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8

If you want to have a general idea of what a subject is about, then casual reading will be sufficient. It will help you understand what the problems are in the field, but not a lot more. Also, in any mathematics book, after you get through the first couple of chapters, it will get hard to read if you haven't been working along the way. The exercises in a ...


7

I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested in Algebraic Geometry and Number Theory, ...


6

What you propose is perfectly fine... although all too often textbooks (intentionally or not) play a sort of passive-aggressive game wherein the main text does not "let on" certain subtleties, on which the "exercises" specifically prank the reader. Or major results are posed as "exercises", for which there's no methodological prototype in the text. This is ...


5

In my opinion Hardy &Wright's book on Number Theory is not the best possible book for someone "who has no prior training in Number Theory", I would suggest the following books. Elementary Number theory by David M. Burton. Number Theory A Historical Approach by John H. Watkins Higher Arithmetic by H. Davenport All the books are ...


5

I think it's pretty hard to find a book which covers martingale theory; usually, books either give just an introduction or they focus on one particular aspect of martingale theory. I'll list some books which might be of interest and sketch (roughly) which parts they cover: David Williams: Probability with Martingales (Basic properties, optional stopping, ...


4

One of the best is An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery.


3

I would recommend An Introduction to the Theory of Numbers By G.H. Hardy and E.M. Wright .


3

Yes, can you can - it's just that the sum of two "points", i.e., the sum of the pure states associated to two points, is not necessarily itself a pure state. As your excerpt says: ... whereas a general state $\rho:A\to \mathbb{R}$ is a distribution of such specific states. So there is certainly not, in general, an addition operation on the points of ...


3

The paper "On The Bound of Proximity of the Binomial Distribution to the Normal One" - Nagaev, Chebotarev (2010) has improvements to $C$ specifically for the Binomial Distribution. Theorem 2 on page 3 gathers together the results in the paper to show that $C$ can be taken to be .4215 in general. The paper notes that an Esseen (1956) paper demonstrates that ...


3

In the classical theory, you see that the set-theoretic points of the complex manifolds $Y(N)$, $Y_1(N)$, etc., are in bijection with certain complex tori plus some additional data (level structure) (up to isomorphism). But they are in fact moduli spaces in the more precise sense (involving Yoneda's lemma) for complex analytic elliptic curves over more ...


3

You have the following result. Theorem. Let $a$ and $b$ be rational numbers which are not integers and such that $a+b$ is not an integer. Then the number $$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}=\int_{0}^{1}u^{a-1}(1-u)^{b-1} \,du$$ is transcendental. Here is, among others, an interesting reference from Michel Waldschmidt: Transcendence of ...


3

Necas' book "Direct Methods in the Theory of Elliptic Problems" is a wonderful guide for the topics that you mentioned above, although the proofs are very abstract and most steps are omitted. Evans' book is more understandable and also includes the topics above, but not in the most general settings of the theorems. If you just start studying the Sobolev ...


2

I think one of the reasons why textbooks encourage you to do a lot of exercises is so that you can see things for yourself. Sometimes even the derivation of important results will be left as an exercise to the reader. One of the best teachers I've ever had almost never presented us with a statement of fact or lectured us while we took notes. Instead, she ...


2

In the complex case, you have $(Su,v)=\overline {(Sv,u)} $ for all $u,v$. Then $$ (Su,v)=\overline {(Sv,u)}=(u,Sv)=(S^*u,v). $$ So $(\, (S-S^*)u,v)=0$ for all $v $, which implies $(S-S^*)u=0$. As this occurs for all $u $, $S-S^*=0$. For the real case, just remove the bars.


2

Here is a good beginners graph theory note (see https://homepages.warwick.ac.uk/~masgax/Graph-Theory-notes.pdf). It was designed for a fourth year undergraduate (equivalent to a Masters) module in Mathematics at the University of Warwick. It gives a good insight to graph theory that will be important in giving you a very good exposure. There are also a ...


2

A delicious classic would be "The Theory of Functions" by Titchmarsh. Despite being old (1939), it is very modern in its rigour, fear not! You may find it freely available on the internet, it is no longer subjected to any copyright.


2

Elements of Abstract Algebra by Allan Clark is the shortest book I've ever seen. This book is a little strange in that it covers Field and Galois Theory before ring theory if I remember correctly. It's also not a "hand holding" book and it expects you to do some work to read through it. This can be good or bad depending on the individual. Serge Lang's books ...


