## How to prove that I am going home after a Schengen visa (I am a minor)

I have already applied for a Schengen visa in Italy when I was a student.
You can show the document to prove that you will return home after the visit by asking your school / university to give you a certificate of study.
This letter will read "Attention of the Italian Embassy" and indicate your full name, date of birth, details of your course, including the dates of the beginning and end of your course, as well as your residential address. It also bears the logo of your school / university.

Here is the list of documents that you should provide when applying for a visa.

• Confirmation of appointment
• Visa application
• A copy of the passport
• Copy of the identity card
• 2 passport size photos
• Introductory letter (To introduce yourself and tell why you want to go to Italy)
• Map of the route
• Travel insurance
• Confirmation of airline tickets (do not pay the ticket until you have obtained the visa)
• Bank statement (6 months)
• Letter of invitation from your sister
• Proof of relationship with the host (it can act of your sister's letter and a copy of his passport)

*** If your parents or sister are paying for your trip, mention it in the introductory letter and show them your sponsor's bank statement and a copy of their passport.

The main thing is that you can show your school and your letter that you will go to Italy on vacation and that you will have to come back to finish your course.

I hope this help ..
Discover more details here: https://www.schengenvisainfo.com/italy-visa/

## How to prove that the time complexity of this algorithm is O (\$ sqrt {N} \$)?

``````        int n;
cin >> n;
int sum = 0;
for (int i = 1; sum <= n; i ++) {
sum + = i;
}
``````

If I assumed that $$N = 100$$, the loop will run $$13$$ steps, which is almost the square root of $$N$$, if $$N = 10000$$, the loop will run $$141$$ steps, which is almost the square root of $$N$$, but I do not know how to prove it, I only know it by intuition

## Complex Analysis – Tried to prove that \$ e \$ is irrational, but ended up proving that it was not a real number.

assume $$e = p / q$$, with integers $$p, q$$. Then, we have:
begin {align} & qe = p \ & (qe) ^ {2i pi} = p ^ {2i pi}, text {from e ^ {2i pi} = 1 :} \ & q ^ {2i pi} = p ^ {2i pi}, text {since the exponents are the same:} \ & q = p end {align}

that would imply $$e = p / q = 1$$. But nowhere in the "proof" does it use the fact that $$p, q$$ are integers. If we take $$p, q$$ complex $$e$$ could not even be a number.

Where is the error?

## ABC is a triangle. AM, CE are the altitude of the triangle. FC = FG. FE = FH How can I prove that HMGA is a cyclic quadrilateral?

Figure: The required triangle and the given conditions

## probability – Assume that \$ A \$ and \$ B \$ are independent events. For a \$ C \$ event such as \$ P (C)> \$ 0, prove that \$ A \$ event donated \$ C \$

assume $$A$$ and $$B$$ are independent events. For an event $$C$$ such as $$P (C)> 0$$ , prove that the event of $$A$$ given $$C$$ is independent of the event of $$B$$ given $$C$$

We have A and B are independent so $$P (AB) = P (A) cdot P (B)$$

We must show that $$P ((A mid C) cap (B mid C)) = P (A mid C) cdot P (B mid C)$$

My procedure was like that
$$P ((A mid C) cap (B mid C)) = P ((A mid C) mid (B mid C)) cdot P (B mid C)$$
$$= frac {P (AB mid C)} {P (B mid C)}$$

I played until I got this
$$frac {P (AC)} {P (C)} cdot frac {P (B mid AC)} {P (B mid C)}$$
Now the first part gives us $$P (A mid C)$$ . I could not get from the second part the missing part that is $$P (B mid C)$$.

Is my procedure correct? If so, how can I find the second part?

## combinatorial – Prove that among any set of 34 different positive integers that are at most 99, there is always a pair of numbers that differs by at most 2.

Okay, so I'm pretty new to this. I think this should be a pretty simple solution, but I do not know how to start.

So here's where I was going

if we have a set {1,2, … 99}, then I start creating groups that have an integer pair other than 2 at most.

group 1 = (1,2)

group 2 = (3,4)

group 49 = (97.98)

Now, I know I do not use 99 but it says at most 99. Another thing I could do is make sure that the groups

group 1 = (1,3)

group 2 = (2,4)

group 3 = (5,7)

group 4 = (6.8)

group 48 = (94.96)

group 49 = (95.97)

group 50 = (98.99)

and I think it works better.

Now, from here, I do not know how to progress, in fact, I think it's a bad way to go, but I do not know how to solve this problem differently. Maybe I could use the pigeon trap principle but I do not know how.

## graphs – Prove that the total distance is minimized (on the longest trip)

Here is the problem: given a tree $$T$$I have to visit each node of the tree once. I can start and finish where I want.

As my specifications are not really clear, I suggest an example: consider the graph (which is a tree – non-oriented weighted acyclic graph) to have nodes as cities and edges as routes between cities. I have to deliver something in each city (visit each node at least once). I can leave any city and finish in the city of my choice.

I read the following result. Find the two most distant cities in the graph (call them $$c1$$ and $$c2$$). Start with one of them ($$c1$$ or $$c2$$), visit all other cities until you reach ($$c2$$ or $$c1$$). This minimizes the total distance to travel.

How should I prove that it is the minimum distance?

I tried the next one. I have the last route and I call the edges, $$m_1, m_2, …, m_i$$ and $$e_1, e_2, …, e_j$$. Or $$m_1, m_2, …, m_i$$ boundaries of the most distant cities ($$c1$$ and $$c2$$) and $$e_1, e_2, …, e_j$$ are all the rest in the graph. As I leave $$c1$$, I travel along the edges labeled m once and all the rest is a digression and come and go twice on these edges before reaching $$c2$$.

