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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0
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1answer
27 views

Number theory text.And advices.

What do you suggest as a must number theory text maybe leaning towards diophantine equations.I think im gonna do a project as an undergraduate.And my topic i think it will be " Aplications of ...
2
votes
1answer
53 views

Proof of Lagrange's four square theorem using Cauchy-Davenport Theorem

Cauchy used the Cauchy-Davenport theorem to prove that $ax^2 + by^2 + c \equiv 0 \pmod p$ has solutions provided that $abc \neq 0$. Lagrange used this result to establish his four squares theorem. I ...
1
vote
1answer
56 views

reference for classifying groups of order $p^2q^2$

In a previous question I asked about the number and structure of groups of order $p^2q^2$ where $p,q$ are primes and with the help of Prof. Derek Holt I understand it now (see here non-abelian groups ...
0
votes
0answers
35 views

Physical side of TQFT [migrated]

How would one go about understanding the physical side of TQFTs? What are the best introductory resources? I know Atiyah axioms but I don't know any QFT.
0
votes
1answer
29 views

Ind- and pro-objects, reference request

Can someone point me to a good exposition of ind- and pro-objects, the intuition behind, and how one "in practice" works with them (i.e. prove things)? The nlab page is nice (especially for the ...
25
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2answers
985 views

Qual question archives?

Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions ...
6
votes
0answers
222 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
70
votes
3answers
20k views

phd qualifying exams

Where can I find phd qualifying exams questions.Is there any website that keeps a collection of such problems? I need it for doing some revision of the basic topics. I know of a book but that do not ...
4
votes
1answer
121 views

What is a number theory book I can read in bed?

I am looking for a good book that is very easy going but not a "pop science" account i.e. something that goes through theory that would be on a basic undergraduate course for someone who finds the ...
0
votes
0answers
33 views

The approximation formula $\left|\alpha -\frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$

I have seen a result about the approximation of irrational numbers and want to find its proof. Suppose $\alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=...
1
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0answers
39 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
3
votes
1answer
91 views

Formalizing splitting into cases

Let $x$ denote a fixed but arbitrary real, and suppose we're trying to solve an equation like $$(x^2-1)^2 = 1.$$ The 'high school' approach is to just shuffle the functions on one side onto the other ...
0
votes
1answer
39 views

How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
4
votes
2answers
68 views

For which classes of matrix can the matrix exponential be easily computed?

We have diagonal matrices $A = \mbox{diag} (\lambda_1, \ldots, \lambda_n)$ for which matrix exponential has simple form $e^A = \mbox{diag} (e^{\lambda_1}, \ldots, e^{\lambda_n})$, and it can be ...
1
vote
0answers
13 views

Looking for an entry level discussion on convex analysis

I have been studying for a qualifier and every so often I come across questions such as: Let $f_n:[a,b] \to \mathbb{R}$ be convex functions and suppose that $f(x) = \lim_{n \to \infty} f_n(x)$ exists ...
2
votes
0answers
28 views

PDE with Stochastic Coefficients

Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ...
1
vote
2answers
1k views

Differential Equations applications in Computer Science

I'm writing a project on differential equations and their applications on several scientific fields (such as electrical circuits, polulation dynamics, oscillations etc) but i'm mainly interested in DE ...
-1
votes
0answers
42 views

Killing fields and the Laplacian on $S^{2n + 1}$

If $X_{ij} = (x_i\partial_j - x_j\partial_i|_{S^n}$, where $(x_1,..,x_{n + 1}) \in \mathbb{R}^{n + 1}$, then it is known that the Laplacian $\Delta$ on $S^n$ is given by $\Delta = \sum_{i \neq j} X_{...
0
votes
0answers
9 views

Input Error estimation

I was wondering what are the methods used to detect the input's error when having the output's error of a model. I thoroughly searched on google, but I failed to find a well explined method, or ...
0
votes
1answer
47 views

Are there any examples of consistent proper axiomatic extensions of classical logic?

By a proper axiomatic extension, I mean a logic with the same set of well formed formulas as classical logic, but with the set of theorems of the logic a proper superset of the theorems of classical ...
2
votes
1answer
67 views

Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$?

Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request. I use the following well-known and somewhat-easy-to-...
4
votes
0answers
66 views

Reference about $\sigma$-linked posets and related notions

In this link, the following list appears: Some chain conditions [of posets], listed from easiest to satisfy to hardest to satisfy: ccc powerfully ccc productively ccc $\sigma$-...
4
votes
1answer
52 views

A topology on the natural numbers

Is there a name of or a reference to the following topology on $X=\mathbb N$: $A\subseteq X$ is closed if and only if $n\in A\wedge m|n\implies m\in A$?
0
votes
0answers
22 views

Green functions

do you know some litterature about green functions for the heat equation ? in particular for the non-linear equation : $\frac{\partial u(x,y,t)}{\partial t}-\frac{\partial^2 \left[ f(u(x,y,t))u(x,y,t)...
1
vote
0answers
20 views

Rappaports algorithm for the convex hull of multiple circles.

