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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4
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0answers
21 views

terminology for a “forward flow” type of random digraph

I am trying to find a characterization of the probability that vertex $1$ is connected to an arbitrary large vertex $N$ in a random digraph. The difference from typical random digraphs is that if ...
1
vote
2answers
26 views

Definition of associative algebra over a field

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product. Question: Does anyone know how a bilinear ...
0
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0answers
14 views

approach for clustering of huge networks

Can you give me some kind of direction on the best approach for clustering huge networks? (so large, that even the list of nodes cannot be stored in RAM) Thanks for anyone who helps.
0
votes
0answers
19 views

For graphs in a recursive graph class: Does m = O(n) hold?

For recursive k-terminal Graph classes - for example definied in this paper - is it true that |E| = O(|V|)? If so, I would be very grateful for a reference! Thanks!
3
votes
0answers
79 views

Inner Functions in Annuli: Not Likely!

The other day someone reminded me of something I'd thought about some years ago. As back then it took me a little while to see why there was any problem; this time I got much farther on a solution ...
1
vote
0answers
37 views

Transformation laws for tensors on general manifolds

I was interested in tensor products rather from the mathematical, abstract point of view, so topics like various tensor products in the context of, for example, Banach spaces, $C^*$-algebras and so ...
3
votes
1answer
29 views

Diameter of Schreier coset graphs.

I'm looking for a source from which to learn about Schreier coset graphs. Especially, examples in which combinatorial properties (specifically, diameter) of Schreier graphs are calculated. Also, is ...
0
votes
0answers
9 views

Sign patterns in kernel and rowspace of a matrix

I'm looking for the reference of the following fact from oriented matroid theory. This must be known; in fact, I think it is in the book "Oriented Matroids" by Björner et al., but I can't locate it. ...
1
vote
0answers
20 views

Why Klein Maskit Combination Theorem is important?

I am learning basics about Kleinian Groups. Recently, I read the proof of Klein Maskit Combination Theorem. I want to know about some good applications of this tool. Please share some references.
3
votes
0answers
30 views

Convex hull of curve

Let $f_1, \ldots, f_n \in \mathbb{R}[t]$ and let $S = \{(f_1(t), \ldots, f_n(t)) : t \in \mathbb{R}\}$. Is there a method of computing the convex hull of this set? How about if each $f_i$ has the same ...
0
votes
0answers
25 views

To use topology and Riemann surface in number theory.

I would like to learn some rudiment topology and Riemann surface in order to apply in number theory. I already know some algebraic topology, like covering space and fundamental group, singular ...
1
vote
1answer
43 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
0
votes
2answers
47 views

I need a good statistics book [duplicate]

I have an upcoming statistics exam and I'm studying it on my own. I was recommended Hogg's Introduction to Mathematical Statistics, but I didn't find it helpful. I just want a book which covers basic ...
0
votes
1answer
23 views

Reiterating the piecewise-and-uniform-limit operation

Probably a hopeless question, but: Let $C$ be the class of constant functions $f$: $[a, b]\longrightarrow\mathbb{R}$. Let $\mathcal{U}(\mathcal{P}(C))$ denote the class of uniform limits of piecewise ...
0
votes
0answers
30 views

The boundary integral equation

in which case we use the single layer potential and the double layer potential for the Laplace equation ? \begin{eqnarray}\tag{1} \Delta u = 0 \; \mathbb{R}^2\backslash\omega\\ u \to 0 \; at \; ...
3
votes
1answer
35 views

Induction and Compact induction of representations

Let $H \leq G$ be a subgroup of a finite group, $G.$ Suppose $(\sigma, W)$ is a representation of $H.$ Then we know that $Ind_H^G \sigma $ and $ind_H^G \sigma $ are isomorphic, where $$Ind_H^G ...
5
votes
0answers
53 views

Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
4
votes
0answers
61 views

Normal curvature of geodesic spheres

I would like to ask the community for a reference on the following property of geodesic spheres. Let $(M,g)$ be a compact Riemannian manifold without conjugate points and $\tilde{M}$ its universal ...
3
votes
1answer
125 views

Construction of a Strongly Regular Graph which has regular Neighbourhood graphs in all iteration.

