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.

linear algebra – kernel degree of a module card $ R ^ n rightarrow R ^ n $ for a Euclidean domain $ R $

Let $ R $ to be a Euclidean domain with the degree function $ d $. Let $ A in R ^ {n times n} $ bean $ n times n $-matrix with entries in $ R $ such as det$ (A) = $ 0. As a map module $ A: R ^ n rightarrow R ^ n $, there is still a kernel element $ v in R $ n since det$ (A) = $ 0.

Supposing $ d (A_ {ij}) leq m $ for everyone $ i, j $is there an explicit limit $ k (m, n) $ such as there is a core element $ v in R $ n satisfactory $ d (v_i) leq k (m, n) $?

elementary theory of numbers – efficiency of the Euclidean algorithm.

According to the algorithm of Euclides, suppose that $ a ge b gt $ 0, we have

$$ a = bq_0 + r_0 qquad 0 lt r_0 lt b $$
$$ b = r_0q_1 + r_1 qquad 0 lt r_1 lt r_0 $$
$$ r_0 = r_1q_2 + r_2 qquad 0 lt r_2 lt r_1 $$
$$. $$
$$. $$
$$ r_ {i-2} = r_ {i-1} q_i + r_i qquad 0 lt r_i lt r_ {i-1} $$
$$ r_ {i-1} = r_iq_ {i + 1} + r_ {i + 1} qquad r_ {i + 1} = 0 $$

prove it $ b gt 2 ^ {i / 2} $

I have not been able to prove it, even if it seems quite logical.

I need a clue please.

pr.probability – Sub-Gaussian Decay of Euclidean Ball Measurement

Let $ X $ to be a random vector $ mathbb {R} ^ d $ satisfying the following property: there is $ C_1, C_2> $ 0 such as
$$ int_0 ^ {+ infty} mathbb {P} ( | X- mu_0 | leq sqrt {t}) exp (-t) dt leq C_1 exp (-C_2 | mu_0 | ^ 2) $$
for all $ mu_0 in mathbb {R} ^ d $. Right here $ | | $ is the Euclidean norm $ mathbb {R} ^ d $.
If the above property is valid, is the following statement true: there is a sequence of vectors $ mu_n $ in $ mathbb {R} ^ d $ and a sequence of real numbers $ t_n to + infty $ ($ t_n $ may depend on $ mu_n $ for example $ t_n = | mu_n | ^ 2/4 $) such as:
$$ lim_ {n to + infty} frac { mathbb {P} ( | X- mu_n | leq1)} { mathbb {P} ( | X- mu_n | leq sqrt {t_n}) exp (-t_n) = 0 $$

If this is not true, is there a counterexample?

Are disjunct varieties separated by disks incorporated into a higher Euclidean space?

Let $ A, B $ two disjoints $ p $ and $ q $ varieties integrated into $ R ^ n $. Can we still find a PL card? $ f: R ^ n longrightarrow R ^ k $ such as $ f (A) $ and $ f (B) $ are contained in two separate separate disjoints $ kDiscs that?

I want to find the number of steps it takes to find the GCD by Euclidean algorithm

Let's say I have two numbers a and b. I want to find the number of steps it takes to find the GCD by Euclidean algorithm.
If I rely on this implementation, how many times will I reach gcd:

    public static int gcd (int a, int b)
{
if (a == 0)
return b;

return gcd (b% a, a);
} 

General Topology – The punctured balloon is not homeomorphic to the Euclidean space

Just another ordinary day with another (big) ordinary math conversation. A friend and I asked this:

Problem. Prove it $ B (0, r) setminus {0 } subseteq mathbb {R} ^ n $ is not homeomorphic to open bullets for $ r> $ 0.

I have not taken a topology course yet and my friend has just started a topology course. So we tried to find as basic a solution as possible.

It seems easy enough, even though we struggled …

  • It is enough to prove that it is not homeomorphic $ mathbb {R} ^ n $.
  • Most elemental topological invariants do not work.

The best we have found is to compute the fundamental group and take generalizations of the fundamental group for $ n> $ 2. This should probably work.

So here's my question: Is there another way to prove this result?

Euclidean geometry – Which solid deals with the truncation of an icosahedron?

I was looking at fun math stuff on youtube, and I came across this video: https://www.youtube.com/watch?v=cwWBpjeyRS0 where the guy mentions how a balloon is obtained by truncating an icosahedron in a certain, so I wondered: what happens if you repeat the process? I mean, if you truncate the form of "football" he's talking about, you'll get an object with a lot of triangular faces, but what if you keep doing it? Does it approach a sphere? Does it become "smooth" at some point?

What would be an Euclidean argument for why the characteristic axiom is still valid in hyperbolic geometry?

What would be an Euclidean argument for why the characteristic axiom is still valid in hyperbolic geometry?

Characteristic Axiom States Let k be a line and a point p not on k, there are at least two lines on p that do not intersect k.

Calculating the Euclidean distance [on hold]

I'm trying to calculate the Euclidean distance from animal tracks to infrastructure (roads). I need help finding the Euclidean distance in QGIS, is it possible? Any help greatly appreciated.