**2**

votes

**0**answers

40 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 ...

**3**

votes

**1**answer

47 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 ...

**2**

votes

**0**answers

41 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

64 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 ...

**4**

votes

**1**answer

149 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? ...

**0**

votes

**0**answers

6 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

281 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

80 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

**2**answers

175 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 ...

**6**

votes

**1**answer

199 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

**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

46 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

**1**answer

146 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. ...

**1**

vote

**1**answer

157 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 ...

**12**

votes

**1**answer

471 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) ...

**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 ...

**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 ...

**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

28 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 ...

**8**

votes

**1**answer

148 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 ...

**3**

votes

**2**answers

130 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

**0**answers

36 views

+100

### Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science.
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...

**0**

votes

**0**answers

42 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 ...

**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", ...

**3**

votes

**0**answers

150 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 : ...

**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 ...

**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 ...

**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
...

**-3**

votes

**0**answers

45 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

**4**answers

373 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 ...

**4**

votes

**3**answers

318 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

93 views

### A question on $J(f)$ and $J(f')$

I was confused by the following question for a long time:
Does there exists a transcendental entire function $f$ such that
$J(f)\cap J(f')=\emptyset$ ?
where $J(f)$, ($J(f')$) is the Julia set of ...

**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 ...

**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 ...

**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. ...

**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 ...

**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
...

**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 ...

**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 ...

**4**

votes

**0**answers

101 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, ...

**3**

votes

**1**answer

145 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

**1**answer

415 views

+500

### Separating pure states on the $2\times 2$ matrix algebra

I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of ...

**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

**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.

**-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**

votes

**0**answers

60 views

### Virasoro-like algebras over the quaternions

The Virasoro algebra is a central extension of the Lie algebra of vector fields on $\mathbb{S}^1$.
This central extension exists and is unique because $H^2(Vect (\mathbb{S}^1))$ is one-dimensional ...

**4**

votes

**1**answer

169 views

### Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...

**3**

votes

**1**answer

99 views

### A relative property gamma and $L(\mathbb F_2)$

Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?

**7**

votes

**1**answer

93 views

### Detecting positive endomaps of the formal reals

A locale is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and ...

**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 ...