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

**3**

votes

**0**answers

82 views

### How to describe morphisms to a weighted projective space (bundle)?

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. ...

**4**

votes

**1**answer

239 views

### Recognize this countably generated abelian group?

I recently came across a construction that, in abstraction, leads to the following family of abelian groups: Fix $1<q<p$ with $q$ and $p$ relatively prime. The group $G_{(p,q)}$ is given by the ...

**1**

vote

**0**answers

53 views

### Does any one have Gerhard Ringel's note: “Cyclic Arrangements of the elements of a group.” [closed]

Which has reference:
G. Ringel,
Cyclic Arrangements of the elements of a group.
Notices of the American Mathematical Society
Vol 21 1974 pages A95-96
I'm having a hard time getting a look at ...

**8**

votes

**1**answer

296 views

### Multiplicity of Laplace eigenvalues

Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...

**1**

vote

**0**answers

67 views

### A reference for “Borel Sets and Circuit Complexity”

Is there any pdf version of M.Sipser's "Borel Sets and Circuit Complexity" or , since I am unable to get this paper, is there other reference closely related to theory in that paper?

**4**

votes

**1**answer

266 views

### Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao:
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...

**8**

votes

**3**answers

337 views

### Isometry group of a compact hyperbolic surface

Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have ...

**4**

votes

**0**answers

66 views

### A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...

**1**

vote

**0**answers

73 views

### reference for weighted blow-up

Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$
(over $\mathbb{C}$).
We think of $X$ as a small neighborhood of $0$ (for studying singularity).
If we can think $X$ as a ...

**2**

votes

**2**answers

285 views

### Ricci flow and isometry group

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...

**6**

votes

**1**answer

222 views

### Subsets of [1..N] with no three-term arithmetic progressions and no large gaps

Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...

**5**

votes

**0**answers

100 views

### Relativistic correction to the Hydrogen atom spectrum, reference? [closed]

Can anyone provide me a reference to the relativistic correction to the Hydrogen atom spectrum worked out? I've tried googling, but I haven't found anything (perhaps I'm just bad at googling). Thanks!
...

**2**

votes

**2**answers

194 views

### Connectedness of units in finite-dimensional commutative complex algebras

In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).
Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...

**9**

votes

**5**answers

445 views

### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...

**5**

votes

**0**answers

167 views

### Constructive approximation of Lipschitz functions

There are a number of theorems in classical functional analysis about approximation of Lipschitz functions by smooth functions. I was wondering if there are any similar constructive and explicit ...

**2**

votes

**1**answer

139 views

### Some notational questions regarding tangent vectors

I am not a specialist in differential geometry, so I have some difficulties in finding the right words for the following natural things:
First of all it seems that there is a lot of nonequivalent ...

**4**

votes

**0**answers

176 views

### Commutative algebraic version of algebraic geometric object

In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...

**3**

votes

**1**answer

113 views

### dirichlet problem in the heisenberg group

Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...

**3**

votes

**0**answers

40 views

### Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...

**1**

vote

**0**answers

51 views

### Generalized “elliptic integrals”

I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape
...

**16**

votes

**2**answers

374 views

### Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...

**1**

vote

**0**answers

76 views

### Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...

**3**

votes

**1**answer

166 views

### Reference for constructing tensor products of finitely presented functors

I need references related to the construction of tensor product between functors
Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote ...

**2**

votes

**0**answers

212 views

### Generalized Ramanujan's identity with hyperbolic cotangent

Three weeks ago I derived an identity, which generalizes Ramanujan's identity with hyperbolic cotangent. Don't you know is it original or not?
$$
\sum_{n=1}^\infty \frac{\coth(\pi ...

**2**

votes

**0**answers

49 views

### Factorization of linear bounded operators in Banach spaces

I am looking for a reference, if any, to the following statement: "Let $X$ and $Y$ be
Banach spaces, $A\in\mathcal{B}(X,X)$ be a linear bounded operator, and $B\in\mathcal{B}(X,Y)$ be surjective. ...

**0**

votes

**0**answers

51 views

### Tau functions for the KP and Toda lattice hierarchies

A statement, which is known to be true can be vaguely stated as "the tau function for the KP and Toda hierarchies are the same". I would just like to know exactly what it means as it is not obvious ...

**4**

votes

**1**answer

80 views

### Weights on cyclic orderings

Are there standard or known weights/metrics on cyclic orders?
Cyclic orderings are different ways of listing elements from a finite set, where you call two lists the same if they differ only by a ...

**2**

votes

**1**answer

85 views

### Looking for a modern source about Ulm Invariants

I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...

**11**

votes

**1**answer

268 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

**9**

votes

**2**answers

244 views

### Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?

When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...

**4**

votes

**2**answers

331 views

### Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...

**11**

votes

**1**answer

217 views

### Have “sturdy squares” been studied before?

Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...

**4**

votes

**0**answers

157 views

### Convergence of spectrum

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...

**3**

votes

**1**answer

152 views

### Intersection of maximal subgroups of PSL(2,q)

Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or ...

**5**

votes

**0**answers

93 views

### How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?

It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...

**16**

votes

**1**answer

710 views

### Reference Request: Conductors of Twists of Hyperelliptic Curves

It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to ...

**7**

votes

**0**answers

219 views

### A question on a result of Colin de Verdiere

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdiere (see here) says that if we fix $\gamma$ and select a finite sequence ...

**2**

votes

**1**answer

113 views

### Join of simplicial categories

Let $\mathcal{C},\mathcal{D}$ be simplicial categories.
Of course, we have the "naïve" join $\mathcal{C} \star \mathcal{D}$, which has
$$
\mathrm{Ob}(\mathcal{C} \star \mathcal{D}) := ...

**5**

votes

**1**answer

81 views

### Hausdorff open image of a Polish space

Let $f\colon X\to Y$ a continuous open and surjective function, where $X$ is Polish.
It is known that $Y$ is Polish if: $f$ is closed or $Y$ is metric.
Suppose that we know that $Y$ is Hausdorff, ...

**2**

votes

**0**answers

61 views

### Alternating elements in free graded-commutative algebras

It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...

**3**

votes

**0**answers

91 views

### Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...

**5**

votes

**0**answers

40 views

### “Quasi-orthogonal” subgroup of a group with length?

For my project in bivariant K-theory for locally convex algebras, I'm looking how to call a particular notion of groups, too simple to be never considered elsewhere.
Let $G$ b a group with length ...

**14**

votes

**3**answers

672 views

### Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?

Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...

**15**

votes

**3**answers

797 views

### Reference for a linear algebra result

I asked the following question (http://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...

**-1**

votes

**1**answer

211 views

### Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation ...

**0**

votes

**1**answer

61 views

### Existence and estimates of a solution of a perturbed first order partial differential equation

My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you ...

**6**

votes

**1**answer

160 views

### Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?

Let
$$
M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast}
$$
be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If ...

**10**

votes

**0**answers

149 views

### Can we find minimal-diameter metrics without computability?

A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...

**4**

votes

**1**answer

73 views

### Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...

**2**

votes

**0**answers

83 views

### Complex scalar field, computation of propagators, four point function [closed]

This is a followup to my previous question here.
In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can ...