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Problem: Find all the values of \( \beta \), for which there is a linear transformation of variables \( (x, y) \rightarrow(z, t) \), bringing the equation to the following canonical form: a) \( u_{t t}= \) \( =u_{z z} \), b) \( u_{t}=u_{z z} \). \[ u_{x x}+4 u_{x y}-\beta u_{y y}-\beta u_{x}+\beta^{2} u_{y}=0 . \]

11.5.4.4 With constant coefficients

6.24 $

Problem: Let \( f(x)=e^{-|x|} \cos \omega x \). Calculate the integral: \[ F(y)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} f(x) e^{i y x} d x \text {. } \]

2.6.1.6 Fourier integral

3 $

Problem: Does function \( f_{n}(x)=\cos \frac{x}{n} \cdot \cos x \) uniformly converge in a) \( [0, \pi] \), b) \( \mathbb{R} \).

2.9.4 Functional sequences and series

2 $

Problem: Find the solution of the comparison: \[ x^{8} \equiv 62(\bmod 169) \text {. } \]

12.2.14 Number theory

3 $

Problem: Find the solution of the comparison: \[ 355 x \equiv 257 \cdot 156(\bmod 338) . \]

12.2.13 Number theory

3 $

Problem: Let \( a_{n} \) be a sequence of positive numbers, for which the following inequality holds. \[ a_{n}^{2}

12.3.1.4 Algebraic

3.75 $

Problem: Prove the equivalence of the formulas: \[ \int_{0}^{a} t^{\beta-1} e^{-\lambda t} d t \sim \Gamma(\beta) \lambda^{-\beta}, \quad \lambda \rightarrow+\infty, \quad a, \beta>0 . \]

2.12.1 Asymptotic analysis

3 $

Problem: Let function \( f(x) \in C(0 ; 1] \) be such that \[ f(x) \sim x^{a-1}(x \rightarrow+0), 0

2.12.2 Asymptotic analysis

6.24 $

Problem: Write down the asymptotic formula for the function: \[ f(x)=\left(\frac{e^{x}-1}{x}\right)^{\frac{1}{x}}, \quad x \rightarrow 0 . \]

2.12.3 Asymptotic analysis

3.75 $

Problem: Write down the asymptotic formula for the function: \[ \begin{array}{l} f(x)=\sqrt{x^{4}+2 x^{2}-2 x}-x^{2} \cos \frac{x}{x^{2}+1}, \\ x \rightarrow+\infty . \end{array} \]

2.12.4 Asymptotic analysis

4.25 $

Problem: Find the asymptotics of the roots of the equation: \[ \cos x=\frac{1}{x}, \quad x>0 . \]

2.12.5 Asymptotic analysis

7.49 $

Problem: Write down the asymptotic representation of the function: \[ F(x)=\int_{x}^{+\infty} \frac{e^{-t}}{\sqrt{t^{3}+2}} d t, \quad x \rightarrow+\infty . \]

2.12.6 Asymptotic analysis

6.24 $

Problem: Write down the asymptotic representation of the function: \[ F(x)=\int_{1}^{x} \sqrt{t^{3}+t} d t, \quad x \rightarrow+\infty . \]

2.12.7 Asymptotic analysis

6.24 $

Problem: Let \( (M, \cdot) \) - be a group, and \( \alpha \) and \( \beta \) - mappings from \( M \) to \( M \). Let's define an operation * on \( M \) by the rule: \( x * y=\alpha(x) \beta(y) \forall x, y \in M \). Prove that the conditions are equivalent: (1) \( (M, *) \) - a group; (2) \( \exists a, b \in M: \alpha(g)=g a, \beta(g)=b g \quad \forall g \in M \).

1.6.9 Fields, Groups, Rings

6.24 $

Problem: Prove that \( \mathbb{Q}(p) / \mathbb{Z}^{+} \cong \mathbb{Z}_{p^{\infty}} \).

1.6.10 Fields, Groups, Rings

4.25 $

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