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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1answer
159 views

Citable Reference for Picard's Theorem in Banach Space

I was wondering if anyone knew of a legitimate citable reference where Picard's Theorem on the existence of solutions to ODEs in Banach space is proven? For some reason I can only find proofs for the ...
9
votes
2answers
529 views

“8 Dice arranged as a Cube” Face-Sum Equals 14 Problem

I found this here: Sum Problem Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same. $\hskip2.7in$ Here is one of 20 736 ...
1
vote
2answers
311 views

Connections of Geometric Group Theory with other areas of mathematics.

I'm a master's student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin's ...
6
votes
3answers
298 views

Algebra Texts that Emphasize Universal Properties/Contructions

I am interested in elementary algebra texts and/or notes that place early and continuous emphasis on universal constructions, functors and other aspects of category theory. One text that takes this ...
2
votes
0answers
81 views

References needed for gradient flow

Can anyone recommend lecture notes or (not too obscure) books that teaches me about gradient flow and what it has to do with PDEs? I did search but usually the material talks about dynamical systems ...
5
votes
4answers
228 views

Mathematics of computation

What is a good introduction to turing machines, complexity classes, P=NP etc from a purely mathematical viewpoint? I want to know how computation relates to provability in mathematics, I need the ...
2
votes
4answers
211 views

A good introduction to Prime Numbers

I'm looking for a good introduction to Primes Numbers, their properties, and some of the better known theorems concerning them. I would prefer references assume knowledge of undergraduate level real ...
20
votes
2answers
1k views

History of Modern Mathematics Available on the Internet

I have been meaning to ask this question for some time, and have been spurred to do so by Georges Elencwajg's fantastic answer to this question and the link contained therein. In my free time I ...
5
votes
0answers
535 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
1
vote
1answer
177 views

random graphs without cycles

Recall that a closed walk (in a undirected graph) is a cycle if its vertices are pairwise distinct. Does there exist random constructions of bipartite graphs without cycles with high probability?
10
votes
1answer
203 views

Diophantine equation $x^y-y^x=11$

How can one find all integer solutions to $x^y-y^x=k$, for a given k? Example case $x^y-y^x=11$
1
vote
0answers
41 views

What's is this optimization function property called?

An optimization function $$f:\bigcup_{n \in \mathbf{N}} S^n \to \mathbf{R}$$ may have the property that given a domain $$P = \prod_{i=1}^N P_i$$ and the solution $$(m_i)_{i=1}^N = \underset{x \in ...
1
vote
2answers
1k views

Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces

Looking for the proof of Eberlein-Smulian Theorem. Searching for the proof is what I break with this morning. Some of my friends recommend Haim Brezis (Functional Analysis, Sobolev Spaces and ...
15
votes
6answers
4k views

A good reference to begin analytic number theory

I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about ...
1
vote
1answer
101 views

Limit sets of representations of once-punctured torus groups and circle packings

Let $\rho\colon\pi_1(T_1)\to PSL(2,\mathbb{C})$ be a faithful representation of the fundamental group of a once-punctured torus. If both the components of the convex core in the quotient manifold are ...
8
votes
3answers
2k views

What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
11
votes
1answer
308 views

What and where in the notebooks of Ramanujan is this series?

The wikipedia page on Ramanujan contains the following series: $$ 1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - ...
9
votes
2answers
709 views

Ramanujan's approximation to factorial

I saw this approximation for the factorial given by Ramanujan as $$\log(n!) \approx n \log n - n + \frac{\log(n(1+4n(1+2n)))}{6} + \frac{\log(\pi)}{2}$$ in wikipedia, which claims the approximation is ...
2
votes
1answer
763 views

Reference of Sobolev space

currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level). Also, Adams 1975 version has been widely cited. So besides these ones, which book in your ...
8
votes
2answers
155 views

Seeking analytic proof for $\sum_{n=r}^\infty \frac{1}{n!}\left[ n-1 \atop r-1 \right] = 1$

In Blom, Holst, Sandell, "Problems and snapshots from the world of probability", section 9.4, a model of records is discussed: Elements are ordered in a sequence of increasing length according to ...
1
vote
0answers
126 views

Literature on Riccati equations (algebraic and differential)

Advise me please some book on algebraic and differential Riccati equations: I'm interested in such questions as theorems of existence, uniqueness and extendibility of solutions of differential ...
4
votes
1answer
89 views

Classifying continuous characters $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$.

I recently saw the following claim: Let $\mathbf{C}$ denote the field of complex numbers together with its usual topology. If $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$ is a continuous ...
2
votes
2answers
4k views

Ellipse fitting methods.

I have set of points and want to fit ellipse to this set. I have found only function which fits ellipse in least squares sense. In this set of points there are some noise points which should not be ...
1
vote
3answers
717 views

Riemann-Stieltjes integral exercises

Recently I was looking for some source for exercises in Riemann-Stieltjes integral and going through several calculus books (Hunt, Edwards, Thomas, Adams) I found no such exercises. Is there any ...
14
votes
1answer
358 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
9
votes
1answer
507 views

Navier-Stokes Equation and turbulence, current status of research?

