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\begin{document}
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 {\large\sf Math Bite: 
 Constructing  Efficient Particle Accelerators is as Easy as 1+1=2 \\
 (Thanks to Vladimir Visnjic)
 }
  
 \vskip 0.2in
 \noindent
  
 The difficulty in ``applied math'' is usually not the
 ``math'' part (it is often trivial), but the ``applied'' part, i.e.
 {\it realizing} that math can help, and separating the
 mathematical wheat from the scientific chaff.
  
 A dramatic illustration of the above is Vladimir Visnjic's
 ingenious construction of two local waves on the particle beam in an
 accelerator
 that promises to save the American
 taxpayer millions of dollars (but, most unfairly, will not earn its
 discoverer even a penny). 
 
Sources at positions $x=x_1,x_2,\dots ,x_n$ with amplitudes
$A_1,A_2,\dots,A_n$, produce waves of two types, $A_i\sin (x-x_i)$ and 
$f(x_i)A_i\sin ((x-x_i)/2)$. Here $f(x)$ is a certain function (specific 
to the accelerator) of which we only need to know that it is nonnegative 
and has zeros. The number of sources, their positions, and amplitudes 
can be chosen freely up to the constraint $\sum A_i=0$. The objective 
is to find the minimal set of sources which creates local waves of both 
types at the same position. 
 
Visnjic's construction hinges on the following
 immediate corollary of the celebrated identities $1+1-2=0$ and
 $1-1+0=0$.   
  
 {\sc Proposition.} 
 {\it 
 Let $H(x)$ be the Heaviside function (i.e., $H(x)=1$ if $x>0$, 
 $H(x)=0$ otherwise).   Then
 $$
 H(x)\sin(x)+ H(x-2 \pi) \sin(x-2\pi)-2
 H(x-4\pi)\sin(x-4\pi)
 $$ 
 vanishes outside $0 < x < 4\pi$.   If $f(0)=f(2\pi)$ and $f(4 \pi)=0$, 
 then 
 $$
 f(0)H(x)\sin(x/2)+ f(2\pi) H(x-2\pi) \sin(x/2-\pi)-
 2f(4\pi) H(x-4\pi)\sin(x/2-2\pi)
 $$ 
 vanishes outside $0 < x < 2\pi$.
 }
  
  \vspace*{.25 in}
  
 {\sf REFERENCE}
 \begin{enumerate}
  
 \item
 Vladimir Visnjic, {\it A Novel Focusing Element for Accelerators},
 Phys. Rev. Letters 21 (1994), 2860.
  
 \end{enumerate}
  
  
 \vskip .5 in
  
 \begin{flushright}
 --- Doron Zeilberger \\
 Temple University \\
 Philadelphia, PA 19122 \\
 \verb+http://www.math.temple.edu/~zeilberg+
 \end{flushright}
  
 \end{document}
  
 
