**13**

votes

**18**answers

9k views

### Good books on theory of distributions

Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.

**32**

votes

**1**answer

6k views

### Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...

**20**

votes

**1**answer

713 views

### Is there an online encyclopedia of Diophantine equations (OEDE)?

Hello all!
I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences.
While trying to solve one Diophantine equation, I reduced the ...

**4**

votes

**2**answers

305 views

### P.J. Hilton notes requested

Does anybody here have the mimeographed notes Homotopy theory and duality, by P.J. Hilton, Cornell University, 1959 ?
I guess that those notes were never published online.
I believe that some ...

**2**

votes

**1**answer

665 views

### First Chern class of canonical bundle ?

This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega_X$. It is clear that in $H^2(X,\mathbb{R})$,
$$c_1(\omega_X) = - c_1(T_X)$$
(Obvious using ...

**0**

votes

**0**answers

19 views

### Which Dihedral Groups are $CI$-Groups?

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...

**2**

votes

**1**answer

77 views

### When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...

**3**

votes

**0**answers

262 views

### How the idea of adjugate matrix has been designed? [on hold]

I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...

**0**

votes

**1**answer

273 views

### Andre-Oort for conjecture [on hold]

Is Andre-Oort conjecture expected to hold for complex analytic topology? If yes, do the recent results on abelian type Shimura varieties cover this case?
Sorry for my ignorance, but several ...

**7**

votes

**2**answers

598 views

### Weil's paper under a pseudonym on deforming singular varieties

I am looking for a paper of Weil that is published under a pseudonym, in which he proves a statement along the lines of: a singular algebraic variety cannot be deformed into a nonsingular one.
Thanks ...

**5**

votes

**1**answer

167 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**1**

vote

**1**answer

72 views

### “Immovable” topological spaces

Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$ x \simeq_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$
Clearly ...

**69**

votes

**22**answers

6k views

### Special rational numbers that appear as answers to natural questions

Motivation:
Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. ...

**2**

votes

**1**answer

100 views

### Looking for Schmickler-Hirzebruch' monograph on elliptic surfaces

I wonder if it is possible to find (and if yes, where?) an electronic copy of the following monograph:
Author: Schmickler-Hirzebruch, Ulrike
Title: Elliptische Flächen über $\mathbb P^1(\mathbb ...

**3**

votes

**0**answers

89 views

### The density of square-free integers represented by a cubic polynomial

Suppose that $f(x)$ is an irreducible cubic polynomial with integral coefficients. Suppose further that for all primes $p$, there exists an integer $n_p$ for which $p^2 \nmid f(n_p)$. Then it is a ...

**21**

votes

**2**answers

434 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**4**

votes

**0**answers

52 views

### Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...

**2**

votes

**1**answer

76 views

### Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary ...

**2**

votes

**0**answers

89 views

### McDiarmid-like inequality for subgassian random variables

Let $X_n$ be a set of $N$ subgaussian random variables, not necessarily independent, with $E\exp(\lambda X_n) \le \exp(\lambda^2/2)$. Let $X=(X_1,\ldots, X_N)$ and $f:\mathbb R^N \rightarrow \mathbb ...

**2**

votes

**1**answer

90 views

### Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$

In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon ...

**1**

vote

**0**answers

19 views

### Constrained absolute orientation of 3D point sets

Let us assume we have two 3D point sets, $P=\{p_i\}$ and $Q=\{q_i\}$, and that we need to recover the transformation that takes $P$ as close to $Q$ as possible. In particular, I am interested in ...

**-4**

votes

**0**answers

61 views

### $\pi$-system and $\lambda$- systems [on hold]

I have some trouble with a theoretic-like exercise about measure theory, and I would like to have some help. The problem is stated in the book Mathematical Statistics, Jun Shao, exercise 5 of section ...

**1**

vote

**1**answer

64 views

### Cancellation of 2-component links

Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link ...

**4**

votes

**1**answer

52 views

### second dual of minimal tensor products of $C^*$-algebras

Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?

