Vanishing function and Manipulate

Within Manipulate, I want to vanish certain controls if other controls satisfy a specific condition. The simplest case is to consider two checkboxes, x and y, where y vanishes if x is selected.

I have heard of the vanishing function ## &[] or Unevaluated[Sequence[]] (from this question), but it doesn’t seem to do what I want. Consider the following code

Manipulate[x,  Control[{{x, 0}, {1, 0}}],  Dynamic@If[x == 0, Control[{{y, 0}, {1, 0}}], ## &[]]] 

which yields

enter image description here

What I want is

enter image description here

but I’m getting

enter image description here

Surprisingly enough, using Grid in the following manner solves my problem

Manipulate[x, Dynamic@Grid[{     {Control[{{x, 0}, {1, 0}}]},     {If[x == 0, Control[{{y, 0}, {1, 0}}], ## &[]]}}]] 

Any idea why? How do I fix my code?

Vanishing of Chow groups in high codimension

Let $ X$ be a smooth affine variety of dimension $ n>2$ over $ \mathbb{C}$ . From the examples I have seen (admittedly very little) it seems to me that these varieties don’t have torsion classes a little bit after degree $ \frac{n+1}{2}$ , where grading is by codimension.

$ \mathbf{Question}_i$ :For $ i>\frac{n+1}{2}$ is is true that $ CH^{i}(X)_{torsion}=0$ .

Note that the answer is affirmative if $ i=n$ since by a theorem of Bloch (see “Torsion algebraic cycles and a theorem of Roitman” ) $ CH^n(X)[l]$ is isomorphic to $ H^{2n-1}_{et}(X,\mathbb{Q}_l/ \mathbb{Z}_l[n])$ and etale cohomology groups of affine scheme over $ \mathbb{C}$ (with torsion coefficient) vanishes after degree $ n$ .

I don’t know if the question in the case $ i=n-1$ . If someone knows an example where this fails please do tell. If this is hard I would be grateful if one could point me to the right literature where these kind of questions are studied.

Vanishing of certain coefficients coming from Coxeter groups

Let $ \left(W\text{, }S\right)$ be a Coxeter system. For every $ w\in W$ let us write $ \left|w\right|$ for the length of $ w$ . Set $ \lambda\left(e\right)=1$ where $ e\in W$ denotes the neutral element of the group and define coefficients $ \lambda\left(w\right)\in\mathbb{Z}$ recursively via \begin{eqnarray} \sum_{v\in W \text{: }\left|v^{-1}w\right|=\left|w\right|-\left|v\right|}\lambda\left(v\right)=0 \end{eqnarray} for any $ w\in W$ . For example this gives us $ \lambda\left(e\right)=1$ , $ \lambda\left(s\right)=\left(-1\right)$ for all $ s\in S$ , $ \lambda\left(st\right)=1$ if $ m_{st}=2$ and $ \lambda\left(st\right)=0$ if $ m_{st}\neq2$ (for the notation see Wikipedia) for all $ s\text{, }t\in S$ with $ s\neq t$ , and so on.

My question is: Does there always exist some $ l\in\mathbb{N}$ such that $ \lambda\left(w\right)=0$ for all $ w\in W$ with $ \left|w\right|\geq l$ ?

In the case that $ \left(W\text{, }S\right)$ is right-angled (i.e. $ m_{st}\in\left\{ 2\text{, }\infty\right\}$ for all $ s\text{, }t\in S$ with $ s\neq t$ ) this is true and we can take $ l=\left|S\right|$ . I’m wondering if in the general case this property also holds. If no, is it possible to characterize the property in an instructive way?

A non vanishing vector field on $S^3$ whose flow does not preserves any transversal foliation

Is there a non vanishing vector field $ X$ on $ S^3$ which does not admit a transversal $ 2$ dimensional foliation? if the answer is negative, is there a non vanishing vector field $ X$ on $ S^3$ which does not admit a transversal foliation whose leaves are invariant under the flow of $ X$ ? If the answer is positive, is there an example of this situation with the extra assumption that $ X$ is invariant under the obvious action of $ S^1$ on $ S^3$ ?

The motivation is described in the following post and the paper linked in that post.(The flow invariant foliation)

Irrational closed orbits of vector fields on $ S^2$ (Limit cycles and trace formula)

The vanishing XLM! Help please?

