Is the geometric curve of Fargues-Fontaine locally Noetherian?

Fix a completion $ mathbb {C} _p $ of an algebraic closure of $ mathbb {Q} _p $. Choose an inclination of it. The Told space (equivalence modulo Frobenius) is naturally parametrized by a regular, separate, netherian scheme of the Krull 1 dimension, noted $ X_ {FF} $. This scheme is naturally defined on $ mathbb {Q} _p $.

Choose an algebraic closure $ mathbb {Q} _p subset highlighting { mathbb {Q} _p} $. What I'm asking is, is it true that the basic change $ X_ {FF} times _ { mathrm {Spec} : mathbb {Q} _p} mathrm {Spec} : overline { mathbb {Q} _p} $ is a Noetherian scheme? The difficulty is of course that $ X_ {FF} $ is not of finite type on $ mathbb {Q} _p $. The discussion here is potentially relevant.

I do not ask this because of some applications of number theory, but rather to better understand the construction.

ap.analysis of pdes – What is the geometric or dynamic meaning of a global attractor with an infinite fractal dimension?

In the article of Efendiev-Otani: attractors of infinite dimension for parabolic equations with p-Laplacian in a heterogeneous medium (Ann.IH Poincaré, AN 28,2011), we obtain that the fractal dimension of the Overall attractor is infinite. However, no geometric or dynamic consequence is obtained.

Original message:
https://www.researchgate.net/post/What_is_the_geometric_or_dynamic_meaning_of_a_global_attractor_with_an_in_infinite_fractal_dimension

Geometric topology – Wild character of the codimension 1 subvarieties of the Euclidean space

This question arose from this pile trading post. I am writing a thesis on the $ s $-Cobordism and Siebenmann's work on the end obstructions. Combined, they give a quick proof of the uniqueness of the smooth structure for $ mathbb {R} ^ n $, $ n geq $ 6. Siebenmann's theorem says pretty much that for $ n geq $ 6 a contractible $ n $-collecteur $ M $ which is simply connected to infinity integrates (smoothly) inside a compact collector. Since this compact variety is contractible, by the $ s $-cobordism, it is diffeomorphic to the standard $ n $-disk $ D ^ n $ (see Minor Conferences on the $ h $-cobordisme for example). It follows that $ M = text {int} D ^ n $ is diffeomorphic to $ mathbb {R} ^ n $.

The problem is that the case $ n = $ 5 is not covered. I am aware of Stallings beautifully written On the piecewise linear structure of the Euclidean space but I'm looking for a way to deal with the $ n = $ 5 case via Siebenmann's end theorem and the good $ s $theorem of -cobordism (see link to the question mse). This brings me to the next question, which is interesting in itself

Since codimension 1 is well integrated and smooth $ S subset mathbb {R} ^ {n + 1} $, is there a diffeomorphism auto $ mathbb {R} ^ {n + 1} rightarrow mathbb {R} ^ {n + 1} $ who wears $ S $ in a region limited to one dimension $ mathbb {R} ^ n times (-1, 1) $ ?

Now if $ M $ is a multiple that is homeomorphic to $ mathbb {R} ^ $ 5, the product $ M times mathbb {R} $ is homeomorphic to $ mathbb {R} ^ $ 6, and therefore also diffeomorphic. Subject to the existence of diffeomorphism in my question, we could find a diffeomorphism $ f: M times mathbb {R} rightarrow mathbb {R} ^ $ 6 that cards $ M times $ 0 in $ mathbb {R} ^ 5 times (-1, 1) $. This would produce a good $ h $-cobordism between $ M $ and $ mathbb {R} ^ $ 5 taking the area between $ f (M times 0) $ and $ mathbb {R} ^ 5 times $ 1 in $ mathbb {R} ^ 5 times mathbb {R} $. Since $ M $ is simply connected, the good $ s $-cobordism theorem applies and shows that $ M $ and $ mathbb {R} ^ $ 5 are really diffeomorphic.

Calculation of the sum of an infinite series as a variant of a geometric series

I came across the following series by calculating the covariance of a two-variable Gaussian random vector transformation via the Hermite polynomials and the Mehler expansion:

$$
S = sum_ {n = 1} ^ { infty} frac { rho ^ n} {n ^ {1/6}}
$$

for $ green rho green <$ 1.
We know that $ S $ must be finished and satisfy
$$
S le rho (1- rho) ^ {- 1}
$$

since the original series is dominated by $ sum_ {n = 1} ^ { infty} rho ^ n $.

However, there is a trap if we use to $ S $ the upper limit $ rho (1- rho) ^ {- 1} $which tends to $ infty $ when $ rho to 1- $. This occurs when the two marginal random variables of the Gaussian vector are almost certainly positively (asymptotically) dependent.

The goal is therefore to obtain a good upper limit, much better than $ rho (1- rho) ^ {- 1} $ when we restrict $ rho $ being away from $ 1 $, to reduce the effect of $ rho to 1- $. In other words, let $ 1- rho = delta $ for some fixed $ delta in (0,1) $, what is the best upper limit for $ S $?

