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0

I liked P.M. Cohn's book, it came in 2 volumes and was easier to read. Jacobson is exhaustive, including specialized topics as constructing irreducible representations of $S_n$, but I always used Jacobson to read a specific topic something like 4 or 5 pages at a time.


3

Hungerford is great. You should prove the things he puts out as exercises. It will give you a strong grasp of Algebra. I used it to pass the Ph. D. qualifier at the University of Texas.


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Perhaps the best way to think about the difference is as one of emphasis. "Proof-based Calculus" would have the goal of doing Calculus but would justify the methods of Calculus (integration and differentiation) by proving their validity. "Analysis" would have the goal of developing the theory of Calculus (and more than just Calculus)* at least partly for ...


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For details a reference is the book of Hartshorne "Algebraic Geometry", page 116. If $S$ is a graded ring, then a graded $S$-module $M$ gives a sheaf $\tilde{M}$ on $Proj(S)$. On the basic open, $\tilde{M}(D(f)) \cong \tilde{M_f}$. Now, it's only pure algebra. There is a general construction with $S$ a graded ring, and $M$ a graded $S-$module. We can ...


0

Well, for finding a deterministic function $g(y)$ to minimize $E[(X-g(Y))^4]$, we can do this: \begin{align} E[(X-g(Y))^4] &= \int_{y \in \mathbb{R}} E[(X-g(Y))^4|Y=y]f_Y(y)dy \\ &\geq \int_{y\in\mathbb{R}} \left(\inf_{\theta}E[(X-\theta)^4|Y=y]\right)f_Y(y)dy \end{align} where equality is achieved by the function $g^*(y) = \arg\inf_{\theta} ...


1

Martin Isaac's book Algebra might be useful to you. It is at the graduate level so it might be useful if you are going through the material a second time.


1

Although I believe that Dummit and Foote's Abstract Algebra combined with Herstein's Topics in Algebra which has excellent exercises is the perfect recipe for the job, I would recommend two more books for the study of finite groups. The first one is Daniel Gorenstein's Finite Groups which is a book of great depth and covers a lot of material about groups. ...


1

I enjoyed Michael Artin's book Algebra. This would still be as you say the "group theory" part of an abstract algebra book, but it is rigorous and touches on the topics you have listed.


3

Try these excellent books: An Introduction to the Theory of Groups by Rotman. The Theory of Groups by Marshall Hall Jr.


0

There is a small list on Wikipedia: https://en.wikipedia.org/wiki/Riemann_hypothesis#Excluded_middle


0

Am not sure what you mean by discreet parameters, degree and rank? That will not be enough. As you said, take $C$ to be a smooth cubic curve. Then $\mathcal{O}_C$ and a line bundle $L$ of degree zero but $L\neq \mathcal{O}_C$ have the same discreet parameters. But ...


1

I hope you can find this useful. All elliptic curve $C(\mathbb {Q})$ is canonically write as $y^2=4x^3-g_2x-g_3$... (1) in which the three roots of the right side are distinct; by a birational transformation $(x,y)\to (\frac x4, \frac y4)$ you have $y^2=x^3-h_2x-h_3$... (2) where $h_2$ and $h_3$ can be supposed rational integers. Being $e_1, e_2, e_3$ ...


0

The proof of that result, usually called (an explicit version of) the Weak Mordell-Weil theorem, can be found in Silverman's Arithmetic of Elliptic Curves book. The proof uses Galois cohomology and some minor arithmetic that can be followed with the knowledge of a few of the main theorems of global class field theory.


1

I can recommend the following book "Probability and Statistics by Example: Volume 2, Markov Chains: A Primer in Random Processes and their Applications" by Yuri Suhov and Mark Kelbert This book covers most of your topics (depends of course how deep you want to dive into), is well written and explains everything with examples and solved exercises. So at ...


3

A good starting point would be Introduction to Probability Models by Sheldon Ross. I think it covers all the topics that you described. I like the book because it's easy to read and has plenty of problems to try out, which makes it ideal for self learning.


0

The interesting question is the action on the largest open subset of the reals on which $\Gamma$ acts properly discontinuously (denote it by $\Omega$). Then $\mathbb{H} \cup \Omega/\Gamma$ is a complex Riemann surface with boundary. $\Omega/\Gamma$ is called the ideal boundary of $S$. It may be empty. It plays an important role in Teichmüller theory, see ...


