Intereting Posts

How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one?
Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$
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The speed of the top of a sliding ladder
$X$ is algebraic over $E$
Integral help with completing the square!!
Prove that $p \in \mathbb{R}$ can be represented as a sum of squares of polynomials from $\mathbb{R}$
How to compare points in multi-dimensional space?
How to use the Mean Value Theorem to prove the following statement:
How to show a sequence of independent random variables do not almost surely converge by definition?
Continuous probability distribution with no first moment but the characteristic function is differentiable
Uniform convergence in distribution
Why isn't the Ito integral just the Riemann-Stieltjes integral?
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Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.

I am a Japanese so it is difficult for me to read English and I may make some grammatical mistakes.

I have little experience with mathematics but plan to self-study mathematics by reading mathematics books in English. I would like to know what the best text is in each of the following subjects,

- set theory
- abstract algebra (algebraic structure, etc.)
- linear algebra
- analysis

Please provide a rationale behind suggestions. I would prefer texts with many examples (as opposed to formulas), and length and price are not a prohibitive consideration.

- probability textbooks
- Long exact sequence for cohomology with compact supports
- If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
- Understanding the Musical Isomorphisms in Vector Spaces
- Definition of definition
- A (possibly) easier version of Bertrand's Postulate

- Lee, Introduction to Smooth Manifolds Solutions
- Foundations book using category theory for student embarking on PhD in mathematical biology?
- ArcTan(2) a rational multiple of $\pi$?
- Connections between SDE and PDE
- History of Commutative Algebra
- Most inspirational mathematical books
- Reference to self-study Abstract Algebra and Category Theory
- Book for field and galois theory.
- Understanding differentials
- Is the set of all conformal structures on $\mathbb{R}^n$ a manifold? Does it have a name?

**Naive Set Theory – Halmos**. This is a good starting place for set theory. It is about 100 pages in length. If you find this work interesting, then you can move on to more formal\advanced texts such as **Hrbacek & Jech – Introduction to Set Theory**.

For Algebra, you will probably get a mix bewteen **Fraleigh** and **Herstein**. Both are excellent but I would recommend you get a copy of Fraleigh since he likes to explain everything in minute detail. You can find lectures of Fraleigh (free online at UCCS). If you want something a little bit more advanced, I would recommend **Algebra – Artin** (there is a free lecture series by Harvard on this text).

For Linear Algebra, ‘**Linear Algebra Done Right**‘ is very popular. **Introduction to Linear Algebra – Strang** is also very good. You will be able to find MIT lectures on this book (by the author himself!), which makes it perfect for self study. Even though a bit dated, ‘**Matrices and Linear Transformations – Cullen**‘ is written very clearly and will guide you to the core of the theory in a very clear way. The notation is just a bit outdated.

Analysis is a difficult one since it largely depends on your ‘mathematical maturity’. Since you say, for self study, I would recommend “**Analysis – Steven Lay**‘. This book has an excellent introduction to logic and proofs. It also covers set theory and functions and how to prove just about anything relating to sets and functions. Moving on from there, there is of course ‘**Rudin’s Principles of Mathematical Analysis**‘. Of course, this is not a book that is suited for self study, but you can find free lectures on youtube (Harvey Mudd) which covers about half of the material in this book. The lectures are very good and will enable you to work through this material.

- Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.
- Integral: $\int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t$
- How to show that the given equation has at least one complex root ,a s.t |a|>12
- Parenthesis vs brackets for matrices
- An example of an easy to understand undecidable problem
- How to show $ a,b,c \in \mathbb R , z \in \mathbb C , az^2 + bz + c = 0 \iff a\bar{z}^2 + b\bar{z} + c = 0$?
- Complement of a totally disconnected compact subset of the plane
- prove the existence of a measure $\mu$
- Bellard's exotic formula for $\pi$
- Show $\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}d\theta=0$
- The unit square stays path-connected when you delete a cycle-free countable family of open segments?
- Product rule for logarithms works on any non-zero value?
- The fundamental group of $U(n)/O(n)$
- Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $
- Solving a literal equation containing fractions.