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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8
votes
1answer
231 views

Connections between linear representations and permutation representations

A finite group $\Gamma$ might be represented by a linear transformation $$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$ or by permutations $$\phi :\Gamma\to\mathrm{Sym}(n).$$ Of course, latter ones can ...
0
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3answers
137 views

Reference request: book on stochastic calculus (not finance)

I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...
5
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0answers
157 views

“Determinant” rather than “trace” in the alternative formula “Lefschetz number”

For a self map $f$ on a topological space $X$ we replace "trace" with "determinant" in the alternative Lefschetz formula $$\Lambda(f)=\sum(-1)^i trace(f^*)|H^i(X,\mathbb{Q})$$ So we have $$\...
11
votes
1answer
292 views

Notions in the literature capturing the “symmetric” or “homogeneous” flavour of $L_p$?

This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise. For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of ...
3
votes
1answer
57 views

Frechet Lie groups and their subgroups

1) Let $G$ be a Fréchet Lie group. Let $H$ be a closed subgroup. Is it always true that the centraliser of $H$ is a Fréchet subgroup of the lie group? 2) Is the closed subgroup theorem valid for ...
2
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1answer
132 views

Quick enumeration for the coloring of the vertices of an n-dimensional cube

The number of ways to color the vertices of an $n$-dimensional cube can be obtained from the Redfield-Pólya theorem by obtaining the cycle index of the relevant permutations, of which there are $n!2^n$...
2
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1answer
63 views

Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations

I recently got interested in representation theory in quantum mechanics and I read the following theorem: Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\...
1
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0answers
179 views

What does it mean for two natural numbers to be *approximately equal*?

This is related to this other question of mine about a paper of Colin and Honda. I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...
3
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0answers
48 views

Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action. There is an associated fibre bundle $E\rightarrow ...
3
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0answers
82 views

Equivariant sheafs and $G$ actions on modules

I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf. The set up ...
1
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0answers
73 views

Name for a Particular (Parabolic) PDE

This is a cross post from MSE. The original question can be found here:https://math.stackexchange.com/questions/3248114/name-for-a-particular-parabolic-pde Consider the following initial value ...
7
votes
1answer
160 views

Quantifier elimination in uncountable elementary “Fraïssé classes”

Let $\mathcal{L}$ be an infinite relation language (this question is trivial in a finite relational language). Suppose that $\mathcal{K}$ is the class of finite models of some $\mathcal{L}$-theory (...
1
vote
1answer
201 views

Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)

This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question. Background I refer to the following concepts: Liouville ...
3
votes
1answer
120 views

Lower bound of the expectation of the product of inner products of random vectors

I encountered the following value in my research: Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$. Denote $$ L = \mathop{\mathrm{E}}_x[ \prod_{1\...
4
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1answer
102 views

What is a name for co-Sobczyk Banach spaces?

Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0$. ...
6
votes
1answer
150 views

Concrete example to illustrate the theory about blocks of groups with cyclic defect groups

I'd like to to have a concrete example to illustrate the theory about blocks of groups with cyclic defect groups. Thus, I am looking for a finite group $G$ and a prime $p$ dividing $|G|$ satisfying ...
1
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0answers
41 views

Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
12
votes
1answer
299 views

Cotangent Complex in Analytic Category

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
5
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0answers
129 views

Analytic sets are rectifiable

I am looking for a reference on the statement that real analytic sets (i.e. sets in the form $u^{-1}(0)$ where $0\not\equiv u:E\subset \mathbb{R}^n\to \mathbb{R}$ is analytic, or finite intersections ...
2
votes
0answers
190 views

All curves over an infinite field embed into the projective space

Let $k$ be an infinite field. Let $X$ be a separated scheme of finite type over $k$. Assume $X$ have relative dimension $\leq 1$. Does there exist a locally closed immersion $X\rightarrow \mathbb{P}^...
3
votes
1answer
138 views

Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein

Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein? I was told that we can use the following method: Step ...
4
votes
1answer
90 views

Quasi-compact quasi-separated induction?