2

You should look at Barr, M. Acyclic models, CRM Monograph Series, Volume 17. American Mathematical Society, Providence, RI (2002), and see if that satisfies you for completeness. A theorem involving crossed complexes rather than chain complexes is in Section 10.4 of the book partially titled Nonabelian Algebraic Topology (2011). This version gives in ...


2

The short answer is given by the Grothendieck-Lefschetz trace formula $$ \#X(\mathbf F_p) = \sum (-1)^i \mathrm{Tr}(\mathrm{Frob}_p \mid H^i_c(X;\mathbf Q_\ell)).$$ For $X$ a compact modular curve associated with a congruence subgroup $\Gamma$, this means that you need to understand the Galois representation $H^1(X,\mathbf Q_\ell)$, as the trace on $H^0$ and ...


2

Depends on what you mean. There is the standard reformulation lifting technique (RLT) typically used for bilinear/polynomial problem leading to large LPs. Googling on that leads to a wealth of material by, e.g., Sherali, Balas etc. Then you have lifting to more exotic cones such as semidefinite relaxation. Same thing there, google semidefinite relaxations ...


2

I'm not sure what is covered in Enderton's book, so I will just assume that it was the basic set theoretic introductory material, and answer according to that. Your mileage may vary according to what you actually know. To study and understand independence proofs you need to be comfortable with several things: Basic logic, namely the completeness theorem ...


2

This is the rather well known theorem about the derivative of an inverse function. Let us put $\;g:=f^{-1}\;$ , for simplicity, and observe that both $$\begin{cases}\;y\to b\implies& \;\;(y=)\;\color{red}{f(x)\to f(a)}\;(=b)\\{}\\f(x)\to f(a)\implies&\;\; (g(f(x))=)\;\color{red}{x\to a}\;(=g(f(a)))\end{cases} \;\;\;\text{(why?)}$$ , so: ...


2

Hint: Use the chain rule $$(f^{-1} \circ f )(x) = x \implies (f^{-1} \circ f ) ' (x) = 1 \implies (f^{-1})'(f(x)) f'(x) = 1$$ Apply it at $a$. Conversely, if $f'(a) \neq 0$ then any sequence of points $y_n = f(x_n) \in Y - \{b\}$ with $\lim y_n = b$ the continuity of $g$ at $b$ yields $\lim x_n = a$ then $$\begin{align}g'(b) = \lim \frac{g(y_n) - ...


2

Dover publishes many number theory titles. At \$10-\$15 each they're a bargain - no need even to look for the Amazon discount. You can get several and jump back and forth among them to get different perspectives on each topic. You can write yourself notes in the margins. Take them to the library to read. This is a standard old undergraduate text: ...


2

I can certainly recommend Elementary Number Theory by Gareth A Jones et al. It will get you started and then you can move onto more advanced texts. It's a very short book (about 300 pages) which means you can easily read through the whole text-a very good choice for self study. As for pre-requisites, a good grasp of algebra will probably do.


2

I highly recommend reading "Prime Obsession" by John Derbyshire. Although the book is centered around the Riemann Hypothesis, it clearly explains the steps taken prior to it and how we have come to where we are now in order for the reader to gain an understanding of the history of primes. The book is split into two main sections. The even chapters provide ...


1

The key observation is the inequality $$\tag{1} |\tau(xy)|\leq \|x\|\,\tau(|y|). $$ (proof of this at the end). Now, for a fixed $x\in M$, write the polar decomposition $x=u|x|$; then $|x|=u^*x$, with $\|u\|\leq1$. This and $(1)$ show that $$\tag{2} \|x\|_1=\sup\{|\tau(xy):\ y\in M,\ \|y\|\leq1\}. $$ For $y$ with $\tau(|y|)=1$, the inequality $(1)$ shows ...


1

Up to sign, this array seems to appear in numerous contexts, but I don't see a name. Your array matches the following three OEIS entries, up to sign: A028246, A163626, and A142071.


1

For 2, if the collection is finite, then you can consider its interval graph, which is a graph representation of the intersections among the intervals in the collection. Now I quote Wikipedia: The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs.


1

Another possibility is the following paper: Samuel Eilenberg and Saunders MacLane. “Acyclic models”. In: Amer. J. Math. 75 (1953), pp. 189–199. ISSN: 0002-9327. JSTOR: 2372628. I read it a few months ago so I may not remember perfectly, but if I recall correctly, everything was done in detail and I didn't have to fill in any significant gap. The most ...



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