This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.
5
votes
2answers
180 views
A nonlinear “Fredholm” integral equation
Consider the integral equation
\begin{eqnarray*}
u \left( x \right) & = & \int_0^{\infty} u \left( t \right) u \left(
\frac{x}{t} \right) \mathrm{d} t
\end{eqnarray*}
where the objective ...
4
votes
1answer
125 views
Differential Geometry without General Topology
I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: "oh, but what's the problem ? First learn General ...
9
votes
0answers
88 views
Basic categories cheat sheet
Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
3
votes
3answers
63 views
Book recommendation for associative algebras
Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment.
...
5
votes
0answers
179 views
Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$
My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$.
I'm a bit ...
3
votes
0answers
80 views
An Axiomatic Treatment of Mathematics from First Principles to the Major Subjects?
I'm looking for a book - more likely, books - that could take me from the axioms of mathematical logic up to the major subjects of mathematics, like analysis, algebra, geometry, etc.
For example, a ...
3
votes
1answer
55 views
About iterative refinement to the solution of the linear equations
I want to know what is iterative refinement for improving the solution to the linear equations? How they improve solutions and what are the various techniques for the iterative refinements?
Any ...
-1
votes
3answers
105 views
Good books on combinatorics
I have a math Ph.D. but my knowledge of combinatorics sucks and I simply don't know how to compute anything more complicated, i.e. what happens when we put restrictions on the allowed configurations ...
10
votes
1answer
144 views
subgroup of connected locally compact group
I need a reference or a short proof for the following property:
A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
6
votes
3answers
128 views
Preparations for reading Algebraic Number Theory by Serge Lang
I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
1
vote
2answers
71 views
resources to study PDE from
I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
12
votes
3answers
552 views
Every power series is the Taylor series of some $C^{\infty}$ function
Do you have some reference to a proof of the so-called Borel theorem, i.e. every power series is the Taylor series of some $C^{\infty}$ function?
16
votes
3answers
813 views
Consequences of Degree Theory
I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
4
votes
0answers
93 views
A formula by R.L.Graham,AMM(1995)
I see this formula in a book, it comes from R.L.Graham,AMM(1995) :
...
7
votes
1answer
335 views
Practical Tips: Mathematical research and discoveries [closed]
How to behave when you have the feeling of working on something innovative? What to do if there is a chance (even the $1\%$) that your
work is leading you to something original?
For example ...
4
votes
2answers
90 views
Reference for a certain notion of holonomy
I am reading a paper that says $L$ is a flat complex $G$-line bundle over $M$ with holonomy $\alpha$. Here $G$ is an abelian Lie group and $\alpha$ is a character of $G$. I have two questions:
If ...
6
votes
1answer
235 views
Automata theory on infinite words: any video lectures?
I am fun of automata theory. Can you suggest good video lectures on the subject?
(there is a good one here, but it is accessible from RWTH University only)
1
vote
2answers
122 views
Where can one find a list of prime numbers?
I am looking for the biggest list of precomputed prime numbers one can find and download. Where should I look?
2
votes
3answers
90 views
Should I put interpunction after formulas?
I am presently doing my first substantial piece of mathematical writing, hence this, probably somewhat silly, question.
How does display-style mathematics interact with punctuation?
More ...
1
vote
1answer
75 views
Heighway dragon and twindragon relation
The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1.
The ...
0
votes
0answers
33 views
Notation: Historic and Modern
I was just curious to find out if anybody here knows of a resource (online, text, whatever) that has a list of the various names of objects, functions, ideas, etc. in mathematics that have accumulated ...
3
votes
0answers
33 views
Nimber of selective compound games
Background/Definitions. Let $\alpha,\beta$ ordinal numbers. The Hessenberg sum $\alpha \# \beta$ is defined recursively as the smallest ordinal which is $>\alpha' \# \beta$ and $> \alpha \# ...
5
votes
1answer
38 views
When the ordinal sum equals the Hessenberg (“natural”) sum
Let $\alpha_1 \geq \ldots \geq \alpha_n$ be ordinal numbers. I am interested in necessary and sufficient conditions for the ordinal sum $\alpha_1 + \ldots + \alpha_n$ to be equal to the Hessenberg ...
4
votes
2answers
130 views
Counterexamples in algebra
I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot.
Zorn's ...
3
votes
1answer
330 views
Right Inverse of a matrix
I'm reading Linear Algebra by Bill Jacob and am having trouble with his development of the theory behind the right inverse of a matrix. I did an internet search but didn't find anything useful. Does ...
