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Substitute the exact smooth solution into the leapfrog advection scheme and expand the centered differences. With $\tau =\Delta t$ and $h=\Delta x$, the normalized local truncation error isBecause $u_t=u_x$, repeated differentiation gives $u_{ttt}=u_{xxx}$. HenceEquivalently the unnormalized one-update residual is $\tau ({\tau }^2-h^2)u_{xxx}/3+O({\tau }^5+\tau h^4)$. Thus the scheme is second order in space and time. When $\tau =h$, the exact translating solution $u(x,t)=F(x+t)$ satisfies the grid recurrence exactly; initialization still matters for the independent computational branch.

Apply the volume mean value property to the two balls and use nonnegativity together with inclusion:For any $w,z\in B(x,r)$, the triangle inequality gives $B(z,r)\subset B(w,3r)\subset B(x,4r)\subset \Omega$. Taking $R=3r$ in the previous bound therefore gives $u(w)\ge u(z)/27$. Take the infimum over $w$ and supremum over $z$ to obtainNo positivity stronger than nonnegativity is needed; if a value vanishes, this argument forces every value in the smaller ball to vanish too.

The mean value property for harmonic functions states that if $u$ is harmonic on a neighborhood of a closed ball, then its average on the ball or its boundary sphere equals its value at the center. To prove it in three dimensions, define $M(r)=(4\pi {)}^{-1}{\int }_{S^2}u(x+r\omega )d\omega$. Differentiation and the divergence theorem giveSince $M(r)\to u(x)$ as $r\to 0$, the spherical average equals $u(x)$. Integrating $4\pi r^{2}M(r)$ from zero to the ball radius gives the volume version,For balls merely contained in the open harmonicity domain, apply the result first to smaller radii and pass to the radius of interest by continuity.

A fundamental solution of a linear differential operator $P(D)$ is a distribution $E$ satisfying $P(D)E={\delta }_0$. For the three-dimensional negative Laplacian,Away from zero the function is harmonic. For a test function $\varphi$, integrate $E(-\Delta \varphi )$ outside a ball of radius $\epsilon$ and use Green's identity. The small inner sphere contributes $\varphi (0)+o(1)$ because the normal derivative of $E$ has total outward-from-the-ball flux $-1$; the other boundary term is $O(\epsilon )$. Letting $\epsilon \to 0$ verifies the distributional equation and its normalization.

Define the continuous map $T(z)=-(z^6-z^5+2z^2+1)/6$. For $|z|\le 1$, the triangle inequality gives $|T(z)|\le (1+1+2+1)/6=5/6$. Thus $T$ maps the closed unit disk into itself. By the Brouwer fixed-point theorem, some $z$ has $T(z)=z$, which rearranges toThis supplies the requested existence and even places a root strictly inside the unit disk.

The planar Brouwer fixed-point theorem says that every continuous map from the closed disk $D$ to itself has a fixed point. If a retraction $r:D\to \partial D$ existed, $x\mapsto -r(x)$ would be a continuous self-map of $D$. At a fixed point $x=-r(x)$, $x$ lies on the boundary, where $r(x)=x$; this would give $x=-x$, impossible on the unit circle.

Conversely, suppose a continuous self-map $f:D\to D$ has no fixed point. For each $x$, take the ray starting at $f(x)$ through $x$, and let $r(x)$ be its exit point on $\partial D$. The denominator $|x-f(x)|$ never vanishes, and solving the quadratic intersection with the circle shows that the exit point depends continuously on $x$. More explicitly, with $v=x-f(x)$,The ray exits beyond or at $x$, so $\lambda \ge 1$, and when $x\in \partial D$ its exit point is $x$. Hence $r$ is a retraction. Nonexistence of a boundary retraction and the planar fixed-point theorem are equivalent.

For equal rates, the change of numeraire from the Euro bank account to the Dollar account has density process $Z_t=Y_t/Y_0$, a strictly positive mean-one martingale under $Q_E$ by part (b). Conditional change of measure givesThus the reciprocal is a genuine martingale under $Q_D$, directly from the Euro-measure martingale property, without recalculating its stochastic differential equation.

