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

**3**

votes

**0**answers

120 views

### Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...

**2**

votes

**0**answers

82 views

### Pure braid groups of the complement of a lattice in the complex plane: generators and relations

Where can I find a presentation (by `natural' generators and relations between them)
of the pure braid groups $PB_n(S)$ (for $n>0$) of $S=\mathbb C\setminus (\mathbb Z\oplus i \mathbb Z)$?
Thanks ...

**1**

vote

**1**answer

215 views

### A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...

**9**

votes

**2**answers

464 views

### Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original.
Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that :
...

**9**

votes

**2**answers

560 views

### Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...

**7**

votes

**2**answers

248 views

### Arveson's extension for normal completely positive maps

My question deals with a version of Arveson's extension theorem (for the standard version, see, e.g., Paulsen's book Completely Bounded Maps and Operator Algebras). Let $\mathcal A$ be a von Neumann ...

**2**

votes

**0**answers

81 views

### Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ ,
$k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ .
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...

**16**

votes

**1**answer

377 views

### Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...

**6**

votes

**1**answer

156 views

### Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.

**2**

votes

**0**answers

86 views

### Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where
...

**9**

votes

**4**answers

775 views

### Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of ...

**17**

votes

**2**answers

688 views

### Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...

**5**

votes

**1**answer

250 views

### A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let ...

**5**

votes

**1**answer

215 views

### Literature about metapolynomials

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form
$$f(x_1,\cdots , x_k ...

**1**

vote

**0**answers

192 views

### The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**3**

votes

**0**answers

135 views

### Compact set covered by two opens

The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs.
If a ...

**0**

votes

**0**answers

76 views

### Transversal theory in ergodic theory

I was taking a glance at the original paper by Donald Ornstein, Bernoulli Shifts with the Same Entropy are Isomorphic, and I came across The Marriage Problem and a paper with the same name by P. ...

**0**

votes

**1**answer

116 views

### Length-preserving Analogue of Riemann's Mapping Theorem

The Riemann mapping theorem (cf e.g. http://en.wikipedia.org/wiki/Riemann_mapping_theorem) essentially guarantees the existence of a biholomorphic mapping of a simply connected, open subset of the ...

**3**

votes

**2**answers

144 views

### Hamilton-Ivey pinching in dimension 4

I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. ...

**3**

votes

**1**answer

168 views

### Vanishing theorems for pluri-canonical bundle

I would like to know if Grauer-Riemenschneider vanishing theorem is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle.
Let me recall
Grauer-Riemenschneider ...

**4**

votes

**2**answers

187 views

### Inherent ambiguity of the context-sensitive language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or $a^nb^nc^nd^ne^nf^n$

What is the definition of ambiguity of context-sensitive
grammar?This is relevant to the definition of inherent ambiguity of
context-sensitive language.And any proof for the inherent ambiguity of ...

**6**

votes

**0**answers

92 views

### Bundles over Function Spaces

Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable ...

**0**

votes

**0**answers

16 views

### Estimating Missing Shortest-Path-Tree Distances from Point Coordinates

Problem:
given
a finite set of points $\mathcal{P}:=\lbrace p_i\in\mathbb{R}^2 \mid i\in \lbrace 0,...,n\rbrace\wedge i\neq j\Leftrightarrow p_i\neq p_j\rbrace$
a function ...

**4**

votes

**2**answers

140 views

### Visibility spaces and Gromov hyperbolicity

I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...

**11**

votes

**3**answers

497 views

### Is there a “unique” homogeneous contact structure on odd-dimensional spheres?

Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in ...

**1**

vote

**1**answer

90 views

### maximal chain in (strong) Bruhat order satisfying constraint

Consider the (strong) Bruhat order, $\leq_B$, on the symmetric group $S_n$. Suppose there are permutations $\pi,\sigma∈S_n$ such that $\pi\geq_B \sigma$. Suppose further that they satisfy the ...

**2**

votes

**1**answer

140 views

### Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...

