Maths 4200-Fall 2013
Some Practice exercises
Chap2-Topology
1. Let A and B two subsets of R and assume that A is open and that B is closed.
Prove that A B is open and B A is closed.
2. Let (X, d) be a metric space. Let E be a subset of X . Prove that E is
arXiv:1302.6097v1 [math.NT] 25 Feb 2013
On Shifted Eisenstein Polynomials
Randell Heyman
Department of Computing, Macquarie University
Sydney, NSW 2109, Australia
[email protected]
Igor E. Shparlinski
Deptartment of Computing, Macquarie University
Sydne
Why Eisenstein Proved the
Eisenstein Criterion
and Why Sch nemann Discovered It First
o
David A. Cox
Abstract. This article explores the history of the Eisenstein irreducibility criterion and explains how Theodor Sch nemann discovered this criterion befor
Maths 4200
Fall 2013
Take-Home Assignment 1 with solutions
Instructions: Your solutions must appear in an organized and legible format to be
given full consideration.
1. If r is rational (r = 0) and x is irrational, prove that r + x and rx are irrational.
Maths 4200-Fall 2013
Take-Home Assignment 2 with solutions
Due Date: In-Class September 16th
Instructions: Your solutions must appear in an organized and legible format to be
given full consideration.
1. (ex 10 pp44) Let X be an innite set. For p X and q
Maths 4200-Fall 2013
Take-Home Assignment 3 with solutions
Due Date: In-Class September 25th
Instructions: Your solutions must appear in an organized and legible format to be
given full consideration.
1. Let (xn )n1 a real-valued sequence. Prove that the
Maths 4200-Fall 2013
Take-Home Assignment 5 with solutions
Due Date: In-Class October 21st
Instructions: Your solutions must appear in an organized and legible format to be
given full consideration.
1. Using the / denition, prove that f (x) =
x is uniform
Mathematics 4200
Midterm 1 with solutions
October 4th 2013
Partial credit will be awarded for your answers, so it is to your advantage to explain your
reasoning and what theorems you are using when you write your solutions. Please answer
the questions in
Mathematics 4200
Midterm 1
October 4th 2013
Partial credit will be awarded for your answers, so it is to your advantage to explain your
reasoning and what theorems you are using when you write your solutions. Please answer
the questions in the space provi
Mathematics 4200
Midterm 2
November 8th 2013
Partial credit will be awarded for your answers, so it is to your advantage to explain your
reasoning and what theorems you are using when you write your solutions. Please answer
the questions in the space prov
Maths 4200-Fall 2013
Some Practice exercises
Continuity and Dierentiability
1. Determine which of the following continuous functions are uniformly on the given
set
(a) f (x) =
(b) f (x) =
ex
x
1
x2
on (0, 2).
on (0, ).
(c) f (x) = x sin(1/x) on (0, 1).
2.
Maths 4200-Fall 2013
Some Practice exercises
Chap3-Sequences
1. Let (sn )n1 be a real-valued sequence. For each of the following, prove or give a
counterexample
(a) (sn )n1 converges to s then (|sn |)n1 converges to |s|
(b) If (|sn |)n1 converges then (sn
Why Eisenstein Proved the Eisenstein Criterion and Why Schnemann Discovered It First
Author(s): David A. Cox
Source: The American Mathematical Monthly, Vol. 118, No. 1 (January 2011), pp. 3-21
Published by: Mathematical Association of America
Stable URL: