User:Leo Trottier/mathjax demo
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Testing labeling
$ \newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} $
$$ \tag{1} e^{i\pi} +1 = 0 $$
We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \tag{2} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral (2) expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align} \tag{3} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, (3) also holds for negative odd integers. The reason for (3) was long a mystery, but it will be explained at the end of the paper.
WITHOUT EQUATION NUMBERING, equations do not overlap with images
Explicit Adams methods
These methods are introduced by J.C. Adams (1883) for solving practical problems of capillary action. They are based on the following idea: suppose that \(k\) values \( y_{n-k+1},\,y_{n-k+2},\ldots, y_{n}\) approximating \(y(t_{n-k+1}),\, y(t_{n-k+2}),\ldots,y(t_{n}) \) are known. Compute the derivatives \[ f_{n+j}=f(t_{n+j},y_{n+j}), \quad j=-k+1,\ldots ,0 \]
and replace in the integrated form of \eqref{eq:diffeq} \[ y(t_{n+1})=y(t_{n})+\int _{t_{n}}^{t_{n+1}}f(t,y(t))\,dt \]
the integrand \(f(t,y(t))\) by the polynomial \(p(t)\) interpolating the values (4). Then evaluate the integral analytically and obtain the next approximation to the solution, \(y_{n+1}\ .\) After advancing the scheme by one step, this procedure can be repeated to obtain \(y_{n+2},\, y_{n+3}\ ,\) and so on. See Figure 1 for an illustration using the logistic growth equation \(\dot y=a y (1-y)\ .\)
Newton's formula for polynomial interpolation can be written as \[ p(t) =f_n + \frac{1}{h}(t-t_n)\nabla f_n + \frac{1}{2 h^2}(t-t_n)(t-t_{n-1})\nabla^2f_n + \ldots \]
WITH EQUATION NUMBERING, equations overlap images
Explicit Adams methods
These methods are introduced by J.C. Adams (1883) for solving practical problems of capillary action. They are based on the following idea: suppose that \(k\) values \( y_{n-k+1},\,y_{n-k+2},\ldots, y_{n}\) approximating \(y(t_{n-k+1}),\, y(t_{n-k+2}),\ldots,y(t_{n}) \) are known. Compute the derivatives \[\tag{4} f_{n+j}=f(t_{n+j},y_{n+j}), \quad j=-k+1,\ldots ,0 \]
and replace in the integrated form of \eqref{eq:diffeq} \[\tag{5} y(t_{n+1})=y(t_{n})+\int _{t_{n}}^{t_{n+1}}f(t,y(t))\,dt \]
the integrand \(f(t,y(t))\) by the polynomial \(p(t)\) interpolating the values (4). Then evaluate the integral analytically and obtain the next approximation to the solution, \(y_{n+1}\ .\) After advancing the scheme by one step, this procedure can be repeated to obtain \(y_{n+2},\, y_{n+3}\ ,\) and so on. See Figure 1 for an illustration using the logistic growth equation \(\dot y=a y (1-y)\ .\)
Newton's formula for polynomial interpolation can be written as \[\tag{6} p(t) =f_n + \frac{1}{h}(t-t_n)\nabla f_n + \frac{1}{2 h^2}(t-t_n)(t-t_{n-1})\nabla^2f_n + \ldots \]
