Tagged Questions
This tag is used if a reference is needed in a paper or textbook on a specific result.
0
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0answers
36 views
Property of Fixed Point Function
Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
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0answers
143 views
Is there a known shorter axiomatization of NF than this?
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...
3
votes
2answers
139 views
The cofinality of the poset $[\kappa]^{<\kappa}$ for a singular cardinal $\kappa$
For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of ...
6
votes
1answer
71 views
Further Developments of Lieb-Schultz-Mattis theorem in Mathematics
The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving ...
0
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0answers
46 views
Article request--Central trinomial coefficients and convolution type identities
R. Witula and D. Slota, Central trinomial coefficients and convolution type identities, Congressus Numerantium 201, 109-126 (2010).
I would like to be able to read the full text of the paper. But I ...
4
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0answers
62 views
Local properties of Galois representations attached to torsion classes
$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}}$
Let $F$ be a number field, and let $\Gamma$ be a congruence subgroup of $\...
25
votes
2answers
689 views
References for Riemann surfaces
I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.
I am ...
2
votes
1answer
64 views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
4
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0answers
50 views
Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality
I am having some issues computing Poincare duality for the quantum homology $QH(M)$ when $(M,\omega)=(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$. I am using the simple novikov ring $\Lambda$ ...
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0answers
12 views
Kushner Stratonovich type equation for a general observation process
In stochastic filtering theory, Kushner Stratonovich equation gives the dynamics of $E[f(X_t, Y_t)|\mathcal{F}_t^Y]$, where $X$ is the hidden or unobserved process, $Y$ is the observed process and $\...
7
votes
2answers
551 views
Eigenvalues of Laplace-Beltrami on half sphere
Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
3
votes
1answer
158 views
Is there a vector field such that one differential form is the Lie derivative of the other?
I'm looking for a reference or answer for the following question:
Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for ...
2
votes
0answers
73 views
Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
6
votes
2answers
1k views
Does this multiplicative function have a name? If so, what is known about it?
It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
5
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0answers
84 views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
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0answers
15 views
Reference request: duality gap of non-convex objective function with convex constraints
I have the following optimization problem:
$\max_{x \in X} f(x) $
s.t $\ \ g_i(x) \leq 0 \ \ \forall i \in \{1,2,\ldots,n\}$
where $X$ is a convex set in $\mathbb{R^n}$ and $g_i : X \to \mathbb{R}...
5
votes
1answer
372 views
English translation of “Les aspects probabilistes du contrôle stochastique”
I am looking for an English translation of "Les aspects probabilistes du contrôle stochastique" written by Nicole El Karoui, or knowledge whether it exists.
Other references with similar content on ...
2
votes
0answers
44 views
One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
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0answers
23 views
(Upper) bounding the MGF of a semi-decoupled non-homogeneous Rademacher chaos of order 4
Let $(\xi_i)_{1\leq i\leq n}$, $(\xi'_i)_{1\leq i\leq n}$ be two independent vectors of independent Rademacher random variables, and
$(a_{ij})_{1\leq i,j\leq n}$, $(b_{ijk})_{1\leq i,j,k\leq n}$, $(c_{...
2
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0answers
47 views
Almost-prime values attained by polynomials, with extra conditions
Many results in sieve theory are of the following form:
Let $f(x) \in \mathbb{Z}[x]$ be polynomial satisfying certain conditions. There are infinitely many integers $n$ such that $f(n)$ is a ...
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0answers
23 views
Weak convexity in graphs
I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question.
As we know, a finite undirected graph ...
6
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0answers
88 views
Is there any survey of dg-categories from the $\infty$-category point of view?
I was reading this question on dg-categories and a comment by David Ben-Zvi says "An excellent pre-$\infty$-categorical overview is Keller's ICM address https://arxiv.org/abs/math/0601185".
I was ...
3
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0answers
26 views
Convergence of the intertwining operator as a vector valued integral
Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \...
3
votes
2answers
196 views
+50
Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
2
votes
0answers
104 views
DeGiorgi oscillation lemma
Where can I find a proof of the following result?
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(...
0
votes
0answers
19 views
Asymptotic Constancy of solutions of delay/integro differential equations
I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference).
