Questions tagged [reference-request]

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

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57 views

On the dual version of an isomorphism of Spectral sequence term (from Cartan and Eilenberg)

So I'm trying to take spectral sequences as a black box for application in Commutative Algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
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6 views

Properties of differentiable functions on non-locally-bounded fields

I was reading some results on the structure of non-locally-bounded topological fields, and I was wondering what is known about differentiable functions on them. In particular, on the complex numbers ...
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1answer
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How to mathematically characterize a feedback loop in odes?

I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of odes. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
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0answers
30 views

Commuting algebra of a Zariski closure

Suppose that $\rho: G \to \text{GL}(V)$ is a rational representation, where $V$ is a finite dimensional complex vector space and $G$ is an algebraic group. Suppose that $H$ is a subgroup of $G$ (not ...
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1answer
62 views

Basis for space of continuous, surjective monotone functions on $\mathbb{R}$

$\DeclareMathOperator\CM{CM}$ I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
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0answers
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Asymptotics of the number of minimal strongly connected digraphs

Is anything known about the number of minimal strongly connected digraphs on $n$ labeled nodes? (``Minimal’’ meaning that on the deletion of any arc, strong connectivity is lost.) Some values are ...
13
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5answers
10k views

Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
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0answers
26 views

Positivity of mixed derivatives of log-density of a diffusion

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ defined by the time-homogeneous SDE $$\mathrm{d}X_t \ = \ b(X_t,t)\mathrm{d}t \ + \ \sigma(X_t,t)\mathrm{d}B_t$$ with $b$, $\sigma$ smooth and $B=(B_t)...
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7answers
7k views

Triangulating surfaces

I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's ...
4
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1answer
260 views

Invariants of symmetric forms with respect to the symplectic group

Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
4
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1answer
112 views

Explicit computation of spinor norm

I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow. Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...
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1answer
169 views
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$L^p$ compactness for a sequence of functions from compactness of cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
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0answers
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When is $G(\mathbb{Z}_p)$ topologically generated by maximal tori?

Let $G/\mathbb{Z}_p$ be a connected reductive group, and let $T_1, \cdots, T_n \subset G_{\mathbb{Q}_p}$ be maximal tori. Suppose that we have precisely one maximal torus in each $G(\mathbb{Q}_p)$-...
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0answers
124 views

Does a generalization of Tietze's extension theorem hold for set-valued functions?

Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
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1answer
493 views

Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$

Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
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0answers
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What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?

Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
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2answers
110 views

Reference request: excess normal bundle and derived pullback

Consider a fiber square $\require{AMScd}$ \begin{CD} X' @>i'>> Y'\\ @V g V V @VV f V\\ X @>>i> Y, \end{CD} where $i$ and $i'$ are regular immersions, and consider the ...
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0answers
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Suggestion for framing a course in Representation theory + Spectral graph theory

I am going to give a course in spectral graph theory to graduate students. I want to learn and teach the connection between the spectral graph theory and the representation theory of finite groups. I ...
5
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1answer
94 views

Quiver and relations of Schur algebras

Assume that the Schur algebra $S(n,r)$ with $n \geq r$ is not representation-finite. Question: For which $n$, $r$ is the quiver and relations of the blocks of $S(n, r)$ explicitly known? I just ...
16
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6answers
631 views

Reference for topological graph theory (research / problem-oriented)

I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
1
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2answers
809 views

Reference for moduli stack of principal G-bundles?

Hi, I'm looking for a reference for the fact that the moduli stack $M_{GL_r,X}$ of $GL_r$-bundles over $X$ is an algebraic (Artin) stack. I'm only interested in the case where $X$ is a curve (for now)...
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1answer
127 views

Literature request: Schatten class difference of semigroups

Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
5
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1answer
101 views

Survey of recent developments of the Gelfand-Kirillov dimension

It is almost two decades since the now classical books by McConnell and Robinson's [ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
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0answers
50 views

additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...
6
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1answer
296 views

Mori's cone theorem

I need the proof (reference) of Mori’s theorem about this implication : Let $X$ be a projective complex manifold. If $X$ contains no rational curves, then $K_K$ is nef.
6
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1answer
429 views

Is Van der Waerden's function elementary

Van der Waerden's function was proved to have elementary upper bound on growth rate. Is the Van der Waerden's function itself elementary in the sense of Kalmar?
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2answers
522 views

Good reference on the algebraic geometry of non-associative rings

I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras. Preferrably an online resource or a book that is ...
14
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1answer
430 views

Identity involving the probability that a random walk stays below a curve

I'm looking for a direct proof of the following identity: Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have $$ \lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
8
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2answers
129 views

Conjecture of van der Holst and Pendavingh related to bound for Colin de Verdière invariant

In their 2009 paper (“On a graph property generalizing planarity and flatness”. In: Combinatorica 29.3 (May 2009), pp. 337–361. issn: 1439-6912. doi: 10.1007/s00493-009-2219-6.), van der Holst and ...
20
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10answers
6k views

Maxwell's equations and differential forms

Is there a textbook that explains Maxwell's equations in differential forms? What I understood so far is that the $E$ and $B$ fields can be assembled to a 2-form $F$, and Maxwell's equations can be ...
3
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0answers
407 views

Access to a classic reference of Dold-Puppe

There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as: A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
6
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5answers
303 views

Reference for : a Fréchet nuclear space is Montel

I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact" Thank you in advance for the help!
3
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2answers
545 views

Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $\operatorname{LG}(2,4)$?

$\DeclareMathOperator\LG{LG}$In the paper The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE (published on Proc. London Math. Soc.), I've found an interesting ...
6
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4answers
332 views

Reference for graduate-level text or monograph with focus on “the continuum”

I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the ...
13
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2answers
1k views

Categorification of probability theory: what does a “probability sheaf” tell us (if anything) about probability theory?

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics ...
13
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0answers
263 views

Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
5
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0answers
117 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
37
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2answers
5k views

What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
3
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1answer
71 views

Directed graph minor theorems

In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition A directed graph is a minor of ...
4
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1answer
195 views

Latest progress on Tarski numbers

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...
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0answers
73 views

Generalization of elementary symmetric polynomials

The elementary symmetric polynomials (ESPs) are defined as - \begin{align*} E_{1}^{1} &= X_1, \\ E_{1}^{2} &= X_1 + X_2, \\ E_{2}^{2} &= X_1 X_2, \\ E_{2}^{3} &= X_1 X_2 + X_1 X_3 + ...
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0answers
155 views

When is an increasing union a colimit?

Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$ $$ X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots $$ of pointed spaces, indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
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1answer
67 views

Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks

Consider the $(m+n) \times (m+n)$ block matrix $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ I need references where they are talking about the relation between the eigenvalues of $M$ ...
2
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0answers
34 views

Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras

I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on. What are some ...
4
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0answers
108 views

Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$

Denote by $\Lambda(n)$ th e von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If i recall ...
5
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1answer
180 views

A graph similar to the Bruhat graph, what is it called?

The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows: the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they ...
8
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1answer
491 views

Learning from unsuccessful attempts at the Poincaré conjecture

This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong. Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
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0answers
47 views

Angular excitations and Schrodinger operators with radial potential in N-dimensions

Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
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0answers
56 views

Reference for the $3$-series of an elliptic formal group law

The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins $$ [3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
8
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2answers
291 views

Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?

I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator ...

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