A paper of David Bachman-Cooper-White describes a proof that a hyperbolic 3-manifold containing large embedded balls has large Heegaard genus. As they say at the end of the introduction,
a proper subset of the authors wish to subtitle this paper “Big balls imply big genus”
whch is indeed t...
From the ground-breaking paper: On the complexity of omega-automata by Muli Safra
Acknowledgements
The author thanks his advisor, Amir Pnueli, for his encouragement and many fruitful discussions on this research.
Moshe Vardi initiated this research by a most illuminating mini-course on ω-au...
From Vector Calculus, Linear Algebra, And Differential Forms. A Unified Approach. by Hubbard:
When a matrix is described, height is
given first, then width: an m x n
matrix is m high and n wide. After
struggling for years to remember which
goes first, one of the authors hit on
a mne...
If I was trying to take the number $123$ in base $10$ and try and convert it into base zero I would do something like this:
$123 = 100 + 20 + 3$
$10^{\log_0(100)} + 10^{\log_0(20)} + 10^{\log_0(3)}$
But $\log_0(x)$ is the same thing as $\dfrac{\log(x)}{\log(0)}$ and the log of zero is undef...
I am being asked to determine whether the following problem
$\begin{align} u_{xyy}(x,y) + u_{xxy}(x,y)=0, && 0<x,\, y<1 \\
u(x,0)=u(x,1)=0, && 0<x<1 \\
u(0,y)=f(y),\,u(1,y) = 0, && 0<y<1 \end{align}$
can be solved by eigenfunction expansion.
Although it doesn't explicitly state them as suf...
