Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
15,781 questions
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Earliest known reference for the stable EHP sequence
For a based space $X$, let $Q(X) = \Omega^\infty \Sigma^\infty (X)$. Let
$D_2(X) = (X\wedge X) \wedge_{\Bbb Z_2} E\Bbb Z_2$ denote the quadratic construction. Then one has a pair of maps
$$
X \overset{...
- 20.3k
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Determining intersection of degree-$d$ hypersurfaces that contain a variety
I imagine the following problem must have been studied before, but since I didn't do any work in this field before, I can't find any reference.
Fix a positive integer $d > 0$ and consider a ...
- 193
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+100
Is there a name for this type of probabilistic predictability of stopping times?
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space, and let $\tau \colon \Omega \to [0,\infty]$ be an $(\mathcal{F}_t)$-stopping time.
We will say ...
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Do we have a version of Hoeffding's inequality for these non-independent variables
I am writing a probabilistic argument (and I am not a probability theory expert), and the following would be useful to me. I tried asking AI but the answers did not seem helpful, so hopefully this is ...
- 13.1k
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117
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Cesàro convergence rates for products
If I have given two sequences $(a_n)_n$ and $(b_n)_n$ in $\mathbb{R}$, rates $\alpha, \beta > 0$ and constants $C_1, C_2 > 0$ such that
\begin{equation}
\frac{1}{n} \sum_{k = 0}^{n - 1} a_k \le ...
- 505
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Large deviations for elements in a fixed Wiener chaos
It is known (see for example Friz-Victoir), for a Gaussian process $X$ that is $\alpha$-Hoelder for $\alpha>1/4$, that the canonical rough path $(\int dX,\int\int dX\otimes dX,\int\int\int dX\...
- 2,345
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On the Golod-Shafarevich pro-$p$ group
Let $p$ be a prime number, and let $G$ be a pro-$p$ group with finite
generator rank $d(G)$ and finite relation rank $r(G)$. If $G$ satisfies the Golod--Shafarevich condition $r(G) < d(G)^2/4$
and ...
- 917
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168
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Projection formula for D-modules
I am currently reading Claude Sabbah’s classic paper
Sabbah, Claude, Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci. 33, No. 4, 643-685 (1997). ZBL0920.14003.”*
The electronic ...
- 121
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Ramsey ultrafilters after adding splitting reals
This is a reference request. Are there any results of the following kind?
Assume $\mathrm{CH}$ and let $\mathcal{U}$ be a Ramsey ultrafilter. Let $c$ be a Cohen real. Then in $V[c]$, can $\mathcal{U}\...
- 151
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106
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Reference request: Integral motivic cohomology of $BG$
Are there references that compute the integral motivic cohomology $\mathrm{H}^{p,q}(BG,\mathbb Z)$, for $G$ finite cyclic and where $BG$ is defined over the rationals.
I am particularly interested in $...
- 3,366
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58
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q-Deformed Laplacian of a graph
Suppose $G$ is a finite graph.
Let $A$ denote the adjacency matrix and $D$ the diagonal matrix whose entries are the degrees of vertices in $G$.
The matrix
$$
L_q = (1 - q^2)I - q A + q^2 D
$$
can be ...
- 2,215
14
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643
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Reference request: determinacy and Lebesgue-measurability locally
I've heard it said many times times that $\boldsymbol{\Pi}^{1}_{n}$-determinacy implies $\boldsymbol{\Sigma}^{1}_{n+1}$-Lebesgue measurability (hence for instance $n$ many Woodin cardinals with a ...
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156
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Nested connected graph structure
I encountered the following graph, more precisely, graph structure: graph $G$ has $n$ nodes indexed from $1$ to $n$; for any $k$ between $1$ and $n$, if we remove nodes $k,k+1,\cdots,n$ and the ...
- 295
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Optimal constant in $L^1-L^2$ inequality on Gauss space
For a differentiable real-valued function on $\mathbb{R}^n$, denoting $\partial_i f$ for the $i$th partial derivative, we can define the functional
$$
T_n(f) = \sum_{i=1}^n \frac{1}{1 + \log(\|\...
- 582
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Reference request: Stability of strong diameter two property under infinite $\ell_1$-sums
Does anyone know a reference for the following result:
If $\{X_i\}_{i \in I}$ is a familiy of Banach spaces with the strong diameter two property, then its $\ell_1$-sum has this property too.
I'm ...
- 393
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98
views
(Weak) Jordan-Chevalley decompositions over non-perfect fields
This is a reference/literature request.