**4**

votes

**0**answers

51 views

### Are free positive operators equivalent to almost-commuting operators?

Set $A:=C_0((0,1]) * C((0,1])$ (the free product C*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^*x=a$ ...

**3**

votes

**1**answer

90 views

### Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...

**0**

votes

**1**answer

69 views

### Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion.
What is the cardinality of ${\frak P}$, and what's ...

**4**

votes

**0**answers

64 views

### Is there a decomposition strengthening of the Sauer-Shelah Lemma?

Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...

**3**

votes

**0**answers

51 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**2**

votes

**1**answer

142 views

### Is there anything similar to the four color theorem for 3-dimensional objects?

From Wikipedia:
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more ...

**4**

votes

**1**answer

77 views

### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...

**3**

votes

**1**answer

68 views

### Closed leaves on foliations of $\mathbb{R}^n$

I want to know if there exists a characterization of k-foliations of $\mathbb{R}^n$ which have all the leaves closed.
Do exists a $k$-foliation of $\mathbb{R}^n$ with a non-closed leaf?
In general, ...

**0**

votes

**1**answer

69 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**1**

vote

**0**answers

59 views

### Endomorphism of Chow goup induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example ...

**10**

votes

**1**answer

185 views

### What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the ...

**0**

votes

**0**answers

43 views

### Construct a locally concave polynomial [on hold]

I'd like to construct a polynomial $f(x)$ such that:
$f(x)$ has roots 0 and $1$ (and possibly others)
$f'(a) = 0$ for given $a$, where $0 < a < 1$
$f(a) = 1$ (ie. there's a local maxima at ...

**-1**

votes

**1**answer

66 views

### Graphs such that contracting an edge decreases the chromatic number [on hold]

Let $G = (V,E)$ be a finite, simple, undirected, connected graph, such that contracting an edge reduces the chromatic number. Does this imply that $G$ is complete?

**2**

votes

**1**answer

114 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a ...

**4**

votes

**1**answer

163 views

### Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...

**-5**

votes

**0**answers

152 views

### Power equivalence of two positive real numbers [on hold]

I posted the same question on Math.Stackexchange but I didn't get any precise answer until now. Thus I asked it here.
Assume $a,b>0$ are two real numbers. Define the sequences $a_n, b_n$ as ...

**4**

votes

**2**answers

132 views

### Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...

**7**

votes

**1**answer

85 views

### Weighted Permutation Sum

I am trying to find out a closed-form formula (or a generating function at least) for the number of permutations $\sigma$ that satisfy $$ S = \sum_{i = 1}^{n} i\sigma(i)$$ for a given value of $S$. We ...

**2**

votes

**1**answer

75 views

### If two knots in $S^3$ are invertible cobordant (from both ends), are they equivalent?

Let $K_1,K_2$ be two knots in $S^3$ and assume that there exists a cobordism $(W;K_1,K_2)$ which is invertible from both ends. Does this imply that $K_1, K_2$ are equivalent? In the paper by D.W. ...

**9**

votes

**2**answers

233 views

### Traces of operators in nuclear spaces

I am currently reading up on nuclear spaces in Yarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:
Let $F$ be a nuclear ...

**1**

vote

**1**answer

149 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...

**5**

votes

**1**answer

111 views

### Closeness graph of a topological space

Let $(X,\tau)$ be a topological space. We say that $x, y \in X$ are close if for every neighborhood $U$ of $x$ and $V$ of $y$ we have $U\cap V \neq \emptyset$. Let $E$ be the set of $\{x,y\}$ where ...

**0**

votes

**1**answer

146 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**-1**

votes

**0**answers

54 views

### Modules over polynomial algebras [on hold]

Let $M$ be a cyclic torsion module over the polynomial ring $k[x_1, \dots, x_n]$.
Let $M = K/J$, where $K$ denotes the polynomial ring $k[x_1, \dots, x_n]$.
Is it true that $J \cap k[x_i] \ne 0$ for ...

**8**

votes

**1**answer

203 views

### Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?

Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...

**3**

votes

**1**answer

422 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...