I would like to use D. Rappaports convex hull algorithm for discs in a JavaScript application, which I am developing. http://www.sciencedirect.com/science/article/pii/092577219290015K I found ...
0
votes
2answers
39 views

Special Relativity-Book

I would a good book to study the Special Relativity. In my course the professor has treated the following topics: $(1)$ Lagrangian and hamiltonian dynamic of a charged particle; $(2)$ Relaticistic ...
1
vote
1answer
66 views

Books about the foundations of (calculus) functions?

I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe ...
1
vote
1answer
38 views

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

I have some questions on solvability of the following elliptic PDE: in $\mathbb{R}^2$, for $f\in L^2$, $$a^{ij} u_{x^i x^j} +b^i u_{x^i} + c u = f.$$ Here $\{a^{ij}(x)\}_{i,j=1,2}$ is symmetric ...
1
vote
1answer
110 views

Book on Riemannian geometry [duplicate]

I am looking for good books on manifolds, Riemannian geometry. Can you help me? I am a undergraduate level student.
1
vote
0answers
24 views

On a stronger property than being an Armendariz ring

A ring $R$ is said to be Armendariz if $f(x), g(x) \in R[x]$ are such that $f(x)g(x) = 0$, where $f(x) = a_nx^n + \dots a_0, g(x) = b_mx^m + \dots + b_0$, then $a_ib_j=0$ for all $i,j$. In other ...
102
votes
9answers
30k views

Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for all....
1
vote
1answer
43 views

self teach algorithms [closed]

What are some good resources to self teach the subject of Algorithms for someone with background in mathematics? That is, does there exists a more theoretical and abstract approach versus practical ...
2
votes
1answer
46 views

Courant-Hilbert's Book: Weyl's asymptotic law for eigenvalues - Planar domains

In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment Now an arbitrary plane domain $D$ can be approximated by unions of ...
4
votes
1answer
68 views

Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
0
votes
2answers
33 views

Curvature of curves on surfaces

Are there ways to know the curvature of a curve $\gamma$ that lives on a surface $\mathcal{S}$starting from the gaussian curvature of $\mathcal{S}$? In general, is it possible bound the curvature of ...
0
votes
0answers
15 views

Reference for the equivalence of categories between the categories of affine group schemes and commutative hopf algebras.

Where can I find the proof or a discussion that the category of affine group schemes is equivalent to the category of commutative hopf algebras? Thanks.
0
votes
0answers
17 views

A “contraction” on tensor product spaces

Let $X$ be a topological vector space. Let $X'$ denote its continuous dual. Consider the (algebraic) tensor product $\mathbb{X} := X \otimes X'$. For simple tensors $x \otimes x' \in \mathbb{X}$ set $...
1
vote
1answer
36 views

Decision theory references for advanced undergrad/early grad students?

I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
13
votes
7answers
1k views

Is there any book/resource which explain the general idea of the proof of Fermat's last theorem?

I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public. I mean, books which is not for mathematicians but for the general ...
15
votes
2answers
939 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
0
votes
0answers
42 views

In which quadrant of the circle does the angle of $90^\circ$ lie?

By definition and with an authoritative reference, in which quadrant or quadrants does $90^\circ$ lie? (There are non-authoritative references which answer the question, and a related question which ...
0
votes
1answer
12 views

Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
0
votes
2answers
83 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...
-1
votes
1answer
20 views

Differential forms vector space over function field?

Let $V$ be a vector space and $V^*$ be its dual space. Then I know that $V^*$ is considered a vector space because we can scale the basis covectors by real numbers and add them together and all of ...
4
votes
1answer
53 views

Absorbing convex hull into Minkowski sum

Let $B \subseteq \mathbb{R}^n$ be compact. Is there some bounded set $D \subseteq \mathbb{R}^n$ with $0 \in D$ such that $$ Conv(B) + D = B + D \quad ? $$ Here $+$ denotes the Minkowski sum and $...
0
votes
1answer
20 views

On terms “Orientation” & “Oriented” in different mathematical areas?

The goal of this question is to help to deal with different meanings of the words such as "orientation" and "oriented" in different mathematical areas. Are different oriented concepts somehow ...
2
votes
1answer
71 views

Oriented matroids with support on a complement

This is an oriented matroids question, so if there is anyone who is familiar with this area of study, I would greatly appreciate your help. Context Suppose $M$ is an $n\times m$ real matrix of rank $...
0
votes
0answers
8 views

Are Oriented Graphs Related to Oriented Matroids?

My professor said that oriented matroids make it easier to investigate things such as connectivity. Recall that an oriented graph is a digraph without multiple edges or loops. Now Are oriented graphs ...
32
votes
5answers
4k views

Famous papers in algebraic geometry

I'm reading the Mathoverflow thread "Do you read the masters?", and it seems the answer is a partial "yes". Some "masters" are mentioned, for example Riemann and Zariski. In particular, a paper by ...