I am however most ...
1
vote
0answers
53 views
Solve simple stochastic variational inequality
Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
5
votes
0answers
142 views
$X$-rays of permutations
Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.
There has been some study (e.g. ...
4
votes
1answer
178 views
Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
4
votes
1answer
109 views
compact embedding for Sobolev spaces
The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$
Is it possible to determine the ...
1
vote
0answers
13 views
Large Deviations Rate of Convergence and Robbins Monro
I am looking for a result/paper (if there is any) on the large deviations rate of convergence of the Robbins-Monro (RM) algorithm. Specifically, given $X_k \rightarrow X$ a.s. in the RM algorithm, I ...
4
votes
0answers
102 views
Nash embedding for complete manifold
I, ask my question as a comment in this post. Without answer I post a more detailed version.
I am looking for a reference about $C^\infty$ Nash isometric embedding for non compact manifold.
My ...
5
votes
1answer
105 views
Pontryagin square, Postnikov square and their consistency formulas
$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
...
12
votes
1answer
483 views
Reference for the algebro-geometric proof of Matsumoto theorem
Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...
4
votes
0answers
57 views
equidistributed parameters on graphs
Let $\mathcal G_n$ be the set of (isomorphism classes of unlabelled) simple graphs on $n$ vertices.
I wonder whether there are any 'interesting' combinatorial parameters $a,b: \mathcal G_n\to \mathbb ...
8
votes
0answers
222 views
Cohomology of complex manifold vs cohomology of its complex submanifold
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...
7
votes
2answers
166 views
A link between hooks and contents: Part II
This is a question in the spirit of an earlier problem.
Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$.
Recall also the notation for the content of a cell $...
0
votes
1answer
221 views
What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?
The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and ...
6
votes
2answers
121 views
A link between hooks, contents and parts of a partition
Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Denote its conjugate partition by $\lambda'$. For example, if $\lambda=(4,3,1)$ then $\lambda'=(3,2,2,1)$.
...
4
votes
1answer
245 views
Thurston's preprint: “On the geometry and dynamics of diffeomorphisms of surfaces”
W. Veech on Teichmüller curves in moduli space, Eisenstein series and applications to triangular billiards says on the second paragraph of page 579:
"Thurston's original construction [8] corresponds ...
5
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0answers
206 views
Grothendieck letter to Jun-Ichi Yamashita on tame topology
I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...
6
votes
1answer
186 views
Reference request: an elementary result on characters of finite abelian groups
The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $...
7
votes
0answers
147 views
Gerhard Frey, “Links between stable elliptic curves and certain diophantine equations”
I am searching for the article by Gerhard Frey, which has indicated a connection between Fermat's Last Theorem and the Taniyama-Shimura Conjecture. The reference is give as
Gerhard Frey, Links ...
6
votes
2answers
455 views
+50
Pages from a known textbook on Euclidean geometry?
Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below?
A teacher provided us xerox ...
1
vote
0answers
95 views
Why is this the dualising sheaf of a singular curve?
If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is
$$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$
Here, $\pi:X\to X'$ is the ...
3
votes
1answer
109 views
Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
1
vote
1answer
141 views
Degree of the projection of a projective variety
Let $X\subseteq\mathbb{P}^n\times\mathbb{P}^m\subseteq \mathbb P^{(n+1)(m+1)-1}$ be a projective variety of dimension $p$ and degree $d$ defined over an algebraically closed field $k$, where $\mathbb{...
1
vote
2answers
159 views
About finite representability of Banach space
Can someone please tell me the brief sketch (or any known reference) of the following results?
Why $\ell_2$ is finitely representable in any infinite-dimensional Banach space?
Why every Banach space ...
12
votes
0answers
178 views
average value of j-invariant at infinity
Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...
6
votes
0answers
171 views
$R(3,6) = 18$, especially proving that $R(3,6)>17$
I'm studying the Ramsey numbers, especially $R(3,6) = 18$
I understand that the proof using the theorem $R(m,n) <R(m-1,n)+R(m,n-1)$ can only prove that $R(3,6)<20$. However by Cariolaro's "On ...
