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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5
votes
1answer
56 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
150 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
61 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
59 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.
1
vote
0answers
30 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
53 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
47 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
815 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
59 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
122 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 ...
1
vote
0answers
42 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
2
votes
0answers
71 views

Fields of Research in Algebra [closed]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
3
votes
0answers
24 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
0
votes
1answer
30 views

Reference for solving linear operator equations

I'm interested in solving an equation of the form $$ Ax = b $$ for some bounded linear operator $A: H_1 \mapsto H_2$ where $H_1, H_2$ are some Hilbert spaces. I've seen in this math.SE post in ...
2
votes
0answers
39 views

Chandrasekhar history [closed]

I have misplaced an article by Chandrasekhar from the mid 90's in which he drew an analogy between himself and his neat desk and a co-laborer 'down the hall' who was more chaotic with managing his ...
4
votes
0answers
31 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
4
votes
1answer
49 views

When can you take the limit of a parameter before solving the differential equation?

Short example: consider the differential equation \begin{align*} f'(x)=\frac{k^2}{k^2+k+1}xf(x) \end{align*} where $k$ is a parameter. Wolfram Alpha tells me that the solution to this equation is ...
3
votes
0answers
34 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
0
votes
0answers
17 views

Reference for General state space Markov chain

What is a good reference for general state space Markov chains? Is there a reference which assumes only familiarity with finite/countable state space Markov chains and then extends the results (e.g., ...
0
votes
1answer
62 views

What are the topics that must be covered in a beginning graph theory course? [closed]

Good day to everyone. It will be my first time to make a syllabus on elementary graph theory. My question will be: What are the topics that must be covered in a beginning graph theory course? Also ...
0
votes
0answers
21 views

Non-Linear Constrained Optimisation Over Zonotopes: Reference Request

Background I am investigating, numerically, the problem of a chess team attempting to maximise its probability of winning a team match . Each of our N players independently chooses one of two ...
0
votes
1answer
32 views

Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
2
votes
1answer
39 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
2
votes
0answers
28 views

Do complete Hopf algebras have an antipode?

I am reading Quillen's paper on rational homotopy theory. In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures ...
0
votes
1answer
104 views

What's the image of $(a, b)$ under a typical $f$: $\mathbb{R}\longrightarrow\mathbb{R}$?

This question is about "typical" or "generic" functions $f$: $\mathbb{R}\longrightarrow\mathbb{R}$, where I leave the two terms undefined, but they are meant to be in the style of "A generic ...
0
votes
0answers
40 views

How to tell complex structures apart

Complex structures are rigid, yet weirdly flexible. For example, the Riemannian mapping theorem says that every non-empty simply connected open subset of $\mathbb{C}$ that is not $\mathbb{C}$ is ...
3
votes
0answers
74 views

Reference request: fixed point and first-order logics

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in a fixed-point logic to have an equivalent first-order formula. ...
2
votes
1answer
111 views

Proof: $\mathbb{Z}[\zeta_6]$ is a PID.

I am reading through A First Course in Modular Forms. In Proposition 2.2.3 they claim that $\mathbb{Z}[\zeta_6]$ is known to be a principal ideal domain. Does anyone have a reference for the proof of ...
3
votes
2answers
159 views

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
0
votes
0answers
6 views

Successive divisibility of a sequence? Progressive divisibility? terminology or reference

Perhaps I say that an (infinite) sequence $(r_n)$ of positive integers is progressively divisible iff $r_n \mid r_{n+1}$ for all $n$. Is there some other terminology that is in use for this? I am ...
10
votes
2answers
282 views

Algebraically flavoured functional analysis book

I'm looking for a book on functional analysis that would suit someone who is more algebraically/geometrically oriented and seeks to learn the subject with the goal of using it later for geometric ...
3
votes
0answers
28 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
1
vote
0answers
17 views

Are there any resources on this notion of a “directed” semigroup?

Given two additive semigroups $G$ and $H$, and a semigroup action $ \& : G \times H \to G$, then intuitively $H$ can be thought of as a set of "directions" in which we can move in the space $G$. ...
0
votes
1answer
36 views

Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
0
votes
2answers
71 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
0
votes
0answers
4 views

Is the space of smooth sections of a smooth bundle a Fréchet Manifold?

I'm not very prepared on these concepts and i'm wondering if there're some good references addressing this problem... My aim is to present the problem of linearization for the Euler-Lagrange operator ...
2
votes
0answers
28 views

finding an invariant measure

Are there any methods for finding an (infinite, absolutely continuous with respect to lebesgue measure) invariant measure $d\mu=f(u,v)dudv$ for something like the following? $$ T:Q\to Q, \ ...
0
votes
0answers
44 views

Proof of Sokhotski-Plemelj theorem

Sokhotski-Plemelj theorem states $$ \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\varphi(\zeta) \,d\zeta}{\zeta-z}+\frac{1}{2}\varphi(z), \, \\ \phi_e(z)=\frac{1}{2\pi ...
0
votes
1answer
50 views

Books on multivariable calculus

I'm looking for a book that covers the following subjects: multivariable functions, extremes of multivariable functions, integration, implicit function theorems, functions defined by integrals, vector ...