Hello! My name is Muireann, pronounced like Marin County with a soft ending, I'm an exchange student from Ireland! I grew up and live in Cork and study in the University there (UCC). I'm a double major in Mathematical Sciences and Physics.
I began university thinking I'd only like Algebra and Applied Maths but have found interest in subjects like analysis. I took Complex Analysis M185 last semester. I'm really liking the approach to this class so far it is very different to anything I've taken before, so I am really looking forward to it.
My other interests are music and sport. I play piano and saxophone, I teach piano at home and currently trying to teach myself the trumpet. I also ran the half marathon across the golden gate last November!
Study Methods
Revision QuestionsI find the best way to learn analysis is starting with definitions. So, every analysis class I make revision questions from every book. This helps me to form proofs ect later on. Maybe some of you will find them useful too. I will link them here:
Lebesgue Outer Measure (Pugh Chp6 and Tao Chp 7)
m105_fourier_series_revision_questions.pdf
Tool Box As I go through revising for exams and for problem sets. I try to put together a toolbox. Just definitions and theorems we have covered without the proofs. I'll add that here when it is complete
Class Notes
Class Notes will be uploaded in a pdf format
Homework Problems
Homework will be uploaded here in a pdf format
Interesting Links / Problems
Here I will post any useful links or resources I find myself using. As well as any interesting problems I have that a class mate could shed some light on.
HW4 Lebesgue Integral summary
Up until now we had only been solving Reimann Integral. The Lebesgue integral is in some sense a generalization of the Riemann integral. This was only possible with 'Reimann Integrable' functions, i.e not all functions could be integrated. A classic example is f(x) = 1, x is a rational number and zero otherwise on the interval [0,1].
The steps for Lebesgue Integral
1. subdivide the range of function into infinitely many intervals
2. construct a simple function by taking a function whose values are those finitely many numbers
3. Take limit of these simple functions, when more points are added in the range of original functions.
HW6 Summary of results for Lebesgue Measure
Outer Measure
1. From Pugh's approach, he defines the outer measure of a set using Intervals, Rectangles, and boxes.
Lebesgue outer measure of a set $A \subset \R$ is
$m^{*}A$ = $inf${ $\Sigma_{k}$ $\vert$ $I_{k}$ | : {$I_{k}$ is a covering of A by open intervals}
The important theorem for outer measure is proving its properties:
a) The outer measure of the empty set is 0, $m^{*}\emptyset$ = 0
b) If A $\subset$ B then $m^{*}A$ $\leq$ $m^{*}B$
c) if A = $\cup$ $A_{n}$ then $m^{*}A$ $\leq$ $\Sigma$ $m^{*}A_{n}$
Another definition we continued to use throughout Lebesgue Theory was If $Z \subset \R^{n}$ has outer measure zero then it is a zero set
2. The second topic we learnt about was Measurability
First defining (Lebesgue) measurable
A set $E \subset \R$ is Lebesgue measurable if the division $E|E^{c}$ of $\R$ is so “clean” that for each “test set” $X \subset \R$ we have
$m^{*}X$ = $m^{*}$( X $\cap$ E) + $m^{*}$(X $\cap$ $E^{c}$)
The fact an empty set is measurable and that if a set is measurable then so is its compliment was needed for futher chapters as we will see…
In this section we were introduced to $\sigma$ - algebra, which is a collection of sets that includes the empty set, is closed under complement and is closed under countable union.
3. Mesomorphism
measure space, differences between mesemorphism, meseomorphism, and mesisometry
4. Regularity
Theorem 11 Lebesgue measure is regular in the sense that each measurable set E can be sandwiched between an $F_{\sigma}-set$ and a $G_{\delta}-set$ , F $\subset$ E $\subset$ G , such that G\F is a zero set. Conversely, if there is such an F $\subset$ E $\subset$ G, E is measurable.
Affine motions..
Final Essay- Fast Fourier Transforms (FFT) I took a perspective based on my own background and how I visualize and use FT and FFT's in general final_essay_what_is_fast_fourier_transform.pdf
