- #1

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**Summary:**Diffusion Equation, Sequence Space, Banach Space, Linear Algebra, Quadratic Forms, Population Distribution, Sylow Subgroups, Lotka-Volterra, Ring Theory, Field Extension.

**1.**Let ##u(x,t)## satisfy the one dimensional diffusion equation ##u_t=Du_{xx}## in a space-time rectangle ##R=\{0\leq x\leq l,0\leq t\leq T\}##, then the maximum value of ##u(x,t)## is assumed either on the initial line ##(t=0)## or on the boundary lines ##(x=0 \,or\,x=l )##. ##D > 0.##

**2.**(solved by @Office_Shredder, second proof possible ) Show that ## M= \{ (a_n)\in \ell^2(\mathbb{C})\,|\,\forall \,n: |a_n| \leq n^{-1} \} \subseteq \ell^2(\mathbb{C})## is bounded and compact.

**3.**(solved by @Office_Shredder ) Show by two different methods that the normed space ##\mathcal{C}:=(C^1([0,1]),\|.\|_\infty)## is not a Banach space.

**4.**(solved by @PeroK ) Let

$$

A:=\begin{bmatrix}5&0&1&6\\3&3&5&2\\0&0&3&0\\6&0&3&0\end{bmatrix} \in \mathbb{M}_4(\mathbb{Z}_7)

$$

**(a)**Determine the characteristic polynomial ##\chi_A(x)## of ##A.##

**(b)**Determine bases of the eigenspaces.

**(c)**Determine a matrix ##S\in\operatorname{GL}_4(\mathbb{Z}_7)## such that ##S^{-1}AS## is a diagonal matrix. Which one?

**(d)**Calculate ##A^{31}.##

**5.**(solved by @etotheipi ) Let ##f(x,y)=34x^2+24xy+41y^2+20x+110y+50.## Determine the Euclidean normal form of the conic section $$Q_f=\{(x,y)^\tau\in\mathbb{R}^2\,|\,f(x,y)=0\}.$$ What are its foci and vertices in the normal form?

**6.**(solved by @benorin ) Let ##u(x,t)## be a solution of the one dimensional diffusion equation ##u_t=Du_{xx}.## Assume that

$$

C:=\int_{-\infty}^{\infty} u(x,t)\,dx

$$

is independent of ##t,## which corresponds to a constant

*population*, and ##u## is small at infinity, which means that

$$

\lim_{x \to \pm\infty} xu(x,t) = 0 = \lim_{x \to \pm \infty} x^2 \dfrac{\partial}{\partial x}u(x,t)

$$

If

$$

\sigma^2(t)=\dfrac{1}{C}\int_{-\infty}^{+\infty} x^2u(x,t)\,dx

$$

then

$$

\sigma^2(t)=2Dt\,+\,\sigma^2(0)

$$

In the special case of an initial population (i.e. for ##t = 0##) which is concentrated near ##x = 0## (like a ##\delta ##-function) then we get ##\sigma^2(t)\approx 2Dt.##

**7.**(solved by @fishturtle1 ) Let ##G## be a group of order ##351.## Show that ##G## has a non trivial normal subgroup.

**8.**Show that the diffusional Lotka-Volterra system ##(a>0)##

\begin{align}

u_t&\, = \,u(1-v)+D\Delta u\\

v_t&\, = \,av(u-1)+D\Delta v

\end{align}

with equal diffusion coefficient ##D>0## and homogeneous Neumann boundary conditions

$$

\dfrac{\partial u}{\partial n}(x,t)=0=\dfrac{\partial v}{\partial n}(x,t)

$$

for ##x\in \partial\Omega\, , \,\Omega\subseteq \mathbb{R}^n## of finite volume and ##n## outward normal, ##\Delta ## the Laplace operator, tends to a spatially uniform state for ##t \to \infty,## i.e.

$$

\lim_{t \to \infty} \nabla u = \lim_{t \to \infty} \nabla v = 0

$$

**Hint:**Consider the

*energy*of the system ##s=a(u-\log u)+(v - \log v).##

**9. (a)**(solved by @disregardthat ) Let ##R## be a Notherian local commutative ring with ##1## and maximal ideal ##M.## If ##A \trianglelefteq R## is an ideal in ##R## such that ##A/MA\cong_R \{0\},## then ##A=(0).##

**(solved by @disregardthat ) Let ##R## be an integral domain, and ##\dim R_P=0## for all ##P\in \operatorname{Spec}(R)##, then ##R## is a field. The dimension is the Krull dimension.**

**9.**(b)**10.**(solved by @Office_Shredder ) Let ##\alpha\in \mathbb{C}## a root of the polynomial ##f(x)=x^3-3x-1\in \mathbb{Q}[x]##. Show that ##f(x)## is irreducible, and that there is an automorphism ##\sigma\in \operatorname{Aut}(\mathbb{Q}(\alpha)/\mathbb{Q})## with ##\sigma(\alpha)=2-\alpha^2.## If ##\alpha ## is chosen closest to zero, what is ##+\sqrt{12-3\alpha^2}## in the splitting field of ##f(x)##? This means in terms of a polynomial in ##\alpha,## not numerical.

**High Schoolers only**

11.(solved by @songoku ) Determine all ##a\in \mathbb{R}## such that

11.

$$

x(x+1)(x+2)(x+3)=a

$$

has no real solution, a unique real solution, exactly two, three, or four real solutions, more than four real solutions.

**12.**(solved by @songoku ) An international conference has ##30## scientists who speak English, Russian or Spanish. The number of people who speak exactly two languages is more than twice as big, but less than thrice as much as the number of people who speak only one language, which are as many as speak all three languages. Those who speak only English are more than those who speak only Russian, but less than those who speak only Spanish. The number of those who speak only English is less than thrice the number of people who speak only Russian. How many people do speak only English, Russian, Spanish, and how many all three languages? (The conference language is French.)

**13.**(solved by @songoku ) Calculate (manually!)

$$

z=\dfrac{65533^3+65534^3+65535^3+65536^3+65537^3+65538^3+65539^3}{32765\cdot 32766+32767\cdot 32768+32768\cdot 32769+32770\cdot 32771}

$$

**14.**(solved by @songoku ) Show that (##n\in \mathbb{N}_0##)

$$

f_n(x)=1+x+\dfrac{x^2}{2!}+\ldots+\dfrac{x^n}{n!}

$$

has at most one real zero.

**15.**(solved by @songoku ) Find all ##\lambda\in \mathbb{R}## such that

$$

\sin^4x-\cos^4x=\lambda(\tan^4x-\cot^4x)

$$

has no, exactly one, exactly two, more than two real solutions in ##\left(0,\dfrac{\pi}{2}\right)##

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