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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10 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 ...
4
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0answers
28 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 ...
-1
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0answers
43 views

A book on functions! [on hold]

I am looking for a good book on algebra which covers functions very comprehensively. It should cover functions of the type: linear: $f(x) = ax + b$ quadratic: $f(x) = ax^2 + bx + c$ cubic: $f(x) = ...
3
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0answers
29 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 ...
-1
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0answers
20 views

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

Notation and Defination: $G$ is a Strongly Regular Graph( not a complete , cycle graph) with Every two adjacent vertices have $\lambda$ common neighbours. Every two non-adjacent vertices have ...
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0answers
60 views

Engelking or Munkres for General Topology? [on hold]

I am a last-year Bachelors student and when I finish my Bachelors, I will have one year free time just before I start Masters in Pure Mathematics. I like to be better in General Topology before ...
2
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0answers
20 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
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1answer
73 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 ...
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0answers
38 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 ...
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0answers
73 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?

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
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2answers
81 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 ...
9
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1answer
185 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 ...
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0answers
16 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
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1answer
33 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
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0answers
29 views

ROAD MAP FOR PDE

I am not a specialist in PDE, but I am looking for a "ROAD MAP FOR PDE" : I mean, a general picture about main ideas during the last 100 years until today, which allow to solve all kind of PDE. The ...
1
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1answer
54 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
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1answer
26 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 ...
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1answer
43 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
56 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
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0answers
85 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
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3answers
79 views

How to discover other fields of mathematics? [on hold]

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 ...
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0answers
75 views
+200

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 ...
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0answers
52 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
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1answer
47 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? ...
2
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0answers
23 views

The dual space of weighted compact supported function?

Let $\Omega\subset \mathbb R^N$ be open bounded. It is well know that the dual space of $C_c(\Omega)$, i.e., compacted supported continuous function, can be identified by finite Radon measure ...
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46 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 ...
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0answers
63 views

Bourbaki's “Éléments de mathématique” and Grothendieck's EGA, SGA, FGA free available? [on hold]

Are the following documents available online (say as a pdf), in a free and legal/official way: Bourbaki's "Éléments de mathématique" Grothendieck's "EGA, SGA, FGA" Surely, there are some pirated ...
2
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0answers
49 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, ...
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0answers
26 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
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0answers
39 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 ...
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2answers
110 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
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1answer
30 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
22 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
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1answer
43 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
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1answer
35 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, ...
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3answers
584 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
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0answers
34 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
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2answers
55 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
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0answers
22 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
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0answers
50 views

(Theoretical) Complex Analysis Textbooks

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 ...
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0answers
39 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
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0answers
69 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
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0answers
20 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 $a_1 < a_2 < \dots < a_m$ ...
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1answer
28 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
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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
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0answers
27 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
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1answer
45 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 ...
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0answers
84 views

Comments about “Topics in Algebra” by I.N. Herstein and “Abstract Algebra” by Dummit/Foote? [closed]

Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been ...
3
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0answers
26 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 ...
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0answers
16 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., ...