**1**

vote

**0**answers

145 views

### Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty...
My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...

**18**

votes

**3**answers

607 views

### Number of primitive $n$th roots with positive versus negative real parts

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...

**4**

votes

**1**answer

54 views

### Reference: Hardy space regularity of the Jacobian determinant

I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes.
Theorem:
For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...

**1**

vote

**1**answer

64 views

### Unitization via “End points compactification”

We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...

**47**

votes

**8**answers

4k views

### How does one find out what's happening in contemporary mathematics research?

How does one find out what's happening in contemporary mathematics research?
EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside ...

**1**

vote

**0**answers

69 views

### Empty node in cactus construction

Is there a necessary condition for not having empty node in the construction of the cactus of the minimum cuts of a graph?
In particular is there a necessary condition for not having empty k-junction ...

**3**

votes

**1**answer

163 views

### Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$

The following excerpt is from Connes' Noncommutative Geometry
Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel ...

**3**

votes

**1**answer

232 views

### An integral estimate over rotations of the dyadic grid

I'm currently reading the paper Rectifiable Sets and the Traveling Salesman Problem (link) by Peter Jones (Invent. math. 102, 1-15 (1990)), and am having trouble understanding an integral estimate ...

**3**

votes

**2**answers

140 views

### Asymptotics of the number of elements in the intersection of two growing sets

Let $[n]:=\{1,\dots,n\}$ and $0\leq p_n\leq n$. Fix any subset $A_n$ of $[n]$ with $p_n$ elements. The number of subsets $B$ of $[n]$ with $p_n$ elements that are disjoint from $A$ is ...

**2**

votes

**1**answer

91 views

### Estimate for the travelling salesman problem for balls inside a grid

This question is probably easy but I only have "tedious case checking" proof strategy in sight, and I'm sure there should be a reference lying around...
The question concerns the TSP problem (with ...

**2**

votes

**1**answer

103 views

### How to classify the complex function with same natural boundary in complex plane? [closed]

There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with ...

**0**

votes

**0**answers

51 views

### References for LWP of a NLS Equation

I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...

**6**

votes

**1**answer

94 views

### Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...

**2**

votes

**0**answers

117 views

### Dehn functions of Thompson's group $F$

It's well know that the first order Dehn function of $F$ is quadratic. Is a similar result known for its second-order, or even higher-order, Dehn function?
The second-order Dehn function of a group ...

**5**

votes

**7**answers

475 views

### Source for roots of matrix polynomials?

A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...

**2**

votes

**1**answer

273 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**3**

votes

**1**answer

92 views

### Doubling of variables method for parabolic equations

Does anyone have a reference that explains the technique of doubling of variables as introduced by Kruzkov? It seems to be a necessary tool for contraction estimates when we have weak solutions. ...

**1**

vote

**1**answer

85 views

### Getting a comparison principle for parabolic equation when solution is not that smooth

Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to
$$\frac{\partial}{\partial t}b(u) - \Delta u = f$$
where $b$ is continuous, increasing and locally ...

**4**

votes

**0**answers

195 views

### Singularities in Yang Mills Flow

In "The Yang Mills flow in four dimensions", M. Struwe proves that this flow converges, up to bubbling phenomena. And he has conjectured that this explosion in finite time should happen as proven for ...

**7**

votes

**6**answers

381 views

### Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)
Let $A$ be a set ...

**4**

votes

**0**answers

57 views

### Non-reflexive Orlicz spaces

I am looking for a good reference on Orlicz spaces; I would appreciate any books treating this topic from the Banach-space perspective. For example, I would like to find a reference to the following ...

**0**

votes

**0**answers

95 views

### projecting Laplacian onto tangent and normal bundles of submanifold

If I have a simple linear differential equation involving covariant derivatives such as $\nabla^2 g_{\mu\nu}+ 2g_{\mu\nu}=0$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) ...

**9**

votes

**1**answer

158 views

### Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...