- Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function

$$

f(x)=\sum |L_n| z^n,

$$

where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\# L_n\) grows. - If \(\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\) is the standard surface area form for the unit sphere

\(S\)centered at the origin, find

$$

\int_S x^4\omega.

$$ - For \(a\geq 0\), find the image of the half-plane \(\Im z>a\) under the mapping \(z\mapsto z^2\).
- Find the line integral

$$

\int_\gamma \bar{z}^2 dz

$$

over the arc \(\gamma=R\exp(i\phi), \phi\in[0,\phi_*]\) (oriented by increasing \(\phi\)). - Estimate (with reasonable precision, say 5%) the integral

$$

\int_0^1 x^{500}(1-x)^{1500} dx.

$$ - Find (using residues) the integrals

$$

\int_{-\infty}^\infty \frac{e^{iks}}{1+s^4}ds,

$$ - Find the integral of \(\exp(ax+by)\) over the area given by inequalities

$$

\begin{array}{rl}

x&\geq 0\\

y&\geq 0\\

x+y&\geq 1\\

x+y&\leq 2\\

\end{array}

$$ -
Solve the quasilinear PDE

$$

u_t+xu_x=u,

$$

subject to the initial condition \(u(x,0)=\phi(x)\). Does the solution remain smooth at all times, if \(\phi\) is smooth? - Consider the solution to the wave equation

$$

u_{tt}=u_{xx}

$$

on the unit interval with the initial conditions

$$

u(x,0)=\phi(x), u_t(x,0)=0,

$$

where the function \(\phi\) is sketched below, and the fixed boundary condition on the right end \(u_x(1,t)=0\), and free boundary condition on the left end, \(u_x(0,t)=0\).Sketch the wave at times \(t=.25, .5, .75, 1\).

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