complex analysis – evidence of the existence of a maximal analytical continuation of a holomorphic germ

The following happens Lectures on Riemann's surfaces by O. Forster:

I do not understand the second paragraph of the proof. Why is the card? $ F: Z to Y $ well defined if we choose another curve $ a $ to $ x $, we could get another one $ eta $

For the relevant definitions:

Proofing Techniques – Prove that every complete prefix-free language is maximal

I am practicing a problem where I have to prove that every prefix-free language is maximal.

I know that a prefix-free language A is a maximum if it is not a proper subset of a prefix-free language, where a prefix-free language is full if

$$ sum_ {x in A} 2 ^ {- | x |} = 1 $$

Also, I know that a language A 0 {0, 1} * is a prefix-free if no element of A is a prefix of another element of A and that the force inequality says that for each prefix-free language A .

$$ sum_ {x in A} 2 ^ {- | x |} leqslant 1 $$

I'm pretty sure that a full free prefix language is maximal because it belongs to the maximum prefix-free set. But I do not know how to prove it formally. Should I deal with Kraft's inequality and what I know about the relationship between maximum and prefix-free quantities?

Any help would be appreciated!

ag.algebraic geometry – Are maximal tori conjugated locally localizable?

To let $ S $ to be and leave a scheme $ G to S $ to be a reductive group scheme. Then $ G $ admits that there is a maximal torus etale-local, and two maximal tori are conjugate etale-local, according to Theorem 3.2.6 and Corollary 3.2.7 in [B. Conrad, Reductuve group schemes]

As stated in the book under Proposition 3.1.9 and at the beginning of Chapter 4, the presence of maximum Tori Zariski locally can be achieved (cf. [SGA3, XIV, 3.20]). My question is the following

To let $ T $ and $ T # $ be maximum tori $ G $ over defined $ S $are they conjugated Zariski-local?

In my case, $ T $ is split $ S $, and $ S $ is affine with $ Pic (S) = 0 $,

I would appreciate comments!

Determine the probability that X + Y is maximal for a given X

Consider some random variables $ X_1 $ , , , $ X_n $ they are independent and distributed identically to CDF $ F (x) $and some random variables $ Y_1. , , Y_n $ which are independent and identical to CDF distributed $ G (y) $, Suppose that everyone $ X_i $ is independent of everyone $ Y_i $,

These pairs of random variables give random sums. $ X_1 + U_1 $ , , , $ X_n + U_n $, What is the probability that the first random sum is the highest? That's clear $ 1 / n $ through symmetry. Equally clear is the likelihood that the first random sum is the highest Fewer when $ 1 / n $ if I assume that the first random sum falls below a certain limit. This is,

$$ P (X_1 + U_1 geq X_j + U_j hspace {0,1cm} for all i hspace {0,1cm} | hspace {0,1cm} U_i leq u) leq 1 / n $$

Presumably this is true with strict inequality, though $ n> 1 $, Is that true? I've spent a lot of time showing this, and would appreciate any help, ideas, or hints.

Thank you in advance!

algebraic geometry – Is the Zariski topology on a variety $V$ a maximal Noetherian topology?

Let $K$ be an algebraically closed field. By a variety $V$ definable over $K$, I mean a quasi-projective or an algebraic variety in sense of Weil. It is the set of points in an affine or a projective space over some bigger algebraically closed field $L$.

Now consider the Zariski topology $tau$ on $V$ together with Krull dimension on closed sets. Is it possible to enrich this topology in the naive sense by adding new closed sets, so that the enriched topology $tau^prime$ is also a Noetherian topology?

In other words, is the Zariski topology on $V$ a maximal noetherian topology?