# All Questions

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### embdedding standard models of PA into nonstandard models [migrated]

Maybe it's well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the ...
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### Notions of consistency / heterogeneity in sets of vector values?

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
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### How to prove a certain theorem about algebraic function fields [on hold]

This question concerns a point from the book David Goldschmidt, Algebraic Functions and Projective Curves, 2001 (link). Let $K$ be a finitely generated extension of $k$ of transcendence degree ...
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### Does a (non-closed) differential 1-form define a curve? [on hold]

I am trying to understand under what conditions the following procedure properly defines a curve. Take a manifold $M$ with a (non-closed) 1-form $B$ and an exact 1-form $dA$. Define the functions ...
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### When does a symmetric Poisson manifold decompose into homogeneous pieces?

When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct ...
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### Polyhedra containing hexagones only [on hold]

It is well-known that Euler's theorem gives raison d'être to polyhedra containing exactly 12 pentagons if they are connected by 3 in a vertex. The number of hexagons may be arbitrary (in fact >1). In ...
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### why the split extension of a quasicyclic $2$-group $C$ by the cyclic group is not finite by abelien [on hold]

Let $G$ be split extension of a quasicyclic $2$-group $C$ by the cyclic group of ordre 2 generated by the inversion automorphism of $C$ it is clear that $G$ is abelien by (finite cyclic) but why $G$ ...
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### Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...
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### Toral rank conjecture

In 1985, S.~Halperin conjectured in the topological context of maximal free torus actions on topological manifolds, that: If $X$ is a topological space, then $$\dim H^*(X;\mathbb Q)\geq 2^{rk(X)}.$$ ...
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### Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g}$. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...
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### Lower boundedness of the Ricci curvature [on hold]

I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The ...
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### Geometry of rings and semi-rings

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings. One direction would be the following. Consider $\mathbb{N}$ (with the ...
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### Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
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### Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...