What is the currect research status of solving Navier-Stokes Equation, any up-to-date review/good paper on this topic? Or direct numerical simulation is still the best way to understand the ...
0
votes
2answers
153 views

On Reflexive Banach Spaces

My Functional Analysis lecturer gave me a topic for my assignment, the title is "On Reflexive Banach Spaces". I am a looking for several good references to start my work, that is why I brought this ...
1
vote
0answers
130 views

Reference: Wittgenstein teaching mathematics

Can anyone give me any reference concerning L.Wittgenstein teaching school kids mathematics? I have been wondering what kind of mathematics he taught and how he lectured the material.
0
votes
1answer
166 views

How to choose a proper method for PDE?

In most of the books I've seen, the authors explain various methods for solving partial differential equations, sometimes give recommendations about usage and limitations of the approach, but I never ...
0
votes
1answer
39 views

The “theory of compound partitions”

I was reading a sort of mini-bio on Sylvester the other day and a "Theory of Compound Partitions" was mentioned in the discussion of his research interests. I wanted to ask, is this the same or the ...
2
votes
0answers
243 views

Equivalence books for Bourbaki's Element of Mathematics

I find that the arrangement of basic topics in Bourbaki's book is quite elegant, I want to learn mathematics following this order. But one problem is the books is too old and sometimes too complex for ...
3
votes
2answers
237 views

Number of ways to cover $2n$ place ring with $j$ dominos

In Blom, Holst, Sandell, "Problems and snapshots from the world of probability" there is a claim that the number of ways of placing $j$ dominos in a ring with $2n$ places in such a way, that each ...
4
votes
2answers
515 views

Two-sided Laplace transform

I want to study more formally the properties of the two sided Laplace transform $$ \hat f(z)=\int_{-\infty}^{\infty} f(t)e^{zt}dt $$ as a kind of generalization of the Fourier transform. I found some ...
3
votes
1answer
682 views

References for Kolmogorov's strong law of a large numbers

On the Wikipedia law of large numbers site, they mention "Kolmogorov's strong law of large numbers", which works even if the random variables are not identically distributed. Where can I find this ...
8
votes
5answers
1k views

Can anybody recommend me a topology textbook? [duplicate]

Possible Duplicate: choosing a topology text Introductory book on Topology I'm a graduate student in Math. But I never learnt Topology during my undergraduate study. Next semester, I am ...
5
votes
1answer
132 views

Is there an algorithm to determine whether rational matrices generate a finite group?

This is inspired by this question. Given finitely many invertible rational $n\times n$ matrices $A_{1},\ldots, A_{k}\in\operatorname{GL}(n,\mathbb{Q})$, is there an algorithm (a practical one) to ...
1
vote
3answers
143 views

What are some books that I should read on 3D mathematics?

I'm a first-grade highschool student who has been making games in 2D most of the time, but I started working on a 3D project for a change. I'm using a high-level engine that abstracts most of the math ...
0
votes
1answer
104 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
2
votes
0answers
56 views

Finite-dimensional representations of the Lie algebra of vector fields on a circle

I have just began to study infinite-dimensional Lie algebras and I am curious whether the Lie algebra $L$ spanned by the vector fields $z^n \partial/\partial z$, $n=0,1,2,3,\dots$ admits any ...
8
votes
0answers
155 views

Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
2
votes
1answer
113 views

Database for size of Ш

Are there are any references that record the cardinality of Ш for elliptic curves for which Ш is known? Also their corresponding conductors. EDIT: Following the Qiaochu Yuan's comment's I should ...
0
votes
1answer
245 views

Inequalities involving determinants and minors of positive definite matrices

Lately I've been dealing with positive definite matrices in my research (in the context of them being covariance matrices), and, am wondering if anyone knows of a comprehensive list of inequalities ...
7
votes
1answer
1k views

Computing the integrals of the form $\exp(P(x))$, $P(x)$ a polynomial

Wikipedia notes that Exponentials of other even polynomials can easily be solved using series. For example the solution to the integral of the exponential of a quartic polynomial is: ...
2
votes
1answer
203 views

Does anybody know of a site that has a set of all theorems?

I mean, if there exists a site that his function is to show and save theorems with their proofs?
5
votes
3answers
322 views

Linear Algebra Journal

Is there any journal which has significant material on the teaching of linear algebra. I am investigating the most effective way to teach a course on Linear Algebra. What are the most important things ...
1
vote
1answer
478 views

Nearest point projection in uniformly convex Banach spaces

Let $X$ be a uniformly convex Banach space, $x\in X$ and $C\subset X$ closed and convex, then there is a unique $y\in C$ with $$\lVert x-y\rVert=\inf_{z\in C}\lVert x-z \rVert.$$ Is there a good book ...
19
votes
3answers
10k views

How to cite preprints from arXiv?

Obviously when writing a math research paper it is good to cite one's references. However, with the advent of arXiv, oftentimes a paper is only available on arXiv while is awaits the long process of ...
16
votes
13answers
3k views

Creative Thinking Questions?

Math is often intimidating to the average man due to its complex appearance. To show that math requires creative thinking, not just memorization, I was wondering if anyone had any math problems that ...
8
votes
1answer
4k views

Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title ...
7
votes
2answers
284 views

Spectra of restrictions of bounded operators

Suppose $T$ is a bounded operator on a Banach Space $X$ and $Y$ is a non-trivial closed invariant subspace for $T$. It is fairly easy to show that for the point spectrum one has ...