Name 3
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MATH 403
EXAM 2
November 11, 2014
This exam contains 5 problems.
You will have 75 minutes to take the exam.
No books1 notes, or calculators are allowed on the exam.
Problem 2 3 4 5 TOTAL
5 5 2
1
Possible 1 l 1
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Name & Section:
Homework 1, MATH 403, Fall 2015
Due Friday August 28 by 4 PM.
Only starred problems are graded.
(1)* Prove that
3 is irrational.
(2)* Browder 1.10.6
(3)* Browder 1.10.18
(4)* In addition for honors students: rst read Browder p. 5 about the
MATH 403 ANALYSIS I - SPRING 2010
SOLUTIONS to HOMEWORK 5
Problem 1. Let f : (X, d) (Y, ). Show that the following are equaivalent:
(a) f is continuous.
(b) f (A) f (A) for every A X .
(c) f 1 (B ) (f 1 (B ) for every B Y .
Solution: (a) = (b) Assume that
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lVIath 403 Homework 5
due Tuesday. 10/ 14
)r any set X the dismte meme on X is define as p(:i:,y) = 1 if :1: 3% U and
Mr, 31) :- ii if r z y. Verify that p is a distance function.
3 tt (X. p) be any metric space. W can define a new metric on X as
I . .
IV-K'tmrldhu'! w 7- .
W. t 7- :, . .
.n 4., 7 , .' 4 .- -u- r - 7 ' vigw' u' U 5 .-wMAaH
.u. x «.4. m»~mu¢.«<vw 7 .- .-77_.,h,_._* _
7 .,.W_.
/ " V" ' .\ u . r. . ,v- . nll;.- r 7 .
a A .10 >5 . n r 7 "
- (75 . - V
gum» .-».7. .v 7.
_. + a
I? .I 1. . I .I (IF.
I. r. [IL-n I I
I . I I
r I :I III: . I I III 5
\J
1 I : IIImv.-I_IiIII. I
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- IIIPII. - - - .
OIerTInErI, IIuIeSruI'ing
I
I
f .1! \ll
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MATH 403H
PROBLEM SET #17
Vector spaces
Definition. A nonempty set L with two operations, addition + and
multiplication by scalars R, is called vector space if it satises the following
three groups of axioms:
(1) L is an abelian group under addition.
(2)
PROBLEM SET #4
Minimality of the Cantor sets
Cantor sets can be obtained not only as subsets of [0, 1], but in many
other ways as well.
Definition. Let (X, d) and (Y, ) be two metric spaces. Then
the map f is continuous at the point x0 if for any > 0 the
PROBLEM SET #6
Points of discontinuity and the Baire Category Theorem
Let us consider the following two functions:
Q (x) =
1
0
if x is rational,
if x is irrational,
called the Dirichlet function, and
1
if x = 0
r(x) = 1/q if x = 0 is rational, x = p/q, (p
PROBLEM SET #7
Contraction mapping principle
Definition. Let (X, d) be a metric space. A map f : X X is said to
be contracting if there exists < 1 such that for any x, y X
d(f (x), f (y) d(x, y).
(1)
Notice that the inmum of numbers satisfying (1) also sa
MATH 403H
PROBLEM SET #8
Definition. A subset C of a metric space (X, d) is said to be compact
if any open cover of K has a nite subcover, that is whenever cfw_Oi | i I
is a collection of open sets of X indexed by I such that C iI Oi then
there is a nite
MATH 403H
PROBLEM SET #10
Connectedness
Definition. A subset E of a metric space X is said to be disconnected
if there are disjoint open subsets U1 and U2 in X such that
E U1 U2 , E U1 = , E U2 = .
A set E is connected if it is not disconnected.
If we con
MATH 403H
PROBLEM SET #9
Continuity and compactness
66. Let X be a compact metric space and f a continuous map of X onto
a metric space Y . Prove that Y = f (X) is itself compact.
Solution. Let cfw_V be an open cover of Y and U = f 1 (V ). Then U
are ope
MATH 403H
PROBLEM SET #11
Connected and path-connected sets
82. The continuous image of a path-connected space X is path-connected.
Solution. Let f : X Y be continuous and surjective; take any two points
y1 , y2 Y . Then by surjectivity the sets f 1 (yi )
MATH 403H
PROBLEM SET #12
Universality of the Hilbert cube
We proved in 29 that any perfect complete metric space X contains a
closed subset homeomorphic to the Cantor set. This means that the Cantor
set is in some sense a minimal nontrivial compact metri
MATH 403H
PROBLEM SET #13
108. The Sierpinski Carpet S is obtained from the unit square by removing the middle-ninth square (1/3, 2/3) (1/3, 2/3), then removing
from each square (i/3, (i + 1)/3) (j/3, (j + 1)/3) its middle-ninth and so
on. Compute the box
MATH 403H
PROBLEM SET #14
The spaces of continuous functions
Let (X, d) and (Y, ) be two metric spaces, fn : X Y be a sequence of
functions. When does this sequence converge to a limit? We can consider
dierent kinds of convergence.
Definition. Pointwise c
MATH 403H
PROBLEM SET #16
Stone-Weierstrass Approximation Theorem
Our goal is to prove that any f C 0 ([a, b], R) can be uniformly approximated by polynomials.
126. Prove that for any continuous function f : [a, b] R that for any
> 0 there exists a piece
MATH 403H
PROBLEM SET #15
120. Let X be a compact metric space X, and A be a dense in X. Prove
that for any > 0 there is a nite subset cfw_a1 , . . . , ak A which is -dense
in X in the sense that each x X lies within distance of at least one point
aj , =
PROBLEM SET #5
33. Prove that a map f : X Y is continuous if and only if for each open
set U Y its inverse image f 1 (U ) is open in X.
Solution. Let x0 f 1 (U ). Then f (x0 ) U , and since U is open, there
exists an > 0 such that y U if (f (x0 ), y) < .
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Mathematical problems of boundary layer theory
This content has been downloaded from IOPscience. Please scroll down to see the full text.
1968 Russ. Math. Surv. 23 1
(http:/iopscience.iop.org
MATH 403 ANALYSIS I - Spring 2010
Extra Problems
Problem 1. Let d1 and d2 be metrics on X. Show that d1 , d1 + d2 ,
maxcfw_d1 , d2 are also metrics. Are the functions mincfw_d1 , d2 , d1 d2 metrics on
X?
Problem 2. Let (X, d) be a metric space. Show that
TransitiontoTurbulenceandSingularityinBoundaryLayerTheory
F.Gargano,M.Sammartino,V.Sciacca
DipartimentodiMatematicaedApplicazioni,UniversityofPalermo
Introduction to zero viscosity limit of Navier Stokes equations
We test numerical scheme in the case of a
M403 Analysis, Spring 2010
Final Exam Review Problems
On the final exam you will be asked to solve problems using definitions and
theorems discussed in class. You will also be asked to give proofs of two of
theorems listed below.
1. Proposition 7.12 A set