5

After trying many Ansätze in the form of series, I stumbled upon the fact (which I find rather remarkable) that ($\dagger$) admits an exact solution in closed form. To express this, let me first introduce the following notations:
$\kappa := \beta T$ (reproduction number), which I assume $>1$;
$\Gamma := -W(-\kappa \exp(-\kappa))/\kappa$ the solution in $...

4

$\newcommand\ep{\epsilon}$$\newcommand\de{\delta}$We shall be assuming that $\ep\in(0,1/e]$. Note that $l(x):=(\ln x)/x$ is decreasing in $x\ge e$.
So, for $x\ge e$ we have
$$(\ln x)/x\le\ep\iff x\ge x_\ep,$$
where $x_\ep\in[e,\infty)$ is the root of the equation
$$l(x_\ep)=\ep.$$
Letting
$$y:=y_\ep:=\frac1\ep\,\ln\frac1\ep\ge e,$$
we have
$$l(y)=\ep\...

4

If you want the smallest, try
$$x = -LambertW(-\epsilon)/\epsilon = 1+\epsilon+{\frac{3}{2}}{\epsilon}^{2}+{\frac{8}{3}}{\epsilon}^{3}+{\frac{125}{24}}{\epsilon}^{4}
+O \left( {\epsilon}^{5} \right)
$$

3

The ABP estimate indeed holds in your setting. The key is that the concave envelope of $u$ is in $C^{1,\,1}$, so the area formula is valid for its gradient. Assuming for simplicity that $L = \Delta$, that $\Omega = B_1$ and that $\sup_{\partial B_1} u = 0$, the way I would argue is:
Let $\Gamma$ be the concave envelope (the infimum of linear functions ...

1

The equations of motion cannot be written as
$$ \ddot{r}_{ij} = f_{ij}(r_{12},\ldots, r_{n-1,n}, \dot{r}_{12},\ldots, \dot{r}_{n-1,n}) $$
Consider $n$ equally massive particles arranged equidistantly around a circle. If they are all initially
stationary, the system will collapse to the centre of the circle. But if they are rotating at the correct speed, ...

1

To expand on my comment: suppose you have $U$ not simply connected, and $\omega$ closed but not exact, and suppose there exists a closed loop $\gamma: [0,1]\to U$ such that $\omega(\dot{\gamma})$ is signed. (This is in particular the case with the "example" in your comment, where you can take $\gamma$ to be any circle centered at the origin.)
Suppose $\...

1

There are many well-known ODE systems from various fields of science that can be transformed into the Lorenz system or generalized Lorenz system after a change of variables. After such a transformation the parameters of the original system may become negative in terms of the Lorenz system. The reason for looking for such a transformation is simple: to ...

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