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

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0
votes
1answer
89 views

Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions? $D(s) = \sum_{0<n<N}a_n/n^s$ The ...
2
votes
0answers
104 views

Unitary representation of finite-dimensional unitary group

the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...
3
votes
4answers
204 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite ...
4
votes
2answers
347 views

How many metrics of constant curvature exist on a Riemannian surface?

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...
1
vote
1answer
45 views

How to prove that the convolution operator associated to a discrete measure on a LCA group has natural spectrum?

Let $\mu$ be a Borel measure with finite variation on a locally compact abelian group $G$, let $\Gamma$ denote the dual group of $G$, and let $\hat \mu: \Gamma \to \mathbb{C}$ be the Fourier-Stieltjes ...
6
votes
1answer
137 views

Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code

It is well known that every Borel set has the property of Baire. That is, for every Borel set $B$, there is an open set $U$ and a sequence of dense open sets $D_n$ such that for every $x\in \cap_n ...
2
votes
0answers
109 views

Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to ...
2
votes
1answer
180 views

What is the applications of the dg-enhancements of derived categories of sheaves

Let $X$ be a scheme and let $D^b_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies. We know that the category $D^b_{\text{coh}}(X)$ has some ...
3
votes
1answer
141 views

Sobolev spaces on compact manifolds

Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent ...
4
votes
1answer
233 views

Direct limit of compact topological spaces

Let $\{X_n\}_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f_i\colon X_i \to X_{i+1}$ with the necessary compatibility conditions. Is there any ...
3
votes
1answer
64 views

Domain of square root of a self-adjoint positive operator [closed]

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that ...
4
votes
2answers
256 views

Yang–Baxter explanation

What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...
7
votes
1answer
257 views

“Thin film evolution” (Reference request)

Ok this is my first$^*$ question on overflow, my apologies if this is not the right place to ask what follows! I observed the following phenomenon: I put a (vitamin) tablet into water, then after a ...
3
votes
0answers
121 views

Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...
4
votes
1answer
281 views
+100

Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
2
votes
0answers
89 views

Lifting of Commuting Maps of Vector Bundles

Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
2
votes
0answers
49 views

Torsionfree finitely generated compact Iwasawa module

The following fact falls under the category of Iwasawa modules. Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p ...
0
votes
0answers
37 views

Rate-Distortion theory: What is the distribution of distortion on an optimal encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error ...
8
votes
3answers
327 views

Between Tietze's and Dugundji's Extension Theorems

The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
5
votes
1answer
111 views

Decay of solutions to Schrodinger equation with local minimum in potential

Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by $$ L = - \partial_x^2 + V $$ where $V$ is a potential with the following properties: $V$ is non-negative, ...
0
votes
0answers
101 views

How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...
4
votes
1answer
133 views

Embedding of classical into intuitionistic linear logic

Following on from this recent question, there is another construction that is well-known, but I don’t know a good primary source for: the Kolmogorov-style double-negation embedding of classical into ...
2
votes
1answer
106 views

Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact). Question 1. Is it true that every point of $X$ has a ...
9
votes
0answers
137 views

Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis? Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and ...
1
vote
1answer
219 views

Locus where morphism is étale is open on target

Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is finite, flat and locally of finite presentation. Then I can prove that the set $$U:= \{ y\in Y : X_y \to y \hspace{1mm} \text{is ...
3
votes
0answers
121 views

Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type. I find there are plenty of research already been done and ...
2
votes
0answers
68 views

What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question. Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...
2
votes
0answers
51 views

Classes of knots that have known Bridge spectra

Bridge spectra is a knot invariant first defined by Doll, who established some basic properties. Tomova has shown that high distance knots have bridge spectra $(n,n-1,\ldots,2,1,0)$. Zupan has ...
0
votes
0answers
116 views

Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies. Do we have the following ...
4
votes
1answer
81 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
6
votes
0answers
41 views

A “lower-central” filtration of Steenrod algebra?

$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...
1
vote
1answer
143 views

Dieudonné modules -reference request

I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of Reference for Dieudonné modules, ...
4
votes
2answers
90 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
2
votes
1answer
178 views

Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...
2
votes
0answers
117 views

Irreducibility of a general fibre

Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact? If $A$ is algebraically ...
2
votes
0answers
116 views

Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? When is the metric induced by the ...
3
votes
1answer
104 views

Classification of Hopf-Galois Extensions as Torsors

Faithfully flat Hopf-Galois extensions of rings: $A\to B$, with $H$ coacting on $B$ such that $B\otimes_AB\simeq B\otimes H$, are often thought of as being accessible substitutes for $G$-torsors in ...
4
votes
0answers
166 views

Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
-2
votes
0answers
131 views

About “covering” subgroups

Let $H$ be a subgroup of $S_n$ (the symmetric group with n elements). In the paper I read (cf. Thm 3.17 there), the authors define $H$ to be a covering if the following condition holds: for all ...
14
votes
5answers
2k views

Proof that no differentiable space-filling curve exists

Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists? Or piecewise differentiable? Must every continuous space-filling curve be nowhere ...
20
votes
0answers
315 views

Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ Also, a particular ...
8
votes
6answers
1k views

number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. ...
3
votes
0answers
123 views

Counting number of points in a lattice with bounded length

I am interested in counting number of lattices using the following theorem. The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, ...
6
votes
2answers
150 views

Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture)

What is a good source for Silver's proof (or a more modern version) that Con($\exists \omega_1$-Erdos cardinal) implies Con(Chang's Conjecture)? Silver's original proof seems to have never been ...
3
votes
0answers
150 views

Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?

It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e. $$ ...
15
votes
5answers
725 views

Thales' semicircle theorem in higher dimensions

Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle. Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle ...
1
vote
0answers
81 views

Square-free sieve over number fields

I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
4
votes
0answers
57 views

Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...
1
vote
3answers
250 views

orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$. I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates. For ...
6
votes
2answers
355 views

Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...