This question is about my Bitcoin core wallet, I got XLM with the airdrop when the Bitcoin Core wallet added XLM to the wallet. Got it months ago now and its been fine just sitting there until today, when its simply vanished without a trace. I first thought this must be a security issue, but that seems unlikely because the bitcoin I have in the wallet is untouched. There’s also only one 1 entry in the XLM transaction list which is the airdrop received. It’s like the 250 XLM just vanished from the ledger, ceased to exist, is there any network issues or something? I haven’t been able to find any info, like maybe I’m the only one affected. Any ideas?

Vanishing of L-function of elliptic curve over $\mathbb{Q}$

For an elliptic curve $ E$ over $ \mathbb{Q}$ , it is not very difficult to show $ L(E,1)\not=0$ (when the analytic rank$ =0$ ) by computing the several Fourier coefficients but seem to be difficult to determine whether one has $ L(E,1)=0$ (when the analytic rank$ \not=0$ ). Is there a small constant $ c$ such that, if we have $ \mid L(E,1)\mid <c$ , one can obtain $ L(E,1)=0$ ?

Does an Inverted Magic Circle prevent a Conjured Creature from vanishing at the end of its spell’s duration?

Magic Circle (which takes a minute to cast but lasts an hour) can be used in an inverted way :

When you cast this spell, you can elect to cause its magic to operate in the reverse direction, preventing a creature of the specified type from leaving the cylinder and protecting targets outside it.

Many planar entity conjuration spells, such as Conjure Woodland Beings, last up to exactly an hour, at which point the creature vanishes :

A summoned creature disappears when it drops to 0 hit points or when the spell ends.

Imagine the following scenario : Bob the Druid casts Conjure Woodland Beings, which makes DM-decided fey creatures (let’s say 2 quicklings) appear, and maintains concentration on it. After 30 minutes, Dylan the Wizard decides to cast an Inverted Magic Circle around one of the feys, which Bob orders to stand still. After that, Bob drops concentration on Conjure Woodland Beings. What I’m wondering here is what takes priority between “disappears when the spell ends” and “preventing a creature from leaving the cylinder”.

Does an Inverted Magic Circle prevent a Conjured Creature to vanish at the end of its spell’s duration ? If so, that means that the now-trapped fey is no longer friendly to / controlled by Bob, but remains there until the Magic Circle itself ends (total of 1h31). If not, that means that the fey disappears as soon as Bob’s concentration ends, regardless of the Magic Circle (total of 31min).

Polynomials (or analytic functions) vanishing on a real algebraic set

I have seen the following result stated several times in the literature, without proof:

Assume $ P\in\mathbb{R}[X_{1},\ldots,X_{n}]$ is an irreducible polynomial of $ n$ variables, with real (or complex ?) coefficients, and let $ $ V=\{z\in\mathbb{C}^{n},~P(z)=0\},\quad I(V)=\{Q\in\mathbb{C}[X_{1},\ldots,X_{n}],~Q=0\text{ on }V\}. $ $ Assume that $ V$ contains a real point $ a\in\mathbb{R}^{n}$ which is regular, that is $ $ \text{dim }V=n-\text{rank }J_{a}(V), $ $ where $ J_{a}(V)$ denotes the Jacobian of a family of generators of $ I(V)$ , evaluated at $ a$ .

Then the set $ V_{\mathbb{R}}=V\cap\mathbb{R}^{n}$ of real points of $ V$ is Zariski dense in $ V$ , or equivalently any polynomial that vanishes on $ V_{\mathbb{R}}$ must vanish on $ V$ .

I am also interested by the analytic version of this result (if true) where $ V$ is still the zero set of a polynomial $ P$ but $ I(V)$ is the ideal of analytic functions vanishing on $ V$ , and the vanishing of an analytic function $ f$ on $ V_{\mathbb{R}}$ implies the vanishing of $ f$ on $ V$ .

Could someone provide a proof for the algebraic or analytic case ?

Many thanks in advance.

Cohomology of modular curves: vanishing and decomposition

Let $ \pi:E\to Y$ be a universal elliptic curve over an open modular curve $ Y$ . Take a prime $ \ell$ and take $ \mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$ where the dual, $ (-)^\vee$ , is taken in the category of constructible $ \ell$ -adic sheaves. For integers $ r\geq0$ , define $ S^r=\mathrm{Sym}^r\mathcal{H}$ . From time to time while I read papers, I found that the first (etale) cohomology groups of $ Y_{\overline{\mathbb{Q}}}$ with coefficients $ S^r$ or $ S^r(1)$ , i.e. $ H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r)$ or $ H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r(1))$ , has been studied extensively so that authors usually do not give any reference or proofs on the vanishing or decompositions of them.

Could anyone guide me to the right statements and references including their proofs? It would be greatly helpful for me. Thank you in advance.