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If $x$ is represented in floating-point as $y \times 10^{-d}$, $0.1 < y \le 1$, $d \in \mathbb N$, note that $$ x \ln(x) = y (\ln(y) - d \ln(10)) \times 10^{-d} $$ which shouldn't be a problem to eval …

To solve $f_n(x)=x$ (where $f_n$ is the $n$-fold composition of $f$), write $f_n(x)-x$ as a rational function, take the numerator (which is a polynomial, I think of degree $2^n-2$), and find its roots …

We have $f(u) < u$ for $0 < u < 1$ and $f(u) > u$ for $u > 1$, so the fixed point $1$ is unstable. Similarly $f(u) < u$ for $u < -1$ and $f(u) > u$ for $-1 < u < 0$ implies $-1$ is unstable. Since $| …

Any probability measure $\mu_1$ absolutely continuous with respect to $\mu_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below …

I suspect there may be periodic solutions. For $A = 1$, numerically plotting the solution with initial conditions $$x(0)=0, \dot{x}(0) = 0.442091320614410, \ddot{x}(0) = 0.774949154843236$$ I get thi …

If $H$ is continuous, $T(x)$ defined by $\int_0^{T(x)} H(s)\; ds = x$ satisfies the differential equation $$ \dfrac{dT}{dx} = \dfrac{1}{H(T)}$$ and standard numerical methods for differential equatio …

If you mean $f'' + 2 f (1-f^2)$, that can be solved either numerically (by standard techniques, available e.g. using Maple's dsolve(..., numeric)) or symbolically. The solutions of the differential e …

It's easy to come up with a differential equation that has a known general solution, expressed in the form $F(t,y) = constant$ where $F$ is differentiable: $$ \dfrac{\partial F}{\partial t} + \dfrac{ …

Not an answer, but this may help with asymptotics: According to Maple the o.g.f. for $H^s_n$ is $$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it …

$\Gamma$ is compact, has empty interior, and its complement is connected, so Mergelyan's theorem says that any continuous function on $\Gamma$ can be uniformly approximated by polynomials.

Not an answer, but I'll just note that there are some special cases that reduce to ODE's. With $u(x,t) = v(x+at)$, the differential equation becomes $$ (1+a) v' - a^2 (v'')^2 = v \tag{1}$$ In part …

The distance (in operator norm) from square matrix $A$ to the set of singular matrices is the minimum of the singular values of $A$. This is easy to see from the singular value decomposition.

It's easy to see why this is: $\cos(\alpha) \sim 1 - \alpha^2/2$ for $\alpha$ near $0$, so an error of $\delta$ in $\cos(\alpha)$ can produce an error of about $\sqrt{2\delta}$ in $\alpha$ as computed …

If $B_j$, $j = 1 \ldots \infty$, are independent Bernoulli(1/2) random variables, then $X = 2 \sum_{j=1}^\infty 3^{-j} B_j$ has a Cantor distribution. You could let $U$ be uniform on $[0,1]$ and take …

What you're saying, I think, is that (for a particular $t_1$) the map $\Psi$ from $x(0)$ to $x(t_1)$ is affine: $\Psi(x) = \Phi x + c$ where $\Phi$ is linear. If this was true for all $t_1$, the diff …