9
votes
2answers
50 views
Origin of well-ordering proof of uniqueness in the FToArithmetic
In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
28
votes
1answer
684 views
Is there an atlas of Algebraic Groups and corresponding Coordinate rings?
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel ...
11
votes
0answers
163 views
Any proof to $\pi^{e}$'s irrationality?
I've searched for this for a while but get nothing...
There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
5
votes
1answer
32 views
deg functions and maps
For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman.
For algebraic functions on elliptic ...
0
votes
1answer
83 views
How to solve this quasilinear parabolic evolution equation (result of curve shortening flow)?
In the classic paper by Hamilton and Gage (see http://intlpress.com/JDG/archive/1986/23-1-69.pdf), they give the PDE problem:
Find $k:S^1 \times [0,T) \to \mathbb{R} \text{ s.t }$
...
4
votes
1answer
81 views
The manuscript Summa Logicae (William of Ockham)
The Summa Logicae (Latin, in English it's the Sum of Logic) is a textbook on logic by William of Ockham. There are articles about the Summa Logicae in Wikipedia and in Logicmuseum.
It was published ...
1
vote
0answers
54 views
Kolmogorov's paper defining Bayesian sufficiency
I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper
Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
0
votes
1answer
33 views
A matrix has a real logarithm if it has a positive spectrum.
The title is a proposition I read in my notes that's left with no proof. Where can I read one?
2
votes
4answers
117 views
Any advise and suggestions about Real analysis and measure theory?
I am about to take a real analysis course and i wanna ask if anyone can provide some good texts or reference or any other source. The lectuer indeed suggested the Rudin real and complex analysis but i ...
3
votes
2answers
63 views
What's a good book for a beginner in high school math competitions?
Also, I want to make it clear: Beginner.
I'm getting really frustrated trying to study for math competitions:
On the one hand, there are books teaching the high school curriculum,
but that's it. I ...
4
votes
0answers
68 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
7
votes
6answers
2k views
Book recommendation on plane Euclidean geometry
I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
1
vote
0answers
37 views
Are there books on algorithms for architecture?
I need books on algorithms for organic/nonlinear and linear architect anyone recommend a book?
12
votes
4answers
487 views
Good book on evaluating difficult definite integrals (without elementary antiderivatives)?
I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manupulating the integral somehow (e.g. contour integration, interchanging order of ...
4
votes
2answers
79 views
Good book on representation theory after reading Rotman
I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview ...
2
votes
1answer
41 views
Exceptional values in a combinatorial game
Consider the following combinatorial game: We have two heaps of sizes $n_1 \leq n_2$ (with $n_1,n_2 \in \mathbb{N}$). A move leaves the sizes $m_1,m_2$, where $0 \leq m_1 \leq n_1 \leq m_2 \leq n_2$, ...
5
votes
1answer
36 views
Connection between $\mathbb{Q}_p[G]$ and $\mathbb{Z}_p[G]$
In this post there was the comment, that having $\mathbb{Q}_p[G]$ modules, it is possible to construct $\mathbb{Z}_p[G]$ modules. How is it possible to find out when there is a bijection between ...
1
vote
1answer
40 views
Dual basis existence and uniqueness.
In Wikipedia, on Dual Basis they say:
"Algebraically, a dual set always exists, and gives an injection from $V$ into $V^*$. However, a dual basis exists if and only if a vector space is finite ...
8
votes
0answers
58 views
Expression of basis vectors of permutation modules in different bases.
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
7
votes
1answer
57 views
Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$
Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
8
votes
4answers
270 views
A Math book with an inspiring ethos?
I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. Its provides a complete exposition ...
6
votes
3answers
85 views
What questions become answerable/computable given an uncountable character set?
Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities.
This is a subject I was interested in ...
1
vote
1answer
41 views
What is a “distinguished automorphism” of a field?
Math people:
The title is the question. The reason I am asking is that I am trying to determine exactly what fields can be used for an inner product. I posed that question at ...
6
votes
1answer
137 views
Two questions re: $\sum_{n=1}^{\infty}n^{-p_{n}}$
Edit Motivation for question: I looked up the decimal expansion of:
$$\sum _{n=1}^{\infty } \sum _{k=n}^{\infty } k^{-2 k},$$
which matches the first seven digits of the function in question. I would ...
5
votes
1answer
52 views
Does this have a name: If an odd prime $p$ does not divide $a$, then $p$ divides $a^n + 1$ or $a^n - 1$
After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in ...
