The Math Major
CSU Fresno Mathematics Department
October 27, 1996
Vol 1. No. 5
Editor: Dr. Larry Cusick.
The Math Department Colloquia are a series of talks intended for a general audience. Students, in particular, are encouraged to attend.
- Date: Monday, November 4th
- Speaker: Dr. Bela Bajnok from the Gettysburg College Department of Mathematics
- Title: The Boolean Lattice Model of Communication Networks
- Time: 2:30-3:20pm
- Location: Science 143
- Abstract: Imagine a large network of computers (or telephones) where a message needs
to be sent from a source node to a target node, but a certain number of
intermediate nodes are unavailable (malfunctioning or busy). The field of
Network Reliability is concerned with determining whether the network is
capable of carrying out the task of information transmission. In this talk
we answer two questions (one where independent nodes fail and one where
chains of nodes break down) in the Boolean Lattice, a lattice often used
for modeling communication networks. No previous knowledge of either
Network Reliability or the mathematics involved is assumed.
Spring Math Courses
The math department will offer 11 upper division and 4 graduate level mathematics courses this spring. In this issue and the next you will find a short description of each in the words of the instructor.
- 101: (Statistical Methods--Tannenbaum) This course introduces the student to the most important and commonly used techniques for data collection, data description and data interpretation. The emphasis is two fold: (a) the mathematical foundations of statistical methods and (b) their applications in the real world (economics, biology, engineering and social science).
- 107: (Introduction to Probability and Statistics--Harbertson) This course presents those aspects of mathematical probability theory that are needed to use probability models in real situations, particularly in statistics and modeling stochastic events (events which involve uncertainty). Included are discussion and examples for the definition of probability spaces, elementary probability calculation, and presentation of the important probability distributions that arise in applied mathematics, engineering and science.
- 116: (Theory of Numbers--Woo) Number theory illustrates the beauty of pure mathematics and also provides a familiar context for mathematical explorations.
This course involves students formulating conjectures about natural numbers and verifying hypotheses through inductive and deductive proofs. Students should have a clear understanding of elementary number theory after completing this course.
- 124: (Applied Matrix Analysis--Nur) To introduce the algebra of matrices and to present applied problems whose solutions are facilitated by the use of matrices. The presentation of the subject is basic and should be accessible to any one who has had three semesters of calculus. The purpose of the course is to present the beauty of linear algebra and its value.
- 151: (Principles of Algebra--Zane) Professor Zane only teaches Spring semester, and was unavailable to write something. But, those of you who have had him in the past know his courses are both lively and thorough. Abstract algebra brings together all areas of mathematics using the unifying notion of symmetry. Applications are to geometry, combinatorics and solving equations. (Ed.)
- 152: (Linear Algebra--Duncan) The emphasis of this course will be on concepts and proof, although practical computational techniques will be covered. There will be no assigned text.
- 161: (Principles of Geometry--Cusick) In this course we will build Euclidean and Non-Euclidean (Hyperbolic) plane geometries up from scratch using axioms. The course will be proof oriented and we will use a Mathematica (computer) package written by the instructor that draws pictures and computes accurately in both geometries.
Next week: maths 171A and up.
Math Club News
The Math Club will meet on Tuesday October 29 in PB428 at 6:30pm. The topic of discussion will be: Visitations to Reedley and Kerman High Schools by math majors. Stop by if you are interested. If you are interested, but cannot attend, call math major Tina Attashian at 435-9356 or e-mail: ta100@csufresno.edu.
Problem Corner
Problem 1.4 Suppose X = {x1, x2, ..., xn} is a set of distinct positive integers. The weight of X is defined to be the product w(X) = x1 x2... xn. A partition of X is defined to be a collection of mutually disjoint subsets of X whose union is X. If X1, ..., Xk is a partition of X, then we define the weight of the partition as the sum w(X1, ..., Xk) = w(X1) + ... +w(Xk). Which partition of a set of positive integers X has minimum weight?
Solution to Problem 1.4: First we observe that if a and b are two integers larger than 1, then ab >= a + b. Now, suppose X = {x1, x2, ..., xn} and 1 not in X. By our observation, w(X) >= w(A) + w(B) for any 2 subset partition A, B of X. Now proceed inductively to conclude that the minimum weight partition is the set of singletons {x1}, {x2}, ... {xn}. If, on the other hand, 1 in X, then we may write X = {1} U X' where 1 not in X' = {x1, ..., xn}. Thus w(X) = w(X') >= x1 + ... + xn. Since w({1, x}) = x for any x in X', a partition of minimum weight of X is one in the form {1, x1}, {x2}, ..., {xn}.
Correct solutions to problem 1.4 were received from Anar Ahmedov and David Yoshihara.
New Problem
Problem 1.4 A list of consecutive positive integers beginning with 1 is written on a black board. One of the numbers is erased. The average (arithmetic mean) of the remaining numbers is 35 7/17. What number was erased?
Solutions may be delivered to the
math department office (for Dr. Cusick) or by e-mail at larry_cusick@csufresno.edu. no later than Thursday November 7, 3pm. There is a $75 dollar first prize and a $50 second prize to be awarded at the end of the semester to the student(s) who submit the most correct solutions.
CSU Fresno Math Department Home Page
California State University, Fresno