This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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2
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0answers
61 views

Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
1
vote
1answer
99 views

Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes ...
3
votes
1answer
60 views

Unsure on which sources to choose related to Calculus

I tried to get into Spivak's Calculus only to find that I've never been taught the type of Math presented there. First chapters talk about the properties of numbers, then mathematical induction, ...
1
vote
1answer
38 views

I've been working on Spivak and I'm on chapter 7. What are some good books to supplement Spivak for someone beginning to learn pure mathematics.

If I have too much difficulty with a concept/problem, then I'll just press on and solidify my understanding when the concept arises later by going back to it. This seems to be a lucrative method at ...
0
votes
0answers
14 views

Sheaves of Simplicial Rings?

Could someone provide me a reference for a treatment of sheaves of simplicial commutative rings? As in simplicial sheaves with a ring structure.
0
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0answers
18 views

Mapping cylinder of chain complexes via $-\otimes \Delta$

An instructor gave me a homework set where the mapping cylinder of a chain map $C_\bullet \xrightarrow{f} D_\bullet$ is defined as $(\Delta^1_\bullet \otimes C_\bullet) \oplus_{C_\bullet} D_\bullet$, ...
1
vote
0answers
61 views

Cardinallity increasing constructions

If we start with $\mathbb{Z}$ we can through localization get $\mathbb{Q}$, but that has the same cardinallity as $\mathbb{Z}$, so it doesn't increase cardinality for infinite sets, which is what I am ...
1
vote
1answer
44 views

Comparison of capacity of sets in $\mathbb{R}^n$

This is mainly in reference to this MSE post. Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \...
1
vote
1answer
26 views

minimum distance of a linear codes

My question is about computing the minimum distance (weight) of a linear code. Assume that we have the generating matrix of the code. Then we can easily compute the weights of each row and of course ...
0
votes
0answers
28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
0
votes
2answers
47 views

Order Theory and Lattice Theory Synonymous?

Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?
1
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0answers
47 views

How to see that diagonal and tranvections matrices generate $GL_n(\mathbb{Z})$?

I'm trying to see how diagonal and transvection matrices generate $GL_n(\mathbb{Z})$. Is there any book that I can find a more detailed description of this problem? Thanks!
0
votes
0answers
49 views

classification of equilibrium points of 3d systems of ode's

I'm trying to find information about the classification of equilibrium points of 3d systems of differential equations, The qualitative analysis. I wonder if someone could refer me to some book or ...
2
votes
1answer
43 views

Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
2
votes
1answer
36 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
0
votes
0answers
23 views

Implicitization Problem on Graphs?

I learnt the implicitization problem for varieties in introduction course on Algebraic Geometry. I am trying to understand how to formulate a similar implicitization problem on graphs where the ...
1
vote
0answers
38 views

Green's formulae, Stokes theorem, Gauss theorem, divergence theorem and Gauss-Green theorem?

I am getting really confused about the Green's formulae, the Divergence theorem and all those related equalities. For example, How is this formula exactly called? $$\int_\Omega \frac{\partial u}{\...
6
votes
1answer
57 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...
0
votes
0answers
23 views

Any book where Isometric Isomorphisms between Banach Spaces are explained?

When we work in Functional Analysis we usually say things like $(L^p)^*= (L^p)'$ when we want to expres that there exists an isometric isomorphism between these two spaces. All books that deal with ...
1
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0answers
58 views

What can I do to learn special functions? [closed]

I want to learn special functions but I'm finding the book by Ranjan Roy far too advanced for me. Please help.
0
votes
0answers
10 views

Infinite sum of Legendre Polynomials

The infinite sum of a single Legendre Polynomial has a well known expression. Are there any explicit formulas for the infinite sum of the product of two Legendre Polynomials? I'm interested on ...
0
votes
0answers
31 views

Coxeter's “Introduction to Geometry” recommendation

What mathematical background does one need for Coxeter's "Introduction to Geometry"? Is the text suitable for self-study?
1
vote
0answers
60 views

Surveys in probability? In the current literature sense

I mostly come from an economics background so when I want to find where the current state of knowledge is in specific fields I look for surveys. These are basically primers so that a researcher can be ...
1
vote
0answers
34 views

Theorems of euclidean geometry as invariable properties of geometric configurations

Is there some book, or systematic theory, that proves theorems of euclidean geometry by viewing them as invariable properties of certain geometric configurations ? So that from an easy special case, ...
1
vote
0answers
34 views

Geometry text recommendation (for self-study)

I had taken a first year geometry course over the summer in order to 'skip' it during the school year, and looking back it, I can say that it was one of the worst decisions I've made. Although I did ...
1
vote
1answer
48 views

Book(s) about Affine geometry.

