MathDB

2003 SNSB Admission

Part of SNSB Admissions

Subcontests

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Complex properties of the sine function

Let be a function ξ:R→R \xi:\mathbb{R}\to\mathbb{R} of class C∞ C^{\infty } such that ∣dnξdxn(x0)∣≤1=dξdx(0), \left| \frac{d^n\xi }{dx^n} \left( x_0 \right) \right|\le 1=\frac{d\xi}{dx}(0) , for any real numbers x0, x_0, and all natural numbers n, n, and let be the function h:C⟶C,h(z)=1+∑n∈N(znn!⋅dnξdxn(0)). h:\mathbb{C}\longrightarrow\mathbb{C} , h(z)=1+\sum_{n\in\mathbb{N}} \left(\frac{z^n}{n!}\cdot\frac{d^n\xi }{dx^n} \left( 0 \right)\right) .
a) Show that h h is well-defined and analytic. b) Prove that h∣R=ξ∣R. h\bigg|_{\mathbb{R}} =\xi\bigg|_{\mathbb{R}} .
c) Demonstrate that ddt(ξcos⁡)(t0)=4∑p∈Z(−1)pξ((1+2p)π2)((1+2p)π−2t0)2, \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =4\sum_{p\in\mathbb{Z}}\frac{(-1)^p\xi\left( \frac{(1+2p)\pi}{2} \right)}{\left( (1+2p)\pi -2t_0\right)^2} , for any t0∈(−π2,π2) t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) and that ∑p∈Z(−1)p(ξ((1+2p)π2))21+2p=π2. \sum_{p\in\mathbb{Z}} \frac{(-1)^p\left(\xi\left( \frac{(1+2p)\pi}{2} \right)\right)^2}{1+2p} =\frac{\pi }{2} .
d) Deduce that ξ((1+2p)π2)=(−1)p, \xi\left( \frac{(1+2p)\pi}{2} \right)=(-1)^p, for any integer p, p, and that ddt(ξcos⁡)(t0)=ddt(sin⁡cos⁡)(t0), \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =\frac{d}{dt}\left( \frac{\sin }{\cos} \right)\left( t_0 \right) , for any t0∈(−π2,π2). t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) .
e) Conclude that ξ∣R=sin⁡∣R. \xi\bigg|_\mathbb{R} =\sin\bigg|_\mathbb{R} .
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Show that some 4-dimensional hyperplanes are not homeomorphic

Prove that the sets {(x1,x2,x3,x4)∈R4∣x12+x22+x32=x42}, \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , {(x1,x2,x3,x4)∈R4∣x12+x22=x32+x42}, \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, are not homeomorphic on the Euclidean topology induced on them.

Again, Lambda set

Consider Λ={λ∈Hol[C⟶C]∣z∈C  ⟹  ∣λ(z)∣≤e∣Im(z)∣}. \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} .
Prove that g∈Λ g\in\Lambda implies g′∈Λ. g'\in\Lambda .
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Show that a certain ring of fractions is factorial

Let be a prime number p, p, the quotient ring R=Z[X,Y]/(pX,pY), R=\mathbb{Z}[X,Y]/(pX,pY), and a prime ideal I⊃pA I\supset pA that is not maximal. Show that the ring {r/i∣r∈R,i∈I} \left\{ r/i|r\in R, i\in I \right\} is factorial.

Again, the set Lambda of certain holomorphic functions

Let be the set Λ={λ∈Hol[C⟶C]∣z∈C  ⟹  ∣λ(z)∣≤e∣Im(z)∣}. \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . Show that:
(1)sin⁡∈Λ \text{(1)}\sin\in\Lambda
(2)∑p∈Z1(1+2p)2=π24 \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4}
(3)f∈Λ  ⟹  ∣f′(0)∣≤1 \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1
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Standard Galois problem

Let be the polynomial f=X4+X2∈Z2[X] f=X^4+X^2\in\mathbb{Z}_2[X] Find: a) its degree.. b) the splitting field of f f c) the Galois group of f f (Galois group of its splitting field)

Properties of some holomorphic functions

Let be a natural number n, n, denote with C C the square in the complex plane whose vertices are the affixes of 2nπ(±1±i), 2n\pi\left( \pm 1\pm i \right) , and consider the set Λ={λ∈Hol[C⟶C]∣z∈C  ⟹  ∣λ(z)∣≤e∣Im(z)∣} \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} Prove the following implications.
a) \exists \alpha\in\mathbb{R}_{>0}  \forall z\in\partial C  \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} b) \forall f\in\Lambda \frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} c) \forall f\in\Lambda  \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4}
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