I believe I've encountered the statement below, but I've lost my reference and am unable to find another one. So, I'm posting this question to see if someone can give a reference, or at least confirm ...
6
votes
2answers
320 views

Practical example of Hamiltonian reduction

I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
10
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0answers
183 views

Holomorphic versus algebraic $\mathbb C^*$-actions

I believe that the following is true: Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point. Where can I find a proof of ...
4
votes
1answer
173 views

The category of complexes over a dg-algebra is Grothendieck (it has a generator)

Let $A$ be a dg-algebra over some commutative ring $k$. We have an abelian category $\mathrm{C}(A)$ of (right) $A$-dg-modules. I've read in a few sources that $\mathrm{C}(A)$ is a Grothendieck abelian ...
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0answers
34 views

Positive Ricci curvature on biquotients

I am working with biquotients and positive curvatures and I was able to give a relatively simple proof for the following: Theorem: Let $G$ be a compact connected Lie group with a bi-invariant metric $...
8
votes
2answers
584 views

Errata for Bott and Tu's book “Differential Forms in Algebraic Topology”

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel. Is there a good list of errata for Bott and Tu available? ...
5
votes
1answer
764 views

Origin of Hecke operators

What is the original paper in which Erich Hecke had first introduced the Hecke operators?
4
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0answers
72 views

$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$

The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in ...
0
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0answers
82 views

The best error term for the second moment

Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
5
votes
1answer
94 views

The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook): Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....
5
votes
0answers
34 views

Polarization type of the complement abelian subvariety

Assume that $P$ is a Prym variety of a ramified double cover (hence not principally polarized). Let $A,B\subset P$ a complementary pair. Assume that the type of the polarization of $A$ is given by $\...
2
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0answers
53 views

Bridging between Rosethal Inequalities and log convex tails

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$. Then we have the classical "Rosenthal-type ...
2
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0answers
35 views

Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra. Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...
8
votes
0answers
106 views

Monadic second-order theories of the reals

I’m looking for a survey of monadic second-order theories of the reals. I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
8
votes
1answer
381 views

Original reference for Adams Riemann-Roch theorem

Let $f\colon Y\to X$ be a proper morphism between smooth quasiprojective $k$-algebraic varieties. Denote by $\psi^j$ the $j$-th Adams operation on the Grothendieck group of vector bundles and $\theta^...
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0answers
76 views

Unramified local Langlands

Where can I find a complete proof of the unramified local Langlands correspondence for arbitrary reductive groups? Second sentence included because the automated system would not accept my question.
4
votes
1answer
146 views

Cohomogy of local systems over CW-complexes

Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorname{...
4
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0answers
223 views

Do we know what the impulse to “introduce” the Jordan canonical form was?

Mo-ers, Do you know how it was that the study of the Jordan canonical form began? There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
1
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0answers
30 views

Dynkin diagram of Basic classical simple Lie superalgebras

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...
4
votes
1answer
79 views

Reference request: When is the variance in the central limit theorem for Markov chains positive?

I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/...
7
votes
1answer
334 views

Rationally connected Kähler manifolds are projective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin: https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf She writes in this remark the following: ...
-1
votes
2answers
142 views

Directed colimit and homology

I am looking for a reference or a proof of the following fact: Let $X_{1}\subset X_{2}\subset\dots $ be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each $X_{i}\...
3
votes
1answer
136 views

A pair of spaces equivalent to a pair of CW-complexes

Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that $Z-A$ is homeomorhic to $X-Y$ and $Z/A$ homeomorphic to $X/Y$ and The closure of $Z-A$ ...
-1
votes
1answer
83 views

Reference Request: Carnot Group Not Containing Group of Isometries [closed]

This question is a follow-up to this post, from which I quote: Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...
5
votes
0answers
110 views

Closed embedding of CW-complexes

Suppose that $i: X\rightarrow Y$ is a closed embedding such that $X$ and $Y $ are (retracts) of CW-complexes. Does it follow that $i$ is a cofibration ? Remark: There is a similar question here, ...
5
votes
0answers
52 views

Functional Equation of Zeta Function on Statistical Model

I've been studying [1] because I was interested in his ideas on the zeta function. I'll define it here (c.f. p. 31): The Kullback-Leibler distance is defined as $$ K(w)=\int q(x)f(x, w)dx\quad f(x,w)...
2
votes
1answer
370 views

Reference on Grothendieck trace formula

I need to refer to the so-called Grothendieck trace formula, but after checking tens of Google pages, I still cannot find a proper reference on this topic. Could anyone tell me some good book/papers ...
3
votes
0answers
72 views

Algebra of block matrices with scalar diagonals

I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
6
votes
0answers
253 views

Restriction of a cofibration to closed subspaces

Let $i: X\rightarrow Y$ be a cofibration between CW-complexes, more precisely a cellular embedding. Let $A$ be a closed subspace of $Y$ and $Z=i^{-1}(A)$. Let $$j: Z\rightarrow A$$ be the restriction ...