**3**

votes

**0**answers

63 views

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

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 ...

**2**

votes

**0**answers

54 views

### Terminology in combinatorics

I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics.
A finite graph $G$ has the following ...

**1**

vote

**0**answers

74 views

### Possible $\mathsf{NP}$ complete problem from number theory

A candidate $\mathsf{NP}$ complete variant of factoring was posted in http://cstheory.stackexchange.com/questions/4769/an-np-complete-variant-of-factoring, where decision problem ...

**3**

votes

**1**answer

61 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

**0**answers

7 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

**5**

votes

**1**answer

310 views

### Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: ...

**0**

votes

**1**answer

82 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 ...

**0**

votes

**0**answers

44 views

### Skolemization for induction hypothesis

As part of my investigation for a logical system that is limited to $\Pi_2$ sentences I try to prove that FOL + PA is a conservative extension over that system.
Some four and half years ago (I only ...

**4**

votes

**0**answers

74 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 ...

**0**

votes

**1**answer

47 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 ...

**1**

vote

**0**answers

20 views

### Radicals of co-quasitriangular map

Let $B=\mathcal{O}_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", ...

**1**

vote

**1**answer

148 views

### Is this kind of scheme integral?

Let $X\rightarrow Spec(R)$ an irreducible projective scheme over a dvr. Suppose the generic fiber $X_{\eta}$ is smooth (over the field $Frac(R)$) and irreducible. Is it true that $X$ is integral (i.e. ...

**4**

votes

**2**answers

176 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 ...

**4**

votes

**1**answer

152 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

**1**answer

158 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 ...

**-3**

votes

**0**answers

46 views

### Indecomposable commutative rings [on hold]

Let $R$ be an indecomposable commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?

**6**

votes

**1**answer

201 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 ...

**4**

votes

**2**answers

85 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 ...

**1**

vote

**1**answer

48 views

### Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...

**1**

vote

**0**answers

32 views

### About a close strucure on profunctors

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

**0**

votes

**0**answers

82 views

### Computing Hochschild Cohomology [on hold]

Let A be my noncommutative ring.
I have computed a $A^e$ projective resolution and taken $Hom_{A^e}(.,A)$ so I am ready to compute kernels and images to find the hochschild cohomology groups. ...

**12**

votes

**1**answer

474 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

72 views

### Product of a Schubert polynomial and a double Schubert polynomial

Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as
...

**1**

vote

**0**answers

21 views

### Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph.
Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...

**3**

votes

**0**answers

153 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 : ...

**0**

votes

**1**answer

90 views

### Symmetries of non-Riemannian curvature tensor

The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...

**1**

vote

**1**answer

52 views

### Sampling from random unimodular matrices of a particular type?

Is there a nice way to parametrize unimodular matrices of form
$$\begin{bmatrix}
a1& a2& 0& 0\\
b1& b2& a1& a2\\
c1& c2& b1& b2\\
0& 0& c1& c2
...

**1**

vote

**0**answers

83 views

### Finding a lower bound in terms of given integers

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks

**4**

votes

**2**answers

77 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 ...

**2**

votes

**1**answer

60 views

### Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...

**0**

votes

**0**answers

29 views

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

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, doi and apparently the same as in W. T. ...

**1**

vote

**0**answers

32 views

### Optimization question: maximize quadratic objective with semidefinite constraints

I recently encountered the following optimization problem:
$\max \|AX\|_F^2$
subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$.
Matrices $A$ and $X$ are ...

**15**

votes

**3**answers

196 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 ...

**5**

votes

**0**answers

69 views

### Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...

**3**

votes

**1**answer

113 views

### Are automorphism groups of polarized varieties of finite type

It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.
In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme ...

**-4**

votes

**0**answers

29 views

### Calculating matix derivatives with MATLAB or MATHEMATICA? [on hold]

I'd like to calculate the following derivative,
\begin{equation}
\frac{d||(f(C)\cdot f(C)^{+}-I)\cdot u||^2)}{dC}
\end{equation}
Where $C$ is a matrix of dimension $n\times k$ (s.t $k < n$).
And ...

**1**

vote

**1**answer

32 views

### roots in a root system which have nonzero coefficients with respect to each simple root

If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with ...

**4**

votes

**0**answers

102 views

### Manifold approximations to $BO(3)$

We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$
Similarly $BO(2)$ can be approximated by closed, ...

**0**

votes

**1**answer

254 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.

**2**

votes

**1**answer

88 views

### Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category.
Let $P$ be the partially ordered subset of full subcategories of $C$.
Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...

**0**

votes

**0**answers

69 views

### power expansion and matrix inverse

Consider the vector-valued function ($s$ complex):
$$
f(s) = (I-A/s)^{-1} v.
$$
Here $A$ is a real square matrix, $v$ a non null column vector. It is known that $A$ has one simple $0$ eigenvalue, and ...

**3**

votes

**0**answers

37 views

### $n!$ computation in $\mathsf{BSS}$ model

It is well known that if $n!$ cannot be computed in $\mathsf{polylog}(n)$ ring ($\Bbb Z$) operations, then $\mathsf P\neq\mathsf{NP}$ in $\mathsf{BSS}$ model.
Suppose if we assume $\mathsf ...

**1**

vote

**2**answers

158 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 ...

**-1**

votes

**0**answers

18 views

### probability for having the exact same result in a given number of test with a given number of possible combinations [on hold]

The questions was: What is the probability for having exact the same page at least twice when printing my whole SSD (as bytes) on paper.
We have 1117bytes per page leading to ~6,3e2834 possible ...

**4**

votes

**3**answers

319 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**

votes

**1**answer

62 views

### A question about Lebesgue density [on hold]

Is there a set $ A \subseteq \mathbf{R} $ such that the upper lebesgue density of $ A $ and the upper lebesgue density of $ \mathbf{R} \setminus A $ are equal to $ 1 $ at a fixed point?
I would say ...

**3**

votes

**1**answer

76 views

### almost diagonal Positive semidefinite Matrix

Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices.
A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in ...

**1**

vote

**0**answers

53 views

### Examples for differential operators first order

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function ...

**0**

votes

**0**answers

47 views

### Limit inferior of Borel functions [on hold]

Suppose $X$ is separabile metric and $F \colon X \times \mathbb{R}_+ \to [ 0 , 1 ]$ is Borel. Let $ f ( x ) = \liminf_{\varepsilon \to 0} F ( x , \varepsilon )$.
Is $f$ Borel?

**8**

votes

**2**answers

182 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 ...