Given a field $K$ and an endomorphism $x \colon V \to V$ of a finite-dimensional $K$-vector space it is well-known that the Jordan-Chevalley decomposition of ...
- 31
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Local Frobenius algebras, reference
I found this statement on wikipedia:
Commutative, local Frobenius algebras are precisely the zero-dimensional local Gorenstein rings containing their residue field and finite-dimensional over it.
Can ...
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The fourth moment of the Riemann zeta function without absolute values
Are there any results known about the asymptotics/bounds for
$$\int_0^T\zeta(\tfrac{1}{2}+it)^4\;dt,$$
where we don't have the absolute value on the inside?
One could use the triangle inequality to ...
- 191
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Reference request: Formula of Seifert invariant of Dehn surgery along torus knots
Is there a literature that contains an explicit formula of Seifert invariants
of 3-manifold $S^3_{T_{p,q}}(\frac{s}{r})$, the $\frac{s}{r}$-Dehn surgery on the $(p,q)$-torus knot $T_{p,q}$ ?
As for ...
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Is total homology endo-functor, on bounded derived category of finitely generated modules over commutative Noetherian ring, a triangulated functor?
Consider the bounded derived category $D^b(\operatorname{mod } R)$ of finitely generated modules over a commutative Noetherian ring $R$ and the homology functor $H_*: D^b(\operatorname{mod } R) \to D^...
- 583
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Shimura reciprocity for Drinfeld modular varieties?
let's suppose we have a function field $F$ and some Drinfeld modular variety of rank $r$ over $F$, with some level structure $Y^{(r)}(N)$. Then the field of constants of $Y^{(r)}(N)$ is some class ...
- 2,251
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117
views
Is there a real analytic tubular neighbourhood theorem?
The tubular neighbourhood theorem, stating that an embedded submanifold has a neighbourhood that is a diffeomorphic image of an open subset of the normal bundle, is a staple result about smooth ...
- 9,076
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0
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38
views
Justifying the Robbins-Monro procedure using Dvoretzky's theorem on stochastic approximation
A colleague and I are trying to understand some results in stochastic approximation theory with a view to gaining quantitative information about rates of convergence of certain processes. We have done ...
- 1,169
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3
answers
707
views
What literature can I read about the Janibekov effect and the intermediate axis theorem?
I have been studying mathematics for 2 years, and I have already read Terence Tao's publication. Please suggest books on related topics, such as Euler's equations, mathematical modeling, mathematical ...
- 43
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219
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Recursive pointfree approach to algebraic topology
$\newcommand\seq[1]{\langle#1\rangle}$A large number of important topological results require simplicial-algebraic machinery (or comparable) to prove. This machinery is ingenious, impressively so even,...
- 1,739
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Are two symplectic fibration(resp. Hamiltonian fibration) are smoothly fibration isomorphic if it holds continuously?
Let $P, P'$ be symplectic(resp. Hamiltonian) fibrations over a base $B$.
If two $P, P'$ are continuously symplectic fibration(resp. Hamiltonian fibration) isomorphic, then are they also smoothly ...
- 363
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61
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Projection onto translation invariant subspaces
I'm currently looking at a subspace of $A \subset \ell^p(\mathbb{Z}^n)$ which is generated by some finitely supported elements and their translations. My question is an old one (but the answer is ...
- 4,736
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370
views
Minkowski sum of finite ellipses
Let $(E_n)_n$ be any finite collection of centred ellipses in $\mathbb{R}^2$. Suppose that $E_n$ are pairwise non-homothetic (i.e. there is no positive constant $c>0$ such that $E_n = c E_m$). Now ...
- 287
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412
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Selberg's result on primes in short intervals
A result of Selberg (A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943) says essentially
$$\int ...
- 1,676
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Injectivity of derivations from the middle transvectant in the free Lie algebra on $\operatorname{Sym}^m$ for $\mathrm{SL}_2$
Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that
$$
\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le ...
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K-flat complexes and unbounded derived tensor product over a non-commutative sheaf of rings
$\def\R{\mathscr{R}}
\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.
When $\R$ is commutative, there is much literature on ...
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Reference for freeness of the ring generated by roots of unity
The following fact is well-known, and not hard to prove, but I do not know an explicit reference.
Let $R$ be the subring of complex numbers generated by all roots of unity. Then $R$ is free as an ...
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Coefficient comparison between a double sum and a single sum
For a formal Laurent series $F(q)$, denote its coefficient of $q^j$ by $[q^j](F)$.
