# Tagged Questions

This tag is used if a reference is needed in a paper or textbook on a specific result.

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44 views

### Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...

**0**

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27 views

### A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...

**2**

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**0**answers

62 views

### A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form
$$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...

**-4**

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27 views

### Illustration of the search for the right proportion that symbolizes the golden ratio [on hold]

My teacher asked us to illustrate, using a scientific poster the search for the right proportion that symbolizes the golden ratio. The problem statement is the following:
What about today that ...

**1**

vote

**1**answer

204 views

### Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...

**5**

votes

**2**answers

494 views

### mod 5 partition identity proof

I am looking for a proof that:
$$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$
The left hand side expands into:
...

**1**

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**1**answer

89 views

### A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...

**4**

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**1**answer

303 views

### Algebraic Geometry needed for Kahler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kahler-Einstein / Extremal Kahler metric. I was wondering how much Algebraic ...

**5**

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**1**answer

173 views

### Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold.
Given a smooth $(n-1)$-dimensional smooth ...

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175 views

### Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes?
At this point I am still ...

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56 views

### Reference for generalized model categories [on hold]

I would like to know some references for "Stanculescu's notion" of Generalized model category.
Motivation The idea is find some works that use the notion and increase my knowledge about it.
...

**-1**

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**1**answer

146 views

### Distinguished triangle and short exact sequence [on hold]

Forgive me for asking an elementary question.
Given coherent sheaves $A$, $B$, $C$ and morphisms $B\xrightarrow{f} C\xrightarrow{g} A$ which give rise to the distinguished triangle $A[-1] \rightarrow ...

**9**

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**1**answer

185 views

### Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...

**4**

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64 views

### Understanding homotopy t-structure

The following question came up while reading Hoyois'
From algebraic cobordism to motivic cohomology.
Let $S$ be a Noetherian scheme of finite Krull dimension and let $SH(S)$ denote the homotopy ...

**0**

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**1**answer

73 views

### In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made:
Let $u_m$ converges weakly to $u$ in ...

**1**

vote

**1**answer

94 views

### The space of loops as a Banach space [on hold]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...

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**2**answers

51 views

### Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...

**2**

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**1**answer

69 views

### Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...

**3**

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**2**answers

84 views

### References about the matrix generators of the finite subgroups of the orthogonal group O(4)

"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...

**1**

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70 views

### Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...

**4**

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73 views

### Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...

**5**

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**2**answers

304 views

### What is this distance about?

For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
This question by Ritesh Ahuja (positive answered by Iosif Pinelis) says that $d$ is a metric. My ...

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54 views

### Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...

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64 views

### Subquotients of Jantzen Filtration for Kac-Moody algebras

Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and ...

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**1**answer

136 views

### Nonsolvable finite quotients of matrix groups

Suppose that $\Gamma$ is a finitely generated nonsolvable subgroup of $GL(n, R)$. Is it in the literature that $\Gamma$ has a nonsolvable finite quotient? I know how to prove it (the hardest ...

**2**

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**1**answer

121 views

### reference request: simple facts about vector-valued $L^p$ spaces [on hold]

I learned basic results (regarding weak convergence) about Banach-space valued functions of a single real variable when learning PDE. (See e.g. Appendix E in Evans's Partial Differential Equations) I ...

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116 views

### Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$

Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots ...

**1**

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**0**answers

128 views

### Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?

**11**

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**1**answer

311 views

### Definition of ind-schemes

What is the correct definition of an ind-scheme?
I ask this because there are (at least) two definitions in the literature, and they really differ.
Definition 1. An ind-scheme is a directed colimit ...

**5**

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**1**answer

95 views

### Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...

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135 views

### Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...

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59 views

### Weak $H^1$-limit of “almost conformal” maps

Let us consider a sequence of maps $\phi_n : (M, g) \to \mathbb{R}^k$, where $M$ is a surface. Let $\phi_n \to \phi$ weakly in the sense of $H^1$-norm, and let $\phi$ be non-trivial. Let $\mathcal{H}$ ...

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132 views

### books, lecture notes, for studying pullback rings [closed]

Does anyone have suggestions for books, or lecture notes, (or videos) for studying pullback rings? ( whole book or a section that have an easy and basic approach.)
I know definition;
...

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53 views

### Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, ...

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100 views

### Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...

**4**

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**2**answers

179 views

### Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies
$$p_t(x, y) = \frac{1}{(4\pi ...

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21 views

### Linear Program for Single Source Shortest Paths Tree

This question originates in quick, however wrong, idea to calculate a shortest paths tree in the presence of negative cycles. The essential motivation was that a linear program would determine binary ...

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**0**answers

71 views

### Harmonic map into $S^n \times \mathbb{R}$ [closed]

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...

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98 views

### Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field.
First, some notation: let $G$ be a ...

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vote

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150 views

### Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.
Fix an uncountable universe ...

**2**

votes

**1**answer

320 views

### Reference request: English translation of Brieskorn 1970 paper

Is there any english (or french) translation of the following paper by Brieskorn (1970)?
Brieskorn, E., "Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2 ...

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115 views

### Is there any work on “super Fukaya categories”?

There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic ...

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**0**answers

42 views

### Techniques for the analysis of interacting particle systems with a finite number of particles, which do not resort to limiting arguments?

I am interested in pointers to (keywords/authors) recent research on the analysis of interacting particle systems with a finite number of particles which do not resort to limiting arguments converting ...

**2**

votes

**1**answer

140 views

### Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...

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60 views

### Hausdorff limits of fibers of affine maps

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and let
$$
F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m
$$
be a polynomial map. I would like to know under what conditions the preimages $F^{-1}(y)$ of ...

**0**

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67 views

### If the fibers of a submersion are connected, does it mean that any 2 sections are homotopic (locally on the base)?

Is the following fact known? If yes - what is the reference?
Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers.
Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any ...

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vote

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28 views

### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...

**4**

votes

**1**answer

104 views

### Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...

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63 views

### Euclidean metric in spherical coordinates [closed]

I apologize if this question is not research level, but it has been asked on MSE before, and not received really satisfactory answers, for example, see here, and googling does not reveal anything ...

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52 views

### Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...