For a Dollar investor, the Euro bank account has Dollar value $e^{r_{E}t}/Y_t$. After Dollar discounting, it is $e^{(r_E-r_D)t}/Y_t$. Its conditional mean growth exponent is $r_E-r_D+{\sigma }^2-\mu$, so the martingale requirement givesWith equal rates, $Y^{-1}$ is a martingale under $Q_D$ by part (a), whereas $Y$ has positive drift ${\sigma }^2$ and is not a martingale when $\sigma \ne 0$. In the degenerate zero-volatility case, both exchange rates are constant and both are martingales; the asserted nonmartingale conclusion implicitly requires a genuinely risky exchange rate.

The Dollar bank account is a tradable asset whose Euro value is $Y_t{e}^{r_{D}t}$. Under the Euro equivalent martingale measure, discount this value by $e^{r_{E}t}$. The result is a martingale precisely when $\mu +r_D-r_E=0$, givingWhen the rates coincide, $\mu =0$ and part (a) proves that the exchange rate itself is a martingale under $Q_E$.

The independent Gaussian increments of Brownian motion give $E[Y_t\mid {\mathcal{F}}_s]=Y_s{e}^{\mu (t-s)}$; the exponential is integrable at every finite time. ThereforeSimilarly $E[Y_t^{-1}\mid {\mathcal{F}}_s]=Y_s^{-1}{e}^{({\sigma }^2-\mu )(t-s)}$, so $\boxed{Y^{-1}\text{ is a martingale}⟺\mu ={\sigma }^2}$. This uses actual conditional expectations, not only the local stochastic differential equation.

Combining the eigenvalue cases givesThis agrees with the interior drift: the switching phase spends proportions $\beta /(\alpha +\beta )$ and $\alpha /(\alpha +\beta )$ in $C$ and $W$, giving level drift $(\lambda \beta -\mu \alpha )/(\alpha +\beta )$. The harmonic calculation establishes recurrence even at the zero-drift boundary, rather than inferring it from drift alone.

Here $\vartheta <1$ is equivalent to $\mu \alpha <\lambda \beta$, which gives $b<1$ from its explicit formula. Any nonnegative solution is of the formsince its second component at level zero must be one. Letting $i\to \infty$ forces $a\ge 0$ for nonnegativity. The choice $a=0$ is nonnegative, and subtracting it from any other such solution gives $a[(1,1{)}^T-{\vartheta }^i(b,1{)}^T]\ge 0$. Therefore minimality selectsIn particular the return probability after leaving $W_0$ is $b<1$, so the irreducible chain is transient. These formulas are the hitting probabilities for a velocity-switching Markov ladder.

If $\vartheta >1$, the general level solution is a constant multiple of $(1,1{)}^T$ plus a multiple of ${\vartheta }^i(b,1{)}^T$. Boundedness of probabilities removes the growing component, and $h_{W_0}=1$ then fixes the constant to one. If $\vartheta =1$, the nonzero Jordan part produces linear growth unless the initial vector lies in the one-dimensional eigenspace; the same conclusion follows. ThusStarting at $W_0$, the chain leaves to $C_0$ and returns with probability one. An irreducible chain with a recurrent state has all states recurrent, so the chain is recurrent in both cases.

Since one eigenvalue is one, the other is the determinant. Direct calculation givesIndeed the first row requires $(1+\alpha /\lambda -\vartheta )b=\alpha /\lambda$, and the second row then follows from the characteristic equation. At $\vartheta =1$, $b=1$ and the eigenvectors coincide; the nonidentity matrix has a repeated eigenvalue and a Jordan block.

Condition on the first jump, putting $h_{W_0}=1$. The hitting probabilities obeyThe relevant characterization is that a hitting probability is the minimal nonnegative harmonic extension of its target-boundary value. Thus among nonnegative solutions of these equations with $h_{W_0}=1$, choose the componentwise least one. Other harmonic solutions need not be probabilities. This also identifies the solution without imposing an unjustified boundary condition at infinity.

