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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16
votes
4answers
2k views

What resources are there for learning Russian math terminology?

I have been studying Russian for about a year now as a bit of personal edification, but I would like to be able to talk about and write about mathematics in Russian, as well. I find it enhances my ...
7
votes
1answer
230 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
1
vote
1answer
296 views

Intermediate textbook in Linear Algebra

I am looking for a Linear Algebra textbook that for those who just finished elementary Linear Algebra. I just finished Introductory to Linear Algebra by Strang and read to Least Squares and ...
0
votes
2answers
67 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
3
votes
1answer
77 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
3
votes
1answer
90 views
+50

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
0
votes
1answer
25 views
0
votes
4answers
72 views

In need of a group theory textbook.

I am in need of a group theory textbook for a good summer review. I have already studied from various books (mostly "group theory" part from basic algebra books) and the lecture notes of my teacher, ...
6
votes
1answer
127 views

What proof uses both the Riemann Hypothesis and its negation?

Some time ago I happened to see a proof that was remarkable in that it used both the Riemann Hypothesis and its negation. That is, it considered the two cases: RH is true, and RH is false, obtaining, ...
0
votes
0answers
32 views
+50

Mixing up seating charts: Measuring “mixedness” over time

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
5
votes
0answers
34 views

Mathematically rigorous books on programming and computer science

I would like a mathematical approach to programming languages and computer science, not just the theoretical aspects of computer science. Is there any such text out there? After all, the world has a ...
3
votes
1answer
58 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
39
votes
4answers
8k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
1
vote
0answers
27 views

Proof of formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
4
votes
2answers
41 views

About Mordell's Theorem (Elliptic Curves)

I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman. One of the key lemmas used in the proof of the theorem is: Let ...
0
votes
2answers
31 views

Reference request for this topics

I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience.    1.2. Stationary measure.    1.3. ...
-6
votes
0answers
41 views

Which harvard course should I take? [on hold]

apologies if I'm on the wrong website or anything. I'll be starting math at Harvard next year but am at a loss as to which course I should start with. I have a good background in competition math (so ...
1
vote
1answer
29 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
0
votes
0answers
40 views

Does the restriction of the Thom class of a submanifold to the cohomology of the submanifold give the Chern class of the normal bundle?

Let $X$ be a compact complex manifold of (complex) dimension $d$, let $i\colon Y\hookrightarrow X$ be a regular complex submanifold of (complex) codimension $k$, let $\mathcal{N}_{Y/X}$ be the normal ...
7
votes
0answers
141 views
+50

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
3
votes
2answers
101 views

One-dimensional Noetherian UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone ...
4
votes
1answer
150 views
+50

What is known about these arithmetical functions?

Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and $$ \alpha_N(n)=\prod_p p^{c_p \bmod N}. $$ The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
1
vote
1answer
115 views

Extremely difficult: Polynomials $f,g,h$ such that…

I've been trying to solve this for hours and got nowhere, so I can only assume it's a really difficult problem. Problem: Find polynomials $f,g,h$ with integer coefficients such that: ...
0
votes
0answers
22 views

Looking for reference to solve a problem (representation theory, I think?)

this question appeared on an algebra qualifying exam: Let $R$ be the group algebra $\mathbf{C}[S_3]$. How many nonisomorphic, irreducible, left modules does $R$ have and why? Would I be able to ...
0
votes
0answers
47 views

How to prove it and how to solve it

Tomorrow I will begin my studies, real analysis, however I have some difficulties in making statements so I thought before starting the study in real analysis, learn how to do demonstrations properly. ...
3
votes
1answer
49 views

Book recommendation for network theory

I'm looking for a mathematically rigorous book on Network theory covering topics like entropy, degree distribution, centrality, and regular, random, small-world and scale-free networks. I'm familiar ...
3
votes
2answers
437 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
0
votes
0answers
39 views

Why do we call a linear mapping “linear mapping”? [migrated]

What are the historical reasons that created the term "linear mapping"?
8
votes
4answers
9k views

Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
2
votes
1answer
42 views

Desk Reference Question: Linear Algebra and its Applications by Lay or Strang? Or Handbook of Linear Algebra?

