## 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] http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

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 $$Pic (S) = 0$$,

## 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.

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?