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### Questions (100)

 10 Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group 8 Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i}$ 7 Show $f''+vf' +\alpha^2 f(1-f)=0$ has solutions satisfying $\lim_{x \to - \infty}f=0$ and $\lim_{x \to \infty}f=1$ given $v\leq -2\alpha < 0$ 7 Ethical problems in mathematics 6 $M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$

### Reputation (2,124)

 +5 $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable +5 If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ +5 Let $z=re^{i2\pi\theta}$ and $w=\lambda z+cz^2\bar{z}+\mathcal{O}(z^4)=\tilde{r}e^{i2\pi\psi}$, what is an $\mathcal{O}(r^3)$ approximation of $\psi$? +5 Am not getting the right answer for $I = \int\limits_{S_\epsilon} \frac{x \,dy\,dz + y \,dx\,dz + z \,dx\,dy}{(x^2+y^2+z^2)^{\frac32}}$

 6 Solve $y^{\prime \prime}-(y^{\prime})^2-y^{\prime}=0$ 4 Prove by induction that $\sum_{k=0}^{n}(-1)^{n+k} k^{2} = \frac{n(n+1)}{2}$ 4 Is there a convention, law or axiom for associate operators when is a lack of brackets? 4 prove by induction that $P\left(\bigcup\limits_{i=1}^{n} E_i\right) = 1-\prod\limits_{i=1}^{n}(1-P(E_i))$, $E_1,E_2,\ldots , E_i$ independent 4 Homogeneous equation

### Tags (81)

 16 differential-equations × 15 5 abstract-algebra × 6 11 algebra-precalculus × 6 4 probability × 7 6 calculus × 18 4 sums-of-squares 6 induction × 2 4 operator-theory 5 integration × 11 3 real-analysis × 12

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