Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,545
questions
3
votes
0
answers
119
views
Jacobson-style Galois theory on perfect closure
Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he ...
0
votes
1
answer
360
views
First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
1
vote
1
answer
313
views
$L^p$ compactness for a sequence of functions from compactness of product with 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 $$\{f_{n}\psi_m(...
1
vote
1
answer
113
views
Restricted partition problem into parts with a given set of prime factors
I need a reference for the following question:
Let $\mathcal{P}$ be a finite set of $k$ primes and let $f(n)$ be the number of partitions of $n$ into parts whose prime factors are restricted to the ...
7
votes
0
answers
431
views
Reference for the multiprojective Nullstellensatz?
Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here.
I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the ...
8
votes
0
answers
380
views
Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
5
votes
1
answer
178
views
Hadwiger number of a graph: Question about the original article from 1943
I am analyzing Hadwiger's original article (Hadwiger, Hugo (1943), "Uber eine Klassifikation der Streckenkomplexe", Vierteljschr. Naturforsch. Ges. Zurich, 88: 133–143) for my work related ...
1
vote
1
answer
96
views
Reference to log-transition-density of a diffusion process
Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely ...
1
vote
0
answers
25
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 ...
4
votes
0
answers
65
views
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 ...
-3
votes
1
answer
312
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\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 ...
4
votes
1
answer
2k
views
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 ...
4
votes
0
answers
87
views
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 ...
1
vote
0
answers
126
views
On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
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 ...
12
votes
2
answers
608
views
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 ...
6
votes
1
answer
1k
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\...
5
votes
0
answers
163
views
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)$-...
9
votes
0
answers
250
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 ...
3
votes
2
answers
336
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 ...
6
votes
1
answer
192
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 ...
4
votes
0
answers
174
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}$ ...
1
vote
1
answer
409
views
$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(...
6
votes
1
answer
490
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.
9
votes
1
answer
200
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 $e^{-tA},e^{-t(A+B)}$...
6
votes
2
answers
383
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 ...
9
votes
2
answers
236
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 ...
15
votes
0
answers
415
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 ...
0
votes
0
answers
173
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 + ...
9
votes
0
answers
352
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 $X_\xi$ ...
1
vote
1
answer
4k
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 $M$ ...
4
votes
1
answer
238
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 ...
2
votes
0
answers
53
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 ...
7
votes
2
answers
602
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 ...
1
vote
0
answers
125
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....
2
votes
0
answers
139
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 - ...
4
votes
1
answer
191
views
Computation of the Lusztig a-function
See for example https://www.sciencedirect.com/science/article/pii/0021869387901542 for the definition of the Lusztig a-function.
Question 1: Is there a table for the values of Lusztig's a-function ...
9
votes
1
answer
655
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 ...
9
votes
1
answer
487
views
Proof of a 'well-known' result of Shimura on periods of modular forms
It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic....
5
votes
2
answers
370
views
Conjecture about minimal number of edge crossings in complete bipartite graphs
I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$.
The Wikipedia article https://en....
5
votes
2
answers
218
views
Reference for Cochran-Orr-Teichner's filtrations on knot concordance
I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance:
Tim D. Cochran, Kent E. Orr, and Peter Teichner....
2
votes
0
answers
203
views
Poincaré Recurrence Theorem for flows
Someone knows a book that have the proof of the Poincaré Recurrence Theorem for flows when the set have finite volume and when don't?
2
votes
0
answers
209
views
Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions
In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := - \Delta_g + |A|^2$ (usually called Jacobi ...
2
votes
0
answers
68
views
Linearly dependent points and the uniform position theorem
One proof of the uniform position theorem (as stated in p. 109 or p. 113 in Section III.1 of "Geometry of Algebraic Curves") uses a monodromy argument. While this gives us something even ...
9
votes
2
answers
286
views
Schur Weyl duality for the supergroup $\text{GL}(m|n)$
Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...
13
votes
1
answer
686
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
$$
\lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
2
votes
3
answers
247
views
Diophantine equation of a factorial type
I'm interested in nontrivial solutions of Diophantine equations of the type
$$a^2b^3 = \frac{c!}{(c-k)!} $$
For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$
Does anyone have any ...
3
votes
1
answer
149
views
Reference request - random regular graphs vs random graphs w/ degree sequence
There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. ...
1
vote
0
answers
99
views
Dislocations and Random Matrix Theory
Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help.
By ...
4
votes
1
answer
346
views
Blow up the diagonal of a symmetric product space
Consider the complex blow-up of the diagonal $\triangle\subset Sym^2(\mathbb{C}^n)$. It is known that the blown-up space is smooth when $n=1,2$. I was wondering if that is still smooth for $n\geq3$? ...
3
votes
1
answer
294
views
Reference for Function-Valued Random Variables?
Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following:
Let $X$ be a topological space, $Y$ be a random variable with ...