted voting:

Voting Games I

Voting Games II

Apportionment:

Apportionment I

Apportionment II

Example 1

You may find this glossary of terms that come up in fairness and equity questions of use.

You can find my current (or most recent) teaching schedule at York here.

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Updated 8/27/2008