The arrows in the PDF give $C_i\to C_{i+1}$ at rate $\lambda$, $W_i\to W_{i-1}$ at rate $\mu$ for $i\ge 1$, and switches $C_i\to W_i$ and $W_i\to C_i$ at rates $\alpha$ and $\beta$. All rates are positive. One can travel upwards in level in the $C$ phase and downwards in the $W$ phase, switching at either end. Hence any state reaches any other through a finite positive-rate path: the continuous-time Markov chain is irreducible. Its total jump rates are uniformly bounded, so it is also nonexplosive.

Let $c$ be the center of the containing disk and take arclength around the closed curve. The periodic function $\Phi (s)=|\gamma (s)-c{|}^2/2$ has a maximum. At a maximum $s_0$,By the Cauchy-Schwarz inequality and the containing radius, $1\le |\gamma (s_0)-c||{\gamma }^{\prime \prime }(s_0)|\le r|k(s_0)|$. ThereforeClosedness provides an interior maximum on the periodic parameter domain. Without it, the straight segment $\gamma (s)=(s,0)$ for $-r<s<r$ lies in the disk but has signed curvature identically zero, giving the required counterexample.

The planar planar isoperimetric inequality is $4\pi |\Omega |\le |\partial \Omega {|}^2$, with equality exactly for a disk. For one boundary component of length $L$, take its unit-speed periodic parametrization $\gamma =(x,y)$ and subtract its mean position. Green's area formula and the Cauchy-Schwarz inequality giveApply the periodic Wirtinger inequality to each mean-zero coordinate: $\int |\gamma {|}^2\le (L/2\pi {)}^2\int |{\gamma }^{\prime }{|}^2$. Since unit speed gives the final integral $L$, it follows that $|A_{\mathrm{signed}}|\le L^2/(4\pi )$. For several boundary components, orient each with the domain on its left. Green's formula, followed by the triangle inequality, givesThis proves the stated inequality, including disconnected domains or holes. Equality permits only one component; equality in the two integral inequalities makes its coordinates first sine-cosine harmonics with the corresponding orthogonal rotation relation. Unit speed then makes the curve a circle, so its domain is a disk. Conversely disks attain equality.

Parametrize the regular curve by arclength $s$. Its unit tangent is $T={\gamma }^{\prime }$, and its curvature is $\kappa =|T^{\prime }|$. Where $\kappa >0$, define the principal normal $N=T^{\prime }/\kappa$ and binormal $B=T\times N$. The torsion of a space curve is $\tau =-B^{\prime }\cdot N$. Orthonormality and differentiation give the Frenet-Serret formulasFor a general parameter, $\kappa =|\dot{\gamma }\times \ddot{\gamma }|/|\dot{\gamma }{|}^3$ and $\tau =\det (\dot{\gamma },\ddot{\gamma },\stackrel{...}{\gamma })/|\dot{\gamma }\times \ddot{\gamma }{|}^2$. The normal and torsion formulas require nonzero curvature; at zero-curvature points this particular Frenet frame need not be defined.

Units have norm $\pm 1$. Norm $-1$ is impossible modulo three. The least positive unit greater than one isIndeed its norm is one. A smaller positive norm-one unit would have $0<y=({\epsilon }^{\prime }-1/{\epsilon }^{\prime })/(2\sqrt{30})<2$, forcing $y=1$, but then $x^2=31$. Multiplying an arbitrary positive unit by powers of $\epsilon$ reduces it to $[1,\epsilon )$, so minimality proves that all units are $\pm {\epsilon }^n$.

The equation with right side $+5$ is impossible modulo three, since it would require $x^2\equiv 2(\mathrm{mod}3)$. For norm $-5$, let $\beta =5+\sqrt{30}$, so ${\beta }^2=5\epsilon$ and $\beta /\epsilon =\sqrt{30}-5$. For unit reduction for quadratic norm equations, change the sign of $x+y\sqrt{30}$ if necessary to make it positive, then multiply by a power of the fundamental unit to place it in $[\beta /\epsilon ,\beta )$. Its conjugate is $-5$ divided by itself, soThe maximum on this interval occurs at its endpoints. Thus $y=1$, and $x^2=25$; only $x=-5$ lies in the half-open interval. Reversing the reduction gives all solutions:Every displayed element has norm $-5$, and the reduction proves completeness, including both signs of $x$ and $y$.