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 Questions//**I 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) // {{ :math105-s22:s:mheaney: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 {{ :math105-s22:s:mheaney:final_essay_what_is_fast_fourier_transform.pdf |}}