I would like a good working desk reference for linear algebra for someone in applied mathematics (no proving abstract theorems). I saw the Handbook of Linear Algebra as well, but I am concerned it may ...
34
votes
6answers
27k views

Good books for self-studying algebra?

I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra ...
2
votes
0answers
64 views

Categories that differ in morphisms

Let $C_1$ and $C_2$ be two (small) categories defined over the same set of objects: that is, $C_1$ and $C_2$ differ only in their hom-sets. Specifically, $Hom(C_1) \neq Hom(C_2)$, and $Hom(C_1) \cap ...
3
votes
0answers
57 views

How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula (or an algorithm including formulas) to compute some parts of the drawing, but I get lost due to the ...
0
votes
4answers
258 views

Recommended math background for game theory

I recently got interested in some game theory applications to poker. I want to try some of them out programmatically, but a lot of the math is a bit confusing. I learn math on my own fairly quick and ...
2
votes
1answer
29 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
7
votes
0answers
142 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
2
votes
0answers
34 views

Set of functions from a finite field to the integers

Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied? Does anyone know a reference or keywords to search for? ...
-2
votes
0answers
33 views

“Elementary Set Theory - Leung, Chen” - Solution manual? [on hold]

I'm trying to study some ST on me own :-) ! I have found a very nice book with lots of problems but without any solution to the problems. Do you guys know whether someone have made a solution manual ...
5
votes
1answer
89 views

Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
9
votes
1answer
103 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
0
votes
1answer
23 views

Reference help about fuzzy logic and fuzzy set

Good afternoon, I'm trying to learn about Fuzzy alone, I was using some texts on the Internet about it, but it was very difficult to learn. I want to learn about Fuzzy set as shaper of uncertainty. ...
1
vote
1answer
18 views

Reference for the statement “bilinear form $a$ is symmetric if and only if the operator $S$ is self-adjoint”

Thanks to Riesz representation theorem, a continues bilinear (sesquilinear) form on Hilbert space $$a: \mathcal H\times \mathcal H\rightarrow\mathbb R \ \ (\text{or} \ \ \mathbb C)$$ can be ...
1
vote
3answers
47 views

Need visualization advice for learning partial derivatives and calculus with more than one variable.

Okay so I just recently started learning calculus with more than one variable and whilst I'm coming to grips with many of the ideas and stuff I'm finding it difficult to visualize certain things for ...
1
vote
1answer
12 views

Is optimal bound for Alcuin's triangular city problem known?

Alcuin's triangular city problem is Problem 28 from Propositiones ad Acuendos Juvenes. There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 ...
1
vote
1answer
44 views

Schauder estimates of weak solutions of a elliptic PDEs of 2nd order

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $0<\sigma<1$, $L$ be a 2nd order strictly elliptic linear differential operator of divergence form: \begin{eqnarray} ...
0
votes
1answer
19 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
1
vote
1answer
25 views

What is this 2-D array of numbers called?

Define $c(n,r)$ ($n\in\Bbb N;r\in\Bbb Z$) by setting $c(0,-1)=-1$, $c(0,0)=1$, and $c(0,r)=0$ otherwise, with all further $c(n, r)$ given recursively by $$c(n+1,r)=rc(n,r-1)-(r+1)c(n,r)$$(rather in ...
2
votes
1answer
318 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
0answers
16 views

Quartic spline explanation

I need to use a quartic spline (natural spline) for a robotics application. However, most of the documents explaining splines focus on quadratic and cubic splines. Could you please give me some ...
3
votes
1answer
85 views

Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute. I want a reference and a detailed proof of this. Please also give a formal ...