48,707 research outputs found
On Longest Repeat Queries Using GPU
Repeat finding in strings has important applications in subfields such as
computational biology. The challenge of finding the longest repeats covering
particular string positions was recently proposed and solved by \.{I}leri et
al., using a total of the optimal time and space, where is the
string size. However, their solution can only find the \emph{leftmost} longest
repeat for each of the string position. It is also not known how to
parallelize their solution. In this paper, we propose a new solution for
longest repeat finding, which although is theoretically suboptimal in time but
is conceptually simpler and works faster and uses less memory space in practice
than the optimal solution. Further, our solution can find \emph{all} longest
repeats of every string position, while still maintaining a faster processing
speed and less memory space usage. Moreover, our solution is
\emph{parallelizable} in the shared memory architecture (SMA), enabling it to
take advantage of the modern multi-processor computing platforms such as the
general-purpose graphics processing units (GPU). We have implemented both the
sequential and parallel versions of our solution. Experiments with both
biological and non-biological data show that our sequential and parallel
solutions are faster than the optimal solution by a factor of 2--3.5 and 6--14,
respectively, and use less memory space.Comment: 14 page
A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent
Let be a compact Riemannian spin manifold of
dimension , let denote the spinor bundle on , and
let be the Atiyah-Singer Dirac operator acting on spinors
. We study the existence of solutions of the nonlinear
Dirac equation with critical exponent
where and is a subcritical nonlinearity
in the sense that as . A model
nonlinearity is with ,
. In particular we study the nonlinear Dirac equation This equation is a spinorial analogue of the Brezis-Nirenberg
problem. As corollary of our main results we obtain the existence of least
energy solutions of (BND) and (NLD) for every ,
even if is an eigenvalue of . For some classes of nonlinearities
we also obtain solutions of (NLD) for every , except
for non-positive eigenvalues. If (mod 4) we obtain solutions of
(NLD) for every , except for a finite number of
non-positive eigenvalues. In certain parameter ranges we obtain multiple
solutions of (NLD) and (BND), some near the trivial branch, others away from
it.
The proofs of our results are based on variational methods using the strongly
indefinite energy functional associated to (NLD).Comment: 42 page
Existence and convexity of local solutions to degenerate hessian equations
In this work, we prove the existence of local convex solution to the
degenerate Hessian equationComment: corrections some typos in this versio
- …