QUESTION. For integers $r\geq1$, is this true?
$$[q^{2r}]\sum_{n\geq1}\frac{q^n}{1-q^{2n}}\sum_{k=1}^n\frac{q^k}{1+q^...
- 43.9k
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(Cohomological) Brauer group of $BG$ — reference request
Let $G$ be a smooth connected linear algebraic group over an algebraically closed field. Write $\operatorname{Br}'(BG)$ for the cohomological Brauer group of $BG$, i.e. the group of $\mathbb{G}_m$-...
- 363
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Measurability of $t \mapsto \int_A f(t, \omega)\mathbb{Q}_t(\mathrm{d}\omega)$ when $(t, \omega) \mapsto f(t, \omega)$ is not measurable in $t$
I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...
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Like circle packing but not exactly
I am looking for bibliography on the following problem.
Given $N\in\mathbb{N}$ find $N$ points $p_1,...,p_N\in\mathbb{R}^2$ which
(1) maximize $\min_{i,j} |p_i-p_j|$
(2) subject to the constraint $\...
- 85
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Reference request for a proof of Cayley's tree counting formula via the representation theory of the symmetric group
Inspired by a recent project Euler problem, I came up with a proof (sketched below) of Cayley's tree formula using the representation theory of $S_n$. I would like to ask for a reference in the ...
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Ramanujan's work on the central factorial numbers
Background
The central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this ...
- 5,065
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References or examples for "quotient tensors" on manifolds
On a smooth manifold $M$ with altas $\mathcal A=\{\phi:U_\phi\subseteq M\to \Bbb R^n\}_\phi$, we can define a tensor $T$ on $M$ as the collection $\{T_\phi\}_{\phi\in\cal A}$ such that whenever $U_\...
- 1,511
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For an infinite set S, I believe that all automorphisms of Sym(S) are inner. I would like a reference for this
For an infinite set S, I believe that all automorphisms of Sym(S) are inner. I would like a reference for this.
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3
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363
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Weakening of the Idoneal Number condition
This is about the following Math.SE Q&A: "What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?".
...
- 26.7k
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Possible asymptotic behavior of recurrence function
I was wondering and tried to find what are the results known related to recurrence function of a minimal subshift $\Omega \subseteq A^{\mathbb{Z}}$, where $A$ is finite non empty subset.
If I am not ...
- 755
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217
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Book recommendation for smooth ergodic theory
I'm interested in smooth ergodic theory.
Please teach me some recommended books for it.
Actually, now I have been reading the supplement of Katok's book, Introduction to the Modern Theory of Dynamical ...
- 81
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452
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Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?
The groundbreaking work of Maynard and Tao showed the following fundamental result:
For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at ...
- 43
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74
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Current research regarding higher order fractional p-Laplacians
I am currently working on the generalised tangent point energy for surfaces (here denoted by $\mathcal{E}_s^p$, this notation is non-standard but tailored to this particular post) and, more ...
- 21
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1
answer
310
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Testing equal count between pairs of sets
For each fixed positive integer $N\in\mathbb{N}$, let's define two sets
\begin{align}
A_N:=&\{(a,b)\in\mathbb{N}^2: N=a(2b-1)+(2a-1)(b-1)\}, \\
B_N:=&\{(c,d)\in\mathbb{N}^2: N=c(2d-1)+(d-1)(d-...
- 43.9k
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0
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78
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Martingale central limit theorem: simple version reference
I don't know if it is better to ask here or on MSE, if that's the case I can post the question there. I would need a simple version of the martingale central limit theorem. And, by simple, I mean the ...
- 911
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150
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Reference-request in deformation theory
Let $X$ be a smooth projective variety, and $Z\subset X$ a smooth closed subvariety of $X$. The first order deformations of $Z$ in $X$ are parameterized by $H^0(Z, N_{Z/X})$, while the first order ...
- 39.5k
0
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0
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47
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The second term of the Minakshisundaram-Pleijel expansion for the Green function in Euclidean domains
Let $Q$ be unbounded convex domain in $\mathrm R^n$ and $G(x,y,t)$ be the Green function of the first (or second or third) boundary value problem for the heat equation $u_t-\Delta u=0$ in the cylinder ...
- 2,750
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0
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Name for "continued fraction" matrices and the group they generate
Let $R$ be a ring. Is there a standard name for matrices of the form
$$
\begin{pmatrix}a & 1\\ 1 & 0\end{pmatrix}\in \mathbb{M}_2(R)?
$$
When $R=\mathbb{Z}$, these matrices arise naturally in ...
- 19.4k