0

Here is one specialized solution. Let $h=g$. The the original equation becomes: $$(3n-3)(f^2+2g^2)=(9n+1)(2fg+g^2)\tag{1}$$ We solve (1) for $f$ and obtain (assuming $g>0$): $$f=A(n)g\tag{2}$$ $$A(n)=\frac{9n+1\pm\sqrt{10(9n^2+3n-2)}}{3(n-1)}\tag{3}$$ (a) For $n=2$, We have $A(2)=-1/3,13$. If $g=3 j$ where $j$ is arbitrary positive polynomial with ...


0

An excellent book on the matter is "Networks, Crowds, and Markets: Reasoning about a Highly Connected World". A more brief one, but also a high quality work, is "Structure and Dynamics of Information in Networks" by David Kempe. Cheers.


2

A closed form expression is provided in the following paper. SV Amari, RB Misra, Closed-form expressions for distribution of sum of exponential random variables, IEEE Transactions on Reliability, 46 (4), 519-522.


0

I ended up ordering Handbook of Linear Algebra, 2nd Ed by Hogben. Large tome, but great index and binding.


1

does this theorem have a common name? It is sometimes called König's Theorem (1936), for example in lecture notes here. However, this name is ambiguous.


0

There are different resources for reading on game theory 1) There are online lectures: Game theory 101 which is fairly popular. 2) There is a book: Non Cooperative Game Theory by Tamer Basar, which covers the topic like a subject. 3) There is a nice blog called : www.mindyourdecisions.com Overall to start with you should study the following: 1) Normal ...


1

I have used the book Lie Groups, Lie Algebras, and Cohomology by Anthony W. Knapp. It has many examples and is very detailed. It may be a good start for you.


3

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 ...


1

My experience tells me if you are a beginner first start with video lectures. Take a look here. fuzzy And then these are the text books which are used as references for fuzzy systems course in the university where I study. 1- FUZZY SETS AND FUZZY LOGIC Theory and Applications 2- Advances in Fuzzy Clustering and its Applications 3- FUZZY MODELS AND ALGORITHMS ...


0

You can download Graphing Calculator 3D and use to plot any multi-variable formula in 3D. It's very easy to use.


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.


0

The best resource for "practical" packing problems I know of is Eric Friedmans' "Packing Palace." He has an answer to this problem for equilateral triangles and squares. Sadly, he does not address rectangles in triangles.


2

A reference I have found very useful is the book "Second Order Elliptic Equations and Elliptic Systems'' by Ya-Zhe Chen and Lan-Cheng Wu (English translation). The proofs are complete and well organized. In chapter 9, theorem 2.6 the theorem is proven when $k=\nabla \cdot F$ and $F$ is $C^{0,\alpha}$. The result also holds when $k\in L^p$ and $p>n$ and ...


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

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.


0

I myself have been recently looking for a citation and a proof, but here (I think) is the idea behind it. First, to clear up the question, I will assume you are referencing the brief tidbit mentioned in the commutative diagram page on Wikipedia under "verifying commutativity" (https://en.m.wikipedia.org/wiki/Commutative_diagram). This means you are talking ...


0

I was also struggling to find any resources that explicitly even stated the theorem itself, rather than some special case or implication - including the Wikipedia article - until I came across this: http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes12.pdf which gives a bit of an explanation, a statement of the theorem and a proof, starting ...


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

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 ...


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, ...


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.


0

You might find this video lecture series useful: Group Theory by LadislauFernandes. He has quite an extensive set of videso on introductory group theory. I haven't watched everything but from what I have, the quality is ok but a teensy bit slow.


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 ...


1

This answer is intended to complement David's very nice solution by showing that the partial sums of the Fourier integrals are at most be pointwise unbounded almost everywhere (a.e.), when $f\in L^{1}(\mathbb{R})$; there exists a subsequence which converges to $t$ pointwise a.e. Combining this with David's solution, we see that the partial sums of the ...


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 ...


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: ...


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, ...


3

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


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.


4

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


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 ...


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 ...



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