Notation and Definition: $G$ is a Strongly Regular Graph (not complete or a cycle) and is denoted by $\mathrm{SRG}(n,r, \lambda, \mu)$ if it has the following properties: Every two adjacent ...
2
votes
1answer
46 views

Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that ...
2
votes
1answer
85 views

Old calculus books?

this is really a question about math and not books. I am mainly wondering if reading really old calculus books is still beneficial for undrgraduate students today. I was told that the material covered ...
1
vote
0answers
42 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
5
votes
1answer
319 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
2
votes
2answers
93 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
10
votes
2answers
223 views

Proof of infinitude of prime elements?

All proofs of infinitude of primes which I know of essentially prove that there are infinitely many irreducible elements of $\Bbb Z$, and with this goal in mind we can very easily extend this proof to ...
0
votes
0answers
18 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
1
vote
1answer
52 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
1
vote
1answer
61 views

Number of Automorphisms of a Irregular Graph.

I have been looking for results on number of graph automorphisms of irregular graph(upper and lower bound). I searched , but could not find anything which can be used directly. Say, $G$ is $k$ ...
2
votes
1answer
29 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
5
votes
1answer
55 views

A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
2
votes
1answer
147 views

Solution of Graph Isomorphism in current literature.

As of 2008, the best algorithm for graph isomorphism (Babai & Luks 1983) has run time $2^{O(\sqrt(n log n))}$ for graphs with n vertices. Does this algorithm gives a yes / no answer or provide ...
5
votes
0answers
105 views

Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

I would like to know what are the mathematical reasons for which Bernhard Riemann believed that his hypothesis is true, and I would like to know if those mathematical reasons were cited in his ...
3
votes
3answers
90 views

How to discover other fields of mathematics? [closed]

I am currently an undergraduate and thinking about applying to graduate school for math. The problem is that I don't know what field I want to go. Taking graduate classes even more confuse me because ...
14
votes
1answer
167 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
2
votes
0answers
59 views

Literature concearning characteristic classes?

It sounds the literature about characteristic classes is not very abundant (am I wrong?). Whenever I look for books dealing with this matter I'm always lead to the same material like the classical ...
3
votes
1answer
52 views

Emil Artin on visualization of matrices

Someone called my attention to the fact that Emil Artin made very important remarks on the visual representation of matrices in some of his books. Could anyone tell me which precise book that is? ...
-1
votes
0answers
54 views

Learning Math for Computer Science

Apologies if this has been already asked. I have gone through a lot of different questions but they don't adapt to my personal situation. I have a 2 years diploma in software development and I am ...
2
votes
0answers
53 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
1
vote
1answer
58 views

Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
0
votes
0answers
42 views

Can a positive definite kernel expanded as the product form with an arbitrary orthonormal system?

Notations mostly follow https://en.wikipedia.org/wiki/Mercer%27s_theorem. Mercer's theorem uses the eigenfunctions $\{e_j\}$ of the integral operator as the expansion function. I wonder if we could ...
1
vote
2answers
119 views

Looking for a gentle intro to Linear Algebra

Does anyone know of any gentle, introductory books to LA that assume little prerequisites, even in the way of vectors and matrices? I want something that will give intuition and reasonable proofs, and ...
2
votes
1answer
31 views

Reference for a combinatorial theorem

Is there a reference for this theorem https://en.wikipedia.org/wiki/Schur%27s_theorem#Combinatorics? I am unable to locate a reference. Google search does not spot this particular theorem well.
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0answers
28 views

Calculus book for computer science students

I'm going to teach calculus I and II to undergraduate computer science students and I would like to know if someone here knows some book or site with easy calculus applications in computer science. ...
1
vote
1answer
51 views

A good book on basic (Euclidean) geometry.

We were studying demonstrative geometry, so I thought if I read Euclid's Elements it would give me the proper conceptual basis to understand the theorems. But then I learned that Euclid's method of ...
1
vote
1answer
46 views

Intermediary self-learning-readable book(s) for Real Analysis (incl. Measure Theory,…)?

After studying a very readable book, Advanced Calculus by Fitzpatrick, I thought I start more advanced of real analysis by the same author so I started Real Analysis by Fitzpatrick (and Royden). Well, ...
22
votes
4answers
811 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
3
votes
0answers
36 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
1
vote
2answers
58 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
1
vote
0answers
24 views

How to solve differential equations for linear operators?

I want to solve the differential equation $$ BA = \frac{\partial}{\partial t} A $$ for $A$. Here $A : H_1 \mapsto H_2$ and $B : H_2 \mapsto H_2$ are operators and $H_1, H_2$ are some Hilbert spaces. ...
2
votes
1answer
118 views

(Theoretical) Complex Analysis Textbooks [closed]

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...