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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Name of a link invariant?

Below I will describe a link invariant, denoted by me as $inv(L)$. Has anyone encountered this invariant in the literature? If so, what is its name? Also, any references to papers or books that ...
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110 views

W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved. I would like to know whether the ...
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105 views

Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
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113 views

What are the recent advancements in mathematics that an undergraduate can understand?

I am an undergraduate student of mathematics. I am interested to know the conjectures that are proved in the last, say 5 or 10 years. Or any other development. Preferably in pure mathematics. One of ...
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95 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
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166 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
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210 views

Test for equivalence of algebraic expressions

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
4
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114 views

Icosahedral symmetry

Is there a clear and correct reference for full icosahedral symmetry that includes a presentation and its action on vertices, edges, and faces? The Wikipedia article for icoshedral symmetry gives ...
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92 views

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas?

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas? It seems that it have a big gap between this graduate textbooks and the easier ones.
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73 views

What's the name for this unimodal sequence?

Let $a_0, a_1, \ldots, a_n$ be an increasing sequence of positive numbers, and consider the sequence $s_1,\ldots,s_n$, where $$ s_k \;=\; \frac{a_0+\cdots + a_k}{k}. $$ So $$ s_1 \;=\; a_0+a_1,\quad ...
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135 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
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388 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
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130 views

Differentiable manifolds, Serge Lang

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
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62 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 \# ...
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74 views

What about non finitely generated groups?

Finite groups and finitely generated groups are intensively studied, but are there interesting investigations on non finitely generated groups? I already know some references for abelian groups, so I ...
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45 views

Cardinal characteristics — generalisation?

I'm just reading about what is called cardinal characteristics of the continuum. For example there are the bounding number $\mathfrak b$ and the dominating number $\mathfrak d$ etc. which are defined ...
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199 views

Rewriting the advection-diffusion equation

This is mostly a reference request question, although I certainly appreciate any insights and/or comments. Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the ...
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28 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
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75 views

Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?

I am trying to find a translation of this paper either in English or French (preferably English). I am not very optimistic, but i thought of asking in case somebody is more resourceful :)
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97 views

References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
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81 views

Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
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362 views

Number of labelled graphs with $n$ nodes, $k$ edges and $t$ triangles

How many labelled undirected graphs are there with precisely $n$ vertices, $k$ edges, and $t$ triangle subgraphs? (By triangle I mean a graph with three vertices and three edges.) (Clarification: I ...
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55 views

Reference Request: Background/Extention on Bill Thurston's Lecture 'The Mystery of Three-Manifolds'

I found Thurston's lecture The Mystery of Three- Manifolds fascinating but, well, mystifying. For those with insufficient time to watch the video, he establishes correspondence (in his very intuitive ...
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103 views

Connection between modular forms and line bundles

I need a good reference about the connection between modular forms and line bundles. I found only Milne's note that treats briefly this argument. I've already checked, but without finding anything, ...
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197 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
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1k views

Proving that $\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \operatorname{Pic}^0(X \times Y)$

Let $k$ be a field of arbitrary characteristic and let $X$ and $Y$ be projective varieties over $k$. I have come across the formula $$\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong ...
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165 views

Where else has Proposition B1.3.17 in the Elephant been proved?

This is a sort of reference request. Proposition B1.3.17 in Johnstone's Elephant reads: Proposition 1.3.17 Let $\mathcal{S}$ and $\mathcal{T}$ be categories with pullbacks, $F \colon \mathcal{S} ...
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136 views

Cutest proof that PA proves its finitely axiomatised subtheories are consistent?

Back in 1952, Mostowski proved that PA proves the consistency of its finitely axiomatised sub-theories. Any pointers to particularly nice later proofs of this lovely result? Or indeed particular nice ...
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141 views

Regular graph colorings

[I worked this question over and posted it to MO, too.] Call a coloring $C:V(G) \rightarrow \lbrace 1,\dots,|V(G)| \rbrace$ of the vertices of a graph $G$ regular when every vertex of color $c_i$ has ...
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210 views

Characteristic Class exercises

I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes. The classic examples I can think of are tangent bundles and tautological line ...
4
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0answers
210 views

Riemannian Connection (Very basic question)

We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
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69 views

Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following: By Weil’s theorem $[17]$, any local algebraic group is birationally equivalent to an algebraic group. Where $[17]$ A.Weil. On ...
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202 views

Proving NP-completeness (hardness) exercises

I am looking for a list of exercises that can be done to practice polynomial time reductions to prove NP-hardness of problems. I know there are hundreds (thousands?) of problems proven to be NP-hard. ...
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149 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
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137 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...
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85 views

Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
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210 views

Clarification: intersection of a finite number of subgroups of finite index and Poincaré

From Scott's book Group Theory $1.7.10.$ (Poincaré) The intersection of a finite number of subgroups of finite index is of finite index. My question is: Did Poincaré prove the Theorem as stated ...
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391 views

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
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254 views

Reference for Martingale version of Riesz representation theorem / Riemann–Stieltjes integral

(Please let me know if this is more appropriate as a MathOverflow question.) I can work out most of the following martingale generalization to the Riesz representation theorem and the ...
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220 views

How does a Riemannian metric “naturally induce” distances in the Grassmannian bundle

When reading about Pesin theory I've run into the necessity of defining a metric on the Grassmannian bundle of a compact Riemannian manifold $M$. More specifically a fiber at $x \in M$ in the ...
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0answers
75 views

References about the symmetric products of a stack

I would like to know references about a construction of the symmetric product (or the moduli space of effective divisors) $X^{(d)}$ of a stack $X$. I am currently thinking about the following case: ...
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298 views

Understanding orientability of vector bundles

I'm having trouble understanding how orientability of vector bundles work. The book I'm reading, Spivak's A comprehensive introduction to differential geometry, is not very clear on this. Edit: ...
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291 views

“Reverse thinking” exam questions-reference request

I am interested in exam questions that are "backwards" from how they are usually asked. For example: Brian and Megan have the following question on their exam: Find the volume of the solid ...
3
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0answers
31 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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40 views
+50

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
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57 views
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Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
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46 views
+50

What book or website has nice, colorful diagrams illustrating real quadratic integer rings?

I'm sure you all have seen diagrams, colorful or not, illustrating prime numbers in $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, with some of them helpfully pointing out the inert and splitting primes ...
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31 views

Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
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39 views

Showing irreducibility of $F(X)$ by looking at $F(1/X)$

I was reading Knapp's Basic Algebra, p. 455, and he is showing that the polynomial $F(X) = X^3 - X - \frac13 \in \mathbb{Q}[X]$ is irreducible. To do so, he notes that $G(X):= -3X^3F(\frac1X) = X^3 - ...
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53 views

When is a modular curve defined over Q?

Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) ...