Mathematics Bear

This page contains formulas used by mathematically inclined bears.  Very few of these bears are found in the wild.  It is believed that a small family live near Meyrin, Switzerland.

Albert Einstein’s famous formula [latex]E=mc^2[/latex] can be used ….

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At first, we sample [latex]f(x)[/latex] in the [latex]N[/latex] ([latex]N[/latex]is odd) equidistant points around [latex]x^*[/latex]:
\[
   f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\]
where [latex]h[/latex] is some step.
Then we interpolate points [latex]\{(x_k,f_k)\}[/latex] by polynomial 
\begin{equation} \label{eq:poly}
   P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j}
\end{equation}
Its coefficients [latex]\{a_j\}[/latex] are found as a solution of system of linear equations:
\begin{equation} \label{eq:sys}
   \left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\end{equation}
Here are references to existing equations: ([latex]ref{eq:poly}[/latex]), ([latex]ref{eq:sys}[/latex]).
Here is reference to non-existing equation ([latex]ref{eq:unknown}[/latex]).