axiom of choice – Does ZF + BPI alone prove the equivalence between “Baire theorem for compact Hausdorff spaces” and “Rasiowa-Sikorski Lemma for Forcing Posets”?

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $mathbb{P}$ (i.e. $mathbb{P}$ is a reflexive transitive relation) and for any countable family of dense subsets of $mathbb{P}$ there is a generic filter which intersects all dense subsets of the countable family. It is well-known that this statement is equivalent to the Baire Category Theorem for Complete Metric Spaces – and thus it is also equivalent to the Principle of Dependent Choices.

A masters student of mine has found in the literature the following statement: “Rasiowa-Sikorski Lemma is equivalent to the Baire Category Theorem for Compact Hausdorff Spaces, modulo the Boolean Prime Ideal Theorem”. We understood this as the assertion that the theory ZF + BPI alone is able to prove the equivalence between the Baire Category Theorem for Compact Hausdorff Spaces and the Rasiowa-Sikorski Lemma.

Well, I asked my student to verify such claim, and at first glance I suggested him to follow the results 3.1 to 3.4 of Chapter II of Kunen’s book, where there are proofs for some equivalences of Martin’s Axiom at $kappa$, MA($kappa$): the idea was to discard the hypothesis “c.c.c.” and adapt the reasoning, arguing for $kappa = omega$. It turns out that it was not a good suggestion, because in 3.1 a kind of Downward-Lowenheim-Skolem argument is done, to show that it is equivalent to work with a restricted form of the forcing axiom, considering only partial orders of bounded cardinality. However, such argument seems to require the Axiom of Choice, or some part of it other than BPI.

Does any of you know if it is indeed possible to prove the equivalence between “Baire Category Theorem for Compact Hausdorff Spaces” and “Rasiowa-Sikorski Lemma for forcing posets” from ZF + BPI alone ? Any suggestions or references would be appreciated.

group theory – How many ways can you paint a Decagon with q colors? Solve with Burnside’s lemma

How many ways can you paint a Decagon with q colors?
I need to solve it with Burnside’s lemma.
So far I managed to find only 2 symmetries, the identity, and this one.
I believe I miss the method, can someone solve, and try to explain how did he solve it?
enter image description here

ct.category theory – Yoneda Lemma for monoidal functors

Let $(mathcal V,otimes,I)$ be a closed symmetric monoidal category, and let $mathcal C$ be a $mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $Hom(F,G)$ of natural transformations between two $mathcal V$-enriched functors $F,Gcolonmathcal Ctomathcal V$ when $F$ is representable: it is in bijection with the set of maps $I to G(Y)$ in $mathcal V$ where $Y$ is an object representing $F$.

Now suppose that $mathcal C$ itself is a monoidal category, and that our two functors $F$ and $G$ are monoidal functors. Is there a similarly nice description of the set $Hom^otimes(F,G)$ of monoidal natural transformations between the functors, again in the case that $F$ is representable?

My suspicion is that the following might be true (possibly with extra conditions on $mathcal C$). The fact that $F$ is a (lax) monoidal functor induces the structure of a comonoid on the representing object $Y$, and so there is an induced comonoid structure on $G(Y)$. My guess would be that monoidal natural transformations $Fto G$ are in bijection with morphisms of comonoids $Ito G(Y)$, but I can’t prove this in general. (I can prove this in the case that $mathcal V$ is the category of sets with cartesian product, but only for trivial reasons: every map in Set is a morphism of comonoids, and every natural transformation between monoidal Set-valued functors is a monoidal transformation.)

I would be especially interested in any references where this might be addressed.

A lemma in Gerschgorin theorem

Let :

  • $n geq 2$
  • $A in mathcal{M}_n( mathbb{C} )$
  • $forall i leq n , |a_{i,i}| > sum_{j ne i}^{n} |a_{i,j}| $
  • $V in mathcal{M}_{n,1}( mathbb{C} ), V ne 0 , Y=AV$
  • $V=(v_i)_i$ and $Y=(y_i)_i$

We want to prove that :
$$forall i leq n , |a_{i,i}| |v_i| – sum_{j ne i}^{n} |a_{i,j}| |v_j| leq |y_i|$$

My attempt :
$sum_{j ne k} a_{i,j} z_j + a_{i,k} z_k =0$

Pumping lemma for an involved non context free language

Hi I’m trying to show $C={wzzw^R|w,zin{0,1}^+}$ is not a context-free language.( I have this believe because $C={ww|win{0,1}^+}$ is not a context free language.) I’m really struggling to come up with a string that captures the essence of irregularity of this language: I tried strings like $s=1^p0^p1^p0^p1^p1^p$ but there are too many cases to deal with and most of the examples I saw only use 1 or 2 cases, so I believe the direction I’m going is wrong. Can you provide a hint on which string to pick as the ‘pumping string’? Thank you.

ag.algebraic geometry – Reference request: Kleiman’s proof of Snapper’s Lemma

On page 4 of Nitin Nitsure’s paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as

a special case of Snapper’s Lemma, see “An Intersection Theory for Divisors (preprint 1994)” by Steven Kleiman for a proof.

Kleiman’s paper (or book?) mentioned by Nitsure must have changed its title, never gone into publication, or elsehow disappeared, I am unable to find it. Does anybody have a link, or an alternative source for the proof?

Thanks in advance.

Poincarés Lemma, show that is 1-form in Rn is exata and the other items

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ct.category theory – Lemma of HTT

In Lemma of HTT, the proof given relies on Lemma However it seems that Lurie applies Lemma, which requires the given simplicial set to be contractable, to an arbitrary $kappa$-small simplicial set.

This seeming incongruity was pointed out in this question on mathoverflow 2 years ago. However an answer was never given, and therefore I thought I might re-ask this in a new question (Let me know if there is a better way to re-ask an unanswered question).

Is there either
a) a way to salvage the proof given, or
b) a new proof which avoids the issue, or
c) is the proof actually correct (and we’re all being daft)

Use pumping lemma (non regular language) to solve

{0^m 1^n 0^m | m,n >=0}

vv^R :v: {a,b}*

complex analysis – Generalization of Schwarz’s Lemma

I am reading Lectures on Riemann Surfaces by Otto Forster. He says: (p.110)

The following lemma may be viewed as a generalization of Schwarz’s lemma. Let $D,D’$ be a pair of open subsets of $mathbb{C}$, where $D$ is a relatively compact subset of $D’$. For any $varepsilon>0$, there is a closed vector space $Asubset L^2(D,mathcal{O})$, of finite codimension, with
$$ lVert frVert_{L^2(D’)}leq varepsilon lVert frVert_{L^2(D)}$$

He has already shown that $L^2(D,mathcal{O})$, the space of holomorphic functions on $D$, forms a Hilbert space under the inner product $iint foverline{g}dxdy$ thus the “closed” comment.

What does he mean when he says this generalizes Schwarz’s lemma? How is this related to Schwarz’s lemma?