Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial $\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left$ in $n^2$ indeterminates $X_{i,j}$. This is known to be irreducible over $\mathbb Z$, but is there a "nice" ring in which $\mathbb Z$ embeds and where this ....

Amir Amir: 17 hours ago


On generalized Tanaka connection

Many authors used the Tanaka connection in their papers such as to define new Tanaka connection so-called Generalized Tanaka connection $^*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\phi,g)$ as follows: $$^*\Gamma_{ik}^j = \Gamma_{ik}^j + \eta_j\phi_k^i-\nabla_j\xi^i\eta_k+\xi^i\nabla_j\eta_k.$$ My main question: Why we need to define another new connection such as Tanaka connection even ....

Asher Asher: 17 hours ago


Finding all cycles of a certain length in a graph

Hello, I'm looking for a formula or algorithm to find the number of cycles of a certain length $k$ in a graph. I know that $(A^k)_{ii}$ gives me the number of cycles from vertex $i$ to itself ($A$ is the adjacency matrix), but these are cycles that might contain the ....

Josiah Josiah: 18 hours ago


For which Ramsey type results density versions are wrong?

I look for examples of Ramsey-type statements, for which the density counterparts do not hold. Example: usual Ramsey theorem. If all edges of a complete graph $K_n$ are colored in $c$ colors, there is a monochromatic, say, triangle if $n>n_0(c)$ is large enough. But if we choose more than $\frac1c ....

Emilia Emilia: 18 hours ago


Finding integer points on an N-d convex hull

Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is there a fast way to compute the integer points on the surface of the convex hull? Or ....

Elena Elena: 19 hours ago


Exactly how is 'the diagonal is representable' used for algebraic stacks...

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$? Once we know that given a stack $\mathcal{X}$ we have a smooth representable $X_0 \to \mathcal{X}$ where $X_0$ is a scheme, then we can talk about the algebraic groupoid $X_1 :=X_0\times_\mathcal{X} X_0 ....

Harper Harper: 19 hours ago


Finitely generated sheaf of algebras over geometric points

I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}_{X}$-algebras over $X$. Let's suppose that for every geometric point $p$, the pullback of $\mathcal{F}$ to $X_{p}$ is a finitely ....

Jeremiah Jeremiah: 20 hours ago


Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is the least (by inclusion order) c.t.m of $ZFC$ which $M\subseteq N$ and $G\in ....

Autumn Autumn: 21 hours ago


English translation of book by Jean-Pierre Serre?

I need the english translation of the article (or book) by jean-Pierre Serre in French on the topic Homologie singulière des espaces fibrés I am interested in understanding about this conjecture: there exist infinitely many geodesics between two points on a closed Riemannian manifold. Does there exist a translation? Links ....

Layla Layla: 21 hours ago


Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?

I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length of a PA-proof with all cuts eliminated grows (in the worst case) as fast as $f_{\epsilon_0}$ in the fast-growing hierarchy. ....

Lucy Lucy: 22 hours ago