Because of the scaling term $ n ^ {- 1/6} $ which induces a divergent series $ sum_ {n = 1} ^ { infty} n ^ {- 1/6} $, probably not much improvement should be expected.
I googled but I did not find any enlightening technique for that. Any pointer or help is appreciated. Thank you.

geometric topology – any two monotonous polygons can be separated with a single proof of translation

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geometric probability – Determination of sample spaces for Sylvester's four-point problem

I have no knowledge of geometric probability, so I can apologize if the following is wrong or does not make sense:

Hypotheses:

  • no three sampled points are collinear
  • the geometric probability (naive) in Euclidean space is equal to the ratio of lengths or surfaces
  • probabilities do not change under isometric transformations and / or uniform scaling
  • the probability of a set $ P4: = lbrace p_1, p_2, p_3, p_4 rbrace $ four points being in convex configuration in the Euclidean plane is independent of the order in which the elements are drawn by the sampling process; this in turn means that we assume that
  • $ lbrace p_1, p_2, p_3 rbrace $ looks like the corners of $ T_ {max} $, the triangle of the largest area
  • $ lbrace p_1, p_2, p_3 rbrace $ uniformly drawn from the boundary of their circumscribed circle, which has implications for the probability of meeting $ T_ {max} $ with specific values ​​for the pair of smaller central angles.
  • the smallest circle surrounding $ P4 $ is the unit circle centered at the origin
  • the longest side of $ T_ {max} $ is divided in two by the non-negative part of the x-axis

According to the above assumptions, the probability of encountering a convex quadrilateral is equal to the blue zone divided by the red plus blue zone in the images below:

probability of 4 four points in convex configuration if the largest triangle is acute

the sampling area in case of acute toxicity $ T_ {max} $ is equal to the entire disk of the unit

four point probability in convex configuration if the largest triangle is obtuse

the sampling area in case of acute toxicity $ T_ {max} $ is equal to the disk drive with a notch

The notch in the case of obtuse $ T_ {max} $ is due to the assumption that the first three points look like $ T_ {max} $ which implies that the points apart $ T_ {max} $ that generate a deltoid configuration would be in contradiction with the maximality of the zone of $ T_ {max} $

If the above makes sense, the probability that four points are in a convex configuration can be calculated by integrating the blue zone ratios over the entire sampling area defined by the $ T_ {max} $ multiplied by the probability of $ T_ {max} $ resulting in uniform sampling at the boundary of the unit circle.

Questions:

  • Have similar ways of defining sample spaces for Sylvester's four-point problem already been described or studied?
  • what are the objections to the proposed proposal compared to the proposed definition of the available sampling space on $ T_ {max} $?

Note:

in case of obtuse $ T_ {max} $ the surface of the sample space can be calculated based on the angles $ alpha $ and $ beta $ which are adjacent to the longer side of $ T_ {max} $ as follows, keeping in mind that this longest edge is the diameter of the unit circle:

  • the surface of the lower half-blue disk is equal to $ frac { pi} {2} $
  • The area $ A _ { alpha} $ of the union of $ T_ {max} $ with the blue region opposite to the angle $ alpha $ equals $ alpha + sin ( alpha) cos ( alpha) $ and the like $ A _ { beta} $for angle $ beta $
  • the domain of $ A_ {T_ {max}} $ of $ T_ {max} $ equals $ frac {1} { cot ( alpha) + cot ( beta)} $

The area of ​​the sampling area for obtuse $ T_ {max} $ then equal $ frac { pi} {2} + A _ { alpha} + A _ { beta} -A_ {T_ {max}} $

How to prove this statement of the geometric series

If we have $ sum_ {n = 0} ^ { infty} r ^ {i} $, that converges towards $ S = frac {1} {1-r} $, shows CA $ S – s_n = frac {r ^ {n + 1}} {1-r} $, or $ s_n $ is the sum of the first n terms.

My approach,

$ S – s_n = frac {1} {1-r} – frac {1-r ^ n} {1-r} = frac {r ^ n} {1-r} $

So, there is no $ r ^ {n + 1} $ in my answer as in the question. If we have $ s_n $, that means the sum of the first $ n $ terms, so there should not be any $ n + 1 $.

Where am I wrong?

geometric measurement theory – Hausdorff dimension of the set of levels of the basic function of Conway 13

Yesterday I discussed the function of Conway's base 13 (https://en.wikipedia.org/wiki/Conway_base_13_function) (and the fancy properties it has). During this discussion, the other person explained that he represented the sets of levels as fractals. This made me curious and motivated the following question:

Do we know what is the Hausdorff dimension of the level sets of Conway's base 13 function?

If you're worried that it's not measurable (that is), see is Conway's base 13 function measurable?

sequences and series – In a geometric progression, $ S_2 = $ 7 and $ S_6 = $ 91. Estimate $ S_4 $.

In a geometric progression, $ S_2 = $ 7 and $ S_6 = $ 91. Assess $ S_4 $. Alternatives: 28, 32, 35, 49, 84.

Here is what I have tried until now:

$$
S_2 = frac {a_1 (1-r ^ 2)} {1-r} implies 1-r = frac {a_1 (1-r ^ 2)} {7} \
S_6 = frac {a_1 (1-r ^ 6)} {1-r} implies 1-r = frac {a_1 (1-r ^ 6)} {91}
$$

Then:
$$
frac {1-r ^ 2} {1} = frac {1-r ^ 6} {13} \
r ^ 6 – 13r ^ 2 + 12 = 0
$$

Now, I can not solve this equation, there may be an easier way …

Ag.algebraic geometry – Reading geometric properties of an appropriate scheme from its refinement

Let $ k to be a field, $ X rightarrow mathrm {Spec} , k $ to be an appropriate morphism.

  • Yes $ X $ is geometrically reduced, so $ mathcal {O} _X (X) $ is the direct product of finely separable finite extensions of $ k.
  • Yes $ X $ is geometrically connected, then $ mathcal {O} _X (X) $ is an irreducible geometrically $ k-algebra.

Are the reverse statements true?