## discrete mathematics – Demonstrate that x and y in the extended Euclidean algorithm will not overflow an integer (Si

We are given a and b <= 1e8.

The extended Euclid algorithm always finds a solution for ax + by = gcd (a, b) (assuming it exists) that can always be stored in an Int.

How to prove that x and y do not overflow an int?

code: https://cp-algorithms.com/algebra/extended-euclid-algorithm.html

## Number Theory – Trigonometry / Euclidean Geometry for Natural Numbers?

Let $$d (a, b) = 1 – frac {2 gcd (a, b) ^ 3} {ab (a + b)}$$ to be a metric on natural numbers.

The metric space $$X = {x_0, x_1, cdots, x_n }, n> 2$$ is isometric embeddable in $$mathbb {R} ^ n$$ if and only if the matrix:
$$M (x_0, x_1, cdots, x_n) = (1/2 (d (x_0, x_i) ^ 2 + d (x_0, x_j) ^ 2-d (x_i, x_j) ^ 2)) _ {1 i, j the n}$$
is semi-definite positive.

So my question is:

Is the above matrix for $$d$$ as above semi-defined positive for all
choice of $$x_i in mathbb {N}$$? (Perhaps it is possible to prove this using the quadratic method
forms and then transform it into $$sum_ {i} a_ {ii} y_i ^ 2$$ showing then
this $$a_ {ii} 0 ge$$?

If this is the case, it would make Euclidean geometry natural numbers.
For example, for three points / natural numbers (distinct in pairs), we would have:

1) a triangle
2) sine law
3) cosine law
4) All other theorems concerning triangles

Then in the limit three consecutive numbers / prime numbers would construct an equilateral triangle of side length $$1$$. One could therefore imagine prime numbers ("at the limit") as a simplex of infinite dimension, which would be a fun thing to think about.

Related question:
https://math.stackexchange.com/questions/3385102/is-this-metric-matrix-positive-semidefinite

See Theorem 2.4 in https://books.google.de/books?id=7_DuCAAAQBAJ&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false for the isometric insertion of $$( mathbb {N}, d)$$ in a Hilbert space.

## abstract algebra – Is \$ mathbb {Z}[{ sqrt 8 } ] Will \$ form a Euclidean domain? Yes No

is $$mathbb {Z} ({ sqrt 8})$$ will form Euclidean domain? Yes No

I have some confusion, what is the difference between the Euclidean domain and the Euclidean norms?

My attempt: I think so

I know it $$d (a + b sqrt 8) = | a ^ 2 – 8b ^ 2 |$$ as I can show that it is Euclidean domain by the same motive $$mathbb {Z} ({ sqrt 2})$$ is the Euclidean domain

## Euclidean Geometry – Quadrilaterals that have congruent opposite sides are parallelograms

Quadrilaterals with opposite congruent sides are parallelograms.

The following is a proof.

AB = CD, BC = AD, AC = AC

so ABC = CDA (SSS)

mBAC = mDCA (alternative theorem of the inner angle)

Therefore

AB // DC

mACB = mCAD (alternative theorem of the inner angle)

(Q.E.D)

but, I do not know why D, B is opposed by the AC line for the alternative theorem of the inner angle.

How to prove it?

How to prove that ABCD is convex?

## theory of representation – A characterization of reflections on the Euclidean space

In the book "Introduction to Lie Algebras and the Theory of Representation" by J. Humphreys page 42, the author proves the following lemma. Let $$E$$ to be a Euclidean space of finite dimension on $$mathbb R$$ and $$mathbb phi subseteq E$$ to be such that $$mathbb phi$$ bays $$E.$$ Suppose any thought $$sigma_ alpha$$,
$$alpha in mathbb phi$$ leaves $$mathbb phi$$ invariant. Yes $$sigma in GL (E)$$ leaves $$mathbb phi$$ invariant, fixes poinwise a hyperplane $$P$$ and sends a non-zero $$alpha$$ at $$– alpha,$$ then $$sigma$$ it's nothing but the reflection $$sigma_ alpha$$ around the hyperplan $$P.$$ $$sigma_ alpha$$ is defined as $$sigma_ alpha ( beta): = beta- frac {2 langle beta, alpha row} { langle alpha, alpha row alpha.$$

He argues that or $$tau: = sigma sigma_ alpha ^ {- 1},$$ $$tau mathbb phi = mathbb phi.$$ How is this true? Is it true that $$sigma mathbb phi subseteq mathbb phi$$ implies that $$sigma mathbb phi = mathbb phi$$? Also, how to prove the rest of the lemma?

## 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.

## 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 $$k$$Discs that?