We know that $$m_1, m_2, …, m_i$$ and $$e_1, e_2, …, e_j$$ taken together, they include all the edges of the graph (as it is a tree, there is only one path between two nodes). So, the distance I was traveling could be given as $$2 (e_1 + e_2 + …. + e_j) + (m_1 + m_2 + … + m_i)$$.

I have to prove that this sum is lower than all the other roads I can take to reach all the cities. My intuition says that it must be the shortest route. I think I have to use the fact that $$(m_1 + m_2 + … + m_i)$$ is the maximum between two nodes in the graph (does this call the diameter?) and leads to a contradiction.

This is the kind of image I have in mind (the red edges are in ($$m_1, m_2, …, m_i$$) and the gray ones are all the rest),

This graphic is always a tree (please ignore the point of the arrow in the edges that indicates how I decide to travel). I do not know where to go from here. I would appreciate a proof that is simple to understand (this is not a duty or anything related to the courses.)

## sweet question – With what equation could extraterrestrials prove their superior intelligence?

Suppose that humanity apparently receives a mysterious message from an extraterrestrial civilization much more advanced than ours. How could such a civilization ensure to humanity the validity of the message through the use of mathematics? Let us suppose that humanity will consider the message as a lie, unless it contains mathematics which are currently unknown to us, but which can be proved. To clarify, the details are unknown, but humanity still knows the concept. It will not be very useful to give humanity a mathematics they can not understand. It must be simple enough for them to understand and verify. (I'm asking this question about mathematics, because the anecdote of "aliens" is fun, but does not matter for the question.I'm interested in mathematics.)

For the less "science fiction" among us, what's an element, an algorithm, an equation, etc., mathematics that we are still far from discovering, but that we are able to validate?

For example, suppose the message contains the recipe for travel faster than light. Assuming we can validate it, FTL moves are currently considered impossible, but verifiable experiments using exotic equations would make us take this message seriously. What else could cause the same result?

Bonus points for not revealing the secret of potentially dangerous equations or the end of the world. For example, for a humanity before the Second World War, give them the recipe for a nuclear disaster.

Bonus points for humanity will be able to take advantage of it to improve various aspects of the technology.

Here are some ideas to get you started.

## co.combinatorics – Prove that \$ lambda mapsto chi ^ lambda (C) / f ^ lambda \$ is a polynomial

Let $$lambda$$ to be a partition of $$n$$ and $$chi ^ lambda$$ to be the character of $$S_n$$ associated with it. Given any class of conjugation $$C$$, I want to prove that
$$lambda mapsto frac { chi ^ lambda (C)} {f ^ lambda} = frac { chi ^ lambda (C)} { chi ^ lambda ( text {id}) } = frac { chi ^ lambda (C)} { dim chi ^ lambda}$$
is a polynomial

This has been asserted without evidence in [1] (Proposition 2.9). The article refers to [2] for proof. In [2] the evidence is not simple at all (at least for an undergraduate student). I think that since we want to prove something weaker, it should be easier.

[1] Eskin, Alex and Andrei Okounkov. "Asymptotic numbers of ramified liners of a torus and volumes of holomorphic differential module spaces." Inventiones Mathematicae 145.1 (2001): 59-103.

[2] Kerov, Serguei. "Polynomial functions on the set of Young diagrams." CR Acad. Sci. Paris Ser. I'm maths 319 (1994): 121-126.

## Functional Analysis – How to prove the uniformly limited binary function?

Suppose that $$a_1 < 1$$, $$a_3 <1,$$ $$a_1 + a_2 + a_5> 1$$, $$a_3 + a_4 + a_5> 1,$$ $$a_1 + a_2 + a_3 + a_4 + a_5> 2,$$ and $$b_1, b_2> 0$$. For $$x, y> 0,$$ define a function
$$H (u, v) = frac {u ^ { frac {1} {2}} int_0 ^ { infty} int_0 ^ { infty} frac {1} {x ^ {a_1} ~ (1 + x) ^ {a_2 + 1} ~ y ^ {a_3} ~ (1 + y) ^ {a_4} ~ (1 + x + y) ^ {a_5}} exp big {- frac {u} {1 + x} – frac {b_1 v} {1 + x + y} big } dx dy} { int_0 ^ { infty} int_0 ^ { infty} frac {1} {x ^ {a_1} ~ (1 + x) ^ {a_2} ~ y ^ {a_3} ~ (1 + y) ^ { a_4} ~ (1 + x + y) ^ {a_5}} exp big {- frac {u} {1 + x} – frac {b_2 v} {1 + x + y} big } dx dy}.$$
then $$H (u, v)$$ is uniformly delimited $$u, v$$, that is, there is a constant C such that
$$H (u, v) the C.$$

How to prove it? Fedor Petrov gave the answer[Limitéuniforme(miseàjour)pourlecasoù[UniformlyBounded(updating)forthecasewhen[Limitéuniforme(miseàjour)pourlecasoù[UniformlyBounded(updating)forthecasewhen$$b_1 = b_2.$$ I still did not know how to prove the general case. I need your help.

Maybe the following result is useful:

Yes $$a_1 < 1$$ and $$a_1 + a_2> 1,$$ then
$$f (u) equiv int_0 ^ { infty} frac {1} {x ^ {a_1} ~ (1 + x) ^ {a_2}} exp big {- frac {u} { 1 + x} big } dt approx C_1 min {C_2, u ^ {1-a_1-a_2} }$$
for some positive constants $$C_1$$ and $$C_2$$.