**1**

vote

**10**answers

3k views

### What is an explicit example of a sequence converging to two different points? [closed]

In principle a sequence in a non-Hausdorff space can converge to two points simultaneously.
Can anyone give me an explicit example of the above?
Or tell me any method of generating such kinds of ...

**34**

votes

**5**answers

4k views

### Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?

**11**

votes

**5**answers

880 views

### Asymptotics of a Bernoulli-number-like function

Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = ...

**25**

votes

**3**answers

3k views

### What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...

**13**

votes

**1**answer

1k views

### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset ...

**0**

votes

**2**answers

532 views

### Principal Bundle with given Curvature

Hey,
For personal exercising purposes I try to give a proof, that a U(1)-principal-bundle has curvature $\alpha$ iff the cohomology class of $\alpha$ is integral:
By the Cech-deRham-isomorphism a ...

**6**

votes

**2**answers

549 views

### Next (Restricted) B-Smooth Number Problem?

Given a bound, $B$, and a list of (small) primes $(p_0, p_1, \dots, p_{n-1})$ is there an efficient algorithm to find the next number greater than $B$ that can be expressed as a product of primes from ...

**0**

votes

**0**answers

17 views

### Is there a $q-$L'Hospital's Rule?

Let $\binom{n}{j}_q$ be a $q-$binomial coefficient and $(x;q)_n = (1-x)(1-qx)...(1-q^{n-1}x).$ I am interested in a simple proof of the limit relation
$$\lim_ {q\to1}\frac{\sum\limits_{j = 0}^{2n} ...

**0**

votes

**0**answers

5 views

### Automorphism of a restricted irregular graph class

Motivation:
This query is motivated by this question . It has relation to the complexity analysis of this post.
I have been informed Highly Irregular Graph has number of automorphism $\leq n ...

**0**

votes

**0**answers

47 views

### $C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.
I ...

**1**

vote

**0**answers

196 views

### A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

I keep stumbling on the same algebraic structure, and I have no clue how to understand or characterize it at all. It's basically the merger of the Dirichlet ring and the ordinary convolution ring. ...

**0**

votes

**1**answer

59 views

### The weird projection from SO(2n)/B to maximal isotropic grassmannian

Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e_i, e_{2n+1-j}\rangle=\delta_{ij}$)
$$F_1\subset F_2\subset\cdots ...

**2**

votes

**2**answers

122 views

### map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$?

Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows ...

**0**

votes

**1**answer

182 views

### A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$.
(a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$.
(b) If we consider the piece of curve on the region ...

**4**

votes

**4**answers

291 views

### Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:
Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...

**0**

votes

**1**answer

54 views

### How to write a given rank matrix with some constraints?

I want to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct.
If I want to write with only rows or columns distinct, I could just pick $m$ or $n$ ...

**-2**

votes

**0**answers

45 views

### How many techniques are there to test colliniarity of n points? [on hold]

How many techniques are there to test coliniariry of n points?
For example, suppose we have 4 points A, B , C, D. How many ways can it be tested that they are collinear?

**-4**

votes

**0**answers

37 views

### is graph coloring problem in general np-complete?(solvable) [on hold]

graph coloring problem
Hi,
i tried to find an algorithm for this problem and i want to make sure. i found it with this knowledge.
1.is graph coloring problem in general to find the ...

**0**

votes

**0**answers

84 views

### Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$
Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...

**6**

votes

**0**answers

193 views

### How algebraic is the holonomy map?

Let $G$ be either a compact, simple, simply-connected Lie group, or a simply-connected complex reductive group (so either $SU(n)$ or $SL(n,\mathbb{C})$, for instance), or even complex affine. I don't ...

**0**

votes

**0**answers

27 views

### Box counting dimension of the graphs of functions on $\mathbb R \rightarrow \mathbb R$ [on hold]

Generally speaking box counting techniques are applied to fractals defined by some iterative process, but what about functions? Has the concept of box counting dimension been investigated on the graph ...

**3**

votes

**0**answers

38 views

### Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:
...

**5**

votes

**1**answer

273 views

### soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs

I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? ...

**1**

vote

**0**answers

41 views

### Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is that if it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
...

**0**

votes

**0**answers

86 views

### reference for groupoid cohomology

In nLab (groupoid cohomology) says:
"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."
Are there references for ...

**7**

votes

**1**answer

248 views

### Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty.
Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern ...

**0**

votes

**0**answers

41 views

### Decomposition of polynomial ring as $S_n$-module [migrated]

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...

**15**

votes

**3**answers

212 views

### Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$

$\newcommand{\Prb}[1]{\mathcal{P}_{#1}}$
I have the following number theory problem, related to Odlyzko's improvement on Shor’s factoring algorithm (see this cstheory.sx question for details).
Let ...

**6**

votes

**0**answers

130 views

### When did people know that all real polynomials of degree greater than 2 are reducible? [migrated]

Admittedly, this may not be a research level question, but I am deeply curious about this.
Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...

**3**

votes

**1**answer

82 views

### About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.
...

**1**

vote

**0**answers

41 views

### asymptotic behavior of the solution of an ordinary differential equation

I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem.
Consider the following autonomous system of two non-linear ordinary differential ...

**5**

votes

**2**answers

204 views

### Does the ring generated by the odd power sum symmetric functions have a name?

Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd ...

**12**

votes

**1**answer

518 views

### Why is it so hard to prove Toeplitz' conjecture?

I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...

**4**

votes

**0**answers

122 views

### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

**8**

votes

**3**answers

255 views

### “Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem.
Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...

**6**

votes

**4**answers

379 views

### SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...

**1**

vote

**1**answer

121 views

### Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface?

The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory.
Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that ...

**4**

votes

**3**answers

343 views

### Do cotangent bundles have “bounded geometry”?

I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and ...

**1**

vote

**1**answer

172 views

### reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...

**0**

votes

**0**answers

64 views

### Proofs needed for observations regarding prime-partitionable numbers

Below is the definition of a prime-partitionable integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1987) 263-272 and is apparently the same in ...

**3**

votes

**3**answers

183 views

+50

### Terminology for polygons

As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.
So I converge to the ...

**3**

votes

**1**answer

175 views

### History of unstable formulas [on hold]

There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property.
While intuitively it makes sense that ...

**17**

votes

**2**answers

1k views

### Distributing points evenly on a sphere

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible.
I have found some related questions on stackoverflow but ...

**1**

vote

**1**answer

62 views

### examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set ...

**2**

votes

**2**answers

175 views

### Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)

In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...

**0**

votes

**1**answer

268 views

### Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.

**8**

votes

**2**answers

187 views

### Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation:
$$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$
where, $[x]$ is the nearest integer to $x$ not exceeding ...

**4**

votes

**2**answers

99 views

### Reference for (co)limit-preserving functor $X\mapsto R^X$

Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor ...

**4**

votes

**2**answers

83 views

### Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...

**3**

votes

**0**answers

162 views

### When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. Say I have functions $p_1,\ldots,p_m : \mathbb{F}^n \to \mathbb{F}$ each with high degree $\approx n$ as (multilinear) polynomials, and another function $s : ...