A quick look on Stack Exchange enabled me to discover "Geometry" from Michele Audin which is very close from what I'm expecting but there isn't the correction of the exercices. To be more specific, I'...
2
votes
0answers
13 views

Reference/Literature on Eichler integrals

Has anyone reference or literature advices on Eichler integrals? I want to know what the Eichler integral associated to a modular form is. It should help to understand this, page 65, 4th sentence in ...
0
votes
0answers
44 views

What are the prerequsites for this book?

I am considering reading the book Mathematics made Difficult by Carl E Linderholm. PDF: http://i7-dungeon.sourceforge.net/math_hard.pdf . How much math do I need to know in order to benefit from ...
12
votes
0answers
188 views

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
0
votes
1answer
48 views

A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
2
votes
0answers
28 views

Seeking proof to a Hyperbolic polygon conjecture

In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been ...
2
votes
1answer
65 views

Good book on Spherical Trigonometry

Possible approach/content: Modern Practical (Navigation/Geodesy) unifies with Euclidean/Hyperbolic Trigonometry
0
votes
1answer
47 views

Book Recommendation Please! [Casella Berger] Statistical Inference

I would appreciate your 2 cent on book recommendation. I have basic exposure to probability theory back in college (e.g. calc, stats, probability undergrad level) but haven't dealt with them for a ...
1
vote
3answers
139 views

Good introduction to cardinals?

is there a good text book to cardinals? I am more interested in how cardinal works than cardinality. Because it seems in undergrad they cut off at proof of $\mathbb{R}$ is uncountable and does not go ...
1
vote
1answer
31 views

Which functions arise from a probability measure in this way?

Given a probability measure $\mathbf{P}$ on the interval $I=[0,1]$, we get a corresponding function $f:(\mathbb{R}_{>0})^2 \rightarrow \mathbb{R}_{>0}$ as follows: $$f(x,y) = \int_\mathbf{P} x^q ...
0
votes
1answer
27 views

Text Suggestion for the given topics.

Can anyone suggest me a suitable text(s) for the given topics ? Sample Surveys and Design of Experiments: Sampling and non-sampling errors. Conventional sampling techniques (SRSWR/SRSWOR, ...
0
votes
0answers
9 views

Source proof of equivalence between relation algebra and three variable first order logic

I am reading up on relation algebra and a lot about first order logic. Being a bit of a heavy torsion on the latter one as I am not used to it. Either way I read on wikipedia that up to 3 variable ...
0
votes
0answers
41 views

How can I study probability?

I want to have a deep understanding of probability. I've tried William Feller's first book on Probability, and E.T Jaynes' Probability theory - the logic of science (which is very different from most ...
6
votes
0answers
80 views

Rigorous justification for “complex” change of variable in integration

Suppose that I have $X_1,\ldots,X_n$ i.i.d. $\sim $ $X$ and $Y_1,\ldots,Y_n$ i.i.d. $\sim$ $Y$ for some continuous $X$ and $Y$. Consider the r.v.'s $\bar{X}=\frac{1}{n}\sum_jX_j$ and $\bar{Y}=\frac{1}{...
1
vote
2answers
43 views

Reference request: Binary quadratic forms

I am currently a first year grad student doing an independent study on topics in algebraic number theory and am currently looking at some of the properties of the polynomial $n^2 + n + A$, where $A \...
8
votes
3answers
87 views

Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?

I want to use the convergents of the continued fraction for $$c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $$ - but of course a numeric software is very limited here, so I hope there exists a systematic ...
0
votes
1answer
70 views

If $|t| = |W(-\ln z)| = 1$ and $t^n =1$ then $z^{z^{z^{…}}}$ is convergent

Let $z \in \mathbb{C}$ and $W$ be the Lambert W function. In this post I was told if $|t| = |W(-\ln z)| = 1$ and $t^n =1$ for some $n \in \mathbb{N}$ than the iterated exponential $z^{z^{z^{...}}}$ ...
1
vote
1answer
29 views

Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...
0
votes
1answer
30 views

Passing from classical formulation to weak formulation for a general PDE

I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{...
2
votes
1answer
52 views

Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five ...
2
votes
0answers
49 views

GRE Math Resources

I am planning to give the MATH GRE. I intend to go to physics graduate schools but I may have to given the MATH GRE according to my college's requirement. I very much like maths, but I'm overloaded ...
0
votes
1answer
33 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
1
vote
0answers
24 views

What are some methods of proving undefinability results? (Reference)

I'm trying to prove some results regarding undefinability of functions from the natural numbers in certain structures, but besides texts on elemental logic and number theory, i haven't found anything ...
0
votes
0answers
18 views

Ask for an example of the following type of parabolic PDE with analytic solution

I aim to find an example of the following type semi-linear PDE with analytic solution to test my numerical method: $$\begin{cases} u_t=Lu+g(t,x,u,u_x), &(t,x)\in [0,T]\times \mathbb{R}\\ u(0,x)=f(...
0
votes
0answers
9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...