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Thank you, Mr.Papapetros, firstly for your nice solution and secondly for giving the exact solution of the problem (ii). I didn't notice the "mistake" that Tolaso's solution contained, which you pointed out! I'm glad that we had such a long and interesting discussion and slightly different...

Thank you, Tolis, for your solutions! I'm looking forward to seeing your solution to the first integral, which, i believe, is the most interesting of the three and probably the most difficult. Let me point out some things about the third integral: (1) A solution i've seen is based exactly on the sub...

Compute the following indefinite integrals (i) \( \displaystyle \int \frac{ \tan{x} }{ \sqrt{ a+b\tan^{2}{x} } } \mathrm{d}x, \; b>a \) (ii) \( \displaystyle \int \frac{ \sin{x} }{ \sqrt{ 2 + \sin(2x) } } \mathrm{d}x \) (iii) \( \displaystyle \int \frac{1}{x\sqrt[3]{x^{2}+1}} \mathrm{d}x \)

Let \( X \) be a topological space and let \( \mathscr{C}_{X} \) be the sheaf of continuous function on \( X \). Show that \( \left( X , \mathscr{C}_{X} \right) \) is a locally ringed space and describe (for each \( x \in X \)) the maximal ideal \( \mathfrak{m}_{x} \) of \( \mathscr{C}_{X,x} \).

Let \( \displaystyle f : X \longrightarrow Y \) be a continuous mapping between topological spaces. Show that \( \displaystyle f \) induces naturally a morphism \( \displaystyle \left( f , f^{\#} \right) \) of ringed spaces, and prove that it is, in fact, a morphism of locally ringed spaces.

Suppose that \( X \) is a connected and compact Riemann surface and let \( \displaystyle f : X \longrightarrow \mathbb{C} \) be a holomorphic function. Show that \( \displaystyle f \) is constant. Let \( \displaystyle f : \mathbb{C} \longrightarrow \mathbb{C} \) be a holomorphic bounded function. S...

Prove that the set \( \displaystyle \left\{ (t^2,t^3+1) \in \mathbb{C}^{2} \, \big| \, t \in \mathbb{C} \right\} \) defines an (affine) algebraic curve.

Describe examples of pre-sheaves \( \mathcal{F} \) for which the sheafification \( \mathcal{F} \longrightarrow \mathcal{F}^{+} \) is not
(i) injective
(ii) surjective
Also, find an example such that \( \mathcal{F} \neq 0 \), but \( \mathcal{F}^{+} = 0 \).

Recall that, given a non-constant polynomial $ P(x,y) \in \mathbb{C}[x,y] $ without repeated factors, the complex algebraic curve $ C $ in $ \mathbb{C}^2 $ determined by $ P $ is defined as \[ C = \left\{ (x,y) \in \mathbb{C}^2 \ | \ P(x,y) = 0 \right\} \hskip -2pt . \] Show that any complex algebra...

Suppose that \( \mathcal{C} \) is a category with a zero object. Suppose that the morphism \( a \) is the kernel of a morphism \( \displaystyle \gamma : X \longrightarrow Y \). Show that if coker(\( a \)) exists, then \( a = \) ker(coker \( a \)).