# All Questions

**0**

votes

**0**answers

67 views

### Guessing degrees [on hold]

Supposing we have a polynomial $f(x)\in\Bbb Z[x]$, given $t\in[1,\mathsf{deg}(f)]\cap\Bbb N_{}$ is it possible to guess deterministically or probabilistically (with one or two-sided error) if there is ...

**-2**

votes

**1**answer

80 views

### Are limits computable?

This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition:
There cannot exist a Turing Machine $M$ which, given a ...

**0**

votes

**0**answers

6 views

### Can the Laplace operator be represented as a sum of second order derivational operators

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.
Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ ...

**-1**

votes

**0**answers

41 views

### Is it possible to classify the indecomposable representations wild quiver $\mathbb{F_2}$ using infinite sets of parameters?

I have read several other questions about wild representation type but may I ask..what actually can be done and what have been done (such as partial results) about classifying indecomposable ...

**0**

votes

**0**answers

9 views

### Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...

**0**

votes

**0**answers

9 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**27**

votes

**5**answers

6k views

### Computing the Galois group of a polynomial

Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of ...

**10**

votes

**0**answers

70 views

### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...

**0**

votes

**0**answers

19 views

### understanding the proof of gauss lemma [on hold]

I was reading the Gauss Lemma from the do carmos Rienmannian geometry book which says that
Let $p \in M$ and let $v \in T_pM$ such that
$\exp _p v$ is defined Let $w\in T_pM$ is identified with ...

**5**

votes

**2**answers

198 views

### Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...

**0**

votes

**0**answers

13 views

### Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...

**12**

votes

**3**answers

519 views

### Generalization's of Greene's Theorem for the Robinson-Schensted correspondence

One important property of the Robinson-Schensted correspondence (RS) is that the longest increasing subsequence of the permutation $\sigma$ is $\lambda_1$, the first entry of the shape ...

**1**

vote

**1**answer

24 views

### Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials
\begin{align}
& p_1(x_1,x_2,\ldots,x_n)=0, \\
& p_2(x_1,x_2,\ldots,x_n)=0, \\
& \vdots \\
& p_m(x_1,x_2, ...

**5**

votes

**0**answers

38 views

### A Collatz-like question about permutations

An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.
Consider all permutations $\pi$ on the natural numbers such that ...

**1**

vote

**1**answer

63 views

### Commutator subgroups as normal supplmements

The following question has been asked about a week ago on MathUnderflow (no answers).
Let $F$ be a free group and let $N$ be a normal subgroup of $F$ such that
\begin{equation*} \tag{*}
F = [F,F] ...

**11**

votes

**1**answer

294 views

### On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at ...

**1**

vote

**0**answers

24 views

### Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws

Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital ...

**4**

votes

**1**answer

81 views

### Number of representations as sums of squares in rings of integers of number fields

Let $K$ be some number field, $\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O_K$, and consider the quantity $r_{n,K}(\alpha)$, which denotes ...

**4**

votes

**0**answers

33 views

### Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...

**7**

votes

**3**answers

560 views

### A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...

**-2**

votes

**0**answers

80 views

### Every totally disconnected space is Hausdorff space [on hold]

I saw in a websit the following theorem, without any proof, is this theorem true
Blockquote Theorem:Every totally disconnected space is Hausdorff space.
It was in ...

**0**

votes

**0**answers

14 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**0**

votes

**0**answers

14 views

### Clifford semigroups!

I am trying to prove that if a maximal group image G(S) of a Clifford semigroup S, is abelian-by-finite does the Clifford semigroup S is abelian-by- finte?
I am assuming that S is an E-unitary ...

**0**

votes

**0**answers

35 views

### Is there a general theory for the structure of the (semi)group generated by morphisms of an affine space $F_p^3$?

Consider an affine space $\mathbb{F}_p^3$, and assume we have a handful of morphisms $f_i : \mathbb{F}_p^3 \rightarrow \mathbb{F}_p^3$ given by $$f_i(x, y, z) =(P_i(x, y, z), Q_i(x, y, z), R_i(x, y, ...

**11**

votes

**1**answer

392 views

### Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...

**1**

vote

**0**answers

15 views

### Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...

**4**

votes

**0**answers

53 views

### Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here.
The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...

**17**

votes

**3**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 ...

**5**

votes

**2**answers

525 views

### How is differential geometry used in immediate industrial applications and what are some source to know about it?

Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess ...

**-2**

votes

**0**answers

27 views

### Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil L-function (also called zeta function).
Is the rank of the abelian group ...

**3**

votes

**2**answers

59 views

### Inscribed parallelotope in a $d$-simplex

The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we construct a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N ...

**5**

votes

**0**answers

156 views

+100

### Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...

**6**

votes

**1**answer

323 views

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

Let $\binom{n}{j}_q$ be a $q$-binomial coefficient and $(x;q)_n = (1-x)(1-qx)\cdots(1-q^{n-1}x).$
Consider the sum $$f(n,m,r,k)= \sum\limits_{j = 0}^{2n} {( - 1)}^{ j}q^{mj^2+rj}
...

**1**

vote

**1**answer

96 views

### Solutions to system of polynomial equations over finite fields

If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower ...

**38**

votes

**11**answers

4k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...

**8**

votes

**1**answer

382 views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

**16**

votes

**1**answer

602 views

### What were the main ideas and gaps in Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...

**2**

votes

**1**answer

324 views

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

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference.
I keep stumbling on the ...

**1**

vote

**2**answers

38 views

### How to determine whether the following sum is nonzero for a given multivariate polynomial?

My research field is combinatorics. I am not very good at Algebra. So I want to ask for a given real multivariate polynomial $f(x_1,x_2,\cdots,x_n)$, is there any algebraic method to compute whether ...

**3**

votes

**1**answer

91 views

### Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$.
Let $\rho$ be the standard 7-dimensional complex representation
$$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$
We ...

**1**

vote

**1**answer

94 views

### Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:
...

**-4**

votes

**0**answers

32 views

### Non standerd Algebraic topology3 [on hold]

. A space X is n-connected if π k (X, x 0 ) ∼
= ∗ for all k ≤ n and all x 0 ∈ X.

**3**

votes

**0**answers

55 views

### Infimum of Gaussian process

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about ...

**2**

votes

**0**answers

16 views

### s-arc transitivity and the Moore bound

Given a vertex $x$ of a graph, call the subgraph induced by all vertices of distance $\le n$ from $x$ the $n$-neighbourhood of $x$, denoted $N_n(x)$.
Let $G$ be a regular graph of diameter $n+1$. For ...

**0**

votes

**1**answer

34 views

### Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects.
Let $({\cal C},\otimes,*)$ be a semisimple ...

**1**

vote

**0**answers

84 views

### An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?

**1**

vote

**1**answer

89 views

### Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to ...

**-1**

votes

**0**answers

30 views

### Spectral radius of principal submatrices for the case of hermitian matrix

A principal submatrix of a matrix $\mathbf{A}\in\mathbb{C}^{N\times N}$ is any submatrix from $\mathbf{A}$ for which the same rows and columns have been eliminated.
Assume $\mathbf{A}$ is hermitian, ...

**103**

votes

**46**answers

31k views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

**0**

votes

**0**answers

10 views

### Distance minimization and submodular functions

I have a question about square distance minimization and submodular functions. Suppose I have a random variable $X = (X_1,...,X_n)$ over $\mathbb{R}^n$. Let $x$ be a realization of this random ...